Research ArticleAn Algorithm for Identifying the Isomorphism ofPlanar Multiple Joint and Gear Train Kinematic Chains

Yanhuo Zou and Peng He

Quanzhou Institute of Equipment Manufacturing Haixi Institutes Chinese Academy of Science Quanzhou 362200 China

Correspondence should be addressed to Yanhuo Zou zouyhfjirsmaccn

Received 21 September 2015 Revised 4 February 2016 Accepted 1 March 2016

Academic Editor Ibrahim Zeid

Copyright copy 2016 Y Zou and P He This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Isomorphism identification of kinematic chains is one of the most important and challenging mathematical problems in the fieldof mechanism structure synthesis In this paper a new algorithm to identify the isomorphism of planar multiple joint and geartrain kinematic chains has been presented Firstly the topological model (TM) and the corresponding weighted adjacency matrix(WAM) are introduced to describe the two types of kinematic chains respectively Then the equivalent circuit model (ECM) ofTM is established and solved by using circuit analysis method The solved node voltage sequence (NVS) is used to determine thecorrespondence of vertices in two isomorphism identification kinematic chains so an algorithm to identify two specific types ofisomorphic kinematic chains has been obtained Lastly some typical examples are carried out to prove that it is an accurate efficientand easy mathematical algorithm to be realized by computer

1 Introduction

Isomorphism identification is one of indispensable steps inthe structural synthesis of kinematic chains so a lot of studieshave been done in this field Until now many methods havebeen presented for planar simple joint kinematic chainssuch as the characteristic polynomial methods which weredeveloped in the papers [1ndash8] for detecting isomorphic kine-matic chains by typical topological graphs Although mostof these methods are efficient counterexamples have beenfound against them [8]Themax-(min-) code [9 10] degree-code [11] and standard-codemethods [12 13] which are basedon code number may be effective but they are inadequate inefficiency The Hamming number approach [14ndash16] is intro-duced for isomorphism identification but the counterexam-ples also are found against them The adjacent chain tablemethod [17] seems hard to be realized by computerThe fuzzylogic method [18] also displays some new ideas for isomor-phism identification and structural analysis of simple jointkinematic chains but it needsmore storage space Themethodof eigenvectors and eigenvalues of adjacencymatrices [19ndash22]detects isomorphism through computation of eigenvectorsand eigenvalues as well as some permutation operations The

method of finding a unique representation of graphs to iden-tify isomorphism kinematic chains is presented by Ding andHuang [23 24] In addition some unconventional methodssuch as genetic algorithm [25] and artificial neural networkapproach [26] are also introduced to pursue the issue

So far isomorphism identification of kinematic chainsstudied most is about those planar simple joint chains andidentification technique for this kind of chains is compara-tively mature However besides simple joint chains planarkinematic chains also includemultiple joint kinematic chainsand gear train kinematic chainsThe two special types of kine-matic chains can be used in a lot of mechanism design prob-lems As we know isomorphism identification methods usedin the simple joint chains are always not satisfying these twospecific types of kinematic chains In the field of isomorphismidentification of multiple joint kinematic chains adjacentchain table method was introduced in the paper [27] Songet al [28] presented the spanning tree method of identifyingisomorphism and topological symmetry for multiple jointkinematic chains but this method was not easily achievedby computer Liu and Yu [29] presented a representation andisomorphism identification of planarmultiple joint kinematic

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 5310582 15 pageshttpdxdoiorg10115520165310582

2 Mathematical Problems in Engineering

Link5

5

LinkLink Link6

e

e

6

Simplejoint

(a)

Link1 2 9

Link Link Link

Link9

aa

1

Multiplejoint

2Link

(b)

Figure 1 The structural of simple joint and multiple joint

f

e

h a

b

j

i k

cd ①

②

③

④

⑤

⑥

⑦⑧

⑨

⑩

(a)

d

h

a

i j

k

c

b

fe

①

②

③

④

⑤

⑥

⑦

⑧

⑨

⑩

(b)

a

1

2

2

2

2

3

3

b

①

②

(c)

Figure 2 A multiple joint kinematic chain and its TM described

chains based on the converted adjacent matrix but convertedadjacent matrix not only increased the order of matrix butalso added the complexity of isomorphism identificationprocess In the structural synthesis process of gear trainkinematic chains correctly identifying graph isomorphismis an important step which generates new structural typesPapers [30ndash32] try to do the structural synthesis of gear trainkinematic chains One of the main reasons for the incorrectidentification or inconsistency in results is that there is a lackof efficient isomorphism tests The purpose of paper [33] isto present an efficient methodology through using the loopand hamming number concept to detect the isomorphismof gear train kinematic chains in an unambiguous way Sofrom the researches above finding a useful method to solvethe problem of isomorphism identification of planar multiplejoint and gear train kinematic chains is the existing researchcontent and further study is necessary

In this paper a new algorithm based on the equivalentcircuit analysis for isomorphism identification of planarmultiple joint and gear train kinematic chains is presentedFirstly weighted-double-color-contracted-graph (WDCCG)and corresponding weighted adjacency matrix (WAM) areintroduced to describe the two special types of kinematicchains respectively Corresponding equivalent circuit model(ECM) of the WDCCG is established which uses circuitanalysis method to obtain the node voltages of every vertexinWDCCGThe solved node voltage sequence (NVS) is usedto determine correspondence vertices of two isomorphism

identification kinematic chains Then a new algorithm todetermine isomorphism of the two types of kinematic chainsis proposed

2 Topological Model of Kinematic Chains

21 Topological Model of Multiple Joint Kinematic ChainsBefore further discussion the following common conceptsare introduced as follows

Simple Joint If two links are connected by a revolute jointthen a simple joint is formed For example a simple joint 119890is formed by connecting links 5 and 6 which is shown inFigure 1(a)

Multiple Joint If there are more than two links connected byrevolute joints at the same location then a multiple joint isformed Amultiple joint composed by 119896 links contains (119896minus1)revolute joints For example a multiple joint a is formed byconnecting links 1 2 and 9 as shown in Figure 1(b) and itcontains 2 revolute joints

A planar simple joint kinematic chain is a kinematic chainonly composed of simple joints if there is a kinematic chaincontaining multiple joint called a multiple joint kinematicchain for example a multiple joint kinematic chain which isshown in Figure 2(a)

In a multiple joint kinematic chain because a multiplejoint 119894 composed of 119896 links contains (119896 minus 1) simple joints

Mathematical Problems in Engineering 3

B

A

b

D

C

c

a

①

② ③

④

⑤⑥

(a)

A b

D

B

c

C

a

①

②

③

④

⑤

(b)a

11 1

1

223

3

3

3①

②

③

④

⑤

⑥

(c)

Figure 3 A gear train kinematic chain and its TM described

defined multiple joint factor 119901119894of a multiple joint 119894 is as

follows

119901119894= 119896 minus 2 (1)

And the total multiple joint factors119875 of a kinematic chaincan be obtained by adding all multiple joint factors of akinematic chain together as follows

119875 = sum119901119894 (2)

So as for a multiple joint kinematic chain as shown inFigure 2(a) it has twomultiple joints 119886 and 119887 these joints 119888 119889119890 119891 ℎ 119894 119895 and 119896 are simple joints Multiple joint 119886 is formedby connecting four links 6 7 8 and 10 so its correspondingmultiple joint factor is 119901

119886= 2 Multiple joint 119887 is formed

by connecting three links 4 5 and 10 so its correspondingmultiple joint factor is 119901

119886= 1 And the total multiple joint

factors of this kinematic chain is 119875 = 3As we know topological model (TM) of multiple joint

kinematic chain can be represented by the double colorgraph (DCG) as shown in Figure 2(b) [23] But it has toomany vertices (number of vertices equal to the numberof links and joints of a kinematic chain) so more storagespace is needed in the process of structural analysis bycomputer In this paper a weighted double color contractedgraph (WDCCG) is introduced to represent the topologicalstructure of kinematic chains The WDCCG model whichwiped off all of the simple joints and simple joint links formthe multiple joint kinematic chain can improve the efficiencyof isomorphism identification The graph is established asfollows solid vertices ldquoerdquo denotemultiple joint links and hol-low vertices ldquoIrdquo denote multiple joints If two multiple jointlinks are connected by simple joints and simple joint linksconnect two corresponding solid vertices with a weightededge (weighted value equal to the number of simple jointsbetween two vertices) If one multiple joint link is connectedwith one multiple joint or two multiple joints are connected

by simple joints and simple joint links connect them witha weighted edge as well (weighted value also equal to thenumber of simple joints between two vertices) For exampleFigure 2(c) is the WDCCG of multiple joint kinematic chainas shown in Figure 2(a)

22 Topological Model of Gear Train Kinematic Chain Inthis paper topological model (TM) of gear train kinematicchains can be represented by two topological graphs Firstlydouble color graph (DCG) is established as follows representthe gears and planet carrier with solid vertices ldquoerdquo gearjoints with hollow vertices ldquoΔrdquo and revolute joint with hollowvertices ldquoIrdquo And connect corresponding solid vertex andhollow vertex with an edge when a gear or a planet carrier isconnected with a gear joint or a revolute joint For exampleFigure 3(b) is the DCG corresponding to gear train kinematicchain as shown in Figure 3(a) But it also has too manyvertices then a weighted double color contracted graph(WDCCG) is introduced to represent gear train kinematicchain Because the gear train kinematic chain contains twotypes of kinematic pairs namely revolute joint and gearjoints the WDCCG model wiped off all of the simple jointsand can be established as follows solid vertices ldquoerdquo denotegears of chain hollow vertices ldquoΔrdquo denote gear joints andhollow vertices ldquoIrdquo denote multiple revolute joints If twogears are connected directly by a revolute joint connecttwo corresponding solid vertices with a 2-weighted edge Iftwo gears are connected directly by a gear joint connecttwo corresponding solid vertices with a 3-weighted edgeIf a gear is connected with a multiple revolute joint alsoconnect corresponding solid vertex and hollow vertex witha 1-weighted edge For example Figure 3(c) is the WDCCGof the gear train kinematic chain as shown in Figure 3(a)

23 Weighted Adjacency Matrix Obviously a multiple jointor gear train kinematic chain and its WDCCG are one-to-one correspondent to each otherTheWDCCGof a kinematic

4 Mathematical Problems in Engineering

chain can be represented by a vertex-vertex weighted adja-cencymatrix (WAM)while the elements ofWAMare definedas follows

119860 = [119889119894119895]119899times119899

=

(119889119894119895) = (119898

1198941198951 1198981198941198952 119898

119894119895119896) 119894 = 119895

(119889119894119895) 119894 = 119895

(3)

where 119899 is the number of vertices in WDCCGWhen 119894 = 119895 if vertex is hollow vertex then 119889

119894119895= 0

otherwise the vertex is solid vertex then 119889119894119895= 1

When 119894 = 119895 if two vertices are not connected directlythen 119889

119894119895= 0 otherwise the two vertices are connected

directly by 119896 weighted edges then 119889119894119895= 1198981198941198951 1198981198941198952 119898

119894119895119896

(1198981198941198951ge 1198981198941198952ge sdot sdot sdot ge 119898

119894119895119896 119898119894119895119896

is the weighted value of 119896thedge)

For example the WAM 119860 of a multiple joint kinematicchain as shown in Figure 2(c) can be expressed as follows

119860 =

1 2 119886 119887

1

2

119886

119887

[

[

[

[

[

[

(1)

(1)

(2)

(2 3)

(1)

(1)

(3)

(2)

(2)

(3)

(0)

(2)

(2 3)

(2)

(2)

(0)

]

]

]

]

]

]

(4)

The WAM 119860 corresponding to a gear train kinematicchain as shown in Figure 3(c) can be expressed as follows

119860

=

1 2 3 4 5 6 119886

1

2

3

4

5

6

119886

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

(1)

(0)

(0)

(0)

(3)

(0)

(1)

(0)

(1)

(0)

(0)

(3)

(2)

(1)

(0)

(0)

(1)

(0)

(0)

(3)

(1)

(0)

(0)

(0)

(1)

(2)

(3)

(1)

(3)

(3)

(0)

(2)

(1)

(0)

(0)

(0)

(2)

(3)

(3)

(0)

(1)

(0)

(1)

(1)

(1)

(1)

(0)

(0)

(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(5)

3 Isomorphism Identification Algorithm

31 Equivalent Circuit Model As we know when two multi-ple joint kinematic chains or two gear trains kinematic chainsare isomorphic their WDCCGs are exactly the same theirvertices and edges are in one-to-one correspondence witheach other and the vertices and edges in correspondencekeep the same incidence and weighted relation Accordingto this definition the necessary and sufficient condition forisomorphism of the two types of kinematic chains is that theyhave the same WAM that is 119860 = 1198601015840

Equivalent circuit model (ECM) of WDCCG is estab-lished by its each edge which is replaced by conductanceequal to the weighted value of corresponding edge119898

119894119895

Theorem 1 If two WDCCGs corresponding to the same typekinematic chains are isomorphic then their correspondingECMs are identical

For example two isomorphism multiple joint kinematicchains are shown in Figures 4(a) and 4(b) their WDCCGsare shown in Figures 4(c) and 4(d) respectively According tothe description above their corresponding ECMs are shownin Figures 5(a) and 5(b) respectively There are two identicalcircuits and nodes of these two ECMs are in one-to-onecorrespondence

Complete excitation (CE) of an ECM is established asfollows in an ECM with 119899 nodes firstly set a node (119899 + 1) tobe a reference node then connect the reference node (119899 + 1)with every other node 119899

119894(119894 = 1 2 119899) of the ECM whose

conductance values are equal to the sumof weighted values ofthe node edges Secondly apply the same current source 1 Abetween node (119899+1) and other nodes 119899

119894(119894 = 1 2 119899) of the

ECM respectively in directions of the currents going fromnode (119899 + 1) to other nodes 119899

119894 (119894 = 1 2 119899) respectively

According to circuit theory two identical circuits underthe same CE have the same response And their node voltagescan be solved by the nodal method of circuit analysis Thenodal method of circuit analysis can be expressed as followsin a circuit network choose a node as a reference nodethe voltage difference between each node and the referencenode is known as the node voltage of the node For a circuitnetwork with 119899 nodes which has (119899 minus 1) node voltages if thenode 119899 is the reference node the nodal voltage equation canbe expressed as

[

[

[

[

[

[

[

11986611

11986612

sdot sdot sdot 1198661(119899minus1)

11986621

11986622

sdot sdot sdot 1198662(119899minus1)

d

119866(119899minus1)1

119866(119899minus1)2

sdot sdot sdot 119866(119899minus1)(119899minus1)

]

]

]

]

]

]

]

[

[

[

[

[

[

[

V1198991

V1198992

V119899(119899minus1)

]

]

]

]

]

]

]

=

[

[

[

[

[

[

[

1198681199041198991

1198681199041198992

119868119904119899(119899minus1)

]

]

]

]

]

]

]

(6)

where119866119894119894(119894 = 1 2 119899 minus 1) is called the self admittance of a

node 119894 the value of which is the sum of the admittance of allbranches that are connected to the node 119894119866

119894119895(119894 = 1 2 119899minus

1 119895 = 1 2 119899minus1) is called themutual admittance of node 119894and node 119895 whose value is the sum of the branch admittancebetween node 119894 and node 119895 and 119868

119904119899119894(119894 = 1 2 119899 minus 1) is

called the algebraic sum of the current source for the inflownode 119894 (inflow is positive outflow is negative)

For example an ECM with 4 nodes is shown in Fig-ure 6(a) Apply the CE of reference node 5 as shown inFigure 6(b) By nodalmethodof circuit analysis equations (6)node voltages V

1 V2 V3 and V

4can be expressed as follows

Mathematical Problems in Engineering 5

①

②

③

④

⑤

⑥

⑦

⑧

⑨

⑩

a

b

(a)

①

②

③

④

⑤

⑥

⑦

⑧ ⑨⑩

b

a

(b)

①

②③

a

b

1

1

1

2

223

3

(c)

①

②③

a

b1

1

1

2

2

2

3

3

(d)

Figure 4 Two multiple joint kinematic chains and their WDCCG

①

②③

2 2

2

11

1

a

3

3b

(a)

①

②③

2

2

2

1

1

1a

3

3

b

(b)

Figure 5 ECM corresponding to two WDCCGs of Figures 4(c) and 4(d) respectively

G1

G2

G3

G4 G5

① ② ③

④

(a)

G6

G7

G8

G9

①

②

③

④

⑤

1

1

1

1

(b)

Figure 6 An ECM with 4 nodes and its CE by reference node 5

6 Mathematical Problems in Engineering

[

[

[

[

[

[

1198661+ 1198664+ 1198666

minus1198664

0 minus1198661

minus1198664

1198662+ 1198664+ 1198665+ 1198667

minus1198665

minus1198662

0 minus1198665

1198663+ 1198665+ 1198668

minus1198663

minus1198661

minus1198662

minus1198663

1198661+ 1198662+ 1198663+ 1198669

]

]

]

]

]

]

[

[

[

[

[

[

V1

V2

V3

V4

]

]

]

]

]

]

=

[

[

[

[

[

[

1

1

1

1

]

]

]

]

]

]

(7)

Then the node voltages V1 V2 V3 and V

4can be obtained

by solving the above equations (7) Therefore the followingconclusion can be obtained

Theorem 2 If two WDCCGs corresponding to the same typeof kinematic chains are isomorphic their two correspondingECMs 119873 and 1198731015840 can be established and excited by the sameCE as above respectively Then the node voltages of eachcorrespondence node of two ECMs119873 and1198731015840 are the same

For example two identical ECMs119873 and1198731015840 with 5 nodesare shown in Figures 5(a) and 5(b) Apply the same CE ofreference node 6 which is shown in Figures 7(a) and 7(b)respectively And their corresponding WAMs 119860 and 1198601015840 canbe obtained as follows

119860 =

1 2 3 119886 119887

1

2

3

119886

119887

[

[

[

[

[

[

[

[

[

(1)

(2)

(0)

(1)

(2)

(2)

(1)

(1)

(0)

(3)

(0)

(1)

(1)

(2 3)

(1)

(1)

(0)

(2 3)

(0)

(0)

(2)

(3)

(1)

(0)

(0)

]

]

]

]

]

]

]

]

]

1198601015840=

1 2 3 119886 119887

1

2

3

119886

119887

[

[

[

[

[

[

[

[

[

(1)

(1)

(2)

(3)

(0)

(1)

(1)

(0)

(1)

(2 3)

(2)

(0)

(1)

(2)

(1)

(3)

(1)

(2)

(0)

(0)

(0)

(2 3)

(1)

(0)

(0)

]

]

]

]

]

]

]

]

]

(8)

By nodal method of circuit analysis equations of nodevoltages V

1 V2 V3 V119886 and V

119887can be expressed respectively

as follows

[

[

[

[

[

[

[

[

[

10 minus2 0 minus1 minus2

minus2 12 minus1 0 minus3

0 minus1 14 minus5 minus1

minus1 0 minus5 12 0

minus2 minus3 minus1 0 12

]

]

]

]

]

]

]

]

]

[

[

[

[

[

[

[

[

[

V1

V2

V3

V119886

V119887

]

]

]

]

]

]

]

]

]

=

[

[

[

[

[

[

[

[

[

1

1

1

1

1

]

]

]

]

]

]

]

]

]

(9)

[

[

[

[

[

[

[

[

[

12 minus1 minus2 minus3 0

minus1 14 0 minus1 minus5

minus2 0 10 minus2 minus1

minus3 minus1 minus2 12 0

0 minus5 minus1 0 12

]

]

]

]

]

]

]

]

]

[

[

[

[

[

[

[

[

[

V1

V2

V3

V119886

V119887

]

]

]

]

]

]

]

]

]

=

[

[

[

[

[

[

[

[

[

1

1

1

1

1

]

]

]

]

]

]

]

]

]

(10)

Then node voltages V1 V2 V3 V119886 and V

119887of two ECMs can

be obtained by solving the above (9) and (10) respectively asshown in Table 1

From Theorem 2 node voltages of each correspondencenode in these two ECMs are the same as follows V

119886= V119887=

01627 V119887= V119886= 01691 V

1= V3= 01839 V

2= V1= 01691

and V3= V2= 01537 When arranging the obtained node

voltages in descending order with the same node voltagesstaying together this set is called the node voltage sequence(NVS) and denoted as

NVS = lfloor(V1 V2 V

1198991

) (V(1198991+1) V(1198991+2) V

119899)rfloor

(V1ge V2ge V1198991

V(1198991+1)ge V(1198991+2)ge sdot sdot sdot ge V

119899)

(11)

where the variables V1 V2 V

1198991

represent node voltages ofhollow vertex The variables V

(1198991+1) V(1198991+2) V

119899represent

node voltages of solid vertexIf two identical ECMs are stimulated by the same CE as

above then the NVSs in correspondence are the same Forexample two identical ECMs as showed in Figures 5(a) and5(b) the NVS and vertex codes can be obtained respectivelyas follows

NVS (119886) = [(V119887 V119886) (V1 V2 V3)]

= [(01691 01627) (01839 01691 01537)]

NVS (119887) = [(V119886 V119887) (V3 V1 V2)]

= [(01691 01537) (01839 01691 01627)]

(12)

If the NVSs of two ECMs are not the same then the twoWDCCGs are not isomorphic On the contrary if they arethe same correspondence vertices in these two WDCCGscan be determined by the element values of NVS Thenperform the row exchanges of WAM 119860

1015840 to determine if theyare isomorphic Because the NVS can reduce the number ofrow exchanges to several ones or only just one this is a veryeffective method For example by the NVS(119886) and NVS(119887)of two ECMs shown in Figures 5(a) and 5(b) the verticesin correspondence are 119886-119887 119887-119886 1-3 2-1 3-2 then throughexchanging the row of WAM 119860

1015840 we obtain 119860 = 1198601015840 which

are isomorphic

Mathematical Problems in Engineering 7

①

②

③

⑥

1 1

1

11

6

6

7

6

5

a

b

(a)

①

②

③

⑥

1 1

1

1

1

6

6

6

7

5

a

b

(b)

Figure 7 Two ECMs with the same CE respectively

Table 1 Node voltages of two ECMs with the same CE

Vertex type Hollow vertex Solid vertex

Figure 5(a) Vertex code 119886 119887 1 2 3Node voltage 01627 01691 01839 01691 01537

Figure 5(b) Vertex code 119886 119887 1 2 3Node voltage 01691 01627 01691 01537 01839

For example two gear train kinematic chains are shownin Figures 8(a) and 8(b) and their WDCCGs as shownin Figures 8(c) and 8(d) respectively Their correspondingWAMs are 119860 and 1198601015840 as follows

119860

=

1 2 3 4 5 6 119887

1

2

3

4

5

6

119887

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

(1)

(2)

(0)

(0)

(3)

(0)

(1)

(2)

(1)

(3)

(0)

(0)

(0)

(0)

(0)

(3)

(1)

(3)

(0)

(0)

(1)

(0)

(0)

(3)

(1)

(0)

(2)

(0)

(3)

(0)

(0)

(0)

(1)

(0)

(1)

(0)

(0)

(0)

(2)

(0)

(1)

(1)

(1)

(0)

(1)

(0)

(1)

(1)

(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

1198601015840

=

1 2 3 4 5 6 119887

1

2

3

4

5

6

119887

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

(1)

(0)

(0)

(3)

(2)

(0)

(1)

(0)

(1)

(0)

(0)

(3)

(3)

(1)

(0)

(0)

(1)

(0)

(0)

(2)

(1)

(3)

(0)

(0)

(1)

(0)

(0)

(1)

(2)

(3)

(0)

(0)

(1)

(0)

(0)

(0)

(3)

(2)

(0)

(0)

(1)

(0)

(1)

(1)

(1)

(1)

(0)

(0)

(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(13)

According to the abovemethod their correspondingNVSand vertex codes can be obtained as follows

NVS (119886) = [(V119887) (V6 V5 V4 V1 V2 V3)] = [(02330)

(02744 02257 02068 01908 01907 01737)]

NVS (119887) = [(V119887) (V3 V4 V5 V1 V6 V2)] = [(02330)

(02744 02257 02068 01908 01907 01737)]

(14)

By the NVS(119886) and NVS(119887) the vertices of these twoWDCCGs in correspondence can be obtained as follows 119887-1198871-1 2-6 3-2 4-5 5-4 6-3 and through exchanging the rowof WAM 119860

1015840 one time we obtain isomorphic 119860 = 1198601015840

32 Algorithm Here a direct algorithm is given to determineisomorphism of planar multiple joint and gear train kine-matic chains and it can be expressed as follows

Step 1 Input the vertex codes and WAMs 119860 1198601015840 of twoWDCCGs corresponding to the same type of kinematicchains

Step 2 Assign vertices codes of 119860 and 1198601015840 in two groups thehollow vertexes to be the first group and the solid vertexes tobe another group

Step 3 Calculate NVS(119886) NVS(119887) and their vertex codesof the two ECMs with the same CE respectively CompareNVS(119886) and NVS(119887) if equivalence cannot be seen it isdetermined that they are not isomorphic and program stopsotherwise it is determined that they could be isomorphicThen check the NVS(119886) and NVS(119887) to see if the values intwo vertex code groups are all distinct if yes go to Step 4 ifno go to Step 6

8 Mathematical Problems in Engineering

①

②

③

④

⑤

⑥

A

a

b

C

B

c

(a)

① ②③ ④

⑤

⑥

A

a

b

C

B

c

(b)

①

②

③

④

⑤

⑥

1

b

1

1

1

2

3

3

3 2

(c)

①

②

③

④

⑤

⑥

3

3

3

2

2

b

1

1

1

1

(d)

Figure 8 Two gear train kinematic chains and their WDCCGs

Step 4 Take the NVS(119886) NVS(119887) and their vertex codes todetermine the correspondence of vertices according to thevalues that are in correspondence

Step 5 According to the corresponding vertices obtained andthe vertex codes to form new WAM 119860

1015840

1 rewrite WAM 119860

1015840 If119860 = 119860

1015840

1 it is determined that they are isomorphic otherwise

they are not isomorphic and program stops

Step 6 According to the corresponding vertices obtained theNVS(119886) and NVS(119887) have the same values in the correspond-ing two vertex code groups Just adjust the position order ofvertices of the same group in1198601015840 to obtain vertex codes to fromnew adjacency matrix 1198601015840

1 If 119860 = 119860

1015840

1 it is determined that

they are isomorphic and program stops if 119860 = 1198601015840

1 keep on

adjusting the position order of the aforementioned verticesin 1198601015840 of the vertex code groups for all possibilities And if119860 = 119860

1015840

1always it is determined that they are not isomorphic

and program stops

4 Examples

Example 1 Two 10-link multiple joint kinematic chainsare shown in Figures 9(a) and 9(b) Their correspondingWDCCGs are depicted in Figures 9(c) and 9(d) respectively

The direct process to determine isomorphism of these twokinematic chains can be expressed as follows

Step 1 Input their corresponding vertex codes andWAMs1198601198601015840 as follows

119860 =

1 119886 119887 119888 119889

1

119886

119887

119888

119889

[

[

[

[

[

[

[

[

[

(1)

(2)

(0)

(1)

(2)

(2)

(0)

(2)

(0)

(3)

(0)

(2)

(0)

(2)

(2)

(1)

(0)

(2)

(0)

(3)

(2)

(3)

(2)

(3)

(0)

]

]

]

]

]

]

]

]

]

1198601015840=

1 119886 119887 119888 119889

1

119886

119887

119888

119889

[

[

[

[

[

[

[

[

[

(1)

(0)

(2)

(2)

(1)

(0)

(0)

(2)

(2)

(2)

(2)

(2)

(0)

(0)

(3)

(2)

(2)

(0)

(0)

(3)

(1)

(2)

(3)

(3)

(0)

]

]

]

]

]

]

]

]

]

(15)

Step 2 Evidently 119860 = 1198601015840 Assign vertices codes of 119860 and 1198601015840

in two groups (a) [(119886 119887 119888 119889) (1)] and (b) [(119886 119887 119888 119889) (1)]

Mathematical Problems in Engineering 9

①

②

③

④

⑤

⑥

⑦

⑧

⑨

⑩

c

ba

d

(a)

①

②

③ ④

⑤

⑥⑦

⑧

⑨⑩

c b

a

d

(b)

①

cb

a

2

2

2

2

2

3

3

1d

(c)

①

2

2

2

2

2

3

3

d

b

ac

1

(d)

Figure 9 Two 10-link multiple joint kinematic chains and their WDCCGs

①

⑪

②

③

④

⑤

⑥

⑦⑧

⑨

⑩

ab

(a)

①

②

③④

⑤⑥

⑦

⑧

⑨

⑩

a

b

⑪

(b)

①

② ③

④

⑤

⑥ 1

1

11

12

2

2

2

2a

b

1

1 1

(c)

① ②

③

④

⑤

⑥

1

1

1

1

1

11

12

2

2

2

2b a

(d)

Figure 10 Two 11-link multiple joint kinematic chains and their WDCCGs

Step 3 Calculate NVS(119886) NVS(119887) and their vertex codes oftwo ECMs with the same CE respectively as follows

NVS (119886) = [(V119888 V119887 V119886 V119889) (V1)]

= [(01551 01546 01451 01275) (01700)]

NVS (119887) = [(V119886 V119887 V119888 V119889) (V1)]

= [(01538 01459 01459 01308) (01714)]

(16)

Compare NVS(119886) and NVS(119887) and equivalence cannot beseen It is determined that they are not isomorphic andprogram stops

Example 2 Two 11-link multiple joint kinematic chains areshown in Figures 10(a) and 10(b)Their correspondingWDC-CGs are described in Figures 10(c) and 10(d) respectivelyThe direct process to determine isomorphism of these twokinematic chains is as follows

10 Mathematical Problems in Engineering

Step 1 Input their corresponding vertex codes andWAMs1198601198601015840 as follows

119860

=

1 2 3 4 5 6 119886 119887

1

2

3

4

5

6

119886

119887

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

(1)

(2)

(0)

(1)

(1)

(0)

(0)

(2)

(2)

(1)

(1)

(0)

(1)

(0)

(0)

(0)

(0)

(1)

(1)

(1)

(0)

(2)

(2)

(0)

(1)

(0)

(1)

(1)

(0)

(1)

(0)

(0)

(1)

(1)

(0)

(0)

(1)

(0)

(1)

(0)

(0)

(2)

(0)

(1)

(0)

(1)

(0)

(1)

(0)

(0)

(1)

(0)

(1)

(0)

(0)

(2)

(2)

(0)

(0)

(0)

(0)

(1)

(2)

(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

1198601015840

=

1 2 3 4 5 6 119886 119887

1

2

3

4

5

6

119886

119887

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

(1)

(2)

(0)

(1)

(0)

(0)

(0)

(1)

(2)

(1)

(0)

(1)

(1)

(0)

(2)

(0)

(0)

(0)

(1)

(1)

(2)

(1)

(0)

(2)

(1)

(1)

(1)

(1)

(0)

(0)

(0)

(0)

(0)

(1)

(2)

(0)

(1)

(1)

(0)

(0)

(0)

(0)

(1)

(0)

(1)

(1)

(1)

(0)

(0)

(2)

(0)

(0)

(0)

(1)

(0)

(2)

(1)

(0)

(2)

(0)

(0)

(0)

(2)

(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(17)

Step 2 Evidently 119860 = 1198601015840 Assign vertices codes of 119860 and 1198601015840

in two groups (a) [(119886 119887) (1 2 3 4 5 6)] and (b) [(119886 119887) (1 23 4 5 6)]

Step 3 Calculate NVS(119886) NVS(119887) and their vertex codes oftwo ECMs with the same CE respectively as follows

NVS (119886) = [(V119886 V119887) (V5 V4 V2 V6 V1 V3)]

= [(02084 02056)

(0743 02727 02348 02347 02023 01995)]

NVS (119887) = [(V119887 V119888) (V6 V4 V5 V1 V3 V2)]

= [(02084 02056)

(0743 02727 02348 02347 02023 01995)]

(18)

Compare NVS(119886) and NVS(119887) and equivalence can beseen So the node voltages in two vertex code groups are alldistinct go to Step 4

Step 4 Take the NVS(119886) NVS(119887) and their vertex codes todetermine the corresponding vertices of two WDCCGs asfollows (119886-119887 119887-119886 1-3 2-5 3-2 4-4 5-6 6-1)

Step 5 According to the corresponding vertices obtained andthe vertex codes to form new WAM 119860

1015840

1 rewrite WAM 119860

1015840Then we have 119860 = 1198601015840

1 so they are isomorphic and program

stops

Example 3 Two 6-gear train kinematic chains are depicted inFigures 11(a) and 11(b) and their correspondingWDCCGs aredescribed in Figures 11(c) and 11(d) respectively The directprocess to determine isomorphism of these two kinematicchains is as follows

Step 1 Input their corresponding vertex codes andWAMs1198601198601015840 as follows

119860 =

1 2 3 4 5 6 7 119886 119888

1

2

3

4

5

6

7

119886

119888

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

(1)

(3)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(3)

(1)

(2)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(2)

(1)

(3)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(3)

(1)

(0)

(0)

(0)

(1)

(1)

(0)

(0)

(0)

(0)

(1)

(3)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(3)

(1)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(1)

(1)

(0)

(1)

(0)

(1)

(0)

(1)

(1)

(0)

(0)

(1)

(0)

(0)

(1)

(1)

(0)

(1)

(0)

(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

1198601015840=

1 2 3 4 5 6 7 119886 119888

1

2

3

4

5

6

7

119886

119888

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

(1)

(3)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(3)

(1)

(2)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(2)

(1)

(3)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(3)

(1)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(1)

(3)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(3)

(1)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(1)

(1)

(0)

(1)

(0)

(1)

(1)

(0)

(1)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(1)

(1)

(0)

(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(19)

Mathematical Problems in Engineering 11

①

② ③ ④ ⑤

⑥⑦

A

B

b

a

C

c

(a)

①

②③

④

⑤

⑥ ⑦

A

Bb

a

C

c

(b)

①

②

③

④ ⑤

⑥

⑦

3

3

3

c

2

11

1

1

1

1

1

1

a

(c)

①

②

③

④

⑤

⑥

⑦

c

a

1

1

1

1

1

1

1

2

3

3

3

(d)

Figure 11 Two 6-gear train kinematic chains and their WDCCG

Step 2 Evidently 119860 = 1198601015840 Assign vertices codes of 119860 and 1198601015840

in two groups (a) [(119886 119888) (1 2 3 4 5 6 7)] and (b) [(119886 119888) (12 3 4 5 6 7)]

Step 3 Calculate NVS(119886) NVS(119887) and their vertex codes oftwo ECMs with the same CE respectively as follows

NVS (119886) = [(V119887 V119886) (V7 V5 V6 V1 V4 V3 V2)]= [(02591 02547)

(03784 02515 02511 02309 02122 02029 01961)]

NVS (119887) = [(V119887 V119886) (V7 V6 V5 V1 V4 V3 V2)]= [(03156 02607)

(03940 02601 02551 02394 02367 02110 02001)]

(20)

Compare NVS(119886) and NVS(119887) and equivalence cannot beseen So it is determined that they are not isomorphic andprogram stops

Example 4 Two 9-gear train kinematic chains are shown inFigures 12(a) and 12(b) Their corresponding WDCCGs aredepicted in Figures 12(c) and 12(d) respectively The directprocess to determine isomorphism of these two kinematicchains is as follows

Step 1 Input their correspondingWAMs119860 and1198601015840 as follows

119860 =

1 2 3 4 5 6 7 8 9 119886 119887 119888

1

2

3

4

5

6

7

8

9

119886

119887

119888

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

(1)

(0)

(3)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(3)

(1)

(3)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(0)

(3)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(3)

(1)

(1)

(1)

(0)

(0)

(0)

(0)

(1)

(3)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(3)

(1)

(3)

(2)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(3)

(1)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(2)

(0)

(1)

(0)

(1)

(1)

(0)

(0)

(0)

(0)

(3)

(0)

(0)

(0)

(0)

(1)

(0)

(1)

(0)

(0)

(0)

(1)

(0)

(1)

(0)

(1)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(1)

(1)

(0)

(0)

(0)

(1)

(1)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

12 Mathematical Problems in Engineering

① ②

③

a

A

c

B

d

C

D

E

b

④

⑤

⑥

⑦

⑧

⑨

(a)

①

②③a

A

b E

c

DC

d

B④

⑤

⑥ ⑦⑧⑨

(b)

①

1 1

c

1 1

1

1

1

b3

3

3

2

1

1

1

1a

33

②

③

④

⑤⑥

⑦

⑧

⑨

(c)

1

1

1

1

3

3

1

1

1

1

1

1

1

2

3

3

3

b

c

a

①②

③

④

⑤

⑥

⑦

⑧

⑨

(d)

Figure 12 Two 9-gear train kinematic chains and their WDCCGs

1198601015840=

1 2 3 4 5 6 7 8 9 119886 119887 119888

1

2

3

4

5

6

7

8

9

119886

119887

119888

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

(1)

(0)

(0)

(3)

(0)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(1)

(0)

(3)

(0)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(1)

(2)

(0)

(0)

(0)

(0)

(0)

(1)

(0)

(1)

(3)

(3)

(2)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(3)

(3)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(3)

(1)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(3)

(0)

(1)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(3)

(1)

(1)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(3)

(1)

(0)

(0)

(1)

(1)

(1)

(1)

(0)

(1)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(1)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(1)

(0)

(0)

(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(21)

Step 2 Evidently 119860 = 1198601015840 Assign vertices codes of 119860 and 1198601015840

in two groups (a) [(119886 119887 119888) (1 2 3 4 5 6 7 8 9)] and (b) [(119886119887 119888) (1 2 3 4 5 6 7 8 9)]

Step 3 Calculate NVS(119886) NVS(119887) and their vertex codes oftwo ECMs with the same CE respectively as follows

NVS (119886) = [(V119888 V119887 V119886) (V9 V4 V8 V1 V2 V5 V7 V3 V6)]

= [(02845 02815 02075) (02470 02307 02295 02294 02294 02154 02154 01846 01719)]

NVS (119887) = [(V119887 V119888 V119886) (V9 V8 V3 V6 V7 V1 V2 V5 V4)]

= [(02845 02815 02075) (02470 02307 02295 02294 02294 02154 02154 01846 01719)]

(22)

Mathematical Problems in Engineering 13

Compare NVS(119886) and NVS(119887) and equivalence can be seenAnd the node voltages in two vertex code groups are not alldistinct so go to Step 6

Step 4 The vertices (1 2) in Figure 12(a) vertices (6 7) inFigure 12(b) the vertices (5 7) in Figure 12(a) and vertices (12) in Figure 12(b) have the same values in the correspondinggroups so just adjust the position order of the vertices in 1198601015840to obtain a new adjacency matrix 1198601015840 And the correspondingvertices of two WDCCGs are (1-6 2-7 3-5 4-8 5-1 6-4 7-28-3 9-9 119886-119886 119887-119888 119888-119887) so we have 119860 = 119860

1015840

1 It is determined

that they are isomorphic and program stops

5 Algorithm Complexity Analysis

This paper presents a relatively good algorithm for determi-nation of kinematic chains isomorphism with multiple jointand gear train It is feasible for introducing the WDCCGand WAM to describe these two types of kinematic chainsthey dramatically reduce the number of vertices and order ofcorresponding adjacency matrix 119860

The analysis of computational complexity of this isomor-phic identification algorithm is as follows when there areno equivalent node voltages in NVS time consumption isbasically dedicated to solving the linear equation sets If akinematic chain possesses 119899 vertices the order of the nodevoltage equations is 119899 minus 1 Since the adjoint circuit is alinear resistance circuit the node voltage equations forma linear algebra equation set of orders 119899 minus 1 which has acomputational complexity of 119874((119899 minus 1)3) for its solutionSince it needs to solve at most 2119899 adjoint circuits in ourproposed method the computational complexity is of order119874(2119899(119899 minus 1)

3) lt 119874(119899

4) Thus the proposed method has a

computational complexity of a polynomial orderWhen thereare equivalent node voltages in NVS the correspondence ofonly a part of the vertices cannot be determinedThus in theworst situation the proposed method is more efficient thanthose where rowcolumn exchanges are performed blindly

In this paper this algorithm is tested and comparedwith the internationally recognized faster method for planarsimple joint kinematic chains isomorphic identification [20]and they are tested in the same software and hardwareenvironment The test results are shown in Figure 13 (theaverage time of each vertex is 20 times of the kinematic chainwithmultiple joints) test results show that when the vertex ofthe kinematic chain is increased the algorithm proposed inthis paper is more efficient than the algorithm proposed byHe et alThe proposed algorithm by us is very suitable for theisomorphism of the topological graph of kinematic chainsand the algorithm is very efficient at the same time

6 Conclusions

In this paper an algorithm for topological isomorphism iden-tification of planar multiple joint and gear train kinematicchains is introduced

4 6 8 10 12 14 16 18 20 22

The number of vertices

Algorithm proposed by He et al

Algorithm proposed in this paper

00001

0001

001

01

1

The d

ecisi

on ti

me (

s)

Figure 13 The test results of the algorithm proposed in this paperand the algorithm proposed by He et al

Firstly a weighted-double-color-contracted-graph(WDCCG) and a weighted adjacency matrix (WAM) areintroduced to describe the planarmultiple joint and gear trainkinematic chains respectively They dramatically reduce thenumber of vertices and order of the represented adjacencymatrix

Secondly isomorphism identification method of thesetwo types of kinematic chains is carried out by the equivalentcircuit models (ECMs) of WDCCGs with the same completeexcitation (CE) and then uses the solved node voltagesequence (NVS) to determine the corresponding vertices oftwo WDCCGs

Finally an algorithm to identify isomorphic kinematicchains is obtained It is an efficient and easy method tobe realized by computer And this algorithm also can beused in the isomorphism identification of planar simple jointkinematic chains with minor changes

Notations

119860 1198601015840 The incidence matrix of kinematic chain119889119894119895 The elements of incidence matrix

119866119894 Conductance value of edge in the ECM

119898119894119895119896 The weighted value of 119896th edge

1198731198731015840 Variable of equivalent circuit model119899 The number of vertices in WDCCG119899119894 Node 119894 in the ECM

V119894 Node voltages of the node 119894

CE Complete excitationDCG Double color graphECM Equivalent circuit modelNVS Node voltage sequenceTM Topological modelWAM Weighted adjacency matrixWDCCG Weighted double color contracted graph

14 Mathematical Problems in Engineering

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

Thiswork is supported by the Science andTechnology Projectof Quanzhou City 2014G48 and G20140047

References

[1] J J Uicker Jr and A Raicu ldquoAmethod for the identification andrecognition of equivalence of kinematic chainsrdquoMechanismandMachine Theory vol 10 no 5 pp 375ndash383 1975

[2] H S Yan and A S Hall ldquoLinkage characteristic polynomialsdefinitions coefficients by inspectionrdquo Journal of MechanicalDesign vol 103 no 3 pp 578ndash584 1981

[3] H S Yan and A S Hall ldquoLinkage characteristic polynomialsassembly theorems uniquenessrdquo Journal of Mechanical Designvol 104 no 1 pp 11ndash20 1982

[4] T S Mruthyunjaya ldquoA computerized methodology for struc-tural synthesis of kinematic chains Part 1 FormulationrdquoMech-anism and Machine Theory vol 19 no 6 pp 487ndash495 1984

[5] T S Mruthyunjaya ldquoA computerized methodology for struc-tural synthesis of kinematic chains part 2mdashapplication toseveral fully or partially known casesrdquoMechanism andMachineTheory vol 19 no 6 pp 497ndash505 1984

[6] T S Mruthyunjaya ldquoA computerized methodology for struc-tural synthesis of kinematic chains part 3mdashapplication to thenew case of 10-link three- freedom chainsrdquo Mechanism andMachine Theory vol 19 no 6 pp 507ndash530 1984

[7] T S Mruthyunjaya and H R Balasubramanian ldquoIn questof a reliable and efficient computational test for detection ofisomorphism in kinematic chainsrdquo Mechanism and MachineTheory vol 22 no 2 pp 131ndash139 1987

[8] W J Sohn and F Freudenstein ldquoAn application of dual graphs tothe automatic generation of the kinematic structures of mecha-nismsrdquo Journal of Mechanisms Transmissions and Automationvol 108 no 3 pp 392ndash398 1986

[9] A Ambekar and V Agrawal ldquoOn canonical numbering ofkinematic chains and isomorphism problem MAX coderdquo inProceedings of the ASMEMechanisms Conference pp 1615ndash16221986

[10] A G Ambekar and V P Agrawal ldquoCanonical numberingof kinematic chains and isomorphism problem min coderdquoMechanism andMachineTheory vol 22 no 5 pp 453ndash461 1987

[11] C S Tang and T Liu ldquoDegree code a new mechanismidentifierrdquo Journal of Mechanical Design vol 115 no 3 pp 627ndash630 1993

[12] J K Shin and S Krishnamurty ldquoOn identification and canonicalnumbering of pin jointed kinematic chainsrdquo Journal of Mechan-ical Design vol 116 pp 182ndash188 1994

[13] J K Shin and S Krishnamurty ldquoDevelopment of a standardcode for colored graphs and its application to kinematic chainsrdquoJournal of Mechanical Design vol 116 no 1 pp 189ndash196 1994

[14] A C Rao and D Varada Raju ldquoApplication of the hammingnumber technique to detect isomorphism among kinematicchains and inversionsrdquoMechanism andMachineTheory vol 26no 1 pp 55ndash75 1991

[15] A C Rao and C N Rao ldquoLoop based pseudo hammingvalues-1 Testing isomorphism and rating kinematic chainsrdquoMechanism andMachineTheory vol 28 no 1 pp 113ndash127 1992

[16] A C Rao and C N Rao ldquoLoop based pseudo hamming values-II inversions preferred frames and actuatorsrdquo Mechanism andMachine Theory vol 28 no 1 pp 129ndash143 1993

[17] J-K Chu and W-Q Cao ldquoIdentification of isomorphismamong kinematic chains and inversions using linkrsquos adjacent-chain-tablerdquoMechanism and Machine Theory vol 29 no 1 pp53ndash58 1994

[18] A C Rao ldquoApplication of fuzzy logic for the study of iso-morphism inversions symmetry parallelism and mobility inkinematic chainsrdquoMechanism and Machine Theory vol 35 no8 pp 1103ndash1116 2000

[19] P R He W J Zhang Q Li and F X Wu ldquoA new method fordetection of graph isomorphism based on the quadratic formrdquoJournal of Mechanical Design vol 125 no 3 pp 640ndash642 2003

[20] P R He W J Zhang and Q Li ldquoSome further developmenton the eigensystem approach for graph isomorphismdetectionrdquoJournal of the Franklin Institute vol 342 no 6 pp 657ndash6732005

[21] Z Y Chang C Zhang Y H Yang and YWang ldquoA newmethodto mechanism kinematic chain isomorphism identificationrdquoMechanism andMachineTheory vol 37 no 4 pp 411ndash417 2002

[22] J P Cubillo and J BWan ldquoComments onmechanismkinematicchain isomorphism identification using adjacent matricesrdquoMechanism andMachineTheory vol 40 no 2 pp 131ndash139 2005

[23] H F Ding and Z Huang ldquoA unique representation of the kine-matic chain and the atlas databaserdquo Mechanism and MachineTheory vol 42 no 6 pp 637ndash651 2007

[24] H F Ding and Z Huang ldquoA new theory for the topologicalstructure analysis of kinematic chains and its applicationsrdquoMechanism and Machine Theory vol 42 no 10 pp 1264ndash12792007

[25] A C Rao ldquoA genetic algorithm for topological characteristicsof kinematic chainsrdquo Journal of Mechanical Design vol 122 no2 pp 228ndash231 2000

[26] F G Kong Q Li and W J Zhang ldquoAn artificial neuralnetwork approach tomechanism kinematic chain isomorphismidentificationrdquo Mechanism and Machine Theory vol 34 no 2pp 271ndash283 1999

[27] J K Chu The Structural and Dimensional Characteristics ofPlanar Linkages Press of Beihang University of China BeijingChina 1992

[28] L Song J Yang X Zhang and W Q Cao ldquoSpanning treemethod of identifying isomorphism and topological symmetryto planar kinematic chain with multiple jointrdquo Chinese Journalof Mechanical Engineering vol 14 no 1 pp 27ndash31 2001

[29] J G Liu and D J Yu ldquoRepresentations amp isomorphismidentification of planar kinematic chains with multiple jointsbased on the converted adjacent matrixrdquo Chinese Journal ofMechanical Engineering vol 48 pp 15ndash21 2012

[30] R Ravisankar andT SMruthyunjaya ldquoComputerized synthesisof the structure of geared kinematic chainsrdquo Mechanism andMachine Theory vol 20 no 5 pp 367ndash387 1985

[31] J U Kim and B M Kwak ldquoApplication of edge permutationgroup to structural synthesis of epicyclic gear trainsrdquo Mecha-nism and Machine Theory vol 25 no 5 pp 563ndash573 1990

Mathematical Problems in Engineering 15

[32] G Chatterjee and L-W Tsai ldquoComputer-aided sketching ofepicyclic-type automatic transmission gear trainsrdquo Journal ofMechanical Design vol 118 no 3 pp 405ndash411 1996

[33] V V N R Prasadraju and A C Rao ldquoA new technique basedon loops to investigate displacement isomorphism in planetarygear trainsrdquo Journal of Mechanical Design vol 12 pp 666ndash6752002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

Link5

5

LinkLink Link6

e

e

6

Simplejoint

(a)

Link1 2 9

Link Link Link

Link9

aa

1

Multiplejoint

2Link

(b)

Figure 1 The structural of simple joint and multiple joint

f

e

h a

b

j

i k

cd ①

②

③

④

⑤

⑥

⑦⑧

⑨

⑩

(a)

d

h

a

i j

k

c

b

fe

①

②

③

④

⑤

⑥

⑦

⑧

⑨

⑩

(b)

a

1

2

2

2

2

3

3

b

①

②

(c)

Figure 2 A multiple joint kinematic chain and its TM described

chains based on the converted adjacent matrix but convertedadjacent matrix not only increased the order of matrix butalso added the complexity of isomorphism identificationprocess In the structural synthesis process of gear trainkinematic chains correctly identifying graph isomorphismis an important step which generates new structural typesPapers [30ndash32] try to do the structural synthesis of gear trainkinematic chains One of the main reasons for the incorrectidentification or inconsistency in results is that there is a lackof efficient isomorphism tests The purpose of paper [33] isto present an efficient methodology through using the loopand hamming number concept to detect the isomorphismof gear train kinematic chains in an unambiguous way Sofrom the researches above finding a useful method to solvethe problem of isomorphism identification of planar multiplejoint and gear train kinematic chains is the existing researchcontent and further study is necessary

In this paper a new algorithm based on the equivalentcircuit analysis for isomorphism identification of planarmultiple joint and gear train kinematic chains is presentedFirstly weighted-double-color-contracted-graph (WDCCG)and corresponding weighted adjacency matrix (WAM) areintroduced to describe the two special types of kinematicchains respectively Corresponding equivalent circuit model(ECM) of the WDCCG is established which uses circuitanalysis method to obtain the node voltages of every vertexinWDCCGThe solved node voltage sequence (NVS) is usedto determine correspondence vertices of two isomorphism

identification kinematic chains Then a new algorithm todetermine isomorphism of the two types of kinematic chainsis proposed

2 Topological Model of Kinematic Chains

21 Topological Model of Multiple Joint Kinematic ChainsBefore further discussion the following common conceptsare introduced as follows

Simple Joint If two links are connected by a revolute jointthen a simple joint is formed For example a simple joint 119890is formed by connecting links 5 and 6 which is shown inFigure 1(a)

Multiple Joint If there are more than two links connected byrevolute joints at the same location then a multiple joint isformed Amultiple joint composed by 119896 links contains (119896minus1)revolute joints For example a multiple joint a is formed byconnecting links 1 2 and 9 as shown in Figure 1(b) and itcontains 2 revolute joints

A planar simple joint kinematic chain is a kinematic chainonly composed of simple joints if there is a kinematic chaincontaining multiple joint called a multiple joint kinematicchain for example a multiple joint kinematic chain which isshown in Figure 2(a)

In a multiple joint kinematic chain because a multiplejoint 119894 composed of 119896 links contains (119896 minus 1) simple joints

Mathematical Problems in Engineering 3

B

A

b

D

C

c

a

①

② ③

④

⑤⑥

(a)

A b

D

B

c

C

a

①

②

③

④

⑤

(b)a

11 1

1

223

3

3

3①

②

③

④

⑤

⑥

(c)

Figure 3 A gear train kinematic chain and its TM described

defined multiple joint factor 119901119894of a multiple joint 119894 is as

follows

119901119894= 119896 minus 2 (1)

And the total multiple joint factors119875 of a kinematic chaincan be obtained by adding all multiple joint factors of akinematic chain together as follows

119875 = sum119901119894 (2)

So as for a multiple joint kinematic chain as shown inFigure 2(a) it has twomultiple joints 119886 and 119887 these joints 119888 119889119890 119891 ℎ 119894 119895 and 119896 are simple joints Multiple joint 119886 is formedby connecting four links 6 7 8 and 10 so its correspondingmultiple joint factor is 119901

119886= 2 Multiple joint 119887 is formed

by connecting three links 4 5 and 10 so its correspondingmultiple joint factor is 119901

119886= 1 And the total multiple joint

factors of this kinematic chain is 119875 = 3As we know topological model (TM) of multiple joint

kinematic chain can be represented by the double colorgraph (DCG) as shown in Figure 2(b) [23] But it has toomany vertices (number of vertices equal to the numberof links and joints of a kinematic chain) so more storagespace is needed in the process of structural analysis bycomputer In this paper a weighted double color contractedgraph (WDCCG) is introduced to represent the topologicalstructure of kinematic chains The WDCCG model whichwiped off all of the simple joints and simple joint links formthe multiple joint kinematic chain can improve the efficiencyof isomorphism identification The graph is established asfollows solid vertices ldquoerdquo denotemultiple joint links and hol-low vertices ldquoIrdquo denote multiple joints If two multiple jointlinks are connected by simple joints and simple joint linksconnect two corresponding solid vertices with a weightededge (weighted value equal to the number of simple jointsbetween two vertices) If one multiple joint link is connectedwith one multiple joint or two multiple joints are connected

by simple joints and simple joint links connect them witha weighted edge as well (weighted value also equal to thenumber of simple joints between two vertices) For exampleFigure 2(c) is the WDCCG of multiple joint kinematic chainas shown in Figure 2(a)

22 Topological Model of Gear Train Kinematic Chain Inthis paper topological model (TM) of gear train kinematicchains can be represented by two topological graphs Firstlydouble color graph (DCG) is established as follows representthe gears and planet carrier with solid vertices ldquoerdquo gearjoints with hollow vertices ldquoΔrdquo and revolute joint with hollowvertices ldquoIrdquo And connect corresponding solid vertex andhollow vertex with an edge when a gear or a planet carrier isconnected with a gear joint or a revolute joint For exampleFigure 3(b) is the DCG corresponding to gear train kinematicchain as shown in Figure 3(a) But it also has too manyvertices then a weighted double color contracted graph(WDCCG) is introduced to represent gear train kinematicchain Because the gear train kinematic chain contains twotypes of kinematic pairs namely revolute joint and gearjoints the WDCCG model wiped off all of the simple jointsand can be established as follows solid vertices ldquoerdquo denotegears of chain hollow vertices ldquoΔrdquo denote gear joints andhollow vertices ldquoIrdquo denote multiple revolute joints If twogears are connected directly by a revolute joint connecttwo corresponding solid vertices with a 2-weighted edge Iftwo gears are connected directly by a gear joint connecttwo corresponding solid vertices with a 3-weighted edgeIf a gear is connected with a multiple revolute joint alsoconnect corresponding solid vertex and hollow vertex witha 1-weighted edge For example Figure 3(c) is the WDCCGof the gear train kinematic chain as shown in Figure 3(a)

23 Weighted Adjacency Matrix Obviously a multiple jointor gear train kinematic chain and its WDCCG are one-to-one correspondent to each otherTheWDCCGof a kinematic

4 Mathematical Problems in Engineering

chain can be represented by a vertex-vertex weighted adja-cencymatrix (WAM)while the elements ofWAMare definedas follows

119860 = [119889119894119895]119899times119899

=

(119889119894119895) = (119898

1198941198951 1198981198941198952 119898

119894119895119896) 119894 = 119895

(119889119894119895) 119894 = 119895

(3)

where 119899 is the number of vertices in WDCCGWhen 119894 = 119895 if vertex is hollow vertex then 119889

119894119895= 0

otherwise the vertex is solid vertex then 119889119894119895= 1

When 119894 = 119895 if two vertices are not connected directlythen 119889

119894119895= 0 otherwise the two vertices are connected

directly by 119896 weighted edges then 119889119894119895= 1198981198941198951 1198981198941198952 119898

119894119895119896

(1198981198941198951ge 1198981198941198952ge sdot sdot sdot ge 119898

119894119895119896 119898119894119895119896

is the weighted value of 119896thedge)

For example the WAM 119860 of a multiple joint kinematicchain as shown in Figure 2(c) can be expressed as follows

119860 =

1 2 119886 119887

1

2

119886

119887

[

[

[

[

[

[

(1)

(1)

(2)

(2 3)

(1)

(1)

(3)

(2)

(2)

(3)

(0)

(2)

(2 3)

(2)

(2)

(0)

]

]

]

]

]

]

(4)

The WAM 119860 corresponding to a gear train kinematicchain as shown in Figure 3(c) can be expressed as follows

119860

=

1 2 3 4 5 6 119886

1

2

3

4

5

6

119886

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

(1)

(0)

(0)

(0)

(3)

(0)

(1)

(0)

(1)

(0)

(0)

(3)

(2)

(1)

(0)

(0)

(1)

(0)

(0)

(3)

(1)

(0)

(0)

(0)

(1)

(2)

(3)

(1)

(3)

(3)

(0)

(2)

(1)

(0)

(0)

(0)

(2)

(3)

(3)

(0)

(1)

(0)

(1)

(1)

(1)

(1)

(0)

(0)

(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(5)

3 Isomorphism Identification Algorithm

31 Equivalent Circuit Model As we know when two multi-ple joint kinematic chains or two gear trains kinematic chainsare isomorphic their WDCCGs are exactly the same theirvertices and edges are in one-to-one correspondence witheach other and the vertices and edges in correspondencekeep the same incidence and weighted relation Accordingto this definition the necessary and sufficient condition forisomorphism of the two types of kinematic chains is that theyhave the same WAM that is 119860 = 1198601015840

Equivalent circuit model (ECM) of WDCCG is estab-lished by its each edge which is replaced by conductanceequal to the weighted value of corresponding edge119898

119894119895

Theorem 1 If two WDCCGs corresponding to the same typekinematic chains are isomorphic then their correspondingECMs are identical

For example two isomorphism multiple joint kinematicchains are shown in Figures 4(a) and 4(b) their WDCCGsare shown in Figures 4(c) and 4(d) respectively According tothe description above their corresponding ECMs are shownin Figures 5(a) and 5(b) respectively There are two identicalcircuits and nodes of these two ECMs are in one-to-onecorrespondence

Complete excitation (CE) of an ECM is established asfollows in an ECM with 119899 nodes firstly set a node (119899 + 1) tobe a reference node then connect the reference node (119899 + 1)with every other node 119899

119894(119894 = 1 2 119899) of the ECM whose

conductance values are equal to the sumof weighted values ofthe node edges Secondly apply the same current source 1 Abetween node (119899+1) and other nodes 119899

119894(119894 = 1 2 119899) of the

ECM respectively in directions of the currents going fromnode (119899 + 1) to other nodes 119899

119894 (119894 = 1 2 119899) respectively

According to circuit theory two identical circuits underthe same CE have the same response And their node voltagescan be solved by the nodal method of circuit analysis Thenodal method of circuit analysis can be expressed as followsin a circuit network choose a node as a reference nodethe voltage difference between each node and the referencenode is known as the node voltage of the node For a circuitnetwork with 119899 nodes which has (119899 minus 1) node voltages if thenode 119899 is the reference node the nodal voltage equation canbe expressed as

[

[

[

[

[

[

[

11986611

11986612

sdot sdot sdot 1198661(119899minus1)

11986621

11986622

sdot sdot sdot 1198662(119899minus1)

d

119866(119899minus1)1

119866(119899minus1)2

sdot sdot sdot 119866(119899minus1)(119899minus1)

]

]

]

]

]

]

]

[

[

[

[

[

[

[

V1198991

V1198992

V119899(119899minus1)

]

]

]

]

]

]

]

=

[

[

[

[

[

[

[

1198681199041198991

1198681199041198992

119868119904119899(119899minus1)

]

]

]

]

]

]

]

(6)

where119866119894119894(119894 = 1 2 119899 minus 1) is called the self admittance of a

node 119894 the value of which is the sum of the admittance of allbranches that are connected to the node 119894119866

119894119895(119894 = 1 2 119899minus

1 119895 = 1 2 119899minus1) is called themutual admittance of node 119894and node 119895 whose value is the sum of the branch admittancebetween node 119894 and node 119895 and 119868

119904119899119894(119894 = 1 2 119899 minus 1) is

called the algebraic sum of the current source for the inflownode 119894 (inflow is positive outflow is negative)

For example an ECM with 4 nodes is shown in Fig-ure 6(a) Apply the CE of reference node 5 as shown inFigure 6(b) By nodalmethodof circuit analysis equations (6)node voltages V

1 V2 V3 and V

4can be expressed as follows

Mathematical Problems in Engineering 5

①

②

③

④

⑤

⑥

⑦

⑧

⑨

⑩

a

b

(a)

①

②

③

④

⑤

⑥

⑦

⑧ ⑨⑩

b

a

(b)

①

②③

a

b

1

1

1

2

223

3

(c)

①

②③

a

b1

1

1

2

2

2

3

3

(d)

Figure 4 Two multiple joint kinematic chains and their WDCCG

①

②③

2 2

2

11

1

a

3

3b

(a)

①

②③

2

2

2

1

1

1a

3

3

b

(b)

Figure 5 ECM corresponding to two WDCCGs of Figures 4(c) and 4(d) respectively

G1

G2

G3

G4 G5

① ② ③

④

(a)

G6

G7

G8

G9

①

②

③

④

⑤

1

1

1

1

(b)

Figure 6 An ECM with 4 nodes and its CE by reference node 5

6 Mathematical Problems in Engineering

[

[

[

[

[

[

1198661+ 1198664+ 1198666

minus1198664

0 minus1198661

minus1198664

1198662+ 1198664+ 1198665+ 1198667

minus1198665

minus1198662

0 minus1198665

1198663+ 1198665+ 1198668

minus1198663

minus1198661

minus1198662

minus1198663

1198661+ 1198662+ 1198663+ 1198669

]

]

]

]

]

]

[

[

[

[

[

[

V1

V2

V3

V4

]

]

]

]

]

]

=

[

[

[

[

[

[

1

1

1

1

]

]

]

]

]

]

(7)

Then the node voltages V1 V2 V3 and V

4can be obtained

by solving the above equations (7) Therefore the followingconclusion can be obtained

Theorem 2 If two WDCCGs corresponding to the same typeof kinematic chains are isomorphic their two correspondingECMs 119873 and 1198731015840 can be established and excited by the sameCE as above respectively Then the node voltages of eachcorrespondence node of two ECMs119873 and1198731015840 are the same

For example two identical ECMs119873 and1198731015840 with 5 nodesare shown in Figures 5(a) and 5(b) Apply the same CE ofreference node 6 which is shown in Figures 7(a) and 7(b)respectively And their corresponding WAMs 119860 and 1198601015840 canbe obtained as follows

119860 =

1 2 3 119886 119887

1

2

3

119886

119887

[

[

[

[

[

[

[

[

[

(1)

(2)

(0)

(1)

(2)

(2)

(1)

(1)

(0)

(3)

(0)

(1)

(1)

(2 3)

(1)

(1)

(0)

(2 3)

(0)

(0)

(2)

(3)

(1)

(0)

(0)

]

]

]

]

]

]

]

]

]

1198601015840=

1 2 3 119886 119887

1

2

3

119886

119887

[

[

[

[

[

[

[

[

[

(1)

(1)

(2)

(3)

(0)

(1)

(1)

(0)

(1)

(2 3)

(2)

(0)

(1)

(2)

(1)

(3)

(1)

(2)

(0)

(0)

(0)

(2 3)

(1)

(0)

(0)

]

]

]

]

]

]

]

]

]

(8)

By nodal method of circuit analysis equations of nodevoltages V

1 V2 V3 V119886 and V

119887can be expressed respectively

as follows

[

[

[

[

[

[

[

[

[

10 minus2 0 minus1 minus2

minus2 12 minus1 0 minus3

0 minus1 14 minus5 minus1

minus1 0 minus5 12 0

minus2 minus3 minus1 0 12

]

]

]

]

]

]

]

]

]

[

[

[

[

[

[

[

[

[

V1

V2

V3

V119886

V119887

]

]

]

]

]

]

]

]

]

=

[

[

[

[

[

[

[

[

[

1

1

1

1

1

]

]

]

]

]

]

]

]

]

(9)

[

[

[

[

[

[

[

[

[

12 minus1 minus2 minus3 0

minus1 14 0 minus1 minus5

minus2 0 10 minus2 minus1

minus3 minus1 minus2 12 0

0 minus5 minus1 0 12

]

]

]

]

]

]

]

]

]

[

[

[

[

[

[

[

[

[

V1

V2

V3

V119886

V119887

]

]

]

]

]

]

]

]

]

=

[

[

[

[

[

[

[

[

[

1

1

1

1

1

]

]

]

]

]

]

]

]

]

(10)

Then node voltages V1 V2 V3 V119886 and V

119887of two ECMs can

be obtained by solving the above (9) and (10) respectively asshown in Table 1

From Theorem 2 node voltages of each correspondencenode in these two ECMs are the same as follows V

119886= V119887=

01627 V119887= V119886= 01691 V

1= V3= 01839 V

2= V1= 01691

and V3= V2= 01537 When arranging the obtained node

voltages in descending order with the same node voltagesstaying together this set is called the node voltage sequence(NVS) and denoted as

NVS = lfloor(V1 V2 V

1198991

) (V(1198991+1) V(1198991+2) V

119899)rfloor

(V1ge V2ge V1198991

V(1198991+1)ge V(1198991+2)ge sdot sdot sdot ge V

119899)

(11)

where the variables V1 V2 V

1198991

represent node voltages ofhollow vertex The variables V

(1198991+1) V(1198991+2) V

119899represent

node voltages of solid vertexIf two identical ECMs are stimulated by the same CE as

above then the NVSs in correspondence are the same Forexample two identical ECMs as showed in Figures 5(a) and5(b) the NVS and vertex codes can be obtained respectivelyas follows

NVS (119886) = [(V119887 V119886) (V1 V2 V3)]

= [(01691 01627) (01839 01691 01537)]

NVS (119887) = [(V119886 V119887) (V3 V1 V2)]

= [(01691 01537) (01839 01691 01627)]

(12)

If the NVSs of two ECMs are not the same then the twoWDCCGs are not isomorphic On the contrary if they arethe same correspondence vertices in these two WDCCGscan be determined by the element values of NVS Thenperform the row exchanges of WAM 119860

1015840 to determine if theyare isomorphic Because the NVS can reduce the number ofrow exchanges to several ones or only just one this is a veryeffective method For example by the NVS(119886) and NVS(119887)of two ECMs shown in Figures 5(a) and 5(b) the verticesin correspondence are 119886-119887 119887-119886 1-3 2-1 3-2 then throughexchanging the row of WAM 119860

1015840 we obtain 119860 = 1198601015840 which

are isomorphic

Mathematical Problems in Engineering 7

①

②

③

⑥

1 1

1

11

6

6

7

6

5

a

b

(a)

①

②

③

⑥

1 1

1

1

1

6

6

6

7

5

a

b

(b)

Figure 7 Two ECMs with the same CE respectively

Table 1 Node voltages of two ECMs with the same CE

Vertex type Hollow vertex Solid vertex

Figure 5(a) Vertex code 119886 119887 1 2 3Node voltage 01627 01691 01839 01691 01537

Figure 5(b) Vertex code 119886 119887 1 2 3Node voltage 01691 01627 01691 01537 01839

For example two gear train kinematic chains are shownin Figures 8(a) and 8(b) and their WDCCGs as shownin Figures 8(c) and 8(d) respectively Their correspondingWAMs are 119860 and 1198601015840 as follows

119860

=

1 2 3 4 5 6 119887

1

2

3

4

5

6

119887

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

(1)

(2)

(0)

(0)

(3)

(0)

(1)

(2)

(1)

(3)

(0)

(0)

(0)

(0)

(0)

(3)

(1)

(3)

(0)

(0)

(1)

(0)

(0)

(3)

(1)

(0)

(2)

(0)

(3)

(0)

(0)

(0)

(1)

(0)

(1)

(0)

(0)

(0)

(2)

(0)

(1)

(1)

(1)

(0)

(1)

(0)

(1)

(1)

(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

1198601015840

=

1 2 3 4 5 6 119887

1

2

3

4

5

6

119887

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

(1)

(0)

(0)

(3)

(2)

(0)

(1)

(0)

(1)

(0)

(0)

(3)

(3)

(1)

(0)

(0)

(1)

(0)

(0)

(2)

(1)

(3)

(0)

(0)

(1)

(0)

(0)

(1)

(2)

(3)

(0)

(0)

(1)

(0)

(0)

(0)

(3)

(2)

(0)

(0)

(1)

(0)

(1)

(1)

(1)

(1)

(0)

(0)

(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(13)

According to the abovemethod their correspondingNVSand vertex codes can be obtained as follows

NVS (119886) = [(V119887) (V6 V5 V4 V1 V2 V3)] = [(02330)

(02744 02257 02068 01908 01907 01737)]

NVS (119887) = [(V119887) (V3 V4 V5 V1 V6 V2)] = [(02330)

(02744 02257 02068 01908 01907 01737)]

(14)

By the NVS(119886) and NVS(119887) the vertices of these twoWDCCGs in correspondence can be obtained as follows 119887-1198871-1 2-6 3-2 4-5 5-4 6-3 and through exchanging the rowof WAM 119860

1015840 one time we obtain isomorphic 119860 = 1198601015840

32 Algorithm Here a direct algorithm is given to determineisomorphism of planar multiple joint and gear train kine-matic chains and it can be expressed as follows

Step 1 Input the vertex codes and WAMs 119860 1198601015840 of twoWDCCGs corresponding to the same type of kinematicchains

Step 2 Assign vertices codes of 119860 and 1198601015840 in two groups thehollow vertexes to be the first group and the solid vertexes tobe another group

Step 3 Calculate NVS(119886) NVS(119887) and their vertex codesof the two ECMs with the same CE respectively CompareNVS(119886) and NVS(119887) if equivalence cannot be seen it isdetermined that they are not isomorphic and program stopsotherwise it is determined that they could be isomorphicThen check the NVS(119886) and NVS(119887) to see if the values intwo vertex code groups are all distinct if yes go to Step 4 ifno go to Step 6

8 Mathematical Problems in Engineering

①

②

③

④

⑤

⑥

A

a

b

C

B

c

(a)

① ②③ ④

⑤

⑥

A

a

b

C

B

c

(b)

①

②

③

④

⑤

⑥

1

b

1

1

1

2

3

3

3 2

(c)

①

②

③

④

⑤

⑥

3

3

3

2

2

b

1

1

1

1

(d)

Figure 8 Two gear train kinematic chains and their WDCCGs

Step 4 Take the NVS(119886) NVS(119887) and their vertex codes todetermine the correspondence of vertices according to thevalues that are in correspondence

Step 5 According to the corresponding vertices obtained andthe vertex codes to form new WAM 119860

1015840

1 rewrite WAM 119860

1015840 If119860 = 119860

1015840

1 it is determined that they are isomorphic otherwise

they are not isomorphic and program stops

Step 6 According to the corresponding vertices obtained theNVS(119886) and NVS(119887) have the same values in the correspond-ing two vertex code groups Just adjust the position order ofvertices of the same group in1198601015840 to obtain vertex codes to fromnew adjacency matrix 1198601015840

1 If 119860 = 119860

1015840

1 it is determined that

they are isomorphic and program stops if 119860 = 1198601015840

1 keep on

adjusting the position order of the aforementioned verticesin 1198601015840 of the vertex code groups for all possibilities And if119860 = 119860

1015840

1always it is determined that they are not isomorphic

and program stops

4 Examples

Example 1 Two 10-link multiple joint kinematic chainsare shown in Figures 9(a) and 9(b) Their correspondingWDCCGs are depicted in Figures 9(c) and 9(d) respectively

The direct process to determine isomorphism of these twokinematic chains can be expressed as follows

Step 1 Input their corresponding vertex codes andWAMs1198601198601015840 as follows

119860 =

1 119886 119887 119888 119889

1

119886

119887

119888

119889

[

[

[

[

[

[

[

[

[

(1)

(2)

(0)

(1)

(2)

(2)

(0)

(2)

(0)

(3)

(0)

(2)

(0)

(2)

(2)

(1)

(0)

(2)

(0)

(3)

(2)

(3)

(2)

(3)

(0)

]

]

]

]

]

]

]

]

]

1198601015840=

1 119886 119887 119888 119889

1

119886

119887

119888

119889

[

[

[

[

[

[

[

[

[

(1)

(0)

(2)

(2)

(1)

(0)

(0)

(2)

(2)

(2)

(2)

(2)

(0)

(0)

(3)

(2)

(2)

(0)

(0)

(3)

(1)

(2)

(3)

(3)

(0)

]

]

]

]

]

]

]

]

]

(15)

Step 2 Evidently 119860 = 1198601015840 Assign vertices codes of 119860 and 1198601015840

in two groups (a) [(119886 119887 119888 119889) (1)] and (b) [(119886 119887 119888 119889) (1)]

Mathematical Problems in Engineering 9

①

②

③

④

⑤

⑥

⑦

⑧

⑨

⑩

c

ba

d

(a)

①

②

③ ④

⑤

⑥⑦

⑧

⑨⑩

c b

a

d

(b)

①

cb

a

2

2

2

2

2

3

3

1d

(c)

①

2

2

2

2

2

3

3

d

b

ac

1

(d)

Figure 9 Two 10-link multiple joint kinematic chains and their WDCCGs

①

⑪

②

③

④

⑤

⑥

⑦⑧

⑨

⑩

ab

(a)

①

②

③④

⑤⑥

⑦

⑧

⑨

⑩

a

b

⑪

(b)

①

② ③

④

⑤

⑥ 1

1

11

12

2

2

2

2a

b

1

1 1

(c)

① ②

③

④

⑤

⑥

1

1

1

1

1

11

12

2

2

2

2b a

(d)

Figure 10 Two 11-link multiple joint kinematic chains and their WDCCGs

Step 3 Calculate NVS(119886) NVS(119887) and their vertex codes oftwo ECMs with the same CE respectively as follows

NVS (119886) = [(V119888 V119887 V119886 V119889) (V1)]

= [(01551 01546 01451 01275) (01700)]

NVS (119887) = [(V119886 V119887 V119888 V119889) (V1)]

= [(01538 01459 01459 01308) (01714)]

(16)

Compare NVS(119886) and NVS(119887) and equivalence cannot beseen It is determined that they are not isomorphic andprogram stops

Example 2 Two 11-link multiple joint kinematic chains areshown in Figures 10(a) and 10(b)Their correspondingWDC-CGs are described in Figures 10(c) and 10(d) respectivelyThe direct process to determine isomorphism of these twokinematic chains is as follows

10 Mathematical Problems in Engineering

Step 1 Input their corresponding vertex codes andWAMs1198601198601015840 as follows

119860

=

1 2 3 4 5 6 119886 119887

1

2

3

4

5

6

119886

119887

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

(1)

(2)

(0)

(1)

(1)

(0)

(0)

(2)

(2)

(1)

(1)

(0)

(1)

(0)

(0)

(0)

(0)

(1)

(1)

(1)

(0)

(2)

(2)

(0)

(1)

(0)

(1)

(1)

(0)

(1)

(0)

(0)

(1)

(1)

(0)

(0)

(1)

(0)

(1)

(0)

(0)

(2)

(0)

(1)

(0)

(1)

(0)

(1)

(0)

(0)

(1)

(0)

(1)

(0)

(0)

(2)

(2)

(0)

(0)

(0)

(0)

(1)

(2)

(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

1198601015840

=

1 2 3 4 5 6 119886 119887

1

2

3

4

5

6

119886

119887

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

(1)

(2)

(0)

(1)

(0)

(0)

(0)

(1)

(2)

(1)

(0)

(1)

(1)

(0)

(2)

(0)

(0)

(0)

(1)

(1)

(2)

(1)

(0)

(2)

(1)

(1)

(1)

(1)

(0)

(0)

(0)

(0)

(0)

(1)

(2)

(0)

(1)

(1)

(0)

(0)

(0)

(0)

(1)

(0)

(1)

(1)

(1)

(0)

(0)

(2)

(0)

(0)

(0)

(1)

(0)

(2)

(1)

(0)

(2)

(0)

(0)

(0)

(2)

(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(17)

Step 2 Evidently 119860 = 1198601015840 Assign vertices codes of 119860 and 1198601015840

in two groups (a) [(119886 119887) (1 2 3 4 5 6)] and (b) [(119886 119887) (1 23 4 5 6)]

Step 3 Calculate NVS(119886) NVS(119887) and their vertex codes oftwo ECMs with the same CE respectively as follows

NVS (119886) = [(V119886 V119887) (V5 V4 V2 V6 V1 V3)]

= [(02084 02056)

(0743 02727 02348 02347 02023 01995)]

NVS (119887) = [(V119887 V119888) (V6 V4 V5 V1 V3 V2)]

= [(02084 02056)

(0743 02727 02348 02347 02023 01995)]

(18)

Compare NVS(119886) and NVS(119887) and equivalence can beseen So the node voltages in two vertex code groups are alldistinct go to Step 4

Step 4 Take the NVS(119886) NVS(119887) and their vertex codes todetermine the corresponding vertices of two WDCCGs asfollows (119886-119887 119887-119886 1-3 2-5 3-2 4-4 5-6 6-1)

Step 5 According to the corresponding vertices obtained andthe vertex codes to form new WAM 119860

1015840

1 rewrite WAM 119860

1015840Then we have 119860 = 1198601015840

1 so they are isomorphic and program

stops

Example 3 Two 6-gear train kinematic chains are depicted inFigures 11(a) and 11(b) and their correspondingWDCCGs aredescribed in Figures 11(c) and 11(d) respectively The directprocess to determine isomorphism of these two kinematicchains is as follows

Step 1 Input their corresponding vertex codes andWAMs1198601198601015840 as follows

119860 =

1 2 3 4 5 6 7 119886 119888

1

2

3

4

5

6

7

119886

119888

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

(1)

(3)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(3)

(1)

(2)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(2)

(1)

(3)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(3)

(1)

(0)

(0)

(0)

(1)

(1)

(0)

(0)

(0)

(0)

(1)

(3)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(3)

(1)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(1)

(1)

(0)

(1)

(0)

(1)

(0)

(1)

(1)

(0)

(0)

(1)

(0)

(0)

(1)

(1)

(0)

(1)

(0)

(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

1198601015840=

1 2 3 4 5 6 7 119886 119888

1

2

3

4

5

6

7

119886

119888

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

(1)

(3)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(3)

(1)

(2)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(2)

(1)

(3)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(3)

(1)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(1)

(3)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(3)

(1)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(1)

(1)

(0)

(1)

(0)

(1)

(1)

(0)

(1)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(1)

(1)

(0)

(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(19)

Mathematical Problems in Engineering 11

①

② ③ ④ ⑤

⑥⑦

A

B

b

a

C

c

(a)

①

②③

④

⑤

⑥ ⑦

A

Bb

a

C

c

(b)

①

②

③

④ ⑤

⑥

⑦

3

3

3

c

2

11

1

1

1

1

1

1

a

(c)

①

②

③

④

⑤

⑥

⑦

c

a

1

1

1

1

1

1

1

2

3

3

3

(d)

Figure 11 Two 6-gear train kinematic chains and their WDCCG

Step 2 Evidently 119860 = 1198601015840 Assign vertices codes of 119860 and 1198601015840

in two groups (a) [(119886 119888) (1 2 3 4 5 6 7)] and (b) [(119886 119888) (12 3 4 5 6 7)]

Step 3 Calculate NVS(119886) NVS(119887) and their vertex codes oftwo ECMs with the same CE respectively as follows

NVS (119886) = [(V119887 V119886) (V7 V5 V6 V1 V4 V3 V2)]= [(02591 02547)

(03784 02515 02511 02309 02122 02029 01961)]

NVS (119887) = [(V119887 V119886) (V7 V6 V5 V1 V4 V3 V2)]= [(03156 02607)

(03940 02601 02551 02394 02367 02110 02001)]

(20)

Compare NVS(119886) and NVS(119887) and equivalence cannot beseen So it is determined that they are not isomorphic andprogram stops

Example 4 Two 9-gear train kinematic chains are shown inFigures 12(a) and 12(b) Their corresponding WDCCGs aredepicted in Figures 12(c) and 12(d) respectively The directprocess to determine isomorphism of these two kinematicchains is as follows

Step 1 Input their correspondingWAMs119860 and1198601015840 as follows

119860 =

1 2 3 4 5 6 7 8 9 119886 119887 119888

1

2

3

4

5

6

7

8

9

119886

119887

119888

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

(1)

(0)

(3)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(3)

(1)

(3)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(0)

(3)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(3)

(1)

(1)

(1)

(0)

(0)

(0)

(0)

(1)

(3)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(3)

(1)

(3)

(2)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(3)

(1)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(2)

(0)

(1)

(0)

(1)

(1)

(0)

(0)

(0)

(0)

(3)

(0)

(0)

(0)

(0)

(1)

(0)

(1)

(0)

(0)

(0)

(1)

(0)

(1)

(0)

(1)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(1)

(1)

(0)

(0)

(0)

(1)

(1)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

12 Mathematical Problems in Engineering

① ②

③

a

A

c

B

d

C

D

E

b

④

⑤

⑥

⑦

⑧

⑨

(a)

①

②③a

A

b E

c

DC

d

B④

⑤

⑥ ⑦⑧⑨

(b)

①

1 1

c

1 1

1

1

1

b3

3

3

2

1

1

1

1a

33

②

③

④

⑤⑥

⑦

⑧

⑨

(c)

1

1

1

1

3

3

1

1

1

1

1

1

1

2

3

3

3

b

c

a

①②

③

④

⑤

⑥

⑦

⑧

⑨

(d)

Figure 12 Two 9-gear train kinematic chains and their WDCCGs

1198601015840=

1 2 3 4 5 6 7 8 9 119886 119887 119888

1

2

3

4

5

6

7

8

9

119886

119887

119888

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

(1)

(0)

(0)

(3)

(0)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(1)

(0)

(3)

(0)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(1)

(2)

(0)

(0)

(0)

(0)

(0)

(1)

(0)

(1)

(3)

(3)

(2)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(3)

(3)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(3)

(1)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(3)

(0)

(1)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(3)

(1)

(1)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(3)

(1)

(0)

(0)

(1)

(1)

(1)

(1)

(0)

(1)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(1)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(1)

(0)

(0)

(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(21)

Step 2 Evidently 119860 = 1198601015840 Assign vertices codes of 119860 and 1198601015840

in two groups (a) [(119886 119887 119888) (1 2 3 4 5 6 7 8 9)] and (b) [(119886119887 119888) (1 2 3 4 5 6 7 8 9)]

Step 3 Calculate NVS(119886) NVS(119887) and their vertex codes oftwo ECMs with the same CE respectively as follows

NVS (119886) = [(V119888 V119887 V119886) (V9 V4 V8 V1 V2 V5 V7 V3 V6)]

= [(02845 02815 02075) (02470 02307 02295 02294 02294 02154 02154 01846 01719)]

NVS (119887) = [(V119887 V119888 V119886) (V9 V8 V3 V6 V7 V1 V2 V5 V4)]

= [(02845 02815 02075) (02470 02307 02295 02294 02294 02154 02154 01846 01719)]

(22)

Mathematical Problems in Engineering 13

Compare NVS(119886) and NVS(119887) and equivalence can be seenAnd the node voltages in two vertex code groups are not alldistinct so go to Step 6

Step 4 The vertices (1 2) in Figure 12(a) vertices (6 7) inFigure 12(b) the vertices (5 7) in Figure 12(a) and vertices (12) in Figure 12(b) have the same values in the correspondinggroups so just adjust the position order of the vertices in 1198601015840to obtain a new adjacency matrix 1198601015840 And the correspondingvertices of two WDCCGs are (1-6 2-7 3-5 4-8 5-1 6-4 7-28-3 9-9 119886-119886 119887-119888 119888-119887) so we have 119860 = 119860

1015840

1 It is determined

that they are isomorphic and program stops

5 Algorithm Complexity Analysis

This paper presents a relatively good algorithm for determi-nation of kinematic chains isomorphism with multiple jointand gear train It is feasible for introducing the WDCCGand WAM to describe these two types of kinematic chainsthey dramatically reduce the number of vertices and order ofcorresponding adjacency matrix 119860

The analysis of computational complexity of this isomor-phic identification algorithm is as follows when there areno equivalent node voltages in NVS time consumption isbasically dedicated to solving the linear equation sets If akinematic chain possesses 119899 vertices the order of the nodevoltage equations is 119899 minus 1 Since the adjoint circuit is alinear resistance circuit the node voltage equations forma linear algebra equation set of orders 119899 minus 1 which has acomputational complexity of 119874((119899 minus 1)3) for its solutionSince it needs to solve at most 2119899 adjoint circuits in ourproposed method the computational complexity is of order119874(2119899(119899 minus 1)

3) lt 119874(119899

4) Thus the proposed method has a

computational complexity of a polynomial orderWhen thereare equivalent node voltages in NVS the correspondence ofonly a part of the vertices cannot be determinedThus in theworst situation the proposed method is more efficient thanthose where rowcolumn exchanges are performed blindly

In this paper this algorithm is tested and comparedwith the internationally recognized faster method for planarsimple joint kinematic chains isomorphic identification [20]and they are tested in the same software and hardwareenvironment The test results are shown in Figure 13 (theaverage time of each vertex is 20 times of the kinematic chainwithmultiple joints) test results show that when the vertex ofthe kinematic chain is increased the algorithm proposed inthis paper is more efficient than the algorithm proposed byHe et alThe proposed algorithm by us is very suitable for theisomorphism of the topological graph of kinematic chainsand the algorithm is very efficient at the same time

6 Conclusions

In this paper an algorithm for topological isomorphism iden-tification of planar multiple joint and gear train kinematicchains is introduced

4 6 8 10 12 14 16 18 20 22

The number of vertices

Algorithm proposed by He et al

Algorithm proposed in this paper

00001

0001

001

01

1

The d

ecisi

on ti

me (

s)

Figure 13 The test results of the algorithm proposed in this paperand the algorithm proposed by He et al

Firstly a weighted-double-color-contracted-graph(WDCCG) and a weighted adjacency matrix (WAM) areintroduced to describe the planarmultiple joint and gear trainkinematic chains respectively They dramatically reduce thenumber of vertices and order of the represented adjacencymatrix

Secondly isomorphism identification method of thesetwo types of kinematic chains is carried out by the equivalentcircuit models (ECMs) of WDCCGs with the same completeexcitation (CE) and then uses the solved node voltagesequence (NVS) to determine the corresponding vertices oftwo WDCCGs

Finally an algorithm to identify isomorphic kinematicchains is obtained It is an efficient and easy method tobe realized by computer And this algorithm also can beused in the isomorphism identification of planar simple jointkinematic chains with minor changes

Notations

119860 1198601015840 The incidence matrix of kinematic chain119889119894119895 The elements of incidence matrix

119866119894 Conductance value of edge in the ECM

119898119894119895119896 The weighted value of 119896th edge

1198731198731015840 Variable of equivalent circuit model119899 The number of vertices in WDCCG119899119894 Node 119894 in the ECM

V119894 Node voltages of the node 119894

CE Complete excitationDCG Double color graphECM Equivalent circuit modelNVS Node voltage sequenceTM Topological modelWAM Weighted adjacency matrixWDCCG Weighted double color contracted graph

14 Mathematical Problems in Engineering

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

Thiswork is supported by the Science andTechnology Projectof Quanzhou City 2014G48 and G20140047

References

[1] J J Uicker Jr and A Raicu ldquoAmethod for the identification andrecognition of equivalence of kinematic chainsrdquoMechanismandMachine Theory vol 10 no 5 pp 375ndash383 1975

[2] H S Yan and A S Hall ldquoLinkage characteristic polynomialsdefinitions coefficients by inspectionrdquo Journal of MechanicalDesign vol 103 no 3 pp 578ndash584 1981

[3] H S Yan and A S Hall ldquoLinkage characteristic polynomialsassembly theorems uniquenessrdquo Journal of Mechanical Designvol 104 no 1 pp 11ndash20 1982

[4] T S Mruthyunjaya ldquoA computerized methodology for struc-tural synthesis of kinematic chains Part 1 FormulationrdquoMech-anism and Machine Theory vol 19 no 6 pp 487ndash495 1984

[5] T S Mruthyunjaya ldquoA computerized methodology for struc-tural synthesis of kinematic chains part 2mdashapplication toseveral fully or partially known casesrdquoMechanism andMachineTheory vol 19 no 6 pp 497ndash505 1984

[6] T S Mruthyunjaya ldquoA computerized methodology for struc-tural synthesis of kinematic chains part 3mdashapplication to thenew case of 10-link three- freedom chainsrdquo Mechanism andMachine Theory vol 19 no 6 pp 507ndash530 1984

[7] T S Mruthyunjaya and H R Balasubramanian ldquoIn questof a reliable and efficient computational test for detection ofisomorphism in kinematic chainsrdquo Mechanism and MachineTheory vol 22 no 2 pp 131ndash139 1987

[8] W J Sohn and F Freudenstein ldquoAn application of dual graphs tothe automatic generation of the kinematic structures of mecha-nismsrdquo Journal of Mechanisms Transmissions and Automationvol 108 no 3 pp 392ndash398 1986

[9] A Ambekar and V Agrawal ldquoOn canonical numbering ofkinematic chains and isomorphism problem MAX coderdquo inProceedings of the ASMEMechanisms Conference pp 1615ndash16221986

[10] A G Ambekar and V P Agrawal ldquoCanonical numberingof kinematic chains and isomorphism problem min coderdquoMechanism andMachineTheory vol 22 no 5 pp 453ndash461 1987

[11] C S Tang and T Liu ldquoDegree code a new mechanismidentifierrdquo Journal of Mechanical Design vol 115 no 3 pp 627ndash630 1993

[12] J K Shin and S Krishnamurty ldquoOn identification and canonicalnumbering of pin jointed kinematic chainsrdquo Journal of Mechan-ical Design vol 116 pp 182ndash188 1994

[13] J K Shin and S Krishnamurty ldquoDevelopment of a standardcode for colored graphs and its application to kinematic chainsrdquoJournal of Mechanical Design vol 116 no 1 pp 189ndash196 1994

[14] A C Rao and D Varada Raju ldquoApplication of the hammingnumber technique to detect isomorphism among kinematicchains and inversionsrdquoMechanism andMachineTheory vol 26no 1 pp 55ndash75 1991

[15] A C Rao and C N Rao ldquoLoop based pseudo hammingvalues-1 Testing isomorphism and rating kinematic chainsrdquoMechanism andMachineTheory vol 28 no 1 pp 113ndash127 1992

[16] A C Rao and C N Rao ldquoLoop based pseudo hamming values-II inversions preferred frames and actuatorsrdquo Mechanism andMachine Theory vol 28 no 1 pp 129ndash143 1993

[17] J-K Chu and W-Q Cao ldquoIdentification of isomorphismamong kinematic chains and inversions using linkrsquos adjacent-chain-tablerdquoMechanism and Machine Theory vol 29 no 1 pp53ndash58 1994

[18] A C Rao ldquoApplication of fuzzy logic for the study of iso-morphism inversions symmetry parallelism and mobility inkinematic chainsrdquoMechanism and Machine Theory vol 35 no8 pp 1103ndash1116 2000

[19] P R He W J Zhang Q Li and F X Wu ldquoA new method fordetection of graph isomorphism based on the quadratic formrdquoJournal of Mechanical Design vol 125 no 3 pp 640ndash642 2003

[20] P R He W J Zhang and Q Li ldquoSome further developmenton the eigensystem approach for graph isomorphismdetectionrdquoJournal of the Franklin Institute vol 342 no 6 pp 657ndash6732005

[21] Z Y Chang C Zhang Y H Yang and YWang ldquoA newmethodto mechanism kinematic chain isomorphism identificationrdquoMechanism andMachineTheory vol 37 no 4 pp 411ndash417 2002

[22] J P Cubillo and J BWan ldquoComments onmechanismkinematicchain isomorphism identification using adjacent matricesrdquoMechanism andMachineTheory vol 40 no 2 pp 131ndash139 2005

[23] H F Ding and Z Huang ldquoA unique representation of the kine-matic chain and the atlas databaserdquo Mechanism and MachineTheory vol 42 no 6 pp 637ndash651 2007

[24] H F Ding and Z Huang ldquoA new theory for the topologicalstructure analysis of kinematic chains and its applicationsrdquoMechanism and Machine Theory vol 42 no 10 pp 1264ndash12792007

[25] A C Rao ldquoA genetic algorithm for topological characteristicsof kinematic chainsrdquo Journal of Mechanical Design vol 122 no2 pp 228ndash231 2000

[26] F G Kong Q Li and W J Zhang ldquoAn artificial neuralnetwork approach tomechanism kinematic chain isomorphismidentificationrdquo Mechanism and Machine Theory vol 34 no 2pp 271ndash283 1999

[27] J K Chu The Structural and Dimensional Characteristics ofPlanar Linkages Press of Beihang University of China BeijingChina 1992

[28] L Song J Yang X Zhang and W Q Cao ldquoSpanning treemethod of identifying isomorphism and topological symmetryto planar kinematic chain with multiple jointrdquo Chinese Journalof Mechanical Engineering vol 14 no 1 pp 27ndash31 2001

[29] J G Liu and D J Yu ldquoRepresentations amp isomorphismidentification of planar kinematic chains with multiple jointsbased on the converted adjacent matrixrdquo Chinese Journal ofMechanical Engineering vol 48 pp 15ndash21 2012

[30] R Ravisankar andT SMruthyunjaya ldquoComputerized synthesisof the structure of geared kinematic chainsrdquo Mechanism andMachine Theory vol 20 no 5 pp 367ndash387 1985

[31] J U Kim and B M Kwak ldquoApplication of edge permutationgroup to structural synthesis of epicyclic gear trainsrdquo Mecha-nism and Machine Theory vol 25 no 5 pp 563ndash573 1990

Mathematical Problems in Engineering 15

[32] G Chatterjee and L-W Tsai ldquoComputer-aided sketching ofepicyclic-type automatic transmission gear trainsrdquo Journal ofMechanical Design vol 118 no 3 pp 405ndash411 1996

[33] V V N R Prasadraju and A C Rao ldquoA new technique basedon loops to investigate displacement isomorphism in planetarygear trainsrdquo Journal of Mechanical Design vol 12 pp 666ndash6752002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

B

A

b

D

C

c

a

①

② ③

④

⑤⑥

(a)

A b

D

B

c

C

a

①

②

③

④

⑤

(b)a

11 1

1

223

3

3

3①

②

③

④

⑤

⑥

(c)

Figure 3 A gear train kinematic chain and its TM described

defined multiple joint factor 119901119894of a multiple joint 119894 is as

follows

119901119894= 119896 minus 2 (1)

And the total multiple joint factors119875 of a kinematic chaincan be obtained by adding all multiple joint factors of akinematic chain together as follows

119875 = sum119901119894 (2)

So as for a multiple joint kinematic chain as shown inFigure 2(a) it has twomultiple joints 119886 and 119887 these joints 119888 119889119890 119891 ℎ 119894 119895 and 119896 are simple joints Multiple joint 119886 is formedby connecting four links 6 7 8 and 10 so its correspondingmultiple joint factor is 119901

119886= 2 Multiple joint 119887 is formed

by connecting three links 4 5 and 10 so its correspondingmultiple joint factor is 119901

119886= 1 And the total multiple joint

factors of this kinematic chain is 119875 = 3As we know topological model (TM) of multiple joint

kinematic chain can be represented by the double colorgraph (DCG) as shown in Figure 2(b) [23] But it has toomany vertices (number of vertices equal to the numberof links and joints of a kinematic chain) so more storagespace is needed in the process of structural analysis bycomputer In this paper a weighted double color contractedgraph (WDCCG) is introduced to represent the topologicalstructure of kinematic chains The WDCCG model whichwiped off all of the simple joints and simple joint links formthe multiple joint kinematic chain can improve the efficiencyof isomorphism identification The graph is established asfollows solid vertices ldquoerdquo denotemultiple joint links and hol-low vertices ldquoIrdquo denote multiple joints If two multiple jointlinks are connected by simple joints and simple joint linksconnect two corresponding solid vertices with a weightededge (weighted value equal to the number of simple jointsbetween two vertices) If one multiple joint link is connectedwith one multiple joint or two multiple joints are connected

by simple joints and simple joint links connect them witha weighted edge as well (weighted value also equal to thenumber of simple joints between two vertices) For exampleFigure 2(c) is the WDCCG of multiple joint kinematic chainas shown in Figure 2(a)

22 Topological Model of Gear Train Kinematic Chain Inthis paper topological model (TM) of gear train kinematicchains can be represented by two topological graphs Firstlydouble color graph (DCG) is established as follows representthe gears and planet carrier with solid vertices ldquoerdquo gearjoints with hollow vertices ldquoΔrdquo and revolute joint with hollowvertices ldquoIrdquo And connect corresponding solid vertex andhollow vertex with an edge when a gear or a planet carrier isconnected with a gear joint or a revolute joint For exampleFigure 3(b) is the DCG corresponding to gear train kinematicchain as shown in Figure 3(a) But it also has too manyvertices then a weighted double color contracted graph(WDCCG) is introduced to represent gear train kinematicchain Because the gear train kinematic chain contains twotypes of kinematic pairs namely revolute joint and gearjoints the WDCCG model wiped off all of the simple jointsand can be established as follows solid vertices ldquoerdquo denotegears of chain hollow vertices ldquoΔrdquo denote gear joints andhollow vertices ldquoIrdquo denote multiple revolute joints If twogears are connected directly by a revolute joint connecttwo corresponding solid vertices with a 2-weighted edge Iftwo gears are connected directly by a gear joint connecttwo corresponding solid vertices with a 3-weighted edgeIf a gear is connected with a multiple revolute joint alsoconnect corresponding solid vertex and hollow vertex witha 1-weighted edge For example Figure 3(c) is the WDCCGof the gear train kinematic chain as shown in Figure 3(a)

23 Weighted Adjacency Matrix Obviously a multiple jointor gear train kinematic chain and its WDCCG are one-to-one correspondent to each otherTheWDCCGof a kinematic

4 Mathematical Problems in Engineering

chain can be represented by a vertex-vertex weighted adja-cencymatrix (WAM)while the elements ofWAMare definedas follows

119860 = [119889119894119895]119899times119899

=

(119889119894119895) = (119898

1198941198951 1198981198941198952 119898

119894119895119896) 119894 = 119895

(119889119894119895) 119894 = 119895

(3)

where 119899 is the number of vertices in WDCCGWhen 119894 = 119895 if vertex is hollow vertex then 119889

119894119895= 0

otherwise the vertex is solid vertex then 119889119894119895= 1

When 119894 = 119895 if two vertices are not connected directlythen 119889

119894119895= 0 otherwise the two vertices are connected

directly by 119896 weighted edges then 119889119894119895= 1198981198941198951 1198981198941198952 119898

119894119895119896

(1198981198941198951ge 1198981198941198952ge sdot sdot sdot ge 119898

119894119895119896 119898119894119895119896

is the weighted value of 119896thedge)

For example the WAM 119860 of a multiple joint kinematicchain as shown in Figure 2(c) can be expressed as follows

119860 =

1 2 119886 119887

1

2

119886

119887

[

[

[

[

[

[

(1)

(1)

(2)

(2 3)

(1)

(1)

(3)

(2)

(2)

(3)

(0)

(2)

(2 3)

(2)

(2)

(0)

]

]

]

]

]

]

(4)

The WAM 119860 corresponding to a gear train kinematicchain as shown in Figure 3(c) can be expressed as follows

119860

=

1 2 3 4 5 6 119886

1

2

3

4

5

6

119886

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

(1)

(0)

(0)

(0)

(3)

(0)

(1)

(0)

(1)

(0)

(0)

(3)

(2)

(1)

(0)

(0)

(1)

(0)

(0)

(3)

(1)

(0)

(0)

(0)

(1)

(2)

(3)

(1)

(3)

(3)

(0)

(2)

(1)

(0)

(0)

(0)

(2)

(3)

(3)

(0)

(1)

(0)

(1)

(1)

(1)

(1)

(0)

(0)

(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(5)

3 Isomorphism Identification Algorithm

31 Equivalent Circuit Model As we know when two multi-ple joint kinematic chains or two gear trains kinematic chainsare isomorphic their WDCCGs are exactly the same theirvertices and edges are in one-to-one correspondence witheach other and the vertices and edges in correspondencekeep the same incidence and weighted relation Accordingto this definition the necessary and sufficient condition forisomorphism of the two types of kinematic chains is that theyhave the same WAM that is 119860 = 1198601015840

Equivalent circuit model (ECM) of WDCCG is estab-lished by its each edge which is replaced by conductanceequal to the weighted value of corresponding edge119898

119894119895

Theorem 1 If two WDCCGs corresponding to the same typekinematic chains are isomorphic then their correspondingECMs are identical

For example two isomorphism multiple joint kinematicchains are shown in Figures 4(a) and 4(b) their WDCCGsare shown in Figures 4(c) and 4(d) respectively According tothe description above their corresponding ECMs are shownin Figures 5(a) and 5(b) respectively There are two identicalcircuits and nodes of these two ECMs are in one-to-onecorrespondence

Complete excitation (CE) of an ECM is established asfollows in an ECM with 119899 nodes firstly set a node (119899 + 1) tobe a reference node then connect the reference node (119899 + 1)with every other node 119899

119894(119894 = 1 2 119899) of the ECM whose

conductance values are equal to the sumof weighted values ofthe node edges Secondly apply the same current source 1 Abetween node (119899+1) and other nodes 119899

119894(119894 = 1 2 119899) of the

ECM respectively in directions of the currents going fromnode (119899 + 1) to other nodes 119899

119894 (119894 = 1 2 119899) respectively

According to circuit theory two identical circuits underthe same CE have the same response And their node voltagescan be solved by the nodal method of circuit analysis Thenodal method of circuit analysis can be expressed as followsin a circuit network choose a node as a reference nodethe voltage difference between each node and the referencenode is known as the node voltage of the node For a circuitnetwork with 119899 nodes which has (119899 minus 1) node voltages if thenode 119899 is the reference node the nodal voltage equation canbe expressed as

[

[

[

[

[

[

[

11986611

11986612

sdot sdot sdot 1198661(119899minus1)

11986621

11986622

sdot sdot sdot 1198662(119899minus1)

d

119866(119899minus1)1

119866(119899minus1)2

sdot sdot sdot 119866(119899minus1)(119899minus1)

]

]

]

]

]

]

]

[

[

[

[

[

[

[

V1198991

V1198992

V119899(119899minus1)

]

]

]

]

]

]

]

=

[

[

[

[

[

[

[

1198681199041198991

1198681199041198992

119868119904119899(119899minus1)

]

]

]

]

]

]

]

(6)

where119866119894119894(119894 = 1 2 119899 minus 1) is called the self admittance of a

node 119894 the value of which is the sum of the admittance of allbranches that are connected to the node 119894119866

119894119895(119894 = 1 2 119899minus

1 119895 = 1 2 119899minus1) is called themutual admittance of node 119894and node 119895 whose value is the sum of the branch admittancebetween node 119894 and node 119895 and 119868

119904119899119894(119894 = 1 2 119899 minus 1) is

called the algebraic sum of the current source for the inflownode 119894 (inflow is positive outflow is negative)

For example an ECM with 4 nodes is shown in Fig-ure 6(a) Apply the CE of reference node 5 as shown inFigure 6(b) By nodalmethodof circuit analysis equations (6)node voltages V

1 V2 V3 and V

4can be expressed as follows

Mathematical Problems in Engineering 5

①

②

③

④

⑤

⑥

⑦

⑧

⑨

⑩

a

b

(a)

①

②

③

④

⑤

⑥

⑦

⑧ ⑨⑩

b

a

(b)

①

②③

a

b

1

1

1

2

223

3

(c)

①

②③

a

b1

1

1

2

2

2

3

3

(d)

Figure 4 Two multiple joint kinematic chains and their WDCCG

①

②③

2 2

2

11

1

a

3

3b

(a)

①

②③

2

2

2

1

1

1a

3

3

b

(b)

Figure 5 ECM corresponding to two WDCCGs of Figures 4(c) and 4(d) respectively

G1

G2

G3

G4 G5

① ② ③

④

(a)

G6

G7

G8

G9

①

②

③

④

⑤

1

1

1

1

(b)

Figure 6 An ECM with 4 nodes and its CE by reference node 5

6 Mathematical Problems in Engineering

[

[

[

[

[

[

1198661+ 1198664+ 1198666

minus1198664

0 minus1198661

minus1198664

1198662+ 1198664+ 1198665+ 1198667

minus1198665

minus1198662

0 minus1198665

1198663+ 1198665+ 1198668

minus1198663

minus1198661

minus1198662

minus1198663

1198661+ 1198662+ 1198663+ 1198669

]

]

]

]

]

]

[

[

[

[

[

[

V1

V2

V3

V4

]

]

]

]

]

]

=

[

[

[

[

[

[

1

1

1

1

]

]

]

]

]

]

(7)

Then the node voltages V1 V2 V3 and V

4can be obtained

by solving the above equations (7) Therefore the followingconclusion can be obtained

Theorem 2 If two WDCCGs corresponding to the same typeof kinematic chains are isomorphic their two correspondingECMs 119873 and 1198731015840 can be established and excited by the sameCE as above respectively Then the node voltages of eachcorrespondence node of two ECMs119873 and1198731015840 are the same

For example two identical ECMs119873 and1198731015840 with 5 nodesare shown in Figures 5(a) and 5(b) Apply the same CE ofreference node 6 which is shown in Figures 7(a) and 7(b)respectively And their corresponding WAMs 119860 and 1198601015840 canbe obtained as follows

119860 =

1 2 3 119886 119887

1

2

3

119886

119887

[

[

[

[

[

[

[

[

[

(1)

(2)

(0)

(1)

(2)

(2)

(1)

(1)

(0)

(3)

(0)

(1)

(1)

(2 3)

(1)

(1)

(0)

(2 3)

(0)

(0)

(2)

(3)

(1)

(0)

(0)

]

]

]

]

]

]

]

]

]

1198601015840=

1 2 3 119886 119887

1

2

3

119886

119887

[

[

[

[

[

[

[

[

[

(1)

(1)

(2)

(3)

(0)

(1)

(1)

(0)

(1)

(2 3)

(2)

(0)

(1)

(2)

(1)

(3)

(1)

(2)

(0)

(0)

(0)

(2 3)

(1)

(0)

(0)

]

]

]

]

]

]

]

]

]

(8)

By nodal method of circuit analysis equations of nodevoltages V

1 V2 V3 V119886 and V

119887can be expressed respectively

as follows

[

[

[

[

[

[

[

[

[

10 minus2 0 minus1 minus2

minus2 12 minus1 0 minus3

0 minus1 14 minus5 minus1

minus1 0 minus5 12 0

minus2 minus3 minus1 0 12

]

]

]

]

]

]

]

]

]

[

[

[

[

[

[

[

[

[

V1

V2

V3

V119886

V119887

]

]

]

]

]

]

]

]

]

=

[

[

[

[

[

[

[

[

[

1

1

1

1

1

]

]

]

]

]

]

]

]

]

(9)

[

[

[

[

[

[

[

[

[

12 minus1 minus2 minus3 0

minus1 14 0 minus1 minus5

minus2 0 10 minus2 minus1

minus3 minus1 minus2 12 0

0 minus5 minus1 0 12

]

]

]

]

]

]

]

]

]

[

[

[

[

[

[

[

[

[

V1

V2

V3

V119886

V119887

]

]

]

]

]

]

]

]

]

=

[

[

[

[

[

[

[

[

[

1

1

1

1

1

]

]

]

]

]

]

]

]

]

(10)

Then node voltages V1 V2 V3 V119886 and V

119887of two ECMs can

be obtained by solving the above (9) and (10) respectively asshown in Table 1

From Theorem 2 node voltages of each correspondencenode in these two ECMs are the same as follows V

119886= V119887=

01627 V119887= V119886= 01691 V

1= V3= 01839 V

2= V1= 01691

and V3= V2= 01537 When arranging the obtained node

voltages in descending order with the same node voltagesstaying together this set is called the node voltage sequence(NVS) and denoted as

NVS = lfloor(V1 V2 V

1198991

) (V(1198991+1) V(1198991+2) V

119899)rfloor

(V1ge V2ge V1198991

V(1198991+1)ge V(1198991+2)ge sdot sdot sdot ge V

119899)

(11)

where the variables V1 V2 V

1198991

represent node voltages ofhollow vertex The variables V

(1198991+1) V(1198991+2) V

119899represent

node voltages of solid vertexIf two identical ECMs are stimulated by the same CE as

above then the NVSs in correspondence are the same Forexample two identical ECMs as showed in Figures 5(a) and5(b) the NVS and vertex codes can be obtained respectivelyas follows

NVS (119886) = [(V119887 V119886) (V1 V2 V3)]

= [(01691 01627) (01839 01691 01537)]

NVS (119887) = [(V119886 V119887) (V3 V1 V2)]

= [(01691 01537) (01839 01691 01627)]

(12)

If the NVSs of two ECMs are not the same then the twoWDCCGs are not isomorphic On the contrary if they arethe same correspondence vertices in these two WDCCGscan be determined by the element values of NVS Thenperform the row exchanges of WAM 119860

1015840 to determine if theyare isomorphic Because the NVS can reduce the number ofrow exchanges to several ones or only just one this is a veryeffective method For example by the NVS(119886) and NVS(119887)of two ECMs shown in Figures 5(a) and 5(b) the verticesin correspondence are 119886-119887 119887-119886 1-3 2-1 3-2 then throughexchanging the row of WAM 119860

1015840 we obtain 119860 = 1198601015840 which

are isomorphic

Mathematical Problems in Engineering 7

①

②

③

⑥

1 1

1

11

6

6

7

6

5

a

b

(a)

①

②

③

⑥

1 1

1

1

1

6

6

6

7

5

a

b

(b)

Figure 7 Two ECMs with the same CE respectively

Table 1 Node voltages of two ECMs with the same CE

Vertex type Hollow vertex Solid vertex

Figure 5(a) Vertex code 119886 119887 1 2 3Node voltage 01627 01691 01839 01691 01537

Figure 5(b) Vertex code 119886 119887 1 2 3Node voltage 01691 01627 01691 01537 01839

For example two gear train kinematic chains are shownin Figures 8(a) and 8(b) and their WDCCGs as shownin Figures 8(c) and 8(d) respectively Their correspondingWAMs are 119860 and 1198601015840 as follows

119860

=

1 2 3 4 5 6 119887

1

2

3

4

5

6

119887

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

(1)

(2)

(0)

(0)

(3)

(0)

(1)

(2)

(1)

(3)

(0)

(0)

(0)

(0)

(0)

(3)

(1)

(3)

(0)

(0)

(1)

(0)

(0)

(3)

(1)

(0)

(2)

(0)

(3)

(0)

(0)

(0)

(1)

(0)

(1)

(0)

(0)

(0)

(2)

(0)

(1)

(1)

(1)

(0)

(1)

(0)

(1)

(1)

(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

1198601015840

=

1 2 3 4 5 6 119887

1

2

3

4

5

6

119887

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

(1)

(0)

(0)

(3)

(2)

(0)

(1)

(0)

(1)

(0)

(0)

(3)

(3)

(1)

(0)

(0)

(1)

(0)

(0)

(2)

(1)

(3)

(0)

(0)

(1)

(0)

(0)

(1)

(2)

(3)

(0)

(0)

(1)

(0)

(0)

(0)

(3)

(2)

(0)

(0)

(1)

(0)

(1)

(1)

(1)

(1)

(0)

(0)

(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(13)

According to the abovemethod their correspondingNVSand vertex codes can be obtained as follows

NVS (119886) = [(V119887) (V6 V5 V4 V1 V2 V3)] = [(02330)

(02744 02257 02068 01908 01907 01737)]

NVS (119887) = [(V119887) (V3 V4 V5 V1 V6 V2)] = [(02330)

(02744 02257 02068 01908 01907 01737)]

(14)

By the NVS(119886) and NVS(119887) the vertices of these twoWDCCGs in correspondence can be obtained as follows 119887-1198871-1 2-6 3-2 4-5 5-4 6-3 and through exchanging the rowof WAM 119860

1015840 one time we obtain isomorphic 119860 = 1198601015840

32 Algorithm Here a direct algorithm is given to determineisomorphism of planar multiple joint and gear train kine-matic chains and it can be expressed as follows

Step 1 Input the vertex codes and WAMs 119860 1198601015840 of twoWDCCGs corresponding to the same type of kinematicchains

Step 2 Assign vertices codes of 119860 and 1198601015840 in two groups thehollow vertexes to be the first group and the solid vertexes tobe another group

Step 3 Calculate NVS(119886) NVS(119887) and their vertex codesof the two ECMs with the same CE respectively CompareNVS(119886) and NVS(119887) if equivalence cannot be seen it isdetermined that they are not isomorphic and program stopsotherwise it is determined that they could be isomorphicThen check the NVS(119886) and NVS(119887) to see if the values intwo vertex code groups are all distinct if yes go to Step 4 ifno go to Step 6

8 Mathematical Problems in Engineering

①

②

③

④

⑤

⑥

A

a

b

C

B

c

(a)

① ②③ ④

⑤

⑥

A

a

b

C

B

c

(b)

①

②

③

④

⑤

⑥

1

b

1

1

1

2

3

3

3 2

(c)

①

②

③

④

⑤

⑥

3

3

3

2

2

b

1

1

1

1

(d)

Figure 8 Two gear train kinematic chains and their WDCCGs

Step 4 Take the NVS(119886) NVS(119887) and their vertex codes todetermine the correspondence of vertices according to thevalues that are in correspondence

Step 5 According to the corresponding vertices obtained andthe vertex codes to form new WAM 119860

1015840

1 rewrite WAM 119860

1015840 If119860 = 119860

1015840

1 it is determined that they are isomorphic otherwise

they are not isomorphic and program stops

Step 6 According to the corresponding vertices obtained theNVS(119886) and NVS(119887) have the same values in the correspond-ing two vertex code groups Just adjust the position order ofvertices of the same group in1198601015840 to obtain vertex codes to fromnew adjacency matrix 1198601015840

1 If 119860 = 119860

1015840

1 it is determined that

they are isomorphic and program stops if 119860 = 1198601015840

1 keep on

adjusting the position order of the aforementioned verticesin 1198601015840 of the vertex code groups for all possibilities And if119860 = 119860

1015840

1always it is determined that they are not isomorphic

and program stops

4 Examples

Example 1 Two 10-link multiple joint kinematic chainsare shown in Figures 9(a) and 9(b) Their correspondingWDCCGs are depicted in Figures 9(c) and 9(d) respectively

The direct process to determine isomorphism of these twokinematic chains can be expressed as follows

Step 1 Input their corresponding vertex codes andWAMs1198601198601015840 as follows

119860 =

1 119886 119887 119888 119889

1

119886

119887

119888

119889

[

[

[

[

[

[

[

[

[

(1)

(2)

(0)

(1)

(2)

(2)

(0)

(2)

(0)

(3)

(0)

(2)

(0)

(2)

(2)

(1)

(0)

(2)

(0)

(3)

(2)

(3)

(2)

(3)

(0)

]

]

]

]

]

]

]

]

]

1198601015840=

1 119886 119887 119888 119889

1

119886

119887

119888

119889

[

[

[

[

[

[

[

[

[

(1)

(0)

(2)

(2)

(1)

(0)

(0)

(2)

(2)

(2)

(2)

(2)

(0)

(0)

(3)

(2)

(2)

(0)

(0)

(3)

(1)

(2)

(3)

(3)

(0)

]

]

]

]

]

]

]

]

]

(15)

Step 2 Evidently 119860 = 1198601015840 Assign vertices codes of 119860 and 1198601015840

in two groups (a) [(119886 119887 119888 119889) (1)] and (b) [(119886 119887 119888 119889) (1)]

Mathematical Problems in Engineering 9

①

②

③

④

⑤

⑥

⑦

⑧

⑨

⑩

c

ba

d

(a)

①

②

③ ④

⑤

⑥⑦

⑧

⑨⑩

c b

a

d

(b)

①

cb

a

2

2

2

2

2

3

3

1d

(c)

①

2

2

2

2

2

3

3

d

b

ac

1

(d)

Figure 9 Two 10-link multiple joint kinematic chains and their WDCCGs

①

⑪

②

③

④

⑤

⑥

⑦⑧

⑨

⑩

ab

(a)

①

②

③④

⑤⑥

⑦

⑧

⑨

⑩

a

b

⑪

(b)

①

② ③

④

⑤

⑥ 1

1

11

12

2

2

2

2a

b

1

1 1

(c)

① ②

③

④

⑤

⑥

1

1

1

1

1

11

12

2

2

2

2b a

(d)

Figure 10 Two 11-link multiple joint kinematic chains and their WDCCGs

Step 3 Calculate NVS(119886) NVS(119887) and their vertex codes oftwo ECMs with the same CE respectively as follows

NVS (119886) = [(V119888 V119887 V119886 V119889) (V1)]

= [(01551 01546 01451 01275) (01700)]

NVS (119887) = [(V119886 V119887 V119888 V119889) (V1)]

= [(01538 01459 01459 01308) (01714)]

(16)

Compare NVS(119886) and NVS(119887) and equivalence cannot beseen It is determined that they are not isomorphic andprogram stops

Example 2 Two 11-link multiple joint kinematic chains areshown in Figures 10(a) and 10(b)Their correspondingWDC-CGs are described in Figures 10(c) and 10(d) respectivelyThe direct process to determine isomorphism of these twokinematic chains is as follows

10 Mathematical Problems in Engineering

Step 1 Input their corresponding vertex codes andWAMs1198601198601015840 as follows

119860

=

1 2 3 4 5 6 119886 119887

1

2

3

4

5

6

119886

119887

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

(1)

(2)

(0)

(1)

(1)

(0)

(0)

(2)

(2)

(1)

(1)

(0)

(1)

(0)

(0)

(0)

(0)

(1)

(1)

(1)

(0)

(2)

(2)

(0)

(1)

(0)

(1)

(1)

(0)

(1)

(0)

(0)

(1)

(1)

(0)

(0)

(1)

(0)

(1)

(0)

(0)

(2)

(0)

(1)

(0)

(1)

(0)

(1)

(0)

(0)

(1)

(0)

(1)

(0)

(0)

(2)

(2)

(0)

(0)

(0)

(0)

(1)

(2)

(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

1198601015840

=

1 2 3 4 5 6 119886 119887

1

2

3

4

5

6

119886

119887

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

(1)

(2)

(0)

(1)

(0)

(0)

(0)

(1)

(2)

(1)

(0)

(1)

(1)

(0)

(2)

(0)

(0)

(0)

(1)

(1)

(2)

(1)

(0)

(2)

(1)

(1)

(1)

(1)

(0)

(0)

(0)

(0)

(0)

(1)

(2)

(0)

(1)

(1)

(0)

(0)

(0)

(0)

(1)

(0)

(1)

(1)

(1)

(0)

(0)

(2)

(0)

(0)

(0)

(1)

(0)

(2)

(1)

(0)

(2)

(0)

(0)

(0)

(2)

(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(17)

Step 2 Evidently 119860 = 1198601015840 Assign vertices codes of 119860 and 1198601015840

in two groups (a) [(119886 119887) (1 2 3 4 5 6)] and (b) [(119886 119887) (1 23 4 5 6)]

Step 3 Calculate NVS(119886) NVS(119887) and their vertex codes oftwo ECMs with the same CE respectively as follows

NVS (119886) = [(V119886 V119887) (V5 V4 V2 V6 V1 V3)]

= [(02084 02056)

(0743 02727 02348 02347 02023 01995)]

NVS (119887) = [(V119887 V119888) (V6 V4 V5 V1 V3 V2)]

= [(02084 02056)

(0743 02727 02348 02347 02023 01995)]

(18)

Compare NVS(119886) and NVS(119887) and equivalence can beseen So the node voltages in two vertex code groups are alldistinct go to Step 4

Step 4 Take the NVS(119886) NVS(119887) and their vertex codes todetermine the corresponding vertices of two WDCCGs asfollows (119886-119887 119887-119886 1-3 2-5 3-2 4-4 5-6 6-1)

Step 5 According to the corresponding vertices obtained andthe vertex codes to form new WAM 119860

1015840

1 rewrite WAM 119860

1015840Then we have 119860 = 1198601015840

1 so they are isomorphic and program

stops

Example 3 Two 6-gear train kinematic chains are depicted inFigures 11(a) and 11(b) and their correspondingWDCCGs aredescribed in Figures 11(c) and 11(d) respectively The directprocess to determine isomorphism of these two kinematicchains is as follows

Step 1 Input their corresponding vertex codes andWAMs1198601198601015840 as follows

119860 =

1 2 3 4 5 6 7 119886 119888

1

2

3

4

5

6

7

119886

119888

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

(1)

(3)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(3)

(1)

(2)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(2)

(1)

(3)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(3)

(1)

(0)

(0)

(0)

(1)

(1)

(0)

(0)

(0)

(0)

(1)

(3)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(3)

(1)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(1)

(1)

(0)

(1)

(0)

(1)

(0)

(1)

(1)

(0)

(0)

(1)

(0)

(0)

(1)

(1)

(0)

(1)

(0)

(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

1198601015840=

1 2 3 4 5 6 7 119886 119888

1

2

3

4

5

6

7

119886

119888

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

(1)

(3)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(3)

(1)

(2)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(2)

(1)

(3)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(3)

(1)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(1)

(3)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(3)

(1)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(1)

(1)

(0)

(1)

(0)

(1)

(1)

(0)

(1)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(1)

(1)

(0)

(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(19)

Mathematical Problems in Engineering 11

①

② ③ ④ ⑤

⑥⑦

A

B

b

a

C

c

(a)

①

②③

④

⑤

⑥ ⑦

A

Bb

a

C

c

(b)

①

②

③

④ ⑤

⑥

⑦

3

3

3

c

2

11

1

1

1

1

1

1

a

(c)

①

②

③

④

⑤

⑥

⑦

c

a

1

1

1

1

1

1

1

2

3

3

3

(d)

Figure 11 Two 6-gear train kinematic chains and their WDCCG

Step 2 Evidently 119860 = 1198601015840 Assign vertices codes of 119860 and 1198601015840

in two groups (a) [(119886 119888) (1 2 3 4 5 6 7)] and (b) [(119886 119888) (12 3 4 5 6 7)]

Step 3 Calculate NVS(119886) NVS(119887) and their vertex codes oftwo ECMs with the same CE respectively as follows

NVS (119886) = [(V119887 V119886) (V7 V5 V6 V1 V4 V3 V2)]= [(02591 02547)

(03784 02515 02511 02309 02122 02029 01961)]

NVS (119887) = [(V119887 V119886) (V7 V6 V5 V1 V4 V3 V2)]= [(03156 02607)

(03940 02601 02551 02394 02367 02110 02001)]

(20)

Compare NVS(119886) and NVS(119887) and equivalence cannot beseen So it is determined that they are not isomorphic andprogram stops

Example 4 Two 9-gear train kinematic chains are shown inFigures 12(a) and 12(b) Their corresponding WDCCGs aredepicted in Figures 12(c) and 12(d) respectively The directprocess to determine isomorphism of these two kinematicchains is as follows

Step 1 Input their correspondingWAMs119860 and1198601015840 as follows

119860 =

1 2 3 4 5 6 7 8 9 119886 119887 119888

1

2

3

4

5

6

7

8

9

119886

119887

119888

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

(1)

(0)

(3)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(3)

(1)

(3)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(0)

(3)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(3)

(1)

(1)

(1)

(0)

(0)

(0)

(0)

(1)

(3)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(3)

(1)

(3)

(2)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(3)

(1)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(2)

(0)

(1)

(0)

(1)

(1)

(0)

(0)

(0)

(0)

(3)

(0)

(0)

(0)

(0)

(1)

(0)

(1)

(0)

(0)

(0)

(1)

(0)

(1)

(0)

(1)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(1)

(1)

(0)

(0)

(0)

(1)

(1)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

12 Mathematical Problems in Engineering

① ②

③

a

A

c

B

d

C

D

E

b

④

⑤

⑥

⑦

⑧

⑨

(a)

①

②③a

A

b E

c

DC

d

B④

⑤

⑥ ⑦⑧⑨

(b)

①

1 1

c

1 1

1

1

1

b3

3

3

2

1

1

1

1a

33

②

③

④

⑤⑥

⑦

⑧

⑨

(c)

1

1

1

1

3

3

1

1

1

1

1

1

1

2

3

3

3

b

c

a

①②

③

④

⑤

⑥

⑦

⑧

⑨

(d)

Figure 12 Two 9-gear train kinematic chains and their WDCCGs

1198601015840=

1 2 3 4 5 6 7 8 9 119886 119887 119888

1

2

3

4

5

6

7

8

9

119886

119887

119888

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

(1)

(0)

(0)

(3)

(0)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(1)

(0)

(3)

(0)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(1)

(2)

(0)

(0)

(0)

(0)

(0)

(1)

(0)

(1)

(3)

(3)

(2)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(3)

(3)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(3)

(1)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(3)

(0)

(1)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(3)

(1)

(1)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(3)

(1)

(0)

(0)

(1)

(1)

(1)

(1)

(0)

(1)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(1)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(1)

(0)

(0)

(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(21)

Step 2 Evidently 119860 = 1198601015840 Assign vertices codes of 119860 and 1198601015840

in two groups (a) [(119886 119887 119888) (1 2 3 4 5 6 7 8 9)] and (b) [(119886119887 119888) (1 2 3 4 5 6 7 8 9)]

Step 3 Calculate NVS(119886) NVS(119887) and their vertex codes oftwo ECMs with the same CE respectively as follows

NVS (119886) = [(V119888 V119887 V119886) (V9 V4 V8 V1 V2 V5 V7 V3 V6)]

= [(02845 02815 02075) (02470 02307 02295 02294 02294 02154 02154 01846 01719)]

NVS (119887) = [(V119887 V119888 V119886) (V9 V8 V3 V6 V7 V1 V2 V5 V4)]

= [(02845 02815 02075) (02470 02307 02295 02294 02294 02154 02154 01846 01719)]

(22)

Mathematical Problems in Engineering 13

Compare NVS(119886) and NVS(119887) and equivalence can be seenAnd the node voltages in two vertex code groups are not alldistinct so go to Step 6

Step 4 The vertices (1 2) in Figure 12(a) vertices (6 7) inFigure 12(b) the vertices (5 7) in Figure 12(a) and vertices (12) in Figure 12(b) have the same values in the correspondinggroups so just adjust the position order of the vertices in 1198601015840to obtain a new adjacency matrix 1198601015840 And the correspondingvertices of two WDCCGs are (1-6 2-7 3-5 4-8 5-1 6-4 7-28-3 9-9 119886-119886 119887-119888 119888-119887) so we have 119860 = 119860

1015840

1 It is determined

that they are isomorphic and program stops

5 Algorithm Complexity Analysis

This paper presents a relatively good algorithm for determi-nation of kinematic chains isomorphism with multiple jointand gear train It is feasible for introducing the WDCCGand WAM to describe these two types of kinematic chainsthey dramatically reduce the number of vertices and order ofcorresponding adjacency matrix 119860

The analysis of computational complexity of this isomor-phic identification algorithm is as follows when there areno equivalent node voltages in NVS time consumption isbasically dedicated to solving the linear equation sets If akinematic chain possesses 119899 vertices the order of the nodevoltage equations is 119899 minus 1 Since the adjoint circuit is alinear resistance circuit the node voltage equations forma linear algebra equation set of orders 119899 minus 1 which has acomputational complexity of 119874((119899 minus 1)3) for its solutionSince it needs to solve at most 2119899 adjoint circuits in ourproposed method the computational complexity is of order119874(2119899(119899 minus 1)

3) lt 119874(119899

4) Thus the proposed method has a

computational complexity of a polynomial orderWhen thereare equivalent node voltages in NVS the correspondence ofonly a part of the vertices cannot be determinedThus in theworst situation the proposed method is more efficient thanthose where rowcolumn exchanges are performed blindly

In this paper this algorithm is tested and comparedwith the internationally recognized faster method for planarsimple joint kinematic chains isomorphic identification [20]and they are tested in the same software and hardwareenvironment The test results are shown in Figure 13 (theaverage time of each vertex is 20 times of the kinematic chainwithmultiple joints) test results show that when the vertex ofthe kinematic chain is increased the algorithm proposed inthis paper is more efficient than the algorithm proposed byHe et alThe proposed algorithm by us is very suitable for theisomorphism of the topological graph of kinematic chainsand the algorithm is very efficient at the same time

6 Conclusions

In this paper an algorithm for topological isomorphism iden-tification of planar multiple joint and gear train kinematicchains is introduced

4 6 8 10 12 14 16 18 20 22

The number of vertices

Algorithm proposed by He et al

Algorithm proposed in this paper

00001

0001

001

01

1

The d

ecisi

on ti

me (

s)

Figure 13 The test results of the algorithm proposed in this paperand the algorithm proposed by He et al

Firstly a weighted-double-color-contracted-graph(WDCCG) and a weighted adjacency matrix (WAM) areintroduced to describe the planarmultiple joint and gear trainkinematic chains respectively They dramatically reduce thenumber of vertices and order of the represented adjacencymatrix

Secondly isomorphism identification method of thesetwo types of kinematic chains is carried out by the equivalentcircuit models (ECMs) of WDCCGs with the same completeexcitation (CE) and then uses the solved node voltagesequence (NVS) to determine the corresponding vertices oftwo WDCCGs

Finally an algorithm to identify isomorphic kinematicchains is obtained It is an efficient and easy method tobe realized by computer And this algorithm also can beused in the isomorphism identification of planar simple jointkinematic chains with minor changes

Notations

119860 1198601015840 The incidence matrix of kinematic chain119889119894119895 The elements of incidence matrix

119866119894 Conductance value of edge in the ECM

119898119894119895119896 The weighted value of 119896th edge

1198731198731015840 Variable of equivalent circuit model119899 The number of vertices in WDCCG119899119894 Node 119894 in the ECM

V119894 Node voltages of the node 119894

CE Complete excitationDCG Double color graphECM Equivalent circuit modelNVS Node voltage sequenceTM Topological modelWAM Weighted adjacency matrixWDCCG Weighted double color contracted graph

14 Mathematical Problems in Engineering

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

Thiswork is supported by the Science andTechnology Projectof Quanzhou City 2014G48 and G20140047

References

[1] J J Uicker Jr and A Raicu ldquoAmethod for the identification andrecognition of equivalence of kinematic chainsrdquoMechanismandMachine Theory vol 10 no 5 pp 375ndash383 1975

[2] H S Yan and A S Hall ldquoLinkage characteristic polynomialsdefinitions coefficients by inspectionrdquo Journal of MechanicalDesign vol 103 no 3 pp 578ndash584 1981

[3] H S Yan and A S Hall ldquoLinkage characteristic polynomialsassembly theorems uniquenessrdquo Journal of Mechanical Designvol 104 no 1 pp 11ndash20 1982

[4] T S Mruthyunjaya ldquoA computerized methodology for struc-tural synthesis of kinematic chains Part 1 FormulationrdquoMech-anism and Machine Theory vol 19 no 6 pp 487ndash495 1984

[5] T S Mruthyunjaya ldquoA computerized methodology for struc-tural synthesis of kinematic chains part 2mdashapplication toseveral fully or partially known casesrdquoMechanism andMachineTheory vol 19 no 6 pp 497ndash505 1984

[6] T S Mruthyunjaya ldquoA computerized methodology for struc-tural synthesis of kinematic chains part 3mdashapplication to thenew case of 10-link three- freedom chainsrdquo Mechanism andMachine Theory vol 19 no 6 pp 507ndash530 1984

[7] T S Mruthyunjaya and H R Balasubramanian ldquoIn questof a reliable and efficient computational test for detection ofisomorphism in kinematic chainsrdquo Mechanism and MachineTheory vol 22 no 2 pp 131ndash139 1987

[8] W J Sohn and F Freudenstein ldquoAn application of dual graphs tothe automatic generation of the kinematic structures of mecha-nismsrdquo Journal of Mechanisms Transmissions and Automationvol 108 no 3 pp 392ndash398 1986

[9] A Ambekar and V Agrawal ldquoOn canonical numbering ofkinematic chains and isomorphism problem MAX coderdquo inProceedings of the ASMEMechanisms Conference pp 1615ndash16221986

[10] A G Ambekar and V P Agrawal ldquoCanonical numberingof kinematic chains and isomorphism problem min coderdquoMechanism andMachineTheory vol 22 no 5 pp 453ndash461 1987

[11] C S Tang and T Liu ldquoDegree code a new mechanismidentifierrdquo Journal of Mechanical Design vol 115 no 3 pp 627ndash630 1993

[12] J K Shin and S Krishnamurty ldquoOn identification and canonicalnumbering of pin jointed kinematic chainsrdquo Journal of Mechan-ical Design vol 116 pp 182ndash188 1994

[13] J K Shin and S Krishnamurty ldquoDevelopment of a standardcode for colored graphs and its application to kinematic chainsrdquoJournal of Mechanical Design vol 116 no 1 pp 189ndash196 1994

[14] A C Rao and D Varada Raju ldquoApplication of the hammingnumber technique to detect isomorphism among kinematicchains and inversionsrdquoMechanism andMachineTheory vol 26no 1 pp 55ndash75 1991

[15] A C Rao and C N Rao ldquoLoop based pseudo hammingvalues-1 Testing isomorphism and rating kinematic chainsrdquoMechanism andMachineTheory vol 28 no 1 pp 113ndash127 1992

[16] A C Rao and C N Rao ldquoLoop based pseudo hamming values-II inversions preferred frames and actuatorsrdquo Mechanism andMachine Theory vol 28 no 1 pp 129ndash143 1993

[17] J-K Chu and W-Q Cao ldquoIdentification of isomorphismamong kinematic chains and inversions using linkrsquos adjacent-chain-tablerdquoMechanism and Machine Theory vol 29 no 1 pp53ndash58 1994

[18] A C Rao ldquoApplication of fuzzy logic for the study of iso-morphism inversions symmetry parallelism and mobility inkinematic chainsrdquoMechanism and Machine Theory vol 35 no8 pp 1103ndash1116 2000

[19] P R He W J Zhang Q Li and F X Wu ldquoA new method fordetection of graph isomorphism based on the quadratic formrdquoJournal of Mechanical Design vol 125 no 3 pp 640ndash642 2003

[20] P R He W J Zhang and Q Li ldquoSome further developmenton the eigensystem approach for graph isomorphismdetectionrdquoJournal of the Franklin Institute vol 342 no 6 pp 657ndash6732005

[21] Z Y Chang C Zhang Y H Yang and YWang ldquoA newmethodto mechanism kinematic chain isomorphism identificationrdquoMechanism andMachineTheory vol 37 no 4 pp 411ndash417 2002

[22] J P Cubillo and J BWan ldquoComments onmechanismkinematicchain isomorphism identification using adjacent matricesrdquoMechanism andMachineTheory vol 40 no 2 pp 131ndash139 2005

[23] H F Ding and Z Huang ldquoA unique representation of the kine-matic chain and the atlas databaserdquo Mechanism and MachineTheory vol 42 no 6 pp 637ndash651 2007

[24] H F Ding and Z Huang ldquoA new theory for the topologicalstructure analysis of kinematic chains and its applicationsrdquoMechanism and Machine Theory vol 42 no 10 pp 1264ndash12792007

[25] A C Rao ldquoA genetic algorithm for topological characteristicsof kinematic chainsrdquo Journal of Mechanical Design vol 122 no2 pp 228ndash231 2000

[26] F G Kong Q Li and W J Zhang ldquoAn artificial neuralnetwork approach tomechanism kinematic chain isomorphismidentificationrdquo Mechanism and Machine Theory vol 34 no 2pp 271ndash283 1999

[27] J K Chu The Structural and Dimensional Characteristics ofPlanar Linkages Press of Beihang University of China BeijingChina 1992

[28] L Song J Yang X Zhang and W Q Cao ldquoSpanning treemethod of identifying isomorphism and topological symmetryto planar kinematic chain with multiple jointrdquo Chinese Journalof Mechanical Engineering vol 14 no 1 pp 27ndash31 2001

[29] J G Liu and D J Yu ldquoRepresentations amp isomorphismidentification of planar kinematic chains with multiple jointsbased on the converted adjacent matrixrdquo Chinese Journal ofMechanical Engineering vol 48 pp 15ndash21 2012

[30] R Ravisankar andT SMruthyunjaya ldquoComputerized synthesisof the structure of geared kinematic chainsrdquo Mechanism andMachine Theory vol 20 no 5 pp 367ndash387 1985

[31] J U Kim and B M Kwak ldquoApplication of edge permutationgroup to structural synthesis of epicyclic gear trainsrdquo Mecha-nism and Machine Theory vol 25 no 5 pp 563ndash573 1990

Mathematical Problems in Engineering 15

[32] G Chatterjee and L-W Tsai ldquoComputer-aided sketching ofepicyclic-type automatic transmission gear trainsrdquo Journal ofMechanical Design vol 118 no 3 pp 405ndash411 1996

[33] V V N R Prasadraju and A C Rao ldquoA new technique basedon loops to investigate displacement isomorphism in planetarygear trainsrdquo Journal of Mechanical Design vol 12 pp 666ndash6752002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

chain can be represented by a vertex-vertex weighted adja-cencymatrix (WAM)while the elements ofWAMare definedas follows

119860 = [119889119894119895]119899times119899

=

(119889119894119895) = (119898

1198941198951 1198981198941198952 119898

119894119895119896) 119894 = 119895

(119889119894119895) 119894 = 119895

(3)

where 119899 is the number of vertices in WDCCGWhen 119894 = 119895 if vertex is hollow vertex then 119889

119894119895= 0

otherwise the vertex is solid vertex then 119889119894119895= 1

When 119894 = 119895 if two vertices are not connected directlythen 119889

119894119895= 0 otherwise the two vertices are connected

directly by 119896 weighted edges then 119889119894119895= 1198981198941198951 1198981198941198952 119898

119894119895119896

(1198981198941198951ge 1198981198941198952ge sdot sdot sdot ge 119898

119894119895119896 119898119894119895119896

is the weighted value of 119896thedge)

For example the WAM 119860 of a multiple joint kinematicchain as shown in Figure 2(c) can be expressed as follows

119860 =

1 2 119886 119887

1

2

119886

119887

[

[

[

[

[

[

(1)

(1)

(2)

(2 3)

(1)

(1)

(3)

(2)

(2)

(3)

(0)

(2)

(2 3)

(2)

(2)

(0)

]

]

]

]

]

]

(4)

The WAM 119860 corresponding to a gear train kinematicchain as shown in Figure 3(c) can be expressed as follows

119860

=

1 2 3 4 5 6 119886

1

2

3

4

5

6

119886

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

(1)

(0)

(0)

(0)

(3)

(0)

(1)

(0)

(1)

(0)

(0)

(3)

(2)

(1)

(0)

(0)

(1)

(0)

(0)

(3)

(1)

(0)

(0)

(0)

(1)

(2)

(3)

(1)

(3)

(3)

(0)

(2)

(1)

(0)

(0)

(0)

(2)

(3)

(3)

(0)

(1)

(0)

(1)

(1)

(1)

(1)

(0)

(0)

(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(5)

3 Isomorphism Identification Algorithm

31 Equivalent Circuit Model As we know when two multi-ple joint kinematic chains or two gear trains kinematic chainsare isomorphic their WDCCGs are exactly the same theirvertices and edges are in one-to-one correspondence witheach other and the vertices and edges in correspondencekeep the same incidence and weighted relation Accordingto this definition the necessary and sufficient condition forisomorphism of the two types of kinematic chains is that theyhave the same WAM that is 119860 = 1198601015840

Equivalent circuit model (ECM) of WDCCG is estab-lished by its each edge which is replaced by conductanceequal to the weighted value of corresponding edge119898

119894119895

Theorem 1 If two WDCCGs corresponding to the same typekinematic chains are isomorphic then their correspondingECMs are identical

For example two isomorphism multiple joint kinematicchains are shown in Figures 4(a) and 4(b) their WDCCGsare shown in Figures 4(c) and 4(d) respectively According tothe description above their corresponding ECMs are shownin Figures 5(a) and 5(b) respectively There are two identicalcircuits and nodes of these two ECMs are in one-to-onecorrespondence

Complete excitation (CE) of an ECM is established asfollows in an ECM with 119899 nodes firstly set a node (119899 + 1) tobe a reference node then connect the reference node (119899 + 1)with every other node 119899

119894(119894 = 1 2 119899) of the ECM whose

conductance values are equal to the sumof weighted values ofthe node edges Secondly apply the same current source 1 Abetween node (119899+1) and other nodes 119899

119894(119894 = 1 2 119899) of the

ECM respectively in directions of the currents going fromnode (119899 + 1) to other nodes 119899

119894 (119894 = 1 2 119899) respectively

According to circuit theory two identical circuits underthe same CE have the same response And their node voltagescan be solved by the nodal method of circuit analysis Thenodal method of circuit analysis can be expressed as followsin a circuit network choose a node as a reference nodethe voltage difference between each node and the referencenode is known as the node voltage of the node For a circuitnetwork with 119899 nodes which has (119899 minus 1) node voltages if thenode 119899 is the reference node the nodal voltage equation canbe expressed as

[

[

[

[

[

[

[

11986611

11986612

sdot sdot sdot 1198661(119899minus1)

11986621

11986622

sdot sdot sdot 1198662(119899minus1)

d

119866(119899minus1)1

119866(119899minus1)2

sdot sdot sdot 119866(119899minus1)(119899minus1)

]

]

]

]

]

]

]

[

[

[

[

[

[

[

V1198991

V1198992

V119899(119899minus1)

]

]

]

]

]

]

]

=

[

[

[

[

[

[

[

1198681199041198991

1198681199041198992

119868119904119899(119899minus1)

]

]

]

]

]

]

]

(6)

where119866119894119894(119894 = 1 2 119899 minus 1) is called the self admittance of a

node 119894 the value of which is the sum of the admittance of allbranches that are connected to the node 119894119866

119894119895(119894 = 1 2 119899minus

1 119895 = 1 2 119899minus1) is called themutual admittance of node 119894and node 119895 whose value is the sum of the branch admittancebetween node 119894 and node 119895 and 119868

119904119899119894(119894 = 1 2 119899 minus 1) is

called the algebraic sum of the current source for the inflownode 119894 (inflow is positive outflow is negative)

For example an ECM with 4 nodes is shown in Fig-ure 6(a) Apply the CE of reference node 5 as shown inFigure 6(b) By nodalmethodof circuit analysis equations (6)node voltages V

1 V2 V3 and V

4can be expressed as follows

Mathematical Problems in Engineering 5

①

②

③

④

⑤

⑥

⑦

⑧

⑨

⑩

a

b

(a)

①

②

③

④

⑤

⑥

⑦

⑧ ⑨⑩

b

a

(b)

①

②③

a

b

1

1

1

2

223

3

(c)

①

②③

a

b1

1

1

2

2

2

3

3

(d)

Figure 4 Two multiple joint kinematic chains and their WDCCG

①

②③

2 2

2

11

1

a

3

3b

(a)

①

②③

2

2

2

1

1

1a

3

3

b

(b)

Figure 5 ECM corresponding to two WDCCGs of Figures 4(c) and 4(d) respectively

G1

G2

G3

G4 G5

① ② ③

④

(a)

G6

G7

G8

G9

①

②

③

④

⑤

1

1

1

1

(b)

Figure 6 An ECM with 4 nodes and its CE by reference node 5

6 Mathematical Problems in Engineering

[

[

[

[

[

[

1198661+ 1198664+ 1198666

minus1198664

0 minus1198661

minus1198664

1198662+ 1198664+ 1198665+ 1198667

minus1198665

minus1198662

0 minus1198665

1198663+ 1198665+ 1198668

minus1198663

minus1198661

minus1198662

minus1198663

1198661+ 1198662+ 1198663+ 1198669

]

]

]

]

]

]

[

[

[

[

[

[

V1

V2

V3

V4

]

]

]

]

]

]

=

[

[

[

[

[

[

1

1

1

1

]

]

]

]

]

]

(7)

Then the node voltages V1 V2 V3 and V

4can be obtained

by solving the above equations (7) Therefore the followingconclusion can be obtained

Theorem 2 If two WDCCGs corresponding to the same typeof kinematic chains are isomorphic their two correspondingECMs 119873 and 1198731015840 can be established and excited by the sameCE as above respectively Then the node voltages of eachcorrespondence node of two ECMs119873 and1198731015840 are the same

For example two identical ECMs119873 and1198731015840 with 5 nodesare shown in Figures 5(a) and 5(b) Apply the same CE ofreference node 6 which is shown in Figures 7(a) and 7(b)respectively And their corresponding WAMs 119860 and 1198601015840 canbe obtained as follows

119860 =

1 2 3 119886 119887

1

2

3

119886

119887

[

[

[

[

[

[

[

[

[

(1)

(2)

(0)

(1)

(2)

(2)

(1)

(1)

(0)

(3)

(0)

(1)

(1)

(2 3)

(1)

(1)

(0)

(2 3)

(0)

(0)

(2)

(3)

(1)

(0)

(0)

]

]

]

]

]

]

]

]

]

1198601015840=

1 2 3 119886 119887

1

2

3

119886

119887

[

[

[

[

[

[

[

[

[

(1)

(1)

(2)

(3)

(0)

(1)

(1)

(0)

(1)

(2 3)

(2)

(0)

(1)

(2)

(1)

(3)

(1)

(2)

(0)

(0)

(0)

(2 3)

(1)

(0)

(0)

]

]

]

]

]

]

]

]

]

(8)

By nodal method of circuit analysis equations of nodevoltages V

1 V2 V3 V119886 and V

119887can be expressed respectively

as follows

[

[

[

[

[

[

[

[

[

10 minus2 0 minus1 minus2

minus2 12 minus1 0 minus3

0 minus1 14 minus5 minus1

minus1 0 minus5 12 0

minus2 minus3 minus1 0 12

]

]

]

]

]

]

]

]

]

[

[

[

[

[

[

[

[

[

V1

V2

V3

V119886

V119887

]

]

]

]

]

]

]

]

]

=

[

[

[

[

[

[

[

[

[

1

1

1

1

1

]

]

]

]

]

]

]

]

]

(9)

[

[

[

[

[

[

[

[

[

12 minus1 minus2 minus3 0

minus1 14 0 minus1 minus5

minus2 0 10 minus2 minus1

minus3 minus1 minus2 12 0

0 minus5 minus1 0 12

]

]

]

]

]

]

]

]

]

[

[

[

[

[

[

[

[

[

V1

V2

V3

V119886

V119887

]

]

]

]

]

]

]

]

]

=

[

[

[

[

[

[

[

[

[

1

1

1

1

1

]

]

]

]

]

]

]

]

]

(10)

Then node voltages V1 V2 V3 V119886 and V

119887of two ECMs can

be obtained by solving the above (9) and (10) respectively asshown in Table 1

From Theorem 2 node voltages of each correspondencenode in these two ECMs are the same as follows V

119886= V119887=

01627 V119887= V119886= 01691 V

1= V3= 01839 V

2= V1= 01691

and V3= V2= 01537 When arranging the obtained node

voltages in descending order with the same node voltagesstaying together this set is called the node voltage sequence(NVS) and denoted as

NVS = lfloor(V1 V2 V

1198991

) (V(1198991+1) V(1198991+2) V

119899)rfloor

(V1ge V2ge V1198991

V(1198991+1)ge V(1198991+2)ge sdot sdot sdot ge V

119899)

(11)

where the variables V1 V2 V

1198991

represent node voltages ofhollow vertex The variables V

(1198991+1) V(1198991+2) V

119899represent

node voltages of solid vertexIf two identical ECMs are stimulated by the same CE as

above then the NVSs in correspondence are the same Forexample two identical ECMs as showed in Figures 5(a) and5(b) the NVS and vertex codes can be obtained respectivelyas follows

NVS (119886) = [(V119887 V119886) (V1 V2 V3)]

= [(01691 01627) (01839 01691 01537)]

NVS (119887) = [(V119886 V119887) (V3 V1 V2)]

= [(01691 01537) (01839 01691 01627)]

(12)

If the NVSs of two ECMs are not the same then the twoWDCCGs are not isomorphic On the contrary if they arethe same correspondence vertices in these two WDCCGscan be determined by the element values of NVS Thenperform the row exchanges of WAM 119860

1015840 to determine if theyare isomorphic Because the NVS can reduce the number ofrow exchanges to several ones or only just one this is a veryeffective method For example by the NVS(119886) and NVS(119887)of two ECMs shown in Figures 5(a) and 5(b) the verticesin correspondence are 119886-119887 119887-119886 1-3 2-1 3-2 then throughexchanging the row of WAM 119860

1015840 we obtain 119860 = 1198601015840 which

are isomorphic

Mathematical Problems in Engineering 7

①

②

③

⑥

1 1

1

11

6

6

7

6

5

a

b

(a)

①

②

③

⑥

1 1

1

1

1

6

6

6

7

5

a

b

(b)

Figure 7 Two ECMs with the same CE respectively

Table 1 Node voltages of two ECMs with the same CE

Vertex type Hollow vertex Solid vertex

Figure 5(a) Vertex code 119886 119887 1 2 3Node voltage 01627 01691 01839 01691 01537

Figure 5(b) Vertex code 119886 119887 1 2 3Node voltage 01691 01627 01691 01537 01839

For example two gear train kinematic chains are shownin Figures 8(a) and 8(b) and their WDCCGs as shownin Figures 8(c) and 8(d) respectively Their correspondingWAMs are 119860 and 1198601015840 as follows

119860

=

1 2 3 4 5 6 119887

1

2

3

4

5

6

119887

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

(1)

(2)

(0)

(0)

(3)

(0)

(1)

(2)

(1)

(3)

(0)

(0)

(0)

(0)

(0)

(3)

(1)

(3)

(0)

(0)

(1)

(0)

(0)

(3)

(1)

(0)

(2)

(0)

(3)

(0)

(0)

(0)

(1)

(0)

(1)

(0)

(0)

(0)

(2)

(0)

(1)

(1)

(1)

(0)

(1)

(0)

(1)

(1)

(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

1198601015840

=

1 2 3 4 5 6 119887

1

2

3

4

5

6

119887

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

(1)

(0)

(0)

(3)

(2)

(0)

(1)

(0)

(1)

(0)

(0)

(3)

(3)

(1)

(0)

(0)

(1)

(0)

(0)

(2)

(1)

(3)

(0)

(0)

(1)

(0)

(0)

(1)

(2)

(3)

(0)

(0)

(1)

(0)

(0)

(0)

(3)

(2)

(0)

(0)

(1)

(0)

(1)

(1)

(1)

(1)

(0)

(0)

(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(13)

According to the abovemethod their correspondingNVSand vertex codes can be obtained as follows

NVS (119886) = [(V119887) (V6 V5 V4 V1 V2 V3)] = [(02330)

(02744 02257 02068 01908 01907 01737)]

NVS (119887) = [(V119887) (V3 V4 V5 V1 V6 V2)] = [(02330)

(02744 02257 02068 01908 01907 01737)]

(14)

By the NVS(119886) and NVS(119887) the vertices of these twoWDCCGs in correspondence can be obtained as follows 119887-1198871-1 2-6 3-2 4-5 5-4 6-3 and through exchanging the rowof WAM 119860

1015840 one time we obtain isomorphic 119860 = 1198601015840

32 Algorithm Here a direct algorithm is given to determineisomorphism of planar multiple joint and gear train kine-matic chains and it can be expressed as follows

Step 1 Input the vertex codes and WAMs 119860 1198601015840 of twoWDCCGs corresponding to the same type of kinematicchains

Step 2 Assign vertices codes of 119860 and 1198601015840 in two groups thehollow vertexes to be the first group and the solid vertexes tobe another group

Step 3 Calculate NVS(119886) NVS(119887) and their vertex codesof the two ECMs with the same CE respectively CompareNVS(119886) and NVS(119887) if equivalence cannot be seen it isdetermined that they are not isomorphic and program stopsotherwise it is determined that they could be isomorphicThen check the NVS(119886) and NVS(119887) to see if the values intwo vertex code groups are all distinct if yes go to Step 4 ifno go to Step 6

8 Mathematical Problems in Engineering

①

②

③

④

⑤

⑥

A

a

b

C

B

c

(a)

① ②③ ④

⑤

⑥

A

a

b

C

B

c

(b)

①

②

③

④

⑤

⑥

1

b

1

1

1

2

3

3

3 2

(c)

①

②

③

④

⑤

⑥

3

3

3

2

2

b

1

1

1

1

(d)

Figure 8 Two gear train kinematic chains and their WDCCGs

Step 4 Take the NVS(119886) NVS(119887) and their vertex codes todetermine the correspondence of vertices according to thevalues that are in correspondence

Step 5 According to the corresponding vertices obtained andthe vertex codes to form new WAM 119860

1015840

1 rewrite WAM 119860

1015840 If119860 = 119860

1015840

1 it is determined that they are isomorphic otherwise

they are not isomorphic and program stops

Step 6 According to the corresponding vertices obtained theNVS(119886) and NVS(119887) have the same values in the correspond-ing two vertex code groups Just adjust the position order ofvertices of the same group in1198601015840 to obtain vertex codes to fromnew adjacency matrix 1198601015840

1 If 119860 = 119860

1015840

1 it is determined that

they are isomorphic and program stops if 119860 = 1198601015840

1 keep on

adjusting the position order of the aforementioned verticesin 1198601015840 of the vertex code groups for all possibilities And if119860 = 119860

1015840

1always it is determined that they are not isomorphic

and program stops

4 Examples

Example 1 Two 10-link multiple joint kinematic chainsare shown in Figures 9(a) and 9(b) Their correspondingWDCCGs are depicted in Figures 9(c) and 9(d) respectively

The direct process to determine isomorphism of these twokinematic chains can be expressed as follows

Step 1 Input their corresponding vertex codes andWAMs1198601198601015840 as follows

119860 =

1 119886 119887 119888 119889

1

119886

119887

119888

119889

[

[

[

[

[

[

[

[

[

(1)

(2)

(0)

(1)

(2)

(2)

(0)

(2)

(0)

(3)

(0)

(2)

(0)

(2)

(2)

(1)

(0)

(2)

(0)

(3)

(2)

(3)

(2)

(3)

(0)

]

]

]

]

]

]

]

]

]

1198601015840=

1 119886 119887 119888 119889

1

119886

119887

119888

119889

[

[

[

[

[

[

[

[

[

(1)

(0)

(2)

(2)

(1)

(0)

(0)

(2)

(2)

(2)

(2)

(2)

(0)

(0)

(3)

(2)

(2)

(0)

(0)

(3)

(1)

(2)

(3)

(3)

(0)

]

]

]

]

]

]

]

]

]

(15)

Step 2 Evidently 119860 = 1198601015840 Assign vertices codes of 119860 and 1198601015840

in two groups (a) [(119886 119887 119888 119889) (1)] and (b) [(119886 119887 119888 119889) (1)]

Mathematical Problems in Engineering 9

①

②

③

④

⑤

⑥

⑦

⑧

⑨

⑩

c

ba

d

(a)

①

②

③ ④

⑤

⑥⑦

⑧

⑨⑩

c b

a

d

(b)

①

cb

a

2

2

2

2

2

3

3

1d

(c)

①

2

2

2

2

2

3

3

d

b

ac

1

(d)

Figure 9 Two 10-link multiple joint kinematic chains and their WDCCGs

①

⑪

②

③

④

⑤

⑥

⑦⑧

⑨

⑩

ab

(a)

①

②

③④

⑤⑥

⑦

⑧

⑨

⑩

a

b

⑪

(b)

①

② ③

④

⑤

⑥ 1

1

11

12

2

2

2

2a

b

1

1 1

(c)

① ②

③

④

⑤

⑥

1

1

1

1

1

11

12

2

2

2

2b a

(d)

Figure 10 Two 11-link multiple joint kinematic chains and their WDCCGs

Step 3 Calculate NVS(119886) NVS(119887) and their vertex codes oftwo ECMs with the same CE respectively as follows

NVS (119886) = [(V119888 V119887 V119886 V119889) (V1)]

= [(01551 01546 01451 01275) (01700)]

NVS (119887) = [(V119886 V119887 V119888 V119889) (V1)]

= [(01538 01459 01459 01308) (01714)]

(16)

Compare NVS(119886) and NVS(119887) and equivalence cannot beseen It is determined that they are not isomorphic andprogram stops

Example 2 Two 11-link multiple joint kinematic chains areshown in Figures 10(a) and 10(b)Their correspondingWDC-CGs are described in Figures 10(c) and 10(d) respectivelyThe direct process to determine isomorphism of these twokinematic chains is as follows

10 Mathematical Problems in Engineering

Step 1 Input their corresponding vertex codes andWAMs1198601198601015840 as follows

119860

=

1 2 3 4 5 6 119886 119887

1

2

3

4

5

6

119886

119887

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

(1)

(2)

(0)

(1)

(1)

(0)

(0)

(2)

(2)

(1)

(1)

(0)

(1)

(0)

(0)

(0)

(0)

(1)

(1)

(1)

(0)

(2)

(2)

(0)

(1)

(0)

(1)

(1)

(0)

(1)

(0)

(0)

(1)

(1)

(0)

(0)

(1)

(0)

(1)

(0)

(0)

(2)

(0)

(1)

(0)

(1)

(0)

(1)

(0)

(0)

(1)

(0)

(1)

(0)

(0)

(2)

(2)

(0)

(0)

(0)

(0)

(1)

(2)

(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

1198601015840

=

1 2 3 4 5 6 119886 119887

1

2

3

4

5

6

119886

119887

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

(1)

(2)

(0)

(1)

(0)

(0)

(0)

(1)

(2)

(1)

(0)

(1)

(1)

(0)

(2)

(0)

(0)

(0)

(1)

(1)

(2)

(1)

(0)

(2)

(1)

(1)

(1)

(1)

(0)

(0)

(0)

(0)

(0)

(1)

(2)

(0)

(1)

(1)

(0)

(0)

(0)

(0)

(1)

(0)

(1)

(1)

(1)

(0)

(0)

(2)

(0)

(0)

(0)

(1)

(0)

(2)

(1)

(0)

(2)

(0)

(0)

(0)

(2)

(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(17)

Step 2 Evidently 119860 = 1198601015840 Assign vertices codes of 119860 and 1198601015840

in two groups (a) [(119886 119887) (1 2 3 4 5 6)] and (b) [(119886 119887) (1 23 4 5 6)]

Step 3 Calculate NVS(119886) NVS(119887) and their vertex codes oftwo ECMs with the same CE respectively as follows

NVS (119886) = [(V119886 V119887) (V5 V4 V2 V6 V1 V3)]

= [(02084 02056)

(0743 02727 02348 02347 02023 01995)]

NVS (119887) = [(V119887 V119888) (V6 V4 V5 V1 V3 V2)]

= [(02084 02056)

(0743 02727 02348 02347 02023 01995)]

(18)

Compare NVS(119886) and NVS(119887) and equivalence can beseen So the node voltages in two vertex code groups are alldistinct go to Step 4

Step 4 Take the NVS(119886) NVS(119887) and their vertex codes todetermine the corresponding vertices of two WDCCGs asfollows (119886-119887 119887-119886 1-3 2-5 3-2 4-4 5-6 6-1)

Step 5 According to the corresponding vertices obtained andthe vertex codes to form new WAM 119860

1015840

1 rewrite WAM 119860

1015840Then we have 119860 = 1198601015840

1 so they are isomorphic and program

stops

Example 3 Two 6-gear train kinematic chains are depicted inFigures 11(a) and 11(b) and their correspondingWDCCGs aredescribed in Figures 11(c) and 11(d) respectively The directprocess to determine isomorphism of these two kinematicchains is as follows

Step 1 Input their corresponding vertex codes andWAMs1198601198601015840 as follows

119860 =

1 2 3 4 5 6 7 119886 119888

1

2

3

4

5

6

7

119886

119888

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

(1)

(3)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(3)

(1)

(2)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(2)

(1)

(3)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(3)

(1)

(0)

(0)

(0)

(1)

(1)

(0)

(0)

(0)

(0)

(1)

(3)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(3)

(1)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(1)

(1)

(0)

(1)

(0)

(1)

(0)

(1)

(1)

(0)

(0)

(1)

(0)

(0)

(1)

(1)

(0)

(1)

(0)

(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

1198601015840=

1 2 3 4 5 6 7 119886 119888

1

2

3

4

5

6

7

119886

119888

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

(1)

(3)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(3)

(1)

(2)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(2)

(1)

(3)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(3)

(1)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(1)

(3)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(3)

(1)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(1)

(1)

(0)

(1)

(0)

(1)

(1)

(0)

(1)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(1)

(1)

(0)

(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(19)

Mathematical Problems in Engineering 11

①

② ③ ④ ⑤

⑥⑦

A

B

b

a

C

c

(a)

①

②③

④

⑤

⑥ ⑦

A

Bb

a

C

c

(b)

①

②

③

④ ⑤

⑥

⑦

3

3

3

c

2

11

1

1

1

1

1

1

a

(c)

①

②

③

④

⑤

⑥

⑦

c

a

1

1

1

1

1

1

1

2

3

3

3

(d)

Figure 11 Two 6-gear train kinematic chains and their WDCCG

Step 2 Evidently 119860 = 1198601015840 Assign vertices codes of 119860 and 1198601015840

in two groups (a) [(119886 119888) (1 2 3 4 5 6 7)] and (b) [(119886 119888) (12 3 4 5 6 7)]

Step 3 Calculate NVS(119886) NVS(119887) and their vertex codes oftwo ECMs with the same CE respectively as follows

NVS (119886) = [(V119887 V119886) (V7 V5 V6 V1 V4 V3 V2)]= [(02591 02547)

(03784 02515 02511 02309 02122 02029 01961)]

NVS (119887) = [(V119887 V119886) (V7 V6 V5 V1 V4 V3 V2)]= [(03156 02607)

(03940 02601 02551 02394 02367 02110 02001)]

(20)

Compare NVS(119886) and NVS(119887) and equivalence cannot beseen So it is determined that they are not isomorphic andprogram stops

Example 4 Two 9-gear train kinematic chains are shown inFigures 12(a) and 12(b) Their corresponding WDCCGs aredepicted in Figures 12(c) and 12(d) respectively The directprocess to determine isomorphism of these two kinematicchains is as follows

Step 1 Input their correspondingWAMs119860 and1198601015840 as follows

119860 =

1 2 3 4 5 6 7 8 9 119886 119887 119888

1

2

3

4

5

6

7

8

9

119886

119887

119888

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

(1)

(0)

(3)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(3)

(1)

(3)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(0)

(3)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(3)

(1)

(1)

(1)

(0)

(0)

(0)

(0)

(1)

(3)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(3)

(1)

(3)

(2)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(3)

(1)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(2)

(0)

(1)

(0)

(1)

(1)

(0)

(0)

(0)

(0)

(3)

(0)

(0)

(0)

(0)

(1)

(0)

(1)

(0)

(0)

(0)

(1)

(0)

(1)

(0)

(1)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(1)

(1)

(0)

(0)

(0)

(1)

(1)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

12 Mathematical Problems in Engineering

① ②

③

a

A

c

B

d

C

D

E

b

④

⑤

⑥

⑦

⑧

⑨

(a)

①

②③a

A

b E

c

DC

d

B④

⑤

⑥ ⑦⑧⑨

(b)

①

1 1

c

1 1

1

1

1

b3

3

3

2

1

1

1

1a

33

②

③

④

⑤⑥

⑦

⑧

⑨

(c)

1

1

1

1

3

3

1

1

1

1

1

1

1

2

3

3

3

b

c

a

①②

③

④

⑤

⑥

⑦

⑧

⑨

(d)

Figure 12 Two 9-gear train kinematic chains and their WDCCGs

1198601015840=

1 2 3 4 5 6 7 8 9 119886 119887 119888

1

2

3

4

5

6

7

8

9

119886

119887

119888

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

(1)

(0)

(0)

(3)

(0)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(1)

(0)

(3)

(0)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(1)

(2)

(0)

(0)

(0)

(0)

(0)

(1)

(0)

(1)

(3)

(3)

(2)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(3)

(3)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(3)

(1)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(3)

(0)

(1)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(3)

(1)

(1)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(3)

(1)

(0)

(0)

(1)

(1)

(1)

(1)

(0)

(1)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(1)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(1)

(0)

(0)

(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(21)

Step 2 Evidently 119860 = 1198601015840 Assign vertices codes of 119860 and 1198601015840

in two groups (a) [(119886 119887 119888) (1 2 3 4 5 6 7 8 9)] and (b) [(119886119887 119888) (1 2 3 4 5 6 7 8 9)]

Step 3 Calculate NVS(119886) NVS(119887) and their vertex codes oftwo ECMs with the same CE respectively as follows

NVS (119886) = [(V119888 V119887 V119886) (V9 V4 V8 V1 V2 V5 V7 V3 V6)]

= [(02845 02815 02075) (02470 02307 02295 02294 02294 02154 02154 01846 01719)]

NVS (119887) = [(V119887 V119888 V119886) (V9 V8 V3 V6 V7 V1 V2 V5 V4)]

= [(02845 02815 02075) (02470 02307 02295 02294 02294 02154 02154 01846 01719)]

(22)

Mathematical Problems in Engineering 13

Compare NVS(119886) and NVS(119887) and equivalence can be seenAnd the node voltages in two vertex code groups are not alldistinct so go to Step 6

Step 4 The vertices (1 2) in Figure 12(a) vertices (6 7) inFigure 12(b) the vertices (5 7) in Figure 12(a) and vertices (12) in Figure 12(b) have the same values in the correspondinggroups so just adjust the position order of the vertices in 1198601015840to obtain a new adjacency matrix 1198601015840 And the correspondingvertices of two WDCCGs are (1-6 2-7 3-5 4-8 5-1 6-4 7-28-3 9-9 119886-119886 119887-119888 119888-119887) so we have 119860 = 119860

1015840

1 It is determined

that they are isomorphic and program stops

5 Algorithm Complexity Analysis

This paper presents a relatively good algorithm for determi-nation of kinematic chains isomorphism with multiple jointand gear train It is feasible for introducing the WDCCGand WAM to describe these two types of kinematic chainsthey dramatically reduce the number of vertices and order ofcorresponding adjacency matrix 119860

The analysis of computational complexity of this isomor-phic identification algorithm is as follows when there areno equivalent node voltages in NVS time consumption isbasically dedicated to solving the linear equation sets If akinematic chain possesses 119899 vertices the order of the nodevoltage equations is 119899 minus 1 Since the adjoint circuit is alinear resistance circuit the node voltage equations forma linear algebra equation set of orders 119899 minus 1 which has acomputational complexity of 119874((119899 minus 1)3) for its solutionSince it needs to solve at most 2119899 adjoint circuits in ourproposed method the computational complexity is of order119874(2119899(119899 minus 1)

3) lt 119874(119899

4) Thus the proposed method has a

computational complexity of a polynomial orderWhen thereare equivalent node voltages in NVS the correspondence ofonly a part of the vertices cannot be determinedThus in theworst situation the proposed method is more efficient thanthose where rowcolumn exchanges are performed blindly

In this paper this algorithm is tested and comparedwith the internationally recognized faster method for planarsimple joint kinematic chains isomorphic identification [20]and they are tested in the same software and hardwareenvironment The test results are shown in Figure 13 (theaverage time of each vertex is 20 times of the kinematic chainwithmultiple joints) test results show that when the vertex ofthe kinematic chain is increased the algorithm proposed inthis paper is more efficient than the algorithm proposed byHe et alThe proposed algorithm by us is very suitable for theisomorphism of the topological graph of kinematic chainsand the algorithm is very efficient at the same time

6 Conclusions

In this paper an algorithm for topological isomorphism iden-tification of planar multiple joint and gear train kinematicchains is introduced

4 6 8 10 12 14 16 18 20 22

The number of vertices

Algorithm proposed by He et al

Algorithm proposed in this paper

00001

0001

001

01

1

The d

ecisi

on ti

me (

s)

Figure 13 The test results of the algorithm proposed in this paperand the algorithm proposed by He et al

Firstly a weighted-double-color-contracted-graph(WDCCG) and a weighted adjacency matrix (WAM) areintroduced to describe the planarmultiple joint and gear trainkinematic chains respectively They dramatically reduce thenumber of vertices and order of the represented adjacencymatrix

Secondly isomorphism identification method of thesetwo types of kinematic chains is carried out by the equivalentcircuit models (ECMs) of WDCCGs with the same completeexcitation (CE) and then uses the solved node voltagesequence (NVS) to determine the corresponding vertices oftwo WDCCGs

Finally an algorithm to identify isomorphic kinematicchains is obtained It is an efficient and easy method tobe realized by computer And this algorithm also can beused in the isomorphism identification of planar simple jointkinematic chains with minor changes

Notations

119860 1198601015840 The incidence matrix of kinematic chain119889119894119895 The elements of incidence matrix

119866119894 Conductance value of edge in the ECM

119898119894119895119896 The weighted value of 119896th edge

1198731198731015840 Variable of equivalent circuit model119899 The number of vertices in WDCCG119899119894 Node 119894 in the ECM

V119894 Node voltages of the node 119894

CE Complete excitationDCG Double color graphECM Equivalent circuit modelNVS Node voltage sequenceTM Topological modelWAM Weighted adjacency matrixWDCCG Weighted double color contracted graph

14 Mathematical Problems in Engineering

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

Thiswork is supported by the Science andTechnology Projectof Quanzhou City 2014G48 and G20140047

References

[1] J J Uicker Jr and A Raicu ldquoAmethod for the identification andrecognition of equivalence of kinematic chainsrdquoMechanismandMachine Theory vol 10 no 5 pp 375ndash383 1975

[2] H S Yan and A S Hall ldquoLinkage characteristic polynomialsdefinitions coefficients by inspectionrdquo Journal of MechanicalDesign vol 103 no 3 pp 578ndash584 1981

[3] H S Yan and A S Hall ldquoLinkage characteristic polynomialsassembly theorems uniquenessrdquo Journal of Mechanical Designvol 104 no 1 pp 11ndash20 1982

[4] T S Mruthyunjaya ldquoA computerized methodology for struc-tural synthesis of kinematic chains Part 1 FormulationrdquoMech-anism and Machine Theory vol 19 no 6 pp 487ndash495 1984

[5] T S Mruthyunjaya ldquoA computerized methodology for struc-tural synthesis of kinematic chains part 2mdashapplication toseveral fully or partially known casesrdquoMechanism andMachineTheory vol 19 no 6 pp 497ndash505 1984

[6] T S Mruthyunjaya ldquoA computerized methodology for struc-tural synthesis of kinematic chains part 3mdashapplication to thenew case of 10-link three- freedom chainsrdquo Mechanism andMachine Theory vol 19 no 6 pp 507ndash530 1984

[7] T S Mruthyunjaya and H R Balasubramanian ldquoIn questof a reliable and efficient computational test for detection ofisomorphism in kinematic chainsrdquo Mechanism and MachineTheory vol 22 no 2 pp 131ndash139 1987

[8] W J Sohn and F Freudenstein ldquoAn application of dual graphs tothe automatic generation of the kinematic structures of mecha-nismsrdquo Journal of Mechanisms Transmissions and Automationvol 108 no 3 pp 392ndash398 1986

[9] A Ambekar and V Agrawal ldquoOn canonical numbering ofkinematic chains and isomorphism problem MAX coderdquo inProceedings of the ASMEMechanisms Conference pp 1615ndash16221986

[10] A G Ambekar and V P Agrawal ldquoCanonical numberingof kinematic chains and isomorphism problem min coderdquoMechanism andMachineTheory vol 22 no 5 pp 453ndash461 1987

[11] C S Tang and T Liu ldquoDegree code a new mechanismidentifierrdquo Journal of Mechanical Design vol 115 no 3 pp 627ndash630 1993

[12] J K Shin and S Krishnamurty ldquoOn identification and canonicalnumbering of pin jointed kinematic chainsrdquo Journal of Mechan-ical Design vol 116 pp 182ndash188 1994

[13] J K Shin and S Krishnamurty ldquoDevelopment of a standardcode for colored graphs and its application to kinematic chainsrdquoJournal of Mechanical Design vol 116 no 1 pp 189ndash196 1994

[14] A C Rao and D Varada Raju ldquoApplication of the hammingnumber technique to detect isomorphism among kinematicchains and inversionsrdquoMechanism andMachineTheory vol 26no 1 pp 55ndash75 1991

[15] A C Rao and C N Rao ldquoLoop based pseudo hammingvalues-1 Testing isomorphism and rating kinematic chainsrdquoMechanism andMachineTheory vol 28 no 1 pp 113ndash127 1992

[16] A C Rao and C N Rao ldquoLoop based pseudo hamming values-II inversions preferred frames and actuatorsrdquo Mechanism andMachine Theory vol 28 no 1 pp 129ndash143 1993

[17] J-K Chu and W-Q Cao ldquoIdentification of isomorphismamong kinematic chains and inversions using linkrsquos adjacent-chain-tablerdquoMechanism and Machine Theory vol 29 no 1 pp53ndash58 1994

[18] A C Rao ldquoApplication of fuzzy logic for the study of iso-morphism inversions symmetry parallelism and mobility inkinematic chainsrdquoMechanism and Machine Theory vol 35 no8 pp 1103ndash1116 2000

[19] P R He W J Zhang Q Li and F X Wu ldquoA new method fordetection of graph isomorphism based on the quadratic formrdquoJournal of Mechanical Design vol 125 no 3 pp 640ndash642 2003

[20] P R He W J Zhang and Q Li ldquoSome further developmenton the eigensystem approach for graph isomorphismdetectionrdquoJournal of the Franklin Institute vol 342 no 6 pp 657ndash6732005

[21] Z Y Chang C Zhang Y H Yang and YWang ldquoA newmethodto mechanism kinematic chain isomorphism identificationrdquoMechanism andMachineTheory vol 37 no 4 pp 411ndash417 2002

[22] J P Cubillo and J BWan ldquoComments onmechanismkinematicchain isomorphism identification using adjacent matricesrdquoMechanism andMachineTheory vol 40 no 2 pp 131ndash139 2005

[23] H F Ding and Z Huang ldquoA unique representation of the kine-matic chain and the atlas databaserdquo Mechanism and MachineTheory vol 42 no 6 pp 637ndash651 2007

[24] H F Ding and Z Huang ldquoA new theory for the topologicalstructure analysis of kinematic chains and its applicationsrdquoMechanism and Machine Theory vol 42 no 10 pp 1264ndash12792007

[25] A C Rao ldquoA genetic algorithm for topological characteristicsof kinematic chainsrdquo Journal of Mechanical Design vol 122 no2 pp 228ndash231 2000

[26] F G Kong Q Li and W J Zhang ldquoAn artificial neuralnetwork approach tomechanism kinematic chain isomorphismidentificationrdquo Mechanism and Machine Theory vol 34 no 2pp 271ndash283 1999

[27] J K Chu The Structural and Dimensional Characteristics ofPlanar Linkages Press of Beihang University of China BeijingChina 1992

[28] L Song J Yang X Zhang and W Q Cao ldquoSpanning treemethod of identifying isomorphism and topological symmetryto planar kinematic chain with multiple jointrdquo Chinese Journalof Mechanical Engineering vol 14 no 1 pp 27ndash31 2001

[29] J G Liu and D J Yu ldquoRepresentations amp isomorphismidentification of planar kinematic chains with multiple jointsbased on the converted adjacent matrixrdquo Chinese Journal ofMechanical Engineering vol 48 pp 15ndash21 2012

[30] R Ravisankar andT SMruthyunjaya ldquoComputerized synthesisof the structure of geared kinematic chainsrdquo Mechanism andMachine Theory vol 20 no 5 pp 367ndash387 1985

[31] J U Kim and B M Kwak ldquoApplication of edge permutationgroup to structural synthesis of epicyclic gear trainsrdquo Mecha-nism and Machine Theory vol 25 no 5 pp 563ndash573 1990

Mathematical Problems in Engineering 15

[32] G Chatterjee and L-W Tsai ldquoComputer-aided sketching ofepicyclic-type automatic transmission gear trainsrdquo Journal ofMechanical Design vol 118 no 3 pp 405ndash411 1996

[33] V V N R Prasadraju and A C Rao ldquoA new technique basedon loops to investigate displacement isomorphism in planetarygear trainsrdquo Journal of Mechanical Design vol 12 pp 666ndash6752002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

①

②

③

④

⑤

⑥

⑦

⑧

⑨

⑩

a

b

(a)

①

②

③

④

⑤

⑥

⑦

⑧ ⑨⑩

b

a

(b)

①

②③

a

b

1

1

1

2

223

3

(c)

①

②③

a

b1

1

1

2

2

2

3

3

(d)

Figure 4 Two multiple joint kinematic chains and their WDCCG

①

②③

2 2

2

11

1

a

3

3b

(a)

①

②③

2

2

2

1

1

1a

3

3

b

(b)

Figure 5 ECM corresponding to two WDCCGs of Figures 4(c) and 4(d) respectively

G1

G2

G3

G4 G5

① ② ③

④

(a)

G6

G7

G8

G9

①

②

③

④

⑤

1

1

1

1

(b)

Figure 6 An ECM with 4 nodes and its CE by reference node 5

6 Mathematical Problems in Engineering

[

[

[

[

[

[

1198661+ 1198664+ 1198666

minus1198664

0 minus1198661

minus1198664

1198662+ 1198664+ 1198665+ 1198667

minus1198665

minus1198662

0 minus1198665

1198663+ 1198665+ 1198668

minus1198663

minus1198661

minus1198662

minus1198663

1198661+ 1198662+ 1198663+ 1198669

]

]

]

]

]

]

[

[

[

[

[

[

V1

V2

V3

V4

]

]

]

]

]

]

=

[

[

[

[

[

[

1

1

1

1

]

]

]

]

]

]

(7)

Then the node voltages V1 V2 V3 and V

4can be obtained

by solving the above equations (7) Therefore the followingconclusion can be obtained

Theorem 2 If two WDCCGs corresponding to the same typeof kinematic chains are isomorphic their two correspondingECMs 119873 and 1198731015840 can be established and excited by the sameCE as above respectively Then the node voltages of eachcorrespondence node of two ECMs119873 and1198731015840 are the same

For example two identical ECMs119873 and1198731015840 with 5 nodesare shown in Figures 5(a) and 5(b) Apply the same CE ofreference node 6 which is shown in Figures 7(a) and 7(b)respectively And their corresponding WAMs 119860 and 1198601015840 canbe obtained as follows

119860 =

1 2 3 119886 119887

1

2

3

119886

119887

[

[

[

[

[

[

[

[

[

(1)

(2)

(0)

(1)

(2)

(2)

(1)

(1)

(0)

(3)

(0)

(1)

(1)

(2 3)

(1)

(1)

(0)

(2 3)

(0)

(0)

(2)

(3)

(1)

(0)

(0)

]

]

]

]

]

]

]

]

]

1198601015840=

1 2 3 119886 119887

1

2

3

119886

119887

[

[

[

[

[

[

[

[

[

(1)

(1)

(2)

(3)

(0)

(1)

(1)

(0)

(1)

(2 3)

(2)

(0)

(1)

(2)

(1)

(3)

(1)

(2)

(0)

(0)

(0)

(2 3)

(1)

(0)

(0)

]

]

]

]

]

]

]

]

]

(8)

By nodal method of circuit analysis equations of nodevoltages V

1 V2 V3 V119886 and V

119887can be expressed respectively

as follows

[

[

[

[

[

[

[

[

[

10 minus2 0 minus1 minus2

minus2 12 minus1 0 minus3

0 minus1 14 minus5 minus1

minus1 0 minus5 12 0

minus2 minus3 minus1 0 12

]

]

]

]

]

]

]

]

]

[

[

[

[

[

[

[

[

[

V1

V2

V3

V119886

V119887

]

]

]

]

]

]

]

]

]

=

[

[

[

[

[

[

[

[

[

1

1

1

1

1

]

]

]

]

]

]

]

]

]

(9)

[

[

[

[

[

[

[

[

[

12 minus1 minus2 minus3 0

minus1 14 0 minus1 minus5

minus2 0 10 minus2 minus1

minus3 minus1 minus2 12 0

0 minus5 minus1 0 12

]

]

]

]

]

]

]

]

]

[

[

[

[

[

[

[

[

[

V1

V2

V3

V119886

V119887

]

]

]

]

]

]

]

]

]

=

[

[

[

[

[

[

[

[

[

1

1

1

1

1

]

]

]

]

]

]

]

]

]

(10)

Then node voltages V1 V2 V3 V119886 and V

119887of two ECMs can

be obtained by solving the above (9) and (10) respectively asshown in Table 1

From Theorem 2 node voltages of each correspondencenode in these two ECMs are the same as follows V

119886= V119887=

01627 V119887= V119886= 01691 V

1= V3= 01839 V

2= V1= 01691

and V3= V2= 01537 When arranging the obtained node

voltages in descending order with the same node voltagesstaying together this set is called the node voltage sequence(NVS) and denoted as

NVS = lfloor(V1 V2 V

1198991

) (V(1198991+1) V(1198991+2) V

119899)rfloor

(V1ge V2ge V1198991

V(1198991+1)ge V(1198991+2)ge sdot sdot sdot ge V

119899)

(11)

where the variables V1 V2 V

1198991

represent node voltages ofhollow vertex The variables V

(1198991+1) V(1198991+2) V

119899represent

node voltages of solid vertexIf two identical ECMs are stimulated by the same CE as

above then the NVSs in correspondence are the same Forexample two identical ECMs as showed in Figures 5(a) and5(b) the NVS and vertex codes can be obtained respectivelyas follows

NVS (119886) = [(V119887 V119886) (V1 V2 V3)]

= [(01691 01627) (01839 01691 01537)]

NVS (119887) = [(V119886 V119887) (V3 V1 V2)]

= [(01691 01537) (01839 01691 01627)]

(12)

If the NVSs of two ECMs are not the same then the twoWDCCGs are not isomorphic On the contrary if they arethe same correspondence vertices in these two WDCCGscan be determined by the element values of NVS Thenperform the row exchanges of WAM 119860

1015840 to determine if theyare isomorphic Because the NVS can reduce the number ofrow exchanges to several ones or only just one this is a veryeffective method For example by the NVS(119886) and NVS(119887)of two ECMs shown in Figures 5(a) and 5(b) the verticesin correspondence are 119886-119887 119887-119886 1-3 2-1 3-2 then throughexchanging the row of WAM 119860

1015840 we obtain 119860 = 1198601015840 which

are isomorphic

Mathematical Problems in Engineering 7

①

②

③

⑥

1 1

1

11

6

6

7

6

5

a

b

(a)

①

②

③

⑥

1 1

1

1

1

6

6

6

7

5

a

b

(b)

Figure 7 Two ECMs with the same CE respectively

Table 1 Node voltages of two ECMs with the same CE

Vertex type Hollow vertex Solid vertex

Figure 5(a) Vertex code 119886 119887 1 2 3Node voltage 01627 01691 01839 01691 01537

Figure 5(b) Vertex code 119886 119887 1 2 3Node voltage 01691 01627 01691 01537 01839

For example two gear train kinematic chains are shownin Figures 8(a) and 8(b) and their WDCCGs as shownin Figures 8(c) and 8(d) respectively Their correspondingWAMs are 119860 and 1198601015840 as follows

119860

=

1 2 3 4 5 6 119887

1

2

3

4

5

6

119887

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

(1)

(2)

(0)

(0)

(3)

(0)

(1)

(2)

(1)

(3)

(0)

(0)

(0)

(0)

(0)

(3)

(1)

(3)

(0)

(0)

(1)

(0)

(0)

(3)

(1)

(0)

(2)

(0)

(3)

(0)

(0)

(0)

(1)

(0)

(1)

(0)

(0)

(0)

(2)

(0)

(1)

(1)

(1)

(0)

(1)

(0)

(1)

(1)

(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

1198601015840

=

1 2 3 4 5 6 119887

1

2

3

4

5

6

119887

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

(1)

(0)

(0)

(3)

(2)

(0)

(1)

(0)

(1)

(0)

(0)

(3)

(3)

(1)

(0)

(0)

(1)

(0)

(0)

(2)

(1)

(3)

(0)

(0)

(1)

(0)

(0)

(1)

(2)

(3)

(0)

(0)

(1)

(0)

(0)

(0)

(3)

(2)

(0)

(0)

(1)

(0)

(1)

(1)

(1)

(1)

(0)

(0)

(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(13)

According to the abovemethod their correspondingNVSand vertex codes can be obtained as follows

NVS (119886) = [(V119887) (V6 V5 V4 V1 V2 V3)] = [(02330)

(02744 02257 02068 01908 01907 01737)]

NVS (119887) = [(V119887) (V3 V4 V5 V1 V6 V2)] = [(02330)

(02744 02257 02068 01908 01907 01737)]

(14)

By the NVS(119886) and NVS(119887) the vertices of these twoWDCCGs in correspondence can be obtained as follows 119887-1198871-1 2-6 3-2 4-5 5-4 6-3 and through exchanging the rowof WAM 119860

1015840 one time we obtain isomorphic 119860 = 1198601015840

32 Algorithm Here a direct algorithm is given to determineisomorphism of planar multiple joint and gear train kine-matic chains and it can be expressed as follows

Step 1 Input the vertex codes and WAMs 119860 1198601015840 of twoWDCCGs corresponding to the same type of kinematicchains

Step 2 Assign vertices codes of 119860 and 1198601015840 in two groups thehollow vertexes to be the first group and the solid vertexes tobe another group

Step 3 Calculate NVS(119886) NVS(119887) and their vertex codesof the two ECMs with the same CE respectively CompareNVS(119886) and NVS(119887) if equivalence cannot be seen it isdetermined that they are not isomorphic and program stopsotherwise it is determined that they could be isomorphicThen check the NVS(119886) and NVS(119887) to see if the values intwo vertex code groups are all distinct if yes go to Step 4 ifno go to Step 6

8 Mathematical Problems in Engineering

①

②

③

④

⑤

⑥

A

a

b

C

B

c

(a)

① ②③ ④

⑤

⑥

A

a

b

C

B

c

(b)

①

②

③

④

⑤

⑥

1

b

1

1

1

2

3

3

3 2

(c)

①

②

③

④

⑤

⑥

3

3

3

2

2

b

1

1

1

1

(d)

Figure 8 Two gear train kinematic chains and their WDCCGs

Step 4 Take the NVS(119886) NVS(119887) and their vertex codes todetermine the correspondence of vertices according to thevalues that are in correspondence

Step 5 According to the corresponding vertices obtained andthe vertex codes to form new WAM 119860

1015840

1 rewrite WAM 119860

1015840 If119860 = 119860

1015840

1 it is determined that they are isomorphic otherwise

they are not isomorphic and program stops

Step 6 According to the corresponding vertices obtained theNVS(119886) and NVS(119887) have the same values in the correspond-ing two vertex code groups Just adjust the position order ofvertices of the same group in1198601015840 to obtain vertex codes to fromnew adjacency matrix 1198601015840

1 If 119860 = 119860

1015840

1 it is determined that

they are isomorphic and program stops if 119860 = 1198601015840

1 keep on

adjusting the position order of the aforementioned verticesin 1198601015840 of the vertex code groups for all possibilities And if119860 = 119860

1015840

1always it is determined that they are not isomorphic

and program stops

4 Examples

Example 1 Two 10-link multiple joint kinematic chainsare shown in Figures 9(a) and 9(b) Their correspondingWDCCGs are depicted in Figures 9(c) and 9(d) respectively

The direct process to determine isomorphism of these twokinematic chains can be expressed as follows

Step 1 Input their corresponding vertex codes andWAMs1198601198601015840 as follows

119860 =

1 119886 119887 119888 119889

1

119886

119887

119888

119889

[

[

[

[

[

[

[

[

[

(1)

(2)

(0)

(1)

(2)

(2)

(0)

(2)

(0)

(3)

(0)

(2)

(0)

(2)

(2)

(1)

(0)

(2)

(0)

(3)

(2)

(3)

(2)

(3)

(0)

]

]

]

]

]

]

]

]

]

1198601015840=

1 119886 119887 119888 119889

1

119886

119887

119888

119889

[

[

[

[

[

[

[

[

[

(1)

(0)

(2)

(2)

(1)

(0)

(0)

(2)

(2)

(2)

(2)

(2)

(0)

(0)

(3)

(2)

(2)

(0)

(0)

(3)

(1)

(2)

(3)

(3)

(0)

]

]

]

]

]

]

]

]

]

(15)

Step 2 Evidently 119860 = 1198601015840 Assign vertices codes of 119860 and 1198601015840

in two groups (a) [(119886 119887 119888 119889) (1)] and (b) [(119886 119887 119888 119889) (1)]

Mathematical Problems in Engineering 9

①

②

③

④

⑤

⑥

⑦

⑧

⑨

⑩

c

ba

d

(a)

①

②

③ ④

⑤

⑥⑦

⑧

⑨⑩

c b

a

d

(b)

①

cb

a

2

2

2

2

2

3

3

1d

(c)

①

2

2

2

2

2

3

3

d

b

ac

1

(d)

Figure 9 Two 10-link multiple joint kinematic chains and their WDCCGs

①

⑪

②

③

④

⑤

⑥

⑦⑧

⑨

⑩

ab

(a)

①

②

③④

⑤⑥

⑦

⑧

⑨

⑩

a

b

⑪

(b)

①

② ③

④

⑤

⑥ 1

1

11

12

2

2

2

2a

b

1

1 1

(c)

① ②

③

④

⑤

⑥

1

1

1

1

1

11

12

2

2

2

2b a

(d)

Figure 10 Two 11-link multiple joint kinematic chains and their WDCCGs

Step 3 Calculate NVS(119886) NVS(119887) and their vertex codes oftwo ECMs with the same CE respectively as follows

NVS (119886) = [(V119888 V119887 V119886 V119889) (V1)]

= [(01551 01546 01451 01275) (01700)]

NVS (119887) = [(V119886 V119887 V119888 V119889) (V1)]

= [(01538 01459 01459 01308) (01714)]

(16)

Compare NVS(119886) and NVS(119887) and equivalence cannot beseen It is determined that they are not isomorphic andprogram stops

Example 2 Two 11-link multiple joint kinematic chains areshown in Figures 10(a) and 10(b)Their correspondingWDC-CGs are described in Figures 10(c) and 10(d) respectivelyThe direct process to determine isomorphism of these twokinematic chains is as follows

10 Mathematical Problems in Engineering

Step 1 Input their corresponding vertex codes andWAMs1198601198601015840 as follows

119860

=

1 2 3 4 5 6 119886 119887

1

2

3

4

5

6

119886

119887

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

(1)

(2)

(0)

(1)

(1)

(0)

(0)

(2)

(2)

(1)

(1)

(0)

(1)

(0)

(0)

(0)

(0)

(1)

(1)

(1)

(0)

(2)

(2)

(0)

(1)

(0)

(1)

(1)

(0)

(1)

(0)

(0)

(1)

(1)

(0)

(0)

(1)

(0)

(1)

(0)

(0)

(2)

(0)

(1)

(0)

(1)

(0)

(1)

(0)

(0)

(1)

(0)

(1)

(0)

(0)

(2)

(2)

(0)

(0)

(0)

(0)

(1)

(2)

(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

1198601015840

=

1 2 3 4 5 6 119886 119887

1

2

3

4

5

6

119886

119887

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

(1)

(2)

(0)

(1)

(0)

(0)

(0)

(1)

(2)

(1)

(0)

(1)

(1)

(0)

(2)

(0)

(0)

(0)

(1)

(1)

(2)

(1)

(0)

(2)

(1)

(1)

(1)

(1)

(0)

(0)

(0)

(0)

(0)

(1)

(2)

(0)

(1)

(1)

(0)

(0)

(0)

(0)

(1)

(0)

(1)

(1)

(1)

(0)

(0)

(2)

(0)

(0)

(0)

(1)

(0)

(2)

(1)

(0)

(2)

(0)

(0)

(0)

(2)

(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(17)

Step 2 Evidently 119860 = 1198601015840 Assign vertices codes of 119860 and 1198601015840

in two groups (a) [(119886 119887) (1 2 3 4 5 6)] and (b) [(119886 119887) (1 23 4 5 6)]

Step 3 Calculate NVS(119886) NVS(119887) and their vertex codes oftwo ECMs with the same CE respectively as follows

NVS (119886) = [(V119886 V119887) (V5 V4 V2 V6 V1 V3)]

= [(02084 02056)

(0743 02727 02348 02347 02023 01995)]

NVS (119887) = [(V119887 V119888) (V6 V4 V5 V1 V3 V2)]

= [(02084 02056)

(0743 02727 02348 02347 02023 01995)]

(18)

Compare NVS(119886) and NVS(119887) and equivalence can beseen So the node voltages in two vertex code groups are alldistinct go to Step 4

Step 4 Take the NVS(119886) NVS(119887) and their vertex codes todetermine the corresponding vertices of two WDCCGs asfollows (119886-119887 119887-119886 1-3 2-5 3-2 4-4 5-6 6-1)

Step 5 According to the corresponding vertices obtained andthe vertex codes to form new WAM 119860

1015840

1 rewrite WAM 119860

1015840Then we have 119860 = 1198601015840

1 so they are isomorphic and program

stops

Example 3 Two 6-gear train kinematic chains are depicted inFigures 11(a) and 11(b) and their correspondingWDCCGs aredescribed in Figures 11(c) and 11(d) respectively The directprocess to determine isomorphism of these two kinematicchains is as follows

Step 1 Input their corresponding vertex codes andWAMs1198601198601015840 as follows

119860 =

1 2 3 4 5 6 7 119886 119888

1

2

3

4

5

6

7

119886

119888

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

(1)

(3)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(3)

(1)

(2)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(2)

(1)

(3)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(3)

(1)

(0)

(0)

(0)

(1)

(1)

(0)

(0)

(0)

(0)

(1)

(3)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(3)

(1)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(1)

(1)

(0)

(1)

(0)

(1)

(0)

(1)

(1)

(0)

(0)

(1)

(0)

(0)

(1)

(1)

(0)

(1)

(0)

(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

1198601015840=

1 2 3 4 5 6 7 119886 119888

1

2

3

4

5

6

7

119886

119888

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

(1)

(3)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(3)

(1)

(2)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(2)

(1)

(3)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(3)

(1)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(1)

(3)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(3)

(1)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(1)

(1)

(0)

(1)

(0)

(1)

(1)

(0)

(1)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(1)

(1)

(0)

(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(19)

Mathematical Problems in Engineering 11

①

② ③ ④ ⑤

⑥⑦

A

B

b

a

C

c

(a)

①

②③

④

⑤

⑥ ⑦

A

Bb

a

C

c

(b)

①

②

③

④ ⑤

⑥

⑦

3

3

3

c

2

11

1

1

1

1

1

1

a

(c)

①

②

③

④

⑤

⑥

⑦

c

a

1

1

1

1

1

1

1

2

3

3

3

(d)

Figure 11 Two 6-gear train kinematic chains and their WDCCG

Step 2 Evidently 119860 = 1198601015840 Assign vertices codes of 119860 and 1198601015840

in two groups (a) [(119886 119888) (1 2 3 4 5 6 7)] and (b) [(119886 119888) (12 3 4 5 6 7)]

Step 3 Calculate NVS(119886) NVS(119887) and their vertex codes oftwo ECMs with the same CE respectively as follows

NVS (119886) = [(V119887 V119886) (V7 V5 V6 V1 V4 V3 V2)]= [(02591 02547)

(03784 02515 02511 02309 02122 02029 01961)]

NVS (119887) = [(V119887 V119886) (V7 V6 V5 V1 V4 V3 V2)]= [(03156 02607)

(03940 02601 02551 02394 02367 02110 02001)]

(20)

Compare NVS(119886) and NVS(119887) and equivalence cannot beseen So it is determined that they are not isomorphic andprogram stops

Example 4 Two 9-gear train kinematic chains are shown inFigures 12(a) and 12(b) Their corresponding WDCCGs aredepicted in Figures 12(c) and 12(d) respectively The directprocess to determine isomorphism of these two kinematicchains is as follows

Step 1 Input their correspondingWAMs119860 and1198601015840 as follows

119860 =

1 2 3 4 5 6 7 8 9 119886 119887 119888

1

2

3

4

5

6

7

8

9

119886

119887

119888

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

(1)

(0)

(3)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(3)

(1)

(3)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(0)

(3)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(3)

(1)

(1)

(1)

(0)

(0)

(0)

(0)

(1)

(3)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(3)

(1)

(3)

(2)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(3)

(1)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(2)

(0)

(1)

(0)

(1)

(1)

(0)

(0)

(0)

(0)

(3)

(0)

(0)

(0)

(0)

(1)

(0)

(1)

(0)

(0)

(0)

(1)

(0)

(1)

(0)

(1)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(1)

(1)

(0)

(0)

(0)

(1)

(1)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

12 Mathematical Problems in Engineering

① ②

③

a

A

c

B

d

C

D

E

b

④

⑤

⑥

⑦

⑧

⑨

(a)

①

②③a

A

b E

c

DC

d

B④

⑤

⑥ ⑦⑧⑨

(b)

①

1 1

c

1 1

1

1

1

b3

3

3

2

1

1

1

1a

33

②

③

④

⑤⑥

⑦

⑧

⑨

(c)

1

1

1

1

3

3

1

1

1

1

1

1

1

2

3

3

3

b

c

a

①②

③

④

⑤

⑥

⑦

⑧

⑨

(d)

Figure 12 Two 9-gear train kinematic chains and their WDCCGs

1198601015840=

1 2 3 4 5 6 7 8 9 119886 119887 119888

1

2

3

4

5

6

7

8

9

119886

119887

119888

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

(1)

(0)

(0)

(3)

(0)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(1)

(0)

(3)

(0)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(1)

(2)

(0)

(0)

(0)

(0)

(0)

(1)

(0)

(1)

(3)

(3)

(2)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(3)

(3)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(3)

(1)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(3)

(0)

(1)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(3)

(1)

(1)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(3)

(1)

(0)

(0)

(1)

(1)

(1)

(1)

(0)

(1)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(1)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(1)

(0)

(0)

(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(21)

Step 2 Evidently 119860 = 1198601015840 Assign vertices codes of 119860 and 1198601015840

in two groups (a) [(119886 119887 119888) (1 2 3 4 5 6 7 8 9)] and (b) [(119886119887 119888) (1 2 3 4 5 6 7 8 9)]

Step 3 Calculate NVS(119886) NVS(119887) and their vertex codes oftwo ECMs with the same CE respectively as follows

NVS (119886) = [(V119888 V119887 V119886) (V9 V4 V8 V1 V2 V5 V7 V3 V6)]

= [(02845 02815 02075) (02470 02307 02295 02294 02294 02154 02154 01846 01719)]

NVS (119887) = [(V119887 V119888 V119886) (V9 V8 V3 V6 V7 V1 V2 V5 V4)]

= [(02845 02815 02075) (02470 02307 02295 02294 02294 02154 02154 01846 01719)]

(22)

Mathematical Problems in Engineering 13

Compare NVS(119886) and NVS(119887) and equivalence can be seenAnd the node voltages in two vertex code groups are not alldistinct so go to Step 6

Step 4 The vertices (1 2) in Figure 12(a) vertices (6 7) inFigure 12(b) the vertices (5 7) in Figure 12(a) and vertices (12) in Figure 12(b) have the same values in the correspondinggroups so just adjust the position order of the vertices in 1198601015840to obtain a new adjacency matrix 1198601015840 And the correspondingvertices of two WDCCGs are (1-6 2-7 3-5 4-8 5-1 6-4 7-28-3 9-9 119886-119886 119887-119888 119888-119887) so we have 119860 = 119860

1015840

1 It is determined

that they are isomorphic and program stops

5 Algorithm Complexity Analysis

This paper presents a relatively good algorithm for determi-nation of kinematic chains isomorphism with multiple jointand gear train It is feasible for introducing the WDCCGand WAM to describe these two types of kinematic chainsthey dramatically reduce the number of vertices and order ofcorresponding adjacency matrix 119860

The analysis of computational complexity of this isomor-phic identification algorithm is as follows when there areno equivalent node voltages in NVS time consumption isbasically dedicated to solving the linear equation sets If akinematic chain possesses 119899 vertices the order of the nodevoltage equations is 119899 minus 1 Since the adjoint circuit is alinear resistance circuit the node voltage equations forma linear algebra equation set of orders 119899 minus 1 which has acomputational complexity of 119874((119899 minus 1)3) for its solutionSince it needs to solve at most 2119899 adjoint circuits in ourproposed method the computational complexity is of order119874(2119899(119899 minus 1)

3) lt 119874(119899

4) Thus the proposed method has a

computational complexity of a polynomial orderWhen thereare equivalent node voltages in NVS the correspondence ofonly a part of the vertices cannot be determinedThus in theworst situation the proposed method is more efficient thanthose where rowcolumn exchanges are performed blindly

In this paper this algorithm is tested and comparedwith the internationally recognized faster method for planarsimple joint kinematic chains isomorphic identification [20]and they are tested in the same software and hardwareenvironment The test results are shown in Figure 13 (theaverage time of each vertex is 20 times of the kinematic chainwithmultiple joints) test results show that when the vertex ofthe kinematic chain is increased the algorithm proposed inthis paper is more efficient than the algorithm proposed byHe et alThe proposed algorithm by us is very suitable for theisomorphism of the topological graph of kinematic chainsand the algorithm is very efficient at the same time

6 Conclusions

In this paper an algorithm for topological isomorphism iden-tification of planar multiple joint and gear train kinematicchains is introduced

4 6 8 10 12 14 16 18 20 22

The number of vertices

Algorithm proposed by He et al

Algorithm proposed in this paper

00001

0001

001

01

1

The d

ecisi

on ti

me (

s)

Figure 13 The test results of the algorithm proposed in this paperand the algorithm proposed by He et al

Firstly a weighted-double-color-contracted-graph(WDCCG) and a weighted adjacency matrix (WAM) areintroduced to describe the planarmultiple joint and gear trainkinematic chains respectively They dramatically reduce thenumber of vertices and order of the represented adjacencymatrix

Secondly isomorphism identification method of thesetwo types of kinematic chains is carried out by the equivalentcircuit models (ECMs) of WDCCGs with the same completeexcitation (CE) and then uses the solved node voltagesequence (NVS) to determine the corresponding vertices oftwo WDCCGs

Finally an algorithm to identify isomorphic kinematicchains is obtained It is an efficient and easy method tobe realized by computer And this algorithm also can beused in the isomorphism identification of planar simple jointkinematic chains with minor changes

Notations

119860 1198601015840 The incidence matrix of kinematic chain119889119894119895 The elements of incidence matrix

119866119894 Conductance value of edge in the ECM

119898119894119895119896 The weighted value of 119896th edge

1198731198731015840 Variable of equivalent circuit model119899 The number of vertices in WDCCG119899119894 Node 119894 in the ECM

V119894 Node voltages of the node 119894

CE Complete excitationDCG Double color graphECM Equivalent circuit modelNVS Node voltage sequenceTM Topological modelWAM Weighted adjacency matrixWDCCG Weighted double color contracted graph

14 Mathematical Problems in Engineering

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

Thiswork is supported by the Science andTechnology Projectof Quanzhou City 2014G48 and G20140047

References

[1] J J Uicker Jr and A Raicu ldquoAmethod for the identification andrecognition of equivalence of kinematic chainsrdquoMechanismandMachine Theory vol 10 no 5 pp 375ndash383 1975

[2] H S Yan and A S Hall ldquoLinkage characteristic polynomialsdefinitions coefficients by inspectionrdquo Journal of MechanicalDesign vol 103 no 3 pp 578ndash584 1981

[3] H S Yan and A S Hall ldquoLinkage characteristic polynomialsassembly theorems uniquenessrdquo Journal of Mechanical Designvol 104 no 1 pp 11ndash20 1982

[4] T S Mruthyunjaya ldquoA computerized methodology for struc-tural synthesis of kinematic chains Part 1 FormulationrdquoMech-anism and Machine Theory vol 19 no 6 pp 487ndash495 1984

[5] T S Mruthyunjaya ldquoA computerized methodology for struc-tural synthesis of kinematic chains part 2mdashapplication toseveral fully or partially known casesrdquoMechanism andMachineTheory vol 19 no 6 pp 497ndash505 1984

[6] T S Mruthyunjaya ldquoA computerized methodology for struc-tural synthesis of kinematic chains part 3mdashapplication to thenew case of 10-link three- freedom chainsrdquo Mechanism andMachine Theory vol 19 no 6 pp 507ndash530 1984

[7] T S Mruthyunjaya and H R Balasubramanian ldquoIn questof a reliable and efficient computational test for detection ofisomorphism in kinematic chainsrdquo Mechanism and MachineTheory vol 22 no 2 pp 131ndash139 1987

[8] W J Sohn and F Freudenstein ldquoAn application of dual graphs tothe automatic generation of the kinematic structures of mecha-nismsrdquo Journal of Mechanisms Transmissions and Automationvol 108 no 3 pp 392ndash398 1986

[9] A Ambekar and V Agrawal ldquoOn canonical numbering ofkinematic chains and isomorphism problem MAX coderdquo inProceedings of the ASMEMechanisms Conference pp 1615ndash16221986

[10] A G Ambekar and V P Agrawal ldquoCanonical numberingof kinematic chains and isomorphism problem min coderdquoMechanism andMachineTheory vol 22 no 5 pp 453ndash461 1987

[11] C S Tang and T Liu ldquoDegree code a new mechanismidentifierrdquo Journal of Mechanical Design vol 115 no 3 pp 627ndash630 1993

[12] J K Shin and S Krishnamurty ldquoOn identification and canonicalnumbering of pin jointed kinematic chainsrdquo Journal of Mechan-ical Design vol 116 pp 182ndash188 1994

[13] J K Shin and S Krishnamurty ldquoDevelopment of a standardcode for colored graphs and its application to kinematic chainsrdquoJournal of Mechanical Design vol 116 no 1 pp 189ndash196 1994

[14] A C Rao and D Varada Raju ldquoApplication of the hammingnumber technique to detect isomorphism among kinematicchains and inversionsrdquoMechanism andMachineTheory vol 26no 1 pp 55ndash75 1991

[15] A C Rao and C N Rao ldquoLoop based pseudo hammingvalues-1 Testing isomorphism and rating kinematic chainsrdquoMechanism andMachineTheory vol 28 no 1 pp 113ndash127 1992

[16] A C Rao and C N Rao ldquoLoop based pseudo hamming values-II inversions preferred frames and actuatorsrdquo Mechanism andMachine Theory vol 28 no 1 pp 129ndash143 1993

[17] J-K Chu and W-Q Cao ldquoIdentification of isomorphismamong kinematic chains and inversions using linkrsquos adjacent-chain-tablerdquoMechanism and Machine Theory vol 29 no 1 pp53ndash58 1994

[18] A C Rao ldquoApplication of fuzzy logic for the study of iso-morphism inversions symmetry parallelism and mobility inkinematic chainsrdquoMechanism and Machine Theory vol 35 no8 pp 1103ndash1116 2000

[19] P R He W J Zhang Q Li and F X Wu ldquoA new method fordetection of graph isomorphism based on the quadratic formrdquoJournal of Mechanical Design vol 125 no 3 pp 640ndash642 2003

[20] P R He W J Zhang and Q Li ldquoSome further developmenton the eigensystem approach for graph isomorphismdetectionrdquoJournal of the Franklin Institute vol 342 no 6 pp 657ndash6732005

[21] Z Y Chang C Zhang Y H Yang and YWang ldquoA newmethodto mechanism kinematic chain isomorphism identificationrdquoMechanism andMachineTheory vol 37 no 4 pp 411ndash417 2002

[22] J P Cubillo and J BWan ldquoComments onmechanismkinematicchain isomorphism identification using adjacent matricesrdquoMechanism andMachineTheory vol 40 no 2 pp 131ndash139 2005

[23] H F Ding and Z Huang ldquoA unique representation of the kine-matic chain and the atlas databaserdquo Mechanism and MachineTheory vol 42 no 6 pp 637ndash651 2007

[24] H F Ding and Z Huang ldquoA new theory for the topologicalstructure analysis of kinematic chains and its applicationsrdquoMechanism and Machine Theory vol 42 no 10 pp 1264ndash12792007

[25] A C Rao ldquoA genetic algorithm for topological characteristicsof kinematic chainsrdquo Journal of Mechanical Design vol 122 no2 pp 228ndash231 2000

[26] F G Kong Q Li and W J Zhang ldquoAn artificial neuralnetwork approach tomechanism kinematic chain isomorphismidentificationrdquo Mechanism and Machine Theory vol 34 no 2pp 271ndash283 1999

[27] J K Chu The Structural and Dimensional Characteristics ofPlanar Linkages Press of Beihang University of China BeijingChina 1992

[28] L Song J Yang X Zhang and W Q Cao ldquoSpanning treemethod of identifying isomorphism and topological symmetryto planar kinematic chain with multiple jointrdquo Chinese Journalof Mechanical Engineering vol 14 no 1 pp 27ndash31 2001

[29] J G Liu and D J Yu ldquoRepresentations amp isomorphismidentification of planar kinematic chains with multiple jointsbased on the converted adjacent matrixrdquo Chinese Journal ofMechanical Engineering vol 48 pp 15ndash21 2012

[30] R Ravisankar andT SMruthyunjaya ldquoComputerized synthesisof the structure of geared kinematic chainsrdquo Mechanism andMachine Theory vol 20 no 5 pp 367ndash387 1985

[31] J U Kim and B M Kwak ldquoApplication of edge permutationgroup to structural synthesis of epicyclic gear trainsrdquo Mecha-nism and Machine Theory vol 25 no 5 pp 563ndash573 1990

Mathematical Problems in Engineering 15

[32] G Chatterjee and L-W Tsai ldquoComputer-aided sketching ofepicyclic-type automatic transmission gear trainsrdquo Journal ofMechanical Design vol 118 no 3 pp 405ndash411 1996

[33] V V N R Prasadraju and A C Rao ldquoA new technique basedon loops to investigate displacement isomorphism in planetarygear trainsrdquo Journal of Mechanical Design vol 12 pp 666ndash6752002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

[

[

[

[

[

[

1198661+ 1198664+ 1198666

minus1198664

0 minus1198661

minus1198664

1198662+ 1198664+ 1198665+ 1198667

minus1198665

minus1198662

0 minus1198665

1198663+ 1198665+ 1198668

minus1198663

minus1198661

minus1198662

minus1198663

1198661+ 1198662+ 1198663+ 1198669

]

]

]

]

]

]

[

[

[

[

[

[

V1

V2

V3

V4

]

]

]

]

]

]

=

[

[

[

[

[

[

1

1

1

1

]

]

]

]

]

]

(7)

Then the node voltages V1 V2 V3 and V

4can be obtained

by solving the above equations (7) Therefore the followingconclusion can be obtained

Theorem 2 If two WDCCGs corresponding to the same typeof kinematic chains are isomorphic their two correspondingECMs 119873 and 1198731015840 can be established and excited by the sameCE as above respectively Then the node voltages of eachcorrespondence node of two ECMs119873 and1198731015840 are the same

For example two identical ECMs119873 and1198731015840 with 5 nodesare shown in Figures 5(a) and 5(b) Apply the same CE ofreference node 6 which is shown in Figures 7(a) and 7(b)respectively And their corresponding WAMs 119860 and 1198601015840 canbe obtained as follows

119860 =

1 2 3 119886 119887

1

2

3

119886

119887

[

[

[

[

[

[

[

[

[

(1)

(2)

(0)

(1)

(2)

(2)

(1)

(1)

(0)

(3)

(0)

(1)

(1)

(2 3)

(1)

(1)

(0)

(2 3)

(0)

(0)

(2)

(3)

(1)

(0)

(0)

]

]

]

]

]

]

]

]

]

1198601015840=

1 2 3 119886 119887

1

2

3

119886

119887

[

[

[

[

[

[

[

[

[

(1)

(1)

(2)

(3)

(0)

(1)

(1)

(0)

(1)

(2 3)

(2)

(0)

(1)

(2)

(1)

(3)

(1)

(2)

(0)

(0)

(0)

(2 3)

(1)

(0)

(0)

]

]

]

]

]

]

]

]

]

(8)

By nodal method of circuit analysis equations of nodevoltages V

1 V2 V3 V119886 and V

119887can be expressed respectively

as follows

[

[

[

[

[

[

[

[

[

10 minus2 0 minus1 minus2

minus2 12 minus1 0 minus3

0 minus1 14 minus5 minus1

minus1 0 minus5 12 0

minus2 minus3 minus1 0 12

]

]

]

]

]

]

]

]

]

[

[

[

[

[

[

[

[

[

V1

V2

V3

V119886

V119887

]

]

]

]

]

]

]

]

]

=

[

[

[

[

[

[

[

[

[

1

1

1

1

1

]

]

]

]

]

]

]

]

]

(9)

[

[

[

[

[

[

[

[

[

12 minus1 minus2 minus3 0

minus1 14 0 minus1 minus5

minus2 0 10 minus2 minus1

minus3 minus1 minus2 12 0

0 minus5 minus1 0 12

]

]

]

]

]

]

]

]

]

[

[

[

[

[

[

[

[

[

V1

V2

V3

V119886

V119887

]

]

]

]

]

]

]

]

]

=

[

[

[

[

[

[

[

[

[

1

1

1

1

1

]

]

]

]

]

]

]

]

]

(10)

Then node voltages V1 V2 V3 V119886 and V

119887of two ECMs can

be obtained by solving the above (9) and (10) respectively asshown in Table 1

From Theorem 2 node voltages of each correspondencenode in these two ECMs are the same as follows V

119886= V119887=

01627 V119887= V119886= 01691 V

1= V3= 01839 V

2= V1= 01691

and V3= V2= 01537 When arranging the obtained node

voltages in descending order with the same node voltagesstaying together this set is called the node voltage sequence(NVS) and denoted as

NVS = lfloor(V1 V2 V

1198991

) (V(1198991+1) V(1198991+2) V

119899)rfloor

(V1ge V2ge V1198991

V(1198991+1)ge V(1198991+2)ge sdot sdot sdot ge V

119899)

(11)

where the variables V1 V2 V

1198991

represent node voltages ofhollow vertex The variables V

(1198991+1) V(1198991+2) V

119899represent

node voltages of solid vertexIf two identical ECMs are stimulated by the same CE as

above then the NVSs in correspondence are the same Forexample two identical ECMs as showed in Figures 5(a) and5(b) the NVS and vertex codes can be obtained respectivelyas follows

NVS (119886) = [(V119887 V119886) (V1 V2 V3)]

= [(01691 01627) (01839 01691 01537)]

NVS (119887) = [(V119886 V119887) (V3 V1 V2)]

= [(01691 01537) (01839 01691 01627)]

(12)

If the NVSs of two ECMs are not the same then the twoWDCCGs are not isomorphic On the contrary if they arethe same correspondence vertices in these two WDCCGscan be determined by the element values of NVS Thenperform the row exchanges of WAM 119860

1015840 to determine if theyare isomorphic Because the NVS can reduce the number ofrow exchanges to several ones or only just one this is a veryeffective method For example by the NVS(119886) and NVS(119887)of two ECMs shown in Figures 5(a) and 5(b) the verticesin correspondence are 119886-119887 119887-119886 1-3 2-1 3-2 then throughexchanging the row of WAM 119860

1015840 we obtain 119860 = 1198601015840 which

are isomorphic

Mathematical Problems in Engineering 7

①

②

③

⑥

1 1

1

11

6

6

7

6

5

a

b

(a)

①

②

③

⑥

1 1

1

1

1

6

6

6

7

5

a

b

(b)

Figure 7 Two ECMs with the same CE respectively

Table 1 Node voltages of two ECMs with the same CE

Vertex type Hollow vertex Solid vertex

Figure 5(a) Vertex code 119886 119887 1 2 3Node voltage 01627 01691 01839 01691 01537

Figure 5(b) Vertex code 119886 119887 1 2 3Node voltage 01691 01627 01691 01537 01839

For example two gear train kinematic chains are shownin Figures 8(a) and 8(b) and their WDCCGs as shownin Figures 8(c) and 8(d) respectively Their correspondingWAMs are 119860 and 1198601015840 as follows

119860

=

1 2 3 4 5 6 119887

1

2

3

4

5

6

119887

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

(1)

(2)

(0)

(0)

(3)

(0)

(1)

(2)

(1)

(3)

(0)

(0)

(0)

(0)

(0)

(3)

(1)

(3)

(0)

(0)

(1)

(0)

(0)

(3)

(1)

(0)

(2)

(0)

(3)

(0)

(0)

(0)

(1)

(0)

(1)

(0)

(0)

(0)

(2)

(0)

(1)

(1)

(1)

(0)

(1)

(0)

(1)

(1)

(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

1198601015840

=

1 2 3 4 5 6 119887

1

2

3

4

5

6

119887

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

(1)

(0)

(0)

(3)

(2)

(0)

(1)

(0)

(1)

(0)

(0)

(3)

(3)

(1)

(0)

(0)

(1)

(0)

(0)

(2)

(1)

(3)

(0)

(0)

(1)

(0)

(0)

(1)

(2)

(3)

(0)

(0)

(1)

(0)

(0)

(0)

(3)

(2)

(0)

(0)

(1)

(0)

(1)

(1)

(1)

(1)

(0)

(0)

(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(13)

According to the abovemethod their correspondingNVSand vertex codes can be obtained as follows

NVS (119886) = [(V119887) (V6 V5 V4 V1 V2 V3)] = [(02330)

(02744 02257 02068 01908 01907 01737)]

NVS (119887) = [(V119887) (V3 V4 V5 V1 V6 V2)] = [(02330)

(02744 02257 02068 01908 01907 01737)]

(14)

By the NVS(119886) and NVS(119887) the vertices of these twoWDCCGs in correspondence can be obtained as follows 119887-1198871-1 2-6 3-2 4-5 5-4 6-3 and through exchanging the rowof WAM 119860

1015840 one time we obtain isomorphic 119860 = 1198601015840

32 Algorithm Here a direct algorithm is given to determineisomorphism of planar multiple joint and gear train kine-matic chains and it can be expressed as follows

Step 1 Input the vertex codes and WAMs 119860 1198601015840 of twoWDCCGs corresponding to the same type of kinematicchains

Step 2 Assign vertices codes of 119860 and 1198601015840 in two groups thehollow vertexes to be the first group and the solid vertexes tobe another group

Step 3 Calculate NVS(119886) NVS(119887) and their vertex codesof the two ECMs with the same CE respectively CompareNVS(119886) and NVS(119887) if equivalence cannot be seen it isdetermined that they are not isomorphic and program stopsotherwise it is determined that they could be isomorphicThen check the NVS(119886) and NVS(119887) to see if the values intwo vertex code groups are all distinct if yes go to Step 4 ifno go to Step 6

8 Mathematical Problems in Engineering

①

②

③

④

⑤

⑥

A

a

b

C

B

c

(a)

① ②③ ④

⑤

⑥

A

a

b

C

B

c

(b)

①

②

③

④

⑤

⑥

1

b

1

1

1

2

3

3

3 2

(c)

①

②

③

④

⑤

⑥

3

3

3

2

2

b

1

1

1

1

(d)

Figure 8 Two gear train kinematic chains and their WDCCGs

Step 4 Take the NVS(119886) NVS(119887) and their vertex codes todetermine the correspondence of vertices according to thevalues that are in correspondence

Step 5 According to the corresponding vertices obtained andthe vertex codes to form new WAM 119860

1015840

1 rewrite WAM 119860

1015840 If119860 = 119860

1015840

1 it is determined that they are isomorphic otherwise

they are not isomorphic and program stops

Step 6 According to the corresponding vertices obtained theNVS(119886) and NVS(119887) have the same values in the correspond-ing two vertex code groups Just adjust the position order ofvertices of the same group in1198601015840 to obtain vertex codes to fromnew adjacency matrix 1198601015840

1 If 119860 = 119860

1015840

1 it is determined that

they are isomorphic and program stops if 119860 = 1198601015840

1 keep on

adjusting the position order of the aforementioned verticesin 1198601015840 of the vertex code groups for all possibilities And if119860 = 119860

1015840

1always it is determined that they are not isomorphic

and program stops

4 Examples

Example 1 Two 10-link multiple joint kinematic chainsare shown in Figures 9(a) and 9(b) Their correspondingWDCCGs are depicted in Figures 9(c) and 9(d) respectively

The direct process to determine isomorphism of these twokinematic chains can be expressed as follows

Step 1 Input their corresponding vertex codes andWAMs1198601198601015840 as follows

119860 =

1 119886 119887 119888 119889

1

119886

119887

119888

119889

[

[

[

[

[

[

[

[

[

(1)

(2)

(0)

(1)

(2)

(2)

(0)

(2)

(0)

(3)

(0)

(2)

(0)

(2)

(2)

(1)

(0)

(2)

(0)

(3)

(2)

(3)

(2)

(3)

(0)

]

]

]

]

]

]

]

]

]

1198601015840=

1 119886 119887 119888 119889

1

119886

119887

119888

119889

[

[

[

[

[

[

[

[

[

(1)

(0)

(2)

(2)

(1)

(0)

(0)

(2)

(2)

(2)

(2)

(2)

(0)

(0)

(3)

(2)

(2)

(0)

(0)

(3)

(1)

(2)

(3)

(3)

(0)

]

]

]

]

]

]

]

]

]

(15)

Step 2 Evidently 119860 = 1198601015840 Assign vertices codes of 119860 and 1198601015840

in two groups (a) [(119886 119887 119888 119889) (1)] and (b) [(119886 119887 119888 119889) (1)]

Mathematical Problems in Engineering 9

①

②

③

④

⑤

⑥

⑦

⑧

⑨

⑩

c

ba

d

(a)

①

②

③ ④

⑤

⑥⑦

⑧

⑨⑩

c b

a

d

(b)

①

cb

a

2

2

2

2

2

3

3

1d

(c)

①

2

2

2

2

2

3

3

d

b

ac

1

(d)

Figure 9 Two 10-link multiple joint kinematic chains and their WDCCGs

①

⑪

②

③

④

⑤

⑥

⑦⑧

⑨

⑩

ab

(a)

①

②

③④

⑤⑥

⑦

⑧

⑨

⑩

a

b

⑪

(b)

①

② ③

④

⑤

⑥ 1

1

11

12

2

2

2

2a

b

1

1 1

(c)

① ②

③

④

⑤

⑥

1

1

1

1

1

11

12

2

2

2

2b a

(d)

Figure 10 Two 11-link multiple joint kinematic chains and their WDCCGs

Step 3 Calculate NVS(119886) NVS(119887) and their vertex codes oftwo ECMs with the same CE respectively as follows

NVS (119886) = [(V119888 V119887 V119886 V119889) (V1)]

= [(01551 01546 01451 01275) (01700)]

NVS (119887) = [(V119886 V119887 V119888 V119889) (V1)]

= [(01538 01459 01459 01308) (01714)]

(16)

Compare NVS(119886) and NVS(119887) and equivalence cannot beseen It is determined that they are not isomorphic andprogram stops

Example 2 Two 11-link multiple joint kinematic chains areshown in Figures 10(a) and 10(b)Their correspondingWDC-CGs are described in Figures 10(c) and 10(d) respectivelyThe direct process to determine isomorphism of these twokinematic chains is as follows

10 Mathematical Problems in Engineering

Step 1 Input their corresponding vertex codes andWAMs1198601198601015840 as follows

119860

=

1 2 3 4 5 6 119886 119887

1

2

3

4

5

6

119886

119887

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

(1)

(2)

(0)

(1)

(1)

(0)

(0)

(2)

(2)

(1)

(1)

(0)

(1)

(0)

(0)

(0)

(0)

(1)

(1)

(1)

(0)

(2)

(2)

(0)

(1)

(0)

(1)

(1)

(0)

(1)

(0)

(0)

(1)

(1)

(0)

(0)

(1)

(0)

(1)

(0)

(0)

(2)

(0)

(1)

(0)

(1)

(0)

(1)

(0)

(0)

(1)

(0)

(1)

(0)

(0)

(2)

(2)

(0)

(0)

(0)

(0)

(1)

(2)

(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

1198601015840

=

1 2 3 4 5 6 119886 119887

1

2

3

4

5

6

119886

119887

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

(1)

(2)

(0)

(1)

(0)

(0)

(0)

(1)

(2)

(1)

(0)

(1)

(1)

(0)

(2)

(0)

(0)

(0)

(1)

(1)

(2)

(1)

(0)

(2)

(1)

(1)

(1)

(1)

(0)

(0)

(0)

(0)

(0)

(1)

(2)

(0)

(1)

(1)

(0)

(0)

(0)

(0)

(1)

(0)

(1)

(1)

(1)

(0)

(0)

(2)

(0)

(0)

(0)

(1)

(0)

(2)

(1)

(0)

(2)

(0)

(0)

(0)

(2)

(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(17)

Step 2 Evidently 119860 = 1198601015840 Assign vertices codes of 119860 and 1198601015840

in two groups (a) [(119886 119887) (1 2 3 4 5 6)] and (b) [(119886 119887) (1 23 4 5 6)]

Step 3 Calculate NVS(119886) NVS(119887) and their vertex codes oftwo ECMs with the same CE respectively as follows

NVS (119886) = [(V119886 V119887) (V5 V4 V2 V6 V1 V3)]

= [(02084 02056)

(0743 02727 02348 02347 02023 01995)]

NVS (119887) = [(V119887 V119888) (V6 V4 V5 V1 V3 V2)]

= [(02084 02056)

(0743 02727 02348 02347 02023 01995)]

(18)

Compare NVS(119886) and NVS(119887) and equivalence can beseen So the node voltages in two vertex code groups are alldistinct go to Step 4

Step 4 Take the NVS(119886) NVS(119887) and their vertex codes todetermine the corresponding vertices of two WDCCGs asfollows (119886-119887 119887-119886 1-3 2-5 3-2 4-4 5-6 6-1)

Step 5 According to the corresponding vertices obtained andthe vertex codes to form new WAM 119860

1015840

1 rewrite WAM 119860

1015840Then we have 119860 = 1198601015840

1 so they are isomorphic and program

stops

Example 3 Two 6-gear train kinematic chains are depicted inFigures 11(a) and 11(b) and their correspondingWDCCGs aredescribed in Figures 11(c) and 11(d) respectively The directprocess to determine isomorphism of these two kinematicchains is as follows

Step 1 Input their corresponding vertex codes andWAMs1198601198601015840 as follows

119860 =

1 2 3 4 5 6 7 119886 119888

1

2

3

4

5

6

7

119886

119888

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

(1)

(3)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(3)

(1)

(2)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(2)

(1)

(3)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(3)

(1)

(0)

(0)

(0)

(1)

(1)

(0)

(0)

(0)

(0)

(1)

(3)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(3)

(1)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(1)

(1)

(0)

(1)

(0)

(1)

(0)

(1)

(1)

(0)

(0)

(1)

(0)

(0)

(1)

(1)

(0)

(1)

(0)

(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

1198601015840=

1 2 3 4 5 6 7 119886 119888

1

2

3

4

5

6

7

119886

119888

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

(1)

(3)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(3)

(1)

(2)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(2)

(1)

(3)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(3)

(1)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(1)

(3)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(3)

(1)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(1)

(1)

(0)

(1)

(0)

(1)

(1)

(0)

(1)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(1)

(1)

(0)

(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(19)

Mathematical Problems in Engineering 11

①

② ③ ④ ⑤

⑥⑦

A

B

b

a

C

c

(a)

①

②③

④

⑤

⑥ ⑦

A

Bb

a

C

c

(b)

①

②

③

④ ⑤

⑥

⑦

3

3

3

c

2

11

1

1

1

1

1

1

a

(c)

①

②

③

④

⑤

⑥

⑦

c

a

1

1

1

1

1

1

1

2

3

3

3

(d)

Figure 11 Two 6-gear train kinematic chains and their WDCCG

Step 2 Evidently 119860 = 1198601015840 Assign vertices codes of 119860 and 1198601015840

in two groups (a) [(119886 119888) (1 2 3 4 5 6 7)] and (b) [(119886 119888) (12 3 4 5 6 7)]

Step 3 Calculate NVS(119886) NVS(119887) and their vertex codes oftwo ECMs with the same CE respectively as follows

NVS (119886) = [(V119887 V119886) (V7 V5 V6 V1 V4 V3 V2)]= [(02591 02547)

(03784 02515 02511 02309 02122 02029 01961)]

NVS (119887) = [(V119887 V119886) (V7 V6 V5 V1 V4 V3 V2)]= [(03156 02607)

(03940 02601 02551 02394 02367 02110 02001)]

(20)

Compare NVS(119886) and NVS(119887) and equivalence cannot beseen So it is determined that they are not isomorphic andprogram stops

Example 4 Two 9-gear train kinematic chains are shown inFigures 12(a) and 12(b) Their corresponding WDCCGs aredepicted in Figures 12(c) and 12(d) respectively The directprocess to determine isomorphism of these two kinematicchains is as follows

Step 1 Input their correspondingWAMs119860 and1198601015840 as follows

119860 =

1 2 3 4 5 6 7 8 9 119886 119887 119888

1

2

3

4

5

6

7

8

9

119886

119887

119888

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

(1)

(0)

(3)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(3)

(1)

(3)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(0)

(3)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(3)

(1)

(1)

(1)

(0)

(0)

(0)

(0)

(1)

(3)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(3)

(1)

(3)

(2)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(3)

(1)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(2)

(0)

(1)

(0)

(1)

(1)

(0)

(0)

(0)

(0)

(3)

(0)

(0)

(0)

(0)

(1)

(0)

(1)

(0)

(0)

(0)

(1)

(0)

(1)

(0)

(1)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(1)

(1)

(0)

(0)

(0)

(1)

(1)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

12 Mathematical Problems in Engineering

① ②

③

a

A

c

B

d

C

D

E

b

④

⑤

⑥

⑦

⑧

⑨

(a)

①

②③a

A

b E

c

DC

d

B④

⑤

⑥ ⑦⑧⑨

(b)

①

1 1

c

1 1

1

1

1

b3

3

3

2

1

1

1

1a

33

②

③

④

⑤⑥

⑦

⑧

⑨

(c)

1

1

1

1

3

3

1

1

1

1

1

1

1

2

3

3

3

b

c

a

①②

③

④

⑤

⑥

⑦

⑧

⑨

(d)

Figure 12 Two 9-gear train kinematic chains and their WDCCGs

1198601015840=

1 2 3 4 5 6 7 8 9 119886 119887 119888

1

2

3

4

5

6

7

8

9

119886

119887

119888

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

(1)

(0)

(0)

(3)

(0)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(1)

(0)

(3)

(0)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(1)

(2)

(0)

(0)

(0)

(0)

(0)

(1)

(0)

(1)

(3)

(3)

(2)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(3)

(3)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(3)

(1)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(3)

(0)

(1)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(3)

(1)

(1)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(3)

(1)

(0)

(0)

(1)

(1)

(1)

(1)

(0)

(1)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(1)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(1)

(0)

(0)

(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(21)

Step 2 Evidently 119860 = 1198601015840 Assign vertices codes of 119860 and 1198601015840

in two groups (a) [(119886 119887 119888) (1 2 3 4 5 6 7 8 9)] and (b) [(119886119887 119888) (1 2 3 4 5 6 7 8 9)]

Step 3 Calculate NVS(119886) NVS(119887) and their vertex codes oftwo ECMs with the same CE respectively as follows

NVS (119886) = [(V119888 V119887 V119886) (V9 V4 V8 V1 V2 V5 V7 V3 V6)]

= [(02845 02815 02075) (02470 02307 02295 02294 02294 02154 02154 01846 01719)]

NVS (119887) = [(V119887 V119888 V119886) (V9 V8 V3 V6 V7 V1 V2 V5 V4)]

= [(02845 02815 02075) (02470 02307 02295 02294 02294 02154 02154 01846 01719)]

(22)

Mathematical Problems in Engineering 13

Compare NVS(119886) and NVS(119887) and equivalence can be seenAnd the node voltages in two vertex code groups are not alldistinct so go to Step 6

Step 4 The vertices (1 2) in Figure 12(a) vertices (6 7) inFigure 12(b) the vertices (5 7) in Figure 12(a) and vertices (12) in Figure 12(b) have the same values in the correspondinggroups so just adjust the position order of the vertices in 1198601015840to obtain a new adjacency matrix 1198601015840 And the correspondingvertices of two WDCCGs are (1-6 2-7 3-5 4-8 5-1 6-4 7-28-3 9-9 119886-119886 119887-119888 119888-119887) so we have 119860 = 119860

1015840

1 It is determined

that they are isomorphic and program stops

5 Algorithm Complexity Analysis

This paper presents a relatively good algorithm for determi-nation of kinematic chains isomorphism with multiple jointand gear train It is feasible for introducing the WDCCGand WAM to describe these two types of kinematic chainsthey dramatically reduce the number of vertices and order ofcorresponding adjacency matrix 119860

The analysis of computational complexity of this isomor-phic identification algorithm is as follows when there areno equivalent node voltages in NVS time consumption isbasically dedicated to solving the linear equation sets If akinematic chain possesses 119899 vertices the order of the nodevoltage equations is 119899 minus 1 Since the adjoint circuit is alinear resistance circuit the node voltage equations forma linear algebra equation set of orders 119899 minus 1 which has acomputational complexity of 119874((119899 minus 1)3) for its solutionSince it needs to solve at most 2119899 adjoint circuits in ourproposed method the computational complexity is of order119874(2119899(119899 minus 1)

3) lt 119874(119899

4) Thus the proposed method has a

computational complexity of a polynomial orderWhen thereare equivalent node voltages in NVS the correspondence ofonly a part of the vertices cannot be determinedThus in theworst situation the proposed method is more efficient thanthose where rowcolumn exchanges are performed blindly

In this paper this algorithm is tested and comparedwith the internationally recognized faster method for planarsimple joint kinematic chains isomorphic identification [20]and they are tested in the same software and hardwareenvironment The test results are shown in Figure 13 (theaverage time of each vertex is 20 times of the kinematic chainwithmultiple joints) test results show that when the vertex ofthe kinematic chain is increased the algorithm proposed inthis paper is more efficient than the algorithm proposed byHe et alThe proposed algorithm by us is very suitable for theisomorphism of the topological graph of kinematic chainsand the algorithm is very efficient at the same time

6 Conclusions

In this paper an algorithm for topological isomorphism iden-tification of planar multiple joint and gear train kinematicchains is introduced

4 6 8 10 12 14 16 18 20 22

The number of vertices

Algorithm proposed by He et al

Algorithm proposed in this paper

00001

0001

001

01

1

The d

ecisi

on ti

me (

s)

Figure 13 The test results of the algorithm proposed in this paperand the algorithm proposed by He et al

Firstly a weighted-double-color-contracted-graph(WDCCG) and a weighted adjacency matrix (WAM) areintroduced to describe the planarmultiple joint and gear trainkinematic chains respectively They dramatically reduce thenumber of vertices and order of the represented adjacencymatrix

Secondly isomorphism identification method of thesetwo types of kinematic chains is carried out by the equivalentcircuit models (ECMs) of WDCCGs with the same completeexcitation (CE) and then uses the solved node voltagesequence (NVS) to determine the corresponding vertices oftwo WDCCGs

Finally an algorithm to identify isomorphic kinematicchains is obtained It is an efficient and easy method tobe realized by computer And this algorithm also can beused in the isomorphism identification of planar simple jointkinematic chains with minor changes

Notations

119860 1198601015840 The incidence matrix of kinematic chain119889119894119895 The elements of incidence matrix

119866119894 Conductance value of edge in the ECM

119898119894119895119896 The weighted value of 119896th edge

1198731198731015840 Variable of equivalent circuit model119899 The number of vertices in WDCCG119899119894 Node 119894 in the ECM

V119894 Node voltages of the node 119894

CE Complete excitationDCG Double color graphECM Equivalent circuit modelNVS Node voltage sequenceTM Topological modelWAM Weighted adjacency matrixWDCCG Weighted double color contracted graph

14 Mathematical Problems in Engineering

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

Thiswork is supported by the Science andTechnology Projectof Quanzhou City 2014G48 and G20140047

References

[1] J J Uicker Jr and A Raicu ldquoAmethod for the identification andrecognition of equivalence of kinematic chainsrdquoMechanismandMachine Theory vol 10 no 5 pp 375ndash383 1975

[2] H S Yan and A S Hall ldquoLinkage characteristic polynomialsdefinitions coefficients by inspectionrdquo Journal of MechanicalDesign vol 103 no 3 pp 578ndash584 1981

[3] H S Yan and A S Hall ldquoLinkage characteristic polynomialsassembly theorems uniquenessrdquo Journal of Mechanical Designvol 104 no 1 pp 11ndash20 1982

[4] T S Mruthyunjaya ldquoA computerized methodology for struc-tural synthesis of kinematic chains Part 1 FormulationrdquoMech-anism and Machine Theory vol 19 no 6 pp 487ndash495 1984

[5] T S Mruthyunjaya ldquoA computerized methodology for struc-tural synthesis of kinematic chains part 2mdashapplication toseveral fully or partially known casesrdquoMechanism andMachineTheory vol 19 no 6 pp 497ndash505 1984

[6] T S Mruthyunjaya ldquoA computerized methodology for struc-tural synthesis of kinematic chains part 3mdashapplication to thenew case of 10-link three- freedom chainsrdquo Mechanism andMachine Theory vol 19 no 6 pp 507ndash530 1984

[7] T S Mruthyunjaya and H R Balasubramanian ldquoIn questof a reliable and efficient computational test for detection ofisomorphism in kinematic chainsrdquo Mechanism and MachineTheory vol 22 no 2 pp 131ndash139 1987

[8] W J Sohn and F Freudenstein ldquoAn application of dual graphs tothe automatic generation of the kinematic structures of mecha-nismsrdquo Journal of Mechanisms Transmissions and Automationvol 108 no 3 pp 392ndash398 1986

[9] A Ambekar and V Agrawal ldquoOn canonical numbering ofkinematic chains and isomorphism problem MAX coderdquo inProceedings of the ASMEMechanisms Conference pp 1615ndash16221986

[10] A G Ambekar and V P Agrawal ldquoCanonical numberingof kinematic chains and isomorphism problem min coderdquoMechanism andMachineTheory vol 22 no 5 pp 453ndash461 1987

[11] C S Tang and T Liu ldquoDegree code a new mechanismidentifierrdquo Journal of Mechanical Design vol 115 no 3 pp 627ndash630 1993

[12] J K Shin and S Krishnamurty ldquoOn identification and canonicalnumbering of pin jointed kinematic chainsrdquo Journal of Mechan-ical Design vol 116 pp 182ndash188 1994

[13] J K Shin and S Krishnamurty ldquoDevelopment of a standardcode for colored graphs and its application to kinematic chainsrdquoJournal of Mechanical Design vol 116 no 1 pp 189ndash196 1994

[14] A C Rao and D Varada Raju ldquoApplication of the hammingnumber technique to detect isomorphism among kinematicchains and inversionsrdquoMechanism andMachineTheory vol 26no 1 pp 55ndash75 1991

[15] A C Rao and C N Rao ldquoLoop based pseudo hammingvalues-1 Testing isomorphism and rating kinematic chainsrdquoMechanism andMachineTheory vol 28 no 1 pp 113ndash127 1992

[16] A C Rao and C N Rao ldquoLoop based pseudo hamming values-II inversions preferred frames and actuatorsrdquo Mechanism andMachine Theory vol 28 no 1 pp 129ndash143 1993

[17] J-K Chu and W-Q Cao ldquoIdentification of isomorphismamong kinematic chains and inversions using linkrsquos adjacent-chain-tablerdquoMechanism and Machine Theory vol 29 no 1 pp53ndash58 1994

[18] A C Rao ldquoApplication of fuzzy logic for the study of iso-morphism inversions symmetry parallelism and mobility inkinematic chainsrdquoMechanism and Machine Theory vol 35 no8 pp 1103ndash1116 2000

[19] P R He W J Zhang Q Li and F X Wu ldquoA new method fordetection of graph isomorphism based on the quadratic formrdquoJournal of Mechanical Design vol 125 no 3 pp 640ndash642 2003

[20] P R He W J Zhang and Q Li ldquoSome further developmenton the eigensystem approach for graph isomorphismdetectionrdquoJournal of the Franklin Institute vol 342 no 6 pp 657ndash6732005

[21] Z Y Chang C Zhang Y H Yang and YWang ldquoA newmethodto mechanism kinematic chain isomorphism identificationrdquoMechanism andMachineTheory vol 37 no 4 pp 411ndash417 2002

[22] J P Cubillo and J BWan ldquoComments onmechanismkinematicchain isomorphism identification using adjacent matricesrdquoMechanism andMachineTheory vol 40 no 2 pp 131ndash139 2005

[23] H F Ding and Z Huang ldquoA unique representation of the kine-matic chain and the atlas databaserdquo Mechanism and MachineTheory vol 42 no 6 pp 637ndash651 2007

[24] H F Ding and Z Huang ldquoA new theory for the topologicalstructure analysis of kinematic chains and its applicationsrdquoMechanism and Machine Theory vol 42 no 10 pp 1264ndash12792007

[25] A C Rao ldquoA genetic algorithm for topological characteristicsof kinematic chainsrdquo Journal of Mechanical Design vol 122 no2 pp 228ndash231 2000

[26] F G Kong Q Li and W J Zhang ldquoAn artificial neuralnetwork approach tomechanism kinematic chain isomorphismidentificationrdquo Mechanism and Machine Theory vol 34 no 2pp 271ndash283 1999

[27] J K Chu The Structural and Dimensional Characteristics ofPlanar Linkages Press of Beihang University of China BeijingChina 1992

[28] L Song J Yang X Zhang and W Q Cao ldquoSpanning treemethod of identifying isomorphism and topological symmetryto planar kinematic chain with multiple jointrdquo Chinese Journalof Mechanical Engineering vol 14 no 1 pp 27ndash31 2001

[29] J G Liu and D J Yu ldquoRepresentations amp isomorphismidentification of planar kinematic chains with multiple jointsbased on the converted adjacent matrixrdquo Chinese Journal ofMechanical Engineering vol 48 pp 15ndash21 2012

[30] R Ravisankar andT SMruthyunjaya ldquoComputerized synthesisof the structure of geared kinematic chainsrdquo Mechanism andMachine Theory vol 20 no 5 pp 367ndash387 1985

[31] J U Kim and B M Kwak ldquoApplication of edge permutationgroup to structural synthesis of epicyclic gear trainsrdquo Mecha-nism and Machine Theory vol 25 no 5 pp 563ndash573 1990

Mathematical Problems in Engineering 15

[32] G Chatterjee and L-W Tsai ldquoComputer-aided sketching ofepicyclic-type automatic transmission gear trainsrdquo Journal ofMechanical Design vol 118 no 3 pp 405ndash411 1996

[33] V V N R Prasadraju and A C Rao ldquoA new technique basedon loops to investigate displacement isomorphism in planetarygear trainsrdquo Journal of Mechanical Design vol 12 pp 666ndash6752002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

①

②

③

⑥

1 1

1

11

6

6

7

6

5

a

b

(a)

①

②

③

⑥

1 1

1

1

1

6

6

6

7

5

a

b

(b)

Figure 7 Two ECMs with the same CE respectively

Table 1 Node voltages of two ECMs with the same CE

Vertex type Hollow vertex Solid vertex

Figure 5(a) Vertex code 119886 119887 1 2 3Node voltage 01627 01691 01839 01691 01537

Figure 5(b) Vertex code 119886 119887 1 2 3Node voltage 01691 01627 01691 01537 01839

For example two gear train kinematic chains are shownin Figures 8(a) and 8(b) and their WDCCGs as shownin Figures 8(c) and 8(d) respectively Their correspondingWAMs are 119860 and 1198601015840 as follows

119860

=

1 2 3 4 5 6 119887

1

2

3

4

5

6

119887

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

(1)

(2)

(0)

(0)

(3)

(0)

(1)

(2)

(1)

(3)

(0)

(0)

(0)

(0)

(0)

(3)

(1)

(3)

(0)

(0)

(1)

(0)

(0)

(3)

(1)

(0)

(2)

(0)

(3)

(0)

(0)

(0)

(1)

(0)

(1)

(0)

(0)

(0)

(2)

(0)

(1)

(1)

(1)

(0)

(1)

(0)

(1)

(1)

(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

1198601015840

=

1 2 3 4 5 6 119887

1

2

3

4

5

6

119887

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

(1)

(0)

(0)

(3)

(2)

(0)

(1)

(0)

(1)

(0)

(0)

(3)

(3)

(1)

(0)

(0)

(1)

(0)

(0)

(2)

(1)

(3)

(0)

(0)

(1)

(0)

(0)

(1)

(2)

(3)

(0)

(0)

(1)

(0)

(0)

(0)

(3)

(2)

(0)

(0)

(1)

(0)

(1)

(1)

(1)

(1)

(0)

(0)

(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(13)

According to the abovemethod their correspondingNVSand vertex codes can be obtained as follows

NVS (119886) = [(V119887) (V6 V5 V4 V1 V2 V3)] = [(02330)

(02744 02257 02068 01908 01907 01737)]

NVS (119887) = [(V119887) (V3 V4 V5 V1 V6 V2)] = [(02330)

(02744 02257 02068 01908 01907 01737)]

(14)

By the NVS(119886) and NVS(119887) the vertices of these twoWDCCGs in correspondence can be obtained as follows 119887-1198871-1 2-6 3-2 4-5 5-4 6-3 and through exchanging the rowof WAM 119860

1015840 one time we obtain isomorphic 119860 = 1198601015840

32 Algorithm Here a direct algorithm is given to determineisomorphism of planar multiple joint and gear train kine-matic chains and it can be expressed as follows

Step 1 Input the vertex codes and WAMs 119860 1198601015840 of twoWDCCGs corresponding to the same type of kinematicchains

Step 2 Assign vertices codes of 119860 and 1198601015840 in two groups thehollow vertexes to be the first group and the solid vertexes tobe another group

Step 3 Calculate NVS(119886) NVS(119887) and their vertex codesof the two ECMs with the same CE respectively CompareNVS(119886) and NVS(119887) if equivalence cannot be seen it isdetermined that they are not isomorphic and program stopsotherwise it is determined that they could be isomorphicThen check the NVS(119886) and NVS(119887) to see if the values intwo vertex code groups are all distinct if yes go to Step 4 ifno go to Step 6

8 Mathematical Problems in Engineering

①

②

③

④

⑤

⑥

A

a

b

C

B

c

(a)

① ②③ ④

⑤

⑥

A

a

b

C

B

c

(b)

①

②

③

④

⑤

⑥

1

b

1

1

1

2

3

3

3 2

(c)

①

②

③

④

⑤

⑥

3

3

3

2

2

b

1

1

1

1

(d)

Figure 8 Two gear train kinematic chains and their WDCCGs

Step 4 Take the NVS(119886) NVS(119887) and their vertex codes todetermine the correspondence of vertices according to thevalues that are in correspondence

Step 5 According to the corresponding vertices obtained andthe vertex codes to form new WAM 119860

1015840

1 rewrite WAM 119860

1015840 If119860 = 119860

1015840

1 it is determined that they are isomorphic otherwise

they are not isomorphic and program stops

Step 6 According to the corresponding vertices obtained theNVS(119886) and NVS(119887) have the same values in the correspond-ing two vertex code groups Just adjust the position order ofvertices of the same group in1198601015840 to obtain vertex codes to fromnew adjacency matrix 1198601015840

1 If 119860 = 119860

1015840

1 it is determined that

they are isomorphic and program stops if 119860 = 1198601015840

1 keep on

adjusting the position order of the aforementioned verticesin 1198601015840 of the vertex code groups for all possibilities And if119860 = 119860

1015840

1always it is determined that they are not isomorphic

and program stops

4 Examples

Example 1 Two 10-link multiple joint kinematic chainsare shown in Figures 9(a) and 9(b) Their correspondingWDCCGs are depicted in Figures 9(c) and 9(d) respectively

The direct process to determine isomorphism of these twokinematic chains can be expressed as follows

Step 1 Input their corresponding vertex codes andWAMs1198601198601015840 as follows

119860 =

1 119886 119887 119888 119889

1

119886

119887

119888

119889

[

[

[

[

[

[

[

[

[

(1)

(2)

(0)

(1)

(2)

(2)

(0)

(2)

(0)

(3)

(0)

(2)

(0)

(2)

(2)

(1)

(0)

(2)

(0)

(3)

(2)

(3)

(2)

(3)

(0)

]

]

]

]

]

]

]

]

]

1198601015840=

1 119886 119887 119888 119889

1

119886

119887

119888

119889

[

[

[

[

[

[

[

[

[

(1)

(0)

(2)

(2)

(1)

(0)

(0)

(2)

(2)

(2)

(2)

(2)

(0)

(0)

(3)

(2)

(2)

(0)

(0)

(3)

(1)

(2)

(3)

(3)

(0)

]

]

]

]

]

]

]

]

]

(15)

Step 2 Evidently 119860 = 1198601015840 Assign vertices codes of 119860 and 1198601015840

in two groups (a) [(119886 119887 119888 119889) (1)] and (b) [(119886 119887 119888 119889) (1)]

Mathematical Problems in Engineering 9

①

②

③

④

⑤

⑥

⑦

⑧

⑨

⑩

c

ba

d

(a)

①

②

③ ④

⑤

⑥⑦

⑧

⑨⑩

c b

a

d

(b)

①

cb

a

2

2

2

2

2

3

3

1d

(c)

①

2

2

2

2

2

3

3

d

b

ac

1

(d)

Figure 9 Two 10-link multiple joint kinematic chains and their WDCCGs

①

⑪

②

③

④

⑤

⑥

⑦⑧

⑨

⑩

ab

(a)

①

②

③④

⑤⑥

⑦

⑧

⑨

⑩

a

b

⑪

(b)

①

② ③

④

⑤

⑥ 1

1

11

12

2

2

2

2a

b

1

1 1

(c)

① ②

③

④

⑤

⑥

1

1

1

1

1

11

12

2

2

2

2b a

(d)

Figure 10 Two 11-link multiple joint kinematic chains and their WDCCGs

Step 3 Calculate NVS(119886) NVS(119887) and their vertex codes oftwo ECMs with the same CE respectively as follows

NVS (119886) = [(V119888 V119887 V119886 V119889) (V1)]

= [(01551 01546 01451 01275) (01700)]

NVS (119887) = [(V119886 V119887 V119888 V119889) (V1)]

= [(01538 01459 01459 01308) (01714)]

(16)

Compare NVS(119886) and NVS(119887) and equivalence cannot beseen It is determined that they are not isomorphic andprogram stops

Example 2 Two 11-link multiple joint kinematic chains areshown in Figures 10(a) and 10(b)Their correspondingWDC-CGs are described in Figures 10(c) and 10(d) respectivelyThe direct process to determine isomorphism of these twokinematic chains is as follows

10 Mathematical Problems in Engineering

Step 1 Input their corresponding vertex codes andWAMs1198601198601015840 as follows

119860

=

1 2 3 4 5 6 119886 119887

1

2

3

4

5

6

119886

119887

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

(1)

(2)

(0)

(1)

(1)

(0)

(0)

(2)

(2)

(1)

(1)

(0)

(1)

(0)

(0)

(0)

(0)

(1)

(1)

(1)

(0)

(2)

(2)

(0)

(1)

(0)

(1)

(1)

(0)

(1)

(0)

(0)

(1)

(1)

(0)

(0)

(1)

(0)

(1)

(0)

(0)

(2)

(0)

(1)

(0)

(1)

(0)

(1)

(0)

(0)

(1)

(0)

(1)

(0)

(0)

(2)

(2)

(0)

(0)

(0)

(0)

(1)

(2)

(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

1198601015840

=

1 2 3 4 5 6 119886 119887

1

2

3

4

5

6

119886

119887

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

(1)

(2)

(0)

(1)

(0)

(0)

(0)

(1)

(2)

(1)

(0)

(1)

(1)

(0)

(2)

(0)

(0)

(0)

(1)

(1)

(2)

(1)

(0)

(2)

(1)

(1)

(1)

(1)

(0)

(0)

(0)

(0)

(0)

(1)

(2)

(0)

(1)

(1)

(0)

(0)

(0)

(0)

(1)

(0)

(1)

(1)

(1)

(0)

(0)

(2)

(0)

(0)

(0)

(1)

(0)

(2)

(1)

(0)

(2)

(0)

(0)

(0)

(2)

(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(17)

Step 2 Evidently 119860 = 1198601015840 Assign vertices codes of 119860 and 1198601015840

in two groups (a) [(119886 119887) (1 2 3 4 5 6)] and (b) [(119886 119887) (1 23 4 5 6)]

Step 3 Calculate NVS(119886) NVS(119887) and their vertex codes oftwo ECMs with the same CE respectively as follows

NVS (119886) = [(V119886 V119887) (V5 V4 V2 V6 V1 V3)]

= [(02084 02056)

(0743 02727 02348 02347 02023 01995)]

NVS (119887) = [(V119887 V119888) (V6 V4 V5 V1 V3 V2)]

= [(02084 02056)

(0743 02727 02348 02347 02023 01995)]

(18)

Compare NVS(119886) and NVS(119887) and equivalence can beseen So the node voltages in two vertex code groups are alldistinct go to Step 4

Step 4 Take the NVS(119886) NVS(119887) and their vertex codes todetermine the corresponding vertices of two WDCCGs asfollows (119886-119887 119887-119886 1-3 2-5 3-2 4-4 5-6 6-1)

Step 5 According to the corresponding vertices obtained andthe vertex codes to form new WAM 119860

1015840

1 rewrite WAM 119860

1015840Then we have 119860 = 1198601015840

1 so they are isomorphic and program

stops

Example 3 Two 6-gear train kinematic chains are depicted inFigures 11(a) and 11(b) and their correspondingWDCCGs aredescribed in Figures 11(c) and 11(d) respectively The directprocess to determine isomorphism of these two kinematicchains is as follows

Step 1 Input their corresponding vertex codes andWAMs1198601198601015840 as follows

119860 =

1 2 3 4 5 6 7 119886 119888

1

2

3

4

5

6

7

119886

119888

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

(1)

(3)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(3)

(1)

(2)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(2)

(1)

(3)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(3)

(1)

(0)

(0)

(0)

(1)

(1)

(0)

(0)

(0)

(0)

(1)

(3)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(3)

(1)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(1)

(1)

(0)

(1)

(0)

(1)

(0)

(1)

(1)

(0)

(0)

(1)

(0)

(0)

(1)

(1)

(0)

(1)

(0)

(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

1198601015840=

1 2 3 4 5 6 7 119886 119888

1

2

3

4

5

6

7

119886

119888

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

(1)

(3)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(3)

(1)

(2)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(2)

(1)

(3)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(3)

(1)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(1)

(3)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(3)

(1)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(1)

(1)

(0)

(1)

(0)

(1)

(1)

(0)

(1)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(1)

(1)

(0)

(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(19)

Mathematical Problems in Engineering 11

①

② ③ ④ ⑤

⑥⑦

A

B

b

a

C

c

(a)

①

②③

④

⑤

⑥ ⑦

A

Bb

a

C

c

(b)

①

②

③

④ ⑤

⑥

⑦

3

3

3

c

2

11

1

1

1

1

1

1

a

(c)

①

②

③

④

⑤

⑥

⑦

c

a

1

1

1

1

1

1

1

2

3

3

3

(d)

Figure 11 Two 6-gear train kinematic chains and their WDCCG

Step 2 Evidently 119860 = 1198601015840 Assign vertices codes of 119860 and 1198601015840

in two groups (a) [(119886 119888) (1 2 3 4 5 6 7)] and (b) [(119886 119888) (12 3 4 5 6 7)]

Step 3 Calculate NVS(119886) NVS(119887) and their vertex codes oftwo ECMs with the same CE respectively as follows

NVS (119886) = [(V119887 V119886) (V7 V5 V6 V1 V4 V3 V2)]= [(02591 02547)

(03784 02515 02511 02309 02122 02029 01961)]

NVS (119887) = [(V119887 V119886) (V7 V6 V5 V1 V4 V3 V2)]= [(03156 02607)

(03940 02601 02551 02394 02367 02110 02001)]

(20)

Compare NVS(119886) and NVS(119887) and equivalence cannot beseen So it is determined that they are not isomorphic andprogram stops

Example 4 Two 9-gear train kinematic chains are shown inFigures 12(a) and 12(b) Their corresponding WDCCGs aredepicted in Figures 12(c) and 12(d) respectively The directprocess to determine isomorphism of these two kinematicchains is as follows

Step 1 Input their correspondingWAMs119860 and1198601015840 as follows

119860 =

1 2 3 4 5 6 7 8 9 119886 119887 119888

1

2

3

4

5

6

7

8

9

119886

119887

119888

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

(1)

(0)

(3)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(3)

(1)

(3)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(0)

(3)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(3)

(1)

(1)

(1)

(0)

(0)

(0)

(0)

(1)

(3)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(3)

(1)

(3)

(2)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(3)

(1)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(2)

(0)

(1)

(0)

(1)

(1)

(0)

(0)

(0)

(0)

(3)

(0)

(0)

(0)

(0)

(1)

(0)

(1)

(0)

(0)

(0)

(1)

(0)

(1)

(0)

(1)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(1)

(1)

(0)

(0)

(0)

(1)

(1)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

12 Mathematical Problems in Engineering

① ②

③

a

A

c

B

d

C

D

E

b

④

⑤

⑥

⑦

⑧

⑨

(a)

①

②③a

A

b E

c

DC

d

B④

⑤

⑥ ⑦⑧⑨

(b)

①

1 1

c

1 1

1

1

1

b3

3

3

2

1

1

1

1a

33

②

③

④

⑤⑥

⑦

⑧

⑨

(c)

1

1

1

1

3

3

1

1

1

1

1

1

1

2

3

3

3

b

c

a

①②

③

④

⑤

⑥

⑦

⑧

⑨

(d)

Figure 12 Two 9-gear train kinematic chains and their WDCCGs

1198601015840=

1 2 3 4 5 6 7 8 9 119886 119887 119888

1

2

3

4

5

6

7

8

9

119886

119887

119888

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

(1)

(0)

(0)

(3)

(0)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(1)

(0)

(3)

(0)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(1)

(2)

(0)

(0)

(0)

(0)

(0)

(1)

(0)

(1)

(3)

(3)

(2)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(3)

(3)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(3)

(1)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(3)

(0)

(1)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(3)

(1)

(1)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(3)

(1)

(0)

(0)

(1)

(1)

(1)

(1)

(0)

(1)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(1)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(1)

(0)

(0)

(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(21)

Step 2 Evidently 119860 = 1198601015840 Assign vertices codes of 119860 and 1198601015840

in two groups (a) [(119886 119887 119888) (1 2 3 4 5 6 7 8 9)] and (b) [(119886119887 119888) (1 2 3 4 5 6 7 8 9)]

Step 3 Calculate NVS(119886) NVS(119887) and their vertex codes oftwo ECMs with the same CE respectively as follows

NVS (119886) = [(V119888 V119887 V119886) (V9 V4 V8 V1 V2 V5 V7 V3 V6)]

= [(02845 02815 02075) (02470 02307 02295 02294 02294 02154 02154 01846 01719)]

NVS (119887) = [(V119887 V119888 V119886) (V9 V8 V3 V6 V7 V1 V2 V5 V4)]

= [(02845 02815 02075) (02470 02307 02295 02294 02294 02154 02154 01846 01719)]

(22)

Mathematical Problems in Engineering 13

Compare NVS(119886) and NVS(119887) and equivalence can be seenAnd the node voltages in two vertex code groups are not alldistinct so go to Step 6

Step 4 The vertices (1 2) in Figure 12(a) vertices (6 7) inFigure 12(b) the vertices (5 7) in Figure 12(a) and vertices (12) in Figure 12(b) have the same values in the correspondinggroups so just adjust the position order of the vertices in 1198601015840to obtain a new adjacency matrix 1198601015840 And the correspondingvertices of two WDCCGs are (1-6 2-7 3-5 4-8 5-1 6-4 7-28-3 9-9 119886-119886 119887-119888 119888-119887) so we have 119860 = 119860

1015840

1 It is determined

that they are isomorphic and program stops

5 Algorithm Complexity Analysis

This paper presents a relatively good algorithm for determi-nation of kinematic chains isomorphism with multiple jointand gear train It is feasible for introducing the WDCCGand WAM to describe these two types of kinematic chainsthey dramatically reduce the number of vertices and order ofcorresponding adjacency matrix 119860

The analysis of computational complexity of this isomor-phic identification algorithm is as follows when there areno equivalent node voltages in NVS time consumption isbasically dedicated to solving the linear equation sets If akinematic chain possesses 119899 vertices the order of the nodevoltage equations is 119899 minus 1 Since the adjoint circuit is alinear resistance circuit the node voltage equations forma linear algebra equation set of orders 119899 minus 1 which has acomputational complexity of 119874((119899 minus 1)3) for its solutionSince it needs to solve at most 2119899 adjoint circuits in ourproposed method the computational complexity is of order119874(2119899(119899 minus 1)

3) lt 119874(119899

4) Thus the proposed method has a

computational complexity of a polynomial orderWhen thereare equivalent node voltages in NVS the correspondence ofonly a part of the vertices cannot be determinedThus in theworst situation the proposed method is more efficient thanthose where rowcolumn exchanges are performed blindly

In this paper this algorithm is tested and comparedwith the internationally recognized faster method for planarsimple joint kinematic chains isomorphic identification [20]and they are tested in the same software and hardwareenvironment The test results are shown in Figure 13 (theaverage time of each vertex is 20 times of the kinematic chainwithmultiple joints) test results show that when the vertex ofthe kinematic chain is increased the algorithm proposed inthis paper is more efficient than the algorithm proposed byHe et alThe proposed algorithm by us is very suitable for theisomorphism of the topological graph of kinematic chainsand the algorithm is very efficient at the same time

6 Conclusions

In this paper an algorithm for topological isomorphism iden-tification of planar multiple joint and gear train kinematicchains is introduced

4 6 8 10 12 14 16 18 20 22

The number of vertices

Algorithm proposed by He et al

Algorithm proposed in this paper

00001

0001

001

01

1

The d

ecisi

on ti

me (

s)

Figure 13 The test results of the algorithm proposed in this paperand the algorithm proposed by He et al

Firstly a weighted-double-color-contracted-graph(WDCCG) and a weighted adjacency matrix (WAM) areintroduced to describe the planarmultiple joint and gear trainkinematic chains respectively They dramatically reduce thenumber of vertices and order of the represented adjacencymatrix

Secondly isomorphism identification method of thesetwo types of kinematic chains is carried out by the equivalentcircuit models (ECMs) of WDCCGs with the same completeexcitation (CE) and then uses the solved node voltagesequence (NVS) to determine the corresponding vertices oftwo WDCCGs

Finally an algorithm to identify isomorphic kinematicchains is obtained It is an efficient and easy method tobe realized by computer And this algorithm also can beused in the isomorphism identification of planar simple jointkinematic chains with minor changes

Notations

119860 1198601015840 The incidence matrix of kinematic chain119889119894119895 The elements of incidence matrix

119866119894 Conductance value of edge in the ECM

119898119894119895119896 The weighted value of 119896th edge

1198731198731015840 Variable of equivalent circuit model119899 The number of vertices in WDCCG119899119894 Node 119894 in the ECM

V119894 Node voltages of the node 119894

CE Complete excitationDCG Double color graphECM Equivalent circuit modelNVS Node voltage sequenceTM Topological modelWAM Weighted adjacency matrixWDCCG Weighted double color contracted graph

14 Mathematical Problems in Engineering

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

Thiswork is supported by the Science andTechnology Projectof Quanzhou City 2014G48 and G20140047

References

[1] J J Uicker Jr and A Raicu ldquoAmethod for the identification andrecognition of equivalence of kinematic chainsrdquoMechanismandMachine Theory vol 10 no 5 pp 375ndash383 1975

[2] H S Yan and A S Hall ldquoLinkage characteristic polynomialsdefinitions coefficients by inspectionrdquo Journal of MechanicalDesign vol 103 no 3 pp 578ndash584 1981

[3] H S Yan and A S Hall ldquoLinkage characteristic polynomialsassembly theorems uniquenessrdquo Journal of Mechanical Designvol 104 no 1 pp 11ndash20 1982

[4] T S Mruthyunjaya ldquoA computerized methodology for struc-tural synthesis of kinematic chains Part 1 FormulationrdquoMech-anism and Machine Theory vol 19 no 6 pp 487ndash495 1984

[5] T S Mruthyunjaya ldquoA computerized methodology for struc-tural synthesis of kinematic chains part 2mdashapplication toseveral fully or partially known casesrdquoMechanism andMachineTheory vol 19 no 6 pp 497ndash505 1984

[6] T S Mruthyunjaya ldquoA computerized methodology for struc-tural synthesis of kinematic chains part 3mdashapplication to thenew case of 10-link three- freedom chainsrdquo Mechanism andMachine Theory vol 19 no 6 pp 507ndash530 1984

[7] T S Mruthyunjaya and H R Balasubramanian ldquoIn questof a reliable and efficient computational test for detection ofisomorphism in kinematic chainsrdquo Mechanism and MachineTheory vol 22 no 2 pp 131ndash139 1987

[8] W J Sohn and F Freudenstein ldquoAn application of dual graphs tothe automatic generation of the kinematic structures of mecha-nismsrdquo Journal of Mechanisms Transmissions and Automationvol 108 no 3 pp 392ndash398 1986

[9] A Ambekar and V Agrawal ldquoOn canonical numbering ofkinematic chains and isomorphism problem MAX coderdquo inProceedings of the ASMEMechanisms Conference pp 1615ndash16221986

[10] A G Ambekar and V P Agrawal ldquoCanonical numberingof kinematic chains and isomorphism problem min coderdquoMechanism andMachineTheory vol 22 no 5 pp 453ndash461 1987

[11] C S Tang and T Liu ldquoDegree code a new mechanismidentifierrdquo Journal of Mechanical Design vol 115 no 3 pp 627ndash630 1993

[12] J K Shin and S Krishnamurty ldquoOn identification and canonicalnumbering of pin jointed kinematic chainsrdquo Journal of Mechan-ical Design vol 116 pp 182ndash188 1994

[13] J K Shin and S Krishnamurty ldquoDevelopment of a standardcode for colored graphs and its application to kinematic chainsrdquoJournal of Mechanical Design vol 116 no 1 pp 189ndash196 1994

[14] A C Rao and D Varada Raju ldquoApplication of the hammingnumber technique to detect isomorphism among kinematicchains and inversionsrdquoMechanism andMachineTheory vol 26no 1 pp 55ndash75 1991

[15] A C Rao and C N Rao ldquoLoop based pseudo hammingvalues-1 Testing isomorphism and rating kinematic chainsrdquoMechanism andMachineTheory vol 28 no 1 pp 113ndash127 1992

[16] A C Rao and C N Rao ldquoLoop based pseudo hamming values-II inversions preferred frames and actuatorsrdquo Mechanism andMachine Theory vol 28 no 1 pp 129ndash143 1993

[17] J-K Chu and W-Q Cao ldquoIdentification of isomorphismamong kinematic chains and inversions using linkrsquos adjacent-chain-tablerdquoMechanism and Machine Theory vol 29 no 1 pp53ndash58 1994

[18] A C Rao ldquoApplication of fuzzy logic for the study of iso-morphism inversions symmetry parallelism and mobility inkinematic chainsrdquoMechanism and Machine Theory vol 35 no8 pp 1103ndash1116 2000

[19] P R He W J Zhang Q Li and F X Wu ldquoA new method fordetection of graph isomorphism based on the quadratic formrdquoJournal of Mechanical Design vol 125 no 3 pp 640ndash642 2003

[20] P R He W J Zhang and Q Li ldquoSome further developmenton the eigensystem approach for graph isomorphismdetectionrdquoJournal of the Franklin Institute vol 342 no 6 pp 657ndash6732005

[21] Z Y Chang C Zhang Y H Yang and YWang ldquoA newmethodto mechanism kinematic chain isomorphism identificationrdquoMechanism andMachineTheory vol 37 no 4 pp 411ndash417 2002

[22] J P Cubillo and J BWan ldquoComments onmechanismkinematicchain isomorphism identification using adjacent matricesrdquoMechanism andMachineTheory vol 40 no 2 pp 131ndash139 2005

[23] H F Ding and Z Huang ldquoA unique representation of the kine-matic chain and the atlas databaserdquo Mechanism and MachineTheory vol 42 no 6 pp 637ndash651 2007

[24] H F Ding and Z Huang ldquoA new theory for the topologicalstructure analysis of kinematic chains and its applicationsrdquoMechanism and Machine Theory vol 42 no 10 pp 1264ndash12792007

[25] A C Rao ldquoA genetic algorithm for topological characteristicsof kinematic chainsrdquo Journal of Mechanical Design vol 122 no2 pp 228ndash231 2000

[26] F G Kong Q Li and W J Zhang ldquoAn artificial neuralnetwork approach tomechanism kinematic chain isomorphismidentificationrdquo Mechanism and Machine Theory vol 34 no 2pp 271ndash283 1999

[27] J K Chu The Structural and Dimensional Characteristics ofPlanar Linkages Press of Beihang University of China BeijingChina 1992

[28] L Song J Yang X Zhang and W Q Cao ldquoSpanning treemethod of identifying isomorphism and topological symmetryto planar kinematic chain with multiple jointrdquo Chinese Journalof Mechanical Engineering vol 14 no 1 pp 27ndash31 2001

[29] J G Liu and D J Yu ldquoRepresentations amp isomorphismidentification of planar kinematic chains with multiple jointsbased on the converted adjacent matrixrdquo Chinese Journal ofMechanical Engineering vol 48 pp 15ndash21 2012

[30] R Ravisankar andT SMruthyunjaya ldquoComputerized synthesisof the structure of geared kinematic chainsrdquo Mechanism andMachine Theory vol 20 no 5 pp 367ndash387 1985

[31] J U Kim and B M Kwak ldquoApplication of edge permutationgroup to structural synthesis of epicyclic gear trainsrdquo Mecha-nism and Machine Theory vol 25 no 5 pp 563ndash573 1990

Mathematical Problems in Engineering 15

[32] G Chatterjee and L-W Tsai ldquoComputer-aided sketching ofepicyclic-type automatic transmission gear trainsrdquo Journal ofMechanical Design vol 118 no 3 pp 405ndash411 1996

[33] V V N R Prasadraju and A C Rao ldquoA new technique basedon loops to investigate displacement isomorphism in planetarygear trainsrdquo Journal of Mechanical Design vol 12 pp 666ndash6752002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

①

②

③

④

⑤

⑥

A

a

b

C

B

c

(a)

① ②③ ④

⑤

⑥

A

a

b

C

B

c

(b)

①

②

③

④

⑤

⑥

1

b

1

1

1

2

3

3

3 2

(c)

①

②

③

④

⑤

⑥

3

3

3

2

2

b

1

1

1

1

(d)

Figure 8 Two gear train kinematic chains and their WDCCGs

Step 4 Take the NVS(119886) NVS(119887) and their vertex codes todetermine the correspondence of vertices according to thevalues that are in correspondence

Step 5 According to the corresponding vertices obtained andthe vertex codes to form new WAM 119860

1015840

1 rewrite WAM 119860

1015840 If119860 = 119860

1015840

1 it is determined that they are isomorphic otherwise

they are not isomorphic and program stops

Step 6 According to the corresponding vertices obtained theNVS(119886) and NVS(119887) have the same values in the correspond-ing two vertex code groups Just adjust the position order ofvertices of the same group in1198601015840 to obtain vertex codes to fromnew adjacency matrix 1198601015840

1 If 119860 = 119860

1015840

1 it is determined that

they are isomorphic and program stops if 119860 = 1198601015840

1 keep on

adjusting the position order of the aforementioned verticesin 1198601015840 of the vertex code groups for all possibilities And if119860 = 119860

1015840

1always it is determined that they are not isomorphic

and program stops

4 Examples

Example 1 Two 10-link multiple joint kinematic chainsare shown in Figures 9(a) and 9(b) Their correspondingWDCCGs are depicted in Figures 9(c) and 9(d) respectively

The direct process to determine isomorphism of these twokinematic chains can be expressed as follows

Step 1 Input their corresponding vertex codes andWAMs1198601198601015840 as follows

119860 =

1 119886 119887 119888 119889

1

119886

119887

119888

119889

[

[

[

[

[

[

[

[

[

(1)

(2)

(0)

(1)

(2)

(2)

(0)

(2)

(0)

(3)

(0)

(2)

(0)

(2)

(2)

(1)

(0)

(2)

(0)

(3)

(2)

(3)

(2)

(3)

(0)

]

]

]

]

]

]

]

]

]

1198601015840=

1 119886 119887 119888 119889

1

119886

119887

119888

119889

[

[

[

[

[

[

[

[

[

(1)

(0)

(2)

(2)

(1)

(0)

(0)

(2)

(2)

(2)

(2)

(2)

(0)

(0)

(3)

(2)

(2)

(0)

(0)

(3)

(1)

(2)

(3)

(3)

(0)

]

]

]

]

]

]

]

]

]

(15)

Step 2 Evidently 119860 = 1198601015840 Assign vertices codes of 119860 and 1198601015840

in two groups (a) [(119886 119887 119888 119889) (1)] and (b) [(119886 119887 119888 119889) (1)]

Mathematical Problems in Engineering 9

①

②

③

④

⑤

⑥

⑦

⑧

⑨

⑩

c

ba

d

(a)

①

②

③ ④

⑤

⑥⑦

⑧

⑨⑩

c b

a

d

(b)

①

cb

a

2

2

2

2

2

3

3

1d

(c)

①

2

2

2

2

2

3

3

d

b

ac

1

(d)

Figure 9 Two 10-link multiple joint kinematic chains and their WDCCGs

①

⑪

②

③

④

⑤

⑥

⑦⑧

⑨

⑩

ab

(a)

①

②

③④

⑤⑥

⑦

⑧

⑨

⑩

a

b

⑪

(b)

①

② ③

④

⑤

⑥ 1

1

11

12

2

2

2

2a

b

1

1 1

(c)

① ②

③

④

⑤

⑥

1

1

1

1

1

11

12

2

2

2

2b a

(d)

Figure 10 Two 11-link multiple joint kinematic chains and their WDCCGs

Step 3 Calculate NVS(119886) NVS(119887) and their vertex codes oftwo ECMs with the same CE respectively as follows

NVS (119886) = [(V119888 V119887 V119886 V119889) (V1)]

= [(01551 01546 01451 01275) (01700)]

NVS (119887) = [(V119886 V119887 V119888 V119889) (V1)]

= [(01538 01459 01459 01308) (01714)]

(16)

Compare NVS(119886) and NVS(119887) and equivalence cannot beseen It is determined that they are not isomorphic andprogram stops

Example 2 Two 11-link multiple joint kinematic chains areshown in Figures 10(a) and 10(b)Their correspondingWDC-CGs are described in Figures 10(c) and 10(d) respectivelyThe direct process to determine isomorphism of these twokinematic chains is as follows

10 Mathematical Problems in Engineering

Step 1 Input their corresponding vertex codes andWAMs1198601198601015840 as follows

119860

=

1 2 3 4 5 6 119886 119887

1

2

3

4

5

6

119886

119887

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

(1)

(2)

(0)

(1)

(1)

(0)

(0)

(2)

(2)

(1)

(1)

(0)

(1)

(0)

(0)

(0)

(0)

(1)

(1)

(1)

(0)

(2)

(2)

(0)

(1)

(0)

(1)

(1)

(0)

(1)

(0)

(0)

(1)

(1)

(0)

(0)

(1)

(0)

(1)

(0)

(0)

(2)

(0)

(1)

(0)

(1)

(0)

(1)

(0)

(0)

(1)

(0)

(1)

(0)

(0)

(2)

(2)

(0)

(0)

(0)

(0)

(1)

(2)

(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

1198601015840

=

1 2 3 4 5 6 119886 119887

1

2

3

4

5

6

119886

119887

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

(1)

(2)

(0)

(1)

(0)

(0)

(0)

(1)

(2)

(1)

(0)

(1)

(1)

(0)

(2)

(0)

(0)

(0)

(1)

(1)

(2)

(1)

(0)

(2)

(1)

(1)

(1)

(1)

(0)

(0)

(0)

(0)

(0)

(1)

(2)

(0)

(1)

(1)

(0)

(0)

(0)

(0)

(1)

(0)

(1)

(1)

(1)

(0)

(0)

(2)

(0)

(0)

(0)

(1)

(0)

(2)

(1)

(0)

(2)

(0)

(0)

(0)

(2)

(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(17)

Step 2 Evidently 119860 = 1198601015840 Assign vertices codes of 119860 and 1198601015840

in two groups (a) [(119886 119887) (1 2 3 4 5 6)] and (b) [(119886 119887) (1 23 4 5 6)]

Step 3 Calculate NVS(119886) NVS(119887) and their vertex codes oftwo ECMs with the same CE respectively as follows

NVS (119886) = [(V119886 V119887) (V5 V4 V2 V6 V1 V3)]

= [(02084 02056)

(0743 02727 02348 02347 02023 01995)]

NVS (119887) = [(V119887 V119888) (V6 V4 V5 V1 V3 V2)]

= [(02084 02056)

(0743 02727 02348 02347 02023 01995)]

(18)

Compare NVS(119886) and NVS(119887) and equivalence can beseen So the node voltages in two vertex code groups are alldistinct go to Step 4

Step 4 Take the NVS(119886) NVS(119887) and their vertex codes todetermine the corresponding vertices of two WDCCGs asfollows (119886-119887 119887-119886 1-3 2-5 3-2 4-4 5-6 6-1)

Step 5 According to the corresponding vertices obtained andthe vertex codes to form new WAM 119860

1015840

1 rewrite WAM 119860

1015840Then we have 119860 = 1198601015840

1 so they are isomorphic and program

stops

Example 3 Two 6-gear train kinematic chains are depicted inFigures 11(a) and 11(b) and their correspondingWDCCGs aredescribed in Figures 11(c) and 11(d) respectively The directprocess to determine isomorphism of these two kinematicchains is as follows

Step 1 Input their corresponding vertex codes andWAMs1198601198601015840 as follows

119860 =

1 2 3 4 5 6 7 119886 119888

1

2

3

4

5

6

7

119886

119888

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

(1)

(3)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(3)

(1)

(2)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(2)

(1)

(3)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(3)

(1)

(0)

(0)

(0)

(1)

(1)

(0)

(0)

(0)

(0)

(1)

(3)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(3)

(1)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(1)

(1)

(0)

(1)

(0)

(1)

(0)

(1)

(1)

(0)

(0)

(1)

(0)

(0)

(1)

(1)

(0)

(1)

(0)

(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

1198601015840=

1 2 3 4 5 6 7 119886 119888

1

2

3

4

5

6

7

119886

119888

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

(1)

(3)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(3)

(1)

(2)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(2)

(1)

(3)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(3)

(1)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(1)

(3)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(3)

(1)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(1)

(1)

(0)

(1)

(0)

(1)

(1)

(0)

(1)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(1)

(1)

(0)

(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(19)

Mathematical Problems in Engineering 11

①

② ③ ④ ⑤

⑥⑦

A

B

b

a

C

c

(a)

①

②③

④

⑤

⑥ ⑦

A

Bb

a

C

c

(b)

①

②

③

④ ⑤

⑥

⑦

3

3

3

c

2

11

1

1

1

1

1

1

a

(c)

①

②

③

④

⑤

⑥

⑦

c

a

1

1

1

1

1

1

1

2

3

3

3

(d)

Figure 11 Two 6-gear train kinematic chains and their WDCCG

Step 2 Evidently 119860 = 1198601015840 Assign vertices codes of 119860 and 1198601015840

in two groups (a) [(119886 119888) (1 2 3 4 5 6 7)] and (b) [(119886 119888) (12 3 4 5 6 7)]

Step 3 Calculate NVS(119886) NVS(119887) and their vertex codes oftwo ECMs with the same CE respectively as follows

NVS (119886) = [(V119887 V119886) (V7 V5 V6 V1 V4 V3 V2)]= [(02591 02547)

(03784 02515 02511 02309 02122 02029 01961)]

NVS (119887) = [(V119887 V119886) (V7 V6 V5 V1 V4 V3 V2)]= [(03156 02607)

(03940 02601 02551 02394 02367 02110 02001)]

(20)

Compare NVS(119886) and NVS(119887) and equivalence cannot beseen So it is determined that they are not isomorphic andprogram stops

Example 4 Two 9-gear train kinematic chains are shown inFigures 12(a) and 12(b) Their corresponding WDCCGs aredepicted in Figures 12(c) and 12(d) respectively The directprocess to determine isomorphism of these two kinematicchains is as follows

Step 1 Input their correspondingWAMs119860 and1198601015840 as follows

119860 =

1 2 3 4 5 6 7 8 9 119886 119887 119888

1

2

3

4

5

6

7

8

9

119886

119887

119888

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

(1)

(0)

(3)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(3)

(1)

(3)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(0)

(3)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(3)

(1)

(1)

(1)

(0)

(0)

(0)

(0)

(1)

(3)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(3)

(1)

(3)

(2)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(3)

(1)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(2)

(0)

(1)

(0)

(1)

(1)

(0)

(0)

(0)

(0)

(3)

(0)

(0)

(0)

(0)

(1)

(0)

(1)

(0)

(0)

(0)

(1)

(0)

(1)

(0)

(1)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(1)

(1)

(0)

(0)

(0)

(1)

(1)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

12 Mathematical Problems in Engineering

① ②

③

a

A

c

B

d

C

D

E

b

④

⑤

⑥

⑦

⑧

⑨

(a)

①

②③a

A

b E

c

DC

d

B④

⑤

⑥ ⑦⑧⑨

(b)

①

1 1

c

1 1

1

1

1

b3

3

3

2

1

1

1

1a

33

②

③

④

⑤⑥

⑦

⑧

⑨

(c)

1

1

1

1

3

3

1

1

1

1

1

1

1

2

3

3

3

b

c

a

①②

③

④

⑤

⑥

⑦

⑧

⑨

(d)

Figure 12 Two 9-gear train kinematic chains and their WDCCGs

1198601015840=

1 2 3 4 5 6 7 8 9 119886 119887 119888

1

2

3

4

5

6

7

8

9

119886

119887

119888

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

(1)

(0)

(0)

(3)

(0)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(1)

(0)

(3)

(0)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(1)

(2)

(0)

(0)

(0)

(0)

(0)

(1)

(0)

(1)

(3)

(3)

(2)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(3)

(3)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(3)

(1)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(3)

(0)

(1)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(3)

(1)

(1)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(3)

(1)

(0)

(0)

(1)

(1)

(1)

(1)

(0)

(1)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(1)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(1)

(0)

(0)

(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(21)

Step 2 Evidently 119860 = 1198601015840 Assign vertices codes of 119860 and 1198601015840

in two groups (a) [(119886 119887 119888) (1 2 3 4 5 6 7 8 9)] and (b) [(119886119887 119888) (1 2 3 4 5 6 7 8 9)]

Step 3 Calculate NVS(119886) NVS(119887) and their vertex codes oftwo ECMs with the same CE respectively as follows

NVS (119886) = [(V119888 V119887 V119886) (V9 V4 V8 V1 V2 V5 V7 V3 V6)]

= [(02845 02815 02075) (02470 02307 02295 02294 02294 02154 02154 01846 01719)]

NVS (119887) = [(V119887 V119888 V119886) (V9 V8 V3 V6 V7 V1 V2 V5 V4)]

= [(02845 02815 02075) (02470 02307 02295 02294 02294 02154 02154 01846 01719)]

(22)

Mathematical Problems in Engineering 13

Compare NVS(119886) and NVS(119887) and equivalence can be seenAnd the node voltages in two vertex code groups are not alldistinct so go to Step 6

Step 4 The vertices (1 2) in Figure 12(a) vertices (6 7) inFigure 12(b) the vertices (5 7) in Figure 12(a) and vertices (12) in Figure 12(b) have the same values in the correspondinggroups so just adjust the position order of the vertices in 1198601015840to obtain a new adjacency matrix 1198601015840 And the correspondingvertices of two WDCCGs are (1-6 2-7 3-5 4-8 5-1 6-4 7-28-3 9-9 119886-119886 119887-119888 119888-119887) so we have 119860 = 119860

1015840

1 It is determined

that they are isomorphic and program stops

5 Algorithm Complexity Analysis

This paper presents a relatively good algorithm for determi-nation of kinematic chains isomorphism with multiple jointand gear train It is feasible for introducing the WDCCGand WAM to describe these two types of kinematic chainsthey dramatically reduce the number of vertices and order ofcorresponding adjacency matrix 119860

The analysis of computational complexity of this isomor-phic identification algorithm is as follows when there areno equivalent node voltages in NVS time consumption isbasically dedicated to solving the linear equation sets If akinematic chain possesses 119899 vertices the order of the nodevoltage equations is 119899 minus 1 Since the adjoint circuit is alinear resistance circuit the node voltage equations forma linear algebra equation set of orders 119899 minus 1 which has acomputational complexity of 119874((119899 minus 1)3) for its solutionSince it needs to solve at most 2119899 adjoint circuits in ourproposed method the computational complexity is of order119874(2119899(119899 minus 1)

3) lt 119874(119899

4) Thus the proposed method has a

computational complexity of a polynomial orderWhen thereare equivalent node voltages in NVS the correspondence ofonly a part of the vertices cannot be determinedThus in theworst situation the proposed method is more efficient thanthose where rowcolumn exchanges are performed blindly

In this paper this algorithm is tested and comparedwith the internationally recognized faster method for planarsimple joint kinematic chains isomorphic identification [20]and they are tested in the same software and hardwareenvironment The test results are shown in Figure 13 (theaverage time of each vertex is 20 times of the kinematic chainwithmultiple joints) test results show that when the vertex ofthe kinematic chain is increased the algorithm proposed inthis paper is more efficient than the algorithm proposed byHe et alThe proposed algorithm by us is very suitable for theisomorphism of the topological graph of kinematic chainsand the algorithm is very efficient at the same time

6 Conclusions

In this paper an algorithm for topological isomorphism iden-tification of planar multiple joint and gear train kinematicchains is introduced

4 6 8 10 12 14 16 18 20 22

The number of vertices

Algorithm proposed by He et al

Algorithm proposed in this paper

00001

0001

001

01

1

The d

ecisi

on ti

me (

s)

Figure 13 The test results of the algorithm proposed in this paperand the algorithm proposed by He et al

Firstly a weighted-double-color-contracted-graph(WDCCG) and a weighted adjacency matrix (WAM) areintroduced to describe the planarmultiple joint and gear trainkinematic chains respectively They dramatically reduce thenumber of vertices and order of the represented adjacencymatrix

Secondly isomorphism identification method of thesetwo types of kinematic chains is carried out by the equivalentcircuit models (ECMs) of WDCCGs with the same completeexcitation (CE) and then uses the solved node voltagesequence (NVS) to determine the corresponding vertices oftwo WDCCGs

Finally an algorithm to identify isomorphic kinematicchains is obtained It is an efficient and easy method tobe realized by computer And this algorithm also can beused in the isomorphism identification of planar simple jointkinematic chains with minor changes

Notations

119860 1198601015840 The incidence matrix of kinematic chain119889119894119895 The elements of incidence matrix

119866119894 Conductance value of edge in the ECM

119898119894119895119896 The weighted value of 119896th edge

1198731198731015840 Variable of equivalent circuit model119899 The number of vertices in WDCCG119899119894 Node 119894 in the ECM

V119894 Node voltages of the node 119894

CE Complete excitationDCG Double color graphECM Equivalent circuit modelNVS Node voltage sequenceTM Topological modelWAM Weighted adjacency matrixWDCCG Weighted double color contracted graph

14 Mathematical Problems in Engineering

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

Thiswork is supported by the Science andTechnology Projectof Quanzhou City 2014G48 and G20140047

References

[1] J J Uicker Jr and A Raicu ldquoAmethod for the identification andrecognition of equivalence of kinematic chainsrdquoMechanismandMachine Theory vol 10 no 5 pp 375ndash383 1975

[2] H S Yan and A S Hall ldquoLinkage characteristic polynomialsdefinitions coefficients by inspectionrdquo Journal of MechanicalDesign vol 103 no 3 pp 578ndash584 1981

[3] H S Yan and A S Hall ldquoLinkage characteristic polynomialsassembly theorems uniquenessrdquo Journal of Mechanical Designvol 104 no 1 pp 11ndash20 1982

[4] T S Mruthyunjaya ldquoA computerized methodology for struc-tural synthesis of kinematic chains Part 1 FormulationrdquoMech-anism and Machine Theory vol 19 no 6 pp 487ndash495 1984

[5] T S Mruthyunjaya ldquoA computerized methodology for struc-tural synthesis of kinematic chains part 2mdashapplication toseveral fully or partially known casesrdquoMechanism andMachineTheory vol 19 no 6 pp 497ndash505 1984

[6] T S Mruthyunjaya ldquoA computerized methodology for struc-tural synthesis of kinematic chains part 3mdashapplication to thenew case of 10-link three- freedom chainsrdquo Mechanism andMachine Theory vol 19 no 6 pp 507ndash530 1984

[7] T S Mruthyunjaya and H R Balasubramanian ldquoIn questof a reliable and efficient computational test for detection ofisomorphism in kinematic chainsrdquo Mechanism and MachineTheory vol 22 no 2 pp 131ndash139 1987

[8] W J Sohn and F Freudenstein ldquoAn application of dual graphs tothe automatic generation of the kinematic structures of mecha-nismsrdquo Journal of Mechanisms Transmissions and Automationvol 108 no 3 pp 392ndash398 1986

[9] A Ambekar and V Agrawal ldquoOn canonical numbering ofkinematic chains and isomorphism problem MAX coderdquo inProceedings of the ASMEMechanisms Conference pp 1615ndash16221986

[10] A G Ambekar and V P Agrawal ldquoCanonical numberingof kinematic chains and isomorphism problem min coderdquoMechanism andMachineTheory vol 22 no 5 pp 453ndash461 1987

[11] C S Tang and T Liu ldquoDegree code a new mechanismidentifierrdquo Journal of Mechanical Design vol 115 no 3 pp 627ndash630 1993

[12] J K Shin and S Krishnamurty ldquoOn identification and canonicalnumbering of pin jointed kinematic chainsrdquo Journal of Mechan-ical Design vol 116 pp 182ndash188 1994

[13] J K Shin and S Krishnamurty ldquoDevelopment of a standardcode for colored graphs and its application to kinematic chainsrdquoJournal of Mechanical Design vol 116 no 1 pp 189ndash196 1994

[14] A C Rao and D Varada Raju ldquoApplication of the hammingnumber technique to detect isomorphism among kinematicchains and inversionsrdquoMechanism andMachineTheory vol 26no 1 pp 55ndash75 1991

[15] A C Rao and C N Rao ldquoLoop based pseudo hammingvalues-1 Testing isomorphism and rating kinematic chainsrdquoMechanism andMachineTheory vol 28 no 1 pp 113ndash127 1992

[16] A C Rao and C N Rao ldquoLoop based pseudo hamming values-II inversions preferred frames and actuatorsrdquo Mechanism andMachine Theory vol 28 no 1 pp 129ndash143 1993

[17] J-K Chu and W-Q Cao ldquoIdentification of isomorphismamong kinematic chains and inversions using linkrsquos adjacent-chain-tablerdquoMechanism and Machine Theory vol 29 no 1 pp53ndash58 1994

[18] A C Rao ldquoApplication of fuzzy logic for the study of iso-morphism inversions symmetry parallelism and mobility inkinematic chainsrdquoMechanism and Machine Theory vol 35 no8 pp 1103ndash1116 2000

[19] P R He W J Zhang Q Li and F X Wu ldquoA new method fordetection of graph isomorphism based on the quadratic formrdquoJournal of Mechanical Design vol 125 no 3 pp 640ndash642 2003

[20] P R He W J Zhang and Q Li ldquoSome further developmenton the eigensystem approach for graph isomorphismdetectionrdquoJournal of the Franklin Institute vol 342 no 6 pp 657ndash6732005

[21] Z Y Chang C Zhang Y H Yang and YWang ldquoA newmethodto mechanism kinematic chain isomorphism identificationrdquoMechanism andMachineTheory vol 37 no 4 pp 411ndash417 2002

[22] J P Cubillo and J BWan ldquoComments onmechanismkinematicchain isomorphism identification using adjacent matricesrdquoMechanism andMachineTheory vol 40 no 2 pp 131ndash139 2005

[23] H F Ding and Z Huang ldquoA unique representation of the kine-matic chain and the atlas databaserdquo Mechanism and MachineTheory vol 42 no 6 pp 637ndash651 2007

[24] H F Ding and Z Huang ldquoA new theory for the topologicalstructure analysis of kinematic chains and its applicationsrdquoMechanism and Machine Theory vol 42 no 10 pp 1264ndash12792007

[25] A C Rao ldquoA genetic algorithm for topological characteristicsof kinematic chainsrdquo Journal of Mechanical Design vol 122 no2 pp 228ndash231 2000

[26] F G Kong Q Li and W J Zhang ldquoAn artificial neuralnetwork approach tomechanism kinematic chain isomorphismidentificationrdquo Mechanism and Machine Theory vol 34 no 2pp 271ndash283 1999

[27] J K Chu The Structural and Dimensional Characteristics ofPlanar Linkages Press of Beihang University of China BeijingChina 1992

[28] L Song J Yang X Zhang and W Q Cao ldquoSpanning treemethod of identifying isomorphism and topological symmetryto planar kinematic chain with multiple jointrdquo Chinese Journalof Mechanical Engineering vol 14 no 1 pp 27ndash31 2001

[29] J G Liu and D J Yu ldquoRepresentations amp isomorphismidentification of planar kinematic chains with multiple jointsbased on the converted adjacent matrixrdquo Chinese Journal ofMechanical Engineering vol 48 pp 15ndash21 2012

[30] R Ravisankar andT SMruthyunjaya ldquoComputerized synthesisof the structure of geared kinematic chainsrdquo Mechanism andMachine Theory vol 20 no 5 pp 367ndash387 1985

[31] J U Kim and B M Kwak ldquoApplication of edge permutationgroup to structural synthesis of epicyclic gear trainsrdquo Mecha-nism and Machine Theory vol 25 no 5 pp 563ndash573 1990

Mathematical Problems in Engineering 15

[32] G Chatterjee and L-W Tsai ldquoComputer-aided sketching ofepicyclic-type automatic transmission gear trainsrdquo Journal ofMechanical Design vol 118 no 3 pp 405ndash411 1996

[33] V V N R Prasadraju and A C Rao ldquoA new technique basedon loops to investigate displacement isomorphism in planetarygear trainsrdquo Journal of Mechanical Design vol 12 pp 666ndash6752002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

①

②

③

④

⑤

⑥

⑦

⑧

⑨

⑩

c

ba

d

(a)

①

②

③ ④

⑤

⑥⑦

⑧

⑨⑩

c b

a

d

(b)

①

cb

a

2

2

2

2

2

3

3

1d

(c)

①

2

2

2

2

2

3

3

d

b

ac

1

(d)

Figure 9 Two 10-link multiple joint kinematic chains and their WDCCGs

①

⑪

②

③

④

⑤

⑥

⑦⑧

⑨

⑩

ab

(a)

①

②

③④

⑤⑥

⑦

⑧

⑨

⑩

a

b

⑪

(b)

①

② ③

④

⑤

⑥ 1

1

11

12

2

2

2

2a

b

1

1 1

(c)

① ②

③

④

⑤

⑥

1

1

1

1

1

11

12

2

2

2

2b a

(d)

Figure 10 Two 11-link multiple joint kinematic chains and their WDCCGs

Step 3 Calculate NVS(119886) NVS(119887) and their vertex codes oftwo ECMs with the same CE respectively as follows

NVS (119886) = [(V119888 V119887 V119886 V119889) (V1)]

= [(01551 01546 01451 01275) (01700)]

NVS (119887) = [(V119886 V119887 V119888 V119889) (V1)]

= [(01538 01459 01459 01308) (01714)]

(16)

Compare NVS(119886) and NVS(119887) and equivalence cannot beseen It is determined that they are not isomorphic andprogram stops

Example 2 Two 11-link multiple joint kinematic chains areshown in Figures 10(a) and 10(b)Their correspondingWDC-CGs are described in Figures 10(c) and 10(d) respectivelyThe direct process to determine isomorphism of these twokinematic chains is as follows

10 Mathematical Problems in Engineering

Step 1 Input their corresponding vertex codes andWAMs1198601198601015840 as follows

119860

=

1 2 3 4 5 6 119886 119887

1

2

3

4

5

6

119886

119887

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

(1)

(2)

(0)

(1)

(1)

(0)

(0)

(2)

(2)

(1)

(1)

(0)

(1)

(0)

(0)

(0)

(0)

(1)

(1)

(1)

(0)

(2)

(2)

(0)

(1)

(0)

(1)

(1)

(0)

(1)

(0)

(0)

(1)

(1)

(0)

(0)

(1)

(0)

(1)

(0)

(0)

(2)

(0)

(1)

(0)

(1)

(0)

(1)

(0)

(0)

(1)

(0)

(1)

(0)

(0)

(2)

(2)

(0)

(0)

(0)

(0)

(1)

(2)

(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

1198601015840

=

1 2 3 4 5 6 119886 119887

1

2

3

4

5

6

119886

119887

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

(1)

(2)

(0)

(1)

(0)

(0)

(0)

(1)

(2)

(1)

(0)

(1)

(1)

(0)

(2)

(0)

(0)

(0)

(1)

(1)

(2)

(1)

(0)

(2)

(1)

(1)

(1)

(1)

(0)

(0)

(0)

(0)

(0)

(1)

(2)

(0)

(1)

(1)

(0)

(0)

(0)

(0)

(1)

(0)

(1)

(1)

(1)

(0)

(0)

(2)

(0)

(0)

(0)

(1)

(0)

(2)

(1)

(0)

(2)

(0)

(0)

(0)

(2)

(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(17)

Step 2 Evidently 119860 = 1198601015840 Assign vertices codes of 119860 and 1198601015840

in two groups (a) [(119886 119887) (1 2 3 4 5 6)] and (b) [(119886 119887) (1 23 4 5 6)]

Step 3 Calculate NVS(119886) NVS(119887) and their vertex codes oftwo ECMs with the same CE respectively as follows

NVS (119886) = [(V119886 V119887) (V5 V4 V2 V6 V1 V3)]

= [(02084 02056)

(0743 02727 02348 02347 02023 01995)]

NVS (119887) = [(V119887 V119888) (V6 V4 V5 V1 V3 V2)]

= [(02084 02056)

(0743 02727 02348 02347 02023 01995)]

(18)

Compare NVS(119886) and NVS(119887) and equivalence can beseen So the node voltages in two vertex code groups are alldistinct go to Step 4

Step 4 Take the NVS(119886) NVS(119887) and their vertex codes todetermine the corresponding vertices of two WDCCGs asfollows (119886-119887 119887-119886 1-3 2-5 3-2 4-4 5-6 6-1)

Step 5 According to the corresponding vertices obtained andthe vertex codes to form new WAM 119860

1015840

1 rewrite WAM 119860

1015840Then we have 119860 = 1198601015840

1 so they are isomorphic and program

stops

Example 3 Two 6-gear train kinematic chains are depicted inFigures 11(a) and 11(b) and their correspondingWDCCGs aredescribed in Figures 11(c) and 11(d) respectively The directprocess to determine isomorphism of these two kinematicchains is as follows

Step 1 Input their corresponding vertex codes andWAMs1198601198601015840 as follows

119860 =

1 2 3 4 5 6 7 119886 119888

1

2

3

4

5

6

7

119886

119888

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

(1)

(3)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(3)

(1)

(2)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(2)

(1)

(3)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(3)

(1)

(0)

(0)

(0)

(1)

(1)

(0)

(0)

(0)

(0)

(1)

(3)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(3)

(1)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(1)

(1)

(0)

(1)

(0)

(1)

(0)

(1)

(1)

(0)

(0)

(1)

(0)

(0)

(1)

(1)

(0)

(1)

(0)

(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

1198601015840=

1 2 3 4 5 6 7 119886 119888

1

2

3

4

5

6

7

119886

119888

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

(1)

(3)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(3)

(1)

(2)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(2)

(1)

(3)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(3)

(1)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(1)

(3)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(3)

(1)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(1)

(1)

(0)

(1)

(0)

(1)

(1)

(0)

(1)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(1)

(1)

(0)

(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(19)

Mathematical Problems in Engineering 11

①

② ③ ④ ⑤

⑥⑦

A

B

b

a

C

c

(a)

①

②③

④

⑤

⑥ ⑦

A

Bb

a

C

c

(b)

①

②

③

④ ⑤

⑥

⑦

3

3

3

c

2

11

1

1

1

1

1

1

a

(c)

①

②

③

④

⑤

⑥

⑦

c

a

1

1

1

1

1

1

1

2

3

3

3

(d)

Figure 11 Two 6-gear train kinematic chains and their WDCCG

Step 2 Evidently 119860 = 1198601015840 Assign vertices codes of 119860 and 1198601015840

in two groups (a) [(119886 119888) (1 2 3 4 5 6 7)] and (b) [(119886 119888) (12 3 4 5 6 7)]

Step 3 Calculate NVS(119886) NVS(119887) and their vertex codes oftwo ECMs with the same CE respectively as follows

NVS (119886) = [(V119887 V119886) (V7 V5 V6 V1 V4 V3 V2)]= [(02591 02547)

(03784 02515 02511 02309 02122 02029 01961)]

NVS (119887) = [(V119887 V119886) (V7 V6 V5 V1 V4 V3 V2)]= [(03156 02607)

(03940 02601 02551 02394 02367 02110 02001)]

(20)

Compare NVS(119886) and NVS(119887) and equivalence cannot beseen So it is determined that they are not isomorphic andprogram stops

Example 4 Two 9-gear train kinematic chains are shown inFigures 12(a) and 12(b) Their corresponding WDCCGs aredepicted in Figures 12(c) and 12(d) respectively The directprocess to determine isomorphism of these two kinematicchains is as follows

Step 1 Input their correspondingWAMs119860 and1198601015840 as follows

119860 =

1 2 3 4 5 6 7 8 9 119886 119887 119888

1

2

3

4

5

6

7

8

9

119886

119887

119888

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

(1)

(0)

(3)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(3)

(1)

(3)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(0)

(3)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(3)

(1)

(1)

(1)

(0)

(0)

(0)

(0)

(1)

(3)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(3)

(1)

(3)

(2)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(3)

(1)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(2)

(0)

(1)

(0)

(1)

(1)

(0)

(0)

(0)

(0)

(3)

(0)

(0)

(0)

(0)

(1)

(0)

(1)

(0)

(0)

(0)

(1)

(0)

(1)

(0)

(1)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(1)

(1)

(0)

(0)

(0)

(1)

(1)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

12 Mathematical Problems in Engineering

① ②

③

a

A

c

B

d

C

D

E

b

④

⑤

⑥

⑦

⑧

⑨

(a)

①

②③a

A

b E

c

DC

d

B④

⑤

⑥ ⑦⑧⑨

(b)

①

1 1

c

1 1

1

1

1

b3

3

3

2

1

1

1

1a

33

②

③

④

⑤⑥

⑦

⑧

⑨

(c)

1

1

1

1

3

3

1

1

1

1

1

1

1

2

3

3

3

b

c

a

①②

③

④

⑤

⑥

⑦

⑧

⑨

(d)

Figure 12 Two 9-gear train kinematic chains and their WDCCGs

1198601015840=

1 2 3 4 5 6 7 8 9 119886 119887 119888

1

2

3

4

5

6

7

8

9

119886

119887

119888

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

(1)

(0)

(0)

(3)

(0)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(1)

(0)

(3)

(0)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(1)

(2)

(0)

(0)

(0)

(0)

(0)

(1)

(0)

(1)

(3)

(3)

(2)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(3)

(3)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(3)

(1)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(3)

(0)

(1)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(3)

(1)

(1)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(3)

(1)

(0)

(0)

(1)

(1)

(1)

(1)

(0)

(1)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(1)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(1)

(0)

(0)

(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(21)

Step 2 Evidently 119860 = 1198601015840 Assign vertices codes of 119860 and 1198601015840

in two groups (a) [(119886 119887 119888) (1 2 3 4 5 6 7 8 9)] and (b) [(119886119887 119888) (1 2 3 4 5 6 7 8 9)]

Step 3 Calculate NVS(119886) NVS(119887) and their vertex codes oftwo ECMs with the same CE respectively as follows

NVS (119886) = [(V119888 V119887 V119886) (V9 V4 V8 V1 V2 V5 V7 V3 V6)]

= [(02845 02815 02075) (02470 02307 02295 02294 02294 02154 02154 01846 01719)]

NVS (119887) = [(V119887 V119888 V119886) (V9 V8 V3 V6 V7 V1 V2 V5 V4)]

= [(02845 02815 02075) (02470 02307 02295 02294 02294 02154 02154 01846 01719)]

(22)

Mathematical Problems in Engineering 13

Compare NVS(119886) and NVS(119887) and equivalence can be seenAnd the node voltages in two vertex code groups are not alldistinct so go to Step 6

Step 4 The vertices (1 2) in Figure 12(a) vertices (6 7) inFigure 12(b) the vertices (5 7) in Figure 12(a) and vertices (12) in Figure 12(b) have the same values in the correspondinggroups so just adjust the position order of the vertices in 1198601015840to obtain a new adjacency matrix 1198601015840 And the correspondingvertices of two WDCCGs are (1-6 2-7 3-5 4-8 5-1 6-4 7-28-3 9-9 119886-119886 119887-119888 119888-119887) so we have 119860 = 119860

1015840

1 It is determined

that they are isomorphic and program stops

5 Algorithm Complexity Analysis

This paper presents a relatively good algorithm for determi-nation of kinematic chains isomorphism with multiple jointand gear train It is feasible for introducing the WDCCGand WAM to describe these two types of kinematic chainsthey dramatically reduce the number of vertices and order ofcorresponding adjacency matrix 119860

The analysis of computational complexity of this isomor-phic identification algorithm is as follows when there areno equivalent node voltages in NVS time consumption isbasically dedicated to solving the linear equation sets If akinematic chain possesses 119899 vertices the order of the nodevoltage equations is 119899 minus 1 Since the adjoint circuit is alinear resistance circuit the node voltage equations forma linear algebra equation set of orders 119899 minus 1 which has acomputational complexity of 119874((119899 minus 1)3) for its solutionSince it needs to solve at most 2119899 adjoint circuits in ourproposed method the computational complexity is of order119874(2119899(119899 minus 1)

3) lt 119874(119899

4) Thus the proposed method has a

computational complexity of a polynomial orderWhen thereare equivalent node voltages in NVS the correspondence ofonly a part of the vertices cannot be determinedThus in theworst situation the proposed method is more efficient thanthose where rowcolumn exchanges are performed blindly

In this paper this algorithm is tested and comparedwith the internationally recognized faster method for planarsimple joint kinematic chains isomorphic identification [20]and they are tested in the same software and hardwareenvironment The test results are shown in Figure 13 (theaverage time of each vertex is 20 times of the kinematic chainwithmultiple joints) test results show that when the vertex ofthe kinematic chain is increased the algorithm proposed inthis paper is more efficient than the algorithm proposed byHe et alThe proposed algorithm by us is very suitable for theisomorphism of the topological graph of kinematic chainsand the algorithm is very efficient at the same time

6 Conclusions

In this paper an algorithm for topological isomorphism iden-tification of planar multiple joint and gear train kinematicchains is introduced

4 6 8 10 12 14 16 18 20 22

The number of vertices

Algorithm proposed by He et al

Algorithm proposed in this paper

00001

0001

001

01

1

The d

ecisi

on ti

me (

s)

Figure 13 The test results of the algorithm proposed in this paperand the algorithm proposed by He et al

Firstly a weighted-double-color-contracted-graph(WDCCG) and a weighted adjacency matrix (WAM) areintroduced to describe the planarmultiple joint and gear trainkinematic chains respectively They dramatically reduce thenumber of vertices and order of the represented adjacencymatrix

Secondly isomorphism identification method of thesetwo types of kinematic chains is carried out by the equivalentcircuit models (ECMs) of WDCCGs with the same completeexcitation (CE) and then uses the solved node voltagesequence (NVS) to determine the corresponding vertices oftwo WDCCGs

Finally an algorithm to identify isomorphic kinematicchains is obtained It is an efficient and easy method tobe realized by computer And this algorithm also can beused in the isomorphism identification of planar simple jointkinematic chains with minor changes

Notations

119860 1198601015840 The incidence matrix of kinematic chain119889119894119895 The elements of incidence matrix

119866119894 Conductance value of edge in the ECM

119898119894119895119896 The weighted value of 119896th edge

1198731198731015840 Variable of equivalent circuit model119899 The number of vertices in WDCCG119899119894 Node 119894 in the ECM

V119894 Node voltages of the node 119894

CE Complete excitationDCG Double color graphECM Equivalent circuit modelNVS Node voltage sequenceTM Topological modelWAM Weighted adjacency matrixWDCCG Weighted double color contracted graph

14 Mathematical Problems in Engineering

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

Thiswork is supported by the Science andTechnology Projectof Quanzhou City 2014G48 and G20140047

References

[1] J J Uicker Jr and A Raicu ldquoAmethod for the identification andrecognition of equivalence of kinematic chainsrdquoMechanismandMachine Theory vol 10 no 5 pp 375ndash383 1975

[2] H S Yan and A S Hall ldquoLinkage characteristic polynomialsdefinitions coefficients by inspectionrdquo Journal of MechanicalDesign vol 103 no 3 pp 578ndash584 1981

[3] H S Yan and A S Hall ldquoLinkage characteristic polynomialsassembly theorems uniquenessrdquo Journal of Mechanical Designvol 104 no 1 pp 11ndash20 1982

[4] T S Mruthyunjaya ldquoA computerized methodology for struc-tural synthesis of kinematic chains Part 1 FormulationrdquoMech-anism and Machine Theory vol 19 no 6 pp 487ndash495 1984

[5] T S Mruthyunjaya ldquoA computerized methodology for struc-tural synthesis of kinematic chains part 2mdashapplication toseveral fully or partially known casesrdquoMechanism andMachineTheory vol 19 no 6 pp 497ndash505 1984

[6] T S Mruthyunjaya ldquoA computerized methodology for struc-tural synthesis of kinematic chains part 3mdashapplication to thenew case of 10-link three- freedom chainsrdquo Mechanism andMachine Theory vol 19 no 6 pp 507ndash530 1984

[7] T S Mruthyunjaya and H R Balasubramanian ldquoIn questof a reliable and efficient computational test for detection ofisomorphism in kinematic chainsrdquo Mechanism and MachineTheory vol 22 no 2 pp 131ndash139 1987

[8] W J Sohn and F Freudenstein ldquoAn application of dual graphs tothe automatic generation of the kinematic structures of mecha-nismsrdquo Journal of Mechanisms Transmissions and Automationvol 108 no 3 pp 392ndash398 1986

[9] A Ambekar and V Agrawal ldquoOn canonical numbering ofkinematic chains and isomorphism problem MAX coderdquo inProceedings of the ASMEMechanisms Conference pp 1615ndash16221986

[10] A G Ambekar and V P Agrawal ldquoCanonical numberingof kinematic chains and isomorphism problem min coderdquoMechanism andMachineTheory vol 22 no 5 pp 453ndash461 1987

[11] C S Tang and T Liu ldquoDegree code a new mechanismidentifierrdquo Journal of Mechanical Design vol 115 no 3 pp 627ndash630 1993

[12] J K Shin and S Krishnamurty ldquoOn identification and canonicalnumbering of pin jointed kinematic chainsrdquo Journal of Mechan-ical Design vol 116 pp 182ndash188 1994

[13] J K Shin and S Krishnamurty ldquoDevelopment of a standardcode for colored graphs and its application to kinematic chainsrdquoJournal of Mechanical Design vol 116 no 1 pp 189ndash196 1994

[14] A C Rao and D Varada Raju ldquoApplication of the hammingnumber technique to detect isomorphism among kinematicchains and inversionsrdquoMechanism andMachineTheory vol 26no 1 pp 55ndash75 1991

[15] A C Rao and C N Rao ldquoLoop based pseudo hammingvalues-1 Testing isomorphism and rating kinematic chainsrdquoMechanism andMachineTheory vol 28 no 1 pp 113ndash127 1992

[16] A C Rao and C N Rao ldquoLoop based pseudo hamming values-II inversions preferred frames and actuatorsrdquo Mechanism andMachine Theory vol 28 no 1 pp 129ndash143 1993

[17] J-K Chu and W-Q Cao ldquoIdentification of isomorphismamong kinematic chains and inversions using linkrsquos adjacent-chain-tablerdquoMechanism and Machine Theory vol 29 no 1 pp53ndash58 1994

[18] A C Rao ldquoApplication of fuzzy logic for the study of iso-morphism inversions symmetry parallelism and mobility inkinematic chainsrdquoMechanism and Machine Theory vol 35 no8 pp 1103ndash1116 2000

[19] P R He W J Zhang Q Li and F X Wu ldquoA new method fordetection of graph isomorphism based on the quadratic formrdquoJournal of Mechanical Design vol 125 no 3 pp 640ndash642 2003

[20] P R He W J Zhang and Q Li ldquoSome further developmenton the eigensystem approach for graph isomorphismdetectionrdquoJournal of the Franklin Institute vol 342 no 6 pp 657ndash6732005

[21] Z Y Chang C Zhang Y H Yang and YWang ldquoA newmethodto mechanism kinematic chain isomorphism identificationrdquoMechanism andMachineTheory vol 37 no 4 pp 411ndash417 2002

[22] J P Cubillo and J BWan ldquoComments onmechanismkinematicchain isomorphism identification using adjacent matricesrdquoMechanism andMachineTheory vol 40 no 2 pp 131ndash139 2005

[23] H F Ding and Z Huang ldquoA unique representation of the kine-matic chain and the atlas databaserdquo Mechanism and MachineTheory vol 42 no 6 pp 637ndash651 2007

[24] H F Ding and Z Huang ldquoA new theory for the topologicalstructure analysis of kinematic chains and its applicationsrdquoMechanism and Machine Theory vol 42 no 10 pp 1264ndash12792007

[25] A C Rao ldquoA genetic algorithm for topological characteristicsof kinematic chainsrdquo Journal of Mechanical Design vol 122 no2 pp 228ndash231 2000

[26] F G Kong Q Li and W J Zhang ldquoAn artificial neuralnetwork approach tomechanism kinematic chain isomorphismidentificationrdquo Mechanism and Machine Theory vol 34 no 2pp 271ndash283 1999

[27] J K Chu The Structural and Dimensional Characteristics ofPlanar Linkages Press of Beihang University of China BeijingChina 1992

[28] L Song J Yang X Zhang and W Q Cao ldquoSpanning treemethod of identifying isomorphism and topological symmetryto planar kinematic chain with multiple jointrdquo Chinese Journalof Mechanical Engineering vol 14 no 1 pp 27ndash31 2001

[29] J G Liu and D J Yu ldquoRepresentations amp isomorphismidentification of planar kinematic chains with multiple jointsbased on the converted adjacent matrixrdquo Chinese Journal ofMechanical Engineering vol 48 pp 15ndash21 2012

[30] R Ravisankar andT SMruthyunjaya ldquoComputerized synthesisof the structure of geared kinematic chainsrdquo Mechanism andMachine Theory vol 20 no 5 pp 367ndash387 1985

[31] J U Kim and B M Kwak ldquoApplication of edge permutationgroup to structural synthesis of epicyclic gear trainsrdquo Mecha-nism and Machine Theory vol 25 no 5 pp 563ndash573 1990

Mathematical Problems in Engineering 15

[32] G Chatterjee and L-W Tsai ldquoComputer-aided sketching ofepicyclic-type automatic transmission gear trainsrdquo Journal ofMechanical Design vol 118 no 3 pp 405ndash411 1996

[33] V V N R Prasadraju and A C Rao ldquoA new technique basedon loops to investigate displacement isomorphism in planetarygear trainsrdquo Journal of Mechanical Design vol 12 pp 666ndash6752002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

Step 1 Input their corresponding vertex codes andWAMs1198601198601015840 as follows

119860

=

1 2 3 4 5 6 119886 119887

1

2

3

4

5

6

119886

119887

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

(1)

(2)

(0)

(1)

(1)

(0)

(0)

(2)

(2)

(1)

(1)

(0)

(1)

(0)

(0)

(0)

(0)

(1)

(1)

(1)

(0)

(2)

(2)

(0)

(1)

(0)

(1)

(1)

(0)

(1)

(0)

(0)

(1)

(1)

(0)

(0)

(1)

(0)

(1)

(0)

(0)

(2)

(0)

(1)

(0)

(1)

(0)

(1)

(0)

(0)

(1)

(0)

(1)

(0)

(0)

(2)

(2)

(0)

(0)

(0)

(0)

(1)

(2)

(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

1198601015840

=

1 2 3 4 5 6 119886 119887

1

2

3

4

5

6

119886

119887

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

(1)

(2)

(0)

(1)

(0)

(0)

(0)

(1)

(2)

(1)

(0)

(1)

(1)

(0)

(2)

(0)

(0)

(0)

(1)

(1)

(2)

(1)

(0)

(2)

(1)

(1)

(1)

(1)

(0)

(0)

(0)

(0)

(0)

(1)

(2)

(0)

(1)

(1)

(0)

(0)

(0)

(0)

(1)

(0)

(1)

(1)

(1)

(0)

(0)

(2)

(0)

(0)

(0)

(1)

(0)

(2)

(1)

(0)

(2)

(0)

(0)

(0)

(2)

(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(17)

Step 2 Evidently 119860 = 1198601015840 Assign vertices codes of 119860 and 1198601015840

in two groups (a) [(119886 119887) (1 2 3 4 5 6)] and (b) [(119886 119887) (1 23 4 5 6)]

Step 3 Calculate NVS(119886) NVS(119887) and their vertex codes oftwo ECMs with the same CE respectively as follows

NVS (119886) = [(V119886 V119887) (V5 V4 V2 V6 V1 V3)]

= [(02084 02056)

(0743 02727 02348 02347 02023 01995)]

NVS (119887) = [(V119887 V119888) (V6 V4 V5 V1 V3 V2)]

= [(02084 02056)

(0743 02727 02348 02347 02023 01995)]

(18)

Compare NVS(119886) and NVS(119887) and equivalence can beseen So the node voltages in two vertex code groups are alldistinct go to Step 4

Step 4 Take the NVS(119886) NVS(119887) and their vertex codes todetermine the corresponding vertices of two WDCCGs asfollows (119886-119887 119887-119886 1-3 2-5 3-2 4-4 5-6 6-1)

Step 5 According to the corresponding vertices obtained andthe vertex codes to form new WAM 119860

1015840

1 rewrite WAM 119860

1015840Then we have 119860 = 1198601015840

1 so they are isomorphic and program

stops

Example 3 Two 6-gear train kinematic chains are depicted inFigures 11(a) and 11(b) and their correspondingWDCCGs aredescribed in Figures 11(c) and 11(d) respectively The directprocess to determine isomorphism of these two kinematicchains is as follows

Step 1 Input their corresponding vertex codes andWAMs1198601198601015840 as follows

119860 =

1 2 3 4 5 6 7 119886 119888

1

2

3

4

5

6

7

119886

119888

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

(1)

(3)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(3)

(1)

(2)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(2)

(1)

(3)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(3)

(1)

(0)

(0)

(0)

(1)

(1)

(0)

(0)

(0)

(0)

(1)

(3)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(3)

(1)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(1)

(1)

(0)

(1)

(0)

(1)

(0)

(1)

(1)

(0)

(0)

(1)

(0)

(0)

(1)

(1)

(0)

(1)

(0)

(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

1198601015840=

1 2 3 4 5 6 7 119886 119888

1

2

3

4

5

6

7

119886

119888

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

(1)

(3)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(3)

(1)

(2)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(2)

(1)

(3)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(3)

(1)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(1)

(3)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(3)

(1)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(1)

(1)

(0)

(1)

(0)

(1)

(1)

(0)

(1)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(1)

(1)

(0)

(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(19)

Mathematical Problems in Engineering 11

①

② ③ ④ ⑤

⑥⑦

A

B

b

a

C

c

(a)

①

②③

④

⑤

⑥ ⑦

A

Bb

a

C

c

(b)

①

②

③

④ ⑤

⑥

⑦

3

3

3

c

2

11

1

1

1

1

1

1

a

(c)

①

②

③

④

⑤

⑥

⑦

c

a

1

1

1

1

1

1

1

2

3

3

3

(d)

Figure 11 Two 6-gear train kinematic chains and their WDCCG

Step 2 Evidently 119860 = 1198601015840 Assign vertices codes of 119860 and 1198601015840

in two groups (a) [(119886 119888) (1 2 3 4 5 6 7)] and (b) [(119886 119888) (12 3 4 5 6 7)]

Step 3 Calculate NVS(119886) NVS(119887) and their vertex codes oftwo ECMs with the same CE respectively as follows

NVS (119886) = [(V119887 V119886) (V7 V5 V6 V1 V4 V3 V2)]= [(02591 02547)

(03784 02515 02511 02309 02122 02029 01961)]

NVS (119887) = [(V119887 V119886) (V7 V6 V5 V1 V4 V3 V2)]= [(03156 02607)

(03940 02601 02551 02394 02367 02110 02001)]

(20)

Compare NVS(119886) and NVS(119887) and equivalence cannot beseen So it is determined that they are not isomorphic andprogram stops

Example 4 Two 9-gear train kinematic chains are shown inFigures 12(a) and 12(b) Their corresponding WDCCGs aredepicted in Figures 12(c) and 12(d) respectively The directprocess to determine isomorphism of these two kinematicchains is as follows

Step 1 Input their correspondingWAMs119860 and1198601015840 as follows

119860 =

1 2 3 4 5 6 7 8 9 119886 119887 119888

1

2

3

4

5

6

7

8

9

119886

119887

119888

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

(1)

(0)

(3)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(3)

(1)

(3)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(0)

(3)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(3)

(1)

(1)

(1)

(0)

(0)

(0)

(0)

(1)

(3)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(3)

(1)

(3)

(2)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(3)

(1)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(2)

(0)

(1)

(0)

(1)

(1)

(0)

(0)

(0)

(0)

(3)

(0)

(0)

(0)

(0)

(1)

(0)

(1)

(0)

(0)

(0)

(1)

(0)

(1)

(0)

(1)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(1)

(1)

(0)

(0)

(0)

(1)

(1)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

12 Mathematical Problems in Engineering

① ②

③

a

A

c

B

d

C

D

E

b

④

⑤

⑥

⑦

⑧

⑨

(a)

①

②③a

A

b E

c

DC

d

B④

⑤

⑥ ⑦⑧⑨

(b)

①

1 1

c

1 1

1

1

1

b3

3

3

2

1

1

1

1a

33

②

③

④

⑤⑥

⑦

⑧

⑨

(c)

1

1

1

1

3

3

1

1

1

1

1

1

1

2

3

3

3

b

c

a

①②

③

④

⑤

⑥

⑦

⑧

⑨

(d)

Figure 12 Two 9-gear train kinematic chains and their WDCCGs

1198601015840=

1 2 3 4 5 6 7 8 9 119886 119887 119888

1

2

3

4

5

6

7

8

9

119886

119887

119888

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

(1)

(0)

(0)

(3)

(0)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(1)

(0)

(3)

(0)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(1)

(2)

(0)

(0)

(0)

(0)

(0)

(1)

(0)

(1)

(3)

(3)

(2)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(3)

(3)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(3)

(1)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(3)

(0)

(1)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(3)

(1)

(1)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(3)

(1)

(0)

(0)

(1)

(1)

(1)

(1)

(0)

(1)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(1)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(1)

(0)

(0)

(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(21)

Step 2 Evidently 119860 = 1198601015840 Assign vertices codes of 119860 and 1198601015840

in two groups (a) [(119886 119887 119888) (1 2 3 4 5 6 7 8 9)] and (b) [(119886119887 119888) (1 2 3 4 5 6 7 8 9)]

Step 3 Calculate NVS(119886) NVS(119887) and their vertex codes oftwo ECMs with the same CE respectively as follows

NVS (119886) = [(V119888 V119887 V119886) (V9 V4 V8 V1 V2 V5 V7 V3 V6)]

= [(02845 02815 02075) (02470 02307 02295 02294 02294 02154 02154 01846 01719)]

NVS (119887) = [(V119887 V119888 V119886) (V9 V8 V3 V6 V7 V1 V2 V5 V4)]

= [(02845 02815 02075) (02470 02307 02295 02294 02294 02154 02154 01846 01719)]

(22)

Mathematical Problems in Engineering 13

Compare NVS(119886) and NVS(119887) and equivalence can be seenAnd the node voltages in two vertex code groups are not alldistinct so go to Step 6

Step 4 The vertices (1 2) in Figure 12(a) vertices (6 7) inFigure 12(b) the vertices (5 7) in Figure 12(a) and vertices (12) in Figure 12(b) have the same values in the correspondinggroups so just adjust the position order of the vertices in 1198601015840to obtain a new adjacency matrix 1198601015840 And the correspondingvertices of two WDCCGs are (1-6 2-7 3-5 4-8 5-1 6-4 7-28-3 9-9 119886-119886 119887-119888 119888-119887) so we have 119860 = 119860

1015840

1 It is determined

that they are isomorphic and program stops

5 Algorithm Complexity Analysis

This paper presents a relatively good algorithm for determi-nation of kinematic chains isomorphism with multiple jointand gear train It is feasible for introducing the WDCCGand WAM to describe these two types of kinematic chainsthey dramatically reduce the number of vertices and order ofcorresponding adjacency matrix 119860

The analysis of computational complexity of this isomor-phic identification algorithm is as follows when there areno equivalent node voltages in NVS time consumption isbasically dedicated to solving the linear equation sets If akinematic chain possesses 119899 vertices the order of the nodevoltage equations is 119899 minus 1 Since the adjoint circuit is alinear resistance circuit the node voltage equations forma linear algebra equation set of orders 119899 minus 1 which has acomputational complexity of 119874((119899 minus 1)3) for its solutionSince it needs to solve at most 2119899 adjoint circuits in ourproposed method the computational complexity is of order119874(2119899(119899 minus 1)

3) lt 119874(119899

4) Thus the proposed method has a

computational complexity of a polynomial orderWhen thereare equivalent node voltages in NVS the correspondence ofonly a part of the vertices cannot be determinedThus in theworst situation the proposed method is more efficient thanthose where rowcolumn exchanges are performed blindly

In this paper this algorithm is tested and comparedwith the internationally recognized faster method for planarsimple joint kinematic chains isomorphic identification [20]and they are tested in the same software and hardwareenvironment The test results are shown in Figure 13 (theaverage time of each vertex is 20 times of the kinematic chainwithmultiple joints) test results show that when the vertex ofthe kinematic chain is increased the algorithm proposed inthis paper is more efficient than the algorithm proposed byHe et alThe proposed algorithm by us is very suitable for theisomorphism of the topological graph of kinematic chainsand the algorithm is very efficient at the same time

6 Conclusions

In this paper an algorithm for topological isomorphism iden-tification of planar multiple joint and gear train kinematicchains is introduced

4 6 8 10 12 14 16 18 20 22

The number of vertices

Algorithm proposed by He et al

Algorithm proposed in this paper

00001

0001

001

01

1

The d

ecisi

on ti

me (

s)

Figure 13 The test results of the algorithm proposed in this paperand the algorithm proposed by He et al

Firstly a weighted-double-color-contracted-graph(WDCCG) and a weighted adjacency matrix (WAM) areintroduced to describe the planarmultiple joint and gear trainkinematic chains respectively They dramatically reduce thenumber of vertices and order of the represented adjacencymatrix

Secondly isomorphism identification method of thesetwo types of kinematic chains is carried out by the equivalentcircuit models (ECMs) of WDCCGs with the same completeexcitation (CE) and then uses the solved node voltagesequence (NVS) to determine the corresponding vertices oftwo WDCCGs

Finally an algorithm to identify isomorphic kinematicchains is obtained It is an efficient and easy method tobe realized by computer And this algorithm also can beused in the isomorphism identification of planar simple jointkinematic chains with minor changes

Notations

119860 1198601015840 The incidence matrix of kinematic chain119889119894119895 The elements of incidence matrix

119866119894 Conductance value of edge in the ECM

119898119894119895119896 The weighted value of 119896th edge

1198731198731015840 Variable of equivalent circuit model119899 The number of vertices in WDCCG119899119894 Node 119894 in the ECM

V119894 Node voltages of the node 119894

CE Complete excitationDCG Double color graphECM Equivalent circuit modelNVS Node voltage sequenceTM Topological modelWAM Weighted adjacency matrixWDCCG Weighted double color contracted graph

14 Mathematical Problems in Engineering

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

Thiswork is supported by the Science andTechnology Projectof Quanzhou City 2014G48 and G20140047

References

[1] J J Uicker Jr and A Raicu ldquoAmethod for the identification andrecognition of equivalence of kinematic chainsrdquoMechanismandMachine Theory vol 10 no 5 pp 375ndash383 1975

[2] H S Yan and A S Hall ldquoLinkage characteristic polynomialsdefinitions coefficients by inspectionrdquo Journal of MechanicalDesign vol 103 no 3 pp 578ndash584 1981

[3] H S Yan and A S Hall ldquoLinkage characteristic polynomialsassembly theorems uniquenessrdquo Journal of Mechanical Designvol 104 no 1 pp 11ndash20 1982

[4] T S Mruthyunjaya ldquoA computerized methodology for struc-tural synthesis of kinematic chains Part 1 FormulationrdquoMech-anism and Machine Theory vol 19 no 6 pp 487ndash495 1984

[5] T S Mruthyunjaya ldquoA computerized methodology for struc-tural synthesis of kinematic chains part 2mdashapplication toseveral fully or partially known casesrdquoMechanism andMachineTheory vol 19 no 6 pp 497ndash505 1984

[6] T S Mruthyunjaya ldquoA computerized methodology for struc-tural synthesis of kinematic chains part 3mdashapplication to thenew case of 10-link three- freedom chainsrdquo Mechanism andMachine Theory vol 19 no 6 pp 507ndash530 1984

[7] T S Mruthyunjaya and H R Balasubramanian ldquoIn questof a reliable and efficient computational test for detection ofisomorphism in kinematic chainsrdquo Mechanism and MachineTheory vol 22 no 2 pp 131ndash139 1987

[8] W J Sohn and F Freudenstein ldquoAn application of dual graphs tothe automatic generation of the kinematic structures of mecha-nismsrdquo Journal of Mechanisms Transmissions and Automationvol 108 no 3 pp 392ndash398 1986

[9] A Ambekar and V Agrawal ldquoOn canonical numbering ofkinematic chains and isomorphism problem MAX coderdquo inProceedings of the ASMEMechanisms Conference pp 1615ndash16221986

[10] A G Ambekar and V P Agrawal ldquoCanonical numberingof kinematic chains and isomorphism problem min coderdquoMechanism andMachineTheory vol 22 no 5 pp 453ndash461 1987

[11] C S Tang and T Liu ldquoDegree code a new mechanismidentifierrdquo Journal of Mechanical Design vol 115 no 3 pp 627ndash630 1993

[12] J K Shin and S Krishnamurty ldquoOn identification and canonicalnumbering of pin jointed kinematic chainsrdquo Journal of Mechan-ical Design vol 116 pp 182ndash188 1994

[13] J K Shin and S Krishnamurty ldquoDevelopment of a standardcode for colored graphs and its application to kinematic chainsrdquoJournal of Mechanical Design vol 116 no 1 pp 189ndash196 1994

[14] A C Rao and D Varada Raju ldquoApplication of the hammingnumber technique to detect isomorphism among kinematicchains and inversionsrdquoMechanism andMachineTheory vol 26no 1 pp 55ndash75 1991

[15] A C Rao and C N Rao ldquoLoop based pseudo hammingvalues-1 Testing isomorphism and rating kinematic chainsrdquoMechanism andMachineTheory vol 28 no 1 pp 113ndash127 1992

[16] A C Rao and C N Rao ldquoLoop based pseudo hamming values-II inversions preferred frames and actuatorsrdquo Mechanism andMachine Theory vol 28 no 1 pp 129ndash143 1993

[17] J-K Chu and W-Q Cao ldquoIdentification of isomorphismamong kinematic chains and inversions using linkrsquos adjacent-chain-tablerdquoMechanism and Machine Theory vol 29 no 1 pp53ndash58 1994

[18] A C Rao ldquoApplication of fuzzy logic for the study of iso-morphism inversions symmetry parallelism and mobility inkinematic chainsrdquoMechanism and Machine Theory vol 35 no8 pp 1103ndash1116 2000

[19] P R He W J Zhang Q Li and F X Wu ldquoA new method fordetection of graph isomorphism based on the quadratic formrdquoJournal of Mechanical Design vol 125 no 3 pp 640ndash642 2003

[20] P R He W J Zhang and Q Li ldquoSome further developmenton the eigensystem approach for graph isomorphismdetectionrdquoJournal of the Franklin Institute vol 342 no 6 pp 657ndash6732005

[21] Z Y Chang C Zhang Y H Yang and YWang ldquoA newmethodto mechanism kinematic chain isomorphism identificationrdquoMechanism andMachineTheory vol 37 no 4 pp 411ndash417 2002

[22] J P Cubillo and J BWan ldquoComments onmechanismkinematicchain isomorphism identification using adjacent matricesrdquoMechanism andMachineTheory vol 40 no 2 pp 131ndash139 2005

[23] H F Ding and Z Huang ldquoA unique representation of the kine-matic chain and the atlas databaserdquo Mechanism and MachineTheory vol 42 no 6 pp 637ndash651 2007

[24] H F Ding and Z Huang ldquoA new theory for the topologicalstructure analysis of kinematic chains and its applicationsrdquoMechanism and Machine Theory vol 42 no 10 pp 1264ndash12792007

[25] A C Rao ldquoA genetic algorithm for topological characteristicsof kinematic chainsrdquo Journal of Mechanical Design vol 122 no2 pp 228ndash231 2000

[26] F G Kong Q Li and W J Zhang ldquoAn artificial neuralnetwork approach tomechanism kinematic chain isomorphismidentificationrdquo Mechanism and Machine Theory vol 34 no 2pp 271ndash283 1999

[27] J K Chu The Structural and Dimensional Characteristics ofPlanar Linkages Press of Beihang University of China BeijingChina 1992

[28] L Song J Yang X Zhang and W Q Cao ldquoSpanning treemethod of identifying isomorphism and topological symmetryto planar kinematic chain with multiple jointrdquo Chinese Journalof Mechanical Engineering vol 14 no 1 pp 27ndash31 2001

[29] J G Liu and D J Yu ldquoRepresentations amp isomorphismidentification of planar kinematic chains with multiple jointsbased on the converted adjacent matrixrdquo Chinese Journal ofMechanical Engineering vol 48 pp 15ndash21 2012

[30] R Ravisankar andT SMruthyunjaya ldquoComputerized synthesisof the structure of geared kinematic chainsrdquo Mechanism andMachine Theory vol 20 no 5 pp 367ndash387 1985

[31] J U Kim and B M Kwak ldquoApplication of edge permutationgroup to structural synthesis of epicyclic gear trainsrdquo Mecha-nism and Machine Theory vol 25 no 5 pp 563ndash573 1990

Mathematical Problems in Engineering 15

[32] G Chatterjee and L-W Tsai ldquoComputer-aided sketching ofepicyclic-type automatic transmission gear trainsrdquo Journal ofMechanical Design vol 118 no 3 pp 405ndash411 1996

[33] V V N R Prasadraju and A C Rao ldquoA new technique basedon loops to investigate displacement isomorphism in planetarygear trainsrdquo Journal of Mechanical Design vol 12 pp 666ndash6752002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

①

② ③ ④ ⑤

⑥⑦

A

B

b

a

C

c

(a)

①

②③

④

⑤

⑥ ⑦

A

Bb

a

C

c

(b)

①

②

③

④ ⑤

⑥

⑦

3

3

3

c

2

11

1

1

1

1

1

1

a

(c)

①

②

③

④

⑤

⑥

⑦

c

a

1

1

1

1

1

1

1

2

3

3

3

(d)

Figure 11 Two 6-gear train kinematic chains and their WDCCG

Step 2 Evidently 119860 = 1198601015840 Assign vertices codes of 119860 and 1198601015840

in two groups (a) [(119886 119888) (1 2 3 4 5 6 7)] and (b) [(119886 119888) (12 3 4 5 6 7)]

Step 3 Calculate NVS(119886) NVS(119887) and their vertex codes oftwo ECMs with the same CE respectively as follows

NVS (119886) = [(V119887 V119886) (V7 V5 V6 V1 V4 V3 V2)]= [(02591 02547)

(03784 02515 02511 02309 02122 02029 01961)]

NVS (119887) = [(V119887 V119886) (V7 V6 V5 V1 V4 V3 V2)]= [(03156 02607)

(03940 02601 02551 02394 02367 02110 02001)]

(20)

Compare NVS(119886) and NVS(119887) and equivalence cannot beseen So it is determined that they are not isomorphic andprogram stops

Example 4 Two 9-gear train kinematic chains are shown inFigures 12(a) and 12(b) Their corresponding WDCCGs aredepicted in Figures 12(c) and 12(d) respectively The directprocess to determine isomorphism of these two kinematicchains is as follows

Step 1 Input their correspondingWAMs119860 and1198601015840 as follows

119860 =

1 2 3 4 5 6 7 8 9 119886 119887 119888

1

2

3

4

5

6

7

8

9

119886

119887

119888

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

(1)

(0)

(3)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(3)

(1)

(3)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(0)

(3)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(3)

(1)

(1)

(1)

(0)

(0)

(0)

(0)

(1)

(3)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(3)

(1)

(3)

(2)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(3)

(1)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(2)

(0)

(1)

(0)

(1)

(1)

(0)

(0)

(0)

(0)

(3)

(0)

(0)

(0)

(0)

(1)

(0)

(1)

(0)

(0)

(0)

(1)

(0)

(1)

(0)

(1)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(1)

(1)

(0)

(0)

(0)

(1)

(1)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

12 Mathematical Problems in Engineering

① ②

③

a

A

c

B

d

C

D

E

b

④

⑤

⑥

⑦

⑧

⑨

(a)

①

②③a

A

b E

c

DC

d

B④

⑤

⑥ ⑦⑧⑨

(b)

①

1 1

c

1 1

1

1

1

b3

3

3

2

1

1

1

1a

33

②

③

④

⑤⑥

⑦

⑧

⑨

(c)

1

1

1

1

3

3

1

1

1

1

1

1

1

2

3

3

3

b

c

a

①②

③

④

⑤

⑥

⑦

⑧

⑨

(d)

Figure 12 Two 9-gear train kinematic chains and their WDCCGs

1198601015840=

1 2 3 4 5 6 7 8 9 119886 119887 119888

1

2

3

4

5

6

7

8

9

119886

119887

119888

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

(1)

(0)

(0)

(3)

(0)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(1)

(0)

(3)

(0)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(1)

(2)

(0)

(0)

(0)

(0)

(0)

(1)

(0)

(1)

(3)

(3)

(2)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(3)

(3)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(3)

(1)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(3)

(0)

(1)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(3)

(1)

(1)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(3)

(1)

(0)

(0)

(1)

(1)

(1)

(1)

(0)

(1)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(1)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(1)

(0)

(0)

(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(21)

Step 2 Evidently 119860 = 1198601015840 Assign vertices codes of 119860 and 1198601015840

in two groups (a) [(119886 119887 119888) (1 2 3 4 5 6 7 8 9)] and (b) [(119886119887 119888) (1 2 3 4 5 6 7 8 9)]

Step 3 Calculate NVS(119886) NVS(119887) and their vertex codes oftwo ECMs with the same CE respectively as follows

NVS (119886) = [(V119888 V119887 V119886) (V9 V4 V8 V1 V2 V5 V7 V3 V6)]

= [(02845 02815 02075) (02470 02307 02295 02294 02294 02154 02154 01846 01719)]

NVS (119887) = [(V119887 V119888 V119886) (V9 V8 V3 V6 V7 V1 V2 V5 V4)]

= [(02845 02815 02075) (02470 02307 02295 02294 02294 02154 02154 01846 01719)]

(22)

Mathematical Problems in Engineering 13

Compare NVS(119886) and NVS(119887) and equivalence can be seenAnd the node voltages in two vertex code groups are not alldistinct so go to Step 6

Step 4 The vertices (1 2) in Figure 12(a) vertices (6 7) inFigure 12(b) the vertices (5 7) in Figure 12(a) and vertices (12) in Figure 12(b) have the same values in the correspondinggroups so just adjust the position order of the vertices in 1198601015840to obtain a new adjacency matrix 1198601015840 And the correspondingvertices of two WDCCGs are (1-6 2-7 3-5 4-8 5-1 6-4 7-28-3 9-9 119886-119886 119887-119888 119888-119887) so we have 119860 = 119860

1015840

1 It is determined

that they are isomorphic and program stops

5 Algorithm Complexity Analysis

This paper presents a relatively good algorithm for determi-nation of kinematic chains isomorphism with multiple jointand gear train It is feasible for introducing the WDCCGand WAM to describe these two types of kinematic chainsthey dramatically reduce the number of vertices and order ofcorresponding adjacency matrix 119860

The analysis of computational complexity of this isomor-phic identification algorithm is as follows when there areno equivalent node voltages in NVS time consumption isbasically dedicated to solving the linear equation sets If akinematic chain possesses 119899 vertices the order of the nodevoltage equations is 119899 minus 1 Since the adjoint circuit is alinear resistance circuit the node voltage equations forma linear algebra equation set of orders 119899 minus 1 which has acomputational complexity of 119874((119899 minus 1)3) for its solutionSince it needs to solve at most 2119899 adjoint circuits in ourproposed method the computational complexity is of order119874(2119899(119899 minus 1)

3) lt 119874(119899

4) Thus the proposed method has a

computational complexity of a polynomial orderWhen thereare equivalent node voltages in NVS the correspondence ofonly a part of the vertices cannot be determinedThus in theworst situation the proposed method is more efficient thanthose where rowcolumn exchanges are performed blindly

In this paper this algorithm is tested and comparedwith the internationally recognized faster method for planarsimple joint kinematic chains isomorphic identification [20]and they are tested in the same software and hardwareenvironment The test results are shown in Figure 13 (theaverage time of each vertex is 20 times of the kinematic chainwithmultiple joints) test results show that when the vertex ofthe kinematic chain is increased the algorithm proposed inthis paper is more efficient than the algorithm proposed byHe et alThe proposed algorithm by us is very suitable for theisomorphism of the topological graph of kinematic chainsand the algorithm is very efficient at the same time

6 Conclusions

In this paper an algorithm for topological isomorphism iden-tification of planar multiple joint and gear train kinematicchains is introduced

4 6 8 10 12 14 16 18 20 22

The number of vertices

Algorithm proposed by He et al

Algorithm proposed in this paper

00001

0001

001

01

1

The d

ecisi

on ti

me (

s)

Figure 13 The test results of the algorithm proposed in this paperand the algorithm proposed by He et al

Firstly a weighted-double-color-contracted-graph(WDCCG) and a weighted adjacency matrix (WAM) areintroduced to describe the planarmultiple joint and gear trainkinematic chains respectively They dramatically reduce thenumber of vertices and order of the represented adjacencymatrix

Secondly isomorphism identification method of thesetwo types of kinematic chains is carried out by the equivalentcircuit models (ECMs) of WDCCGs with the same completeexcitation (CE) and then uses the solved node voltagesequence (NVS) to determine the corresponding vertices oftwo WDCCGs

Finally an algorithm to identify isomorphic kinematicchains is obtained It is an efficient and easy method tobe realized by computer And this algorithm also can beused in the isomorphism identification of planar simple jointkinematic chains with minor changes

Notations

119860 1198601015840 The incidence matrix of kinematic chain119889119894119895 The elements of incidence matrix

119866119894 Conductance value of edge in the ECM

119898119894119895119896 The weighted value of 119896th edge

1198731198731015840 Variable of equivalent circuit model119899 The number of vertices in WDCCG119899119894 Node 119894 in the ECM

V119894 Node voltages of the node 119894

CE Complete excitationDCG Double color graphECM Equivalent circuit modelNVS Node voltage sequenceTM Topological modelWAM Weighted adjacency matrixWDCCG Weighted double color contracted graph

14 Mathematical Problems in Engineering

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

Thiswork is supported by the Science andTechnology Projectof Quanzhou City 2014G48 and G20140047

References

[1] J J Uicker Jr and A Raicu ldquoAmethod for the identification andrecognition of equivalence of kinematic chainsrdquoMechanismandMachine Theory vol 10 no 5 pp 375ndash383 1975

[2] H S Yan and A S Hall ldquoLinkage characteristic polynomialsdefinitions coefficients by inspectionrdquo Journal of MechanicalDesign vol 103 no 3 pp 578ndash584 1981

[3] H S Yan and A S Hall ldquoLinkage characteristic polynomialsassembly theorems uniquenessrdquo Journal of Mechanical Designvol 104 no 1 pp 11ndash20 1982

[4] T S Mruthyunjaya ldquoA computerized methodology for struc-tural synthesis of kinematic chains Part 1 FormulationrdquoMech-anism and Machine Theory vol 19 no 6 pp 487ndash495 1984

[5] T S Mruthyunjaya ldquoA computerized methodology for struc-tural synthesis of kinematic chains part 2mdashapplication toseveral fully or partially known casesrdquoMechanism andMachineTheory vol 19 no 6 pp 497ndash505 1984

[6] T S Mruthyunjaya ldquoA computerized methodology for struc-tural synthesis of kinematic chains part 3mdashapplication to thenew case of 10-link three- freedom chainsrdquo Mechanism andMachine Theory vol 19 no 6 pp 507ndash530 1984

[7] T S Mruthyunjaya and H R Balasubramanian ldquoIn questof a reliable and efficient computational test for detection ofisomorphism in kinematic chainsrdquo Mechanism and MachineTheory vol 22 no 2 pp 131ndash139 1987

[8] W J Sohn and F Freudenstein ldquoAn application of dual graphs tothe automatic generation of the kinematic structures of mecha-nismsrdquo Journal of Mechanisms Transmissions and Automationvol 108 no 3 pp 392ndash398 1986

[9] A Ambekar and V Agrawal ldquoOn canonical numbering ofkinematic chains and isomorphism problem MAX coderdquo inProceedings of the ASMEMechanisms Conference pp 1615ndash16221986

[10] A G Ambekar and V P Agrawal ldquoCanonical numberingof kinematic chains and isomorphism problem min coderdquoMechanism andMachineTheory vol 22 no 5 pp 453ndash461 1987

[11] C S Tang and T Liu ldquoDegree code a new mechanismidentifierrdquo Journal of Mechanical Design vol 115 no 3 pp 627ndash630 1993

[12] J K Shin and S Krishnamurty ldquoOn identification and canonicalnumbering of pin jointed kinematic chainsrdquo Journal of Mechan-ical Design vol 116 pp 182ndash188 1994

[13] J K Shin and S Krishnamurty ldquoDevelopment of a standardcode for colored graphs and its application to kinematic chainsrdquoJournal of Mechanical Design vol 116 no 1 pp 189ndash196 1994

[14] A C Rao and D Varada Raju ldquoApplication of the hammingnumber technique to detect isomorphism among kinematicchains and inversionsrdquoMechanism andMachineTheory vol 26no 1 pp 55ndash75 1991

[15] A C Rao and C N Rao ldquoLoop based pseudo hammingvalues-1 Testing isomorphism and rating kinematic chainsrdquoMechanism andMachineTheory vol 28 no 1 pp 113ndash127 1992

[16] A C Rao and C N Rao ldquoLoop based pseudo hamming values-II inversions preferred frames and actuatorsrdquo Mechanism andMachine Theory vol 28 no 1 pp 129ndash143 1993

[17] J-K Chu and W-Q Cao ldquoIdentification of isomorphismamong kinematic chains and inversions using linkrsquos adjacent-chain-tablerdquoMechanism and Machine Theory vol 29 no 1 pp53ndash58 1994

[18] A C Rao ldquoApplication of fuzzy logic for the study of iso-morphism inversions symmetry parallelism and mobility inkinematic chainsrdquoMechanism and Machine Theory vol 35 no8 pp 1103ndash1116 2000

[19] P R He W J Zhang Q Li and F X Wu ldquoA new method fordetection of graph isomorphism based on the quadratic formrdquoJournal of Mechanical Design vol 125 no 3 pp 640ndash642 2003

[20] P R He W J Zhang and Q Li ldquoSome further developmenton the eigensystem approach for graph isomorphismdetectionrdquoJournal of the Franklin Institute vol 342 no 6 pp 657ndash6732005

[21] Z Y Chang C Zhang Y H Yang and YWang ldquoA newmethodto mechanism kinematic chain isomorphism identificationrdquoMechanism andMachineTheory vol 37 no 4 pp 411ndash417 2002

[22] J P Cubillo and J BWan ldquoComments onmechanismkinematicchain isomorphism identification using adjacent matricesrdquoMechanism andMachineTheory vol 40 no 2 pp 131ndash139 2005

[23] H F Ding and Z Huang ldquoA unique representation of the kine-matic chain and the atlas databaserdquo Mechanism and MachineTheory vol 42 no 6 pp 637ndash651 2007

[24] H F Ding and Z Huang ldquoA new theory for the topologicalstructure analysis of kinematic chains and its applicationsrdquoMechanism and Machine Theory vol 42 no 10 pp 1264ndash12792007

[25] A C Rao ldquoA genetic algorithm for topological characteristicsof kinematic chainsrdquo Journal of Mechanical Design vol 122 no2 pp 228ndash231 2000

[26] F G Kong Q Li and W J Zhang ldquoAn artificial neuralnetwork approach tomechanism kinematic chain isomorphismidentificationrdquo Mechanism and Machine Theory vol 34 no 2pp 271ndash283 1999

[27] J K Chu The Structural and Dimensional Characteristics ofPlanar Linkages Press of Beihang University of China BeijingChina 1992

[28] L Song J Yang X Zhang and W Q Cao ldquoSpanning treemethod of identifying isomorphism and topological symmetryto planar kinematic chain with multiple jointrdquo Chinese Journalof Mechanical Engineering vol 14 no 1 pp 27ndash31 2001

[29] J G Liu and D J Yu ldquoRepresentations amp isomorphismidentification of planar kinematic chains with multiple jointsbased on the converted adjacent matrixrdquo Chinese Journal ofMechanical Engineering vol 48 pp 15ndash21 2012

[30] R Ravisankar andT SMruthyunjaya ldquoComputerized synthesisof the structure of geared kinematic chainsrdquo Mechanism andMachine Theory vol 20 no 5 pp 367ndash387 1985

[31] J U Kim and B M Kwak ldquoApplication of edge permutationgroup to structural synthesis of epicyclic gear trainsrdquo Mecha-nism and Machine Theory vol 25 no 5 pp 563ndash573 1990

Mathematical Problems in Engineering 15

[32] G Chatterjee and L-W Tsai ldquoComputer-aided sketching ofepicyclic-type automatic transmission gear trainsrdquo Journal ofMechanical Design vol 118 no 3 pp 405ndash411 1996

[33] V V N R Prasadraju and A C Rao ldquoA new technique basedon loops to investigate displacement isomorphism in planetarygear trainsrdquo Journal of Mechanical Design vol 12 pp 666ndash6752002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Mathematical Problems in Engineering

① ②

③

a

A

c

B

d

C

D

E

b

④

⑤

⑥

⑦

⑧

⑨

(a)

①

②③a

A

b E

c

DC

d

B④

⑤

⑥ ⑦⑧⑨

(b)

①

1 1

c

1 1

1

1

1

b3

3

3

2

1

1

1

1a

33

②

③

④

⑤⑥

⑦

⑧

⑨

(c)

1

1

1

1

3

3

1

1

1

1

1

1

1

2

3

3

3

b

c

a

①②

③

④

⑤

⑥

⑦

⑧

⑨

(d)

Figure 12 Two 9-gear train kinematic chains and their WDCCGs

1198601015840=

1 2 3 4 5 6 7 8 9 119886 119887 119888

1

2

3

4

5

6

7

8

9

119886

119887

119888

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

(1)

(0)

(0)

(3)

(0)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(1)

(0)

(3)

(0)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(1)

(2)

(0)

(0)

(0)

(0)

(0)

(1)

(0)

(1)

(3)

(3)

(2)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(3)

(3)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(3)

(1)

(0)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(3)

(0)

(1)

(0)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(3)

(1)

(1)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(3)

(1)

(0)

(0)

(1)

(1)

(1)

(1)

(0)

(1)

(0)

(0)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(1)

(1)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(0)

(1)

(1)

(0)

(0)

(0)

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(21)

Step 2 Evidently 119860 = 1198601015840 Assign vertices codes of 119860 and 1198601015840

in two groups (a) [(119886 119887 119888) (1 2 3 4 5 6 7 8 9)] and (b) [(119886119887 119888) (1 2 3 4 5 6 7 8 9)]

Step 3 Calculate NVS(119886) NVS(119887) and their vertex codes oftwo ECMs with the same CE respectively as follows

NVS (119886) = [(V119888 V119887 V119886) (V9 V4 V8 V1 V2 V5 V7 V3 V6)]

= [(02845 02815 02075) (02470 02307 02295 02294 02294 02154 02154 01846 01719)]

NVS (119887) = [(V119887 V119888 V119886) (V9 V8 V3 V6 V7 V1 V2 V5 V4)]

= [(02845 02815 02075) (02470 02307 02295 02294 02294 02154 02154 01846 01719)]

(22)

Mathematical Problems in Engineering 13

Compare NVS(119886) and NVS(119887) and equivalence can be seenAnd the node voltages in two vertex code groups are not alldistinct so go to Step 6

Step 4 The vertices (1 2) in Figure 12(a) vertices (6 7) inFigure 12(b) the vertices (5 7) in Figure 12(a) and vertices (12) in Figure 12(b) have the same values in the correspondinggroups so just adjust the position order of the vertices in 1198601015840to obtain a new adjacency matrix 1198601015840 And the correspondingvertices of two WDCCGs are (1-6 2-7 3-5 4-8 5-1 6-4 7-28-3 9-9 119886-119886 119887-119888 119888-119887) so we have 119860 = 119860

1015840

1 It is determined

that they are isomorphic and program stops

5 Algorithm Complexity Analysis

This paper presents a relatively good algorithm for determi-nation of kinematic chains isomorphism with multiple jointand gear train It is feasible for introducing the WDCCGand WAM to describe these two types of kinematic chainsthey dramatically reduce the number of vertices and order ofcorresponding adjacency matrix 119860

The analysis of computational complexity of this isomor-phic identification algorithm is as follows when there areno equivalent node voltages in NVS time consumption isbasically dedicated to solving the linear equation sets If akinematic chain possesses 119899 vertices the order of the nodevoltage equations is 119899 minus 1 Since the adjoint circuit is alinear resistance circuit the node voltage equations forma linear algebra equation set of orders 119899 minus 1 which has acomputational complexity of 119874((119899 minus 1)3) for its solutionSince it needs to solve at most 2119899 adjoint circuits in ourproposed method the computational complexity is of order119874(2119899(119899 minus 1)

3) lt 119874(119899

4) Thus the proposed method has a

computational complexity of a polynomial orderWhen thereare equivalent node voltages in NVS the correspondence ofonly a part of the vertices cannot be determinedThus in theworst situation the proposed method is more efficient thanthose where rowcolumn exchanges are performed blindly

In this paper this algorithm is tested and comparedwith the internationally recognized faster method for planarsimple joint kinematic chains isomorphic identification [20]and they are tested in the same software and hardwareenvironment The test results are shown in Figure 13 (theaverage time of each vertex is 20 times of the kinematic chainwithmultiple joints) test results show that when the vertex ofthe kinematic chain is increased the algorithm proposed inthis paper is more efficient than the algorithm proposed byHe et alThe proposed algorithm by us is very suitable for theisomorphism of the topological graph of kinematic chainsand the algorithm is very efficient at the same time

6 Conclusions

In this paper an algorithm for topological isomorphism iden-tification of planar multiple joint and gear train kinematicchains is introduced

4 6 8 10 12 14 16 18 20 22

The number of vertices

Algorithm proposed by He et al

Algorithm proposed in this paper

00001

0001

001

01

1

The d

ecisi

on ti

me (

s)

Figure 13 The test results of the algorithm proposed in this paperand the algorithm proposed by He et al

Firstly a weighted-double-color-contracted-graph(WDCCG) and a weighted adjacency matrix (WAM) areintroduced to describe the planarmultiple joint and gear trainkinematic chains respectively They dramatically reduce thenumber of vertices and order of the represented adjacencymatrix

Secondly isomorphism identification method of thesetwo types of kinematic chains is carried out by the equivalentcircuit models (ECMs) of WDCCGs with the same completeexcitation (CE) and then uses the solved node voltagesequence (NVS) to determine the corresponding vertices oftwo WDCCGs

Finally an algorithm to identify isomorphic kinematicchains is obtained It is an efficient and easy method tobe realized by computer And this algorithm also can beused in the isomorphism identification of planar simple jointkinematic chains with minor changes

Notations

119860 1198601015840 The incidence matrix of kinematic chain119889119894119895 The elements of incidence matrix

119866119894 Conductance value of edge in the ECM

119898119894119895119896 The weighted value of 119896th edge

1198731198731015840 Variable of equivalent circuit model119899 The number of vertices in WDCCG119899119894 Node 119894 in the ECM

V119894 Node voltages of the node 119894

CE Complete excitationDCG Double color graphECM Equivalent circuit modelNVS Node voltage sequenceTM Topological modelWAM Weighted adjacency matrixWDCCG Weighted double color contracted graph

14 Mathematical Problems in Engineering

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

Thiswork is supported by the Science andTechnology Projectof Quanzhou City 2014G48 and G20140047

References

[1] J J Uicker Jr and A Raicu ldquoAmethod for the identification andrecognition of equivalence of kinematic chainsrdquoMechanismandMachine Theory vol 10 no 5 pp 375ndash383 1975

[2] H S Yan and A S Hall ldquoLinkage characteristic polynomialsdefinitions coefficients by inspectionrdquo Journal of MechanicalDesign vol 103 no 3 pp 578ndash584 1981

[3] H S Yan and A S Hall ldquoLinkage characteristic polynomialsassembly theorems uniquenessrdquo Journal of Mechanical Designvol 104 no 1 pp 11ndash20 1982

[4] T S Mruthyunjaya ldquoA computerized methodology for struc-tural synthesis of kinematic chains Part 1 FormulationrdquoMech-anism and Machine Theory vol 19 no 6 pp 487ndash495 1984

[5] T S Mruthyunjaya ldquoA computerized methodology for struc-tural synthesis of kinematic chains part 2mdashapplication toseveral fully or partially known casesrdquoMechanism andMachineTheory vol 19 no 6 pp 497ndash505 1984

[6] T S Mruthyunjaya ldquoA computerized methodology for struc-tural synthesis of kinematic chains part 3mdashapplication to thenew case of 10-link three- freedom chainsrdquo Mechanism andMachine Theory vol 19 no 6 pp 507ndash530 1984

[7] T S Mruthyunjaya and H R Balasubramanian ldquoIn questof a reliable and efficient computational test for detection ofisomorphism in kinematic chainsrdquo Mechanism and MachineTheory vol 22 no 2 pp 131ndash139 1987

[8] W J Sohn and F Freudenstein ldquoAn application of dual graphs tothe automatic generation of the kinematic structures of mecha-nismsrdquo Journal of Mechanisms Transmissions and Automationvol 108 no 3 pp 392ndash398 1986

[9] A Ambekar and V Agrawal ldquoOn canonical numbering ofkinematic chains and isomorphism problem MAX coderdquo inProceedings of the ASMEMechanisms Conference pp 1615ndash16221986

[10] A G Ambekar and V P Agrawal ldquoCanonical numberingof kinematic chains and isomorphism problem min coderdquoMechanism andMachineTheory vol 22 no 5 pp 453ndash461 1987

[11] C S Tang and T Liu ldquoDegree code a new mechanismidentifierrdquo Journal of Mechanical Design vol 115 no 3 pp 627ndash630 1993

[12] J K Shin and S Krishnamurty ldquoOn identification and canonicalnumbering of pin jointed kinematic chainsrdquo Journal of Mechan-ical Design vol 116 pp 182ndash188 1994

[13] J K Shin and S Krishnamurty ldquoDevelopment of a standardcode for colored graphs and its application to kinematic chainsrdquoJournal of Mechanical Design vol 116 no 1 pp 189ndash196 1994

[14] A C Rao and D Varada Raju ldquoApplication of the hammingnumber technique to detect isomorphism among kinematicchains and inversionsrdquoMechanism andMachineTheory vol 26no 1 pp 55ndash75 1991

[15] A C Rao and C N Rao ldquoLoop based pseudo hammingvalues-1 Testing isomorphism and rating kinematic chainsrdquoMechanism andMachineTheory vol 28 no 1 pp 113ndash127 1992

[16] A C Rao and C N Rao ldquoLoop based pseudo hamming values-II inversions preferred frames and actuatorsrdquo Mechanism andMachine Theory vol 28 no 1 pp 129ndash143 1993

[17] J-K Chu and W-Q Cao ldquoIdentification of isomorphismamong kinematic chains and inversions using linkrsquos adjacent-chain-tablerdquoMechanism and Machine Theory vol 29 no 1 pp53ndash58 1994

[18] A C Rao ldquoApplication of fuzzy logic for the study of iso-morphism inversions symmetry parallelism and mobility inkinematic chainsrdquoMechanism and Machine Theory vol 35 no8 pp 1103ndash1116 2000

[19] P R He W J Zhang Q Li and F X Wu ldquoA new method fordetection of graph isomorphism based on the quadratic formrdquoJournal of Mechanical Design vol 125 no 3 pp 640ndash642 2003

[20] P R He W J Zhang and Q Li ldquoSome further developmenton the eigensystem approach for graph isomorphismdetectionrdquoJournal of the Franklin Institute vol 342 no 6 pp 657ndash6732005

[21] Z Y Chang C Zhang Y H Yang and YWang ldquoA newmethodto mechanism kinematic chain isomorphism identificationrdquoMechanism andMachineTheory vol 37 no 4 pp 411ndash417 2002

[22] J P Cubillo and J BWan ldquoComments onmechanismkinematicchain isomorphism identification using adjacent matricesrdquoMechanism andMachineTheory vol 40 no 2 pp 131ndash139 2005

[23] H F Ding and Z Huang ldquoA unique representation of the kine-matic chain and the atlas databaserdquo Mechanism and MachineTheory vol 42 no 6 pp 637ndash651 2007

[24] H F Ding and Z Huang ldquoA new theory for the topologicalstructure analysis of kinematic chains and its applicationsrdquoMechanism and Machine Theory vol 42 no 10 pp 1264ndash12792007

[25] A C Rao ldquoA genetic algorithm for topological characteristicsof kinematic chainsrdquo Journal of Mechanical Design vol 122 no2 pp 228ndash231 2000

[26] F G Kong Q Li and W J Zhang ldquoAn artificial neuralnetwork approach tomechanism kinematic chain isomorphismidentificationrdquo Mechanism and Machine Theory vol 34 no 2pp 271ndash283 1999

[27] J K Chu The Structural and Dimensional Characteristics ofPlanar Linkages Press of Beihang University of China BeijingChina 1992

[28] L Song J Yang X Zhang and W Q Cao ldquoSpanning treemethod of identifying isomorphism and topological symmetryto planar kinematic chain with multiple jointrdquo Chinese Journalof Mechanical Engineering vol 14 no 1 pp 27ndash31 2001

[29] J G Liu and D J Yu ldquoRepresentations amp isomorphismidentification of planar kinematic chains with multiple jointsbased on the converted adjacent matrixrdquo Chinese Journal ofMechanical Engineering vol 48 pp 15ndash21 2012

[30] R Ravisankar andT SMruthyunjaya ldquoComputerized synthesisof the structure of geared kinematic chainsrdquo Mechanism andMachine Theory vol 20 no 5 pp 367ndash387 1985

[31] J U Kim and B M Kwak ldquoApplication of edge permutationgroup to structural synthesis of epicyclic gear trainsrdquo Mecha-nism and Machine Theory vol 25 no 5 pp 563ndash573 1990

Mathematical Problems in Engineering 15

[32] G Chatterjee and L-W Tsai ldquoComputer-aided sketching ofepicyclic-type automatic transmission gear trainsrdquo Journal ofMechanical Design vol 118 no 3 pp 405ndash411 1996

[33] V V N R Prasadraju and A C Rao ldquoA new technique basedon loops to investigate displacement isomorphism in planetarygear trainsrdquo Journal of Mechanical Design vol 12 pp 666ndash6752002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 13

Compare NVS(119886) and NVS(119887) and equivalence can be seenAnd the node voltages in two vertex code groups are not alldistinct so go to Step 6

Step 4 The vertices (1 2) in Figure 12(a) vertices (6 7) inFigure 12(b) the vertices (5 7) in Figure 12(a) and vertices (12) in Figure 12(b) have the same values in the correspondinggroups so just adjust the position order of the vertices in 1198601015840to obtain a new adjacency matrix 1198601015840 And the correspondingvertices of two WDCCGs are (1-6 2-7 3-5 4-8 5-1 6-4 7-28-3 9-9 119886-119886 119887-119888 119888-119887) so we have 119860 = 119860

1015840

1 It is determined

that they are isomorphic and program stops

5 Algorithm Complexity Analysis

This paper presents a relatively good algorithm for determi-nation of kinematic chains isomorphism with multiple jointand gear train It is feasible for introducing the WDCCGand WAM to describe these two types of kinematic chainsthey dramatically reduce the number of vertices and order ofcorresponding adjacency matrix 119860

The analysis of computational complexity of this isomor-phic identification algorithm is as follows when there areno equivalent node voltages in NVS time consumption isbasically dedicated to solving the linear equation sets If akinematic chain possesses 119899 vertices the order of the nodevoltage equations is 119899 minus 1 Since the adjoint circuit is alinear resistance circuit the node voltage equations forma linear algebra equation set of orders 119899 minus 1 which has acomputational complexity of 119874((119899 minus 1)3) for its solutionSince it needs to solve at most 2119899 adjoint circuits in ourproposed method the computational complexity is of order119874(2119899(119899 minus 1)

3) lt 119874(119899

4) Thus the proposed method has a

computational complexity of a polynomial orderWhen thereare equivalent node voltages in NVS the correspondence ofonly a part of the vertices cannot be determinedThus in theworst situation the proposed method is more efficient thanthose where rowcolumn exchanges are performed blindly

In this paper this algorithm is tested and comparedwith the internationally recognized faster method for planarsimple joint kinematic chains isomorphic identification [20]and they are tested in the same software and hardwareenvironment The test results are shown in Figure 13 (theaverage time of each vertex is 20 times of the kinematic chainwithmultiple joints) test results show that when the vertex ofthe kinematic chain is increased the algorithm proposed inthis paper is more efficient than the algorithm proposed byHe et alThe proposed algorithm by us is very suitable for theisomorphism of the topological graph of kinematic chainsand the algorithm is very efficient at the same time

6 Conclusions

In this paper an algorithm for topological isomorphism iden-tification of planar multiple joint and gear train kinematicchains is introduced

4 6 8 10 12 14 16 18 20 22

The number of vertices

Algorithm proposed by He et al

Algorithm proposed in this paper

00001

0001

001

01

1

The d

ecisi

on ti

me (

s)

Figure 13 The test results of the algorithm proposed in this paperand the algorithm proposed by He et al

Firstly a weighted-double-color-contracted-graph(WDCCG) and a weighted adjacency matrix (WAM) areintroduced to describe the planarmultiple joint and gear trainkinematic chains respectively They dramatically reduce thenumber of vertices and order of the represented adjacencymatrix

Secondly isomorphism identification method of thesetwo types of kinematic chains is carried out by the equivalentcircuit models (ECMs) of WDCCGs with the same completeexcitation (CE) and then uses the solved node voltagesequence (NVS) to determine the corresponding vertices oftwo WDCCGs

Finally an algorithm to identify isomorphic kinematicchains is obtained It is an efficient and easy method tobe realized by computer And this algorithm also can beused in the isomorphism identification of planar simple jointkinematic chains with minor changes

Notations

119860 1198601015840 The incidence matrix of kinematic chain119889119894119895 The elements of incidence matrix

119866119894 Conductance value of edge in the ECM

119898119894119895119896 The weighted value of 119896th edge

1198731198731015840 Variable of equivalent circuit model119899 The number of vertices in WDCCG119899119894 Node 119894 in the ECM

V119894 Node voltages of the node 119894

CE Complete excitationDCG Double color graphECM Equivalent circuit modelNVS Node voltage sequenceTM Topological modelWAM Weighted adjacency matrixWDCCG Weighted double color contracted graph

14 Mathematical Problems in Engineering

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

Thiswork is supported by the Science andTechnology Projectof Quanzhou City 2014G48 and G20140047

References

[1] J J Uicker Jr and A Raicu ldquoAmethod for the identification andrecognition of equivalence of kinematic chainsrdquoMechanismandMachine Theory vol 10 no 5 pp 375ndash383 1975

[2] H S Yan and A S Hall ldquoLinkage characteristic polynomialsdefinitions coefficients by inspectionrdquo Journal of MechanicalDesign vol 103 no 3 pp 578ndash584 1981

[3] H S Yan and A S Hall ldquoLinkage characteristic polynomialsassembly theorems uniquenessrdquo Journal of Mechanical Designvol 104 no 1 pp 11ndash20 1982

[4] T S Mruthyunjaya ldquoA computerized methodology for struc-tural synthesis of kinematic chains Part 1 FormulationrdquoMech-anism and Machine Theory vol 19 no 6 pp 487ndash495 1984

[5] T S Mruthyunjaya ldquoA computerized methodology for struc-tural synthesis of kinematic chains part 2mdashapplication toseveral fully or partially known casesrdquoMechanism andMachineTheory vol 19 no 6 pp 497ndash505 1984

[6] T S Mruthyunjaya ldquoA computerized methodology for struc-tural synthesis of kinematic chains part 3mdashapplication to thenew case of 10-link three- freedom chainsrdquo Mechanism andMachine Theory vol 19 no 6 pp 507ndash530 1984

[7] T S Mruthyunjaya and H R Balasubramanian ldquoIn questof a reliable and efficient computational test for detection ofisomorphism in kinematic chainsrdquo Mechanism and MachineTheory vol 22 no 2 pp 131ndash139 1987

[8] W J Sohn and F Freudenstein ldquoAn application of dual graphs tothe automatic generation of the kinematic structures of mecha-nismsrdquo Journal of Mechanisms Transmissions and Automationvol 108 no 3 pp 392ndash398 1986

[9] A Ambekar and V Agrawal ldquoOn canonical numbering ofkinematic chains and isomorphism problem MAX coderdquo inProceedings of the ASMEMechanisms Conference pp 1615ndash16221986

[10] A G Ambekar and V P Agrawal ldquoCanonical numberingof kinematic chains and isomorphism problem min coderdquoMechanism andMachineTheory vol 22 no 5 pp 453ndash461 1987

[11] C S Tang and T Liu ldquoDegree code a new mechanismidentifierrdquo Journal of Mechanical Design vol 115 no 3 pp 627ndash630 1993

[12] J K Shin and S Krishnamurty ldquoOn identification and canonicalnumbering of pin jointed kinematic chainsrdquo Journal of Mechan-ical Design vol 116 pp 182ndash188 1994

[13] J K Shin and S Krishnamurty ldquoDevelopment of a standardcode for colored graphs and its application to kinematic chainsrdquoJournal of Mechanical Design vol 116 no 1 pp 189ndash196 1994

[14] A C Rao and D Varada Raju ldquoApplication of the hammingnumber technique to detect isomorphism among kinematicchains and inversionsrdquoMechanism andMachineTheory vol 26no 1 pp 55ndash75 1991

[15] A C Rao and C N Rao ldquoLoop based pseudo hammingvalues-1 Testing isomorphism and rating kinematic chainsrdquoMechanism andMachineTheory vol 28 no 1 pp 113ndash127 1992

[16] A C Rao and C N Rao ldquoLoop based pseudo hamming values-II inversions preferred frames and actuatorsrdquo Mechanism andMachine Theory vol 28 no 1 pp 129ndash143 1993

[17] J-K Chu and W-Q Cao ldquoIdentification of isomorphismamong kinematic chains and inversions using linkrsquos adjacent-chain-tablerdquoMechanism and Machine Theory vol 29 no 1 pp53ndash58 1994

[18] A C Rao ldquoApplication of fuzzy logic for the study of iso-morphism inversions symmetry parallelism and mobility inkinematic chainsrdquoMechanism and Machine Theory vol 35 no8 pp 1103ndash1116 2000

[19] P R He W J Zhang Q Li and F X Wu ldquoA new method fordetection of graph isomorphism based on the quadratic formrdquoJournal of Mechanical Design vol 125 no 3 pp 640ndash642 2003

[20] P R He W J Zhang and Q Li ldquoSome further developmenton the eigensystem approach for graph isomorphismdetectionrdquoJournal of the Franklin Institute vol 342 no 6 pp 657ndash6732005

[21] Z Y Chang C Zhang Y H Yang and YWang ldquoA newmethodto mechanism kinematic chain isomorphism identificationrdquoMechanism andMachineTheory vol 37 no 4 pp 411ndash417 2002

[22] J P Cubillo and J BWan ldquoComments onmechanismkinematicchain isomorphism identification using adjacent matricesrdquoMechanism andMachineTheory vol 40 no 2 pp 131ndash139 2005

[23] H F Ding and Z Huang ldquoA unique representation of the kine-matic chain and the atlas databaserdquo Mechanism and MachineTheory vol 42 no 6 pp 637ndash651 2007

[24] H F Ding and Z Huang ldquoA new theory for the topologicalstructure analysis of kinematic chains and its applicationsrdquoMechanism and Machine Theory vol 42 no 10 pp 1264ndash12792007

[25] A C Rao ldquoA genetic algorithm for topological characteristicsof kinematic chainsrdquo Journal of Mechanical Design vol 122 no2 pp 228ndash231 2000

[26] F G Kong Q Li and W J Zhang ldquoAn artificial neuralnetwork approach tomechanism kinematic chain isomorphismidentificationrdquo Mechanism and Machine Theory vol 34 no 2pp 271ndash283 1999

[27] J K Chu The Structural and Dimensional Characteristics ofPlanar Linkages Press of Beihang University of China BeijingChina 1992

[28] L Song J Yang X Zhang and W Q Cao ldquoSpanning treemethod of identifying isomorphism and topological symmetryto planar kinematic chain with multiple jointrdquo Chinese Journalof Mechanical Engineering vol 14 no 1 pp 27ndash31 2001

[29] J G Liu and D J Yu ldquoRepresentations amp isomorphismidentification of planar kinematic chains with multiple jointsbased on the converted adjacent matrixrdquo Chinese Journal ofMechanical Engineering vol 48 pp 15ndash21 2012

[30] R Ravisankar andT SMruthyunjaya ldquoComputerized synthesisof the structure of geared kinematic chainsrdquo Mechanism andMachine Theory vol 20 no 5 pp 367ndash387 1985

[31] J U Kim and B M Kwak ldquoApplication of edge permutationgroup to structural synthesis of epicyclic gear trainsrdquo Mecha-nism and Machine Theory vol 25 no 5 pp 563ndash573 1990

Mathematical Problems in Engineering 15

[32] G Chatterjee and L-W Tsai ldquoComputer-aided sketching ofepicyclic-type automatic transmission gear trainsrdquo Journal ofMechanical Design vol 118 no 3 pp 405ndash411 1996

[33] V V N R Prasadraju and A C Rao ldquoA new technique basedon loops to investigate displacement isomorphism in planetarygear trainsrdquo Journal of Mechanical Design vol 12 pp 666ndash6752002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

14 Mathematical Problems in Engineering

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

Thiswork is supported by the Science andTechnology Projectof Quanzhou City 2014G48 and G20140047

References

[1] J J Uicker Jr and A Raicu ldquoAmethod for the identification andrecognition of equivalence of kinematic chainsrdquoMechanismandMachine Theory vol 10 no 5 pp 375ndash383 1975

[2] H S Yan and A S Hall ldquoLinkage characteristic polynomialsdefinitions coefficients by inspectionrdquo Journal of MechanicalDesign vol 103 no 3 pp 578ndash584 1981

[3] H S Yan and A S Hall ldquoLinkage characteristic polynomialsassembly theorems uniquenessrdquo Journal of Mechanical Designvol 104 no 1 pp 11ndash20 1982

[4] T S Mruthyunjaya ldquoA computerized methodology for struc-tural synthesis of kinematic chains Part 1 FormulationrdquoMech-anism and Machine Theory vol 19 no 6 pp 487ndash495 1984

[5] T S Mruthyunjaya ldquoA computerized methodology for struc-tural synthesis of kinematic chains part 2mdashapplication toseveral fully or partially known casesrdquoMechanism andMachineTheory vol 19 no 6 pp 497ndash505 1984

[6] T S Mruthyunjaya ldquoA computerized methodology for struc-tural synthesis of kinematic chains part 3mdashapplication to thenew case of 10-link three- freedom chainsrdquo Mechanism andMachine Theory vol 19 no 6 pp 507ndash530 1984

[7] T S Mruthyunjaya and H R Balasubramanian ldquoIn questof a reliable and efficient computational test for detection ofisomorphism in kinematic chainsrdquo Mechanism and MachineTheory vol 22 no 2 pp 131ndash139 1987

[8] W J Sohn and F Freudenstein ldquoAn application of dual graphs tothe automatic generation of the kinematic structures of mecha-nismsrdquo Journal of Mechanisms Transmissions and Automationvol 108 no 3 pp 392ndash398 1986

[9] A Ambekar and V Agrawal ldquoOn canonical numbering ofkinematic chains and isomorphism problem MAX coderdquo inProceedings of the ASMEMechanisms Conference pp 1615ndash16221986

[10] A G Ambekar and V P Agrawal ldquoCanonical numberingof kinematic chains and isomorphism problem min coderdquoMechanism andMachineTheory vol 22 no 5 pp 453ndash461 1987

[11] C S Tang and T Liu ldquoDegree code a new mechanismidentifierrdquo Journal of Mechanical Design vol 115 no 3 pp 627ndash630 1993

[12] J K Shin and S Krishnamurty ldquoOn identification and canonicalnumbering of pin jointed kinematic chainsrdquo Journal of Mechan-ical Design vol 116 pp 182ndash188 1994

[13] J K Shin and S Krishnamurty ldquoDevelopment of a standardcode for colored graphs and its application to kinematic chainsrdquoJournal of Mechanical Design vol 116 no 1 pp 189ndash196 1994

[14] A C Rao and D Varada Raju ldquoApplication of the hammingnumber technique to detect isomorphism among kinematicchains and inversionsrdquoMechanism andMachineTheory vol 26no 1 pp 55ndash75 1991

[15] A C Rao and C N Rao ldquoLoop based pseudo hammingvalues-1 Testing isomorphism and rating kinematic chainsrdquoMechanism andMachineTheory vol 28 no 1 pp 113ndash127 1992

[16] A C Rao and C N Rao ldquoLoop based pseudo hamming values-II inversions preferred frames and actuatorsrdquo Mechanism andMachine Theory vol 28 no 1 pp 129ndash143 1993

[17] J-K Chu and W-Q Cao ldquoIdentification of isomorphismamong kinematic chains and inversions using linkrsquos adjacent-chain-tablerdquoMechanism and Machine Theory vol 29 no 1 pp53ndash58 1994

[18] A C Rao ldquoApplication of fuzzy logic for the study of iso-morphism inversions symmetry parallelism and mobility inkinematic chainsrdquoMechanism and Machine Theory vol 35 no8 pp 1103ndash1116 2000

[19] P R He W J Zhang Q Li and F X Wu ldquoA new method fordetection of graph isomorphism based on the quadratic formrdquoJournal of Mechanical Design vol 125 no 3 pp 640ndash642 2003

[20] P R He W J Zhang and Q Li ldquoSome further developmenton the eigensystem approach for graph isomorphismdetectionrdquoJournal of the Franklin Institute vol 342 no 6 pp 657ndash6732005

[21] Z Y Chang C Zhang Y H Yang and YWang ldquoA newmethodto mechanism kinematic chain isomorphism identificationrdquoMechanism andMachineTheory vol 37 no 4 pp 411ndash417 2002

[22] J P Cubillo and J BWan ldquoComments onmechanismkinematicchain isomorphism identification using adjacent matricesrdquoMechanism andMachineTheory vol 40 no 2 pp 131ndash139 2005

[23] H F Ding and Z Huang ldquoA unique representation of the kine-matic chain and the atlas databaserdquo Mechanism and MachineTheory vol 42 no 6 pp 637ndash651 2007

[24] H F Ding and Z Huang ldquoA new theory for the topologicalstructure analysis of kinematic chains and its applicationsrdquoMechanism and Machine Theory vol 42 no 10 pp 1264ndash12792007

[25] A C Rao ldquoA genetic algorithm for topological characteristicsof kinematic chainsrdquo Journal of Mechanical Design vol 122 no2 pp 228ndash231 2000

[26] F G Kong Q Li and W J Zhang ldquoAn artificial neuralnetwork approach tomechanism kinematic chain isomorphismidentificationrdquo Mechanism and Machine Theory vol 34 no 2pp 271ndash283 1999

[27] J K Chu The Structural and Dimensional Characteristics ofPlanar Linkages Press of Beihang University of China BeijingChina 1992

[28] L Song J Yang X Zhang and W Q Cao ldquoSpanning treemethod of identifying isomorphism and topological symmetryto planar kinematic chain with multiple jointrdquo Chinese Journalof Mechanical Engineering vol 14 no 1 pp 27ndash31 2001

[29] J G Liu and D J Yu ldquoRepresentations amp isomorphismidentification of planar kinematic chains with multiple jointsbased on the converted adjacent matrixrdquo Chinese Journal ofMechanical Engineering vol 48 pp 15ndash21 2012

[30] R Ravisankar andT SMruthyunjaya ldquoComputerized synthesisof the structure of geared kinematic chainsrdquo Mechanism andMachine Theory vol 20 no 5 pp 367ndash387 1985

[31] J U Kim and B M Kwak ldquoApplication of edge permutationgroup to structural synthesis of epicyclic gear trainsrdquo Mecha-nism and Machine Theory vol 25 no 5 pp 563ndash573 1990

Mathematical Problems in Engineering 15

[32] G Chatterjee and L-W Tsai ldquoComputer-aided sketching ofepicyclic-type automatic transmission gear trainsrdquo Journal ofMechanical Design vol 118 no 3 pp 405ndash411 1996

[33] V V N R Prasadraju and A C Rao ldquoA new technique basedon loops to investigate displacement isomorphism in planetarygear trainsrdquo Journal of Mechanical Design vol 12 pp 666ndash6752002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 15

[32] G Chatterjee and L-W Tsai ldquoComputer-aided sketching ofepicyclic-type automatic transmission gear trainsrdquo Journal ofMechanical Design vol 118 no 3 pp 405ndash411 1996

[33] V V N R Prasadraju and A C Rao ldquoA new technique basedon loops to investigate displacement isomorphism in planetarygear trainsrdquo Journal of Mechanical Design vol 12 pp 666ndash6752002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of