Research Article Constraint Study for a Hand Exoskeleton ...

Hindawi Publishing CorporationJournal of RoboticsVolume 2013 Article ID 910961 17 pageshttpdxdoiorg1011552013910961

Research ArticleConstraint Study for a Hand ExoskeletonHuman Hand Kinematics and Dynamics

Fai Chen Chen Silvia Appendino Alessandro Battezzato Alain FavettoMehdi Mousavi and Francesco Pescarmona

Center for Space Human RoboticsPolito Istituto Italiano di Tecnologia Corso Trento 21 10129 Torino Italy

Correspondence should be addressed to Francesco Pescarmona francescopescarmonaiitit

Received 10 May 2013 Revised 26 July 2013 Accepted 30 July 2013

Academic Editor Kazuhiko Terashima

Copyright copy 2013 Fai Chen Chen et alThis 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

In the last few years the number of projects studying the human hand from the robotic point of view has increased rapidly dueto the growing interest in academic and industrial applications Nevertheless the complexity of the human hand given its largenumber of degrees of freedom (DoF) within a significantly reduced space requires an exhaustive analysis before proposing anyapplications The aim of this paper is to provide a complete summary of the kinematic and dynamic characteristics of the humanhand as a preliminary step towards the development of hand devices such as prostheticrobotic hands and exoskeletons imitatingthe human hand shape and functionality A collection of data and constraints relevant to hand movements is presented and thedirect and inverse kinematics are solved for all the fingers as well as the dynamics anthropometric data and dynamics equationsallow performing simulations to understand the behavior of the finger

1 Introduction

The human hand is a complex mechanism it has a widerange of DoFs allowing a great variety of movements Inrecent years as robotics has advanced significant efforts havebeen devoted to the development of hand devices The twomain related application fields are prostheticrobotic handsand exoskeletons On one side robotic hands are developedwith the characteristics complying to those of the humanhand taking advantage of its variety of movements therebyavoiding the use of a large number of end effectors whenperforming tasks with different objects (eg Eurobot [1]Robonaut [2]) On the other side exoskeletons are designedto fit onto the humanhand aiming at enhancing performancein the carrying out of daily activities (eg improving astro-nautsrsquo hand performance during extravehicular activity [3])or supporting the rehabilitation stage of hand injury recovery

There are currently many different projects underwaySchabowsky et al [4] introduced a newly developed HandExoskeleton Rehabilitation Robot (HEXORR) which wasdesigned to provide a full range of motion for all fingersNASA and General Motors presented a prototype of theHuman Grasp Assist device [5] (K-Glove) Worsnopp et al

[6] introduced a finger exoskeleton for hand rehabilitationfollowing strokes to facilitate movement especially pinchAnother project is being developed by Ho et al [7] theirexoskeleton hand is EMG driven again for rehabilitationbut working on all the fingers All of these projects presenta different number of DoFs and different structures but ingeneral they are developed with the objective of mimickingthe main characteristics of the human hand This impliesa complete understanding of these characteristics involvingthe anthropometric dimensions of the human hand its kine-matics and its dynamics

This paper aims at analyzing all the aforementionedaspects It is organized as follows Section 2 presents anthro-pometric data about the hand such as dimensions of the handand phalanges Section 3 contains themain constraints of fin-ger movements explaining each type of constraint in the nat-ural movement of the human hand Section 4 describes thekinematical model of the hand direct and inverse kinematicsare developed step by step Section 5 presents the dynamicsof a single finger Section 6 shows the implementation of thedynamic equations on a practical example Finally conclu-sions are presented in Section 7

2 Journal of Robotics

2 Human Hand Data

The human hand is composed of 5 digits 4 fingers (indexring middle and little fingers) and the thumb The thumb ischaracterized by three articulations and three phalanges Thefingers also comprise three different articulations and fourphalanges Figure 1 shows names and acronyms of each artic-ulation and phalanx The wrist has two functional DoFs TheTMC joint of the thumb holds two DoFs (flexionextensionand adductionabduction) A single DoF (flexionextension)characterizes the MCP and IP joints of the thumb as well asthe PIP andDIP joints of the fingersWhereas the eight bonesof the carpus articulate finelywith each other producing smalldeformations their representation in a single rigid segment isa consistent approximation [8]

The analysis of kinematics and dynamics requires knowl-edge regarding the dimensions of the fingers and of the palmand their respective range of motion (RoM) those data arereported below Tables 1 and 2 show the results of Garrettrsquosstudies [9 10] for finger lengths and palm dimensions meas-urements that were taken from the right hands of 148 menand 211 women In Table 1 crotch to tip is the distance alongthe axis of the digit from the midpoint of its tip to the levelof the corresponding webbed crotch between two digit wristcrease to tip is the distance along the axis of the digit from themidpoint of its tip to the wrist crease baseline

A few researchers have measured the length of each pha-lanx separately A study with a variety of candidates is the oneperformed byHabib andKamal [11]The results of their studyfor each phalanx of index middle ring and little fingers arein Table 3 (I1 means distal phalanx I2 middle phalanx andI3 proximal phalanx of the index finger The same notationis used for other digits) A similar survey is also present inJasuja and Singhrsquos study [12] As shown in Table 3 the averagedimensions of the hand are quite similar to the ones presentedin Table 2 with a maximum difference of 218 The datapresented here is the sample distribution over geograph-ic regions and Air Force Commands which may be quiterepresentative of possible EVA glove users

As mentioned above those data can give us an idea ofthe mean values of the length of each element that composesthe human hand With proper modifications this data canalso be used to compose a model to simulate its movementsin particular it must be taken into account that the distancebetween a joint and a digit webbing is not representative ofthe equivalent link length A simulation was performed andthe results are presented in Section 6

3 Constraints Overview

Hand and digit motions are subject to several constraints thatlimit the range of the natural movements of human fingersConstraints can be roughly divided into three types staticintrafinger and interfinger constraints Intra- and inter-fin-ger constraints are often called dynamic constraints that inthe fingers of the hand are essentially constraints betweenjoint motions However this range of movement is somewhatambiguous because the range depends on various factorsinvolving human hand biomechanics therefore they are

Table 1 Mean finger lengths and palm dimensions of USAF male(M)female (F) flying personnel [9 10] (cm)

Finger length(crotch to tip)

Finger length(wrist crease to tip)

Mean sd lt5 lt95 Mean sd lt5 lt95M

Thumb 587 045 507 657 1270 113 1105 1468Index 753 046 683 819 1852 088 1733 2006Middle 857 051 782 974 1952 092 1810 2104Ring 80 047 744 893 1872 091 1752 2028Little 614 047 544 699 1661 091 1511 1810

FThumb 537 044 468 612 1105 100 951 1283Index 690 052 610 780 1667 089 1521 1814Middle 779 051 701 868 1765 087 1622 1905Ring 731 052 652 822 1676 894 1528 1820Little 546 044 480 624 1464 092 1311 1612

Table 2 Average hand dimensions of USAF male (M)female (F)flying personnel [9 10] (cm)

Joint Mean lengthHand length

M 1972F 1793

Hand breadthM 896F 771

Hand circumferenceM 2159F 1871

Hand thicknessM 329F 276

Hand depthM 619F 517

difficult to be expressed in closed forms (ie equations)and how to model such constraints still needs further inves-tigation The significance of taking finger constraints intoconsideration is that this causes the DoFs of the human handto decrease It is important to underline the fact that differentindividuals can show significant variations regarding the fol-lowing constraints

31 Intrafinger Constraints Intrafinger constraints are con-straints between different joints in the same finger Cobos etal [14] presented several constraints for fingers and thumbEquation (1) presents the relation between the joints of a fin-ger as first proposed by Rijpkema and Girard [15]

120579DIP asymp2

3

120579PIP120579PIP asymp3

4

120579MCP119891119890 (1)

Journal of Robotics 3

Thumb

Index

MiddleRing

LittleDistal phalanx

Middle phalanx

Proximal phalanx

Metacarpal

DIP distal interphalangeal

PIP proximal interphalangeal

MCP metacarpophalangeal

CMC carpometacarpal

Distal phalanx

Proximal phalanx

Metacarpal

IP interphalangeal


TMC trapeziometacarpal

Figure 1 Anatomical details of the hand skeleton

Table 3 Mean length of the hand and of the phalanges of index middle ring and little fingers (cm)

Hand I1 I2 I3 M1 M2 M3 R1 R2 R3 L1 L2 L3Male right hand 1929 232 237 265 260 278 280 229 256 276 196 192 251Male left hand 1936 232 239 261 260 282 275 230 259 278 195 198 249Female right hand 1760 223 224 245 244 255 256 212 234 252 179 174 226Female left hand 1762 220 224 235 224 243 253 213 236 249 177 177 226

In (2) the constraint relations between the joints whenflexing or extending the thumb are

120579IP asymp1

2

120579MCP119891119890

120579MCP119891119890 asymp5

4

120579TMC119891119890

(2)

These constraints are not strict In fact there are indi-viduals who are more able than others to control their DIPjoint and anybody can force a behavior slightly out of theseequations for any joint However in normal conditions thoseconstraints are respected quite faithfully

It is important to underline that the constraints men-tioned above are due to the physiologic nature of the humanhand Additionally other constraints can be considered dueto ergonomics In this case the relations of the constraintsare related to a specific task such as grasping a circular or aprismatic object

Table 4 shows the typical ergonomic constraints relativeto a circular grasping

Therefore the thumb is defined by 2DoFs index finger by2DoFs middle finger by 1DoF ring finger by 1DoF and littlefinger by 3DoFs The ring finger is calculated on the basisof the little and middle fingers joints Therefore the thumbthe index finger and the little finger are the most importantfingers when defining circular grasps [13]

Table 5 shows the ergonomic intrafinger constraints for aprismatic grasp As a result the thumb is defined by 2DoFsindex finger by 2DoFs middle finger by 1DoF ring fingerby 1DoF and little finger by 3DoFs Comparing the circulargrasping and the prismatic one the latter presents fewerconstraints among fingers

32 Interfinger Constraints Interfinger constraints correlatetwo joints belonging to different fingers For instance withthe hand in the open position (at rest) when one bends theindex finger at the MCP joint the MCP joint of the middlefinger bends automatically as well with respective certainproportionality

As for intra-finger constraints different individuals canshow important differences Moreover also in this case onecan forcefully overcome some of these constraints

On the other hand there are constraints that cannot beexplicitly represented in equations A few cases are explainedbelow

Some joints belonging to different fingers seem to beldquonaturallyrdquo interconnected By that we mean that the respec-tive angles vary more or less proportionally that is unless avoluntary counter force is applied imposing a different typeofmotionHowever trying tomove a finger in such an unnat-ural way often results in oddly fatiguing efforts

For instance there is a coupledmovementwhen the indexfinger and little finger are at rest The flexion of the middlefinger is equal to the flexion of the ring finger as described in

120579MCP119891119890Middle asymp 120579MCP119891119890Ring (3)

Another coupledmovement is producedwhen flexing thering finger thereby causing a slight flexion on the middle fin-ger and on the little finger that happens to be the same anglerotation as described in

120579MCP119891119890Middle asymp 120579MCP119891119890Little (4)


Table4Intrafinger

constraintsfor

acirc

ular

grasp

Thum

bIndex

Middle

Ring

Little

Noconstraint

120579CM

C 119891119890Index=120579CM

C 119891119890Middle

120579CM

C 119891119890Middle=(12)120579CM

C 119891119890Ring

120579CM

C 119891119890Ring=(23)120579CM

C 119891119890Little

Noconstraint

120579TM

C 119891119890=(1110)120579MCP119891119890

Noconstraint

120579MCP119886119887119886119889Middle=(15)120579MCP119886119887119886119889Index

120579MCP119886119887119886119889Ring=(12)120579MCP119886119887119886119889Little

Noconstraint

120579MCP119891119890=(45)120579IP119891119890

120579MCP119891119890Index=(43)120579PIP 119891119890Index

120579MCP119891119890Middle=(43)120579PIP 119891119890Middle

120579MCP119891119890Ring=(43)120579PIP 119891119890Ring

120579MCP119891119890Little=(43)120579PIP 119891119890Little

Noconstraint

120579PIP 119891119890Index=(32)120579DIP119891119890Index

120579PIP 119891119890Middle=(32)120579DIP119891119890Middle

120579PIP 119891119890Ring=(32)120579DIP119891119890Ring

120579PIP 119891119890Little=(32)120579DIP119891119890Little


Table5Intrafinger

constraintsfor

apris

maticgrasp

Thum

bIndex

Middle

Ring

Little

Noconstraint

120579CM

C 119891119890Index=120579CM

C 119891119890Middle

120579CM

C 119891119890Middle=(12)120579CM

C 119891119890Ring

120579CM

C 119891119890Ring=(23)120579CM

C 119891119890Little

Noconstraint

120579TM

C 119891119890=(1110)120579MCP119891119890

Noconstraint



Noconstraint

120579MCP119891119890=(65)120579IP119891119890



120579MCP119891119890Ring=(32)120579PIP 119891119890Ring


Noconstraint

120579PIP 119891119890Index=2120579DIP119891119890Index

120579PIP 119891119890Middle=2120579DIP119891119890Middle

120579PIP 119891119890Ring=2120579DIP119891119890Ring

120579PIP 119891119890Little=2120579DIP119891119890Little


There is also a coupled movement in abductionadduc-tion which is generated among ring and little fingers Inmostcases the movement is similar to

120579MCP119886119887119886119889Ring asymp 120579MCP119886119887119886119889Little (5)

Additionally some joints can be tied by more intricaterelations divided into two types The first type occurs whenthere is a unique motion like the flexion in the MCP joint ofthe little finger Equation (6) represents this type of relationbetween the fingers

120579MCP119891119890Ring asymp7

12

120579MCP119891119890Little

120579MCP119891119890Middle asymp2

3

120579MCP119891119890Ring

120579MCP119891119890Ring minus 120579MCP119891119890Middle lt 60∘

120579MCP119891119890Little minus 120579MCP119891119890Ring lt 50∘

(6)

These equations indicate that when there is flexion thering and middle fingers will also flex in a certain range withrespect to the little finger For instance when the little fingeris flexed the middle finger will flex 23 times the ring finger

The second relationship is when there is a simple MCPflexion as in the case of the index finger thereby causing someadditional natural movement Equation (7) represents thistype of relation between the index and the middle fingers


5

120579MCP119891119890Index (7)

This relation is only present when there is a single flexionon the index finger MCP joint so when it occurs the middlefinger MCP joint will flex naturally and passively with thisproportionality

These constraints are important because they express thenatural relation between single joints Thus devices designedto mimic the human hand must comply as precisely as possi-ble with these constraints

33 Static Constraints The normal range of motion (ROM)of human hand joints corresponds to static constraints onjoint angles in the model These constraints are limits onthe values that the 120579 parameters can assume Main staticconstraints (Table 6) were collected by Cobos et al [13] Byapplying these constraints to the inverse kinematics presentedlater in Section 42 some DoFs can be neglected and there-fore the complexity of the system can be reduced

4 Kinematic Model

The kinematic model proposed here is composed of 19 linkscorresponding to the human bones and 24DoFs modeled byrotational joints Two different kinematic configurations areconsidered for the fingers one for the thumb modeled as 3links and 4 joints and another for the other fingers (indexmiddle ring and little fingers) Each of them is modeledusing 4 links and 5 joints as in Figure 2 Note that the CMC

Table 6 Statics constraints [13]

Finger Flexion Extension AbdaddThumb

TMC 50∘ndash90∘ 15∘ 45∘ndash60∘

MCP 75∘ndash80∘ 0∘ 5∘

IP 75∘ndash80∘ 5∘ndash10∘ 5∘

IndexCMC 5∘ 0∘ 0∘

MCP 90∘ 30∘ndash40∘ 60∘

PIP 110∘ 0∘ 0∘

DIP 80∘ndash90∘ 5∘ 0∘

MiddleCMC 5∘ 0∘ 0∘

MCP 90∘ 30∘ndash40∘ 45∘

PIP 110∘ 0∘ 0∘

DIP 80∘ndash90∘ 5∘ 0∘

RingCMC 10∘ 0∘ 0∘

MCP 90∘ 30∘ndash40∘ 45∘

PIP 120∘ 0∘ 0∘

DIP 80∘ndash90∘ 5∘ 0∘

LittleCMC 15∘ 0∘ 0∘

MCP 90∘ 30∘ndash40∘ 50∘

PIP 135∘ 0∘ 0∘

DIP 90∘ 5∘ 0∘

joint represents the deformation of the palm for instancewhen the hand is grasping a ball while the MCP abduc-tionadduction joint is defined before MCP flexionexten-sion

41 Direct Kinematics Direct kinematic equations are usedto obtain the fingertip position and orientation accordingto the joint angles The model equations are calculated bymeans of modified Denavit-Hartenberg (MDH) parametersintroduced by Craig [16] The difference from the Denavit-Hartenberg convention is the fact that on MDH the 119911-axisof the reference frame 119895 called 119911

119895 is coincident with the axis

of joint 119895 In comparison to MD the 119911-axis of the frame 119895 iscoincident with the axis of joint 119895 + 1 The advantage of usingMDH is that there is no need for further transformation ofthe references in order to work with the dynamics of the rigidbodies

411 Direct Kinematics of the Index Middle Ring and LittleFingers Each of the fingers contains four bones metacarpalproximal middle and distal (Figure 1) These bones corre-spond approximately to the links of the serial kinematic chain(Figure 2) Each articulation presented above for these fourfingers corresponds to the joints CMC MCP PIP and DIPThe MCP joint can be split into 2DoFs which carry outthe adductionabduction and flexionextension movementsrespectively All the other joints only allow flexionextensionmovements Having defined a numbering of the fingers from


Distal link

Middle link

Proximal link

Metacarpal link

Carpal zone

Wrist Reference

0

1

23

4

L01L02

L03L04

L11

L12L13

L14

L21

L22L23

L24

L31

L32L33

L34

pt1

pt2pt3

pt4

pt0

L20

L00L10

R0R0

R1i

R2i

R3i

R5i

R5iR4i

R4i

TMCaa

MCPaa

TMCfe

IPfe

MCPfeCMCfe

MCPfePIPfe

DIPfe

R0i = R1i

R2i = R3i

Figure 2 Kinematic configuration of the human hand The thumb is defined by 3 links and 4 degrees of freedom whereas index middlering and little fingers are defined by 4 links and 5DoFs

0 to 4 where finger 0 is the thumb and finger 4 is the littlefinger Table 7 shows the MDH parameters for fingers 1 to 4

Equation (8) shows the direct kinematics from index (119894 =1) to the little finger (119894 = 4)

119876119894=0

0

119879119894sdot0

6119879119894

(120579119895) =0

0

119879119894sdot

5

prod

119895=1

(119895minus1)

119895

119879119894(120579119895) sdot5

6

119879119905119894

119876119894=0

0

119879119894sdot0

1119879119894

(120579CMC119891119890) sdot1

2119879119894

(120579MCP119886119887119886119889) sdot2

3119879119894

(120579MCP119891119890)

sdot3

4119879119894

(120579PIP119891119890) sdot4

5119879119894

(120579DIP119891119890) sdot5

6

119879119905119894

(8)where

(i) 119876119894represents a matrix containing the position and

orientation of the fingertip of each finger(ii) 00

119879119894represents a rototranslation matrix taking into

account the fact that the fingers are slightly fanned outand making it possible to pass from the initial basereference frame (119877

0) to the alignment of the 119894th finger

first reference frame (1198770119894)

(iii) 06

119879119894(120579119895) is a matrix containing the geometrical trans-

formation between the 119894th finger first reference frameand the 119894th fingertip (ft

119894) The matrix is composed of

the concatenation of the transformation matrices ofeach finger link

(iv) (119895minus1)119895

119879119894(120579119895) is a matrix containing the geometrical

transformation between the (119895 minus 1)th reference frameand the 119895th reference frame of the 119894th finger

Table 7 Modified DndashH parameters for fingers 1 to 4

Finger 119894 (119894 = 1 4)Joint 120572

119895minus1

119886119895minus1

119889119895

120579119895

119895 = 1 1205872 0 0 120579CMC119891119890

119895 = 2 minus1205872 1198710119894

0 120579MCP119886119887119886119889

119895 = 3 1205872 0 0 120579MCP119891119890

119895 = 4 0 1198711119894

0 120579PIP119891119890

119895 = 5 0 1198712119894

0 120579DIP119891119890

(v) 56

119879119905119894represents the position 119901

119905119894= [119897119905119894119909119897119905119894119910119897119905119894119911]

119879

ofthe fingertip with respect to the distal (5th) referenceframe

119894 corresponds to index(1) middle(2) ring(3) and little(4) fin-gers 119895 corresponds to each fingerrsquos joint CMC

119891119890 MCP

119886119887119886119889

MCP119891119890

PIP119891119890

and DIP119891119890

119905119894stands for the tip of the 119894th

fingerThe coefficients119876

119894119904119905 which are the elements of the 119904th row

and 119905th column of the matrix expressed in (8) relative to the119894th finger are given in Appendix A

412 Direct Kinematics of the Thumb The thumb presentsthree bones (Figure 1) metacarpal proximal and distalThese bones correspond approximately to the length of eachlink The respective joints are TMC MCP and IP TheTMC joint presents 2DoFs allowing adductionabduction


Table 8 Modified DndashH parameters of the thumb

Joint 120572119895minus1

119886119895minus1

119889119895

120579119895

119895 = 1 0 0 0 120579TMC119886119887119886119889

119895 = 2 1205872 0 0 120579TMC119891119890

119895 = 3 0 11987100

0 120579MCP119891119890

119895 = 4 0 11987110

0 120579IP119891119890

and flexionextension Table 8 shows the MDH parametersEquation (9) with the same notation scheme as (8) showsthe direct kinematics for the thumb

1198760=0

0

1198790sdot0

5

1198790(120579119896)

1198760=0

0

1198790sdot0

11198790

(120579TMC119886119887119886119889) sdot1

21198790

(120579TMC119891119890)

sdot2

31198790

(120579MCP119891119890) sdot3

41198790

(120579IP119891119890) sdot4

51198791199050

(9)

where 119896 corresponds to the thumb joint TMC119886119887119886119889

TMC119891119890

MCP119891119890

and IP119891119890

The position of the fingertip is 1199011199050=

[1198971199050119909

1198971199050119910

1198971199050119911]

119879

The coefficients 1198760119904119905

of the matrix expressedin (9) are given in Appendix A

42 Inverse Kinematics Inverse kinematics is used to obtainthe joint angle values according to the fingertip position andorientation The inverse kinematics will be solved separatelyfor the thumb and for the other fingers The model of thehuman hand is a redundant case therefore several solutionsto the inverse kinematic problem exist In order to find aunique solution it is necessary to take into account theconstraints presented in Section 3

421 Inverse Kinematics of a Finger (Index (119894 = 1) to LittleFinger (119894 = 4)) The angles 120579

119894 CMC119891119890 120579119894 MCP119886119887119886119889 120579119894 MCP119891119890 and120579119894 PIP119891119890 120579119894 DIP119891119890 are obtained from (8) where the matrix 119876

119894

that is the left member of the equation is known Hencealgebraically we solve the inverse kinematics for the CMC

119891119890

MCP119886119887119886119889

and DIP119891119890

joints The procedure here shownsolves the kinematics of the index finger (119894 = 1) howeverit is valid for every other finger except the thumb (119894 = 2 3 4)For sake of brevity in the following the 119894 index is omitted

The duplication of solutions when solving an arctangentfor example is eliminated thanks to the constraints ofSection 33 because the ranges of all physiological angles aresmaller than 180∘ andmost of them are smaller than 90∘ Thisfact is implicitly taken into account in the following

120579CMC119891119890 = atan(11987633

11987613

)

120579MCP119886119887119886119889 = atan[

[

11987613

minus11987623cos (120579CMC119891119890)

]

]

120579DIP119891119890 = atan(12059811205982

)

(10)

Figure 3 Inverse kinematics for the index finger and thumb

where 1205981and 1205982are expressed as

1205981=

(minus1) sdot (11989811987622+ 11989911987621)

(1198982

minus 1198992

)

1205982=

Q21

119898

+

119899

119898

sdot

(minus1) sdot (11989811987622+ 11989911987621)

(1198982

minus 1198992

)

119898 = sin (120579MCP119886119887119886119889) cos (120579MCP119891119890) cos (120579PIP119891119890)

minus sin (120579MCP119886119887119886119889) sin (120579MCP119891119890) sin (120579PIP119891119890)

119899 = sin (120579MCP119886119887119886119889) cos (120579MCP119891119890) sin (120579PIP119891119890)

minus sin (120579MCP119886119887119886119889) sin (120579MCP119891119890) cos (120579PIP119891119890)

(11)

TheMCP119891119890

PIP119891119890

joints are solved through a geometricmethod see Figure 3

Starting from the vector 1198651which contains the position

of the fingertip it is possible to obtain 1with the following

expression

1= 1198651minus [11987131lowast ] (12)

Then the vector 1198751is expressed as

1198751= [119875111990911987511199101198751119911]

119879

(13)

where1198751119909= 11987101cos (120579CMC)

1198751119910= 11987101sin (120579CMC)

1198751119911= 0

(14)


Knowing 1and 119875

1from (12) and (13) the vector 119906

1is

expressed as1199061= 1minus 1198751 (15)

Angles 1205931and 120593

2are expressed as

1205931= acos(

1199032

2

+ 1198712

01

minus 1199032

1

2119903211989701

)

1205932= acos(

1199032

2

+ 1198712

11

minus 1198712

21

2119903211987111

)

(16)

with variables 1199031and 1199032equal to1199031=

100381710038171003817100381710038171

10038171003817100381710038171003817

1199032=10038171003817100381710038171199061

1003817100381710038171003817

(17)

So 120579MCP119891119890 is obtained as

120579MCP119891119890 = 120587 minus 1205931 minus 1205932 (18)

And 120579PIP119891119890 is obtained as

120579PIP119891119890 = 120587 minus 1205933 (19)

where

1205933= acos(

1198712

21

+ 1198712

11

minus 1199032

2

21198712111987111

) (20)

422 Inverse Kinematics of the Thumb A similar procedurecan be applied to the thumb in which119876

0 that is the left side

of (9) is known then the joint angle 120579TMC119886119887119886119889 is obtained alge-braically as follows

120579TMC119886119887119886119889 = atan(11987613

minus11987623

) (21)

where the119876119904119905term in (21) is the element of the 119904th row and 119905th

columnof the1198760matrixThen using the geometricalmethod

(Figure 3) the joint angles 120579MCP119891119890 and 120579IP119891119890 can be obtained

120579MCP119891119890 = 120587 minus 1205741

120579IP119891119890 = 120587 minus 1205742 minus 1205743(22)

where

1205741= acos(

1198712

10

+ 1198712

00

minus 1199032

3

21198711011987100

)

1205742= acos(

1198712

20

+ 1199042

3

minus 1199032

4

2119871201199033

)

1205743= acos(

1199032

3

+ 1198712

10

minus 1198712

00

211990311990411987110

)

1199033=

100381710038171003817100381710038170

10038171003817100381710038171003817

1199034=

10038171003817100381710038171003817

1198650

10038171003817100381710038171003817

0= 1198650minus [11987120lowast ]

(23)

The joint 120579TMC119891119890 is obtained algebraically as follows

120579TMC119891119890 = atan(12059841205983

) (24)

where

1205983= (11987632+ 11987631) cos 120583

1205984=

11986031minus (11987631+ 11987632) cos 120583 sdot sin 120583

cos 120583

120583 = 120579MCP + 120579IP

(25)

5 Dynamics of a Single Finger

This section provides the dynamics equation system of ageneric single finger neglecting the MCP abductionadduc-tion motion The 119894th index is again omitted Based on theconvention of Figure 4 the dynamic model is determinedusing Euler-Lagrange equations It is important to highlightthe fact that on the model the metacarpus is fixed whileonly the finger phalanges are moving parts Thus 119877

2is the

base reference system and all the equations are written withrespect to 119877

2

The following equations are applicable to any 3R planarrobot aftermaking the necessary adaptations to the variables

The equations require the estimation of the kinetic energyand potential energy as shown in the following

In order to simplify the expressed vectors and furtherdevelopments the following notation is used

1205793= 120579MCP119891119890

1205794= 120579PIP119891119890

1205795= 120579DIP119891119890

(26)

The kinetic energy is calculated starting from the positionvectors of the center of mass of each phalanx with respect tothe base reference frame 119877

2 in general the position of the

center of mass of the 119895th phalanx with respect to the 119895 refer-ence frame is

119892= [

[

119887119892119909

119887119892119910

0

]

]

119892 = 1 2 3 (27)

Themass of the 119895th phalanx is equal to119898119892 and the respec-

tive moment of inertia with respect to the axis 119911 is equal to 119868119892

Thus each position vector with respect to the base refer-ence frame 119877

2is presented below where 119888120579

119895and 119904120579

119895stand for

cosine and sine of 120579119895 respectively Consider

1= [

[

11988711199091198881205793minus 11988711199101199041205793

11988711199091199041205793+ 11988711199101198881205793

0

]

]

2= [

[

11987111198881205793

11987111199041205793

0

]

]

+

[[[

[

1198872119909119888 (1205793+ 1205794) minus 1198872119910119904 (1205793+ 1205794)

1198872119909119904 (1205793+ 1205794) + 1198872119910119888 (1205793+ 1205794)

0

]]]

]


Figure 4 Dynamic model of index(1)

3= [

[

11987111198881205793

11987111199041205793

0

]

]

+[[

[

1198712119888 (1205793+ 1205794)

1198712119904 (1205793+ 1205794)

0

]]

]

+

[[[

[

1198873119909119888 (1205793+ 1205794+ 1205795) minus 1198873119910119904 (1205793+ 1205794+ 1205795)

1198873119909119904 (1205793+ 1205794+ 1205795) + 1198873119910119888 (1205793+ 1205794+ 1205795)

0

]]]

]

(28)

The velocities can be obtained by differentiating theposition vectors with respect to time taking into account thefollowing notation of (29) and representing time derivativesthrough dot notation

1205793+ 1205794= 12057934

1205793+1205794=

12057934

1205793+ 1205794+ 1205795= 120579345

1205793+1205794+1205795=

120579345

(29)

The velocities of each center of mass are obtained

V1198661=[[

[

minus1198871119909

12057931199041205793minus 1198871119910

12057931198881205793

1198871119909

12057931198881205793minus 1198871119910

12057931199041205793

0

]]

]

V1198662=[[

[

minus1198711

12057931199041205793minus1198872119909

1205793411990412057934minus 1198872119910

1205793411988812057934

1198711

12057931198881205793+ 1198872119909

1205793411988812057934minus 1198872119910

1205793411990412057934

0

]]

]

V1198663=[[

[

minus1198711

12057931199041205793minus1198712

1205793411990412057934minus1198873119909

120579345119904120579345minus 1198873119910

120579345119888120579345

1198711

12057931198881205793+ 1198712

1205793411988812057934+ 1198873119909

120579345119888120579345minus 1198873119910

120579345119904120579345

0

]]

]

(30)

The kinetic energy is expressed as

119879 =

1

2

1198981(V1198661)

2

+

1

2

1198982(V1198662)

2

+

1

2

1198983(V1198663)

2

+

1

2

1198681(1205793)

2

+

1

2

1198682(12057934)

2

+

1

2

1198683(120579345)

2

(31)

There are two forms of potential energy considered inthis study the gravitational potential energy and the elasticpotential energy Regarding the elastic potential energy thevalue of the stiffness is considered as an average value (con-stant value 119896

119892) in this study as a close approximation of the

nonlinear and anisotropic finger stiffness (ie it varies withthe direction) 119896

1 1198962 and 119896

3are the stiffness values for the

MCP DIP and PIP joints respectively [17]The potential energy for the finger is expressed as

119880 = 1198981119892 (11988711199091199041205793+ 11988711199101198881205793)

+ 1198982119892 (11987111199041205793+ 119887211990911990412057934+ 119887211991011988812057934)

+ 1198983119892 (11987111199041205793+ 119871211990412057934+ 1198873119909119904120579345+ 1198873119910119888120579345)

+

1

2

1198961(1205793)2

+

1

2

1198962(12057934minus 1205793)2

+

1

2

1198963(120579345minus 12057934)2

(32)

Moreover introducing a function of the generalized ve-locities usually referred to as the Rayleigh dissipation func-tion 119865 for the damping forces this function is expressed as

119865 =

1

2

1198881(1205793)

2

+

1

2

1198882(12057934minus1205793)

2

+

1

2

1198883(120579345minus12057934)

2

(33)

where the damping constant (119888119892) stands for the nonconserva-

tive contribution caused by the muscles actuating the fingerNonconservative forces contributed less than 15 to the totalforce response to static displacement Muscle viscosity isdissipative and hence non-conservative resulting in a forcefield with nonzero curl [17] To be more precise values 119888

1

1198882 and 119888

3are the damping values for the MCP DIP and PIP

joints respectively


The Euler-Lagrange equations thus become consideringthe three generalized coordinates 120579

3 12057934 and 120579

345

119889

119889119905

(

120597 (119879 minus 119880)

1205971205793

) minus

120597 (119879 minus 119880)

1205971205793

+

120597119865

1205971205793

= 1205913

119889

119889119905

(

120597 (119879 minus 119880)

12059712057934

) minus

120597 (119879 minus 119880)

12059712057934

+

120597119865

12059712057934

= 1205914

119889

119889119905

(

120597 (119879 minus 119880)

120597120579345

) minus

120597 (119879 minus 119880)

120597120579345

+

120597119865

120597120579345

= 1205915

(34)

in which the 120591119895(119895 = 3 4 5) terms contain the forces applied

through themuscles in order to actuate the phalanges and thecontact forces shown in Figure 5

According to the virtual work principle the equation tocalculate the generalized force can be expressed as

120591119895=

(sum5

119895=3

120575119882119895)

120575120579119895

119895 = 3 4 5 (35)

where 120575119882119895is the virtual work done by the force applied to the

system In the current case it is

1205913= (11986531199101198903119909minus 11986531199091198903119910) + [

[

minus11987111199041205793

11987111198881205793

0

]

]

sdot [

[

1198654119909c12057934minus 1198654119910s12057934

1198654119909s12057934+ 1198654119910c12057934

0

]

]

+ [

[

minus11987111199041205793

11987111198881205793

0

]

]

sdot [

[

1198655119909c120579345minus 1198655119910s120579345

1198655119909s120579345+ 1198655119910c120579345

0

]

]

+ 1198621198981minus 1198621198982

1205914= (11986541199101198904119909minus 11986541199091198904119910) + [

[

minus119871211990412057934

119871211988812057934

0

]

]

sdot [

[

1198655119909c120579345minus 1198655119910s120579345

1198655119909s120579345+ 1198655119910c120579345

0

]

]

+ 1198621198982minus 1198621198983

1205915= (11986551199101198905119909minus 11986551199091198905119910) + 1198621198983

(36)

where the term 119862119898119892

refers to the torque produced by themuscles on the 119895th joint and119865

119895= [1198651198951199091198651198951199100]119879 is the contact

force applied to the 119895th phalanx at the point defined by theposition vector 119890

119895= [1198901198951199091198901198941199100]119879

Calculating each element of the Euler-Lagrange equa-tions the dynamical system (33) becomes

1198601

1205793+ 1198602

12057934+ 1198603

120579345= 1198604

1198611

1205793+ 1198612

12057934+ 1198613

120579345= 1198614

1198621

1205793+ 1198622

12057934+ 1198623

120579345= 1198624

(37)

Figure 5 Contact forces

The terms 1198601 1198602 and 119860

3contain the coefficients of the

accelerations and 1198604the remaining terms the same goes for

the other Euler-Lagrange equationsEquation (37) allows the direct dynamics of the finger to

be solved where given the torques exerted by the muscles oneach phalange the movement of the finger can be calculatedIf on the other side an inverse dynamics problem is set itis simple to rearrange the equation system (33) to obtain thetrend of the unknown muscle torques from the phalangesmotion laws

An expansion of the coefficients of equation (37) can befound in Appendix B equation (B1)

6 No-Load Joint Velocities

A test campaign was carried out in order to evaluate themaximum angular velocities of the finger joints Ten maleand ten female subjects were asked to perform the followingoperation as fast as possible during 10 seconds (ie beforeany fatigue phenomenon arises to deteriorate the subjectperformance) starting from the completely extended handthey have to flex the fingers to a fist and then to extendthem againThemaximumflexion angles for the index fingercorresponding to the fist configuration on each cycle havebeen statically measured on the subjects as 85 deg for theMCP joint 105 deg for the PIP joint and 70 deg for the DIPone In the following those angles will be considered negativein accordance with the notations of Section 41 Tenmale andten female subjects were involved in the investigation thestatistical output of the test is reported in Table 9

There are no significant differences between the maleand the female subjects The results obtained by our testcampaign show velocities in accordance with previous testsfor instance in [18]

The trend of the phalange angles was supposed to follow asinusoidal law in the form 120579

119895= (1198631198952)(cos(120587119905119905

1)minus1) where

119895 = 3 4 5 The term 119863119895is the range of the movement on

the 119895th joint while 1199051= 10snumcycles is the period of each

cycle The time differentiation of this law permits obtainingthe trend of the velocities as well as the maximum angularvelocity of each joint equal to

120579119895 max = | minus 12058711986311989521199051|

Here such experimental motion law has been imple-mented in the dynamicmodeling of the fingerThe simulation


Table 9 Joint velocity tests

Males FP AB AF MP AL NC MS AL DM RC Mean stdev Average freq(Hz)

Omega maxMCP (rads)

Omega maxPIP (rads)

Omega maxDIP (rads)

Cycles in 10 s 27 27 31 25 20 29 23 27 21 31 261 3843 261 12164 15026 10018

Females SA TT NG FC EA GM VC SP MQ DH Mean stdev Average freq(Hz)

Omega maxMCP (rads)

Omega maxPIP (rads)

Omega maxDIP (rads)

Cycles in 10 s 27 24 17 34 29 27 30 24 21 27 26 4784 26 12118 14969 9979

deals with an inverse dynamics case study given the motionlaw of the system (ie the kinematic angles of the phalangesand their time derivatives) the torques exerted by themuscleson each phalanx are calculated In order to evaluate correctlysuch torques in absence of load it is necessary to insert alsothe viscoelastic term that can be preponderant with respectto the dynamical term [19] The values of the stiffness andthe damping coefficients for the MCP joint are obtainedthrough an extrapolation of the values of MCP stiffness anddamping reported in a series of tests in [20] for the no-load case where the applied external force is null Thengiven the fact that the tests in [20] compute translationalstiffness and damping coefficients they are transformed intorotational ones to be coherent with the dynamic modelingThe resulting coefficients are shown in Table 11 reported inthe appendix together with the anthropometric data andnumerical constants imposed in the model to simulate thebehavior of a human index finger

The trends of the muscle torques 1198621198981 1198621198982 and 119862

1198983are

reported in Figure 6 First of all these values can be comparedwith the maximum torque capability of each phalangeHasser [18] reports these maximum values as 119862

119898max=

[4630 2280 775]119879 The current test without external load

correctly produces a set of torques that are quite lower thanthese limits Moreover the results shown in [19] state that theaverage total torque for a complete flexion-extension move-ment of the MCP joint is 1008Nmm for the female subjectsand 1949Nmm for the male ones these values are quite nearto the ones produced here where the joint velocity is higherThe same paper states that the contribute of the viscoelasticterms is preponderant with respect to the dynamical one thesame is true for the current analysis where the coefficients119891119896119895= |119862119896119895||119862119898119895| 119891119888119895= |119862119888119895||119862119898119895| and 119891

119889119895= |119862119889119895||119862119898119895|

that are respectively the torque portion due to the stiffnessterm the damping term and the dynamic term for the 119895thphalange are reported in Table 10 However if [19] shows thatthe stiffness term is about the double of the damping one inthe current analysis the situation is reversed it is presumablydue to the higher velocity investigated here

In conclusion the results of the current simulation are ingood accordance with the literature dataMore precise resultscould be obtained as a consequence of a more accurate esti-mation of the dynamical parameters inserted into the modelin particular the stiffness and the damping coefficients Forexample due to lack of data in the current simulation theseparameters have been set as equal for the MCP PIP andDIP joints in reality the PIP and DIP joints have different

0 20 40 60 80 100minus100

minus50

0

50

100

150

200

250

300

Cycle ()

C(N

mm

)

Cm1

Cm2

Cm3

Figure 6 Trend of the muscle torques

Table 10 Percentages of stiffness damping and inertial torqueswith respect to the total torque

MCP PIP DIP119891119896119895

3805 3848 3858119891119888119895

5758 6059 61119891119889119895

437 093 042

characteristics which should not be neglected for a deeperinvestigation

Finally given the previously described test and the relatedmuscle torque shown in Figure 6 then the mean powerassociated with the opening and closing of the index finger atmaximum velocity in the absence of load can be calculated itis equal to 13W The maximum power for the MCP joint inthe absence of load is 124W themaximumpower for the PIPjoint is 194W and for the DIP joint it is 086W This valueconstitutes a correction of the maximum power estimationfor the finger joints reported by Hasser [18] which statesthat the absence of external load corresponds to zero muscletorque

7 Conclusions

An exhaustive study of the human handwas performed deal-ing with the kinematics of the human hand and showing the


Table 11 Anthropometric data and numerical constants used for simulation

1198981

= 705 g 1198982

= 397 g 1198983

= 270 g1198681

= 1000 gmm2 1198682

= 240 gmm2 1198683

= 120 gmm2

1198961

= 00902Nm 1198962

= 00902Nm 1198963

= 00902Nm1198881

= 00225Nms 1198882

= 00225Nms 1198883

= 00225Nms1198711

= [50 0 0]

119879

mm 1198712

= [20 0 0]

119879

mm 1198713

= [25 0 0]

119879

mm1198871

= [25 0 0]

119879

mm 1198872

= [10 0 0]

119879

mm 1198873

= [125 0 0]

119879

mm1198901

= [25 minus8 0]

119879

mm 1198902

= [10 minus7 0]

119879

mm 1198903

= [12 minus6 0]

119879

mm

matrices with the MDH parameters and detailed equationsfor both the direct and inverse kinematics of the hand Inaddition the dynamics of a single finger was analyzed andfinally written using the same reference systems as per thekinematics A case study was proposed and a simulationwas completed using the provided anthropometric data inorder to investigate the capabilities of the proposed analyticalsystem

The results of the current study can be exploited to con-ceive future hand devices On one side direct and inversekinematics constitute preliminary stages for the developmentof any structure similar to the human hand for instancein robotic or rehabilitation hands projects On the otherside dynamics equations allow the behavior of fingers to besimulated Hence the value of these equations is twofold tomodel the finger itself for the use in a control scheme modelof a human-machine interface such as a hand exoskeleton orto model a finger-like architecture such as a robotic hand

Appendices

A Rototranslation Matrix Elements

The 119894 subscript of the angles is omitted First of all the termsfor the fingers (119894 = 1 to 4) are given Consider

11987611989411= [(119888120579CMC119891119890119888120579MCP119886119887119886119889) 119888120579MCP119891119890

minus 119904120579CMC119891119890119904120579MCP119891119890]

times (119888120579PIP119891119890119888120579DIP119891119890 minus 119904120579PIP119891119890119904120579DIP119891119890)

minus [(119888120579CMC119891119890119888120579MCP119886119887119886119889) 119904120579MCP119891119890

+ 119904120579CMC119891119890119888120579MCP119891119890]

times (119904120579PIP119891119890119888120579DIP119891119890 + 119888120579PIP119891119890119904120579DIP119891119890)

11987611989412= [(119888120579CMC119891119890119888120579MCP119886119887119886119889) 119888120579MCP119891119890

minus 119904120579CMC119891119890119904120579MCP119891119890]

times (minus119888120579PIP119891119890119904120579DIP119891119890 minus 119904120579PIP119891119890119888120579DIP119891119890)

minus [(119888120579CMC119891119890119888120579MCP119886119887119886119889) 119904120579MCP119891119890

+ 119904120579CMC119891119890119888120579MCP119891119890]

times (minus119904120579PIP119891119890119904120579DIP119891119890 + 119888120579PIP119891119890119888120579DIP119891119890)

11987611989413= 119888120579CMC119891119890119904120579MCP119886119887119886119889

11987611989414= [(119888120579CMC119891119890119888120579MCP119886119887119886119889) 119888120579MCP119891119890

minus 119904120579CMC119891119890119904120579MCP119891119890]

times [119871119905119894119909(119888120579PIP119891119890119888120579DIP119891119890 minus 119904120579PIP119891119890119904120579DIP119891119890)

+ 119871119905119894119910(minus119888120579PIP119891119890119904120579DIP119891119890 minus 119904120579PIP119891119890119888120579DIP119891119890)

+ 1198711119894+ 1198712119894119888120579PIP119891119890]

minus [(119888120579CMC119891119890119888120579MCP119886119887119886119889) 119904120579MCP119891119890

+ 119904120579CMC119891119890119888120579MCP119891119890]

times [119871119905119894119909(119904120579PIP119891119890119888120579DIP119891119890 + 119888120579PIP119891119890119904120579DIP119891119890)

+ 119871119905119894119910(minus119904120579PIP119891119890119904120579DIP119891119890 + 119888120579PIP119891119890119888120579DIP119891119890)

+ 1198712119894119904120579PIP119891119890]

+ (119888120579CMC119891119890119904120579MCP119886119887119886119889) 119871 119905119894119911 + 1198710119894119888120579CMC119891119890

11987611989421= 119904120579MCP119886119887119886119889119888120579MCP119891119890


minus 119904120579MCP119886119887119886119889119904120579MCP119891119890


11987611989422= 119904120579MCP119886119887119886119889119888120579MCP119891119890



minus 119904120579MCP119886119887119886119889119904120579MCP119891119890


11987611989423= minus 119888120579MCP119886119887119886119889

11987611989424= 119904120579MCP119886119887119886119889119888120579MCP119891119890



+ 1198711119894+ 1198712119894119888120579PIP119891119890] minus 119904120579MCP119886119887119886119889119904120579MCP119891119890



+ 1198712119894119904120579PIP119891119890] + (minus119888120579MCP119886119887119886119889) 119871 119905119894119911

11987611989431= (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119888120579MCP119891119890

+ 119888120579CMC119891119890119904120579MCP119891119890


+ minus (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119904120579MCP119891119890

+ 119888120579CMC119891119890119888120579MCP119891119890


11987611989432= (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119888120579MCP119891119890

+ 119888120579CMC119891119890119904120579MCP119891119890


+ minus (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119904120579MCP119891119890

+ 119888120579CMC119891119890119888120579MCP119891119890


11987611989433= 119904120579CMC119891119890119904120579MCP119886119887119886119889

11987611989434= (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119888120579MCP119891119890

+ 119888120579119862119872119862119891119890

119904120579119872119862119875119891119890



+ 1198711119894+ 1198712119894119888120579PIP119891119890]

+ minus (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119904120579MCP119891119890

+ 119888120579CMC119891119890119888120579MCP119891119890



+ 1198712119894119904120579PIP119891119890]

+ (119904120579CMC119891119890119904120579MCP119886119887119886119889) 119871 119905119894119911 + 1198710119894119904120579CMC119891119890

(A1)

Now the terms for the rototranslation matrix of thethumb (119894 = 0) are reported Take

119876011= [119888120579TMC119886119887119886119889119888120579TMC119891119890]

times (119888120579MCP119891119890119888120579IP119891119890 minus 119904120579MCP119891119890119904120579IP119891119890)

+ [minus119888120579TMC119886119887119886119889119904120579TMC119891119890]

times (119904120579MCP119891119890119888120579IP119891119890 + 119888120579MCP119891119890119904120579IP119891119890)

119876012= [119888120579TMC119886119887119886119889119888120579TMC119891119890]

times (minus119888120579MCP119891119890119904120579IP119891119890 minus 119904120579MCP119891119890119888120579IP119891119890)

+ [minus119888120579TMC119886119887119886119889119904120579TMC119891119890]

times (minus119904120579MCP119891119890119904120579IP119891119890 + 119888120579MCP119891119890119888120579IP119891119890)

119876013= 119904120579TMC119886119887119886119889

119876014= [119888120579TMC119886119887119886119889119888120579TMC119891119890]

times [1198711199050 119909

(119888120579MCP119891119890119888120579IP119891119890 minus 119904120579MCP119891119890119904120579IP119891119890)

+ 1198711199050 119910

(minus119888120579MCP119891119890119904120579IP119891119890 minus 119904120579MCP119891119890119888120579IP119891119890)

+ 11987110119888120579MCP119891119890 + 11987100]

+ [minus119888120579TMC119886119887119886119889119904120579TMC119891119890]

times [1198711199050 119909

(119904120579MCP119891119890119888120579IP119891119890 + 119888120579MCP119891119890119904120579IP119891119890)

+ 1198711199050 119910

(minus119904120579MCP119891119890119904120579IP119891119890 + 119888120579MCP119891119890119888120579IP119891119890)

+ 11987110119904120579MCP119891119890] + 119871 1199050 119911119904120579TMC119886119887119886119889


119876021= [119904120579TMC119886119887119886119889119888120579TMC119891119890]


+ [minus119904120579TMC119886119887119886119889119904120579TMC119891119890]


119876022= [119904120579TMC119886119887119886119889119888120579TMC119891119890]


+ [minus119904120579TMC119886119887119886119889119904120579TMC119891119890]


119876023= minus 119888120579TMC119886119887119886119889

119876024= [119904120579TMC119886119887119886119889119888120579TMC119891119890]

times [1198711199050 119909


+ 1198711199050 119910


+ 11987110119888120579MCP119891119890 + 11987100]

+ [minus119904120579TMC119886119887119886119889119904120579TMC119891119890]

times [1198711199050 119909

(119904120579MCP119891119890119888120579IP119891119890 + 119888120579MCP119891119890119904120579IP119891119890)

+ 1198711199050 119910


+ 11987110119904120579MCP119891119890] minus 119888120579TMC119886119887119886119889119871 1199050 119911

119876031= 119904120579TMC119891119890 (119888120579MCP119891119890119888120579IP119891119890 minus 119904120579MCP119891119890119904120579IP119891119890)

+ 119888120579TMC119891119890 (119904120579MCP119891119890119888120579IP119891119890 + 119888120579MCP119891119890119904120579IP119891119890)

119876032= 119904120579TMC119891119890 (minus119888120579MCP119891119890119904120579IP119891119890 minus 119904120579MCP119891119890119888120579IP119891119890)

+ 119888120579TMC119891119890 (minus119904120579MCP119891119890119904120579IP119891119890 + 119888120579MCP119891119890119888120579IP119891119890)

119876033= 0

119876034= 119904120579TMC119891119890

times [1198711199050 119909


+ 1198711199050 119910


+ 11987110119888120579MCP119891119890 + 11987100] + 119888120579TMC119891119890

times [1198711199050 119909

(119904120579MCP119891119890119888120579IP119891119890 + 119888120579MCP119891119890119904120579IP119891119890)

+ 1198711199050 119910


+11987110119904120579MCP119891119890]

(A2)

B Dynamic Equation Terms

Consider

1198601= 1198981[(minus11988711199091199041205791minus 11988711199101198881205791) (minus1198871119909s1205791minus 1198871119910c1205791)

+ (11988711199091198881205791minus 11988711199101199041205791) (1198871119909c1205791minus 1198871119910s1205791)]

+ 1198982[(minus11987111199091199041205791minus 11987111199101198881205791) (minus119871

11199091199041205791minus 11987111199101198881205791)

+ (11987111199091198881205791minus 11987111199101199041205791) (11987111199091198881205791minus 11987111199101199041205791)]

+ 1198983[(minus11987111199091199041205791minus 11987111199101198881205791) (minus11987111199091199041205791minus 11987111199101198881205791)

+(11987111199091198881205791minus 11987111199101199041205791)times (119871

11199091198881205791minus 11987111199101199041205791)]+1198681

1198602= 1198982[(minus119887211990911990412057912minus 119887211991011988812057912) (minus119871

11199091199041205791minus 11987111199101198881205791)

+ (+119887211990911988812057912minus 119887211991011990412057912) (11987111199091198881205791minus 11987111199101199041205791)]


+ (+119871211990911988812057912minus 119871211991011990412057912) (11987111199091198881205791minus 11987111199101199041205791)]

1198603= 1198983[(minus1198873119909119904120579123minus 1198873119910119888120579123) (minus11987111199091199041205791minus 11987111199101198881205791)

+ (+1198873119909119888120579123minus 1198873119910119904120579123) (11987111199091198881205791minus 11987111199101199041205791)]

1198604= 1198981[(minus1198871119909

12057911199041205791minus 1198871119910

12057911198881205791) (minus1198871119909

1205791c1205791+ 1198871119910

12057911199041205791)

+ (11988711199091198881205791minus 11988711199101199041205791) (minus1198871119909s1205791minus 11988711199101198881205791)1205791]

+ 1198982[minus (minus119871

1119909

1205792

1

1198881205791+ 1198711119910

1205792

1

1199041205791

minus1198872119909

1205792

12

11988812057912+ 1198872119910

1205792

12

11990412057912)

times (minus11987111199091199041205791minus 11987111199101198881205791)

minus (minus1198711119909

1205792

1

1199041205791minus 1198711119910

1205792

1

1198881205791minus 1198872119909

1205792

12

11990412057912

minus1198872119910

1205792

12

11988812057912) (11987111199091198881205791minus 11987111199101199041205791)]

+ 1198983[minus (minus119871

1119909

1205792

1

1198881205791+ 1198711119910

1205792

1

1199041205791minus1198712119909

1205792

12

11988812057912

+1198712119910

1205792

12

11990412057912minus1198873119909

1205792

123

119888120579123+ 1198873119910

1205792

123

119904120579123)



1205792

1

1199041205791minus 1198711119910

1205792

1

1198881205791minus 1198712119909

1205792

12

11990412057912


minus1198712119910

1205792

12

11988812057912minus 1198873119909

1205792

123

119904120579123minus 1198873119910

1205792

123

119888120579123)

times (11987111199091198881205791minus 11987111199101199041205791)]

minus (1198981119892 (11988711199091198881205791minus 11988711199101199041205791) + (119898

2+ 1198983) 119892

times (11987111199091198881205791minus 11987111199101199041205791) + 11989611205791minus 1198962(12057912minus 1205791))

minus 1198881

1205791+ 1198882(12057912minus1205791)


+ (11987111199091198881205791minus 11987111199101199041205791) (119887211990911988812057912minus 119887211991011990412057912)]


+ (11987111199091198881205791minus 11987111199101199041205791) (119871211990911988812057912minus 119871211991011990412057912)]


+ (+119887211990911988812057912minus 119887211991011990412057912) (119887211990911988812057912minus 119887211991011990412057912)]


+ (+119871211990911988812057912) (119871211990911988812057912minus 119871211991011990412057912)] + 119868

2


+ (+1198873119909119888120579123minus 1198873119910119904120579123) (119871211990911988812057912minus 119871211991011990412057912)]

1198614= 1198982[minus (minus119871

1119909

1205792

1

1198881205791+ 1198711119910

1205792

1

1199041205791minus1198872119909

1205792

12

11988812057912

+1198872119910

1205792

12

11990412057912) (minus119887211990911990412057912minus 119887211991011988812057912)


1205792

1

1199041205791minus 1198711119910

1205792

1

1198881205791minus 1198872119909

1205792

12

11990412057912

minus1198872119910

1205792

12

11988812057912) (119887211990911988812057912minus 119887211991011990412057912)]

+ 1198983[minus (minus119871

1119909

1205792

1

1198881205791+ 1198711119910

1205792

1

1199041205791minus1198712119909

1205792

12

11988812057912

+ 1198712119910

1205792

12

11990412057912minus1198873119909

1205792

123

119888120579123

+1198873119910

1205792

123

119904120579123) (minus119871211990911990412057912minus 119871211991011988812057912)


1205792

1

1199041205791minus 1198711119910

1205792

1

1198881205791minus 1198712119909

1205792

12

11990412057912

minus 1198712119910

1205792

12

11988812057912minus 1198873119909

1205792

123

119904120579123

minus1198873119910

1205792

123

119888120579123) (119871211990911988812057912minus 119871211991011990412057912)]

minus (1198982119892 (119887211990911988812057912minus 119887211991011990412057912)

+ 1198983119892 (119871211990911988812057912minus 119871211991011990412057912)

+1198962(12057912minus 1205791) minus 1198963(120579123minus 12057912))

minus 1198882(12057912minus1205791) + 1198883(120579123minus12057912)


+ (11987111199091198881205791minus 11987111199101199041205791) (1198873119909119888120579123minus 1198873119910119904120579123)]


times (+119871211990911988812057912minus 119871211991011990412057912) (1198873119909119888120579123minus 1198873119910119904120579123)]


+ (+1198873119909119888120579123minus 1198873119910119904120579123)

times (1198873119909119888120579123minus 1198873119910119904120579123)] + 119868

3

1198624= 1198983[minus (minus119871

1119909

1205792

1

1198881205791+ 1198711119910

1205792

1

1199041205791minus1198712119909

1205792

12

11988812057912

+ 1198712119910

1205792

12

11990412057912minus1198873119909

1205792

123

119888120579123

+1198873119910

1205792

123

119904120579123) (minus1198873119909119904120579123minus 1198873119910119888120579123)


1205792

1

1199041205791minus 1198711119910

1205792

1

1198881205791minus 1198712119909

1205792

12

11990412057912

minus 1198712119910

1205791211990412057912minus 1198712119910

1205792

12

11988812057912+ 1198873119909

120579123119888120579123

minus1198873119909

1205792

123

119904120579123minus 1198873119910

1205792

123

119888120579123)

times (1198873119909119888120579123minus 1198873119910119904120579123)]

minus (1198982119892 (119887211990911988812057912minus 119887211991011990412057912)+ 1198983119892 (119871211990911988812057912minus 119871211991011990412057912)

+1198962(12057912minus 1205791) minus 1198963(120579123minus 12057912)) minus 1198883(120579123minus12057912)

(B1)

References

[1] ldquoEurobot Ground Prototyperdquo httpwwwesaintesaHSSEM9NCZGRMG research 0html

[2] ldquoWhat is a Robonautrdquo httprobonautjscnasagovdefaultasp[3] N C Jordan J H Saleh andD J Newman ldquoThe extravehicular

mobility unit a review of environment requirements anddesign changes in the US spacesuitrdquo Acta Astronautica vol 59no 12 pp 1135ndash1145 2006

[4] C N Schabowsky S B Godfrey R J Holley and P S LumldquoDevelopment and pilot testing of HEXORR Hand exoskeletonrehabilitation robotrdquo Journal of NeuroEngineering and Rehabil-itation vol 7 no 1 article 36 2010

[5] httpwwwnasagovmission pagesstationmainrobo-glovehtml

[6] T TWorsnopp M A Peshkin J E Colgate and D G KamperldquoAn actuated finger exoskeleton for hand rehabilitation follow-ing strokerdquo in Proceedings of the 10th IEEE International Confer-ence on Rehabilitation Robotics (ICORR rsquo07) pp 896ndash901Noordwijk The Netherlands June 2007

[7] N S K Ho K Y Tong X L Hu et al ldquoAn EMG-driven exoskel-eton hand robotic training device on chronic stroke subjectstask training system for stroke rehabilitationrdquo in Proceedingsof the IEEE International Conference on Rehabilitation Robotics(Rehab Week Zurich) ETH Zurich Science City SwitzerlandJuly 2011


[8] P Cerveri N Lopomo A Pedotti and G Ferrigno ldquoDerivationof centers and axes of rotation for wrist and fingers in a handkinematicmodelmethods and reliability resultsrdquoAnnals of Bio-medical Engineering vol 33 no 3 pp 402ndash412 2005

[9] J W Garrett ldquoAnthropometry of the hands of male air forceflight personnelrdquo Tech Rep Aerospace Medical Research Lab-oratory Air Force Systems Command Wright-Patterson AirForce Base Ohio USA 1970

[10] J W Garrett ldquoAnthropometry of the hands of female air forceflight personnelrdquo Tech Rep Aerospace Medical Research Lab-oratory Air Force Systems Command Wright-Patterson AirForce Base Ohio USA 1970

[11] S R Habib andNN Kamal ldquoStature estimation fromhand andphalanges lengths of Egyptiansrdquo Journal of Forensic and LegalMedicine vol 17 no 3 pp 156ndash160 2010

[12] O P Jasuja and G Singh ldquoEstimation of stature from hand andphalange lengthrdquo Journal of Indian Academy of Forensic Medi-cine vol 26 no 3 pp 100ndash106 2004

[13] S Cobos M Ferre M A Sanchez-Uran J Ortego and C PenaldquoEfficient human hand kinematics for manipulation tasksrdquo inProceedings of the IEEERSJ International Conference on Intelli-gent Robots and Systems (IROS rsquo08) pp 2246ndash2251 AcropolisConvention Center Nice France September 2008

[14] S Cobos M Ferre M Sanchez and J Ortego ldquoConstraints forrealistic hand manipulationrdquo in Proceedings of the 10th AnnualInternational Workshop on Presence (PRESENCE rsquo07) pp 369ndash370 Barcelona Spain October 2007

[15] H Rijpkema and M Girard ldquoComputer animation of knowl-edge-based human graspingrdquo in Proceedings of the 18th AnnualConference on Computer Graphics and Interactive Techniques(SIGGRAPH rsquo91) pp 339ndash348 1991

[16] J J Craig Introduction to RoboticsMechanics andControl Pear-son Education International 3rd edition 1986

[17] T E Milner and DW Franklin ldquoCharacterization of multijointfinger stiffness dependence on finger posture and force direc-tionrdquo IEEE Transactions on Biomedical Engineering vol 45 no11 pp 1363ndash1375 1998

[18] C J Hasser Force-Reflecting Anthropomorphic Hand MastersArmstrong Lab Crew Systems Directorate Wright-PattersonAir Force Base Ohio USA 1995

[19] A D Deshpande N Gialias and Y Matsuoka ldquoContributionsof intrinsic visco-elastic torques during planar index finger andwrist movementsrdquo IEEE Transactions on Biomedical Engineer-ing vol 59 no 2 pp 586ndash594 2012

[20] A Z Hajian and R D Howe ldquoIdentification of the mechanicalimpedance at the human finger tiprdquo Journal of BiomechanicalEngineering vol 119 no 1 pp 109ndash114 1997

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



2 Human Hand Data

The human hand is composed of 5 digits 4 fingers (indexring middle and little fingers) and the thumb The thumb ischaracterized by three articulations and three phalanges Thefingers also comprise three different articulations and fourphalanges Figure 1 shows names and acronyms of each artic-ulation and phalanx The wrist has two functional DoFs TheTMC joint of the thumb holds two DoFs (flexionextensionand adductionabduction) A single DoF (flexionextension)characterizes the MCP and IP joints of the thumb as well asthe PIP andDIP joints of the fingersWhereas the eight bonesof the carpus articulate finelywith each other producing smalldeformations their representation in a single rigid segment isa consistent approximation [8]

The analysis of kinematics and dynamics requires knowl-edge regarding the dimensions of the fingers and of the palmand their respective range of motion (RoM) those data arereported below Tables 1 and 2 show the results of Garrettrsquosstudies [9 10] for finger lengths and palm dimensions meas-urements that were taken from the right hands of 148 menand 211 women In Table 1 crotch to tip is the distance alongthe axis of the digit from the midpoint of its tip to the levelof the corresponding webbed crotch between two digit wristcrease to tip is the distance along the axis of the digit from themidpoint of its tip to the wrist crease baseline

A few researchers have measured the length of each pha-lanx separately A study with a variety of candidates is the oneperformed byHabib andKamal [11]The results of their studyfor each phalanx of index middle ring and little fingers arein Table 3 (I1 means distal phalanx I2 middle phalanx andI3 proximal phalanx of the index finger The same notationis used for other digits) A similar survey is also present inJasuja and Singhrsquos study [12] As shown in Table 3 the averagedimensions of the hand are quite similar to the ones presentedin Table 2 with a maximum difference of 218 The datapresented here is the sample distribution over geograph-ic regions and Air Force Commands which may be quiterepresentative of possible EVA glove users

As mentioned above those data can give us an idea ofthe mean values of the length of each element that composesthe human hand With proper modifications this data canalso be used to compose a model to simulate its movementsin particular it must be taken into account that the distancebetween a joint and a digit webbing is not representative ofthe equivalent link length A simulation was performed andthe results are presented in Section 6

3 Constraints Overview

Hand and digit motions are subject to several constraints thatlimit the range of the natural movements of human fingersConstraints can be roughly divided into three types staticintrafinger and interfinger constraints Intra- and inter-fin-ger constraints are often called dynamic constraints that inthe fingers of the hand are essentially constraints betweenjoint motions However this range of movement is somewhatambiguous because the range depends on various factorsinvolving human hand biomechanics therefore they are

Table 1 Mean finger lengths and palm dimensions of USAF male(M)female (F) flying personnel [9 10] (cm)

Finger length(crotch to tip)

Finger length(wrist crease to tip)

Mean sd lt5 lt95 Mean sd lt5 lt95M

Thumb 587 045 507 657 1270 113 1105 1468Index 753 046 683 819 1852 088 1733 2006Middle 857 051 782 974 1952 092 1810 2104Ring 80 047 744 893 1872 091 1752 2028Little 614 047 544 699 1661 091 1511 1810

FThumb 537 044 468 612 1105 100 951 1283Index 690 052 610 780 1667 089 1521 1814Middle 779 051 701 868 1765 087 1622 1905Ring 731 052 652 822 1676 894 1528 1820Little 546 044 480 624 1464 092 1311 1612

Table 2 Average hand dimensions of USAF male (M)female (F)flying personnel [9 10] (cm)

Joint Mean lengthHand length

M 1972F 1793

Hand breadthM 896F 771

Hand circumferenceM 2159F 1871

Hand thicknessM 329F 276

Hand depthM 619F 517

difficult to be expressed in closed forms (ie equations)and how to model such constraints still needs further inves-tigation The significance of taking finger constraints intoconsideration is that this causes the DoFs of the human handto decrease It is important to underline the fact that differentindividuals can show significant variations regarding the fol-lowing constraints

31 Intrafinger Constraints Intrafinger constraints are con-straints between different joints in the same finger Cobos etal [14] presented several constraints for fingers and thumbEquation (1) presents the relation between the joints of a fin-ger as first proposed by Rijpkema and Girard [15]

120579DIP asymp2

3

120579PIP120579PIP asymp3

4

120579MCP119891119890 (1)


Thumb

Index

MiddleRing


Middle phalanx

Proximal phalanx

Metacarpal




CMC carpometacarpal

Distal phalanx

Proximal phalanx

Metacarpal

IP interphalangeal







120579IP asymp1

2

120579MCP119891119890

120579MCP119891119890 asymp5

4

120579TMC119891119890

(2)















Table4Intrafinger

constraintsfor

acirc

ular

grasp

Thum

bIndex

Middle

Ring

Little

Noconstraint

120579CM

C 119891119890Index=120579CM

C 119891119890Middle

120579CM

C 119891119890Middle=(12)120579CM

C 119891119890Ring

120579CM

C 119891119890Ring=(23)120579CM

C 119891119890Little

Noconstraint

120579TM

C 119891119890=(1110)120579MCP119891119890

Noconstraint



Noconstraint

120579MCP119891119890=(45)120579IP119891119890



120579MCP119891119890Ring=(43)120579PIP 119891119890Ring


Noconstraint



120579PIP 119891119890Ring=(32)120579DIP119891119890Ring



Table5Intrafinger

constraintsfor

apris

maticgrasp

Thum

bIndex

Middle

Ring

Little

Noconstraint

120579CM

C 119891119890Index=120579CM

C 119891119890Middle

120579CM

C 119891119890Middle=(12)120579CM

C 119891119890Ring

120579CM

C 119891119890Ring=(23)120579CM

C 119891119890Little

Noconstraint

120579TM

C 119891119890=(1110)120579MCP119891119890

Noconstraint



Noconstraint

120579MCP119891119890=(65)120579IP119891119890



120579MCP119891119890Ring=(32)120579PIP 119891119890Ring


Noconstraint



120579PIP 119891119890Ring=2120579DIP119891119890Ring






120579MCP119891119890Ring asymp7

12

120579MCP119891119890Little


3

120579MCP119891119890Ring



(6)




5

120579MCP119891119890Index (7)




4 Kinematic Model





MCP 75∘ndash80∘ 0∘ 5∘



MCP 90∘ 30∘ndash40∘ 60∘

PIP 110∘ 0∘ 0∘

DIP 80∘ndash90∘ 5∘ 0∘


MCP 90∘ 30∘ndash40∘ 45∘

PIP 110∘ 0∘ 0∘

DIP 80∘ndash90∘ 5∘ 0∘


MCP 90∘ 30∘ndash40∘ 45∘

PIP 120∘ 0∘ 0∘

DIP 80∘ndash90∘ 5∘ 0∘


MCP 90∘ 30∘ndash40∘ 50∘

PIP 135∘ 0∘ 0∘

DIP 90∘ 5∘ 0∘







Distal link

Middle link

Proximal link

Metacarpal link

Carpal zone

Wrist Reference

0

1

23

4

L01L02

L03L04

L11

L12L13

L14

L21

L22L23

L24

L31

L32L33

L34

pt1

pt2pt3

pt4

pt0

L20

L00L10

R0R0

R1i

R2i

R3i

R5i

R5iR4i

R4i

TMCaa

MCPaa

TMCfe

IPfe

MCPfeCMCfe

MCPfePIPfe

DIPfe

R0i = R1i

R2i = R3i




119876119894=0

0

119879119894sdot0

6119879119894

(120579119895) =0

0

119879119894sdot

5

prod

119895=1

(119895minus1)

119895

119879119894(120579119895) sdot5

6

119879119905119894

119876119894=0

0

119879119894sdot0

1119879119894

(120579CMC119891119890) sdot1

2119879119894

(120579MCP119886119887119886119889) sdot2

3119879119894

(120579MCP119891119890)

sdot3

4119879119894

(120579PIP119891119890) sdot4

5119879119894

(120579DIP119891119890) sdot5

6

119879119905119894

(8)where







(iii) 06





(iv) (119895minus1)119895




Finger 119894 (119894 = 1 4)Joint 120572

119895minus1

119886119895minus1

119889119895

120579119895

119895 = 1 1205872 0 0 120579CMC119891119890

119895 = 2 minus1205872 1198710119894

0 120579MCP119886119887119886119889

119895 = 3 1205872 0 0 120579MCP119891119890

119895 = 4 0 1198711119894

0 120579PIP119891119890

119895 = 5 0 1198712119894

0 120579DIP119891119890

(v) 56


119905119894= [119897119905119894119909119897119905119894119910119897119905119894119911]

119879



119891119890 MCP

119886119887119886119889

MCP119891119890

PIP119891119890

and DIP119891119890









119886119895minus1

119889119895

120579119895

119895 = 1 0 0 0 120579TMC119886119887119886119889

119895 = 2 1205872 0 0 120579TMC119891119890

119895 = 3 0 11987100

0 120579MCP119891119890

119895 = 4 0 11987110

0 120579IP119891119890


1198760=0

0

1198790sdot0

5

1198790(120579119896)

1198760=0

0

1198790sdot0

11198790

(120579TMC119886119887119886119889) sdot1

21198790

(120579TMC119891119890)

sdot2

31198790

(120579MCP119891119890) sdot3

41198790

(120579IP119891119890) sdot4

51198791199050

(9)


TMC119891119890

MCP119891119890

and IP119891119890


[1198971199050119909

1198971199050119910

1198971199050119911]

119879






119894


119891119890

MCP119886119887119886119889

and DIP119891119890



120579CMC119891119890 = atan(11987633

11987613

)

120579MCP119886119887119886119889 = atan[

[

11987613

minus11987623cos (120579CMC119891119890)

]

]

120579DIP119891119890 = atan(12059811205982

)

(10)



1205981=

(minus1) sdot (11989811987622+ 11989911987621)

(1198982

minus 1198992

)

1205982=

Q21

119898

+

119899

119898

sdot

(minus1) sdot (11989811987622+ 11989911987621)

(1198982

minus 1198992

)





(11)

TheMCP119891119890

PIP119891119890




expression

1= 1198651minus [11987131lowast ] (12)


1198751= [119875111990911987511199101198751119911]

119879

(13)

where1198751119909= 11987101cos (120579CMC)

1198751119910= 11987101sin (120579CMC)

1198751119911= 0

(14)


Knowing 1and 119875


1is


Angles 1205931and 120593

2are expressed as

1205931= acos(

1199032

2

+ 1198712

01

minus 1199032

1

2119903211989701

)

1205932= acos(

1199032

2

+ 1198712

11

minus 1198712

21

2119903211987111

)

(16)


100381710038171003817100381710038171

10038171003817100381710038171003817

1199032=10038171003817100381710038171199061

1003817100381710038171003817

(17)


120579MCP119891119890 = 120587 minus 1205931 minus 1205932 (18)


120579PIP119891119890 = 120587 minus 1205933 (19)

where

1205933= acos(

1198712

21

+ 1198712

11

minus 1199032

2

21198712111987111

) (20)




120579TMC119886119887119886119889 = atan(11987613

minus11987623

) (21)




120579MCP119891119890 = 120587 minus 1205741

120579IP119891119890 = 120587 minus 1205742 minus 1205743(22)

where

1205741= acos(

1198712

10

+ 1198712

00

minus 1199032

3

21198711011987100

)

1205742= acos(

1198712

20

+ 1199042

3

minus 1199032

4

2119871201199033

)

1205743= acos(

1199032

3

+ 1198712

10

minus 1198712

00

211990311990411987110

)

1199033=

100381710038171003817100381710038170

10038171003817100381710038171003817

1199034=

10038171003817100381710038171003817

1198650

10038171003817100381710038171003817

0= 1198650minus [11987120lowast ]

(23)


120579TMC119891119890 = atan(12059841205983

) (24)

where

1205983= (11987632+ 11987631) cos 120583

1205984=


cos 120583

120583 = 120579MCP + 120579IP

(25)



2is the


2




1205793= 120579MCP119891119890

1205794= 120579PIP119891119890

1205795= 120579DIP119891119890

(26)




119892= [

[

119887119892119909

119887119892119910

0

]

]

119892 = 1 2 3 (27)





119895and 119904120579

119895stand for


1= [

[

11988711199091198881205793minus 11988711199101199041205793

11988711199091199041205793+ 11988711199101198881205793

0

]

]

2= [

[

11987111198881205793

11987111199041205793

0

]

]

+

[[[

[

1198872119909119888 (1205793+ 1205794) minus 1198872119910119904 (1205793+ 1205794)

1198872119909119904 (1205793+ 1205794) + 1198872119910119888 (1205793+ 1205794)

0

]]]

]



3= [

[

11987111198881205793

11987111199041205793

0

]

]

+[[

[

1198712119888 (1205793+ 1205794)

1198712119904 (1205793+ 1205794)

0

]]

]

+

[[[

[

1198873119909119888 (1205793+ 1205794+ 1205795) minus 1198873119910119904 (1205793+ 1205794+ 1205795)

1198873119909119904 (1205793+ 1205794+ 1205795) + 1198873119910119888 (1205793+ 1205794+ 1205795)

0

]]]

]

(28)


1205793+ 1205794= 12057934

1205793+1205794=

12057934

1205793+ 1205794+ 1205795= 120579345

1205793+1205794+1205795=

120579345

(29)


V1198661=[[

[

minus1198871119909

12057931199041205793minus 1198871119910

12057931198881205793

1198871119909

12057931198881205793minus 1198871119910

12057931199041205793

0

]]

]

V1198662=[[

[

minus1198711

12057931199041205793minus1198872119909

1205793411990412057934minus 1198872119910

1205793411988812057934

1198711

12057931198881205793+ 1198872119909

1205793411988812057934minus 1198872119910

1205793411990412057934

0

]]

]

V1198663=[[

[

minus1198711

12057931199041205793minus1198712

1205793411990412057934minus1198873119909

120579345119904120579345minus 1198873119910

120579345119888120579345

1198711

12057931198881205793+ 1198712

1205793411988812057934+ 1198873119909

120579345119888120579345minus 1198873119910

120579345119904120579345

0

]]

]

(30)


119879 =

1

2

1198981(V1198661)

2

+

1

2

1198982(V1198662)

2

+

1

2

1198983(V1198663)

2

+

1

2

1198681(1205793)

2

+

1

2

1198682(12057934)

2

+

1

2

1198683(120579345)

2

(31)




1 1198962 and 119896



119880 = 1198981119892 (11988711199091199041205793+ 11988711199101198881205793)

+ 1198982119892 (11987111199041205793+ 119887211990911990412057934+ 119887211991011988812057934)

+ 1198983119892 (11987111199041205793+ 119871211990412057934+ 1198873119909119904120579345+ 1198873119910119888120579345)

+

1

2

1198961(1205793)2

+

1

2

1198962(12057934minus 1205793)2

+

1

2

1198963(120579345minus 12057934)2

(32)


119865 =

1

2

1198881(1205793)

2

+

1

2

1198882(12057934minus1205793)

2

+

1

2

1198883(120579345minus12057934)

2

(33)



1

1198882 and 119888


joints respectively



3 12057934 and 120579

345

119889

119889119905

(

120597 (119879 minus 119880)

1205971205793

) minus

120597 (119879 minus 119880)

1205971205793

+

120597119865

1205971205793

= 1205913

119889

119889119905

(

120597 (119879 minus 119880)

12059712057934

) minus

120597 (119879 minus 119880)

12059712057934

+

120597119865

12059712057934

= 1205914

119889

119889119905

(

120597 (119879 minus 119880)

120597120579345

) minus

120597 (119879 minus 119880)

120597120579345

+

120597119865

120597120579345

= 1205915

(34)




120591119895=

(sum5

119895=3

120575119882119895)

120575120579119895

119895 = 3 4 5 (35)



1205913= (11986531199101198903119909minus 11986531199091198903119910) + [

[

minus11987111199041205793

11987111198881205793

0

]

]

sdot [

[

1198654119909c12057934minus 1198654119910s12057934

1198654119909s12057934+ 1198654119910c12057934

0

]

]

+ [

[

minus11987111199041205793

11987111198881205793

0

]

]

sdot [

[

1198655119909c120579345minus 1198655119910s120579345

1198655119909s120579345+ 1198655119910c120579345

0

]

]

+ 1198621198981minus 1198621198982

1205914= (11986541199101198904119909minus 11986541199091198904119910) + [

[

minus119871211990412057934

119871211988812057934

0

]

]

sdot [

[

1198655119909c120579345minus 1198655119910s120579345

1198655119909s120579345+ 1198655119910c120579345

0

]

]

+ 1198621198982minus 1198621198983

1205915= (11986551199101198905119909minus 11986551199091198905119910) + 1198621198983

(36)

where the term 119862119898119892


119895= [1198651198951199091198651198951199100]119879 is the contact


119895= [1198901198951199091198901198941199100]119879


1198601

1205793+ 1198602

12057934+ 1198603

120579345= 1198604

1198611

1205793+ 1198612

12057934+ 1198613

120579345= 1198614

1198621

1205793+ 1198622

12057934+ 1198623

120579345= 1198624

(37)


The terms 1198601 1198602 and 119860










119895= (1198631198952)(cos(120587119905119905

1)minus1) where




120579119895 max = | minus 12058711986311989521199051|





Omega maxMCP (rads)

Omega maxPIP (rads)

Omega maxDIP (rads)

Cycles in 10 s 27 27 31 25 20 29 23 27 21 31 261 3843 261 12164 15026 10018


Omega maxMCP (rads)

Omega maxPIP (rads)

Omega maxDIP (rads)

Cycles in 10 s 27 24 17 34 29 27 30 24 21 27 26 4784 26 12118 14969 9979



1198983are


119898max=



119889119895= |119862119889119895||119862119898119895|



0 20 40 60 80 100minus100

minus50

0

50

100

150

200

250

300

Cycle ()

C(N

mm

)

Cm1

Cm2

Cm3



MCP PIP DIP119891119896119895

3805 3848 3858119891119888119895

5758 6059 61119891119889119895

437 093 042



7 Conclusions




1198981

= 705 g 1198982

= 397 g 1198983

= 270 g1198681

= 1000 gmm2 1198682

= 240 gmm2 1198683

= 120 gmm2

1198961

= 00902Nm 1198962

= 00902Nm 1198963

= 00902Nm1198881

= 00225Nms 1198882

= 00225Nms 1198883

= 00225Nms1198711

= [50 0 0]

119879

mm 1198712

= [20 0 0]

119879

mm 1198713

= [25 0 0]

119879

mm1198871

= [25 0 0]

119879

mm 1198872

= [10 0 0]

119879

mm 1198873

= [125 0 0]

119879

mm1198901

= [25 minus8 0]

119879

mm 1198902

= [10 minus7 0]

119879

mm 1198903

= [12 minus6 0]

119879

mm



Appendices



11987611989411= [(119888120579CMC119891119890119888120579MCP119886119887119886119889) 119888120579MCP119891119890

minus 119904120579CMC119891119890119904120579MCP119891119890]


minus [(119888120579CMC119891119890119888120579MCP119886119887119886119889) 119904120579MCP119891119890

+ 119904120579CMC119891119890119888120579MCP119891119890]


11987611989412= [(119888120579CMC119891119890119888120579MCP119886119887119886119889) 119888120579MCP119891119890

minus 119904120579CMC119891119890119904120579MCP119891119890]


minus [(119888120579CMC119891119890119888120579MCP119886119887119886119889) 119904120579MCP119891119890

+ 119904120579CMC119891119890119888120579MCP119891119890]


11987611989413= 119888120579CMC119891119890119904120579MCP119886119887119886119889

11987611989414= [(119888120579CMC119891119890119888120579MCP119886119887119886119889) 119888120579MCP119891119890

minus 119904120579CMC119891119890119904120579MCP119891119890]



+ 1198711119894+ 1198712119894119888120579PIP119891119890]

minus [(119888120579CMC119891119890119888120579MCP119886119887119886119889) 119904120579MCP119891119890

+ 119904120579CMC119891119890119888120579MCP119891119890]



+ 1198712119894119904120579PIP119891119890]

+ (119888120579CMC119891119890119904120579MCP119886119887119886119889) 119871 119905119894119911 + 1198710119894119888120579CMC119891119890

11987611989421= 119904120579MCP119886119887119886119889119888120579MCP119891119890


minus 119904120579MCP119886119887119886119889119904120579MCP119891119890


11987611989422= 119904120579MCP119886119887119886119889119888120579MCP119891119890



minus 119904120579MCP119886119887119886119889119904120579MCP119891119890


11987611989423= minus 119888120579MCP119886119887119886119889

11987611989424= 119904120579MCP119886119887119886119889119888120579MCP119891119890



+ 1198711119894+ 1198712119894119888120579PIP119891119890] minus 119904120579MCP119886119887119886119889119904120579MCP119891119890



+ 1198712119894119904120579PIP119891119890] + (minus119888120579MCP119886119887119886119889) 119871 119905119894119911

11987611989431= (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119888120579MCP119891119890

+ 119888120579CMC119891119890119904120579MCP119891119890


+ minus (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119904120579MCP119891119890

+ 119888120579CMC119891119890119888120579MCP119891119890


11987611989432= (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119888120579MCP119891119890

+ 119888120579CMC119891119890119904120579MCP119891119890


+ minus (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119904120579MCP119891119890

+ 119888120579CMC119891119890119888120579MCP119891119890


11987611989433= 119904120579CMC119891119890119904120579MCP119886119887119886119889

11987611989434= (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119888120579MCP119891119890

+ 119888120579119862119872119862119891119890

119904120579119872119862119875119891119890



+ 1198711119894+ 1198712119894119888120579PIP119891119890]

+ minus (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119904120579MCP119891119890

+ 119888120579CMC119891119890119888120579MCP119891119890



+ 1198712119894119904120579PIP119891119890]

+ (119904120579CMC119891119890119904120579MCP119886119887119886119889) 119871 119905119894119911 + 1198710119894119904120579CMC119891119890

(A1)


119876011= [119888120579TMC119886119887119886119889119888120579TMC119891119890]


+ [minus119888120579TMC119886119887119886119889119904120579TMC119891119890]


119876012= [119888120579TMC119886119887119886119889119888120579TMC119891119890]


+ [minus119888120579TMC119886119887119886119889119904120579TMC119891119890]


119876013= 119904120579TMC119886119887119886119889

119876014= [119888120579TMC119886119887119886119889119888120579TMC119891119890]

times [1198711199050 119909


+ 1198711199050 119910


+ 11987110119888120579MCP119891119890 + 11987100]

+ [minus119888120579TMC119886119887119886119889119904120579TMC119891119890]

times [1198711199050 119909

(119904120579MCP119891119890119888120579IP119891119890 + 119888120579MCP119891119890119904120579IP119891119890)

+ 1198711199050 119910


+ 11987110119904120579MCP119891119890] + 119871 1199050 119911119904120579TMC119886119887119886119889


119876021= [119904120579TMC119886119887119886119889119888120579TMC119891119890]


+ [minus119904120579TMC119886119887119886119889119904120579TMC119891119890]


119876022= [119904120579TMC119886119887119886119889119888120579TMC119891119890]


+ [minus119904120579TMC119886119887119886119889119904120579TMC119891119890]


119876023= minus 119888120579TMC119886119887119886119889

119876024= [119904120579TMC119886119887119886119889119888120579TMC119891119890]

times [1198711199050 119909


+ 1198711199050 119910


+ 11987110119888120579MCP119891119890 + 11987100]

+ [minus119904120579TMC119886119887119886119889119904120579TMC119891119890]

times [1198711199050 119909

(119904120579MCP119891119890119888120579IP119891119890 + 119888120579MCP119891119890119904120579IP119891119890)

+ 1198711199050 119910


+ 11987110119904120579MCP119891119890] minus 119888120579TMC119886119887119886119889119871 1199050 119911


+ 119888120579TMC119891119890 (119904120579MCP119891119890119888120579IP119891119890 + 119888120579MCP119891119890119904120579IP119891119890)



119876033= 0

119876034= 119904120579TMC119891119890

times [1198711199050 119909


+ 1198711199050 119910


+ 11987110119888120579MCP119891119890 + 11987100] + 119888120579TMC119891119890

times [1198711199050 119909

(119904120579MCP119891119890119888120579IP119891119890 + 119888120579MCP119891119890119904120579IP119891119890)

+ 1198711199050 119910


+11987110119904120579MCP119891119890]

(A2)


Consider


+ (11988711199091198881205791minus 11988711199101199041205791) (1198871119909c1205791minus 1198871119910s1205791)]


11199091199041205791minus 11987111199101198881205791)

+ (11987111199091198881205791minus 11987111199101199041205791) (11987111199091198881205791minus 11987111199101199041205791)]


+(11987111199091198881205791minus 11987111199101199041205791)times (119871

11199091198881205791minus 11987111199101199041205791)]+1198681


11199091199041205791minus 11987111199101198881205791)

+ (+119887211990911988812057912minus 119887211991011990412057912) (11987111199091198881205791minus 11987111199101199041205791)]


+ (+119871211990911988812057912minus 119871211991011990412057912) (11987111199091198881205791minus 11987111199101199041205791)]


+ (+1198873119909119888120579123minus 1198873119910119904120579123) (11987111199091198881205791minus 11987111199101199041205791)]

1198604= 1198981[(minus1198871119909

12057911199041205791minus 1198871119910

12057911198881205791) (minus1198871119909

1205791c1205791+ 1198871119910

12057911199041205791)


+ 1198982[minus (minus119871

1119909

1205792

1

1198881205791+ 1198711119910

1205792

1

1199041205791

minus1198872119909

1205792

12

11988812057912+ 1198872119910

1205792

12

11990412057912)



1205792

1

1199041205791minus 1198711119910

1205792

1

1198881205791minus 1198872119909

1205792

12

11990412057912

minus1198872119910

1205792

12

11988812057912) (11987111199091198881205791minus 11987111199101199041205791)]

+ 1198983[minus (minus119871

1119909

1205792

1

1198881205791+ 1198711119910

1205792

1

1199041205791minus1198712119909

1205792

12

11988812057912

+1198712119910

1205792

12

11990412057912minus1198873119909

1205792

123

119888120579123+ 1198873119910

1205792

123

119904120579123)



1205792

1

1199041205791minus 1198711119910

1205792

1

1198881205791minus 1198712119909

1205792

12

11990412057912


minus1198712119910

1205792

12

11988812057912minus 1198873119909

1205792

123

119904120579123minus 1198873119910

1205792

123

119888120579123)

times (11987111199091198881205791minus 11987111199101199041205791)]

minus (1198981119892 (11988711199091198881205791minus 11988711199101199041205791) + (119898

2+ 1198983) 119892


minus 1198881

1205791+ 1198882(12057912minus1205791)


+ (11987111199091198881205791minus 11987111199101199041205791) (119887211990911988812057912minus 119887211991011990412057912)]


+ (11987111199091198881205791minus 11987111199101199041205791) (119871211990911988812057912minus 119871211991011990412057912)]


+ (+119887211990911988812057912minus 119887211991011990412057912) (119887211990911988812057912minus 119887211991011990412057912)]


+ (+119871211990911988812057912) (119871211990911988812057912minus 119871211991011990412057912)] + 119868

2


+ (+1198873119909119888120579123minus 1198873119910119904120579123) (119871211990911988812057912minus 119871211991011990412057912)]

1198614= 1198982[minus (minus119871

1119909

1205792

1

1198881205791+ 1198711119910

1205792

1

1199041205791minus1198872119909

1205792

12

11988812057912

+1198872119910

1205792

12

11990412057912) (minus119887211990911990412057912minus 119887211991011988812057912)


1205792

1

1199041205791minus 1198711119910

1205792

1

1198881205791minus 1198872119909

1205792

12

11990412057912

minus1198872119910

1205792

12

11988812057912) (119887211990911988812057912minus 119887211991011990412057912)]

+ 1198983[minus (minus119871

1119909

1205792

1

1198881205791+ 1198711119910

1205792

1

1199041205791minus1198712119909

1205792

12

11988812057912

+ 1198712119910

1205792

12

11990412057912minus1198873119909

1205792

123

119888120579123

+1198873119910

1205792

123

119904120579123) (minus119871211990911990412057912minus 119871211991011988812057912)


1205792

1

1199041205791minus 1198711119910

1205792

1

1198881205791minus 1198712119909

1205792

12

11990412057912

minus 1198712119910

1205792

12

11988812057912minus 1198873119909

1205792

123

119904120579123

minus1198873119910

1205792

123

119888120579123) (119871211990911988812057912minus 119871211991011990412057912)]

minus (1198982119892 (119887211990911988812057912minus 119887211991011990412057912)

+ 1198983119892 (119871211990911988812057912minus 119871211991011990412057912)




+ (11987111199091198881205791minus 11987111199101199041205791) (1198873119909119888120579123minus 1198873119910119904120579123)]




+ (+1198873119909119888120579123minus 1198873119910119904120579123)

times (1198873119909119888120579123minus 1198873119910119904120579123)] + 119868

3

1198624= 1198983[minus (minus119871

1119909

1205792

1

1198881205791+ 1198711119910

1205792

1

1199041205791minus1198712119909

1205792

12

11988812057912

+ 1198712119910

1205792

12

11990412057912minus1198873119909

1205792

123

119888120579123

+1198873119910

1205792

123

119904120579123) (minus1198873119909119904120579123minus 1198873119910119888120579123)


1205792

1

1199041205791minus 1198711119910

1205792

1

1198881205791minus 1198712119909

1205792

12

11990412057912

minus 1198712119910

1205791211990412057912minus 1198712119910

1205792

12

11988812057912+ 1198873119909

120579123119888120579123

minus1198873119909

1205792

123

119904120579123minus 1198873119910

1205792

123

119888120579123)

times (1198873119909119888120579123minus 1198873119910119904120579123)]



(B1)

References
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Thumb

Index

MiddleRing


Middle phalanx

Proximal phalanx

Metacarpal




CMC carpometacarpal

Distal phalanx

Proximal phalanx

Metacarpal

IP interphalangeal







120579IP asymp1

2

120579MCP119891119890

120579MCP119891119890 asymp5

4

120579TMC119891119890

(2)















Table4Intrafinger

constraintsfor

acirc

ular

grasp

Thum

bIndex

Middle

Ring

Little

Noconstraint

120579CM

C 119891119890Index=120579CM

C 119891119890Middle

120579CM

C 119891119890Middle=(12)120579CM

C 119891119890Ring

120579CM

C 119891119890Ring=(23)120579CM

C 119891119890Little

Noconstraint

120579TM

C 119891119890=(1110)120579MCP119891119890

Noconstraint



Noconstraint

120579MCP119891119890=(45)120579IP119891119890



120579MCP119891119890Ring=(43)120579PIP 119891119890Ring


Noconstraint



120579PIP 119891119890Ring=(32)120579DIP119891119890Ring



Table5Intrafinger

constraintsfor

apris

maticgrasp

Thum

bIndex

Middle

Ring

Little

Noconstraint

120579CM

C 119891119890Index=120579CM

C 119891119890Middle

120579CM

C 119891119890Middle=(12)120579CM

C 119891119890Ring

120579CM

C 119891119890Ring=(23)120579CM

C 119891119890Little

Noconstraint

120579TM

C 119891119890=(1110)120579MCP119891119890

Noconstraint



Noconstraint

120579MCP119891119890=(65)120579IP119891119890



120579MCP119891119890Ring=(32)120579PIP 119891119890Ring


Noconstraint



120579PIP 119891119890Ring=2120579DIP119891119890Ring






120579MCP119891119890Ring asymp7

12

120579MCP119891119890Little


3

120579MCP119891119890Ring



(6)




5

120579MCP119891119890Index (7)




4 Kinematic Model





MCP 75∘ndash80∘ 0∘ 5∘



MCP 90∘ 30∘ndash40∘ 60∘

PIP 110∘ 0∘ 0∘

DIP 80∘ndash90∘ 5∘ 0∘


MCP 90∘ 30∘ndash40∘ 45∘

PIP 110∘ 0∘ 0∘

DIP 80∘ndash90∘ 5∘ 0∘


MCP 90∘ 30∘ndash40∘ 45∘

PIP 120∘ 0∘ 0∘

DIP 80∘ndash90∘ 5∘ 0∘


MCP 90∘ 30∘ndash40∘ 50∘

PIP 135∘ 0∘ 0∘

DIP 90∘ 5∘ 0∘







Distal link

Middle link

Proximal link

Metacarpal link

Carpal zone

Wrist Reference

0

1

23

4

L01L02

L03L04

L11

L12L13

L14

L21

L22L23

L24

L31

L32L33

L34

pt1

pt2pt3

pt4

pt0

L20

L00L10

R0R0

R1i

R2i

R3i

R5i

R5iR4i

R4i

TMCaa

MCPaa

TMCfe

IPfe

MCPfeCMCfe

MCPfePIPfe

DIPfe

R0i = R1i

R2i = R3i




119876119894=0

0

119879119894sdot0

6119879119894

(120579119895) =0

0

119879119894sdot

5

prod

119895=1

(119895minus1)

119895

119879119894(120579119895) sdot5

6

119879119905119894

119876119894=0

0

119879119894sdot0

1119879119894

(120579CMC119891119890) sdot1

2119879119894

(120579MCP119886119887119886119889) sdot2

3119879119894

(120579MCP119891119890)

sdot3

4119879119894

(120579PIP119891119890) sdot4

5119879119894

(120579DIP119891119890) sdot5

6

119879119905119894

(8)where







(iii) 06





(iv) (119895minus1)119895




Finger 119894 (119894 = 1 4)Joint 120572

119895minus1

119886119895minus1

119889119895

120579119895

119895 = 1 1205872 0 0 120579CMC119891119890

119895 = 2 minus1205872 1198710119894

0 120579MCP119886119887119886119889

119895 = 3 1205872 0 0 120579MCP119891119890

119895 = 4 0 1198711119894

0 120579PIP119891119890

119895 = 5 0 1198712119894

0 120579DIP119891119890

(v) 56


119905119894= [119897119905119894119909119897119905119894119910119897119905119894119911]

119879



119891119890 MCP

119886119887119886119889

MCP119891119890

PIP119891119890

and DIP119891119890









119886119895minus1

119889119895

120579119895

119895 = 1 0 0 0 120579TMC119886119887119886119889

119895 = 2 1205872 0 0 120579TMC119891119890

119895 = 3 0 11987100

0 120579MCP119891119890

119895 = 4 0 11987110

0 120579IP119891119890


1198760=0

0

1198790sdot0

5

1198790(120579119896)

1198760=0

0

1198790sdot0

11198790

(120579TMC119886119887119886119889) sdot1

21198790

(120579TMC119891119890)

sdot2

31198790

(120579MCP119891119890) sdot3

41198790

(120579IP119891119890) sdot4

51198791199050

(9)


TMC119891119890

MCP119891119890

and IP119891119890


[1198971199050119909

1198971199050119910

1198971199050119911]

119879






119894


119891119890

MCP119886119887119886119889

and DIP119891119890



120579CMC119891119890 = atan(11987633

11987613

)

120579MCP119886119887119886119889 = atan[

[

11987613

minus11987623cos (120579CMC119891119890)

]

]

120579DIP119891119890 = atan(12059811205982

)

(10)



1205981=

(minus1) sdot (11989811987622+ 11989911987621)

(1198982

minus 1198992

)

1205982=

Q21

119898

+

119899

119898

sdot

(minus1) sdot (11989811987622+ 11989911987621)

(1198982

minus 1198992

)





(11)

TheMCP119891119890

PIP119891119890




expression

1= 1198651minus [11987131lowast ] (12)


1198751= [119875111990911987511199101198751119911]

119879

(13)

where1198751119909= 11987101cos (120579CMC)

1198751119910= 11987101sin (120579CMC)

1198751119911= 0

(14)


Knowing 1and 119875


1is


Angles 1205931and 120593

2are expressed as

1205931= acos(

1199032

2

+ 1198712

01

minus 1199032

1

2119903211989701

)

1205932= acos(

1199032

2

+ 1198712

11

minus 1198712

21

2119903211987111

)

(16)


100381710038171003817100381710038171

10038171003817100381710038171003817

1199032=10038171003817100381710038171199061

1003817100381710038171003817

(17)


120579MCP119891119890 = 120587 minus 1205931 minus 1205932 (18)


120579PIP119891119890 = 120587 minus 1205933 (19)

where

1205933= acos(

1198712

21

+ 1198712

11

minus 1199032

2

21198712111987111

) (20)




120579TMC119886119887119886119889 = atan(11987613

minus11987623

) (21)




120579MCP119891119890 = 120587 minus 1205741

120579IP119891119890 = 120587 minus 1205742 minus 1205743(22)

where

1205741= acos(

1198712

10

+ 1198712

00

minus 1199032

3

21198711011987100

)

1205742= acos(

1198712

20

+ 1199042

3

minus 1199032

4

2119871201199033

)

1205743= acos(

1199032

3

+ 1198712

10

minus 1198712

00

211990311990411987110

)

1199033=

100381710038171003817100381710038170

10038171003817100381710038171003817

1199034=

10038171003817100381710038171003817

1198650

10038171003817100381710038171003817

0= 1198650minus [11987120lowast ]

(23)


120579TMC119891119890 = atan(12059841205983

) (24)

where

1205983= (11987632+ 11987631) cos 120583

1205984=


cos 120583

120583 = 120579MCP + 120579IP

(25)



2is the


2




1205793= 120579MCP119891119890

1205794= 120579PIP119891119890

1205795= 120579DIP119891119890

(26)




119892= [

[

119887119892119909

119887119892119910

0

]

]

119892 = 1 2 3 (27)





119895and 119904120579

119895stand for


1= [

[

11988711199091198881205793minus 11988711199101199041205793

11988711199091199041205793+ 11988711199101198881205793

0

]

]

2= [

[

11987111198881205793

11987111199041205793

0

]

]

+

[[[

[

1198872119909119888 (1205793+ 1205794) minus 1198872119910119904 (1205793+ 1205794)

1198872119909119904 (1205793+ 1205794) + 1198872119910119888 (1205793+ 1205794)

0

]]]

]



3= [

[

11987111198881205793

11987111199041205793

0

]

]

+[[

[

1198712119888 (1205793+ 1205794)

1198712119904 (1205793+ 1205794)

0

]]

]

+

[[[

[

1198873119909119888 (1205793+ 1205794+ 1205795) minus 1198873119910119904 (1205793+ 1205794+ 1205795)

1198873119909119904 (1205793+ 1205794+ 1205795) + 1198873119910119888 (1205793+ 1205794+ 1205795)

0

]]]

]

(28)


1205793+ 1205794= 12057934

1205793+1205794=

12057934

1205793+ 1205794+ 1205795= 120579345

1205793+1205794+1205795=

120579345

(29)


V1198661=[[

[

minus1198871119909

12057931199041205793minus 1198871119910

12057931198881205793

1198871119909

12057931198881205793minus 1198871119910

12057931199041205793

0

]]

]

V1198662=[[

[

minus1198711

12057931199041205793minus1198872119909

1205793411990412057934minus 1198872119910

1205793411988812057934

1198711

12057931198881205793+ 1198872119909

1205793411988812057934minus 1198872119910

1205793411990412057934

0

]]

]

V1198663=[[

[

minus1198711

12057931199041205793minus1198712

1205793411990412057934minus1198873119909

120579345119904120579345minus 1198873119910

120579345119888120579345

1198711

12057931198881205793+ 1198712

1205793411988812057934+ 1198873119909

120579345119888120579345minus 1198873119910

120579345119904120579345

0

]]

]

(30)


119879 =

1

2

1198981(V1198661)

2

+

1

2

1198982(V1198662)

2

+

1

2

1198983(V1198663)

2

+

1

2

1198681(1205793)

2

+

1

2

1198682(12057934)

2

+

1

2

1198683(120579345)

2

(31)




1 1198962 and 119896



119880 = 1198981119892 (11988711199091199041205793+ 11988711199101198881205793)

+ 1198982119892 (11987111199041205793+ 119887211990911990412057934+ 119887211991011988812057934)

+ 1198983119892 (11987111199041205793+ 119871211990412057934+ 1198873119909119904120579345+ 1198873119910119888120579345)

+

1

2

1198961(1205793)2

+

1

2

1198962(12057934minus 1205793)2

+

1

2

1198963(120579345minus 12057934)2

(32)


119865 =

1

2

1198881(1205793)

2

+

1

2

1198882(12057934minus1205793)

2

+

1

2

1198883(120579345minus12057934)

2

(33)



1

1198882 and 119888


joints respectively



3 12057934 and 120579

345

119889

119889119905

(

120597 (119879 minus 119880)

1205971205793

) minus

120597 (119879 minus 119880)

1205971205793

+

120597119865

1205971205793

= 1205913

119889

119889119905

(

120597 (119879 minus 119880)

12059712057934

) minus

120597 (119879 minus 119880)

12059712057934

+

120597119865

12059712057934

= 1205914

119889

119889119905

(

120597 (119879 minus 119880)

120597120579345

) minus

120597 (119879 minus 119880)

120597120579345

+

120597119865

120597120579345

= 1205915

(34)




120591119895=

(sum5

119895=3

120575119882119895)

120575120579119895

119895 = 3 4 5 (35)



1205913= (11986531199101198903119909minus 11986531199091198903119910) + [

[

minus11987111199041205793

11987111198881205793

0

]

]

sdot [

[

1198654119909c12057934minus 1198654119910s12057934

1198654119909s12057934+ 1198654119910c12057934

0

]

]

+ [

[

minus11987111199041205793

11987111198881205793

0

]

]

sdot [

[

1198655119909c120579345minus 1198655119910s120579345

1198655119909s120579345+ 1198655119910c120579345

0

]

]

+ 1198621198981minus 1198621198982

1205914= (11986541199101198904119909minus 11986541199091198904119910) + [

[

minus119871211990412057934

119871211988812057934

0

]

]

sdot [

[

1198655119909c120579345minus 1198655119910s120579345

1198655119909s120579345+ 1198655119910c120579345

0

]

]

+ 1198621198982minus 1198621198983

1205915= (11986551199101198905119909minus 11986551199091198905119910) + 1198621198983

(36)

where the term 119862119898119892


119895= [1198651198951199091198651198951199100]119879 is the contact


119895= [1198901198951199091198901198941199100]119879


1198601

1205793+ 1198602

12057934+ 1198603

120579345= 1198604

1198611

1205793+ 1198612

12057934+ 1198613

120579345= 1198614

1198621

1205793+ 1198622

12057934+ 1198623

120579345= 1198624

(37)


The terms 1198601 1198602 and 119860










119895= (1198631198952)(cos(120587119905119905

1)minus1) where




120579119895 max = | minus 12058711986311989521199051|





Omega maxMCP (rads)

Omega maxPIP (rads)

Omega maxDIP (rads)

Cycles in 10 s 27 27 31 25 20 29 23 27 21 31 261 3843 261 12164 15026 10018


Omega maxMCP (rads)

Omega maxPIP (rads)

Omega maxDIP (rads)

Cycles in 10 s 27 24 17 34 29 27 30 24 21 27 26 4784 26 12118 14969 9979



1198983are


119898max=



119889119895= |119862119889119895||119862119898119895|



0 20 40 60 80 100minus100

minus50

0

50

100

150

200

250

300

Cycle ()

C(N

mm

)

Cm1

Cm2

Cm3



MCP PIP DIP119891119896119895

3805 3848 3858119891119888119895

5758 6059 61119891119889119895

437 093 042



7 Conclusions




1198981

= 705 g 1198982

= 397 g 1198983

= 270 g1198681

= 1000 gmm2 1198682

= 240 gmm2 1198683

= 120 gmm2

1198961

= 00902Nm 1198962

= 00902Nm 1198963

= 00902Nm1198881

= 00225Nms 1198882

= 00225Nms 1198883

= 00225Nms1198711

= [50 0 0]

119879

mm 1198712

= [20 0 0]

119879

mm 1198713

= [25 0 0]

119879

mm1198871

= [25 0 0]

119879

mm 1198872

= [10 0 0]

119879

mm 1198873

= [125 0 0]

119879

mm1198901

= [25 minus8 0]

119879

mm 1198902

= [10 minus7 0]

119879

mm 1198903

= [12 minus6 0]

119879

mm



Appendices



11987611989411= [(119888120579CMC119891119890119888120579MCP119886119887119886119889) 119888120579MCP119891119890

minus 119904120579CMC119891119890119904120579MCP119891119890]


minus [(119888120579CMC119891119890119888120579MCP119886119887119886119889) 119904120579MCP119891119890

+ 119904120579CMC119891119890119888120579MCP119891119890]


11987611989412= [(119888120579CMC119891119890119888120579MCP119886119887119886119889) 119888120579MCP119891119890

minus 119904120579CMC119891119890119904120579MCP119891119890]


minus [(119888120579CMC119891119890119888120579MCP119886119887119886119889) 119904120579MCP119891119890

+ 119904120579CMC119891119890119888120579MCP119891119890]


11987611989413= 119888120579CMC119891119890119904120579MCP119886119887119886119889

11987611989414= [(119888120579CMC119891119890119888120579MCP119886119887119886119889) 119888120579MCP119891119890

minus 119904120579CMC119891119890119904120579MCP119891119890]



+ 1198711119894+ 1198712119894119888120579PIP119891119890]

minus [(119888120579CMC119891119890119888120579MCP119886119887119886119889) 119904120579MCP119891119890

+ 119904120579CMC119891119890119888120579MCP119891119890]



+ 1198712119894119904120579PIP119891119890]

+ (119888120579CMC119891119890119904120579MCP119886119887119886119889) 119871 119905119894119911 + 1198710119894119888120579CMC119891119890

11987611989421= 119904120579MCP119886119887119886119889119888120579MCP119891119890


minus 119904120579MCP119886119887119886119889119904120579MCP119891119890


11987611989422= 119904120579MCP119886119887119886119889119888120579MCP119891119890



minus 119904120579MCP119886119887119886119889119904120579MCP119891119890


11987611989423= minus 119888120579MCP119886119887119886119889

11987611989424= 119904120579MCP119886119887119886119889119888120579MCP119891119890



+ 1198711119894+ 1198712119894119888120579PIP119891119890] minus 119904120579MCP119886119887119886119889119904120579MCP119891119890



+ 1198712119894119904120579PIP119891119890] + (minus119888120579MCP119886119887119886119889) 119871 119905119894119911

11987611989431= (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119888120579MCP119891119890

+ 119888120579CMC119891119890119904120579MCP119891119890


+ minus (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119904120579MCP119891119890

+ 119888120579CMC119891119890119888120579MCP119891119890


11987611989432= (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119888120579MCP119891119890

+ 119888120579CMC119891119890119904120579MCP119891119890


+ minus (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119904120579MCP119891119890

+ 119888120579CMC119891119890119888120579MCP119891119890


11987611989433= 119904120579CMC119891119890119904120579MCP119886119887119886119889

11987611989434= (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119888120579MCP119891119890

+ 119888120579119862119872119862119891119890

119904120579119872119862119875119891119890



+ 1198711119894+ 1198712119894119888120579PIP119891119890]

+ minus (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119904120579MCP119891119890

+ 119888120579CMC119891119890119888120579MCP119891119890



+ 1198712119894119904120579PIP119891119890]

+ (119904120579CMC119891119890119904120579MCP119886119887119886119889) 119871 119905119894119911 + 1198710119894119904120579CMC119891119890

(A1)


119876011= [119888120579TMC119886119887119886119889119888120579TMC119891119890]


+ [minus119888120579TMC119886119887119886119889119904120579TMC119891119890]


119876012= [119888120579TMC119886119887119886119889119888120579TMC119891119890]


+ [minus119888120579TMC119886119887119886119889119904120579TMC119891119890]


119876013= 119904120579TMC119886119887119886119889

119876014= [119888120579TMC119886119887119886119889119888120579TMC119891119890]

times [1198711199050 119909


+ 1198711199050 119910


+ 11987110119888120579MCP119891119890 + 11987100]

+ [minus119888120579TMC119886119887119886119889119904120579TMC119891119890]

times [1198711199050 119909

(119904120579MCP119891119890119888120579IP119891119890 + 119888120579MCP119891119890119904120579IP119891119890)

+ 1198711199050 119910


+ 11987110119904120579MCP119891119890] + 119871 1199050 119911119904120579TMC119886119887119886119889


119876021= [119904120579TMC119886119887119886119889119888120579TMC119891119890]


+ [minus119904120579TMC119886119887119886119889119904120579TMC119891119890]


119876022= [119904120579TMC119886119887119886119889119888120579TMC119891119890]


+ [minus119904120579TMC119886119887119886119889119904120579TMC119891119890]


119876023= minus 119888120579TMC119886119887119886119889

119876024= [119904120579TMC119886119887119886119889119888120579TMC119891119890]

times [1198711199050 119909


+ 1198711199050 119910


+ 11987110119888120579MCP119891119890 + 11987100]

+ [minus119904120579TMC119886119887119886119889119904120579TMC119891119890]

times [1198711199050 119909

(119904120579MCP119891119890119888120579IP119891119890 + 119888120579MCP119891119890119904120579IP119891119890)

+ 1198711199050 119910


+ 11987110119904120579MCP119891119890] minus 119888120579TMC119886119887119886119889119871 1199050 119911


+ 119888120579TMC119891119890 (119904120579MCP119891119890119888120579IP119891119890 + 119888120579MCP119891119890119904120579IP119891119890)



119876033= 0

119876034= 119904120579TMC119891119890

times [1198711199050 119909


+ 1198711199050 119910


+ 11987110119888120579MCP119891119890 + 11987100] + 119888120579TMC119891119890

times [1198711199050 119909

(119904120579MCP119891119890119888120579IP119891119890 + 119888120579MCP119891119890119904120579IP119891119890)

+ 1198711199050 119910


+11987110119904120579MCP119891119890]

(A2)


Consider


+ (11988711199091198881205791minus 11988711199101199041205791) (1198871119909c1205791minus 1198871119910s1205791)]


11199091199041205791minus 11987111199101198881205791)

+ (11987111199091198881205791minus 11987111199101199041205791) (11987111199091198881205791minus 11987111199101199041205791)]


+(11987111199091198881205791minus 11987111199101199041205791)times (119871

11199091198881205791minus 11987111199101199041205791)]+1198681


11199091199041205791minus 11987111199101198881205791)

+ (+119887211990911988812057912minus 119887211991011990412057912) (11987111199091198881205791minus 11987111199101199041205791)]


+ (+119871211990911988812057912minus 119871211991011990412057912) (11987111199091198881205791minus 11987111199101199041205791)]


+ (+1198873119909119888120579123minus 1198873119910119904120579123) (11987111199091198881205791minus 11987111199101199041205791)]

1198604= 1198981[(minus1198871119909

12057911199041205791minus 1198871119910

12057911198881205791) (minus1198871119909

1205791c1205791+ 1198871119910

12057911199041205791)


+ 1198982[minus (minus119871

1119909

1205792

1

1198881205791+ 1198711119910

1205792

1

1199041205791

minus1198872119909

1205792

12

11988812057912+ 1198872119910

1205792

12

11990412057912)



1205792

1

1199041205791minus 1198711119910

1205792

1

1198881205791minus 1198872119909

1205792

12

11990412057912

minus1198872119910

1205792

12

11988812057912) (11987111199091198881205791minus 11987111199101199041205791)]

+ 1198983[minus (minus119871

1119909

1205792

1

1198881205791+ 1198711119910

1205792

1

1199041205791minus1198712119909

1205792

12

11988812057912

+1198712119910

1205792

12

11990412057912minus1198873119909

1205792

123

119888120579123+ 1198873119910

1205792

123

119904120579123)



1205792

1

1199041205791minus 1198711119910

1205792

1

1198881205791minus 1198712119909

1205792

12

11990412057912


minus1198712119910

1205792

12

11988812057912minus 1198873119909

1205792

123

119904120579123minus 1198873119910

1205792

123

119888120579123)

times (11987111199091198881205791minus 11987111199101199041205791)]

minus (1198981119892 (11988711199091198881205791minus 11988711199101199041205791) + (119898

2+ 1198983) 119892


minus 1198881

1205791+ 1198882(12057912minus1205791)


+ (11987111199091198881205791minus 11987111199101199041205791) (119887211990911988812057912minus 119887211991011990412057912)]


+ (11987111199091198881205791minus 11987111199101199041205791) (119871211990911988812057912minus 119871211991011990412057912)]


+ (+119887211990911988812057912minus 119887211991011990412057912) (119887211990911988812057912minus 119887211991011990412057912)]


+ (+119871211990911988812057912) (119871211990911988812057912minus 119871211991011990412057912)] + 119868

2


+ (+1198873119909119888120579123minus 1198873119910119904120579123) (119871211990911988812057912minus 119871211991011990412057912)]

1198614= 1198982[minus (minus119871

1119909

1205792

1

1198881205791+ 1198711119910

1205792

1

1199041205791minus1198872119909

1205792

12

11988812057912

+1198872119910

1205792

12

11990412057912) (minus119887211990911990412057912minus 119887211991011988812057912)


1205792

1

1199041205791minus 1198711119910

1205792

1

1198881205791minus 1198872119909

1205792

12

11990412057912

minus1198872119910

1205792

12

11988812057912) (119887211990911988812057912minus 119887211991011990412057912)]

+ 1198983[minus (minus119871

1119909

1205792

1

1198881205791+ 1198711119910

1205792

1

1199041205791minus1198712119909

1205792

12

11988812057912

+ 1198712119910

1205792

12

11990412057912minus1198873119909

1205792

123

119888120579123

+1198873119910

1205792

123

119904120579123) (minus119871211990911990412057912minus 119871211991011988812057912)


1205792

1

1199041205791minus 1198711119910

1205792

1

1198881205791minus 1198712119909

1205792

12

11990412057912

minus 1198712119910

1205792

12

11988812057912minus 1198873119909

1205792

123

119904120579123

minus1198873119910

1205792

123

119888120579123) (119871211990911988812057912minus 119871211991011990412057912)]

minus (1198982119892 (119887211990911988812057912minus 119887211991011990412057912)

+ 1198983119892 (119871211990911988812057912minus 119871211991011990412057912)




+ (11987111199091198881205791minus 11987111199101199041205791) (1198873119909119888120579123minus 1198873119910119904120579123)]




+ (+1198873119909119888120579123minus 1198873119910119904120579123)

times (1198873119909119888120579123minus 1198873119910119904120579123)] + 119868

3

1198624= 1198983[minus (minus119871

1119909

1205792

1

1198881205791+ 1198711119910

1205792

1

1199041205791minus1198712119909

1205792

12

11988812057912

+ 1198712119910

1205792

12

11990412057912minus1198873119909

1205792

123

119888120579123

+1198873119910

1205792

123

119904120579123) (minus1198873119909119904120579123minus 1198873119910119888120579123)


1205792

1

1199041205791minus 1198711119910

1205792

1

1198881205791minus 1198712119909

1205792

12

11990412057912

minus 1198712119910

1205791211990412057912minus 1198712119910

1205792

12

11988812057912+ 1198873119909

120579123119888120579123

minus1198873119909

1205792

123

119904120579123minus 1198873119910

1205792

123

119888120579123)

times (1198873119909119888120579123minus 1198873119910119904120579123)]



(B1)

References
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Table4Intrafinger

constraintsfor

acirc

ular

grasp

Thum

bIndex

Middle

Ring

Little

Noconstraint

120579CM

C 119891119890Index=120579CM

C 119891119890Middle

120579CM

C 119891119890Middle=(12)120579CM

C 119891119890Ring

120579CM

C 119891119890Ring=(23)120579CM

C 119891119890Little

Noconstraint

120579TM

C 119891119890=(1110)120579MCP119891119890

Noconstraint



Noconstraint

120579MCP119891119890=(45)120579IP119891119890



120579MCP119891119890Ring=(43)120579PIP 119891119890Ring


Noconstraint



120579PIP 119891119890Ring=(32)120579DIP119891119890Ring



Table5Intrafinger

constraintsfor

apris

maticgrasp

Thum

bIndex

Middle

Ring

Little

Noconstraint

120579CM

C 119891119890Index=120579CM

C 119891119890Middle

120579CM

C 119891119890Middle=(12)120579CM

C 119891119890Ring

120579CM

C 119891119890Ring=(23)120579CM

C 119891119890Little

Noconstraint

120579TM

C 119891119890=(1110)120579MCP119891119890

Noconstraint



Noconstraint

120579MCP119891119890=(65)120579IP119891119890



120579MCP119891119890Ring=(32)120579PIP 119891119890Ring


Noconstraint



120579PIP 119891119890Ring=2120579DIP119891119890Ring






120579MCP119891119890Ring asymp7

12

120579MCP119891119890Little


3

120579MCP119891119890Ring



(6)




5

120579MCP119891119890Index (7)




4 Kinematic Model





MCP 75∘ndash80∘ 0∘ 5∘



MCP 90∘ 30∘ndash40∘ 60∘

PIP 110∘ 0∘ 0∘

DIP 80∘ndash90∘ 5∘ 0∘


MCP 90∘ 30∘ndash40∘ 45∘

PIP 110∘ 0∘ 0∘

DIP 80∘ndash90∘ 5∘ 0∘


MCP 90∘ 30∘ndash40∘ 45∘

PIP 120∘ 0∘ 0∘

DIP 80∘ndash90∘ 5∘ 0∘


MCP 90∘ 30∘ndash40∘ 50∘

PIP 135∘ 0∘ 0∘

DIP 90∘ 5∘ 0∘







Distal link

Middle link

Proximal link

Metacarpal link

Carpal zone

Wrist Reference

0

1

23

4

L01L02

L03L04

L11

L12L13

L14

L21

L22L23

L24

L31

L32L33

L34

pt1

pt2pt3

pt4

pt0

L20

L00L10

R0R0

R1i

R2i

R3i

R5i

R5iR4i

R4i

TMCaa

MCPaa

TMCfe

IPfe

MCPfeCMCfe

MCPfePIPfe

DIPfe

R0i = R1i

R2i = R3i




119876119894=0

0

119879119894sdot0

6119879119894

(120579119895) =0

0

119879119894sdot

5

prod

119895=1

(119895minus1)

119895

119879119894(120579119895) sdot5

6

119879119905119894

119876119894=0

0

119879119894sdot0

1119879119894

(120579CMC119891119890) sdot1

2119879119894

(120579MCP119886119887119886119889) sdot2

3119879119894

(120579MCP119891119890)

sdot3

4119879119894

(120579PIP119891119890) sdot4

5119879119894

(120579DIP119891119890) sdot5

6

119879119905119894

(8)where







(iii) 06





(iv) (119895minus1)119895




Finger 119894 (119894 = 1 4)Joint 120572

119895minus1

119886119895minus1

119889119895

120579119895

119895 = 1 1205872 0 0 120579CMC119891119890

119895 = 2 minus1205872 1198710119894

0 120579MCP119886119887119886119889

119895 = 3 1205872 0 0 120579MCP119891119890

119895 = 4 0 1198711119894

0 120579PIP119891119890

119895 = 5 0 1198712119894

0 120579DIP119891119890

(v) 56


119905119894= [119897119905119894119909119897119905119894119910119897119905119894119911]

119879



119891119890 MCP

119886119887119886119889

MCP119891119890

PIP119891119890

and DIP119891119890









119886119895minus1

119889119895

120579119895

119895 = 1 0 0 0 120579TMC119886119887119886119889

119895 = 2 1205872 0 0 120579TMC119891119890

119895 = 3 0 11987100

0 120579MCP119891119890

119895 = 4 0 11987110

0 120579IP119891119890


1198760=0

0

1198790sdot0

5

1198790(120579119896)

1198760=0

0

1198790sdot0

11198790

(120579TMC119886119887119886119889) sdot1

21198790

(120579TMC119891119890)

sdot2

31198790

(120579MCP119891119890) sdot3

41198790

(120579IP119891119890) sdot4

51198791199050

(9)


TMC119891119890

MCP119891119890

and IP119891119890


[1198971199050119909

1198971199050119910

1198971199050119911]

119879






119894


119891119890

MCP119886119887119886119889

and DIP119891119890



120579CMC119891119890 = atan(11987633

11987613

)

120579MCP119886119887119886119889 = atan[

[

11987613

minus11987623cos (120579CMC119891119890)

]

]

120579DIP119891119890 = atan(12059811205982

)

(10)



1205981=

(minus1) sdot (11989811987622+ 11989911987621)

(1198982

minus 1198992

)

1205982=

Q21

119898

+

119899

119898

sdot

(minus1) sdot (11989811987622+ 11989911987621)

(1198982

minus 1198992

)





(11)

TheMCP119891119890

PIP119891119890




expression

1= 1198651minus [11987131lowast ] (12)


1198751= [119875111990911987511199101198751119911]

119879

(13)

where1198751119909= 11987101cos (120579CMC)

1198751119910= 11987101sin (120579CMC)

1198751119911= 0

(14)


Knowing 1and 119875


1is


Angles 1205931and 120593

2are expressed as

1205931= acos(

1199032

2

+ 1198712

01

minus 1199032

1

2119903211989701

)

1205932= acos(

1199032

2

+ 1198712

11

minus 1198712

21

2119903211987111

)

(16)


100381710038171003817100381710038171

10038171003817100381710038171003817

1199032=10038171003817100381710038171199061

1003817100381710038171003817

(17)


120579MCP119891119890 = 120587 minus 1205931 minus 1205932 (18)


120579PIP119891119890 = 120587 minus 1205933 (19)

where

1205933= acos(

1198712

21

+ 1198712

11

minus 1199032

2

21198712111987111

) (20)




120579TMC119886119887119886119889 = atan(11987613

minus11987623

) (21)




120579MCP119891119890 = 120587 minus 1205741

120579IP119891119890 = 120587 minus 1205742 minus 1205743(22)

where

1205741= acos(

1198712

10

+ 1198712

00

minus 1199032

3

21198711011987100

)

1205742= acos(

1198712

20

+ 1199042

3

minus 1199032

4

2119871201199033

)

1205743= acos(

1199032

3

+ 1198712

10

minus 1198712

00

211990311990411987110

)

1199033=

100381710038171003817100381710038170

10038171003817100381710038171003817

1199034=

10038171003817100381710038171003817

1198650

10038171003817100381710038171003817

0= 1198650minus [11987120lowast ]

(23)


120579TMC119891119890 = atan(12059841205983

) (24)

where

1205983= (11987632+ 11987631) cos 120583

1205984=


cos 120583

120583 = 120579MCP + 120579IP

(25)



2is the


2




1205793= 120579MCP119891119890

1205794= 120579PIP119891119890

1205795= 120579DIP119891119890

(26)




119892= [

[

119887119892119909

119887119892119910

0

]

]

119892 = 1 2 3 (27)





119895and 119904120579

119895stand for


1= [

[

11988711199091198881205793minus 11988711199101199041205793

11988711199091199041205793+ 11988711199101198881205793

0

]

]

2= [

[

11987111198881205793

11987111199041205793

0

]

]

+

[[[

[

1198872119909119888 (1205793+ 1205794) minus 1198872119910119904 (1205793+ 1205794)

1198872119909119904 (1205793+ 1205794) + 1198872119910119888 (1205793+ 1205794)

0

]]]

]



3= [

[

11987111198881205793

11987111199041205793

0

]

]

+[[

[

1198712119888 (1205793+ 1205794)

1198712119904 (1205793+ 1205794)

0

]]

]

+

[[[

[

1198873119909119888 (1205793+ 1205794+ 1205795) minus 1198873119910119904 (1205793+ 1205794+ 1205795)

1198873119909119904 (1205793+ 1205794+ 1205795) + 1198873119910119888 (1205793+ 1205794+ 1205795)

0

]]]

]

(28)


1205793+ 1205794= 12057934

1205793+1205794=

12057934

1205793+ 1205794+ 1205795= 120579345

1205793+1205794+1205795=

120579345

(29)


V1198661=[[

[

minus1198871119909

12057931199041205793minus 1198871119910

12057931198881205793

1198871119909

12057931198881205793minus 1198871119910

12057931199041205793

0

]]

]

V1198662=[[

[

minus1198711

12057931199041205793minus1198872119909

1205793411990412057934minus 1198872119910

1205793411988812057934

1198711

12057931198881205793+ 1198872119909

1205793411988812057934minus 1198872119910

1205793411990412057934

0

]]

]

V1198663=[[

[

minus1198711

12057931199041205793minus1198712

1205793411990412057934minus1198873119909

120579345119904120579345minus 1198873119910

120579345119888120579345

1198711

12057931198881205793+ 1198712

1205793411988812057934+ 1198873119909

120579345119888120579345minus 1198873119910

120579345119904120579345

0

]]

]

(30)


119879 =

1

2

1198981(V1198661)

2

+

1

2

1198982(V1198662)

2

+

1

2

1198983(V1198663)

2

+

1

2

1198681(1205793)

2

+

1

2

1198682(12057934)

2

+

1

2

1198683(120579345)

2

(31)




1 1198962 and 119896



119880 = 1198981119892 (11988711199091199041205793+ 11988711199101198881205793)

+ 1198982119892 (11987111199041205793+ 119887211990911990412057934+ 119887211991011988812057934)

+ 1198983119892 (11987111199041205793+ 119871211990412057934+ 1198873119909119904120579345+ 1198873119910119888120579345)

+

1

2

1198961(1205793)2

+

1

2

1198962(12057934minus 1205793)2

+

1

2

1198963(120579345minus 12057934)2

(32)


119865 =

1

2

1198881(1205793)

2

+

1

2

1198882(12057934minus1205793)

2

+

1

2

1198883(120579345minus12057934)

2

(33)



1

1198882 and 119888


joints respectively



3 12057934 and 120579

345

119889

119889119905

(

120597 (119879 minus 119880)

1205971205793

) minus

120597 (119879 minus 119880)

1205971205793

+

120597119865

1205971205793

= 1205913

119889

119889119905

(

120597 (119879 minus 119880)

12059712057934

) minus

120597 (119879 minus 119880)

12059712057934

+

120597119865

12059712057934

= 1205914

119889

119889119905

(

120597 (119879 minus 119880)

120597120579345

) minus

120597 (119879 minus 119880)

120597120579345

+

120597119865

120597120579345

= 1205915

(34)




120591119895=

(sum5

119895=3

120575119882119895)

120575120579119895

119895 = 3 4 5 (35)



1205913= (11986531199101198903119909minus 11986531199091198903119910) + [

[

minus11987111199041205793

11987111198881205793

0

]

]

sdot [

[

1198654119909c12057934minus 1198654119910s12057934

1198654119909s12057934+ 1198654119910c12057934

0

]

]

+ [

[

minus11987111199041205793

11987111198881205793

0

]

]

sdot [

[

1198655119909c120579345minus 1198655119910s120579345

1198655119909s120579345+ 1198655119910c120579345

0

]

]

+ 1198621198981minus 1198621198982

1205914= (11986541199101198904119909minus 11986541199091198904119910) + [

[

minus119871211990412057934

119871211988812057934

0

]

]

sdot [

[

1198655119909c120579345minus 1198655119910s120579345

1198655119909s120579345+ 1198655119910c120579345

0

]

]

+ 1198621198982minus 1198621198983

1205915= (11986551199101198905119909minus 11986551199091198905119910) + 1198621198983

(36)

where the term 119862119898119892


119895= [1198651198951199091198651198951199100]119879 is the contact


119895= [1198901198951199091198901198941199100]119879


1198601

1205793+ 1198602

12057934+ 1198603

120579345= 1198604

1198611

1205793+ 1198612

12057934+ 1198613

120579345= 1198614

1198621

1205793+ 1198622

12057934+ 1198623

120579345= 1198624

(37)


The terms 1198601 1198602 and 119860










119895= (1198631198952)(cos(120587119905119905

1)minus1) where




120579119895 max = | minus 12058711986311989521199051|





Omega maxMCP (rads)

Omega maxPIP (rads)

Omega maxDIP (rads)

Cycles in 10 s 27 27 31 25 20 29 23 27 21 31 261 3843 261 12164 15026 10018


Omega maxMCP (rads)

Omega maxPIP (rads)

Omega maxDIP (rads)

Cycles in 10 s 27 24 17 34 29 27 30 24 21 27 26 4784 26 12118 14969 9979



1198983are


119898max=



119889119895= |119862119889119895||119862119898119895|



0 20 40 60 80 100minus100

minus50

0

50

100

150

200

250

300

Cycle ()

C(N

mm

)

Cm1

Cm2

Cm3



MCP PIP DIP119891119896119895

3805 3848 3858119891119888119895

5758 6059 61119891119889119895

437 093 042



7 Conclusions




1198981

= 705 g 1198982

= 397 g 1198983

= 270 g1198681

= 1000 gmm2 1198682

= 240 gmm2 1198683

= 120 gmm2

1198961

= 00902Nm 1198962

= 00902Nm 1198963

= 00902Nm1198881

= 00225Nms 1198882

= 00225Nms 1198883

= 00225Nms1198711

= [50 0 0]

119879

mm 1198712

= [20 0 0]

119879

mm 1198713

= [25 0 0]

119879

mm1198871

= [25 0 0]

119879

mm 1198872

= [10 0 0]

119879

mm 1198873

= [125 0 0]

119879

mm1198901

= [25 minus8 0]

119879

mm 1198902

= [10 minus7 0]

119879

mm 1198903

= [12 minus6 0]

119879

mm



Appendices



11987611989411= [(119888120579CMC119891119890119888120579MCP119886119887119886119889) 119888120579MCP119891119890

minus 119904120579CMC119891119890119904120579MCP119891119890]


minus [(119888120579CMC119891119890119888120579MCP119886119887119886119889) 119904120579MCP119891119890

+ 119904120579CMC119891119890119888120579MCP119891119890]


11987611989412= [(119888120579CMC119891119890119888120579MCP119886119887119886119889) 119888120579MCP119891119890

minus 119904120579CMC119891119890119904120579MCP119891119890]


minus [(119888120579CMC119891119890119888120579MCP119886119887119886119889) 119904120579MCP119891119890

+ 119904120579CMC119891119890119888120579MCP119891119890]


11987611989413= 119888120579CMC119891119890119904120579MCP119886119887119886119889

11987611989414= [(119888120579CMC119891119890119888120579MCP119886119887119886119889) 119888120579MCP119891119890

minus 119904120579CMC119891119890119904120579MCP119891119890]



+ 1198711119894+ 1198712119894119888120579PIP119891119890]

minus [(119888120579CMC119891119890119888120579MCP119886119887119886119889) 119904120579MCP119891119890

+ 119904120579CMC119891119890119888120579MCP119891119890]



+ 1198712119894119904120579PIP119891119890]

+ (119888120579CMC119891119890119904120579MCP119886119887119886119889) 119871 119905119894119911 + 1198710119894119888120579CMC119891119890

11987611989421= 119904120579MCP119886119887119886119889119888120579MCP119891119890


minus 119904120579MCP119886119887119886119889119904120579MCP119891119890


11987611989422= 119904120579MCP119886119887119886119889119888120579MCP119891119890



minus 119904120579MCP119886119887119886119889119904120579MCP119891119890


11987611989423= minus 119888120579MCP119886119887119886119889

11987611989424= 119904120579MCP119886119887119886119889119888120579MCP119891119890



+ 1198711119894+ 1198712119894119888120579PIP119891119890] minus 119904120579MCP119886119887119886119889119904120579MCP119891119890



+ 1198712119894119904120579PIP119891119890] + (minus119888120579MCP119886119887119886119889) 119871 119905119894119911

11987611989431= (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119888120579MCP119891119890

+ 119888120579CMC119891119890119904120579MCP119891119890


+ minus (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119904120579MCP119891119890

+ 119888120579CMC119891119890119888120579MCP119891119890


11987611989432= (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119888120579MCP119891119890

+ 119888120579CMC119891119890119904120579MCP119891119890


+ minus (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119904120579MCP119891119890

+ 119888120579CMC119891119890119888120579MCP119891119890


11987611989433= 119904120579CMC119891119890119904120579MCP119886119887119886119889

11987611989434= (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119888120579MCP119891119890

+ 119888120579119862119872119862119891119890

119904120579119872119862119875119891119890



+ 1198711119894+ 1198712119894119888120579PIP119891119890]

+ minus (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119904120579MCP119891119890

+ 119888120579CMC119891119890119888120579MCP119891119890



+ 1198712119894119904120579PIP119891119890]

+ (119904120579CMC119891119890119904120579MCP119886119887119886119889) 119871 119905119894119911 + 1198710119894119904120579CMC119891119890

(A1)


119876011= [119888120579TMC119886119887119886119889119888120579TMC119891119890]


+ [minus119888120579TMC119886119887119886119889119904120579TMC119891119890]


119876012= [119888120579TMC119886119887119886119889119888120579TMC119891119890]


+ [minus119888120579TMC119886119887119886119889119904120579TMC119891119890]


119876013= 119904120579TMC119886119887119886119889

119876014= [119888120579TMC119886119887119886119889119888120579TMC119891119890]

times [1198711199050 119909


+ 1198711199050 119910


+ 11987110119888120579MCP119891119890 + 11987100]

+ [minus119888120579TMC119886119887119886119889119904120579TMC119891119890]

times [1198711199050 119909

(119904120579MCP119891119890119888120579IP119891119890 + 119888120579MCP119891119890119904120579IP119891119890)

+ 1198711199050 119910


+ 11987110119904120579MCP119891119890] + 119871 1199050 119911119904120579TMC119886119887119886119889


119876021= [119904120579TMC119886119887119886119889119888120579TMC119891119890]


+ [minus119904120579TMC119886119887119886119889119904120579TMC119891119890]


119876022= [119904120579TMC119886119887119886119889119888120579TMC119891119890]


+ [minus119904120579TMC119886119887119886119889119904120579TMC119891119890]


119876023= minus 119888120579TMC119886119887119886119889

119876024= [119904120579TMC119886119887119886119889119888120579TMC119891119890]

times [1198711199050 119909


+ 1198711199050 119910


+ 11987110119888120579MCP119891119890 + 11987100]

+ [minus119904120579TMC119886119887119886119889119904120579TMC119891119890]

times [1198711199050 119909

(119904120579MCP119891119890119888120579IP119891119890 + 119888120579MCP119891119890119904120579IP119891119890)

+ 1198711199050 119910


+ 11987110119904120579MCP119891119890] minus 119888120579TMC119886119887119886119889119871 1199050 119911


+ 119888120579TMC119891119890 (119904120579MCP119891119890119888120579IP119891119890 + 119888120579MCP119891119890119904120579IP119891119890)



119876033= 0

119876034= 119904120579TMC119891119890

times [1198711199050 119909


+ 1198711199050 119910


+ 11987110119888120579MCP119891119890 + 11987100] + 119888120579TMC119891119890

times [1198711199050 119909

(119904120579MCP119891119890119888120579IP119891119890 + 119888120579MCP119891119890119904120579IP119891119890)

+ 1198711199050 119910


+11987110119904120579MCP119891119890]

(A2)


Consider


+ (11988711199091198881205791minus 11988711199101199041205791) (1198871119909c1205791minus 1198871119910s1205791)]


11199091199041205791minus 11987111199101198881205791)

+ (11987111199091198881205791minus 11987111199101199041205791) (11987111199091198881205791minus 11987111199101199041205791)]


+(11987111199091198881205791minus 11987111199101199041205791)times (119871

11199091198881205791minus 11987111199101199041205791)]+1198681


11199091199041205791minus 11987111199101198881205791)

+ (+119887211990911988812057912minus 119887211991011990412057912) (11987111199091198881205791minus 11987111199101199041205791)]


+ (+119871211990911988812057912minus 119871211991011990412057912) (11987111199091198881205791minus 11987111199101199041205791)]


+ (+1198873119909119888120579123minus 1198873119910119904120579123) (11987111199091198881205791minus 11987111199101199041205791)]

1198604= 1198981[(minus1198871119909

12057911199041205791minus 1198871119910

12057911198881205791) (minus1198871119909

1205791c1205791+ 1198871119910

12057911199041205791)


+ 1198982[minus (minus119871

1119909

1205792

1

1198881205791+ 1198711119910

1205792

1

1199041205791

minus1198872119909

1205792

12

11988812057912+ 1198872119910

1205792

12

11990412057912)



1205792

1

1199041205791minus 1198711119910

1205792

1

1198881205791minus 1198872119909

1205792

12

11990412057912

minus1198872119910

1205792

12

11988812057912) (11987111199091198881205791minus 11987111199101199041205791)]

+ 1198983[minus (minus119871

1119909

1205792

1

1198881205791+ 1198711119910

1205792

1

1199041205791minus1198712119909

1205792

12

11988812057912

+1198712119910

1205792

12

11990412057912minus1198873119909

1205792

123

119888120579123+ 1198873119910

1205792

123

119904120579123)



1205792

1

1199041205791minus 1198711119910

1205792

1

1198881205791minus 1198712119909

1205792

12

11990412057912


minus1198712119910

1205792

12

11988812057912minus 1198873119909

1205792

123

119904120579123minus 1198873119910

1205792

123

119888120579123)

times (11987111199091198881205791minus 11987111199101199041205791)]

minus (1198981119892 (11988711199091198881205791minus 11988711199101199041205791) + (119898

2+ 1198983) 119892


minus 1198881

1205791+ 1198882(12057912minus1205791)


+ (11987111199091198881205791minus 11987111199101199041205791) (119887211990911988812057912minus 119887211991011990412057912)]


+ (11987111199091198881205791minus 11987111199101199041205791) (119871211990911988812057912minus 119871211991011990412057912)]


+ (+119887211990911988812057912minus 119887211991011990412057912) (119887211990911988812057912minus 119887211991011990412057912)]


+ (+119871211990911988812057912) (119871211990911988812057912minus 119871211991011990412057912)] + 119868

2


+ (+1198873119909119888120579123minus 1198873119910119904120579123) (119871211990911988812057912minus 119871211991011990412057912)]

1198614= 1198982[minus (minus119871

1119909

1205792

1

1198881205791+ 1198711119910

1205792

1

1199041205791minus1198872119909

1205792

12

11988812057912

+1198872119910

1205792

12

11990412057912) (minus119887211990911990412057912minus 119887211991011988812057912)


1205792

1

1199041205791minus 1198711119910

1205792

1

1198881205791minus 1198872119909

1205792

12

11990412057912

minus1198872119910

1205792

12

11988812057912) (119887211990911988812057912minus 119887211991011990412057912)]

+ 1198983[minus (minus119871

1119909

1205792

1

1198881205791+ 1198711119910

1205792

1

1199041205791minus1198712119909

1205792

12

11988812057912

+ 1198712119910

1205792

12

11990412057912minus1198873119909

1205792

123

119888120579123

+1198873119910

1205792

123

119904120579123) (minus119871211990911990412057912minus 119871211991011988812057912)


1205792

1

1199041205791minus 1198711119910

1205792

1

1198881205791minus 1198712119909

1205792

12

11990412057912

minus 1198712119910

1205792

12

11988812057912minus 1198873119909

1205792

123

119904120579123

minus1198873119910

1205792

123

119888120579123) (119871211990911988812057912minus 119871211991011990412057912)]

minus (1198982119892 (119887211990911988812057912minus 119887211991011990412057912)

+ 1198983119892 (119871211990911988812057912minus 119871211991011990412057912)




+ (11987111199091198881205791minus 11987111199101199041205791) (1198873119909119888120579123minus 1198873119910119904120579123)]




+ (+1198873119909119888120579123minus 1198873119910119904120579123)

times (1198873119909119888120579123minus 1198873119910119904120579123)] + 119868

3

1198624= 1198983[minus (minus119871

1119909

1205792

1

1198881205791+ 1198711119910

1205792

1

1199041205791minus1198712119909

1205792

12

11988812057912

+ 1198712119910

1205792

12

11990412057912minus1198873119909

1205792

123

119888120579123

+1198873119910

1205792

123

119904120579123) (minus1198873119909119904120579123minus 1198873119910119888120579123)


1205792

1

1199041205791minus 1198711119910

1205792

1

1198881205791minus 1198712119909

1205792

12

11990412057912

minus 1198712119910

1205791211990412057912minus 1198712119910

1205792

12

11988812057912+ 1198873119909

120579123119888120579123

minus1198873119909

1205792

123

119904120579123minus 1198873119910

1205792

123

119888120579123)

times (1198873119909119888120579123minus 1198873119910119904120579123)]



(B1)

References
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Table5Intrafinger

constraintsfor

apris

maticgrasp

Thum

bIndex

Middle

Ring

Little

Noconstraint

120579CM

C 119891119890Index=120579CM

C 119891119890Middle

120579CM

C 119891119890Middle=(12)120579CM

C 119891119890Ring

120579CM

C 119891119890Ring=(23)120579CM

C 119891119890Little

Noconstraint

120579TM

C 119891119890=(1110)120579MCP119891119890

Noconstraint



Noconstraint

120579MCP119891119890=(65)120579IP119891119890



120579MCP119891119890Ring=(32)120579PIP 119891119890Ring


Noconstraint



120579PIP 119891119890Ring=2120579DIP119891119890Ring






120579MCP119891119890Ring asymp7

12

120579MCP119891119890Little


3

120579MCP119891119890Ring



(6)




5

120579MCP119891119890Index (7)




4 Kinematic Model





MCP 75∘ndash80∘ 0∘ 5∘



MCP 90∘ 30∘ndash40∘ 60∘

PIP 110∘ 0∘ 0∘

DIP 80∘ndash90∘ 5∘ 0∘


MCP 90∘ 30∘ndash40∘ 45∘

PIP 110∘ 0∘ 0∘

DIP 80∘ndash90∘ 5∘ 0∘


MCP 90∘ 30∘ndash40∘ 45∘

PIP 120∘ 0∘ 0∘

DIP 80∘ndash90∘ 5∘ 0∘


MCP 90∘ 30∘ndash40∘ 50∘

PIP 135∘ 0∘ 0∘

DIP 90∘ 5∘ 0∘







Distal link

Middle link

Proximal link

Metacarpal link

Carpal zone

Wrist Reference

0

1

23

4

L01L02

L03L04

L11

L12L13

L14

L21

L22L23

L24

L31

L32L33

L34

pt1

pt2pt3

pt4

pt0

L20

L00L10

R0R0

R1i

R2i

R3i

R5i

R5iR4i

R4i

TMCaa

MCPaa

TMCfe

IPfe

MCPfeCMCfe

MCPfePIPfe

DIPfe

R0i = R1i

R2i = R3i




119876119894=0

0

119879119894sdot0

6119879119894

(120579119895) =0

0

119879119894sdot

5

prod

119895=1

(119895minus1)

119895

119879119894(120579119895) sdot5

6

119879119905119894

119876119894=0

0

119879119894sdot0

1119879119894

(120579CMC119891119890) sdot1

2119879119894

(120579MCP119886119887119886119889) sdot2

3119879119894

(120579MCP119891119890)

sdot3

4119879119894

(120579PIP119891119890) sdot4

5119879119894

(120579DIP119891119890) sdot5

6

119879119905119894

(8)where







(iii) 06





(iv) (119895minus1)119895




Finger 119894 (119894 = 1 4)Joint 120572

119895minus1

119886119895minus1

119889119895

120579119895

119895 = 1 1205872 0 0 120579CMC119891119890

119895 = 2 minus1205872 1198710119894

0 120579MCP119886119887119886119889

119895 = 3 1205872 0 0 120579MCP119891119890

119895 = 4 0 1198711119894

0 120579PIP119891119890

119895 = 5 0 1198712119894

0 120579DIP119891119890

(v) 56


119905119894= [119897119905119894119909119897119905119894119910119897119905119894119911]

119879



119891119890 MCP

119886119887119886119889

MCP119891119890

PIP119891119890

and DIP119891119890









119886119895minus1

119889119895

120579119895

119895 = 1 0 0 0 120579TMC119886119887119886119889

119895 = 2 1205872 0 0 120579TMC119891119890

119895 = 3 0 11987100

0 120579MCP119891119890

119895 = 4 0 11987110

0 120579IP119891119890


1198760=0

0

1198790sdot0

5

1198790(120579119896)

1198760=0

0

1198790sdot0

11198790

(120579TMC119886119887119886119889) sdot1

21198790

(120579TMC119891119890)

sdot2

31198790

(120579MCP119891119890) sdot3

41198790

(120579IP119891119890) sdot4

51198791199050

(9)


TMC119891119890

MCP119891119890

and IP119891119890


[1198971199050119909

1198971199050119910

1198971199050119911]

119879






119894


119891119890

MCP119886119887119886119889

and DIP119891119890



120579CMC119891119890 = atan(11987633

11987613

)

120579MCP119886119887119886119889 = atan[

[

11987613

minus11987623cos (120579CMC119891119890)

]

]

120579DIP119891119890 = atan(12059811205982

)

(10)



1205981=

(minus1) sdot (11989811987622+ 11989911987621)

(1198982

minus 1198992

)

1205982=

Q21

119898

+

119899

119898

sdot

(minus1) sdot (11989811987622+ 11989911987621)

(1198982

minus 1198992

)





(11)

TheMCP119891119890

PIP119891119890




expression

1= 1198651minus [11987131lowast ] (12)


1198751= [119875111990911987511199101198751119911]

119879

(13)

where1198751119909= 11987101cos (120579CMC)

1198751119910= 11987101sin (120579CMC)

1198751119911= 0

(14)


Knowing 1and 119875


1is


Angles 1205931and 120593

2are expressed as

1205931= acos(

1199032

2

+ 1198712

01

minus 1199032

1

2119903211989701

)

1205932= acos(

1199032

2

+ 1198712

11

minus 1198712

21

2119903211987111

)

(16)


100381710038171003817100381710038171

10038171003817100381710038171003817

1199032=10038171003817100381710038171199061

1003817100381710038171003817

(17)


120579MCP119891119890 = 120587 minus 1205931 minus 1205932 (18)


120579PIP119891119890 = 120587 minus 1205933 (19)

where

1205933= acos(

1198712

21

+ 1198712

11

minus 1199032

2

21198712111987111

) (20)




120579TMC119886119887119886119889 = atan(11987613

minus11987623

) (21)




120579MCP119891119890 = 120587 minus 1205741

120579IP119891119890 = 120587 minus 1205742 minus 1205743(22)

where

1205741= acos(

1198712

10

+ 1198712

00

minus 1199032

3

21198711011987100

)

1205742= acos(

1198712

20

+ 1199042

3

minus 1199032

4

2119871201199033

)

1205743= acos(

1199032

3

+ 1198712

10

minus 1198712

00

211990311990411987110

)

1199033=

100381710038171003817100381710038170

10038171003817100381710038171003817

1199034=

10038171003817100381710038171003817

1198650

10038171003817100381710038171003817

0= 1198650minus [11987120lowast ]

(23)


120579TMC119891119890 = atan(12059841205983

) (24)

where

1205983= (11987632+ 11987631) cos 120583

1205984=


cos 120583

120583 = 120579MCP + 120579IP

(25)



2is the


2




1205793= 120579MCP119891119890

1205794= 120579PIP119891119890

1205795= 120579DIP119891119890

(26)




119892= [

[

119887119892119909

119887119892119910

0

]

]

119892 = 1 2 3 (27)





119895and 119904120579

119895stand for


1= [

[

11988711199091198881205793minus 11988711199101199041205793

11988711199091199041205793+ 11988711199101198881205793

0

]

]

2= [

[

11987111198881205793

11987111199041205793

0

]

]

+

[[[

[

1198872119909119888 (1205793+ 1205794) minus 1198872119910119904 (1205793+ 1205794)

1198872119909119904 (1205793+ 1205794) + 1198872119910119888 (1205793+ 1205794)

0

]]]

]



3= [

[

11987111198881205793

11987111199041205793

0

]

]

+[[

[

1198712119888 (1205793+ 1205794)

1198712119904 (1205793+ 1205794)

0

]]

]

+

[[[

[

1198873119909119888 (1205793+ 1205794+ 1205795) minus 1198873119910119904 (1205793+ 1205794+ 1205795)

1198873119909119904 (1205793+ 1205794+ 1205795) + 1198873119910119888 (1205793+ 1205794+ 1205795)

0

]]]

]

(28)


1205793+ 1205794= 12057934

1205793+1205794=

12057934

1205793+ 1205794+ 1205795= 120579345

1205793+1205794+1205795=

120579345

(29)


V1198661=[[

[

minus1198871119909

12057931199041205793minus 1198871119910

12057931198881205793

1198871119909

12057931198881205793minus 1198871119910

12057931199041205793

0

]]

]

V1198662=[[

[

minus1198711

12057931199041205793minus1198872119909

1205793411990412057934minus 1198872119910

1205793411988812057934

1198711

12057931198881205793+ 1198872119909

1205793411988812057934minus 1198872119910

1205793411990412057934

0

]]

]

V1198663=[[

[

minus1198711

12057931199041205793minus1198712

1205793411990412057934minus1198873119909

120579345119904120579345minus 1198873119910

120579345119888120579345

1198711

12057931198881205793+ 1198712

1205793411988812057934+ 1198873119909

120579345119888120579345minus 1198873119910

120579345119904120579345

0

]]

]

(30)


119879 =

1

2

1198981(V1198661)

2

+

1

2

1198982(V1198662)

2

+

1

2

1198983(V1198663)

2

+

1

2

1198681(1205793)

2

+

1

2

1198682(12057934)

2

+

1

2

1198683(120579345)

2

(31)




1 1198962 and 119896



119880 = 1198981119892 (11988711199091199041205793+ 11988711199101198881205793)

+ 1198982119892 (11987111199041205793+ 119887211990911990412057934+ 119887211991011988812057934)

+ 1198983119892 (11987111199041205793+ 119871211990412057934+ 1198873119909119904120579345+ 1198873119910119888120579345)

+

1

2

1198961(1205793)2

+

1

2

1198962(12057934minus 1205793)2

+

1

2

1198963(120579345minus 12057934)2

(32)


119865 =

1

2

1198881(1205793)

2

+

1

2

1198882(12057934minus1205793)

2

+

1

2

1198883(120579345minus12057934)

2

(33)



1

1198882 and 119888


joints respectively



3 12057934 and 120579

345

119889

119889119905

(

120597 (119879 minus 119880)

1205971205793

) minus

120597 (119879 minus 119880)

1205971205793

+

120597119865

1205971205793

= 1205913

119889

119889119905

(

120597 (119879 minus 119880)

12059712057934

) minus

120597 (119879 minus 119880)

12059712057934

+

120597119865

12059712057934

= 1205914

119889

119889119905

(

120597 (119879 minus 119880)

120597120579345

) minus

120597 (119879 minus 119880)

120597120579345

+

120597119865

120597120579345

= 1205915

(34)




120591119895=

(sum5

119895=3

120575119882119895)

120575120579119895

119895 = 3 4 5 (35)



1205913= (11986531199101198903119909minus 11986531199091198903119910) + [

[

minus11987111199041205793

11987111198881205793

0

]

]

sdot [

[

1198654119909c12057934minus 1198654119910s12057934

1198654119909s12057934+ 1198654119910c12057934

0

]

]

+ [

[

minus11987111199041205793

11987111198881205793

0

]

]

sdot [

[

1198655119909c120579345minus 1198655119910s120579345

1198655119909s120579345+ 1198655119910c120579345

0

]

]

+ 1198621198981minus 1198621198982

1205914= (11986541199101198904119909minus 11986541199091198904119910) + [

[

minus119871211990412057934

119871211988812057934

0

]

]

sdot [

[

1198655119909c120579345minus 1198655119910s120579345

1198655119909s120579345+ 1198655119910c120579345

0

]

]

+ 1198621198982minus 1198621198983

1205915= (11986551199101198905119909minus 11986551199091198905119910) + 1198621198983

(36)

where the term 119862119898119892


119895= [1198651198951199091198651198951199100]119879 is the contact


119895= [1198901198951199091198901198941199100]119879


1198601

1205793+ 1198602

12057934+ 1198603

120579345= 1198604

1198611

1205793+ 1198612

12057934+ 1198613

120579345= 1198614

1198621

1205793+ 1198622

12057934+ 1198623

120579345= 1198624

(37)


The terms 1198601 1198602 and 119860










119895= (1198631198952)(cos(120587119905119905

1)minus1) where




120579119895 max = | minus 12058711986311989521199051|





Omega maxMCP (rads)

Omega maxPIP (rads)

Omega maxDIP (rads)

Cycles in 10 s 27 27 31 25 20 29 23 27 21 31 261 3843 261 12164 15026 10018


Omega maxMCP (rads)

Omega maxPIP (rads)

Omega maxDIP (rads)

Cycles in 10 s 27 24 17 34 29 27 30 24 21 27 26 4784 26 12118 14969 9979



1198983are


119898max=



119889119895= |119862119889119895||119862119898119895|



0 20 40 60 80 100minus100

minus50

0

50

100

150

200

250

300

Cycle ()

C(N

mm

)

Cm1

Cm2

Cm3



MCP PIP DIP119891119896119895

3805 3848 3858119891119888119895

5758 6059 61119891119889119895

437 093 042



7 Conclusions




1198981

= 705 g 1198982

= 397 g 1198983

= 270 g1198681

= 1000 gmm2 1198682

= 240 gmm2 1198683

= 120 gmm2

1198961

= 00902Nm 1198962

= 00902Nm 1198963

= 00902Nm1198881

= 00225Nms 1198882

= 00225Nms 1198883

= 00225Nms1198711

= [50 0 0]

119879

mm 1198712

= [20 0 0]

119879

mm 1198713

= [25 0 0]

119879

mm1198871

= [25 0 0]

119879

mm 1198872

= [10 0 0]

119879

mm 1198873

= [125 0 0]

119879

mm1198901

= [25 minus8 0]

119879

mm 1198902

= [10 minus7 0]

119879

mm 1198903

= [12 minus6 0]

119879

mm



Appendices



11987611989411= [(119888120579CMC119891119890119888120579MCP119886119887119886119889) 119888120579MCP119891119890

minus 119904120579CMC119891119890119904120579MCP119891119890]


minus [(119888120579CMC119891119890119888120579MCP119886119887119886119889) 119904120579MCP119891119890

+ 119904120579CMC119891119890119888120579MCP119891119890]


11987611989412= [(119888120579CMC119891119890119888120579MCP119886119887119886119889) 119888120579MCP119891119890

minus 119904120579CMC119891119890119904120579MCP119891119890]


minus [(119888120579CMC119891119890119888120579MCP119886119887119886119889) 119904120579MCP119891119890

+ 119904120579CMC119891119890119888120579MCP119891119890]


11987611989413= 119888120579CMC119891119890119904120579MCP119886119887119886119889

11987611989414= [(119888120579CMC119891119890119888120579MCP119886119887119886119889) 119888120579MCP119891119890

minus 119904120579CMC119891119890119904120579MCP119891119890]



+ 1198711119894+ 1198712119894119888120579PIP119891119890]

minus [(119888120579CMC119891119890119888120579MCP119886119887119886119889) 119904120579MCP119891119890

+ 119904120579CMC119891119890119888120579MCP119891119890]



+ 1198712119894119904120579PIP119891119890]

+ (119888120579CMC119891119890119904120579MCP119886119887119886119889) 119871 119905119894119911 + 1198710119894119888120579CMC119891119890

11987611989421= 119904120579MCP119886119887119886119889119888120579MCP119891119890


minus 119904120579MCP119886119887119886119889119904120579MCP119891119890


11987611989422= 119904120579MCP119886119887119886119889119888120579MCP119891119890



minus 119904120579MCP119886119887119886119889119904120579MCP119891119890


11987611989423= minus 119888120579MCP119886119887119886119889

11987611989424= 119904120579MCP119886119887119886119889119888120579MCP119891119890



+ 1198711119894+ 1198712119894119888120579PIP119891119890] minus 119904120579MCP119886119887119886119889119904120579MCP119891119890



+ 1198712119894119904120579PIP119891119890] + (minus119888120579MCP119886119887119886119889) 119871 119905119894119911

11987611989431= (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119888120579MCP119891119890

+ 119888120579CMC119891119890119904120579MCP119891119890


+ minus (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119904120579MCP119891119890

+ 119888120579CMC119891119890119888120579MCP119891119890


11987611989432= (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119888120579MCP119891119890

+ 119888120579CMC119891119890119904120579MCP119891119890


+ minus (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119904120579MCP119891119890

+ 119888120579CMC119891119890119888120579MCP119891119890


11987611989433= 119904120579CMC119891119890119904120579MCP119886119887119886119889

11987611989434= (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119888120579MCP119891119890

+ 119888120579119862119872119862119891119890

119904120579119872119862119875119891119890



+ 1198711119894+ 1198712119894119888120579PIP119891119890]

+ minus (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119904120579MCP119891119890

+ 119888120579CMC119891119890119888120579MCP119891119890



+ 1198712119894119904120579PIP119891119890]

+ (119904120579CMC119891119890119904120579MCP119886119887119886119889) 119871 119905119894119911 + 1198710119894119904120579CMC119891119890

(A1)


119876011= [119888120579TMC119886119887119886119889119888120579TMC119891119890]


+ [minus119888120579TMC119886119887119886119889119904120579TMC119891119890]


119876012= [119888120579TMC119886119887119886119889119888120579TMC119891119890]


+ [minus119888120579TMC119886119887119886119889119904120579TMC119891119890]


119876013= 119904120579TMC119886119887119886119889

119876014= [119888120579TMC119886119887119886119889119888120579TMC119891119890]

times [1198711199050 119909


+ 1198711199050 119910


+ 11987110119888120579MCP119891119890 + 11987100]

+ [minus119888120579TMC119886119887119886119889119904120579TMC119891119890]

times [1198711199050 119909

(119904120579MCP119891119890119888120579IP119891119890 + 119888120579MCP119891119890119904120579IP119891119890)

+ 1198711199050 119910


+ 11987110119904120579MCP119891119890] + 119871 1199050 119911119904120579TMC119886119887119886119889


119876021= [119904120579TMC119886119887119886119889119888120579TMC119891119890]


+ [minus119904120579TMC119886119887119886119889119904120579TMC119891119890]


119876022= [119904120579TMC119886119887119886119889119888120579TMC119891119890]


+ [minus119904120579TMC119886119887119886119889119904120579TMC119891119890]


119876023= minus 119888120579TMC119886119887119886119889

119876024= [119904120579TMC119886119887119886119889119888120579TMC119891119890]

times [1198711199050 119909


+ 1198711199050 119910


+ 11987110119888120579MCP119891119890 + 11987100]

+ [minus119904120579TMC119886119887119886119889119904120579TMC119891119890]

times [1198711199050 119909

(119904120579MCP119891119890119888120579IP119891119890 + 119888120579MCP119891119890119904120579IP119891119890)

+ 1198711199050 119910


+ 11987110119904120579MCP119891119890] minus 119888120579TMC119886119887119886119889119871 1199050 119911


+ 119888120579TMC119891119890 (119904120579MCP119891119890119888120579IP119891119890 + 119888120579MCP119891119890119904120579IP119891119890)



119876033= 0

119876034= 119904120579TMC119891119890

times [1198711199050 119909


+ 1198711199050 119910


+ 11987110119888120579MCP119891119890 + 11987100] + 119888120579TMC119891119890

times [1198711199050 119909

(119904120579MCP119891119890119888120579IP119891119890 + 119888120579MCP119891119890119904120579IP119891119890)

+ 1198711199050 119910


+11987110119904120579MCP119891119890]

(A2)


Consider


+ (11988711199091198881205791minus 11988711199101199041205791) (1198871119909c1205791minus 1198871119910s1205791)]


11199091199041205791minus 11987111199101198881205791)

+ (11987111199091198881205791minus 11987111199101199041205791) (11987111199091198881205791minus 11987111199101199041205791)]


+(11987111199091198881205791minus 11987111199101199041205791)times (119871

11199091198881205791minus 11987111199101199041205791)]+1198681


11199091199041205791minus 11987111199101198881205791)

+ (+119887211990911988812057912minus 119887211991011990412057912) (11987111199091198881205791minus 11987111199101199041205791)]


+ (+119871211990911988812057912minus 119871211991011990412057912) (11987111199091198881205791minus 11987111199101199041205791)]


+ (+1198873119909119888120579123minus 1198873119910119904120579123) (11987111199091198881205791minus 11987111199101199041205791)]

1198604= 1198981[(minus1198871119909

12057911199041205791minus 1198871119910

12057911198881205791) (minus1198871119909

1205791c1205791+ 1198871119910

12057911199041205791)


+ 1198982[minus (minus119871

1119909

1205792

1

1198881205791+ 1198711119910

1205792

1

1199041205791

minus1198872119909

1205792

12

11988812057912+ 1198872119910

1205792

12

11990412057912)



1205792

1

1199041205791minus 1198711119910

1205792

1

1198881205791minus 1198872119909

1205792

12

11990412057912

minus1198872119910

1205792

12

11988812057912) (11987111199091198881205791minus 11987111199101199041205791)]

+ 1198983[minus (minus119871

1119909

1205792

1

1198881205791+ 1198711119910

1205792

1

1199041205791minus1198712119909

1205792

12

11988812057912

+1198712119910

1205792

12

11990412057912minus1198873119909

1205792

123

119888120579123+ 1198873119910

1205792

123

119904120579123)



1205792

1

1199041205791minus 1198711119910

1205792

1

1198881205791minus 1198712119909

1205792

12

11990412057912


minus1198712119910

1205792

12

11988812057912minus 1198873119909

1205792

123

119904120579123minus 1198873119910

1205792

123

119888120579123)

times (11987111199091198881205791minus 11987111199101199041205791)]

minus (1198981119892 (11988711199091198881205791minus 11988711199101199041205791) + (119898

2+ 1198983) 119892


minus 1198881

1205791+ 1198882(12057912minus1205791)


+ (11987111199091198881205791minus 11987111199101199041205791) (119887211990911988812057912minus 119887211991011990412057912)]


+ (11987111199091198881205791minus 11987111199101199041205791) (119871211990911988812057912minus 119871211991011990412057912)]


+ (+119887211990911988812057912minus 119887211991011990412057912) (119887211990911988812057912minus 119887211991011990412057912)]


+ (+119871211990911988812057912) (119871211990911988812057912minus 119871211991011990412057912)] + 119868

2


+ (+1198873119909119888120579123minus 1198873119910119904120579123) (119871211990911988812057912minus 119871211991011990412057912)]

1198614= 1198982[minus (minus119871

1119909

1205792

1

1198881205791+ 1198711119910

1205792

1

1199041205791minus1198872119909

1205792

12

11988812057912

+1198872119910

1205792

12

11990412057912) (minus119887211990911990412057912minus 119887211991011988812057912)


1205792

1

1199041205791minus 1198711119910

1205792

1

1198881205791minus 1198872119909

1205792

12

11990412057912

minus1198872119910

1205792

12

11988812057912) (119887211990911988812057912minus 119887211991011990412057912)]

+ 1198983[minus (minus119871

1119909

1205792

1

1198881205791+ 1198711119910

1205792

1

1199041205791minus1198712119909

1205792

12

11988812057912

+ 1198712119910

1205792

12

11990412057912minus1198873119909

1205792

123

119888120579123

+1198873119910

1205792

123

119904120579123) (minus119871211990911990412057912minus 119871211991011988812057912)


1205792

1

1199041205791minus 1198711119910

1205792

1

1198881205791minus 1198712119909

1205792

12

11990412057912

minus 1198712119910

1205792

12

11988812057912minus 1198873119909

1205792

123

119904120579123

minus1198873119910

1205792

123

119888120579123) (119871211990911988812057912minus 119871211991011990412057912)]

minus (1198982119892 (119887211990911988812057912minus 119887211991011990412057912)

+ 1198983119892 (119871211990911988812057912minus 119871211991011990412057912)




+ (11987111199091198881205791minus 11987111199101199041205791) (1198873119909119888120579123minus 1198873119910119904120579123)]




+ (+1198873119909119888120579123minus 1198873119910119904120579123)

times (1198873119909119888120579123minus 1198873119910119904120579123)] + 119868

3

1198624= 1198983[minus (minus119871

1119909

1205792

1

1198881205791+ 1198711119910

1205792

1

1199041205791minus1198712119909

1205792

12

11988812057912

+ 1198712119910

1205792

12

11990412057912minus1198873119909

1205792

123

119888120579123

+1198873119910

1205792

123

119904120579123) (minus1198873119909119904120579123minus 1198873119910119888120579123)


1205792

1

1199041205791minus 1198711119910

1205792

1

1198881205791minus 1198712119909

1205792

12

11990412057912

minus 1198712119910

1205791211990412057912minus 1198712119910

1205792

12

11988812057912+ 1198873119909

120579123119888120579123

minus1198873119909

1205792

123

119904120579123minus 1198873119910

1205792

123

119888120579123)

times (1198873119909119888120579123minus 1198873119910119904120579123)]



(B1)

References
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













120579MCP119891119890Ring asymp7

12

120579MCP119891119890Little


3

120579MCP119891119890Ring



(6)




5

120579MCP119891119890Index (7)




4 Kinematic Model





MCP 75∘ndash80∘ 0∘ 5∘



MCP 90∘ 30∘ndash40∘ 60∘

PIP 110∘ 0∘ 0∘

DIP 80∘ndash90∘ 5∘ 0∘


MCP 90∘ 30∘ndash40∘ 45∘

PIP 110∘ 0∘ 0∘

DIP 80∘ndash90∘ 5∘ 0∘


MCP 90∘ 30∘ndash40∘ 45∘

PIP 120∘ 0∘ 0∘

DIP 80∘ndash90∘ 5∘ 0∘


MCP 90∘ 30∘ndash40∘ 50∘

PIP 135∘ 0∘ 0∘

DIP 90∘ 5∘ 0∘







Distal link

Middle link

Proximal link

Metacarpal link

Carpal zone

Wrist Reference

0

1

23

4

L01L02

L03L04

L11

L12L13

L14

L21

L22L23

L24

L31

L32L33

L34

pt1

pt2pt3

pt4

pt0

L20

L00L10

R0R0

R1i

R2i

R3i

R5i

R5iR4i

R4i

TMCaa

MCPaa

TMCfe

IPfe

MCPfeCMCfe

MCPfePIPfe

DIPfe

R0i = R1i

R2i = R3i




119876119894=0

0

119879119894sdot0

6119879119894

(120579119895) =0

0

119879119894sdot

5

prod

119895=1

(119895minus1)

119895

119879119894(120579119895) sdot5

6

119879119905119894

119876119894=0

0

119879119894sdot0

1119879119894

(120579CMC119891119890) sdot1

2119879119894

(120579MCP119886119887119886119889) sdot2

3119879119894

(120579MCP119891119890)

sdot3

4119879119894

(120579PIP119891119890) sdot4

5119879119894

(120579DIP119891119890) sdot5

6

119879119905119894

(8)where







(iii) 06





(iv) (119895minus1)119895




Finger 119894 (119894 = 1 4)Joint 120572

119895minus1

119886119895minus1

119889119895

120579119895

119895 = 1 1205872 0 0 120579CMC119891119890

119895 = 2 minus1205872 1198710119894

0 120579MCP119886119887119886119889

119895 = 3 1205872 0 0 120579MCP119891119890

119895 = 4 0 1198711119894

0 120579PIP119891119890

119895 = 5 0 1198712119894

0 120579DIP119891119890

(v) 56


119905119894= [119897119905119894119909119897119905119894119910119897119905119894119911]

119879



119891119890 MCP

119886119887119886119889

MCP119891119890

PIP119891119890

and DIP119891119890









119886119895minus1

119889119895

120579119895

119895 = 1 0 0 0 120579TMC119886119887119886119889

119895 = 2 1205872 0 0 120579TMC119891119890

119895 = 3 0 11987100

0 120579MCP119891119890

119895 = 4 0 11987110

0 120579IP119891119890


1198760=0

0

1198790sdot0

5

1198790(120579119896)

1198760=0

0

1198790sdot0

11198790

(120579TMC119886119887119886119889) sdot1

21198790

(120579TMC119891119890)

sdot2

31198790

(120579MCP119891119890) sdot3

41198790

(120579IP119891119890) sdot4

51198791199050

(9)


TMC119891119890

MCP119891119890

and IP119891119890


[1198971199050119909

1198971199050119910

1198971199050119911]

119879






119894


119891119890

MCP119886119887119886119889

and DIP119891119890



120579CMC119891119890 = atan(11987633

11987613

)

120579MCP119886119887119886119889 = atan[

[

11987613

minus11987623cos (120579CMC119891119890)

]

]

120579DIP119891119890 = atan(12059811205982

)

(10)



1205981=

(minus1) sdot (11989811987622+ 11989911987621)

(1198982

minus 1198992

)

1205982=

Q21

119898

+

119899

119898

sdot

(minus1) sdot (11989811987622+ 11989911987621)

(1198982

minus 1198992

)





(11)

TheMCP119891119890

PIP119891119890




expression

1= 1198651minus [11987131lowast ] (12)


1198751= [119875111990911987511199101198751119911]

119879

(13)

where1198751119909= 11987101cos (120579CMC)

1198751119910= 11987101sin (120579CMC)

1198751119911= 0

(14)


Knowing 1and 119875


1is


Angles 1205931and 120593

2are expressed as

1205931= acos(

1199032

2

+ 1198712

01

minus 1199032

1

2119903211989701

)

1205932= acos(

1199032

2

+ 1198712

11

minus 1198712

21

2119903211987111

)

(16)


100381710038171003817100381710038171

10038171003817100381710038171003817

1199032=10038171003817100381710038171199061

1003817100381710038171003817

(17)


120579MCP119891119890 = 120587 minus 1205931 minus 1205932 (18)


120579PIP119891119890 = 120587 minus 1205933 (19)

where

1205933= acos(

1198712

21

+ 1198712

11

minus 1199032

2

21198712111987111

) (20)




120579TMC119886119887119886119889 = atan(11987613

minus11987623

) (21)




120579MCP119891119890 = 120587 minus 1205741

120579IP119891119890 = 120587 minus 1205742 minus 1205743(22)

where

1205741= acos(

1198712

10

+ 1198712

00

minus 1199032

3

21198711011987100

)

1205742= acos(

1198712

20

+ 1199042

3

minus 1199032

4

2119871201199033

)

1205743= acos(

1199032

3

+ 1198712

10

minus 1198712

00

211990311990411987110

)

1199033=

100381710038171003817100381710038170

10038171003817100381710038171003817

1199034=

10038171003817100381710038171003817

1198650

10038171003817100381710038171003817

0= 1198650minus [11987120lowast ]

(23)


120579TMC119891119890 = atan(12059841205983

) (24)

where

1205983= (11987632+ 11987631) cos 120583

1205984=


cos 120583

120583 = 120579MCP + 120579IP

(25)



2is the


2




1205793= 120579MCP119891119890

1205794= 120579PIP119891119890

1205795= 120579DIP119891119890

(26)




119892= [

[

119887119892119909

119887119892119910

0

]

]

119892 = 1 2 3 (27)





119895and 119904120579

119895stand for


1= [

[

11988711199091198881205793minus 11988711199101199041205793

11988711199091199041205793+ 11988711199101198881205793

0

]

]

2= [

[

11987111198881205793

11987111199041205793

0

]

]

+

[[[

[

1198872119909119888 (1205793+ 1205794) minus 1198872119910119904 (1205793+ 1205794)

1198872119909119904 (1205793+ 1205794) + 1198872119910119888 (1205793+ 1205794)

0

]]]

]



3= [

[

11987111198881205793

11987111199041205793

0

]

]

+[[

[

1198712119888 (1205793+ 1205794)

1198712119904 (1205793+ 1205794)

0

]]

]

+

[[[

[

1198873119909119888 (1205793+ 1205794+ 1205795) minus 1198873119910119904 (1205793+ 1205794+ 1205795)

1198873119909119904 (1205793+ 1205794+ 1205795) + 1198873119910119888 (1205793+ 1205794+ 1205795)

0

]]]

]

(28)


1205793+ 1205794= 12057934

1205793+1205794=

12057934

1205793+ 1205794+ 1205795= 120579345

1205793+1205794+1205795=

120579345

(29)


V1198661=[[

[

minus1198871119909

12057931199041205793minus 1198871119910

12057931198881205793

1198871119909

12057931198881205793minus 1198871119910

12057931199041205793

0

]]

]

V1198662=[[

[

minus1198711

12057931199041205793minus1198872119909

1205793411990412057934minus 1198872119910

1205793411988812057934

1198711

12057931198881205793+ 1198872119909

1205793411988812057934minus 1198872119910

1205793411990412057934

0

]]

]

V1198663=[[

[

minus1198711

12057931199041205793minus1198712

1205793411990412057934minus1198873119909

120579345119904120579345minus 1198873119910

120579345119888120579345

1198711

12057931198881205793+ 1198712

1205793411988812057934+ 1198873119909

120579345119888120579345minus 1198873119910

120579345119904120579345

0

]]

]

(30)


119879 =

1

2

1198981(V1198661)

2

+

1

2

1198982(V1198662)

2

+

1

2

1198983(V1198663)

2

+

1

2

1198681(1205793)

2

+

1

2

1198682(12057934)

2

+

1

2

1198683(120579345)

2

(31)




1 1198962 and 119896



119880 = 1198981119892 (11988711199091199041205793+ 11988711199101198881205793)

+ 1198982119892 (11987111199041205793+ 119887211990911990412057934+ 119887211991011988812057934)

+ 1198983119892 (11987111199041205793+ 119871211990412057934+ 1198873119909119904120579345+ 1198873119910119888120579345)

+

1

2

1198961(1205793)2

+

1

2

1198962(12057934minus 1205793)2

+

1

2

1198963(120579345minus 12057934)2

(32)


119865 =

1

2

1198881(1205793)

2

+

1

2

1198882(12057934minus1205793)

2

+

1

2

1198883(120579345minus12057934)

2

(33)



1

1198882 and 119888


joints respectively



3 12057934 and 120579

345

119889

119889119905

(

120597 (119879 minus 119880)

1205971205793

) minus

120597 (119879 minus 119880)

1205971205793

+

120597119865

1205971205793

= 1205913

119889

119889119905

(

120597 (119879 minus 119880)

12059712057934

) minus

120597 (119879 minus 119880)

12059712057934

+

120597119865

12059712057934

= 1205914

119889

119889119905

(

120597 (119879 minus 119880)

120597120579345

) minus

120597 (119879 minus 119880)

120597120579345

+

120597119865

120597120579345

= 1205915

(34)




120591119895=

(sum5

119895=3

120575119882119895)

120575120579119895

119895 = 3 4 5 (35)



1205913= (11986531199101198903119909minus 11986531199091198903119910) + [

[

minus11987111199041205793

11987111198881205793

0

]

]

sdot [

[

1198654119909c12057934minus 1198654119910s12057934

1198654119909s12057934+ 1198654119910c12057934

0

]

]

+ [

[

minus11987111199041205793

11987111198881205793

0

]

]

sdot [

[

1198655119909c120579345minus 1198655119910s120579345

1198655119909s120579345+ 1198655119910c120579345

0

]

]

+ 1198621198981minus 1198621198982

1205914= (11986541199101198904119909minus 11986541199091198904119910) + [

[

minus119871211990412057934

119871211988812057934

0

]

]

sdot [

[

1198655119909c120579345minus 1198655119910s120579345

1198655119909s120579345+ 1198655119910c120579345

0

]

]

+ 1198621198982minus 1198621198983

1205915= (11986551199101198905119909minus 11986551199091198905119910) + 1198621198983

(36)

where the term 119862119898119892


119895= [1198651198951199091198651198951199100]119879 is the contact


119895= [1198901198951199091198901198941199100]119879


1198601

1205793+ 1198602

12057934+ 1198603

120579345= 1198604

1198611

1205793+ 1198612

12057934+ 1198613

120579345= 1198614

1198621

1205793+ 1198622

12057934+ 1198623

120579345= 1198624

(37)


The terms 1198601 1198602 and 119860










119895= (1198631198952)(cos(120587119905119905

1)minus1) where




120579119895 max = | minus 12058711986311989521199051|





Omega maxMCP (rads)

Omega maxPIP (rads)

Omega maxDIP (rads)

Cycles in 10 s 27 27 31 25 20 29 23 27 21 31 261 3843 261 12164 15026 10018


Omega maxMCP (rads)

Omega maxPIP (rads)

Omega maxDIP (rads)

Cycles in 10 s 27 24 17 34 29 27 30 24 21 27 26 4784 26 12118 14969 9979



1198983are


119898max=



119889119895= |119862119889119895||119862119898119895|



0 20 40 60 80 100minus100

minus50

0

50

100

150

200

250

300

Cycle ()

C(N

mm

)

Cm1

Cm2

Cm3



MCP PIP DIP119891119896119895

3805 3848 3858119891119888119895

5758 6059 61119891119889119895

437 093 042



7 Conclusions




1198981

= 705 g 1198982

= 397 g 1198983

= 270 g1198681

= 1000 gmm2 1198682

= 240 gmm2 1198683

= 120 gmm2

1198961

= 00902Nm 1198962

= 00902Nm 1198963

= 00902Nm1198881

= 00225Nms 1198882

= 00225Nms 1198883

= 00225Nms1198711

= [50 0 0]

119879

mm 1198712

= [20 0 0]

119879

mm 1198713

= [25 0 0]

119879

mm1198871

= [25 0 0]

119879

mm 1198872

= [10 0 0]

119879

mm 1198873

= [125 0 0]

119879

mm1198901

= [25 minus8 0]

119879

mm 1198902

= [10 minus7 0]

119879

mm 1198903

= [12 minus6 0]

119879

mm



Appendices



11987611989411= [(119888120579CMC119891119890119888120579MCP119886119887119886119889) 119888120579MCP119891119890

minus 119904120579CMC119891119890119904120579MCP119891119890]


minus [(119888120579CMC119891119890119888120579MCP119886119887119886119889) 119904120579MCP119891119890

+ 119904120579CMC119891119890119888120579MCP119891119890]


11987611989412= [(119888120579CMC119891119890119888120579MCP119886119887119886119889) 119888120579MCP119891119890

minus 119904120579CMC119891119890119904120579MCP119891119890]


minus [(119888120579CMC119891119890119888120579MCP119886119887119886119889) 119904120579MCP119891119890

+ 119904120579CMC119891119890119888120579MCP119891119890]


11987611989413= 119888120579CMC119891119890119904120579MCP119886119887119886119889

11987611989414= [(119888120579CMC119891119890119888120579MCP119886119887119886119889) 119888120579MCP119891119890

minus 119904120579CMC119891119890119904120579MCP119891119890]



+ 1198711119894+ 1198712119894119888120579PIP119891119890]

minus [(119888120579CMC119891119890119888120579MCP119886119887119886119889) 119904120579MCP119891119890

+ 119904120579CMC119891119890119888120579MCP119891119890]



+ 1198712119894119904120579PIP119891119890]

+ (119888120579CMC119891119890119904120579MCP119886119887119886119889) 119871 119905119894119911 + 1198710119894119888120579CMC119891119890

11987611989421= 119904120579MCP119886119887119886119889119888120579MCP119891119890


minus 119904120579MCP119886119887119886119889119904120579MCP119891119890


11987611989422= 119904120579MCP119886119887119886119889119888120579MCP119891119890



minus 119904120579MCP119886119887119886119889119904120579MCP119891119890


11987611989423= minus 119888120579MCP119886119887119886119889

11987611989424= 119904120579MCP119886119887119886119889119888120579MCP119891119890



+ 1198711119894+ 1198712119894119888120579PIP119891119890] minus 119904120579MCP119886119887119886119889119904120579MCP119891119890



+ 1198712119894119904120579PIP119891119890] + (minus119888120579MCP119886119887119886119889) 119871 119905119894119911

11987611989431= (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119888120579MCP119891119890

+ 119888120579CMC119891119890119904120579MCP119891119890


+ minus (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119904120579MCP119891119890

+ 119888120579CMC119891119890119888120579MCP119891119890


11987611989432= (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119888120579MCP119891119890

+ 119888120579CMC119891119890119904120579MCP119891119890


+ minus (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119904120579MCP119891119890

+ 119888120579CMC119891119890119888120579MCP119891119890


11987611989433= 119904120579CMC119891119890119904120579MCP119886119887119886119889

11987611989434= (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119888120579MCP119891119890

+ 119888120579119862119872119862119891119890

119904120579119872119862119875119891119890



+ 1198711119894+ 1198712119894119888120579PIP119891119890]

+ minus (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119904120579MCP119891119890

+ 119888120579CMC119891119890119888120579MCP119891119890



+ 1198712119894119904120579PIP119891119890]

+ (119904120579CMC119891119890119904120579MCP119886119887119886119889) 119871 119905119894119911 + 1198710119894119904120579CMC119891119890

(A1)


119876011= [119888120579TMC119886119887119886119889119888120579TMC119891119890]


+ [minus119888120579TMC119886119887119886119889119904120579TMC119891119890]


119876012= [119888120579TMC119886119887119886119889119888120579TMC119891119890]


+ [minus119888120579TMC119886119887119886119889119904120579TMC119891119890]


119876013= 119904120579TMC119886119887119886119889

119876014= [119888120579TMC119886119887119886119889119888120579TMC119891119890]

times [1198711199050 119909


+ 1198711199050 119910


+ 11987110119888120579MCP119891119890 + 11987100]

+ [minus119888120579TMC119886119887119886119889119904120579TMC119891119890]

times [1198711199050 119909

(119904120579MCP119891119890119888120579IP119891119890 + 119888120579MCP119891119890119904120579IP119891119890)

+ 1198711199050 119910


+ 11987110119904120579MCP119891119890] + 119871 1199050 119911119904120579TMC119886119887119886119889


119876021= [119904120579TMC119886119887119886119889119888120579TMC119891119890]


+ [minus119904120579TMC119886119887119886119889119904120579TMC119891119890]


119876022= [119904120579TMC119886119887119886119889119888120579TMC119891119890]


+ [minus119904120579TMC119886119887119886119889119904120579TMC119891119890]


119876023= minus 119888120579TMC119886119887119886119889

119876024= [119904120579TMC119886119887119886119889119888120579TMC119891119890]

times [1198711199050 119909


+ 1198711199050 119910


+ 11987110119888120579MCP119891119890 + 11987100]

+ [minus119904120579TMC119886119887119886119889119904120579TMC119891119890]

times [1198711199050 119909

(119904120579MCP119891119890119888120579IP119891119890 + 119888120579MCP119891119890119904120579IP119891119890)

+ 1198711199050 119910


+ 11987110119904120579MCP119891119890] minus 119888120579TMC119886119887119886119889119871 1199050 119911


+ 119888120579TMC119891119890 (119904120579MCP119891119890119888120579IP119891119890 + 119888120579MCP119891119890119904120579IP119891119890)



119876033= 0

119876034= 119904120579TMC119891119890

times [1198711199050 119909


+ 1198711199050 119910


+ 11987110119888120579MCP119891119890 + 11987100] + 119888120579TMC119891119890

times [1198711199050 119909

(119904120579MCP119891119890119888120579IP119891119890 + 119888120579MCP119891119890119904120579IP119891119890)

+ 1198711199050 119910


+11987110119904120579MCP119891119890]

(A2)


Consider


+ (11988711199091198881205791minus 11988711199101199041205791) (1198871119909c1205791minus 1198871119910s1205791)]


11199091199041205791minus 11987111199101198881205791)

+ (11987111199091198881205791minus 11987111199101199041205791) (11987111199091198881205791minus 11987111199101199041205791)]


+(11987111199091198881205791minus 11987111199101199041205791)times (119871

11199091198881205791minus 11987111199101199041205791)]+1198681


11199091199041205791minus 11987111199101198881205791)

+ (+119887211990911988812057912minus 119887211991011990412057912) (11987111199091198881205791minus 11987111199101199041205791)]


+ (+119871211990911988812057912minus 119871211991011990412057912) (11987111199091198881205791minus 11987111199101199041205791)]


+ (+1198873119909119888120579123minus 1198873119910119904120579123) (11987111199091198881205791minus 11987111199101199041205791)]

1198604= 1198981[(minus1198871119909

12057911199041205791minus 1198871119910

12057911198881205791) (minus1198871119909

1205791c1205791+ 1198871119910

12057911199041205791)


+ 1198982[minus (minus119871

1119909

1205792

1

1198881205791+ 1198711119910

1205792

1

1199041205791

minus1198872119909

1205792

12

11988812057912+ 1198872119910

1205792

12

11990412057912)



1205792

1

1199041205791minus 1198711119910

1205792

1

1198881205791minus 1198872119909

1205792

12

11990412057912

minus1198872119910

1205792

12

11988812057912) (11987111199091198881205791minus 11987111199101199041205791)]

+ 1198983[minus (minus119871

1119909

1205792

1

1198881205791+ 1198711119910

1205792

1

1199041205791minus1198712119909

1205792

12

11988812057912

+1198712119910

1205792

12

11990412057912minus1198873119909

1205792

123

119888120579123+ 1198873119910

1205792

123

119904120579123)



1205792

1

1199041205791minus 1198711119910

1205792

1

1198881205791minus 1198712119909

1205792

12

11990412057912


minus1198712119910

1205792

12

11988812057912minus 1198873119909

1205792

123

119904120579123minus 1198873119910

1205792

123

119888120579123)

times (11987111199091198881205791minus 11987111199101199041205791)]

minus (1198981119892 (11988711199091198881205791minus 11988711199101199041205791) + (119898

2+ 1198983) 119892


minus 1198881

1205791+ 1198882(12057912minus1205791)


+ (11987111199091198881205791minus 11987111199101199041205791) (119887211990911988812057912minus 119887211991011990412057912)]


+ (11987111199091198881205791minus 11987111199101199041205791) (119871211990911988812057912minus 119871211991011990412057912)]


+ (+119887211990911988812057912minus 119887211991011990412057912) (119887211990911988812057912minus 119887211991011990412057912)]


+ (+119871211990911988812057912) (119871211990911988812057912minus 119871211991011990412057912)] + 119868

2


+ (+1198873119909119888120579123minus 1198873119910119904120579123) (119871211990911988812057912minus 119871211991011990412057912)]

1198614= 1198982[minus (minus119871

1119909

1205792

1

1198881205791+ 1198711119910

1205792

1

1199041205791minus1198872119909

1205792

12

11988812057912

+1198872119910

1205792

12

11990412057912) (minus119887211990911990412057912minus 119887211991011988812057912)


1205792

1

1199041205791minus 1198711119910

1205792

1

1198881205791minus 1198872119909

1205792

12

11990412057912

minus1198872119910

1205792

12

11988812057912) (119887211990911988812057912minus 119887211991011990412057912)]

+ 1198983[minus (minus119871

1119909

1205792

1

1198881205791+ 1198711119910

1205792

1

1199041205791minus1198712119909

1205792

12

11988812057912

+ 1198712119910

1205792

12

11990412057912minus1198873119909

1205792

123

119888120579123

+1198873119910

1205792

123

119904120579123) (minus119871211990911990412057912minus 119871211991011988812057912)


1205792

1

1199041205791minus 1198711119910

1205792

1

1198881205791minus 1198712119909

1205792

12

11990412057912

minus 1198712119910

1205792

12

11988812057912minus 1198873119909

1205792

123

119904120579123

minus1198873119910

1205792

123

119888120579123) (119871211990911988812057912minus 119871211991011990412057912)]

minus (1198982119892 (119887211990911988812057912minus 119887211991011990412057912)

+ 1198983119892 (119871211990911988812057912minus 119871211991011990412057912)




+ (11987111199091198881205791minus 11987111199101199041205791) (1198873119909119888120579123minus 1198873119910119904120579123)]




+ (+1198873119909119888120579123minus 1198873119910119904120579123)

times (1198873119909119888120579123minus 1198873119910119904120579123)] + 119868

3

1198624= 1198983[minus (minus119871

1119909

1205792

1

1198881205791+ 1198711119910

1205792

1

1199041205791minus1198712119909

1205792

12

11988812057912

+ 1198712119910

1205792

12

11990412057912minus1198873119909

1205792

123

119888120579123

+1198873119910

1205792

123

119904120579123) (minus1198873119909119904120579123minus 1198873119910119888120579123)


1205792

1

1199041205791minus 1198711119910

1205792

1

1198881205791minus 1198712119909

1205792

12

11990412057912

minus 1198712119910

1205791211990412057912minus 1198712119910

1205792

12

11988812057912+ 1198873119909

120579123119888120579123

minus1198873119909

1205792

123

119904120579123minus 1198873119910

1205792

123

119888120579123)

times (1198873119909119888120579123minus 1198873119910119904120579123)]



(B1)

References
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Distal link

Middle link

Proximal link

Metacarpal link

Carpal zone

Wrist Reference

0

1

23

4

L01L02

L03L04

L11

L12L13

L14

L21

L22L23

L24

L31

L32L33

L34

pt1

pt2pt3

pt4

pt0

L20

L00L10

R0R0

R1i

R2i

R3i

R5i

R5iR4i

R4i

TMCaa

MCPaa

TMCfe

IPfe

MCPfeCMCfe

MCPfePIPfe

DIPfe

R0i = R1i

R2i = R3i




119876119894=0

0

119879119894sdot0

6119879119894

(120579119895) =0

0

119879119894sdot

5

prod

119895=1

(119895minus1)

119895

119879119894(120579119895) sdot5

6

119879119905119894

119876119894=0

0

119879119894sdot0

1119879119894

(120579CMC119891119890) sdot1

2119879119894

(120579MCP119886119887119886119889) sdot2

3119879119894

(120579MCP119891119890)

sdot3

4119879119894

(120579PIP119891119890) sdot4

5119879119894

(120579DIP119891119890) sdot5

6

119879119905119894

(8)where







(iii) 06





(iv) (119895minus1)119895




Finger 119894 (119894 = 1 4)Joint 120572

119895minus1

119886119895minus1

119889119895

120579119895

119895 = 1 1205872 0 0 120579CMC119891119890

119895 = 2 minus1205872 1198710119894

0 120579MCP119886119887119886119889

119895 = 3 1205872 0 0 120579MCP119891119890

119895 = 4 0 1198711119894

0 120579PIP119891119890

119895 = 5 0 1198712119894

0 120579DIP119891119890

(v) 56


119905119894= [119897119905119894119909119897119905119894119910119897119905119894119911]

119879



119891119890 MCP

119886119887119886119889

MCP119891119890

PIP119891119890

and DIP119891119890









119886119895minus1

119889119895

120579119895

119895 = 1 0 0 0 120579TMC119886119887119886119889

119895 = 2 1205872 0 0 120579TMC119891119890

119895 = 3 0 11987100

0 120579MCP119891119890

119895 = 4 0 11987110

0 120579IP119891119890


1198760=0

0

1198790sdot0

5

1198790(120579119896)

1198760=0

0

1198790sdot0

11198790

(120579TMC119886119887119886119889) sdot1

21198790

(120579TMC119891119890)

sdot2

31198790

(120579MCP119891119890) sdot3

41198790

(120579IP119891119890) sdot4

51198791199050

(9)


TMC119891119890

MCP119891119890

and IP119891119890


[1198971199050119909

1198971199050119910

1198971199050119911]

119879






119894


119891119890

MCP119886119887119886119889

and DIP119891119890



120579CMC119891119890 = atan(11987633

11987613

)

120579MCP119886119887119886119889 = atan[

[

11987613

minus11987623cos (120579CMC119891119890)

]

]

120579DIP119891119890 = atan(12059811205982

)

(10)



1205981=

(minus1) sdot (11989811987622+ 11989911987621)

(1198982

minus 1198992

)

1205982=

Q21

119898

+

119899

119898

sdot

(minus1) sdot (11989811987622+ 11989911987621)

(1198982

minus 1198992

)





(11)

TheMCP119891119890

PIP119891119890




expression

1= 1198651minus [11987131lowast ] (12)


1198751= [119875111990911987511199101198751119911]

119879

(13)

where1198751119909= 11987101cos (120579CMC)

1198751119910= 11987101sin (120579CMC)

1198751119911= 0

(14)


Knowing 1and 119875


1is


Angles 1205931and 120593

2are expressed as

1205931= acos(

1199032

2

+ 1198712

01

minus 1199032

1

2119903211989701

)

1205932= acos(

1199032

2

+ 1198712

11

minus 1198712

21

2119903211987111

)

(16)


100381710038171003817100381710038171

10038171003817100381710038171003817

1199032=10038171003817100381710038171199061

1003817100381710038171003817

(17)


120579MCP119891119890 = 120587 minus 1205931 minus 1205932 (18)


120579PIP119891119890 = 120587 minus 1205933 (19)

where

1205933= acos(

1198712

21

+ 1198712

11

minus 1199032

2

21198712111987111

) (20)




120579TMC119886119887119886119889 = atan(11987613

minus11987623

) (21)




120579MCP119891119890 = 120587 minus 1205741

120579IP119891119890 = 120587 minus 1205742 minus 1205743(22)

where

1205741= acos(

1198712

10

+ 1198712

00

minus 1199032

3

21198711011987100

)

1205742= acos(

1198712

20

+ 1199042

3

minus 1199032

4

2119871201199033

)

1205743= acos(

1199032

3

+ 1198712

10

minus 1198712

00

211990311990411987110

)

1199033=

100381710038171003817100381710038170

10038171003817100381710038171003817

1199034=

10038171003817100381710038171003817

1198650

10038171003817100381710038171003817

0= 1198650minus [11987120lowast ]

(23)


120579TMC119891119890 = atan(12059841205983

) (24)

where

1205983= (11987632+ 11987631) cos 120583

1205984=


cos 120583

120583 = 120579MCP + 120579IP

(25)



2is the


2




1205793= 120579MCP119891119890

1205794= 120579PIP119891119890

1205795= 120579DIP119891119890

(26)




119892= [

[

119887119892119909

119887119892119910

0

]

]

119892 = 1 2 3 (27)





119895and 119904120579

119895stand for


1= [

[

11988711199091198881205793minus 11988711199101199041205793

11988711199091199041205793+ 11988711199101198881205793

0

]

]

2= [

[

11987111198881205793

11987111199041205793

0

]

]

+

[[[

[

1198872119909119888 (1205793+ 1205794) minus 1198872119910119904 (1205793+ 1205794)

1198872119909119904 (1205793+ 1205794) + 1198872119910119888 (1205793+ 1205794)

0

]]]

]



3= [

[

11987111198881205793

11987111199041205793

0

]

]

+[[

[

1198712119888 (1205793+ 1205794)

1198712119904 (1205793+ 1205794)

0

]]

]

+

[[[

[

1198873119909119888 (1205793+ 1205794+ 1205795) minus 1198873119910119904 (1205793+ 1205794+ 1205795)

1198873119909119904 (1205793+ 1205794+ 1205795) + 1198873119910119888 (1205793+ 1205794+ 1205795)

0

]]]

]

(28)


1205793+ 1205794= 12057934

1205793+1205794=

12057934

1205793+ 1205794+ 1205795= 120579345

1205793+1205794+1205795=

120579345

(29)


V1198661=[[

[

minus1198871119909

12057931199041205793minus 1198871119910

12057931198881205793

1198871119909

12057931198881205793minus 1198871119910

12057931199041205793

0

]]

]

V1198662=[[

[

minus1198711

12057931199041205793minus1198872119909

1205793411990412057934minus 1198872119910

1205793411988812057934

1198711

12057931198881205793+ 1198872119909

1205793411988812057934minus 1198872119910

1205793411990412057934

0

]]

]

V1198663=[[

[

minus1198711

12057931199041205793minus1198712

1205793411990412057934minus1198873119909

120579345119904120579345minus 1198873119910

120579345119888120579345

1198711

12057931198881205793+ 1198712

1205793411988812057934+ 1198873119909

120579345119888120579345minus 1198873119910

120579345119904120579345

0

]]

]

(30)


119879 =

1

2

1198981(V1198661)

2

+

1

2

1198982(V1198662)

2

+

1

2

1198983(V1198663)

2

+

1

2

1198681(1205793)

2

+

1

2

1198682(12057934)

2

+

1

2

1198683(120579345)

2

(31)




1 1198962 and 119896



119880 = 1198981119892 (11988711199091199041205793+ 11988711199101198881205793)

+ 1198982119892 (11987111199041205793+ 119887211990911990412057934+ 119887211991011988812057934)

+ 1198983119892 (11987111199041205793+ 119871211990412057934+ 1198873119909119904120579345+ 1198873119910119888120579345)

+

1

2

1198961(1205793)2

+

1

2

1198962(12057934minus 1205793)2

+

1

2

1198963(120579345minus 12057934)2

(32)


119865 =

1

2

1198881(1205793)

2

+

1

2

1198882(12057934minus1205793)

2

+

1

2

1198883(120579345minus12057934)

2

(33)



1

1198882 and 119888


joints respectively



3 12057934 and 120579

345

119889

119889119905

(

120597 (119879 minus 119880)

1205971205793

) minus

120597 (119879 minus 119880)

1205971205793

+

120597119865

1205971205793

= 1205913

119889

119889119905

(

120597 (119879 minus 119880)

12059712057934

) minus

120597 (119879 minus 119880)

12059712057934

+

120597119865

12059712057934

= 1205914

119889

119889119905

(

120597 (119879 minus 119880)

120597120579345

) minus

120597 (119879 minus 119880)

120597120579345

+

120597119865

120597120579345

= 1205915

(34)




120591119895=

(sum5

119895=3

120575119882119895)

120575120579119895

119895 = 3 4 5 (35)



1205913= (11986531199101198903119909minus 11986531199091198903119910) + [

[

minus11987111199041205793

11987111198881205793

0

]

]

sdot [

[

1198654119909c12057934minus 1198654119910s12057934

1198654119909s12057934+ 1198654119910c12057934

0

]

]

+ [

[

minus11987111199041205793

11987111198881205793

0

]

]

sdot [

[

1198655119909c120579345minus 1198655119910s120579345

1198655119909s120579345+ 1198655119910c120579345

0

]

]

+ 1198621198981minus 1198621198982

1205914= (11986541199101198904119909minus 11986541199091198904119910) + [

[

minus119871211990412057934

119871211988812057934

0

]

]

sdot [

[

1198655119909c120579345minus 1198655119910s120579345

1198655119909s120579345+ 1198655119910c120579345

0

]

]

+ 1198621198982minus 1198621198983

1205915= (11986551199101198905119909minus 11986551199091198905119910) + 1198621198983

(36)

where the term 119862119898119892


119895= [1198651198951199091198651198951199100]119879 is the contact


119895= [1198901198951199091198901198941199100]119879


1198601

1205793+ 1198602

12057934+ 1198603

120579345= 1198604

1198611

1205793+ 1198612

12057934+ 1198613

120579345= 1198614

1198621

1205793+ 1198622

12057934+ 1198623

120579345= 1198624

(37)


The terms 1198601 1198602 and 119860










119895= (1198631198952)(cos(120587119905119905

1)minus1) where




120579119895 max = | minus 12058711986311989521199051|





Omega maxMCP (rads)

Omega maxPIP (rads)

Omega maxDIP (rads)

Cycles in 10 s 27 27 31 25 20 29 23 27 21 31 261 3843 261 12164 15026 10018


Omega maxMCP (rads)

Omega maxPIP (rads)

Omega maxDIP (rads)

Cycles in 10 s 27 24 17 34 29 27 30 24 21 27 26 4784 26 12118 14969 9979



1198983are


119898max=



119889119895= |119862119889119895||119862119898119895|



0 20 40 60 80 100minus100

minus50

0

50

100

150

200

250

300

Cycle ()

C(N

mm

)

Cm1

Cm2

Cm3



MCP PIP DIP119891119896119895

3805 3848 3858119891119888119895

5758 6059 61119891119889119895

437 093 042



7 Conclusions




1198981

= 705 g 1198982

= 397 g 1198983

= 270 g1198681

= 1000 gmm2 1198682

= 240 gmm2 1198683

= 120 gmm2

1198961

= 00902Nm 1198962

= 00902Nm 1198963

= 00902Nm1198881

= 00225Nms 1198882

= 00225Nms 1198883

= 00225Nms1198711

= [50 0 0]

119879

mm 1198712

= [20 0 0]

119879

mm 1198713

= [25 0 0]

119879

mm1198871

= [25 0 0]

119879

mm 1198872

= [10 0 0]

119879

mm 1198873

= [125 0 0]

119879

mm1198901

= [25 minus8 0]

119879

mm 1198902

= [10 minus7 0]

119879

mm 1198903

= [12 minus6 0]

119879

mm



Appendices



11987611989411= [(119888120579CMC119891119890119888120579MCP119886119887119886119889) 119888120579MCP119891119890

minus 119904120579CMC119891119890119904120579MCP119891119890]


minus [(119888120579CMC119891119890119888120579MCP119886119887119886119889) 119904120579MCP119891119890

+ 119904120579CMC119891119890119888120579MCP119891119890]


11987611989412= [(119888120579CMC119891119890119888120579MCP119886119887119886119889) 119888120579MCP119891119890

minus 119904120579CMC119891119890119904120579MCP119891119890]


minus [(119888120579CMC119891119890119888120579MCP119886119887119886119889) 119904120579MCP119891119890

+ 119904120579CMC119891119890119888120579MCP119891119890]


11987611989413= 119888120579CMC119891119890119904120579MCP119886119887119886119889

11987611989414= [(119888120579CMC119891119890119888120579MCP119886119887119886119889) 119888120579MCP119891119890

minus 119904120579CMC119891119890119904120579MCP119891119890]



+ 1198711119894+ 1198712119894119888120579PIP119891119890]

minus [(119888120579CMC119891119890119888120579MCP119886119887119886119889) 119904120579MCP119891119890

+ 119904120579CMC119891119890119888120579MCP119891119890]



+ 1198712119894119904120579PIP119891119890]

+ (119888120579CMC119891119890119904120579MCP119886119887119886119889) 119871 119905119894119911 + 1198710119894119888120579CMC119891119890

11987611989421= 119904120579MCP119886119887119886119889119888120579MCP119891119890


minus 119904120579MCP119886119887119886119889119904120579MCP119891119890


11987611989422= 119904120579MCP119886119887119886119889119888120579MCP119891119890



minus 119904120579MCP119886119887119886119889119904120579MCP119891119890


11987611989423= minus 119888120579MCP119886119887119886119889

11987611989424= 119904120579MCP119886119887119886119889119888120579MCP119891119890



+ 1198711119894+ 1198712119894119888120579PIP119891119890] minus 119904120579MCP119886119887119886119889119904120579MCP119891119890



+ 1198712119894119904120579PIP119891119890] + (minus119888120579MCP119886119887119886119889) 119871 119905119894119911

11987611989431= (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119888120579MCP119891119890

+ 119888120579CMC119891119890119904120579MCP119891119890


+ minus (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119904120579MCP119891119890

+ 119888120579CMC119891119890119888120579MCP119891119890


11987611989432= (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119888120579MCP119891119890

+ 119888120579CMC119891119890119904120579MCP119891119890


+ minus (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119904120579MCP119891119890

+ 119888120579CMC119891119890119888120579MCP119891119890


11987611989433= 119904120579CMC119891119890119904120579MCP119886119887119886119889

11987611989434= (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119888120579MCP119891119890

+ 119888120579119862119872119862119891119890

119904120579119872119862119875119891119890



+ 1198711119894+ 1198712119894119888120579PIP119891119890]

+ minus (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119904120579MCP119891119890

+ 119888120579CMC119891119890119888120579MCP119891119890



+ 1198712119894119904120579PIP119891119890]

+ (119904120579CMC119891119890119904120579MCP119886119887119886119889) 119871 119905119894119911 + 1198710119894119904120579CMC119891119890

(A1)


119876011= [119888120579TMC119886119887119886119889119888120579TMC119891119890]


+ [minus119888120579TMC119886119887119886119889119904120579TMC119891119890]


119876012= [119888120579TMC119886119887119886119889119888120579TMC119891119890]


+ [minus119888120579TMC119886119887119886119889119904120579TMC119891119890]


119876013= 119904120579TMC119886119887119886119889

119876014= [119888120579TMC119886119887119886119889119888120579TMC119891119890]

times [1198711199050 119909


+ 1198711199050 119910


+ 11987110119888120579MCP119891119890 + 11987100]

+ [minus119888120579TMC119886119887119886119889119904120579TMC119891119890]

times [1198711199050 119909

(119904120579MCP119891119890119888120579IP119891119890 + 119888120579MCP119891119890119904120579IP119891119890)

+ 1198711199050 119910


+ 11987110119904120579MCP119891119890] + 119871 1199050 119911119904120579TMC119886119887119886119889


119876021= [119904120579TMC119886119887119886119889119888120579TMC119891119890]


+ [minus119904120579TMC119886119887119886119889119904120579TMC119891119890]


119876022= [119904120579TMC119886119887119886119889119888120579TMC119891119890]


+ [minus119904120579TMC119886119887119886119889119904120579TMC119891119890]


119876023= minus 119888120579TMC119886119887119886119889

119876024= [119904120579TMC119886119887119886119889119888120579TMC119891119890]

times [1198711199050 119909


+ 1198711199050 119910


+ 11987110119888120579MCP119891119890 + 11987100]

+ [minus119904120579TMC119886119887119886119889119904120579TMC119891119890]

times [1198711199050 119909

(119904120579MCP119891119890119888120579IP119891119890 + 119888120579MCP119891119890119904120579IP119891119890)

+ 1198711199050 119910


+ 11987110119904120579MCP119891119890] minus 119888120579TMC119886119887119886119889119871 1199050 119911


+ 119888120579TMC119891119890 (119904120579MCP119891119890119888120579IP119891119890 + 119888120579MCP119891119890119904120579IP119891119890)



119876033= 0

119876034= 119904120579TMC119891119890

times [1198711199050 119909


+ 1198711199050 119910


+ 11987110119888120579MCP119891119890 + 11987100] + 119888120579TMC119891119890

times [1198711199050 119909

(119904120579MCP119891119890119888120579IP119891119890 + 119888120579MCP119891119890119904120579IP119891119890)

+ 1198711199050 119910


+11987110119904120579MCP119891119890]

(A2)


Consider


+ (11988711199091198881205791minus 11988711199101199041205791) (1198871119909c1205791minus 1198871119910s1205791)]


11199091199041205791minus 11987111199101198881205791)

+ (11987111199091198881205791minus 11987111199101199041205791) (11987111199091198881205791minus 11987111199101199041205791)]


+(11987111199091198881205791minus 11987111199101199041205791)times (119871

11199091198881205791minus 11987111199101199041205791)]+1198681


11199091199041205791minus 11987111199101198881205791)

+ (+119887211990911988812057912minus 119887211991011990412057912) (11987111199091198881205791minus 11987111199101199041205791)]


+ (+119871211990911988812057912minus 119871211991011990412057912) (11987111199091198881205791minus 11987111199101199041205791)]


+ (+1198873119909119888120579123minus 1198873119910119904120579123) (11987111199091198881205791minus 11987111199101199041205791)]

1198604= 1198981[(minus1198871119909

12057911199041205791minus 1198871119910

12057911198881205791) (minus1198871119909

1205791c1205791+ 1198871119910

12057911199041205791)


+ 1198982[minus (minus119871

1119909

1205792

1

1198881205791+ 1198711119910

1205792

1

1199041205791

minus1198872119909

1205792

12

11988812057912+ 1198872119910

1205792

12

11990412057912)



1205792

1

1199041205791minus 1198711119910

1205792

1

1198881205791minus 1198872119909

1205792

12

11990412057912

minus1198872119910

1205792

12

11988812057912) (11987111199091198881205791minus 11987111199101199041205791)]

+ 1198983[minus (minus119871

1119909

1205792

1

1198881205791+ 1198711119910

1205792

1

1199041205791minus1198712119909

1205792

12

11988812057912

+1198712119910

1205792

12

11990412057912minus1198873119909

1205792

123

119888120579123+ 1198873119910

1205792

123

119904120579123)



1205792

1

1199041205791minus 1198711119910

1205792

1

1198881205791minus 1198712119909

1205792

12

11990412057912


minus1198712119910

1205792

12

11988812057912minus 1198873119909

1205792

123

119904120579123minus 1198873119910

1205792

123

119888120579123)

times (11987111199091198881205791minus 11987111199101199041205791)]

minus (1198981119892 (11988711199091198881205791minus 11988711199101199041205791) + (119898

2+ 1198983) 119892


minus 1198881

1205791+ 1198882(12057912minus1205791)


+ (11987111199091198881205791minus 11987111199101199041205791) (119887211990911988812057912minus 119887211991011990412057912)]


+ (11987111199091198881205791minus 11987111199101199041205791) (119871211990911988812057912minus 119871211991011990412057912)]


+ (+119887211990911988812057912minus 119887211991011990412057912) (119887211990911988812057912minus 119887211991011990412057912)]


+ (+119871211990911988812057912) (119871211990911988812057912minus 119871211991011990412057912)] + 119868

2


+ (+1198873119909119888120579123minus 1198873119910119904120579123) (119871211990911988812057912minus 119871211991011990412057912)]

1198614= 1198982[minus (minus119871

1119909

1205792

1

1198881205791+ 1198711119910

1205792

1

1199041205791minus1198872119909

1205792

12

11988812057912

+1198872119910

1205792

12

11990412057912) (minus119887211990911990412057912minus 119887211991011988812057912)


1205792

1

1199041205791minus 1198711119910

1205792

1

1198881205791minus 1198872119909

1205792

12

11990412057912

minus1198872119910

1205792

12

11988812057912) (119887211990911988812057912minus 119887211991011990412057912)]

+ 1198983[minus (minus119871

1119909

1205792

1

1198881205791+ 1198711119910

1205792

1

1199041205791minus1198712119909

1205792

12

11988812057912

+ 1198712119910

1205792

12

11990412057912minus1198873119909

1205792

123

119888120579123

+1198873119910

1205792

123

119904120579123) (minus119871211990911990412057912minus 119871211991011988812057912)


1205792

1

1199041205791minus 1198711119910

1205792

1

1198881205791minus 1198712119909

1205792

12

11990412057912

minus 1198712119910

1205792

12

11988812057912minus 1198873119909

1205792

123

119904120579123

minus1198873119910

1205792

123

119888120579123) (119871211990911988812057912minus 119871211991011990412057912)]

minus (1198982119892 (119887211990911988812057912minus 119887211991011990412057912)

+ 1198983119892 (119871211990911988812057912minus 119871211991011990412057912)




+ (11987111199091198881205791minus 11987111199101199041205791) (1198873119909119888120579123minus 1198873119910119904120579123)]




+ (+1198873119909119888120579123minus 1198873119910119904120579123)

times (1198873119909119888120579123minus 1198873119910119904120579123)] + 119868

3

1198624= 1198983[minus (minus119871

1119909

1205792

1

1198881205791+ 1198711119910

1205792

1

1199041205791minus1198712119909

1205792

12

11988812057912

+ 1198712119910

1205792

12

11990412057912minus1198873119909

1205792

123

119888120579123

+1198873119910

1205792

123

119904120579123) (minus1198873119909119904120579123minus 1198873119910119888120579123)


1205792

1

1199041205791minus 1198711119910

1205792

1

1198881205791minus 1198712119909

1205792

12

11990412057912

minus 1198712119910

1205791211990412057912minus 1198712119910

1205792

12

11988812057912+ 1198873119909

120579123119888120579123

minus1198873119909

1205792

123

119904120579123minus 1198873119910

1205792

123

119888120579123)

times (1198873119909119888120579123minus 1198873119910119904120579123)]



(B1)

References
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












119886119895minus1

119889119895

120579119895

119895 = 1 0 0 0 120579TMC119886119887119886119889

119895 = 2 1205872 0 0 120579TMC119891119890

119895 = 3 0 11987100

0 120579MCP119891119890

119895 = 4 0 11987110

0 120579IP119891119890


1198760=0

0

1198790sdot0

5

1198790(120579119896)

1198760=0

0

1198790sdot0

11198790

(120579TMC119886119887119886119889) sdot1

21198790

(120579TMC119891119890)

sdot2

31198790

(120579MCP119891119890) sdot3

41198790

(120579IP119891119890) sdot4

51198791199050

(9)


TMC119891119890

MCP119891119890

and IP119891119890


[1198971199050119909

1198971199050119910

1198971199050119911]

119879






119894


119891119890

MCP119886119887119886119889

and DIP119891119890



120579CMC119891119890 = atan(11987633

11987613

)

120579MCP119886119887119886119889 = atan[

[

11987613

minus11987623cos (120579CMC119891119890)

]

]

120579DIP119891119890 = atan(12059811205982

)

(10)



1205981=

(minus1) sdot (11989811987622+ 11989911987621)

(1198982

minus 1198992

)

1205982=

Q21

119898

+

119899

119898

sdot

(minus1) sdot (11989811987622+ 11989911987621)

(1198982

minus 1198992

)





(11)

TheMCP119891119890

PIP119891119890




expression

1= 1198651minus [11987131lowast ] (12)


1198751= [119875111990911987511199101198751119911]

119879

(13)

where1198751119909= 11987101cos (120579CMC)

1198751119910= 11987101sin (120579CMC)

1198751119911= 0

(14)


Knowing 1and 119875


1is


Angles 1205931and 120593

2are expressed as

1205931= acos(

1199032

2

+ 1198712

01

minus 1199032

1

2119903211989701

)

1205932= acos(

1199032

2

+ 1198712

11

minus 1198712

21

2119903211987111

)

(16)


100381710038171003817100381710038171

10038171003817100381710038171003817

1199032=10038171003817100381710038171199061

1003817100381710038171003817

(17)


120579MCP119891119890 = 120587 minus 1205931 minus 1205932 (18)


120579PIP119891119890 = 120587 minus 1205933 (19)

where

1205933= acos(

1198712

21

+ 1198712

11

minus 1199032

2

21198712111987111

) (20)




120579TMC119886119887119886119889 = atan(11987613

minus11987623

) (21)




120579MCP119891119890 = 120587 minus 1205741

120579IP119891119890 = 120587 minus 1205742 minus 1205743(22)

where

1205741= acos(

1198712

10

+ 1198712

00

minus 1199032

3

21198711011987100

)

1205742= acos(

1198712

20

+ 1199042

3

minus 1199032

4

2119871201199033

)

1205743= acos(

1199032

3

+ 1198712

10

minus 1198712

00

211990311990411987110

)

1199033=

100381710038171003817100381710038170

10038171003817100381710038171003817

1199034=

10038171003817100381710038171003817

1198650

10038171003817100381710038171003817

0= 1198650minus [11987120lowast ]

(23)


120579TMC119891119890 = atan(12059841205983

) (24)

where

1205983= (11987632+ 11987631) cos 120583

1205984=


cos 120583

120583 = 120579MCP + 120579IP

(25)



2is the


2




1205793= 120579MCP119891119890

1205794= 120579PIP119891119890

1205795= 120579DIP119891119890

(26)




119892= [

[

119887119892119909

119887119892119910

0

]

]

119892 = 1 2 3 (27)





119895and 119904120579

119895stand for


1= [

[

11988711199091198881205793minus 11988711199101199041205793

11988711199091199041205793+ 11988711199101198881205793

0

]

]

2= [

[

11987111198881205793

11987111199041205793

0

]

]

+

[[[

[

1198872119909119888 (1205793+ 1205794) minus 1198872119910119904 (1205793+ 1205794)

1198872119909119904 (1205793+ 1205794) + 1198872119910119888 (1205793+ 1205794)

0

]]]

]



3= [

[

11987111198881205793

11987111199041205793

0

]

]

+[[

[

1198712119888 (1205793+ 1205794)

1198712119904 (1205793+ 1205794)

0

]]

]

+

[[[

[

1198873119909119888 (1205793+ 1205794+ 1205795) minus 1198873119910119904 (1205793+ 1205794+ 1205795)

1198873119909119904 (1205793+ 1205794+ 1205795) + 1198873119910119888 (1205793+ 1205794+ 1205795)

0

]]]

]

(28)


1205793+ 1205794= 12057934

1205793+1205794=

12057934

1205793+ 1205794+ 1205795= 120579345

1205793+1205794+1205795=

120579345

(29)


V1198661=[[

[

minus1198871119909

12057931199041205793minus 1198871119910

12057931198881205793

1198871119909

12057931198881205793minus 1198871119910

12057931199041205793

0

]]

]

V1198662=[[

[

minus1198711

12057931199041205793minus1198872119909

1205793411990412057934minus 1198872119910

1205793411988812057934

1198711

12057931198881205793+ 1198872119909

1205793411988812057934minus 1198872119910

1205793411990412057934

0

]]

]

V1198663=[[

[

minus1198711

12057931199041205793minus1198712

1205793411990412057934minus1198873119909

120579345119904120579345minus 1198873119910

120579345119888120579345

1198711

12057931198881205793+ 1198712

1205793411988812057934+ 1198873119909

120579345119888120579345minus 1198873119910

120579345119904120579345

0

]]

]

(30)


119879 =

1

2

1198981(V1198661)

2

+

1

2

1198982(V1198662)

2

+

1

2

1198983(V1198663)

2

+

1

2

1198681(1205793)

2

+

1

2

1198682(12057934)

2

+

1

2

1198683(120579345)

2

(31)




1 1198962 and 119896



119880 = 1198981119892 (11988711199091199041205793+ 11988711199101198881205793)

+ 1198982119892 (11987111199041205793+ 119887211990911990412057934+ 119887211991011988812057934)

+ 1198983119892 (11987111199041205793+ 119871211990412057934+ 1198873119909119904120579345+ 1198873119910119888120579345)

+

1

2

1198961(1205793)2

+

1

2

1198962(12057934minus 1205793)2

+

1

2

1198963(120579345minus 12057934)2

(32)


119865 =

1

2

1198881(1205793)

2

+

1

2

1198882(12057934minus1205793)

2

+

1

2

1198883(120579345minus12057934)

2

(33)



1

1198882 and 119888


joints respectively



3 12057934 and 120579

345

119889

119889119905

(

120597 (119879 minus 119880)

1205971205793

) minus

120597 (119879 minus 119880)

1205971205793

+

120597119865

1205971205793

= 1205913

119889

119889119905

(

120597 (119879 minus 119880)

12059712057934

) minus

120597 (119879 minus 119880)

12059712057934

+

120597119865

12059712057934

= 1205914

119889

119889119905

(

120597 (119879 minus 119880)

120597120579345

) minus

120597 (119879 minus 119880)

120597120579345

+

120597119865

120597120579345

= 1205915

(34)




120591119895=

(sum5

119895=3

120575119882119895)

120575120579119895

119895 = 3 4 5 (35)



1205913= (11986531199101198903119909minus 11986531199091198903119910) + [

[

minus11987111199041205793

11987111198881205793

0

]

]

sdot [

[

1198654119909c12057934minus 1198654119910s12057934

1198654119909s12057934+ 1198654119910c12057934

0

]

]

+ [

[

minus11987111199041205793

11987111198881205793

0

]

]

sdot [

[

1198655119909c120579345minus 1198655119910s120579345

1198655119909s120579345+ 1198655119910c120579345

0

]

]

+ 1198621198981minus 1198621198982

1205914= (11986541199101198904119909minus 11986541199091198904119910) + [

[

minus119871211990412057934

119871211988812057934

0

]

]

sdot [

[

1198655119909c120579345minus 1198655119910s120579345

1198655119909s120579345+ 1198655119910c120579345

0

]

]

+ 1198621198982minus 1198621198983

1205915= (11986551199101198905119909minus 11986551199091198905119910) + 1198621198983

(36)

where the term 119862119898119892


119895= [1198651198951199091198651198951199100]119879 is the contact


119895= [1198901198951199091198901198941199100]119879


1198601

1205793+ 1198602

12057934+ 1198603

120579345= 1198604

1198611

1205793+ 1198612

12057934+ 1198613

120579345= 1198614

1198621

1205793+ 1198622

12057934+ 1198623

120579345= 1198624

(37)


The terms 1198601 1198602 and 119860










119895= (1198631198952)(cos(120587119905119905

1)minus1) where




120579119895 max = | minus 12058711986311989521199051|





Omega maxMCP (rads)

Omega maxPIP (rads)

Omega maxDIP (rads)

Cycles in 10 s 27 27 31 25 20 29 23 27 21 31 261 3843 261 12164 15026 10018


Omega maxMCP (rads)

Omega maxPIP (rads)

Omega maxDIP (rads)

Cycles in 10 s 27 24 17 34 29 27 30 24 21 27 26 4784 26 12118 14969 9979



1198983are


119898max=



119889119895= |119862119889119895||119862119898119895|



0 20 40 60 80 100minus100

minus50

0

50

100

150

200

250

300

Cycle ()

C(N

mm

)

Cm1

Cm2

Cm3



MCP PIP DIP119891119896119895

3805 3848 3858119891119888119895

5758 6059 61119891119889119895

437 093 042



7 Conclusions




1198981

= 705 g 1198982

= 397 g 1198983

= 270 g1198681

= 1000 gmm2 1198682

= 240 gmm2 1198683

= 120 gmm2

1198961

= 00902Nm 1198962

= 00902Nm 1198963

= 00902Nm1198881

= 00225Nms 1198882

= 00225Nms 1198883

= 00225Nms1198711

= [50 0 0]

119879

mm 1198712

= [20 0 0]

119879

mm 1198713

= [25 0 0]

119879

mm1198871

= [25 0 0]

119879

mm 1198872

= [10 0 0]

119879

mm 1198873

= [125 0 0]

119879

mm1198901

= [25 minus8 0]

119879

mm 1198902

= [10 minus7 0]

119879

mm 1198903

= [12 minus6 0]

119879

mm



Appendices



11987611989411= [(119888120579CMC119891119890119888120579MCP119886119887119886119889) 119888120579MCP119891119890

minus 119904120579CMC119891119890119904120579MCP119891119890]


minus [(119888120579CMC119891119890119888120579MCP119886119887119886119889) 119904120579MCP119891119890

+ 119904120579CMC119891119890119888120579MCP119891119890]


11987611989412= [(119888120579CMC119891119890119888120579MCP119886119887119886119889) 119888120579MCP119891119890

minus 119904120579CMC119891119890119904120579MCP119891119890]


minus [(119888120579CMC119891119890119888120579MCP119886119887119886119889) 119904120579MCP119891119890

+ 119904120579CMC119891119890119888120579MCP119891119890]


11987611989413= 119888120579CMC119891119890119904120579MCP119886119887119886119889

11987611989414= [(119888120579CMC119891119890119888120579MCP119886119887119886119889) 119888120579MCP119891119890

minus 119904120579CMC119891119890119904120579MCP119891119890]



+ 1198711119894+ 1198712119894119888120579PIP119891119890]

minus [(119888120579CMC119891119890119888120579MCP119886119887119886119889) 119904120579MCP119891119890

+ 119904120579CMC119891119890119888120579MCP119891119890]



+ 1198712119894119904120579PIP119891119890]

+ (119888120579CMC119891119890119904120579MCP119886119887119886119889) 119871 119905119894119911 + 1198710119894119888120579CMC119891119890

11987611989421= 119904120579MCP119886119887119886119889119888120579MCP119891119890


minus 119904120579MCP119886119887119886119889119904120579MCP119891119890


11987611989422= 119904120579MCP119886119887119886119889119888120579MCP119891119890



minus 119904120579MCP119886119887119886119889119904120579MCP119891119890


11987611989423= minus 119888120579MCP119886119887119886119889

11987611989424= 119904120579MCP119886119887119886119889119888120579MCP119891119890



+ 1198711119894+ 1198712119894119888120579PIP119891119890] minus 119904120579MCP119886119887119886119889119904120579MCP119891119890



+ 1198712119894119904120579PIP119891119890] + (minus119888120579MCP119886119887119886119889) 119871 119905119894119911

11987611989431= (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119888120579MCP119891119890

+ 119888120579CMC119891119890119904120579MCP119891119890


+ minus (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119904120579MCP119891119890

+ 119888120579CMC119891119890119888120579MCP119891119890


11987611989432= (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119888120579MCP119891119890

+ 119888120579CMC119891119890119904120579MCP119891119890


+ minus (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119904120579MCP119891119890

+ 119888120579CMC119891119890119888120579MCP119891119890


11987611989433= 119904120579CMC119891119890119904120579MCP119886119887119886119889

11987611989434= (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119888120579MCP119891119890

+ 119888120579119862119872119862119891119890

119904120579119872119862119875119891119890



+ 1198711119894+ 1198712119894119888120579PIP119891119890]

+ minus (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119904120579MCP119891119890

+ 119888120579CMC119891119890119888120579MCP119891119890



+ 1198712119894119904120579PIP119891119890]

+ (119904120579CMC119891119890119904120579MCP119886119887119886119889) 119871 119905119894119911 + 1198710119894119904120579CMC119891119890

(A1)


119876011= [119888120579TMC119886119887119886119889119888120579TMC119891119890]


+ [minus119888120579TMC119886119887119886119889119904120579TMC119891119890]


119876012= [119888120579TMC119886119887119886119889119888120579TMC119891119890]


+ [minus119888120579TMC119886119887119886119889119904120579TMC119891119890]


119876013= 119904120579TMC119886119887119886119889

119876014= [119888120579TMC119886119887119886119889119888120579TMC119891119890]

times [1198711199050 119909


+ 1198711199050 119910


+ 11987110119888120579MCP119891119890 + 11987100]

+ [minus119888120579TMC119886119887119886119889119904120579TMC119891119890]

times [1198711199050 119909

(119904120579MCP119891119890119888120579IP119891119890 + 119888120579MCP119891119890119904120579IP119891119890)

+ 1198711199050 119910


+ 11987110119904120579MCP119891119890] + 119871 1199050 119911119904120579TMC119886119887119886119889


119876021= [119904120579TMC119886119887119886119889119888120579TMC119891119890]


+ [minus119904120579TMC119886119887119886119889119904120579TMC119891119890]


119876022= [119904120579TMC119886119887119886119889119888120579TMC119891119890]


+ [minus119904120579TMC119886119887119886119889119904120579TMC119891119890]


119876023= minus 119888120579TMC119886119887119886119889

119876024= [119904120579TMC119886119887119886119889119888120579TMC119891119890]

times [1198711199050 119909


+ 1198711199050 119910


+ 11987110119888120579MCP119891119890 + 11987100]

+ [minus119904120579TMC119886119887119886119889119904120579TMC119891119890]

times [1198711199050 119909

(119904120579MCP119891119890119888120579IP119891119890 + 119888120579MCP119891119890119904120579IP119891119890)

+ 1198711199050 119910


+ 11987110119904120579MCP119891119890] minus 119888120579TMC119886119887119886119889119871 1199050 119911


+ 119888120579TMC119891119890 (119904120579MCP119891119890119888120579IP119891119890 + 119888120579MCP119891119890119904120579IP119891119890)



119876033= 0

119876034= 119904120579TMC119891119890

times [1198711199050 119909


+ 1198711199050 119910


+ 11987110119888120579MCP119891119890 + 11987100] + 119888120579TMC119891119890

times [1198711199050 119909

(119904120579MCP119891119890119888120579IP119891119890 + 119888120579MCP119891119890119904120579IP119891119890)

+ 1198711199050 119910


+11987110119904120579MCP119891119890]

(A2)


Consider


+ (11988711199091198881205791minus 11988711199101199041205791) (1198871119909c1205791minus 1198871119910s1205791)]


11199091199041205791minus 11987111199101198881205791)

+ (11987111199091198881205791minus 11987111199101199041205791) (11987111199091198881205791minus 11987111199101199041205791)]


+(11987111199091198881205791minus 11987111199101199041205791)times (119871

11199091198881205791minus 11987111199101199041205791)]+1198681


11199091199041205791minus 11987111199101198881205791)

+ (+119887211990911988812057912minus 119887211991011990412057912) (11987111199091198881205791minus 11987111199101199041205791)]


+ (+119871211990911988812057912minus 119871211991011990412057912) (11987111199091198881205791minus 11987111199101199041205791)]


+ (+1198873119909119888120579123minus 1198873119910119904120579123) (11987111199091198881205791minus 11987111199101199041205791)]

1198604= 1198981[(minus1198871119909

12057911199041205791minus 1198871119910

12057911198881205791) (minus1198871119909

1205791c1205791+ 1198871119910

12057911199041205791)


+ 1198982[minus (minus119871

1119909

1205792

1

1198881205791+ 1198711119910

1205792

1

1199041205791

minus1198872119909

1205792

12

11988812057912+ 1198872119910

1205792

12

11990412057912)



1205792

1

1199041205791minus 1198711119910

1205792

1

1198881205791minus 1198872119909

1205792

12

11990412057912

minus1198872119910

1205792

12

11988812057912) (11987111199091198881205791minus 11987111199101199041205791)]

+ 1198983[minus (minus119871

1119909

1205792

1

1198881205791+ 1198711119910

1205792

1

1199041205791minus1198712119909

1205792

12

11988812057912

+1198712119910

1205792

12

11990412057912minus1198873119909

1205792

123

119888120579123+ 1198873119910

1205792

123

119904120579123)



1205792

1

1199041205791minus 1198711119910

1205792

1

1198881205791minus 1198712119909

1205792

12

11990412057912


minus1198712119910

1205792

12

11988812057912minus 1198873119909

1205792

123

119904120579123minus 1198873119910

1205792

123

119888120579123)

times (11987111199091198881205791minus 11987111199101199041205791)]

minus (1198981119892 (11988711199091198881205791minus 11988711199101199041205791) + (119898

2+ 1198983) 119892


minus 1198881

1205791+ 1198882(12057912minus1205791)


+ (11987111199091198881205791minus 11987111199101199041205791) (119887211990911988812057912minus 119887211991011990412057912)]


+ (11987111199091198881205791minus 11987111199101199041205791) (119871211990911988812057912minus 119871211991011990412057912)]


+ (+119887211990911988812057912minus 119887211991011990412057912) (119887211990911988812057912minus 119887211991011990412057912)]


+ (+119871211990911988812057912) (119871211990911988812057912minus 119871211991011990412057912)] + 119868

2


+ (+1198873119909119888120579123minus 1198873119910119904120579123) (119871211990911988812057912minus 119871211991011990412057912)]

1198614= 1198982[minus (minus119871

1119909

1205792

1

1198881205791+ 1198711119910

1205792

1

1199041205791minus1198872119909

1205792

12

11988812057912

+1198872119910

1205792

12

11990412057912) (minus119887211990911990412057912minus 119887211991011988812057912)


1205792

1

1199041205791minus 1198711119910

1205792

1

1198881205791minus 1198872119909

1205792

12

11990412057912

minus1198872119910

1205792

12

11988812057912) (119887211990911988812057912minus 119887211991011990412057912)]

+ 1198983[minus (minus119871

1119909

1205792

1

1198881205791+ 1198711119910

1205792

1

1199041205791minus1198712119909

1205792

12

11988812057912

+ 1198712119910

1205792

12

11990412057912minus1198873119909

1205792

123

119888120579123

+1198873119910

1205792

123

119904120579123) (minus119871211990911990412057912minus 119871211991011988812057912)


1205792

1

1199041205791minus 1198711119910

1205792

1

1198881205791minus 1198712119909

1205792

12

11990412057912

minus 1198712119910

1205792

12

11988812057912minus 1198873119909

1205792

123

119904120579123

minus1198873119910

1205792

123

119888120579123) (119871211990911988812057912minus 119871211991011990412057912)]

minus (1198982119892 (119887211990911988812057912minus 119887211991011990412057912)

+ 1198983119892 (119871211990911988812057912minus 119871211991011990412057912)




+ (11987111199091198881205791minus 11987111199101199041205791) (1198873119909119888120579123minus 1198873119910119904120579123)]




+ (+1198873119909119888120579123minus 1198873119910119904120579123)

times (1198873119909119888120579123minus 1198873119910119904120579123)] + 119868

3

1198624= 1198983[minus (minus119871

1119909

1205792

1

1198881205791+ 1198711119910

1205792

1

1199041205791minus1198712119909

1205792

12

11988812057912

+ 1198712119910

1205792

12

11990412057912minus1198873119909

1205792

123

119888120579123

+1198873119910

1205792

123

119904120579123) (minus1198873119909119904120579123minus 1198873119910119888120579123)


1205792

1

1199041205791minus 1198711119910

1205792

1

1198881205791minus 1198712119909

1205792

12

11990412057912

minus 1198712119910

1205791211990412057912minus 1198712119910

1205792

12

11988812057912+ 1198873119909

120579123119888120579123

minus1198873119909

1205792

123

119904120579123minus 1198873119910

1205792

123

119888120579123)

times (1198873119909119888120579123minus 1198873119910119904120579123)]



(B1)

References
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Knowing 1and 119875


1is


Angles 1205931and 120593

2are expressed as

1205931= acos(

1199032

2

+ 1198712

01

minus 1199032

1

2119903211989701

)

1205932= acos(

1199032

2

+ 1198712

11

minus 1198712

21

2119903211987111

)

(16)


100381710038171003817100381710038171

10038171003817100381710038171003817

1199032=10038171003817100381710038171199061

1003817100381710038171003817

(17)


120579MCP119891119890 = 120587 minus 1205931 minus 1205932 (18)


120579PIP119891119890 = 120587 minus 1205933 (19)

where

1205933= acos(

1198712

21

+ 1198712

11

minus 1199032

2

21198712111987111

) (20)




120579TMC119886119887119886119889 = atan(11987613

minus11987623

) (21)




120579MCP119891119890 = 120587 minus 1205741

120579IP119891119890 = 120587 minus 1205742 minus 1205743(22)

where

1205741= acos(

1198712

10

+ 1198712

00

minus 1199032

3

21198711011987100

)

1205742= acos(

1198712

20

+ 1199042

3

minus 1199032

4

2119871201199033

)

1205743= acos(

1199032

3

+ 1198712

10

minus 1198712

00

211990311990411987110

)

1199033=

100381710038171003817100381710038170

10038171003817100381710038171003817

1199034=

10038171003817100381710038171003817

1198650

10038171003817100381710038171003817

0= 1198650minus [11987120lowast ]

(23)


120579TMC119891119890 = atan(12059841205983

) (24)

where

1205983= (11987632+ 11987631) cos 120583

1205984=


cos 120583

120583 = 120579MCP + 120579IP

(25)



2is the


2




1205793= 120579MCP119891119890

1205794= 120579PIP119891119890

1205795= 120579DIP119891119890

(26)




119892= [

[

119887119892119909

119887119892119910

0

]

]

119892 = 1 2 3 (27)





119895and 119904120579

119895stand for


1= [

[

11988711199091198881205793minus 11988711199101199041205793

11988711199091199041205793+ 11988711199101198881205793

0

]

]

2= [

[

11987111198881205793

11987111199041205793

0

]

]

+

[[[

[

1198872119909119888 (1205793+ 1205794) minus 1198872119910119904 (1205793+ 1205794)

1198872119909119904 (1205793+ 1205794) + 1198872119910119888 (1205793+ 1205794)

0

]]]

]



3= [

[

11987111198881205793

11987111199041205793

0

]

]

+[[

[

1198712119888 (1205793+ 1205794)

1198712119904 (1205793+ 1205794)

0

]]

]

+

[[[

[

1198873119909119888 (1205793+ 1205794+ 1205795) minus 1198873119910119904 (1205793+ 1205794+ 1205795)

1198873119909119904 (1205793+ 1205794+ 1205795) + 1198873119910119888 (1205793+ 1205794+ 1205795)

0

]]]

]

(28)


1205793+ 1205794= 12057934

1205793+1205794=

12057934

1205793+ 1205794+ 1205795= 120579345

1205793+1205794+1205795=

120579345

(29)


V1198661=[[

[

minus1198871119909

12057931199041205793minus 1198871119910

12057931198881205793

1198871119909

12057931198881205793minus 1198871119910

12057931199041205793

0

]]

]

V1198662=[[

[

minus1198711

12057931199041205793minus1198872119909

1205793411990412057934minus 1198872119910

1205793411988812057934

1198711

12057931198881205793+ 1198872119909

1205793411988812057934minus 1198872119910

1205793411990412057934

0

]]

]

V1198663=[[

[

minus1198711

12057931199041205793minus1198712

1205793411990412057934minus1198873119909

120579345119904120579345minus 1198873119910

120579345119888120579345

1198711

12057931198881205793+ 1198712

1205793411988812057934+ 1198873119909

120579345119888120579345minus 1198873119910

120579345119904120579345

0

]]

]

(30)


119879 =

1

2

1198981(V1198661)

2

+

1

2

1198982(V1198662)

2

+

1

2

1198983(V1198663)

2

+

1

2

1198681(1205793)

2

+

1

2

1198682(12057934)

2

+

1

2

1198683(120579345)

2

(31)




1 1198962 and 119896



119880 = 1198981119892 (11988711199091199041205793+ 11988711199101198881205793)

+ 1198982119892 (11987111199041205793+ 119887211990911990412057934+ 119887211991011988812057934)

+ 1198983119892 (11987111199041205793+ 119871211990412057934+ 1198873119909119904120579345+ 1198873119910119888120579345)

+

1

2

1198961(1205793)2

+

1

2

1198962(12057934minus 1205793)2

+

1

2

1198963(120579345minus 12057934)2

(32)


119865 =

1

2

1198881(1205793)

2

+

1

2

1198882(12057934minus1205793)

2

+

1

2

1198883(120579345minus12057934)

2

(33)



1

1198882 and 119888


joints respectively



3 12057934 and 120579

345

119889

119889119905

(

120597 (119879 minus 119880)

1205971205793

) minus

120597 (119879 minus 119880)

1205971205793

+

120597119865

1205971205793

= 1205913

119889

119889119905

(

120597 (119879 minus 119880)

12059712057934

) minus

120597 (119879 minus 119880)

12059712057934

+

120597119865

12059712057934

= 1205914

119889

119889119905

(

120597 (119879 minus 119880)

120597120579345

) minus

120597 (119879 minus 119880)

120597120579345

+

120597119865

120597120579345

= 1205915

(34)




120591119895=

(sum5

119895=3

120575119882119895)

120575120579119895

119895 = 3 4 5 (35)



1205913= (11986531199101198903119909minus 11986531199091198903119910) + [

[

minus11987111199041205793

11987111198881205793

0

]

]

sdot [

[

1198654119909c12057934minus 1198654119910s12057934

1198654119909s12057934+ 1198654119910c12057934

0

]

]

+ [

[

minus11987111199041205793

11987111198881205793

0

]

]

sdot [

[

1198655119909c120579345minus 1198655119910s120579345

1198655119909s120579345+ 1198655119910c120579345

0

]

]

+ 1198621198981minus 1198621198982

1205914= (11986541199101198904119909minus 11986541199091198904119910) + [

[

minus119871211990412057934

119871211988812057934

0

]

]

sdot [

[

1198655119909c120579345minus 1198655119910s120579345

1198655119909s120579345+ 1198655119910c120579345

0

]

]

+ 1198621198982minus 1198621198983

1205915= (11986551199101198905119909minus 11986551199091198905119910) + 1198621198983

(36)

where the term 119862119898119892


119895= [1198651198951199091198651198951199100]119879 is the contact


119895= [1198901198951199091198901198941199100]119879


1198601

1205793+ 1198602

12057934+ 1198603

120579345= 1198604

1198611

1205793+ 1198612

12057934+ 1198613

120579345= 1198614

1198621

1205793+ 1198622

12057934+ 1198623

120579345= 1198624

(37)


The terms 1198601 1198602 and 119860










119895= (1198631198952)(cos(120587119905119905

1)minus1) where




120579119895 max = | minus 12058711986311989521199051|





Omega maxMCP (rads)

Omega maxPIP (rads)

Omega maxDIP (rads)

Cycles in 10 s 27 27 31 25 20 29 23 27 21 31 261 3843 261 12164 15026 10018


Omega maxMCP (rads)

Omega maxPIP (rads)

Omega maxDIP (rads)

Cycles in 10 s 27 24 17 34 29 27 30 24 21 27 26 4784 26 12118 14969 9979



1198983are


119898max=



119889119895= |119862119889119895||119862119898119895|



0 20 40 60 80 100minus100

minus50

0

50

100

150

200

250

300

Cycle ()

C(N

mm

)

Cm1

Cm2

Cm3



MCP PIP DIP119891119896119895

3805 3848 3858119891119888119895

5758 6059 61119891119889119895

437 093 042



7 Conclusions




1198981

= 705 g 1198982

= 397 g 1198983

= 270 g1198681

= 1000 gmm2 1198682

= 240 gmm2 1198683

= 120 gmm2

1198961

= 00902Nm 1198962

= 00902Nm 1198963

= 00902Nm1198881

= 00225Nms 1198882

= 00225Nms 1198883

= 00225Nms1198711

= [50 0 0]

119879

mm 1198712

= [20 0 0]

119879

mm 1198713

= [25 0 0]

119879

mm1198871

= [25 0 0]

119879

mm 1198872

= [10 0 0]

119879

mm 1198873

= [125 0 0]

119879

mm1198901

= [25 minus8 0]

119879

mm 1198902

= [10 minus7 0]

119879

mm 1198903

= [12 minus6 0]

119879

mm



Appendices



11987611989411= [(119888120579CMC119891119890119888120579MCP119886119887119886119889) 119888120579MCP119891119890

minus 119904120579CMC119891119890119904120579MCP119891119890]


minus [(119888120579CMC119891119890119888120579MCP119886119887119886119889) 119904120579MCP119891119890

+ 119904120579CMC119891119890119888120579MCP119891119890]


11987611989412= [(119888120579CMC119891119890119888120579MCP119886119887119886119889) 119888120579MCP119891119890

minus 119904120579CMC119891119890119904120579MCP119891119890]


minus [(119888120579CMC119891119890119888120579MCP119886119887119886119889) 119904120579MCP119891119890

+ 119904120579CMC119891119890119888120579MCP119891119890]


11987611989413= 119888120579CMC119891119890119904120579MCP119886119887119886119889

11987611989414= [(119888120579CMC119891119890119888120579MCP119886119887119886119889) 119888120579MCP119891119890

minus 119904120579CMC119891119890119904120579MCP119891119890]



+ 1198711119894+ 1198712119894119888120579PIP119891119890]

minus [(119888120579CMC119891119890119888120579MCP119886119887119886119889) 119904120579MCP119891119890

+ 119904120579CMC119891119890119888120579MCP119891119890]



+ 1198712119894119904120579PIP119891119890]

+ (119888120579CMC119891119890119904120579MCP119886119887119886119889) 119871 119905119894119911 + 1198710119894119888120579CMC119891119890

11987611989421= 119904120579MCP119886119887119886119889119888120579MCP119891119890


minus 119904120579MCP119886119887119886119889119904120579MCP119891119890


11987611989422= 119904120579MCP119886119887119886119889119888120579MCP119891119890



minus 119904120579MCP119886119887119886119889119904120579MCP119891119890


11987611989423= minus 119888120579MCP119886119887119886119889

11987611989424= 119904120579MCP119886119887119886119889119888120579MCP119891119890



+ 1198711119894+ 1198712119894119888120579PIP119891119890] minus 119904120579MCP119886119887119886119889119904120579MCP119891119890



+ 1198712119894119904120579PIP119891119890] + (minus119888120579MCP119886119887119886119889) 119871 119905119894119911

11987611989431= (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119888120579MCP119891119890

+ 119888120579CMC119891119890119904120579MCP119891119890


+ minus (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119904120579MCP119891119890

+ 119888120579CMC119891119890119888120579MCP119891119890


11987611989432= (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119888120579MCP119891119890

+ 119888120579CMC119891119890119904120579MCP119891119890


+ minus (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119904120579MCP119891119890

+ 119888120579CMC119891119890119888120579MCP119891119890


11987611989433= 119904120579CMC119891119890119904120579MCP119886119887119886119889

11987611989434= (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119888120579MCP119891119890

+ 119888120579119862119872119862119891119890

119904120579119872119862119875119891119890



+ 1198711119894+ 1198712119894119888120579PIP119891119890]

+ minus (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119904120579MCP119891119890

+ 119888120579CMC119891119890119888120579MCP119891119890



+ 1198712119894119904120579PIP119891119890]

+ (119904120579CMC119891119890119904120579MCP119886119887119886119889) 119871 119905119894119911 + 1198710119894119904120579CMC119891119890

(A1)


119876011= [119888120579TMC119886119887119886119889119888120579TMC119891119890]


+ [minus119888120579TMC119886119887119886119889119904120579TMC119891119890]


119876012= [119888120579TMC119886119887119886119889119888120579TMC119891119890]


+ [minus119888120579TMC119886119887119886119889119904120579TMC119891119890]


119876013= 119904120579TMC119886119887119886119889

119876014= [119888120579TMC119886119887119886119889119888120579TMC119891119890]

times [1198711199050 119909


+ 1198711199050 119910


+ 11987110119888120579MCP119891119890 + 11987100]

+ [minus119888120579TMC119886119887119886119889119904120579TMC119891119890]

times [1198711199050 119909

(119904120579MCP119891119890119888120579IP119891119890 + 119888120579MCP119891119890119904120579IP119891119890)

+ 1198711199050 119910


+ 11987110119904120579MCP119891119890] + 119871 1199050 119911119904120579TMC119886119887119886119889


119876021= [119904120579TMC119886119887119886119889119888120579TMC119891119890]


+ [minus119904120579TMC119886119887119886119889119904120579TMC119891119890]


119876022= [119904120579TMC119886119887119886119889119888120579TMC119891119890]


+ [minus119904120579TMC119886119887119886119889119904120579TMC119891119890]


119876023= minus 119888120579TMC119886119887119886119889

119876024= [119904120579TMC119886119887119886119889119888120579TMC119891119890]

times [1198711199050 119909


+ 1198711199050 119910


+ 11987110119888120579MCP119891119890 + 11987100]

+ [minus119904120579TMC119886119887119886119889119904120579TMC119891119890]

times [1198711199050 119909

(119904120579MCP119891119890119888120579IP119891119890 + 119888120579MCP119891119890119904120579IP119891119890)

+ 1198711199050 119910


+ 11987110119904120579MCP119891119890] minus 119888120579TMC119886119887119886119889119871 1199050 119911


+ 119888120579TMC119891119890 (119904120579MCP119891119890119888120579IP119891119890 + 119888120579MCP119891119890119904120579IP119891119890)



119876033= 0

119876034= 119904120579TMC119891119890

times [1198711199050 119909


+ 1198711199050 119910


+ 11987110119888120579MCP119891119890 + 11987100] + 119888120579TMC119891119890

times [1198711199050 119909

(119904120579MCP119891119890119888120579IP119891119890 + 119888120579MCP119891119890119904120579IP119891119890)

+ 1198711199050 119910


+11987110119904120579MCP119891119890]

(A2)


Consider


+ (11988711199091198881205791minus 11988711199101199041205791) (1198871119909c1205791minus 1198871119910s1205791)]


11199091199041205791minus 11987111199101198881205791)

+ (11987111199091198881205791minus 11987111199101199041205791) (11987111199091198881205791minus 11987111199101199041205791)]


+(11987111199091198881205791minus 11987111199101199041205791)times (119871

11199091198881205791minus 11987111199101199041205791)]+1198681


11199091199041205791minus 11987111199101198881205791)

+ (+119887211990911988812057912minus 119887211991011990412057912) (11987111199091198881205791minus 11987111199101199041205791)]


+ (+119871211990911988812057912minus 119871211991011990412057912) (11987111199091198881205791minus 11987111199101199041205791)]


+ (+1198873119909119888120579123minus 1198873119910119904120579123) (11987111199091198881205791minus 11987111199101199041205791)]

1198604= 1198981[(minus1198871119909

12057911199041205791minus 1198871119910

12057911198881205791) (minus1198871119909

1205791c1205791+ 1198871119910

12057911199041205791)


+ 1198982[minus (minus119871

1119909

1205792

1

1198881205791+ 1198711119910

1205792

1

1199041205791

minus1198872119909

1205792

12

11988812057912+ 1198872119910

1205792

12

11990412057912)



1205792

1

1199041205791minus 1198711119910

1205792

1

1198881205791minus 1198872119909

1205792

12

11990412057912

minus1198872119910

1205792

12

11988812057912) (11987111199091198881205791minus 11987111199101199041205791)]

+ 1198983[minus (minus119871

1119909

1205792

1

1198881205791+ 1198711119910

1205792

1

1199041205791minus1198712119909

1205792

12

11988812057912

+1198712119910

1205792

12

11990412057912minus1198873119909

1205792

123

119888120579123+ 1198873119910

1205792

123

119904120579123)



1205792

1

1199041205791minus 1198711119910

1205792

1

1198881205791minus 1198712119909

1205792

12

11990412057912


minus1198712119910

1205792

12

11988812057912minus 1198873119909

1205792

123

119904120579123minus 1198873119910

1205792

123

119888120579123)

times (11987111199091198881205791minus 11987111199101199041205791)]

minus (1198981119892 (11988711199091198881205791minus 11988711199101199041205791) + (119898

2+ 1198983) 119892


minus 1198881

1205791+ 1198882(12057912minus1205791)


+ (11987111199091198881205791minus 11987111199101199041205791) (119887211990911988812057912minus 119887211991011990412057912)]


+ (11987111199091198881205791minus 11987111199101199041205791) (119871211990911988812057912minus 119871211991011990412057912)]


+ (+119887211990911988812057912minus 119887211991011990412057912) (119887211990911988812057912minus 119887211991011990412057912)]


+ (+119871211990911988812057912) (119871211990911988812057912minus 119871211991011990412057912)] + 119868

2


+ (+1198873119909119888120579123minus 1198873119910119904120579123) (119871211990911988812057912minus 119871211991011990412057912)]

1198614= 1198982[minus (minus119871

1119909

1205792

1

1198881205791+ 1198711119910

1205792

1

1199041205791minus1198872119909

1205792

12

11988812057912

+1198872119910

1205792

12

11990412057912) (minus119887211990911990412057912minus 119887211991011988812057912)


1205792

1

1199041205791minus 1198711119910

1205792

1

1198881205791minus 1198872119909

1205792

12

11990412057912

minus1198872119910

1205792

12

11988812057912) (119887211990911988812057912minus 119887211991011990412057912)]

+ 1198983[minus (minus119871

1119909

1205792

1

1198881205791+ 1198711119910

1205792

1

1199041205791minus1198712119909

1205792

12

11988812057912

+ 1198712119910

1205792

12

11990412057912minus1198873119909

1205792

123

119888120579123

+1198873119910

1205792

123

119904120579123) (minus119871211990911990412057912minus 119871211991011988812057912)


1205792

1

1199041205791minus 1198711119910

1205792

1

1198881205791minus 1198712119909

1205792

12

11990412057912

minus 1198712119910

1205792

12

11988812057912minus 1198873119909

1205792

123

119904120579123

minus1198873119910

1205792

123

119888120579123) (119871211990911988812057912minus 119871211991011990412057912)]

minus (1198982119892 (119887211990911988812057912minus 119887211991011990412057912)

+ 1198983119892 (119871211990911988812057912minus 119871211991011990412057912)




+ (11987111199091198881205791minus 11987111199101199041205791) (1198873119909119888120579123minus 1198873119910119904120579123)]




+ (+1198873119909119888120579123minus 1198873119910119904120579123)

times (1198873119909119888120579123minus 1198873119910119904120579123)] + 119868

3

1198624= 1198983[minus (minus119871

1119909

1205792

1

1198881205791+ 1198711119910

1205792

1

1199041205791minus1198712119909

1205792

12

11988812057912

+ 1198712119910

1205792

12

11990412057912minus1198873119909

1205792

123

119888120579123

+1198873119910

1205792

123

119904120579123) (minus1198873119909119904120579123minus 1198873119910119888120579123)


1205792

1

1199041205791minus 1198711119910

1205792

1

1198881205791minus 1198712119909

1205792

12

11990412057912

minus 1198712119910

1205791211990412057912minus 1198712119910

1205792

12

11988812057912+ 1198873119909

120579123119888120579123

minus1198873119909

1205792

123

119904120579123minus 1198873119910

1205792

123

119888120579123)

times (1198873119909119888120579123minus 1198873119910119904120579123)]



(B1)

References
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











3= [

[

11987111198881205793

11987111199041205793

0

]

]

+[[

[

1198712119888 (1205793+ 1205794)

1198712119904 (1205793+ 1205794)

0

]]

]

+

[[[

[

1198873119909119888 (1205793+ 1205794+ 1205795) minus 1198873119910119904 (1205793+ 1205794+ 1205795)

1198873119909119904 (1205793+ 1205794+ 1205795) + 1198873119910119888 (1205793+ 1205794+ 1205795)

0

]]]

]

(28)


1205793+ 1205794= 12057934

1205793+1205794=

12057934

1205793+ 1205794+ 1205795= 120579345

1205793+1205794+1205795=

120579345

(29)


V1198661=[[

[

minus1198871119909

12057931199041205793minus 1198871119910

12057931198881205793

1198871119909

12057931198881205793minus 1198871119910

12057931199041205793

0

]]

]

V1198662=[[

[

minus1198711

12057931199041205793minus1198872119909

1205793411990412057934minus 1198872119910

1205793411988812057934

1198711

12057931198881205793+ 1198872119909

1205793411988812057934minus 1198872119910

1205793411990412057934

0

]]

]

V1198663=[[

[

minus1198711

12057931199041205793minus1198712

1205793411990412057934minus1198873119909

120579345119904120579345minus 1198873119910

120579345119888120579345

1198711

12057931198881205793+ 1198712

1205793411988812057934+ 1198873119909

120579345119888120579345minus 1198873119910

120579345119904120579345

0

]]

]

(30)


119879 =

1

2

1198981(V1198661)

2

+

1

2

1198982(V1198662)

2

+

1

2

1198983(V1198663)

2

+

1

2

1198681(1205793)

2

+

1

2

1198682(12057934)

2

+

1

2

1198683(120579345)

2

(31)




1 1198962 and 119896



119880 = 1198981119892 (11988711199091199041205793+ 11988711199101198881205793)

+ 1198982119892 (11987111199041205793+ 119887211990911990412057934+ 119887211991011988812057934)

+ 1198983119892 (11987111199041205793+ 119871211990412057934+ 1198873119909119904120579345+ 1198873119910119888120579345)

+

1

2

1198961(1205793)2

+

1

2

1198962(12057934minus 1205793)2

+

1

2

1198963(120579345minus 12057934)2

(32)


119865 =

1

2

1198881(1205793)

2

+

1

2

1198882(12057934minus1205793)

2

+

1

2

1198883(120579345minus12057934)

2

(33)



1

1198882 and 119888


joints respectively



3 12057934 and 120579

345

119889

119889119905

(

120597 (119879 minus 119880)

1205971205793

) minus

120597 (119879 minus 119880)

1205971205793

+

120597119865

1205971205793

= 1205913

119889

119889119905

(

120597 (119879 minus 119880)

12059712057934

) minus

120597 (119879 minus 119880)

12059712057934

+

120597119865

12059712057934

= 1205914

119889

119889119905

(

120597 (119879 minus 119880)

120597120579345

) minus

120597 (119879 minus 119880)

120597120579345

+

120597119865

120597120579345

= 1205915

(34)




120591119895=

(sum5

119895=3

120575119882119895)

120575120579119895

119895 = 3 4 5 (35)



1205913= (11986531199101198903119909minus 11986531199091198903119910) + [

[

minus11987111199041205793

11987111198881205793

0

]

]

sdot [

[

1198654119909c12057934minus 1198654119910s12057934

1198654119909s12057934+ 1198654119910c12057934

0

]

]

+ [

[

minus11987111199041205793

11987111198881205793

0

]

]

sdot [

[

1198655119909c120579345minus 1198655119910s120579345

1198655119909s120579345+ 1198655119910c120579345

0

]

]

+ 1198621198981minus 1198621198982

1205914= (11986541199101198904119909minus 11986541199091198904119910) + [

[

minus119871211990412057934

119871211988812057934

0

]

]

sdot [

[

1198655119909c120579345minus 1198655119910s120579345

1198655119909s120579345+ 1198655119910c120579345

0

]

]

+ 1198621198982minus 1198621198983

1205915= (11986551199101198905119909minus 11986551199091198905119910) + 1198621198983

(36)

where the term 119862119898119892


119895= [1198651198951199091198651198951199100]119879 is the contact


119895= [1198901198951199091198901198941199100]119879


1198601

1205793+ 1198602

12057934+ 1198603

120579345= 1198604

1198611

1205793+ 1198612

12057934+ 1198613

120579345= 1198614

1198621

1205793+ 1198622

12057934+ 1198623

120579345= 1198624

(37)


The terms 1198601 1198602 and 119860










119895= (1198631198952)(cos(120587119905119905

1)minus1) where




120579119895 max = | minus 12058711986311989521199051|





Omega maxMCP (rads)

Omega maxPIP (rads)

Omega maxDIP (rads)

Cycles in 10 s 27 27 31 25 20 29 23 27 21 31 261 3843 261 12164 15026 10018


Omega maxMCP (rads)

Omega maxPIP (rads)

Omega maxDIP (rads)

Cycles in 10 s 27 24 17 34 29 27 30 24 21 27 26 4784 26 12118 14969 9979



1198983are


119898max=



119889119895= |119862119889119895||119862119898119895|



0 20 40 60 80 100minus100

minus50

0

50

100

150

200

250

300

Cycle ()

C(N

mm

)

Cm1

Cm2

Cm3



MCP PIP DIP119891119896119895

3805 3848 3858119891119888119895

5758 6059 61119891119889119895

437 093 042



7 Conclusions




1198981

= 705 g 1198982

= 397 g 1198983

= 270 g1198681

= 1000 gmm2 1198682

= 240 gmm2 1198683

= 120 gmm2

1198961

= 00902Nm 1198962

= 00902Nm 1198963

= 00902Nm1198881

= 00225Nms 1198882

= 00225Nms 1198883

= 00225Nms1198711

= [50 0 0]

119879

mm 1198712

= [20 0 0]

119879

mm 1198713

= [25 0 0]

119879

mm1198871

= [25 0 0]

119879

mm 1198872

= [10 0 0]

119879

mm 1198873

= [125 0 0]

119879

mm1198901

= [25 minus8 0]

119879

mm 1198902

= [10 minus7 0]

119879

mm 1198903

= [12 minus6 0]

119879

mm



Appendices



11987611989411= [(119888120579CMC119891119890119888120579MCP119886119887119886119889) 119888120579MCP119891119890

minus 119904120579CMC119891119890119904120579MCP119891119890]


minus [(119888120579CMC119891119890119888120579MCP119886119887119886119889) 119904120579MCP119891119890

+ 119904120579CMC119891119890119888120579MCP119891119890]


11987611989412= [(119888120579CMC119891119890119888120579MCP119886119887119886119889) 119888120579MCP119891119890

minus 119904120579CMC119891119890119904120579MCP119891119890]


minus [(119888120579CMC119891119890119888120579MCP119886119887119886119889) 119904120579MCP119891119890

+ 119904120579CMC119891119890119888120579MCP119891119890]


11987611989413= 119888120579CMC119891119890119904120579MCP119886119887119886119889

11987611989414= [(119888120579CMC119891119890119888120579MCP119886119887119886119889) 119888120579MCP119891119890

minus 119904120579CMC119891119890119904120579MCP119891119890]



+ 1198711119894+ 1198712119894119888120579PIP119891119890]

minus [(119888120579CMC119891119890119888120579MCP119886119887119886119889) 119904120579MCP119891119890

+ 119904120579CMC119891119890119888120579MCP119891119890]



+ 1198712119894119904120579PIP119891119890]

+ (119888120579CMC119891119890119904120579MCP119886119887119886119889) 119871 119905119894119911 + 1198710119894119888120579CMC119891119890

11987611989421= 119904120579MCP119886119887119886119889119888120579MCP119891119890


minus 119904120579MCP119886119887119886119889119904120579MCP119891119890


11987611989422= 119904120579MCP119886119887119886119889119888120579MCP119891119890



minus 119904120579MCP119886119887119886119889119904120579MCP119891119890


11987611989423= minus 119888120579MCP119886119887119886119889

11987611989424= 119904120579MCP119886119887119886119889119888120579MCP119891119890



+ 1198711119894+ 1198712119894119888120579PIP119891119890] minus 119904120579MCP119886119887119886119889119904120579MCP119891119890



+ 1198712119894119904120579PIP119891119890] + (minus119888120579MCP119886119887119886119889) 119871 119905119894119911

11987611989431= (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119888120579MCP119891119890

+ 119888120579CMC119891119890119904120579MCP119891119890


+ minus (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119904120579MCP119891119890

+ 119888120579CMC119891119890119888120579MCP119891119890


11987611989432= (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119888120579MCP119891119890

+ 119888120579CMC119891119890119904120579MCP119891119890


+ minus (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119904120579MCP119891119890

+ 119888120579CMC119891119890119888120579MCP119891119890


11987611989433= 119904120579CMC119891119890119904120579MCP119886119887119886119889

11987611989434= (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119888120579MCP119891119890

+ 119888120579119862119872119862119891119890

119904120579119872119862119875119891119890



+ 1198711119894+ 1198712119894119888120579PIP119891119890]

+ minus (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119904120579MCP119891119890

+ 119888120579CMC119891119890119888120579MCP119891119890



+ 1198712119894119904120579PIP119891119890]

+ (119904120579CMC119891119890119904120579MCP119886119887119886119889) 119871 119905119894119911 + 1198710119894119904120579CMC119891119890

(A1)


119876011= [119888120579TMC119886119887119886119889119888120579TMC119891119890]


+ [minus119888120579TMC119886119887119886119889119904120579TMC119891119890]


119876012= [119888120579TMC119886119887119886119889119888120579TMC119891119890]


+ [minus119888120579TMC119886119887119886119889119904120579TMC119891119890]


119876013= 119904120579TMC119886119887119886119889

119876014= [119888120579TMC119886119887119886119889119888120579TMC119891119890]

times [1198711199050 119909


+ 1198711199050 119910


+ 11987110119888120579MCP119891119890 + 11987100]

+ [minus119888120579TMC119886119887119886119889119904120579TMC119891119890]

times [1198711199050 119909

(119904120579MCP119891119890119888120579IP119891119890 + 119888120579MCP119891119890119904120579IP119891119890)

+ 1198711199050 119910


+ 11987110119904120579MCP119891119890] + 119871 1199050 119911119904120579TMC119886119887119886119889


119876021= [119904120579TMC119886119887119886119889119888120579TMC119891119890]


+ [minus119904120579TMC119886119887119886119889119904120579TMC119891119890]


119876022= [119904120579TMC119886119887119886119889119888120579TMC119891119890]


+ [minus119904120579TMC119886119887119886119889119904120579TMC119891119890]


119876023= minus 119888120579TMC119886119887119886119889

119876024= [119904120579TMC119886119887119886119889119888120579TMC119891119890]

times [1198711199050 119909


+ 1198711199050 119910


+ 11987110119888120579MCP119891119890 + 11987100]

+ [minus119904120579TMC119886119887119886119889119904120579TMC119891119890]

times [1198711199050 119909

(119904120579MCP119891119890119888120579IP119891119890 + 119888120579MCP119891119890119904120579IP119891119890)

+ 1198711199050 119910


+ 11987110119904120579MCP119891119890] minus 119888120579TMC119886119887119886119889119871 1199050 119911


+ 119888120579TMC119891119890 (119904120579MCP119891119890119888120579IP119891119890 + 119888120579MCP119891119890119904120579IP119891119890)



119876033= 0

119876034= 119904120579TMC119891119890

times [1198711199050 119909


+ 1198711199050 119910


+ 11987110119888120579MCP119891119890 + 11987100] + 119888120579TMC119891119890

times [1198711199050 119909

(119904120579MCP119891119890119888120579IP119891119890 + 119888120579MCP119891119890119904120579IP119891119890)

+ 1198711199050 119910


+11987110119904120579MCP119891119890]

(A2)


Consider


+ (11988711199091198881205791minus 11988711199101199041205791) (1198871119909c1205791minus 1198871119910s1205791)]


11199091199041205791minus 11987111199101198881205791)

+ (11987111199091198881205791minus 11987111199101199041205791) (11987111199091198881205791minus 11987111199101199041205791)]


+(11987111199091198881205791minus 11987111199101199041205791)times (119871

11199091198881205791minus 11987111199101199041205791)]+1198681


11199091199041205791minus 11987111199101198881205791)

+ (+119887211990911988812057912minus 119887211991011990412057912) (11987111199091198881205791minus 11987111199101199041205791)]


+ (+119871211990911988812057912minus 119871211991011990412057912) (11987111199091198881205791minus 11987111199101199041205791)]


+ (+1198873119909119888120579123minus 1198873119910119904120579123) (11987111199091198881205791minus 11987111199101199041205791)]

1198604= 1198981[(minus1198871119909

12057911199041205791minus 1198871119910

12057911198881205791) (minus1198871119909

1205791c1205791+ 1198871119910

12057911199041205791)


+ 1198982[minus (minus119871

1119909

1205792

1

1198881205791+ 1198711119910

1205792

1

1199041205791

minus1198872119909

1205792

12

11988812057912+ 1198872119910

1205792

12

11990412057912)



1205792

1

1199041205791minus 1198711119910

1205792

1

1198881205791minus 1198872119909

1205792

12

11990412057912

minus1198872119910

1205792

12

11988812057912) (11987111199091198881205791minus 11987111199101199041205791)]

+ 1198983[minus (minus119871

1119909

1205792

1

1198881205791+ 1198711119910

1205792

1

1199041205791minus1198712119909

1205792

12

11988812057912

+1198712119910

1205792

12

11990412057912minus1198873119909

1205792

123

119888120579123+ 1198873119910

1205792

123

119904120579123)



1205792

1

1199041205791minus 1198711119910

1205792

1

1198881205791minus 1198712119909

1205792

12

11990412057912


minus1198712119910

1205792

12

11988812057912minus 1198873119909

1205792

123

119904120579123minus 1198873119910

1205792

123

119888120579123)

times (11987111199091198881205791minus 11987111199101199041205791)]

minus (1198981119892 (11988711199091198881205791minus 11988711199101199041205791) + (119898

2+ 1198983) 119892


minus 1198881

1205791+ 1198882(12057912minus1205791)


+ (11987111199091198881205791minus 11987111199101199041205791) (119887211990911988812057912minus 119887211991011990412057912)]


+ (11987111199091198881205791minus 11987111199101199041205791) (119871211990911988812057912minus 119871211991011990412057912)]


+ (+119887211990911988812057912minus 119887211991011990412057912) (119887211990911988812057912minus 119887211991011990412057912)]


+ (+119871211990911988812057912) (119871211990911988812057912minus 119871211991011990412057912)] + 119868

2


+ (+1198873119909119888120579123minus 1198873119910119904120579123) (119871211990911988812057912minus 119871211991011990412057912)]

1198614= 1198982[minus (minus119871

1119909

1205792

1

1198881205791+ 1198711119910

1205792

1

1199041205791minus1198872119909

1205792

12

11988812057912

+1198872119910

1205792

12

11990412057912) (minus119887211990911990412057912minus 119887211991011988812057912)


1205792

1

1199041205791minus 1198711119910

1205792

1

1198881205791minus 1198872119909

1205792

12

11990412057912

minus1198872119910

1205792

12

11988812057912) (119887211990911988812057912minus 119887211991011990412057912)]

+ 1198983[minus (minus119871

1119909

1205792

1

1198881205791+ 1198711119910

1205792

1

1199041205791minus1198712119909

1205792

12

11988812057912

+ 1198712119910

1205792

12

11990412057912minus1198873119909

1205792

123

119888120579123

+1198873119910

1205792

123

119904120579123) (minus119871211990911990412057912minus 119871211991011988812057912)


1205792

1

1199041205791minus 1198711119910

1205792

1

1198881205791minus 1198712119909

1205792

12

11990412057912

minus 1198712119910

1205792

12

11988812057912minus 1198873119909

1205792

123

119904120579123

minus1198873119910

1205792

123

119888120579123) (119871211990911988812057912minus 119871211991011990412057912)]

minus (1198982119892 (119887211990911988812057912minus 119887211991011990412057912)

+ 1198983119892 (119871211990911988812057912minus 119871211991011990412057912)




+ (11987111199091198881205791minus 11987111199101199041205791) (1198873119909119888120579123minus 1198873119910119904120579123)]




+ (+1198873119909119888120579123minus 1198873119910119904120579123)

times (1198873119909119888120579123minus 1198873119910119904120579123)] + 119868

3

1198624= 1198983[minus (minus119871

1119909

1205792

1

1198881205791+ 1198711119910

1205792

1

1199041205791minus1198712119909

1205792

12

11988812057912

+ 1198712119910

1205792

12

11990412057912minus1198873119909

1205792

123

119888120579123

+1198873119910

1205792

123

119904120579123) (minus1198873119909119904120579123minus 1198873119910119888120579123)


1205792

1

1199041205791minus 1198711119910

1205792

1

1198881205791minus 1198712119909

1205792

12

11990412057912

minus 1198712119910

1205791211990412057912minus 1198712119910

1205792

12

11988812057912+ 1198873119909

120579123119888120579123

minus1198873119909

1205792

123

119904120579123minus 1198873119910

1205792

123

119888120579123)

times (1198873119909119888120579123minus 1198873119910119904120579123)]



(B1)

References
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











3 12057934 and 120579

345

119889

119889119905

(

120597 (119879 minus 119880)

1205971205793

) minus

120597 (119879 minus 119880)

1205971205793

+

120597119865

1205971205793

= 1205913

119889

119889119905

(

120597 (119879 minus 119880)

12059712057934

) minus

120597 (119879 minus 119880)

12059712057934

+

120597119865

12059712057934

= 1205914

119889

119889119905

(

120597 (119879 minus 119880)

120597120579345

) minus

120597 (119879 minus 119880)

120597120579345

+

120597119865

120597120579345

= 1205915

(34)




120591119895=

(sum5

119895=3

120575119882119895)

120575120579119895

119895 = 3 4 5 (35)



1205913= (11986531199101198903119909minus 11986531199091198903119910) + [

[

minus11987111199041205793

11987111198881205793

0

]

]

sdot [

[

1198654119909c12057934minus 1198654119910s12057934

1198654119909s12057934+ 1198654119910c12057934

0

]

]

+ [

[

minus11987111199041205793

11987111198881205793

0

]

]

sdot [

[

1198655119909c120579345minus 1198655119910s120579345

1198655119909s120579345+ 1198655119910c120579345

0

]

]

+ 1198621198981minus 1198621198982

1205914= (11986541199101198904119909minus 11986541199091198904119910) + [

[

minus119871211990412057934

119871211988812057934

0

]

]

sdot [

[

1198655119909c120579345minus 1198655119910s120579345

1198655119909s120579345+ 1198655119910c120579345

0

]

]

+ 1198621198982minus 1198621198983

1205915= (11986551199101198905119909minus 11986551199091198905119910) + 1198621198983

(36)

where the term 119862119898119892


119895= [1198651198951199091198651198951199100]119879 is the contact


119895= [1198901198951199091198901198941199100]119879


1198601

1205793+ 1198602

12057934+ 1198603

120579345= 1198604

1198611

1205793+ 1198612

12057934+ 1198613

120579345= 1198614

1198621

1205793+ 1198622

12057934+ 1198623

120579345= 1198624

(37)


The terms 1198601 1198602 and 119860










119895= (1198631198952)(cos(120587119905119905

1)minus1) where




120579119895 max = | minus 12058711986311989521199051|





Omega maxMCP (rads)

Omega maxPIP (rads)

Omega maxDIP (rads)

Cycles in 10 s 27 27 31 25 20 29 23 27 21 31 261 3843 261 12164 15026 10018


Omega maxMCP (rads)

Omega maxPIP (rads)

Omega maxDIP (rads)

Cycles in 10 s 27 24 17 34 29 27 30 24 21 27 26 4784 26 12118 14969 9979



1198983are


119898max=



119889119895= |119862119889119895||119862119898119895|



0 20 40 60 80 100minus100

minus50

0

50

100

150

200

250

300

Cycle ()

C(N

mm

)

Cm1

Cm2

Cm3



MCP PIP DIP119891119896119895

3805 3848 3858119891119888119895

5758 6059 61119891119889119895

437 093 042



7 Conclusions




1198981

= 705 g 1198982

= 397 g 1198983

= 270 g1198681

= 1000 gmm2 1198682

= 240 gmm2 1198683

= 120 gmm2

1198961

= 00902Nm 1198962

= 00902Nm 1198963

= 00902Nm1198881

= 00225Nms 1198882

= 00225Nms 1198883

= 00225Nms1198711

= [50 0 0]

119879

mm 1198712

= [20 0 0]

119879

mm 1198713

= [25 0 0]

119879

mm1198871

= [25 0 0]

119879

mm 1198872

= [10 0 0]

119879

mm 1198873

= [125 0 0]

119879

mm1198901

= [25 minus8 0]

119879

mm 1198902

= [10 minus7 0]

119879

mm 1198903

= [12 minus6 0]

119879

mm



Appendices



11987611989411= [(119888120579CMC119891119890119888120579MCP119886119887119886119889) 119888120579MCP119891119890

minus 119904120579CMC119891119890119904120579MCP119891119890]


minus [(119888120579CMC119891119890119888120579MCP119886119887119886119889) 119904120579MCP119891119890

+ 119904120579CMC119891119890119888120579MCP119891119890]


11987611989412= [(119888120579CMC119891119890119888120579MCP119886119887119886119889) 119888120579MCP119891119890

minus 119904120579CMC119891119890119904120579MCP119891119890]


minus [(119888120579CMC119891119890119888120579MCP119886119887119886119889) 119904120579MCP119891119890

+ 119904120579CMC119891119890119888120579MCP119891119890]


11987611989413= 119888120579CMC119891119890119904120579MCP119886119887119886119889

11987611989414= [(119888120579CMC119891119890119888120579MCP119886119887119886119889) 119888120579MCP119891119890

minus 119904120579CMC119891119890119904120579MCP119891119890]



+ 1198711119894+ 1198712119894119888120579PIP119891119890]

minus [(119888120579CMC119891119890119888120579MCP119886119887119886119889) 119904120579MCP119891119890

+ 119904120579CMC119891119890119888120579MCP119891119890]



+ 1198712119894119904120579PIP119891119890]

+ (119888120579CMC119891119890119904120579MCP119886119887119886119889) 119871 119905119894119911 + 1198710119894119888120579CMC119891119890

11987611989421= 119904120579MCP119886119887119886119889119888120579MCP119891119890


minus 119904120579MCP119886119887119886119889119904120579MCP119891119890


11987611989422= 119904120579MCP119886119887119886119889119888120579MCP119891119890



minus 119904120579MCP119886119887119886119889119904120579MCP119891119890


11987611989423= minus 119888120579MCP119886119887119886119889

11987611989424= 119904120579MCP119886119887119886119889119888120579MCP119891119890



+ 1198711119894+ 1198712119894119888120579PIP119891119890] minus 119904120579MCP119886119887119886119889119904120579MCP119891119890



+ 1198712119894119904120579PIP119891119890] + (minus119888120579MCP119886119887119886119889) 119871 119905119894119911

11987611989431= (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119888120579MCP119891119890

+ 119888120579CMC119891119890119904120579MCP119891119890


+ minus (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119904120579MCP119891119890

+ 119888120579CMC119891119890119888120579MCP119891119890


11987611989432= (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119888120579MCP119891119890

+ 119888120579CMC119891119890119904120579MCP119891119890


+ minus (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119904120579MCP119891119890

+ 119888120579CMC119891119890119888120579MCP119891119890


11987611989433= 119904120579CMC119891119890119904120579MCP119886119887119886119889

11987611989434= (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119888120579MCP119891119890

+ 119888120579119862119872119862119891119890

119904120579119872119862119875119891119890



+ 1198711119894+ 1198712119894119888120579PIP119891119890]

+ minus (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119904120579MCP119891119890

+ 119888120579CMC119891119890119888120579MCP119891119890



+ 1198712119894119904120579PIP119891119890]

+ (119904120579CMC119891119890119904120579MCP119886119887119886119889) 119871 119905119894119911 + 1198710119894119904120579CMC119891119890

(A1)


119876011= [119888120579TMC119886119887119886119889119888120579TMC119891119890]


+ [minus119888120579TMC119886119887119886119889119904120579TMC119891119890]


119876012= [119888120579TMC119886119887119886119889119888120579TMC119891119890]


+ [minus119888120579TMC119886119887119886119889119904120579TMC119891119890]


119876013= 119904120579TMC119886119887119886119889

119876014= [119888120579TMC119886119887119886119889119888120579TMC119891119890]

times [1198711199050 119909


+ 1198711199050 119910


+ 11987110119888120579MCP119891119890 + 11987100]

+ [minus119888120579TMC119886119887119886119889119904120579TMC119891119890]

times [1198711199050 119909

(119904120579MCP119891119890119888120579IP119891119890 + 119888120579MCP119891119890119904120579IP119891119890)

+ 1198711199050 119910


+ 11987110119904120579MCP119891119890] + 119871 1199050 119911119904120579TMC119886119887119886119889


119876021= [119904120579TMC119886119887119886119889119888120579TMC119891119890]


+ [minus119904120579TMC119886119887119886119889119904120579TMC119891119890]


119876022= [119904120579TMC119886119887119886119889119888120579TMC119891119890]


+ [minus119904120579TMC119886119887119886119889119904120579TMC119891119890]


119876023= minus 119888120579TMC119886119887119886119889

119876024= [119904120579TMC119886119887119886119889119888120579TMC119891119890]

times [1198711199050 119909


+ 1198711199050 119910


+ 11987110119888120579MCP119891119890 + 11987100]

+ [minus119904120579TMC119886119887119886119889119904120579TMC119891119890]

times [1198711199050 119909

(119904120579MCP119891119890119888120579IP119891119890 + 119888120579MCP119891119890119904120579IP119891119890)

+ 1198711199050 119910


+ 11987110119904120579MCP119891119890] minus 119888120579TMC119886119887119886119889119871 1199050 119911


+ 119888120579TMC119891119890 (119904120579MCP119891119890119888120579IP119891119890 + 119888120579MCP119891119890119904120579IP119891119890)



119876033= 0

119876034= 119904120579TMC119891119890

times [1198711199050 119909


+ 1198711199050 119910


+ 11987110119888120579MCP119891119890 + 11987100] + 119888120579TMC119891119890

times [1198711199050 119909

(119904120579MCP119891119890119888120579IP119891119890 + 119888120579MCP119891119890119904120579IP119891119890)

+ 1198711199050 119910


+11987110119904120579MCP119891119890]

(A2)


Consider


+ (11988711199091198881205791minus 11988711199101199041205791) (1198871119909c1205791minus 1198871119910s1205791)]


11199091199041205791minus 11987111199101198881205791)

+ (11987111199091198881205791minus 11987111199101199041205791) (11987111199091198881205791minus 11987111199101199041205791)]


+(11987111199091198881205791minus 11987111199101199041205791)times (119871

11199091198881205791minus 11987111199101199041205791)]+1198681


11199091199041205791minus 11987111199101198881205791)

+ (+119887211990911988812057912minus 119887211991011990412057912) (11987111199091198881205791minus 11987111199101199041205791)]


+ (+119871211990911988812057912minus 119871211991011990412057912) (11987111199091198881205791minus 11987111199101199041205791)]


+ (+1198873119909119888120579123minus 1198873119910119904120579123) (11987111199091198881205791minus 11987111199101199041205791)]

1198604= 1198981[(minus1198871119909

12057911199041205791minus 1198871119910

12057911198881205791) (minus1198871119909

1205791c1205791+ 1198871119910

12057911199041205791)


+ 1198982[minus (minus119871

1119909

1205792

1

1198881205791+ 1198711119910

1205792

1

1199041205791

minus1198872119909

1205792

12

11988812057912+ 1198872119910

1205792

12

11990412057912)



1205792

1

1199041205791minus 1198711119910

1205792

1

1198881205791minus 1198872119909

1205792

12

11990412057912

minus1198872119910

1205792

12

11988812057912) (11987111199091198881205791minus 11987111199101199041205791)]

+ 1198983[minus (minus119871

1119909

1205792

1

1198881205791+ 1198711119910

1205792

1

1199041205791minus1198712119909

1205792

12

11988812057912

+1198712119910

1205792

12

11990412057912minus1198873119909

1205792

123

119888120579123+ 1198873119910

1205792

123

119904120579123)



1205792

1

1199041205791minus 1198711119910

1205792

1

1198881205791minus 1198712119909

1205792

12

11990412057912


minus1198712119910

1205792

12

11988812057912minus 1198873119909

1205792

123

119904120579123minus 1198873119910

1205792

123

119888120579123)

times (11987111199091198881205791minus 11987111199101199041205791)]

minus (1198981119892 (11988711199091198881205791minus 11988711199101199041205791) + (119898

2+ 1198983) 119892


minus 1198881

1205791+ 1198882(12057912minus1205791)


+ (11987111199091198881205791minus 11987111199101199041205791) (119887211990911988812057912minus 119887211991011990412057912)]


+ (11987111199091198881205791minus 11987111199101199041205791) (119871211990911988812057912minus 119871211991011990412057912)]


+ (+119887211990911988812057912minus 119887211991011990412057912) (119887211990911988812057912minus 119887211991011990412057912)]


+ (+119871211990911988812057912) (119871211990911988812057912minus 119871211991011990412057912)] + 119868

2


+ (+1198873119909119888120579123minus 1198873119910119904120579123) (119871211990911988812057912minus 119871211991011990412057912)]

1198614= 1198982[minus (minus119871

1119909

1205792

1

1198881205791+ 1198711119910

1205792

1

1199041205791minus1198872119909

1205792

12

11988812057912

+1198872119910

1205792

12

11990412057912) (minus119887211990911990412057912minus 119887211991011988812057912)


1205792

1

1199041205791minus 1198711119910

1205792

1

1198881205791minus 1198872119909

1205792

12

11990412057912

minus1198872119910

1205792

12

11988812057912) (119887211990911988812057912minus 119887211991011990412057912)]

+ 1198983[minus (minus119871

1119909

1205792

1

1198881205791+ 1198711119910

1205792

1

1199041205791minus1198712119909

1205792

12

11988812057912

+ 1198712119910

1205792

12

11990412057912minus1198873119909

1205792

123

119888120579123

+1198873119910

1205792

123

119904120579123) (minus119871211990911990412057912minus 119871211991011988812057912)


1205792

1

1199041205791minus 1198711119910

1205792

1

1198881205791minus 1198712119909

1205792

12

11990412057912

minus 1198712119910

1205792

12

11988812057912minus 1198873119909

1205792

123

119904120579123

minus1198873119910

1205792

123

119888120579123) (119871211990911988812057912minus 119871211991011990412057912)]

minus (1198982119892 (119887211990911988812057912minus 119887211991011990412057912)

+ 1198983119892 (119871211990911988812057912minus 119871211991011990412057912)




+ (11987111199091198881205791minus 11987111199101199041205791) (1198873119909119888120579123minus 1198873119910119904120579123)]




+ (+1198873119909119888120579123minus 1198873119910119904120579123)

times (1198873119909119888120579123minus 1198873119910119904120579123)] + 119868

3

1198624= 1198983[minus (minus119871

1119909

1205792

1

1198881205791+ 1198711119910

1205792

1

1199041205791minus1198712119909

1205792

12

11988812057912

+ 1198712119910

1205792

12

11990412057912minus1198873119909

1205792

123

119888120579123

+1198873119910

1205792

123

119904120579123) (minus1198873119909119904120579123minus 1198873119910119888120579123)


1205792

1

1199041205791minus 1198711119910

1205792

1

1198881205791minus 1198712119909

1205792

12

11990412057912

minus 1198712119910

1205791211990412057912minus 1198712119910

1205792

12

11988812057912+ 1198873119909

120579123119888120579123

minus1198873119909

1205792

123

119904120579123minus 1198873119910

1205792

123

119888120579123)

times (1198873119909119888120579123minus 1198873119910119904120579123)]



(B1)

References
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












Omega maxMCP (rads)

Omega maxPIP (rads)

Omega maxDIP (rads)

Cycles in 10 s 27 27 31 25 20 29 23 27 21 31 261 3843 261 12164 15026 10018


Omega maxMCP (rads)

Omega maxPIP (rads)

Omega maxDIP (rads)

Cycles in 10 s 27 24 17 34 29 27 30 24 21 27 26 4784 26 12118 14969 9979



1198983are


119898max=



119889119895= |119862119889119895||119862119898119895|



0 20 40 60 80 100minus100

minus50

0

50

100

150

200

250

300

Cycle ()

C(N

mm

)

Cm1

Cm2

Cm3



MCP PIP DIP119891119896119895

3805 3848 3858119891119888119895

5758 6059 61119891119889119895

437 093 042



7 Conclusions




1198981

= 705 g 1198982

= 397 g 1198983

= 270 g1198681

= 1000 gmm2 1198682

= 240 gmm2 1198683

= 120 gmm2

1198961

= 00902Nm 1198962

= 00902Nm 1198963

= 00902Nm1198881

= 00225Nms 1198882

= 00225Nms 1198883

= 00225Nms1198711

= [50 0 0]

119879

mm 1198712

= [20 0 0]

119879

mm 1198713

= [25 0 0]

119879

mm1198871

= [25 0 0]

119879

mm 1198872

= [10 0 0]

119879

mm 1198873

= [125 0 0]

119879

mm1198901

= [25 minus8 0]

119879

mm 1198902

= [10 minus7 0]

119879

mm 1198903

= [12 minus6 0]

119879

mm



Appendices



11987611989411= [(119888120579CMC119891119890119888120579MCP119886119887119886119889) 119888120579MCP119891119890

minus 119904120579CMC119891119890119904120579MCP119891119890]


minus [(119888120579CMC119891119890119888120579MCP119886119887119886119889) 119904120579MCP119891119890

+ 119904120579CMC119891119890119888120579MCP119891119890]


11987611989412= [(119888120579CMC119891119890119888120579MCP119886119887119886119889) 119888120579MCP119891119890

minus 119904120579CMC119891119890119904120579MCP119891119890]


minus [(119888120579CMC119891119890119888120579MCP119886119887119886119889) 119904120579MCP119891119890

+ 119904120579CMC119891119890119888120579MCP119891119890]


11987611989413= 119888120579CMC119891119890119904120579MCP119886119887119886119889

11987611989414= [(119888120579CMC119891119890119888120579MCP119886119887119886119889) 119888120579MCP119891119890

minus 119904120579CMC119891119890119904120579MCP119891119890]



+ 1198711119894+ 1198712119894119888120579PIP119891119890]

minus [(119888120579CMC119891119890119888120579MCP119886119887119886119889) 119904120579MCP119891119890

+ 119904120579CMC119891119890119888120579MCP119891119890]



+ 1198712119894119904120579PIP119891119890]

+ (119888120579CMC119891119890119904120579MCP119886119887119886119889) 119871 119905119894119911 + 1198710119894119888120579CMC119891119890

11987611989421= 119904120579MCP119886119887119886119889119888120579MCP119891119890


minus 119904120579MCP119886119887119886119889119904120579MCP119891119890


11987611989422= 119904120579MCP119886119887119886119889119888120579MCP119891119890



minus 119904120579MCP119886119887119886119889119904120579MCP119891119890


11987611989423= minus 119888120579MCP119886119887119886119889

11987611989424= 119904120579MCP119886119887119886119889119888120579MCP119891119890



+ 1198711119894+ 1198712119894119888120579PIP119891119890] minus 119904120579MCP119886119887119886119889119904120579MCP119891119890



+ 1198712119894119904120579PIP119891119890] + (minus119888120579MCP119886119887119886119889) 119871 119905119894119911

11987611989431= (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119888120579MCP119891119890

+ 119888120579CMC119891119890119904120579MCP119891119890


+ minus (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119904120579MCP119891119890

+ 119888120579CMC119891119890119888120579MCP119891119890


11987611989432= (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119888120579MCP119891119890

+ 119888120579CMC119891119890119904120579MCP119891119890


+ minus (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119904120579MCP119891119890

+ 119888120579CMC119891119890119888120579MCP119891119890


11987611989433= 119904120579CMC119891119890119904120579MCP119886119887119886119889

11987611989434= (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119888120579MCP119891119890

+ 119888120579119862119872119862119891119890

119904120579119872119862119875119891119890



+ 1198711119894+ 1198712119894119888120579PIP119891119890]

+ minus (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119904120579MCP119891119890

+ 119888120579CMC119891119890119888120579MCP119891119890



+ 1198712119894119904120579PIP119891119890]

+ (119904120579CMC119891119890119904120579MCP119886119887119886119889) 119871 119905119894119911 + 1198710119894119904120579CMC119891119890

(A1)


119876011= [119888120579TMC119886119887119886119889119888120579TMC119891119890]


+ [minus119888120579TMC119886119887119886119889119904120579TMC119891119890]


119876012= [119888120579TMC119886119887119886119889119888120579TMC119891119890]


+ [minus119888120579TMC119886119887119886119889119904120579TMC119891119890]


119876013= 119904120579TMC119886119887119886119889

119876014= [119888120579TMC119886119887119886119889119888120579TMC119891119890]

times [1198711199050 119909


+ 1198711199050 119910


+ 11987110119888120579MCP119891119890 + 11987100]

+ [minus119888120579TMC119886119887119886119889119904120579TMC119891119890]

times [1198711199050 119909

(119904120579MCP119891119890119888120579IP119891119890 + 119888120579MCP119891119890119904120579IP119891119890)

+ 1198711199050 119910


+ 11987110119904120579MCP119891119890] + 119871 1199050 119911119904120579TMC119886119887119886119889


119876021= [119904120579TMC119886119887119886119889119888120579TMC119891119890]


+ [minus119904120579TMC119886119887119886119889119904120579TMC119891119890]


119876022= [119904120579TMC119886119887119886119889119888120579TMC119891119890]


+ [minus119904120579TMC119886119887119886119889119904120579TMC119891119890]


119876023= minus 119888120579TMC119886119887119886119889

119876024= [119904120579TMC119886119887119886119889119888120579TMC119891119890]

times [1198711199050 119909


+ 1198711199050 119910


+ 11987110119888120579MCP119891119890 + 11987100]

+ [minus119904120579TMC119886119887119886119889119904120579TMC119891119890]

times [1198711199050 119909

(119904120579MCP119891119890119888120579IP119891119890 + 119888120579MCP119891119890119904120579IP119891119890)

+ 1198711199050 119910


+ 11987110119904120579MCP119891119890] minus 119888120579TMC119886119887119886119889119871 1199050 119911


+ 119888120579TMC119891119890 (119904120579MCP119891119890119888120579IP119891119890 + 119888120579MCP119891119890119904120579IP119891119890)



119876033= 0

119876034= 119904120579TMC119891119890

times [1198711199050 119909


+ 1198711199050 119910


+ 11987110119888120579MCP119891119890 + 11987100] + 119888120579TMC119891119890

times [1198711199050 119909

(119904120579MCP119891119890119888120579IP119891119890 + 119888120579MCP119891119890119904120579IP119891119890)

+ 1198711199050 119910


+11987110119904120579MCP119891119890]

(A2)


Consider


+ (11988711199091198881205791minus 11988711199101199041205791) (1198871119909c1205791minus 1198871119910s1205791)]


11199091199041205791minus 11987111199101198881205791)

+ (11987111199091198881205791minus 11987111199101199041205791) (11987111199091198881205791minus 11987111199101199041205791)]


+(11987111199091198881205791minus 11987111199101199041205791)times (119871

11199091198881205791minus 11987111199101199041205791)]+1198681


11199091199041205791minus 11987111199101198881205791)

+ (+119887211990911988812057912minus 119887211991011990412057912) (11987111199091198881205791minus 11987111199101199041205791)]


+ (+119871211990911988812057912minus 119871211991011990412057912) (11987111199091198881205791minus 11987111199101199041205791)]


+ (+1198873119909119888120579123minus 1198873119910119904120579123) (11987111199091198881205791minus 11987111199101199041205791)]

1198604= 1198981[(minus1198871119909

12057911199041205791minus 1198871119910

12057911198881205791) (minus1198871119909

1205791c1205791+ 1198871119910

12057911199041205791)


+ 1198982[minus (minus119871

1119909

1205792

1

1198881205791+ 1198711119910

1205792

1

1199041205791

minus1198872119909

1205792

12

11988812057912+ 1198872119910

1205792

12

11990412057912)



1205792

1

1199041205791minus 1198711119910

1205792

1

1198881205791minus 1198872119909

1205792

12

11990412057912

minus1198872119910

1205792

12

11988812057912) (11987111199091198881205791minus 11987111199101199041205791)]

+ 1198983[minus (minus119871

1119909

1205792

1

1198881205791+ 1198711119910

1205792

1

1199041205791minus1198712119909

1205792

12

11988812057912

+1198712119910

1205792

12

11990412057912minus1198873119909

1205792

123

119888120579123+ 1198873119910

1205792

123

119904120579123)



1205792

1

1199041205791minus 1198711119910

1205792

1

1198881205791minus 1198712119909

1205792

12

11990412057912


minus1198712119910

1205792

12

11988812057912minus 1198873119909

1205792

123

119904120579123minus 1198873119910

1205792

123

119888120579123)

times (11987111199091198881205791minus 11987111199101199041205791)]

minus (1198981119892 (11988711199091198881205791minus 11988711199101199041205791) + (119898

2+ 1198983) 119892


minus 1198881

1205791+ 1198882(12057912minus1205791)


+ (11987111199091198881205791minus 11987111199101199041205791) (119887211990911988812057912minus 119887211991011990412057912)]


+ (11987111199091198881205791minus 11987111199101199041205791) (119871211990911988812057912minus 119871211991011990412057912)]


+ (+119887211990911988812057912minus 119887211991011990412057912) (119887211990911988812057912minus 119887211991011990412057912)]


+ (+119871211990911988812057912) (119871211990911988812057912minus 119871211991011990412057912)] + 119868

2


+ (+1198873119909119888120579123minus 1198873119910119904120579123) (119871211990911988812057912minus 119871211991011990412057912)]

1198614= 1198982[minus (minus119871

1119909

1205792

1

1198881205791+ 1198711119910

1205792

1

1199041205791minus1198872119909

1205792

12

11988812057912

+1198872119910

1205792

12

11990412057912) (minus119887211990911990412057912minus 119887211991011988812057912)


1205792

1

1199041205791minus 1198711119910

1205792

1

1198881205791minus 1198872119909

1205792

12

11990412057912

minus1198872119910

1205792

12

11988812057912) (119887211990911988812057912minus 119887211991011990412057912)]

+ 1198983[minus (minus119871

1119909

1205792

1

1198881205791+ 1198711119910

1205792

1

1199041205791minus1198712119909

1205792

12

11988812057912

+ 1198712119910

1205792

12

11990412057912minus1198873119909

1205792

123

119888120579123

+1198873119910

1205792

123

119904120579123) (minus119871211990911990412057912minus 119871211991011988812057912)


1205792

1

1199041205791minus 1198711119910

1205792

1

1198881205791minus 1198712119909

1205792

12

11990412057912

minus 1198712119910

1205792

12

11988812057912minus 1198873119909

1205792

123

119904120579123

minus1198873119910

1205792

123

119888120579123) (119871211990911988812057912minus 119871211991011990412057912)]

minus (1198982119892 (119887211990911988812057912minus 119887211991011990412057912)

+ 1198983119892 (119871211990911988812057912minus 119871211991011990412057912)




+ (11987111199091198881205791minus 11987111199101199041205791) (1198873119909119888120579123minus 1198873119910119904120579123)]




+ (+1198873119909119888120579123minus 1198873119910119904120579123)

times (1198873119909119888120579123minus 1198873119910119904120579123)] + 119868

3

1198624= 1198983[minus (minus119871

1119909

1205792

1

1198881205791+ 1198711119910

1205792

1

1199041205791minus1198712119909

1205792

12

11988812057912

+ 1198712119910

1205792

12

11990412057912minus1198873119909

1205792

123

119888120579123

+1198873119910

1205792

123

119904120579123) (minus1198873119909119904120579123minus 1198873119910119888120579123)


1205792

1

1199041205791minus 1198711119910

1205792

1

1198881205791minus 1198712119909

1205792

12

11990412057912

minus 1198712119910

1205791211990412057912minus 1198712119910

1205792

12

11988812057912+ 1198873119909

120579123119888120579123

minus1198873119909

1205792

123

119904120579123minus 1198873119910

1205792

123

119888120579123)

times (1198873119909119888120579123minus 1198873119910119904120579123)]



(B1)

References
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











1198981

= 705 g 1198982

= 397 g 1198983

= 270 g1198681

= 1000 gmm2 1198682

= 240 gmm2 1198683

= 120 gmm2

1198961

= 00902Nm 1198962

= 00902Nm 1198963

= 00902Nm1198881

= 00225Nms 1198882

= 00225Nms 1198883

= 00225Nms1198711

= [50 0 0]

119879

mm 1198712

= [20 0 0]

119879

mm 1198713

= [25 0 0]

119879

mm1198871

= [25 0 0]

119879

mm 1198872

= [10 0 0]

119879

mm 1198873

= [125 0 0]

119879

mm1198901

= [25 minus8 0]

119879

mm 1198902

= [10 minus7 0]

119879

mm 1198903

= [12 minus6 0]

119879

mm



Appendices



11987611989411= [(119888120579CMC119891119890119888120579MCP119886119887119886119889) 119888120579MCP119891119890

minus 119904120579CMC119891119890119904120579MCP119891119890]


minus [(119888120579CMC119891119890119888120579MCP119886119887119886119889) 119904120579MCP119891119890

+ 119904120579CMC119891119890119888120579MCP119891119890]


11987611989412= [(119888120579CMC119891119890119888120579MCP119886119887119886119889) 119888120579MCP119891119890

minus 119904120579CMC119891119890119904120579MCP119891119890]


minus [(119888120579CMC119891119890119888120579MCP119886119887119886119889) 119904120579MCP119891119890

+ 119904120579CMC119891119890119888120579MCP119891119890]


11987611989413= 119888120579CMC119891119890119904120579MCP119886119887119886119889

11987611989414= [(119888120579CMC119891119890119888120579MCP119886119887119886119889) 119888120579MCP119891119890

minus 119904120579CMC119891119890119904120579MCP119891119890]



+ 1198711119894+ 1198712119894119888120579PIP119891119890]

minus [(119888120579CMC119891119890119888120579MCP119886119887119886119889) 119904120579MCP119891119890

+ 119904120579CMC119891119890119888120579MCP119891119890]



+ 1198712119894119904120579PIP119891119890]

+ (119888120579CMC119891119890119904120579MCP119886119887119886119889) 119871 119905119894119911 + 1198710119894119888120579CMC119891119890

11987611989421= 119904120579MCP119886119887119886119889119888120579MCP119891119890


minus 119904120579MCP119886119887119886119889119904120579MCP119891119890


11987611989422= 119904120579MCP119886119887119886119889119888120579MCP119891119890



minus 119904120579MCP119886119887119886119889119904120579MCP119891119890


11987611989423= minus 119888120579MCP119886119887119886119889

11987611989424= 119904120579MCP119886119887119886119889119888120579MCP119891119890



+ 1198711119894+ 1198712119894119888120579PIP119891119890] minus 119904120579MCP119886119887119886119889119904120579MCP119891119890



+ 1198712119894119904120579PIP119891119890] + (minus119888120579MCP119886119887119886119889) 119871 119905119894119911

11987611989431= (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119888120579MCP119891119890

+ 119888120579CMC119891119890119904120579MCP119891119890


+ minus (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119904120579MCP119891119890

+ 119888120579CMC119891119890119888120579MCP119891119890


11987611989432= (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119888120579MCP119891119890

+ 119888120579CMC119891119890119904120579MCP119891119890


+ minus (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119904120579MCP119891119890

+ 119888120579CMC119891119890119888120579MCP119891119890


11987611989433= 119904120579CMC119891119890119904120579MCP119886119887119886119889

11987611989434= (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119888120579MCP119891119890

+ 119888120579119862119872119862119891119890

119904120579119872119862119875119891119890



+ 1198711119894+ 1198712119894119888120579PIP119891119890]

+ minus (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119904120579MCP119891119890

+ 119888120579CMC119891119890119888120579MCP119891119890



+ 1198712119894119904120579PIP119891119890]

+ (119904120579CMC119891119890119904120579MCP119886119887119886119889) 119871 119905119894119911 + 1198710119894119904120579CMC119891119890

(A1)


119876011= [119888120579TMC119886119887119886119889119888120579TMC119891119890]


+ [minus119888120579TMC119886119887119886119889119904120579TMC119891119890]


119876012= [119888120579TMC119886119887119886119889119888120579TMC119891119890]


+ [minus119888120579TMC119886119887119886119889119904120579TMC119891119890]


119876013= 119904120579TMC119886119887119886119889

119876014= [119888120579TMC119886119887119886119889119888120579TMC119891119890]

times [1198711199050 119909


+ 1198711199050 119910


+ 11987110119888120579MCP119891119890 + 11987100]

+ [minus119888120579TMC119886119887119886119889119904120579TMC119891119890]

times [1198711199050 119909

(119904120579MCP119891119890119888120579IP119891119890 + 119888120579MCP119891119890119904120579IP119891119890)

+ 1198711199050 119910


+ 11987110119904120579MCP119891119890] + 119871 1199050 119911119904120579TMC119886119887119886119889


119876021= [119904120579TMC119886119887119886119889119888120579TMC119891119890]


+ [minus119904120579TMC119886119887119886119889119904120579TMC119891119890]


119876022= [119904120579TMC119886119887119886119889119888120579TMC119891119890]


+ [minus119904120579TMC119886119887119886119889119904120579TMC119891119890]


119876023= minus 119888120579TMC119886119887119886119889

119876024= [119904120579TMC119886119887119886119889119888120579TMC119891119890]

times [1198711199050 119909


+ 1198711199050 119910


+ 11987110119888120579MCP119891119890 + 11987100]

+ [minus119904120579TMC119886119887119886119889119904120579TMC119891119890]

times [1198711199050 119909

(119904120579MCP119891119890119888120579IP119891119890 + 119888120579MCP119891119890119904120579IP119891119890)

+ 1198711199050 119910


+ 11987110119904120579MCP119891119890] minus 119888120579TMC119886119887119886119889119871 1199050 119911


+ 119888120579TMC119891119890 (119904120579MCP119891119890119888120579IP119891119890 + 119888120579MCP119891119890119904120579IP119891119890)



119876033= 0

119876034= 119904120579TMC119891119890

times [1198711199050 119909


+ 1198711199050 119910


+ 11987110119888120579MCP119891119890 + 11987100] + 119888120579TMC119891119890

times [1198711199050 119909

(119904120579MCP119891119890119888120579IP119891119890 + 119888120579MCP119891119890119904120579IP119891119890)

+ 1198711199050 119910


+11987110119904120579MCP119891119890]

(A2)


Consider


+ (11988711199091198881205791minus 11988711199101199041205791) (1198871119909c1205791minus 1198871119910s1205791)]


11199091199041205791minus 11987111199101198881205791)

+ (11987111199091198881205791minus 11987111199101199041205791) (11987111199091198881205791minus 11987111199101199041205791)]


+(11987111199091198881205791minus 11987111199101199041205791)times (119871

11199091198881205791minus 11987111199101199041205791)]+1198681


11199091199041205791minus 11987111199101198881205791)

+ (+119887211990911988812057912minus 119887211991011990412057912) (11987111199091198881205791minus 11987111199101199041205791)]


+ (+119871211990911988812057912minus 119871211991011990412057912) (11987111199091198881205791minus 11987111199101199041205791)]


+ (+1198873119909119888120579123minus 1198873119910119904120579123) (11987111199091198881205791minus 11987111199101199041205791)]

1198604= 1198981[(minus1198871119909

12057911199041205791minus 1198871119910

12057911198881205791) (minus1198871119909

1205791c1205791+ 1198871119910

12057911199041205791)


+ 1198982[minus (minus119871

1119909

1205792

1

1198881205791+ 1198711119910

1205792

1

1199041205791

minus1198872119909

1205792

12

11988812057912+ 1198872119910

1205792

12

11990412057912)



1205792

1

1199041205791minus 1198711119910

1205792

1

1198881205791minus 1198872119909

1205792

12

11990412057912

minus1198872119910

1205792

12

11988812057912) (11987111199091198881205791minus 11987111199101199041205791)]

+ 1198983[minus (minus119871

1119909

1205792

1

1198881205791+ 1198711119910

1205792

1

1199041205791minus1198712119909

1205792

12

11988812057912

+1198712119910

1205792

12

11990412057912minus1198873119909

1205792

123

119888120579123+ 1198873119910

1205792

123

119904120579123)



1205792

1

1199041205791minus 1198711119910

1205792

1

1198881205791minus 1198712119909

1205792

12

11990412057912


minus1198712119910

1205792

12

11988812057912minus 1198873119909

1205792

123

119904120579123minus 1198873119910

1205792

123

119888120579123)

times (11987111199091198881205791minus 11987111199101199041205791)]

minus (1198981119892 (11988711199091198881205791minus 11988711199101199041205791) + (119898

2+ 1198983) 119892


minus 1198881

1205791+ 1198882(12057912minus1205791)


+ (11987111199091198881205791minus 11987111199101199041205791) (119887211990911988812057912minus 119887211991011990412057912)]


+ (11987111199091198881205791minus 11987111199101199041205791) (119871211990911988812057912minus 119871211991011990412057912)]


+ (+119887211990911988812057912minus 119887211991011990412057912) (119887211990911988812057912minus 119887211991011990412057912)]


+ (+119871211990911988812057912) (119871211990911988812057912minus 119871211991011990412057912)] + 119868

2


+ (+1198873119909119888120579123minus 1198873119910119904120579123) (119871211990911988812057912minus 119871211991011990412057912)]

1198614= 1198982[minus (minus119871

1119909

1205792

1

1198881205791+ 1198711119910

1205792

1

1199041205791minus1198872119909

1205792

12

11988812057912

+1198872119910

1205792

12

11990412057912) (minus119887211990911990412057912minus 119887211991011988812057912)


1205792

1

1199041205791minus 1198711119910

1205792

1

1198881205791minus 1198872119909

1205792

12

11990412057912

minus1198872119910

1205792

12

11988812057912) (119887211990911988812057912minus 119887211991011990412057912)]

+ 1198983[minus (minus119871

1119909

1205792

1

1198881205791+ 1198711119910

1205792

1

1199041205791minus1198712119909

1205792

12

11988812057912

+ 1198712119910

1205792

12

11990412057912minus1198873119909

1205792

123

119888120579123

+1198873119910

1205792

123

119904120579123) (minus119871211990911990412057912minus 119871211991011988812057912)


1205792

1

1199041205791minus 1198711119910

1205792

1

1198881205791minus 1198712119909

1205792

12

11990412057912

minus 1198712119910

1205792

12

11988812057912minus 1198873119909

1205792

123

119904120579123

minus1198873119910

1205792

123

119888120579123) (119871211990911988812057912minus 119871211991011990412057912)]

minus (1198982119892 (119887211990911988812057912minus 119887211991011990412057912)

+ 1198983119892 (119871211990911988812057912minus 119871211991011990412057912)




+ (11987111199091198881205791minus 11987111199101199041205791) (1198873119909119888120579123minus 1198873119910119904120579123)]




+ (+1198873119909119888120579123minus 1198873119910119904120579123)

times (1198873119909119888120579123minus 1198873119910119904120579123)] + 119868

3

1198624= 1198983[minus (minus119871

1119909

1205792

1

1198881205791+ 1198711119910

1205792

1

1199041205791minus1198712119909

1205792

12

11988812057912

+ 1198712119910

1205792

12

11990412057912minus1198873119909

1205792

123

119888120579123

+1198873119910

1205792

123

119904120579123) (minus1198873119909119904120579123minus 1198873119910119888120579123)


1205792

1

1199041205791minus 1198711119910

1205792

1

1198881205791minus 1198712119909

1205792

12

11990412057912

minus 1198712119910

1205791211990412057912minus 1198712119910

1205792

12

11988812057912+ 1198873119909

120579123119888120579123

minus1198873119909

1205792

123

119904120579123minus 1198873119910

1205792

123

119888120579123)

times (1198873119909119888120579123minus 1198873119910119904120579123)]



(B1)

References
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










minus 119904120579MCP119886119887119886119889119904120579MCP119891119890


11987611989423= minus 119888120579MCP119886119887119886119889

11987611989424= 119904120579MCP119886119887119886119889119888120579MCP119891119890



+ 1198711119894+ 1198712119894119888120579PIP119891119890] minus 119904120579MCP119886119887119886119889119904120579MCP119891119890



+ 1198712119894119904120579PIP119891119890] + (minus119888120579MCP119886119887119886119889) 119871 119905119894119911

11987611989431= (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119888120579MCP119891119890

+ 119888120579CMC119891119890119904120579MCP119891119890


+ minus (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119904120579MCP119891119890

+ 119888120579CMC119891119890119888120579MCP119891119890


11987611989432= (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119888120579MCP119891119890

+ 119888120579CMC119891119890119904120579MCP119891119890


+ minus (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119904120579MCP119891119890

+ 119888120579CMC119891119890119888120579MCP119891119890


11987611989433= 119904120579CMC119891119890119904120579MCP119886119887119886119889

11987611989434= (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119888120579MCP119891119890

+ 119888120579119862119872119862119891119890

119904120579119872119862119875119891119890



+ 1198711119894+ 1198712119894119888120579PIP119891119890]

+ minus (119904120579CMC119891119890119888120579MCP119886119887119886119889) 119904120579MCP119891119890

+ 119888120579CMC119891119890119888120579MCP119891119890



+ 1198712119894119904120579PIP119891119890]

+ (119904120579CMC119891119890119904120579MCP119886119887119886119889) 119871 119905119894119911 + 1198710119894119904120579CMC119891119890

(A1)


119876011= [119888120579TMC119886119887119886119889119888120579TMC119891119890]


+ [minus119888120579TMC119886119887119886119889119904120579TMC119891119890]


119876012= [119888120579TMC119886119887119886119889119888120579TMC119891119890]


+ [minus119888120579TMC119886119887119886119889119904120579TMC119891119890]


119876013= 119904120579TMC119886119887119886119889

119876014= [119888120579TMC119886119887119886119889119888120579TMC119891119890]

times [1198711199050 119909


+ 1198711199050 119910


+ 11987110119888120579MCP119891119890 + 11987100]

+ [minus119888120579TMC119886119887119886119889119904120579TMC119891119890]

times [1198711199050 119909

(119904120579MCP119891119890119888120579IP119891119890 + 119888120579MCP119891119890119904120579IP119891119890)

+ 1198711199050 119910


+ 11987110119904120579MCP119891119890] + 119871 1199050 119911119904120579TMC119886119887119886119889


119876021= [119904120579TMC119886119887119886119889119888120579TMC119891119890]


+ [minus119904120579TMC119886119887119886119889119904120579TMC119891119890]


119876022= [119904120579TMC119886119887119886119889119888120579TMC119891119890]


+ [minus119904120579TMC119886119887119886119889119904120579TMC119891119890]


119876023= minus 119888120579TMC119886119887119886119889

119876024= [119904120579TMC119886119887119886119889119888120579TMC119891119890]

times [1198711199050 119909


+ 1198711199050 119910


+ 11987110119888120579MCP119891119890 + 11987100]

+ [minus119904120579TMC119886119887119886119889119904120579TMC119891119890]

times [1198711199050 119909

(119904120579MCP119891119890119888120579IP119891119890 + 119888120579MCP119891119890119904120579IP119891119890)

+ 1198711199050 119910


+ 11987110119904120579MCP119891119890] minus 119888120579TMC119886119887119886119889119871 1199050 119911


+ 119888120579TMC119891119890 (119904120579MCP119891119890119888120579IP119891119890 + 119888120579MCP119891119890119904120579IP119891119890)



119876033= 0

119876034= 119904120579TMC119891119890

times [1198711199050 119909


+ 1198711199050 119910


+ 11987110119888120579MCP119891119890 + 11987100] + 119888120579TMC119891119890

times [1198711199050 119909

(119904120579MCP119891119890119888120579IP119891119890 + 119888120579MCP119891119890119904120579IP119891119890)

+ 1198711199050 119910


+11987110119904120579MCP119891119890]

(A2)


Consider


+ (11988711199091198881205791minus 11988711199101199041205791) (1198871119909c1205791minus 1198871119910s1205791)]


11199091199041205791minus 11987111199101198881205791)

+ (11987111199091198881205791minus 11987111199101199041205791) (11987111199091198881205791minus 11987111199101199041205791)]


+(11987111199091198881205791minus 11987111199101199041205791)times (119871

11199091198881205791minus 11987111199101199041205791)]+1198681


11199091199041205791minus 11987111199101198881205791)

+ (+119887211990911988812057912minus 119887211991011990412057912) (11987111199091198881205791minus 11987111199101199041205791)]


+ (+119871211990911988812057912minus 119871211991011990412057912) (11987111199091198881205791minus 11987111199101199041205791)]


+ (+1198873119909119888120579123minus 1198873119910119904120579123) (11987111199091198881205791minus 11987111199101199041205791)]

1198604= 1198981[(minus1198871119909

12057911199041205791minus 1198871119910

12057911198881205791) (minus1198871119909

1205791c1205791+ 1198871119910

12057911199041205791)


+ 1198982[minus (minus119871

1119909

1205792

1

1198881205791+ 1198711119910

1205792

1

1199041205791

minus1198872119909

1205792

12

11988812057912+ 1198872119910

1205792

12

11990412057912)



1205792

1

1199041205791minus 1198711119910

1205792

1

1198881205791minus 1198872119909

1205792

12

11990412057912

minus1198872119910

1205792

12

11988812057912) (11987111199091198881205791minus 11987111199101199041205791)]

+ 1198983[minus (minus119871

1119909

1205792

1

1198881205791+ 1198711119910

1205792

1

1199041205791minus1198712119909

1205792

12

11988812057912

+1198712119910

1205792

12

11990412057912minus1198873119909

1205792

123

119888120579123+ 1198873119910

1205792

123

119904120579123)



1205792

1

1199041205791minus 1198711119910

1205792

1

1198881205791minus 1198712119909

1205792

12

11990412057912


minus1198712119910

1205792

12

11988812057912minus 1198873119909

1205792

123

119904120579123minus 1198873119910

1205792

123

119888120579123)

times (11987111199091198881205791minus 11987111199101199041205791)]

minus (1198981119892 (11988711199091198881205791minus 11988711199101199041205791) + (119898

2+ 1198983) 119892


minus 1198881

1205791+ 1198882(12057912minus1205791)


+ (11987111199091198881205791minus 11987111199101199041205791) (119887211990911988812057912minus 119887211991011990412057912)]


+ (11987111199091198881205791minus 11987111199101199041205791) (119871211990911988812057912minus 119871211991011990412057912)]


+ (+119887211990911988812057912minus 119887211991011990412057912) (119887211990911988812057912minus 119887211991011990412057912)]


+ (+119871211990911988812057912) (119871211990911988812057912minus 119871211991011990412057912)] + 119868

2


+ (+1198873119909119888120579123minus 1198873119910119904120579123) (119871211990911988812057912minus 119871211991011990412057912)]

1198614= 1198982[minus (minus119871

1119909

1205792

1

1198881205791+ 1198711119910

1205792

1

1199041205791minus1198872119909

1205792

12

11988812057912

+1198872119910

1205792

12

11990412057912) (minus119887211990911990412057912minus 119887211991011988812057912)


1205792

1

1199041205791minus 1198711119910

1205792

1

1198881205791minus 1198872119909

1205792

12

11990412057912

minus1198872119910

1205792

12

11988812057912) (119887211990911988812057912minus 119887211991011990412057912)]

+ 1198983[minus (minus119871

1119909

1205792

1

1198881205791+ 1198711119910

1205792

1

1199041205791minus1198712119909

1205792

12

11988812057912

+ 1198712119910

1205792

12

11990412057912minus1198873119909

1205792

123

119888120579123

+1198873119910

1205792

123

119904120579123) (minus119871211990911990412057912minus 119871211991011988812057912)


1205792

1

1199041205791minus 1198711119910

1205792

1

1198881205791minus 1198712119909

1205792

12

11990412057912

minus 1198712119910

1205792

12

11988812057912minus 1198873119909

1205792

123

119904120579123

minus1198873119910

1205792

123

119888120579123) (119871211990911988812057912minus 119871211991011990412057912)]

minus (1198982119892 (119887211990911988812057912minus 119887211991011990412057912)

+ 1198983119892 (119871211990911988812057912minus 119871211991011990412057912)




+ (11987111199091198881205791minus 11987111199101199041205791) (1198873119909119888120579123minus 1198873119910119904120579123)]




+ (+1198873119909119888120579123minus 1198873119910119904120579123)

times (1198873119909119888120579123minus 1198873119910119904120579123)] + 119868

3

1198624= 1198983[minus (minus119871

1119909

1205792

1

1198881205791+ 1198711119910

1205792

1

1199041205791minus1198712119909

1205792

12

11988812057912

+ 1198712119910

1205792

12

11990412057912minus1198873119909

1205792

123

119888120579123

+1198873119910

1205792

123

119904120579123) (minus1198873119909119904120579123minus 1198873119910119888120579123)


1205792

1

1199041205791minus 1198711119910

1205792

1

1198881205791minus 1198712119909

1205792

12

11990412057912

minus 1198712119910

1205791211990412057912minus 1198712119910

1205792

12

11988812057912+ 1198873119909

120579123119888120579123

minus1198873119909

1205792

123

119904120579123minus 1198873119910

1205792

123

119888120579123)

times (1198873119909119888120579123minus 1198873119910119904120579123)]



(B1)

References
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










119876021= [119904120579TMC119886119887119886119889119888120579TMC119891119890]


+ [minus119904120579TMC119886119887119886119889119904120579TMC119891119890]


119876022= [119904120579TMC119886119887119886119889119888120579TMC119891119890]


+ [minus119904120579TMC119886119887119886119889119904120579TMC119891119890]


119876023= minus 119888120579TMC119886119887119886119889

119876024= [119904120579TMC119886119887119886119889119888120579TMC119891119890]

times [1198711199050 119909


+ 1198711199050 119910


+ 11987110119888120579MCP119891119890 + 11987100]

+ [minus119904120579TMC119886119887119886119889119904120579TMC119891119890]

times [1198711199050 119909

(119904120579MCP119891119890119888120579IP119891119890 + 119888120579MCP119891119890119904120579IP119891119890)

+ 1198711199050 119910


+ 11987110119904120579MCP119891119890] minus 119888120579TMC119886119887119886119889119871 1199050 119911


+ 119888120579TMC119891119890 (119904120579MCP119891119890119888120579IP119891119890 + 119888120579MCP119891119890119904120579IP119891119890)



119876033= 0

119876034= 119904120579TMC119891119890

times [1198711199050 119909


+ 1198711199050 119910


+ 11987110119888120579MCP119891119890 + 11987100] + 119888120579TMC119891119890

times [1198711199050 119909

(119904120579MCP119891119890119888120579IP119891119890 + 119888120579MCP119891119890119904120579IP119891119890)

+ 1198711199050 119910


+11987110119904120579MCP119891119890]

(A2)


Consider


+ (11988711199091198881205791minus 11988711199101199041205791) (1198871119909c1205791minus 1198871119910s1205791)]


11199091199041205791minus 11987111199101198881205791)

+ (11987111199091198881205791minus 11987111199101199041205791) (11987111199091198881205791minus 11987111199101199041205791)]


+(11987111199091198881205791minus 11987111199101199041205791)times (119871

11199091198881205791minus 11987111199101199041205791)]+1198681


11199091199041205791minus 11987111199101198881205791)

+ (+119887211990911988812057912minus 119887211991011990412057912) (11987111199091198881205791minus 11987111199101199041205791)]


+ (+119871211990911988812057912minus 119871211991011990412057912) (11987111199091198881205791minus 11987111199101199041205791)]


+ (+1198873119909119888120579123minus 1198873119910119904120579123) (11987111199091198881205791minus 11987111199101199041205791)]

1198604= 1198981[(minus1198871119909

12057911199041205791minus 1198871119910

12057911198881205791) (minus1198871119909

1205791c1205791+ 1198871119910

12057911199041205791)


+ 1198982[minus (minus119871

1119909

1205792

1

1198881205791+ 1198711119910

1205792

1

1199041205791

minus1198872119909

1205792

12

11988812057912+ 1198872119910

1205792

12

11990412057912)



1205792

1

1199041205791minus 1198711119910

1205792

1

1198881205791minus 1198872119909

1205792

12

11990412057912

minus1198872119910

1205792

12

11988812057912) (11987111199091198881205791minus 11987111199101199041205791)]

+ 1198983[minus (minus119871

1119909

1205792

1

1198881205791+ 1198711119910

1205792

1

1199041205791minus1198712119909

1205792

12

11988812057912

+1198712119910

1205792

12

11990412057912minus1198873119909

1205792

123

119888120579123+ 1198873119910

1205792

123

119904120579123)



1205792

1

1199041205791minus 1198711119910

1205792

1

1198881205791minus 1198712119909

1205792

12

11990412057912


minus1198712119910

1205792

12

11988812057912minus 1198873119909

1205792

123

119904120579123minus 1198873119910

1205792

123

119888120579123)

times (11987111199091198881205791minus 11987111199101199041205791)]

minus (1198981119892 (11988711199091198881205791minus 11988711199101199041205791) + (119898

2+ 1198983) 119892


minus 1198881

1205791+ 1198882(12057912minus1205791)


+ (11987111199091198881205791minus 11987111199101199041205791) (119887211990911988812057912minus 119887211991011990412057912)]


+ (11987111199091198881205791minus 11987111199101199041205791) (119871211990911988812057912minus 119871211991011990412057912)]


+ (+119887211990911988812057912minus 119887211991011990412057912) (119887211990911988812057912minus 119887211991011990412057912)]


+ (+119871211990911988812057912) (119871211990911988812057912minus 119871211991011990412057912)] + 119868

2


+ (+1198873119909119888120579123minus 1198873119910119904120579123) (119871211990911988812057912minus 119871211991011990412057912)]

1198614= 1198982[minus (minus119871

1119909

1205792

1

1198881205791+ 1198711119910

1205792

1

1199041205791minus1198872119909

1205792

12

11988812057912

+1198872119910

1205792

12

11990412057912) (minus119887211990911990412057912minus 119887211991011988812057912)


1205792

1

1199041205791minus 1198711119910

1205792

1

1198881205791minus 1198872119909

1205792

12

11990412057912

minus1198872119910

1205792

12

11988812057912) (119887211990911988812057912minus 119887211991011990412057912)]

+ 1198983[minus (minus119871

1119909

1205792

1

1198881205791+ 1198711119910

1205792

1

1199041205791minus1198712119909

1205792

12

11988812057912

+ 1198712119910

1205792

12

11990412057912minus1198873119909

1205792

123

119888120579123

+1198873119910

1205792

123

119904120579123) (minus119871211990911990412057912minus 119871211991011988812057912)


1205792

1

1199041205791minus 1198711119910

1205792

1

1198881205791minus 1198712119909

1205792

12

11990412057912

minus 1198712119910

1205792

12

11988812057912minus 1198873119909

1205792

123

119904120579123

minus1198873119910

1205792

123

119888120579123) (119871211990911988812057912minus 119871211991011990412057912)]

minus (1198982119892 (119887211990911988812057912minus 119887211991011990412057912)

+ 1198983119892 (119871211990911988812057912minus 119871211991011990412057912)




+ (11987111199091198881205791minus 11987111199101199041205791) (1198873119909119888120579123minus 1198873119910119904120579123)]




+ (+1198873119909119888120579123minus 1198873119910119904120579123)

times (1198873119909119888120579123minus 1198873119910119904120579123)] + 119868

3

1198624= 1198983[minus (minus119871

1119909

1205792

1

1198881205791+ 1198711119910

1205792

1

1199041205791minus1198712119909

1205792

12

11988812057912

+ 1198712119910

1205792

12

11990412057912minus1198873119909

1205792

123

119888120579123

+1198873119910

1205792

123

119904120579123) (minus1198873119909119904120579123minus 1198873119910119888120579123)


1205792

1

1199041205791minus 1198711119910

1205792

1

1198881205791minus 1198712119909

1205792

12

11990412057912

minus 1198712119910

1205791211990412057912minus 1198712119910

1205792

12

11988812057912+ 1198873119909

120579123119888120579123

minus1198873119909

1205792

123

119904120579123minus 1198873119910

1205792

123

119888120579123)

times (1198873119909119888120579123minus 1198873119910119904120579123)]



(B1)

References
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










minus1198712119910

1205792

12

11988812057912minus 1198873119909

1205792

123

119904120579123minus 1198873119910

1205792

123

119888120579123)

times (11987111199091198881205791minus 11987111199101199041205791)]

minus (1198981119892 (11988711199091198881205791minus 11988711199101199041205791) + (119898

2+ 1198983) 119892


minus 1198881

1205791+ 1198882(12057912minus1205791)


+ (11987111199091198881205791minus 11987111199101199041205791) (119887211990911988812057912minus 119887211991011990412057912)]


+ (11987111199091198881205791minus 11987111199101199041205791) (119871211990911988812057912minus 119871211991011990412057912)]


+ (+119887211990911988812057912minus 119887211991011990412057912) (119887211990911988812057912minus 119887211991011990412057912)]


+ (+119871211990911988812057912) (119871211990911988812057912minus 119871211991011990412057912)] + 119868

2


+ (+1198873119909119888120579123minus 1198873119910119904120579123) (119871211990911988812057912minus 119871211991011990412057912)]

1198614= 1198982[minus (minus119871

1119909

1205792

1

1198881205791+ 1198711119910

1205792

1

1199041205791minus1198872119909

1205792

12

11988812057912

+1198872119910

1205792

12

11990412057912) (minus119887211990911990412057912minus 119887211991011988812057912)


1205792

1

1199041205791minus 1198711119910

1205792

1

1198881205791minus 1198872119909

1205792

12

11990412057912

minus1198872119910

1205792

12

11988812057912) (119887211990911988812057912minus 119887211991011990412057912)]

+ 1198983[minus (minus119871

1119909

1205792

1

1198881205791+ 1198711119910

1205792

1

1199041205791minus1198712119909

1205792

12

11988812057912

+ 1198712119910

1205792

12

11990412057912minus1198873119909

1205792

123

119888120579123

+1198873119910

1205792

123

119904120579123) (minus119871211990911990412057912minus 119871211991011988812057912)


1205792

1

1199041205791minus 1198711119910

1205792

1

1198881205791minus 1198712119909

1205792

12

11990412057912

minus 1198712119910

1205792

12

11988812057912minus 1198873119909

1205792

123

119904120579123

minus1198873119910

1205792

123

119888120579123) (119871211990911988812057912minus 119871211991011990412057912)]

minus (1198982119892 (119887211990911988812057912minus 119887211991011990412057912)

+ 1198983119892 (119871211990911988812057912minus 119871211991011990412057912)




+ (11987111199091198881205791minus 11987111199101199041205791) (1198873119909119888120579123minus 1198873119910119904120579123)]




+ (+1198873119909119888120579123minus 1198873119910119904120579123)

times (1198873119909119888120579123minus 1198873119910119904120579123)] + 119868

3

1198624= 1198983[minus (minus119871

1119909

1205792

1

1198881205791+ 1198711119910

1205792

1

1199041205791minus1198712119909

1205792

12

11988812057912

+ 1198712119910

1205792

12

11990412057912minus1198873119909

1205792

123

119888120579123

+1198873119910

1205792

123

119904120579123) (minus1198873119909119904120579123minus 1198873119910119888120579123)


1205792

1

1199041205791minus 1198711119910

1205792

1

1198881205791minus 1198712119909

1205792

12

11990412057912

minus 1198712119910

1205791211990412057912minus 1198712119910

1205792

12

11988812057912+ 1198873119909

120579123119888120579123

minus1198873119909

1205792

123

119904120579123minus 1198873119910

1205792

123

119888120579123)

times (1198873119909119888120579123minus 1198873119910119904120579123)]



(B1)

References
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









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