1 Copyright © 1997 by ASME

Proceedings of DETC ‘971997 ASME Design Technical Conferences

September 14-17, 1997, Sacramento, California

DETC97/DAC-3851

INVERSE KINEMATICS FOR PLANAR PARALLEL MANIPULATORS

Robert L. Williams IIBrett H. Shelley

Department of MechanicalEngineering

Ohio UniversityAthens, Ohio

ABSTRACT

This paper presents algebraic inverse position and velocitykinematics solutions for a broad class of three degree-of-freedomplanar in-parallel-actuated manipulators. Given an end-effector poseand rate, all active and passive joint values and rates are calculatedindependently for each serial chain connecting the ground link to theend-effector link. The solutions are independent of joint actuation.Seven serial chains consisting of revolute and prismatic joints areidentified and their inverse solutions presented. To reducecomputations, inverse Jacobian matrices for overall manipulators arederived to give only actuated joint rates. This matrix yields conditionsfor invalid actuation schemes. Simulation examples are given.

INTRODUCTION

Parallel manipulators are robots that consist of separateserial chains that connect the fixed link to the end-effector link. Thefollowing are potential advantages over serial robots: better stiffnessand accuracy, lighter weight, greater load bearing, higher velocitiesand accelerations, and less powerful actuators. A major drawback ofthe parallel robot is reduced workspace.

Parallel robotic devices were proposed over 17 years ago(MacCallion and Pham, 1979). Some configurations have been builtand controlled (e.g. Sumpter and Soni, 1985). Numerous worksanalyze kinematics, dynamics, workspace and control of parallelmanipulators (see Williams, 1988 and references therein). Hunt(1983) conducted preliminary studies of various parallel robotconfigurations. Cox and Tesar (1981) compared the relative merits ofserial and parallel robots.

These past works have focused on only a few differentarchitectures. For example, Aradyfio and Qiao (1985) examine theinverse kinematics solutions for three different 3-dof planar parallelrobots. Williams and Reinholtz (1988a and 1988b) study dynamicsand workspace for a limited number of parallel manipulators.Shirkhodaie and Soni (1987) , Gosselin and Angeles (1988), andPennock and Kassner (1990) each present a kinematic study of one

planar parallel robot. Gosselin et.al. (1996) present the position,workspace, and velocity kinematics of one planar parallel robot.

Recently, more general approaches have been presented.Daniali et.al. (1995) present an in-depth study of actuation schemes,velocity relationships, and singular conditions for general planarparallel robots. Gosselin (1996) presents general parallel computationalgorithms for kinematics and dynamics of planar and spatial parallelrobots. Merlet (1996) solved the forward position kinematics problemfor a class of planar parallel robots (see Fig. 1).

X 0 C 3

x,y

φ

A 1B 1

C 1

A 2B 2

C 2

A 3

B 3Y 0

Figure 1. General Class of Manipulators

The current paper presents a purely algebraic approach tosolve inverse position and velocity kinematics for Merlet’s class of 3-dof planar parallel manipulators. Geometric approaches have beenapplied with success in the past for specific architectures (e.g.Williams and Reinholtz, 1988b and Gosselin and Angeles, 1988). Thealgebraic methods in this paper are suitable when one desireskinematic solutions for control of all members in a broad class ofplanar parallel manipulators. For instance, the current paper wouldenable implementation of modular reconfigurable 3-dof planar parallelmanipulators.

This paper is organized as follows. First, the class ofmanipulators is discussed. Next the general inverse position

2 Copyright © 1997 by ASME

kinematics solutions are presented, in terms of independent solutionsfor all possible serial chains. The general inverse velocity solutions arepresented, singularity conditions are presented for each chain, and theoverall inverse Jacobian is derived. Determination of invalid actuationschemes is presented. Simulation examples are given for position andvelocity.

CLASS OF MANIPULATORS

Overall ManipulatorThe class of manipulators in this paper are planar and

actuated in parallel. Three serial chains with 3-dof each connect thefixed link to the end-effector link. One joint per chain is actuated andthe remaining two are passive. The active and/or passive joints may berevolute (R) and/or prismatic (P) (see Fig. 1). If serial chain i is RPR,there is a revolute joint at Ai, one of the two binary links is variable,the angle at Bi is fixed, and there is a revolute joint at Ci.

Serial ChainsWhen solving the forward position kinematics problem for

this robot class the actuated joint in each serial chain must bespecified. Merlet, (1996) identifies eighteen possible serial chains,given in Table 1. Underlining indicates actuation. Any one of thethree joints in a chain may be actuated unless the remaining passivejoints are PP. Therefore, the combinations RPP, PRP, and PPR areomitted from Table I.

RRR RRR RRR RRP RRP RRPRPR RPR RPR RPP RPP PRRPRR PRR PRP PRP PPR PPR

Table 1. Actuated Serial Chain Combinations

The PPP chain is not used because only two planar P jointsin a chain are independent. For the inverse position problem of thecurrent paper, the actuation scheme does not affect the solutions. Table1 reduces to seven chains, given in Table 2 and Fig. 2.

RRR RRP RPR RPP PRR PRP PPR

Table 2. Serial Chains for Inverse Kinematics

RR

R

RR

P

R

R

P

R P

P

P R

R

P RP

R

PP

Figure 2. Seven Serial Chains

For an overall manipulator, there are three independent serialchains, each having seven possible configurations. There are 343 (73)robots (the number is considerably lower if the ordering of distinctchains is not considered important). A general method is required tosolve inverse kinematics for all.

INVERSE POSITION KINEMATICS SOLUTIONS

The inverse position kinematics problem is stated: Given

the end-effector pose { }x yTφ , calculate the three actuated joint

(R or P) values. The passive joint values may also be determined foruse in velocity, acceleration, dynamics. The inverse position problemmay be solved for each serial chain independently. The solution is notdependent on joint actuation.

The kinematic diagram for the ith chain is Fig. 3, for allseven chains in Fig. 2. Equation 1 is the position vector equation forand Eq. 2 is the angle relationship (ψ i are defined in Fig. 4).

θ3i

θ2i

θ1i

E (x,y)

C i

A i

φ

L 1i

L 2i

L 3i

B i

Figure 3. Kinematic Diagram for all Seven Serial Chains

( ) ( )Ex

yA L e L e L ei i

ji

ji

ji i i i i i=

= + + ++ + +1 2 3

1 1 2 1 2 3θ θ θ θ θ θ(1)

θ θ θ π φ ψ1 2 3i i i i+ + + = + (2)

ψ2

C 3

Y

X

C 1

C 2

ψ3

ψ1

Figure 4. End-Effector Triangle Geometry

The inverse kinematics problem requires three independentinverse solutions, one for each serial chain. The coupledtranscendental Eqs. 1 and 2 must be solved three times, i=1,2,3. Ifjoint k on chain i is revolute, θki is variable; if joint k on chain i is

prismatic, Lki is variable. The remaining terms are fixed.

The given manipulator pose fixes the position andorientation of the end-effector triangle in the plane. Each Ci is:

( )( )

C x L

C y L

ix i i

iy i i

= + +

= + +3

3

cos

sin

φ ψ

φ ψ (3)

3 Copyright © 1997 by ASME

Making use of Eq. 3 and expanding Eq. 1 to x and y components:

A L c L c C

A L s L s Cix i i ix

iy i i iy

+ + =+ + =

1 1 2 12

1 1 2 12(4)

where ( )c i i12 1 2= +cosθ θ and ( )s i i12 1 2= +sin θ θ . Equations 4 and

2 are three equations in three unknowns to solve for each of the threeserial chains in a given manipulator. Inverse position solutions aregiven below for the seven serial chains.

RRR Chain Given { }x yTφ , determine θ θ θ1 2 3i i i, , .

Isolating θ θ1 2i i+ terms in Eqs. 4, squaring and adding yields Eq. 5 in

θ1i . Equation 6 is solved for θ1i using the tangent half-angle

substitution (Mabie and Reinholtz, 1987); the result is Eq. 6. Thereare two solutions (elbow-up and elbow-down). For each θ1i , a unique

θ2i is found by a y to x ratio of Eqs. 4, given in Eq. 7. For each

(θ1i ,θ2i ), a unique θ3i is found from Eq. 2.

Ec Fs G1 1 0+ + =( )( )

( ) ( )

E C A L

F C A L

G L L C A C A

ix ix i

iy iy i

i i ix ix iy iy

= −

= −

= − − − − −

2

2

1

1

22

12 2 2

(5)

θ11

2 2 2

1 22i

F E F G

G E,tan=

− ± + −−

− (6)

( )θ θ2 1 1 1 1 11 2 1 2 1 2 1 22i iy iy i ix ix i ia C A L s C A L c

, , , ,tan ,= − − − − − (7)

RRP Chain Given { }x yTφ , determine θ θ1 2 2i i iL, , .

Equation 2 is rewritten θ ζ θ2 1i i i= − where ζ i is the constant

φ ψ θ π+ − −i i3 . Eqs. 4 are simplified to eliminate θ1i . L i2 is

isolated in the x equation (Eq. 8) and substituted into the y equation togive the Eq. 5 form, where E, F, and G are Eq. 9. Two values for θ1i

are obtained using Eq. 6. θ2i and L i2 may now be evaluated from

θ ζ θ2 1i i i= − and Eq. 8. The two solutions correspond to a shorter

and longer L i2 joint length.

LC A L c

ci

ix ix i

i2

1 1= − −ζ

(8)

( )

E L s

F L c

G C A c A C

i i

i i

iy iy i ix ix

== −

= − + −

1

1

ζζ

ζ

(9)

RPR Chain Given { }x yTφ , determine θ θ1 1 3i i iL, , . The

Ai terms of Eqs. 4 are brought to the RHS, sum of angles formulas are

used, and L i1 is isolated in the x equation (Eq. 10). L i1 is substituted

into the y equation and greatly simplified to the Eq. 5 form, where E,F, and G are Eq. 11. Two θ1i are found from Eq. 6. L i1 may now be

evaluated from Eq. 10, and θ3i from Eq. 2. There are two solutions

but one has negative L i1 .

( )L

C A L c c s s

ci

ix ix i1

2 1 2 1 2

1=

− − −(10)

E C A

F A C

G L s

iy iy

ix ix

i

= −

= −= − 2 2

(11)

RPP Chain Given { }x yTφ , determine θ1 1 2i i iL L, , . A

unique θ1i is found from Eq. 2. Eqs. 4 are then two linear equations

in the unknowns L Li i1 2, with a unique solution.

( ) ( )L

C A s C A c

siix ix iy iy

112 12

2=

− − −(12)

( ) ( )L

C A s C A c

siix ix iy iy

21 1

2=

− − + −(13)

PRR Chain Given { }x yTφ , determine L i i i1 2 3, ,θ θ . The

solution starts with Eq. 11 (and Eq. 6) of the RPR. In this case thevariable to be solved is θ2i , not the constant θ1i . Two solutions exist,

a near (Eq. 21) and far ( π θ− 2i ) solution. L i1 and θ3i are solved

using Eqs. 10 and 2.

( ) ( )C A s L s C A cix ix i iy iy− + = −1 2 2 1 (14)

( ) ( )θ2

1 1 1

2i

iy iy ix ix

i

C A c C A s

L=

− − −

−sin (15)

PRP Chain Given { }x yTφ , determine L Li i i1 2 2, ,θ . A

unique θ2i is found from Eq. 2. The remaining unknowns L i1 and

L i2 are found from Eqs. 12 and 13.

PPR Chain Given { }x yTφ , determine L Li i i1 2 3, ,θ . θ1i and

θ2i are constant so the unknowns L i1 and L i2 are found from Eqs.

12 and 13. A unique θ3i is solved with Eq. 2.

Overall ManipulatorFor the inverse position kinematics problem for all 343

robots, the three serial chains are solved independently. Given the

commanded pose { }x yTφ , Eq. 3 is used. Then the unknown

joint parameters for each serial chain are solved by choosing theproper algorithm from above. The number of inverse position

4 Copyright © 1997 by ASME

solutions is the product of the number of solutions (1 or 2) in eachserial chain.

INVERSE VELOCITY KINEMATICS SOLUTIONS

The inverse velocity kinematics problem is stated: Given the

end-effector rate command { }�

� �X x y zT= ω and all position

variables, calculate the three actuated joint (R or P) rates.Additionally, the passive joint rates may be determined for use inacceleration and dynamics. The inverse velocity problem may besolved for each serial chain independently. The solution is notdependent on actuation scheme.

Velocity relationships for each independent serial chain i areobtained from the first time derivatives of Eqs. 1 and 2 (where theindex i is dropped for notational convenience):

( ) ( ) ( )

( ) ( ) ( )

�

�

� � � � �

� � � �

x

yL je L e L je L e

L je L e

j j j j

j j

= + + + + +

+ + +

+ +

+ + + +

1 1 1 2 1 2 2

3 1 2 3 3

1 1 1 2 1 2

1 2 3 1 2 3

θ θ θ θ θ θ

θ θ θ θ θ θ

θ θ θ

θ θ θ

(16)

ω φ θ θ θz = = + +� � � �

1 2 3 (17)

Separating Eq. 16 into real and imaginary parts yields the x

and y velocities. Note that �L i3 0= because L3i is part of the rigid

triangle link. The general translational and rotational velocities for allseven serial chains are:

( ) ( )( ) ( )

�

� � � � � � � �

�

� � � � � � � �

� � �

x L s L c L s L c L s

y L c L s L c L s L c

z

= − + − + + − + +

= + + + + + + +

= + +

1 1 1 1 1 2 12 1 2 2 12 3 123 1 2 3

1 1 1 1 1 2 12 1 2 2 12 3 123 1 2 3

1 2 3

θ θ θ θ θ θ

θ θ θ θ θ θ

ω θ θ θ

(18)

The velocity relationship for each serial chain can be expressed:

�

�X Ji i= ρ (19)

where { }�

� �X x y zT= ω is the Cartesian end-effector velocity, Ji is

the ith serial chain 3x3 Jacobian matrix, and �ρi is the vector of joint

rates for the ith serial chain (one active and two passive, R and/or P).Since Eqs. 18 are written with respect to {0} coordinates, that is the

frame of expression for �X and Ji. Jacobian matrices for each of theseven chains are determined from Eqs. 18 by zeroing terms that do notapply to a specific chain. Jacobian matrices for each of the sevenserial chains are:

RRR

j j j

j j j11 12 13

21 22 23

1 1 1

RRPj j c

j j s11 12 12

21 22 12

1 1 0

RPRj c j

j s j11 1 13

21 1 23

1 0 1

RPP

j c c

j s s11 1 12

21 1 12

1 0 0

PRRc j j

s j j1 12 13

1 22 23

0 1 1

PRPc j c

s j s1 12 12

1 22 12

0 1 0

PPR

c c j

s s j1 12 13

1 12 23

0 0 1

(Eqs. 20-26)

where: j L s L s L s11 1 1 2 12 3 123= − − − , j L s L s12 2 12 3 123= − − ,

j L s13 3 123= − , j L c L c L c21 1 1 2 12 3 123= + + , j L c L c22 2 12 3 123= + ,

j L c23 3 123= , ( )c123 1 2 3= + +cosθ θ θ , ( )s123 1 2 3= + +sin θ θ θ .

The inverse velocity solution for the ith serial chain is:

�

�ρi iJ X= −1 (27)

providing Ji has full rank. These inverse Jacobians are presentedsymbolically below for the seven serial chains.

RRR ( )( )

1

1 2 2

2 12 2 12 2 3 3

1 1 2 12 1 1 2 12 3 1 23 2 3

1 1 1 1 1 2 2 3 23

L L s

L c L s L L s

L c L c L s L s L L s L s

L c L s L L s L s

− − − − − ++

RRP

( )1

1 2

12 12 2 3 3

12 12 1 2 2 3 3

1 1 1 1 1 2 2 3 23

L c

s c L L c

s c L c L L c

L c L s L L s L s

− − −− + +

+

RPR

( )−+

−− − − − − +

− − − −

1

1 2 2

1 1 3 23

1 1 2 12 1 1 2 12 3 1 23 2 3

1 1 1 2 2 3 23

L L c

s c L c

L c L c L s L s L L s L s

s c L L c L c

RPP 1

0 0

2

2

12 12 1 2 2 3 3

1 1 1 2 2 3 23

s

s

s c L c L L c

s c L L c L c

− + +− − − −

PRR 1

2 2

2 12 2 12 2 3 3

1 1 3 23

1 1 2 2 3 23

L c

L c L s L L s

s c L c

s c L c L c

− −− +

PRP 1

0 02

12 12 2 3 3

2

1 1 2 2 3 23

s

s c L L c

s

s c L c L c

− +

− − −

PPR 1

0 02

12 12 3 3

1 1 3 23

2

s

s c L c

s c L c

s

−− −

(Eqs. 28-34)

The singularity conditions (cases where Ji has less than fullrank) for each chain are determined as follows. First it is assumed thatL1 0≠ and L2 0≠ ; these correspond to degenerate robot chains. The

RRR, RPP, PRP, and PPR serial chains are all singular when s2 0=

5 Copyright © 1997 by ASME

(θ π2 0i = , ). The RRP and PRR serial chains are singular when

c2 0= (θ π2 2i = ± / ). Note that in some cases the relative angle θ 2i

is fixed so it can be designed to avoid the singularity for all motion,where in other cases θ 2i is variable. The RPR serial chain is singular

when L L c1 2 2 0+ = .

Overall ManipulatorThe preceding section is sufficient to solve the inverse

velocity problem. Three appropriate inverse mappings (Eqs. 28-34)must be chosen, one for each independent serial chain. The Cartesian

rates { }�

� �X x y zT= ω are commanded to each chain; all joint rates,

active and passive, are solved.However, the above procedure yields six passive results

which are not required for resolved-rate control. Therefore, for moreefficient real-time computation, a method is developed in this section

to map �X into active joint rates only. This mapping is called theoverall manipulator inverse Jacobian matrix.

In this case, the actuated joints must be considered. The jth

row of the seven serial chain inverse Jacobian matrices (Eqs. 28-34)represents the mapping of the end-effector’s velocity to the jth joint.Therefore, for any robot configuration, its overall inverse Jacobian Mmay be constructed from the rows corresponding to the actuated jointsin each independent chain. The 3x3 matrix is built from actuated row jfor each serial chain i (j can be 1, 2, or 3 for i=1,2,3). Equation 35gives the general overall inverse Jacobian M:

( )( )( )

�

�

�

�

�

ρρρ ω

1

2

3

1 1

2 2

3 3

1

2

3

j

j

j

chain

chain

chain z

Row j

Row j

Row j

x

y

st

nd

rd

=

( � �Ρ = MX ) (35)

where �ρij i is the actuated joint rate (joint j i) for the ith serial chain and

( )Row ji i chainth is the jith row of the ith serial chain inverse Jacobian

matrix Ji−1 . (Note this method is equivalent to the rate kinematics in

Gosselin et.al. (1996) where A BX� �Ρ + = 0 ; the relationship is thus

M A B= − −1 .)For example, consider an RRP-PRP-RRR manipulator. In

this robot, the third, first, and third joints, respectively, are actuated forserial chains 1, 2, and 3 (j1=3, j2=1, j3=3). The manipulator overallinverse Jacobian matrix is given in Eq. 36:

( )( )

( )

�

�

�

/ / /

/ / /

/ / /

�

�

L

L

c c s c L s L s c

s s c s L L c s

c L s s L s L s L s L s

x

y

z

2

1

3

1 2 1 2 2 2 3 23 2

12 2 12 2 2 3 3 2

1 2 2 1 2 2 2 2 3 23 2 2

1

2

3

1 1 1 1 1 1 1 1 1

2 2 2 2 2 2 2 2

3 3 3 3 3 3 3 3θ ω

=

+

− ++

(36)

Invalid Actuation SchemesThe matrix M maps Cartesian end-effector velocity into

actuated joint velocities. If M is invertible, the forward velocitykinematics problem for the overall robot can be solved.

� �X M= −1Ρ (37)

The overall inverse Jacobian matrix M may not be invertiblebased on a manipulator’s actuation scheme. Certain manipulatorsconstructed of three identical chains will produce M matrices withcolumns of zero. For example, for a 3-RPP robot (where j1=j2= j3=1):

M =

0 0 1

0 0 1

0 0 1

(38)

Clearly, M has rank one and is singular. The physical interpretation ofthis mathematical singularity is that when the three actuated joints arelocked, the overall manipulator has an additional uncontrollablefreedom. Due to this problem, certain parallel robots have invalidactuation schemes. This singularity condition is not configurationdependent; the 3-RPP robot has an uncontrollable fourth degree-of-freedom for all motion. Table 3 shows the invalid actuation schemes.The actuated joint in each chain is underlined.

RPP PRP PPR

Table 3. Invalid Serial-Chain Actuation Schemes

Merlet (1996) states that “each joint of a chain may beactuated [provided] the chain obtained when locking the actuatedjoint is not of the PP type.” This rule, derived in the position domain,is verified in the velocity domain as shown in Table 3. Any parallelmanipulator constructed with permutations of the serial chains in Table3 has an invalid actuation scheme (not just for identical chains).

If a manipulator is not always singular, it is still subject tothe serial chain singularity conditions identified previously. For amore complete treatment of planar parallel manipulator singularities,see Daniali et. al. (1995), Sefriou and Gosselin (1995), and Merlet(1989).

SIMULATION EXAMPLES

An RPP-RRR-PRR robot is arbitrarily chosen to demonstratethe inverse position solutions. This robot has 4 solutions (1x2x2), asshown in Fig. 5. The RPP chain has a unique solution, the RRR chainhas elbow-up and elbow-down solutions, and the PRR chain near andfar.

0 5 10 15 20

0

2

4

6

8

10

12

14

16

18

20

Elbow-Up, Near Solution of RPP,RRR,PRR Robot

0 5 10 15 20

0

2

4

6

8

10

12

14

16

18

20

Elbow-Up, Far Solution of RPP,RRR,PRR Robot

0 5 10 15 20

0

2

4

6

8

10

12

14

16

18

20

Elbow-Down, Near Solution of RPP,RRR,PRR Robot

0 5 10 15 20

0

2

4

6

8

10

12

14

16

18

20

Elbow-Down, Far Solution of RPP,RRR,PRR Robot

Figure 5. Four Solutions for RPP-RRR-PRR Manipulator

6 Copyright © 1997 by ASME

An PRP-PRP-PRP robot is arbitrarily chosen to demonstratethe resolved rate simulation in Fig. 6.

0 5 10 15 20

0

2

4

6

8

10

12

14

16

18

20

X-direction

Y-di

rect

ion

Figure 6. Resolved-Rate Simulation of PRP-PRP-PRP

CONCLUSION

This paper has presented the general inverse position andvelocity kinematics solutions for a broad class of planar parallelrobots. The class has 3-dof, three serial chains of 3-dof eachconnecting to the end-effector triangle, one actuator per chain, and Rand P joints. Each serial chain is solved independently given the end-effector pose and rate. Since the actuation scheme does not affect theinverse solution, there are seven possible serial chains, for a total of343 distinct robots (less if the order of chains is unimportant).

The primary contributions of this paper are: 1) generalalgebraic approach, where one computer program solves all possibleinverse position and velocity problems; 2) all possible independent 3-dof planar serial chains with R and P joints are classified and theirinverse position and velocity problems solved in a unified and succinctmanner; 3) overall inverse mapping for actuated joint rates only; and4) identification of invalid actuation schemes, which agrees withpreviously published results. The results of this paper can be used forthe design, simulation, and control of any 3-dof planar parallel robot.

ACKNOWLEDGMENTS

The first author would like to acknowledge the reviewersfor their helpful comments.

REFERENCES

Aradyfio D.D. and Qiao D., 1985, "Kinematic Simulation of Novel RoboticMechanisms Having Closed Chains", ASME Paper 85-DET-81.

Cox D.J. and Tesar D., 1981, "The Dynamic Modeling and Command SignalFormulation for Parallel Multi-Parameter Robotic Devices", DOE Report.

Daniali H.R.M., Zsomber-Murray P.J.Z., and Angeles J., 1995, "SingularityAnalysis of Planar Parallel Manipulators", Mechanism and Machine Theory, Vol. 30, No. 5,pp. 665-678.

Gosselin C.M., 1996, "Parallel Computation Algorithms for the Kinematics andDynamics of Planar and Spatial Parallel Manipulators", Journal of Dynamic Systems,Measurement, and Control, Vol. 118, No. 1, pp. 22-28.

Gosselin C.M., Lemieux S. and Merlet J.-P., 1996, "A New Architecture ofPlanar Three-Degree-of-Freedom Parallel Manipulator", IEEE International Conference onRobotics and Automation, Minneapolis, MN, Vol 4, pp. 3738-3743.

Gosselin C.M. and Angeles J., 1988, "The Optimum Kinematic Design of aPlanar Three-Degree-of-Freedom Parallel Manipulator", ASME Journal of Mechanisms,Transmissions, and Automation in Design, Vol. 110, No. 1, pp. 35-41.

Hunt K.H., 1983, "Structural Kinematics of In-Parallel-Actuated Robot Arms",Journal of Mechanisms, Transmissions, and Automation in Design, Vol. 105, No. 4.

Mabie H.H. and Reinholtz C.F., 1987, Mechanisms and Dynamics ofMachinery, John Wiley & Sons, Inc., New York.

MacCallion H. and Pham D.T., 1979, "The Analysis of a Six-Degree-of-Freedom Workstation for Mechanized Assembly", 5th World Congress on TMM, Montreal.

Merlet J.-P., 1996, "Direct Kinematics of Planar Parallel Manipulators", IEEEInternational Conference on Robotics and Automation, Vol 4, pp. 3744-3749.

Merlet J.-P., 1989, "Singular Configurations of Parallel Manipulators andGrassman Geometry", International Journal of Robotics Research, Vol 8, No. 5, pp. 45-56.

Pennock G.R. and Kassner D.J., 1990, "Kinematic Analysis of a Planar Eight-Bar Linkage: Application to a Platform-type Robot", ASME Mechanisms Conference, DE-25, pp. 37-43.

Sefriou J. and Gosselin C.M., 1995, "On the Quadratic nature of the SingularityCurves of Planar Three-degree-of-freedom Parallel Manipulators", Mechanism and MachineTheory, Vol. 30, No. 4, pp. 533-551.

Shirkhodaie A.H. and Soni A.H., 1987, "Forward and Inverse Synthesis for aRobot with Three Degrees of Freedom", Summer Computer Simulation Conference,Montreal, pp. 851-856.

Sumpter B. and Soni A.H., 1985, "Simulation Algorithm of OklahomaCrawdad Robot", 9th Applied Mechanisms Conference, Kansas City, pp. VI.1-VI.3.

Williams II R.L., 1988, "Planar Robotic Mechanisms: Analysis andConfiguration Comparison", Ph.D. Dissertation, VPI&SU, Blacksburg, VA.

Williams II R.L. and Reinholtz C.F., 1988a, "Forward Dynamic Analysis andPower Requirement Comparison of Parallel Robotic Mechanisms", 20th Biennial ASMEMechanisms Conference, Kissimmee FL, DE Vol. 15-3, pp. 71-78.

Williams II R.L. and Reinholtz C.F., 1988b, "Closed-Form WorkspaceDetermination and Optimization for Parallel Robotic Mechanisms", 20th Biennial ASMEMechanisms Conference, Kissimmee FL, DE Vol. 15-3, pp. 341-351.