Inverse Kinematics for Planar Parallel Manipulators

Proceedings of DETC ‘97 1997 ASME Design Technical Conferences September 14-17, 1997, Sacramento, California

DETC97/DAC-3851

INVERSE KINEMATICS FOR PLANAR PARALLEL MANIPULATORS

Robert L. Williams II Brett H. Shelley Department of Mechanical Engineering Ohio University Athens, Ohio

ABSTRACT planar parallel robot. Gosselin et.al. (1996) present the position, workspace, and velocity kinematics of one planar parallel robot. This paper presents algebraic inverse position and velocity Recently, more general approaches have been presented. kinematics solutions for a broad class of three degree-of-freedom Daniali et.al. (1995) present an in-depth study of actuation schemes, planar in-parallel-actuated manipulators. Given an end-effector pose velocity relationships, and singular conditions for general planar and rate, all active and passive joint values and rates are calculated parallel robots. Gosselin (1996) presents general parallel computation independently for each serial chain connecting the ground link to the algorithms for kinematics and dynamics of planar and spatial parallel end-effector link. The solutions are independent of joint actuation. robots. Merlet (1996) solved the forward position kinematics problem Seven serial chains consisting of revolute and prismatic joints are for a class of planar parallel robots (see Fig. 1). identified and their inverse solutions presented. To reduce computations, inverse Jacobian matrices for overall manipulators are A derived to give only actuated joint rates. This matrix yields conditions 2 B for invalid actuation schemes. Simulation examples are given. 2 C 2 B 3 INTRODUCTION Y 0 x,y

X Parallel manipulators are robots that consist of separate 0 φ C serial chains that connect the fixed link to the end-effector link. The 3 C 1 following are potential advantages over serial robots: better stiffness A 3 and accuracy, lighter weight, greater load bearing, higher velocities A 1 and accelerations, and less powerful actuators. A major drawback of B 1 the parallel robot is reduced workspace. Parallel robotic devices were proposed over 17 years ago Figure 1. General Class of Manipulators (MacCallion and Pham, 1979). Some configurations have been built and controlled (e.g. Sumpter and Soni, 1985). Numerous works The current paper presents a purely algebraic approach to analyze kinematics, dynamics, workspace and control of parallel solve inverse position and velocity kinematics for Merlet’s class of 3- manipulators (see Williams, 1988 and references therein). Hunt dof planar parallel manipulators. Geometric approaches have been (1983) conducted preliminary studies of various parallel robot applied with success in the past for specific architectures (e.g. configurations. Cox and Tesar (1981) compared the relative merits of Williams and Reinholtz, 1988b and Gosselin and Angeles, 1988). The serial and parallel robots. algebraic methods in this paper are suitable when one desires These past works have focused on only a few different kinematic solutions for control of all members in a broad class of architectures. For example, Aradyfio and Qiao (1985) examine the planar parallel manipulators. For instance, the current paper would inverse kinematics solutions for three different 3-dof planar parallel enable implementation of modular reconfigurable 3-dof planar parallel robots. Williams and Reinholtz (1988a and 1988b) study dynamics manipulators. and workspace for a limited number of parallel manipulators. This paper is organized as follows. First, the class of Shirkhodaie and Soni (1987) , Gosselin and Angeles (1988), and manipulators is discussed. Next the general inverse position Pennock and Kassner (1990) each present a kinematic study of one

1 Copyright © 1997 by ASME kinematics solutions are presented, in terms of independent solutions for all possible serial chains. The general inverse velocity solutions are For an overall manipulator, there are three independent serial presented, singularity conditions are presented for each chain, and the chains, each having seven possible configurations. There are 343 (73) overall inverse Jacobian is derived. Determination of invalid actuation robots (the number is considerably lower if the ordering of distinct schemes is presented. Simulation examples are given for position and chains is not considered important). A general method is required to velocity. solve inverse kinematics for all.

CLASS OF MANIPULATORS INVERSE POSITION KINEMATICS SOLUTIONS Overall Manipulator The class of manipulators in this paper are planar and The inverse position kinematics problem is stated: Given T actuated in parallel. Three serial chains with 3-dof each connect the the end-effector pose {}xyφ , calculate the three actuated joint fixed link to the end-effector link. One joint per chain is actuated and the remaining two are passive. The active and/or passive joints may be (R or P) values. The passive joint values may also be determined for revolute (R) and/or prismatic (P) (see Fig. 1). If serial chain i is RPR, use in velocity, acceleration, dynamics. The inverse position problem may be solved for each serial chain independently. The solution is not there is a revolute joint at Ai, one of the two binary links is variable, dependent on joint actuation. the angle at Bi is fixed, and there is a revolute joint at Ci. The kinematic diagram for the ith chain is Fig. 3, for all Serial Chains seven chains in Fig. 2. Equation 1 is the position vector equation for ψ When solving the forward position kinematics problem for and Eq. 2 is the angle relationship ( i are defined in Fig. 4). this robot class the actuated joint in each serial chain must be specified. Merlet, (1996) identifies eighteen possible serial chains, E (x,y) given in Table 1. Underlining indicates actuation. Any one of the θ three joints in a chain may be actuated unless the remaining passive L 3i 3i joints are PP. Therefore, the combinations RPP, PRP, and PPR are φ omitted from Table I. L 2i C i L θ A 1i 2i RRR RRR RRR RRP RRP RRP i B i RPR RPR RPR RPP RPP PRR θ PRR PRR PRP PRP PPR PPR 1i Figure 3. Kinematic Diagram for all Seven Serial Chains Table 1. Actuated Serial Chain Combinations x θ j()θθ+++j ( θθθ ) ==+j1i +12ii + i 123 ii EALe12i Lei Le 3i The PPP chain is not used because only two planar P joints yi (1) in a chain are independent. For the inverse position problem of the θθθπ+++=+φψ current paper, the actuation scheme does not affect the solutions. Table 123iii i (2) 1 reduces to seven chains, given in Table 2 and Fig. 2. Y RRR RRP RPR RPP PRR PRP PPR C 2

Table 2. Serial Chains for Inverse Kinematics ψ ψ 1 2 X

R R P ψ R C 1 C R P 3 3 R R R Figure 4. End-Effector Triangle Geometry

The inverse kinematics problem requires three independent R inverse solutions, one for each serial chain. The coupled P transcendental Eqs. 1 and 2 must be solved three times, i=1,2,3. If P R θ R P joint k on chain i is revolute, ki 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 and R orientation of the end-effector triangle in the plane. Each Ci is: P P R P P =+φ +ψ CxLix3 icos( i ) (3) CyL=+sin(φ +ψ ) Figure 2. Seven Serial Chains iy3 i i

2 Copyright © 1997 by ASME into the y equation and greatly simplified to the Eq. 5 form, where E, θ Making use of Eq. 3 and expanding Eq. 1 to x and y components: F, and G are Eq. 11. Two 1i are found from Eq. 6. L1i may now be evaluated from Eq. 10, and θ from Eq. 2. There are two solutions ALcLcC++ = 3i ix11 i 212 i ix but one has negative L . ++ = (4) 1i ALsLsCiy11 i 212 i iy CALccss−−( −) =+θθ =+θθ =ix ix21212 i where c12cos() 1ii 2 and s12sin() 1ii 2 . Equations 4 and L1i (10) c1 2 are three equations in three unknowns to solve for each of the three serial chains in a given manipulator. Inverse position solutions are EC=− A given below for the seven serial chains. iy iy =− FAix C ix (11) φT θθ θ GLs=− RRR Chain Given {}xy , determine 123iii,,. 22i θθ+ Isolating 12ii terms in Eqs. 4, squaring and adding yields Eq. 5 in φT θ θ θ RPP Chain Given {}xy , determine 112ii,,LL i. A 1i . Equation 6 is solved for 1i using the tangent half-angle θ substitution (Mabie and Reinholtz, 1987); the result is Eq. 6. There unique 1i is found from Eq. 2. Eqs. 4 are then two linear equations θ are two solutions (elbow-up and elbow-down). For each 1i , a unique in the unknowns LL12ii, with a unique solution. θ 2i is found by a y to x ratio of Eqs. 4, given in Eq. 7. For each (θ ,θ ), a unique θ is found from Eq. 2. ()CAsCAc−−−() 1i 2i 3i = ix ix12 iy iy 12 L1i (12) s2 =− ECAL2(ix ix)1 i −−()CAsCAc +( −) Ec++= Fs G 0=− (5) =ix ix11 iy iy 11 FCAL2(iy iy)1 i L2i (13) s2 =2 −−−2 2 −−2 GL2i L1 i( C ix A ix)( C iy A iy ) φT θθ PRR Chain Given {}xy , determine L123iii,,. The  222 −−±FEFG + −  solution starts with Eq. 11 (and Eq. 6) of the RPR. In this case the θ = 2tan 1 (6) 1i12,   variable to be solved is θ , not the constant θ . Two solutions exist,  GE−  2i 1i πθ− θ a near (Eq. 21) and far ( 2i) solution. L1i and 3i are solved using Eqs. 10 and 2. θθ=−−−−−atan2( CALsCALc , ) (7) 211111i12,,,, iy iy i 12 ix ix i 12 i 12 −+=− (CAsLsCAcix ix) 122 i( iy iy ) 1 (14) φT θθ RRP Chain Given {}xy , determine 12ii,,L 2 i. θ =−ζ θ ζ −−− Equation 2 is rewritten 21 where is the constant −(CAcCAsiy iy) 11( ix ix ) ii i i θ =sin1  (15) φ +−ψ θπ − θ 2i  ii3. Eqs. 4 are simplified to eliminate 1i . L2i is  L2i  isolated in the x equation (Eq. 8) and substituted into the y equation to give the Eq. 5 form, where E, F, and G are Eq. 9. Two values for θ 1i {}φT θ θ PRP Chain Given xy , determine LL122iii,, . A are obtained using Eq. 6. 2i and L2i may now be evaluated from θ θ =−ζ θ unique 2i is found from Eq. 2. The remaining unknowns L1i and 21ii i and Eq. 8. The two solutions correspond to a shorter L2i are found from Eqs. 12 and 13. and longer L2i joint length.

T CALc−− PPR Chain Given {}xyφ , determine LL,,θ. θ and L = ix ix11 i (8) 123iii 1i 2i ζ θ ci 2i are constant so the unknowns L1i and L2i are found from Eqs. θ 12 and 13. A unique 3i is solved with Eq. 2. = ζ ELs1ii =− ζ FLc1ii (9) Overall Manipulator GC=−() AcAζ +− C For the inverse position kinematics problem for all 343 iy iy i ix ix robots, the three serial chains are solved independently. Given the {}φT T commanded pose xy , Eq. 3 is used. Then the unknown RPR Chain Given {}xyφ , determine θθ,,L . The 113iii joint parameters for each serial chain are solved by choosing the Ai terms of Eqs. 4 are brought to the RHS, sum of angles formulas are proper algorithm from above. The number of inverse position used, and L1i is isolated in the x equation (Eq. 10). L1i is substituted

3 Copyright © 1997 by ASME solutions is the product of the number of solutions (1 or 2) in each serial chain. cj c cc j 11212 11213 PRP sj12212 s PPR ss11223 j (Eqs. 20-26) 01 0 00 1 INVERSE VELOCITY KINEMATICS SOLUTIONS     =− − − =− − The inverse velocity kinematics problem is stated: Given the where: jLsLsLs11 1 1 2 12 3 123 , jLsLs12 2 12 3 123 , =− =+ + =+ =ωT jLs13 3 123 , jLcLcLc21 1 1 2 12 3 123 , jLcLc22 2 12 3 123 , end-effector rate command Xxy{}z and all position = =++()θθθ =++()θθθ variables, calculate the three actuated joint (R or P) rates. jLc23 3 123 , c123cos 1 2 3 , s123sin 1 2 3 . Additionally, the passive joint rates may be determined for use in acceleration and dynamics. The inverse velocity problem may be The inverse velocity solution for the ith serial chain is: solved for each serial chain independently. The solution is not dependent on actuation scheme. ρ =−1 iiJX (27) Velocity relationships for each independent serial chain i are obtained from the first time derivatives of Eqs. 1 and 2 (where the index i is dropped for notational convenience): providing Ji has full rank. These inverse Jacobians are presented symbolically below for the seven serial chains.  x jjθθ jj()θθ++ () θθ =++11θθθ 12 +++ 12 Lje1112 Le Lje( 122) Le   y (16) Lc2 12 Ls 2 12 LLs 2 3 3  1   jj()θθθ++ () θθθ ++ −− −− −( +) Lje123(θθθ++) + Le 123 RRR Lc1 1 Lc 2 12 Ls 1 1 L 2 s 12 L 3 Ls 1 23 L 2 s 3  31233 LLs 122 +  Lc11 Ls 11 L 1( L 22 s Ls 323)  ω ==φθθθ + + z 123 (17) −−− sc12 12 LLc 2 3 3 1   Separating Eq. 16 into real and imaginary parts yields the x RRP scLcLLc−++ Lc 12 12 1 2 2 3 3 = 12 + and y velocities. Note that L3i 0 because L3i is part of the rigid Lc11 Ls 11 L 1( L 22 s Ls 323) triangle link. The general translational and rotational velocities for all seven serial chains are: RPR − scLc11323 =−θθθθθθ + − + + − + + −1  xLsLcLs1 1 1 1 1 2 12( 1 2) LcLs 2 12 3 123( 1 2 3) −− −− − + Lc1 1 Lc 2 12 Ls 1 1 L 2 s 12 L 3() Ls 1 23 L 2 s 3  LLc+ =+++++θθθθθθ ++ 122−−−− yLc1 1 1 LsLc 1 1 2 12( 1 2) LsLc 2 12 3 123( 1 2 3) (18) scLLcLc11122323 ωθθθ=++ z 123 00 s   2  1 −++ RPP scLcLLc12 12 1 2 2 3 3  The velocity relationship for each serial chain can be expressed: s 2−−−− sc11 LLcLc 122323 = ρ XJii (19)

Lc2 12 Ls 2 12 LLs 2 3 3  1   =ωT PRR −−sc Lc where Xxy{}z is the Cartesian end-effector velocity, Ji is 11 323 Lc22 th scLcLc−+ the i serial chain 3x3 Jacobian matrix, and ρi is the vector of joint 1122323 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 sc12 12 LLc 2 3 3  1   frame of expression for X and Ji. Jacobian matrices for each of the PRP 00 s2  s seven chains are determined from Eqs. 18 by zeroing terms that do not 2−−− sc11 LcLc 22323 apply to a specific chain. Jacobian matrices for each of the seven serial chains are: − scLc12 12 3 3  1   PPR −−sc Lc (Eqs. 28-34) jjj11 12 13  jjc11 12 12  jcj11 1 13  11 323       s2 00 s RRR jjj21 22 23RRP jjs21 22 12 RPR jsj21 1 23  2 111 110 101 The singularity conditions (cases where Ji has less than full rank) for each chain are determined as follows. First it is assumed that     jcc11 1 12 cj11213 j ≠ ≠     L1 0 and L2 0 ; these correspond to degenerate robot chains. The RPP jss21 1 12  PRR sj12223 j = RRR, RPP, PRP, and PPR serial chains are all singular when s2 0 100 01 1

4 Copyright © 1997 by ASME θπ= ( 2i0,). The RRP and PRR serial chains are singular when The overall inverse Jacobian matrix M may not be invertible c = 0 (θπ=± /2). Note that in some cases the relative angle θ based on a manipulator’s actuation scheme. Certain manipulators 2 2i 2i constructed of three identical chains will produce M matrices with is fixed so it can be designed to avoid the singularity for all motion, columns of zero. For example, for a 3-RPP robot (where j =j = j =1): θ 1 2 3 where in other cases 2i is variable. The RPR serial chain is singular += when LLc1220. 001   M = 001 (38) Overall Manipulator 001 The 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 Clearly, M has rank one and is singular. The physical interpretation of this mathematical singularity is that when the three actuated joints are ={}ωT rates Xxyz are commanded to each chain; all joint rates, locked, the overall manipulator has an additional uncontrollable active and passive, are solved. freedom. Due to this problem, certain parallel robots have invalid However, the above procedure yields six passive results actuation schemes. This singularity condition is not configuration which are not required for resolved-rate control. Therefore, for more dependent; the 3-RPP robot has an uncontrollable fourth degree-of- efficient real-time computation, a method is developed in this section freedom for all motion. Table 3 shows the invalid actuation schemes. to map X into active joint rates only. This mapping is called the The actuated joint in each chain is underlined. overall manipulator inverse Jacobian matrix. In this case, the actuated joints must be considered. The jth RPP PRP PPR 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. Table 3. Invalid Serial-Chain Actuation Schemes Therefore, for any robot configuration, its overall inverse Jacobian M may be constructed from the rows corresponding to the actuated joints Merlet (1996) states that “each joint of a chain may be in each independent chain. The 3x3 matrix is built from actuated row j actuated [provided] the chain obtained when locking the actuated for each serial chain i (j can be 1, 2, or 3 for i=1,2,3). Equation 35 joint is not of the PP type.” This rule, derived in the position domain, gives the general overall inverse Jacobian M: is verified in the velocity domain as shown in Table 3. Any parallel manipulator constructed with permutations of the serial chains in Table     3 has an invalid actuation scheme (not just for identical chains). ρ Row( j ) st  x   1 j1   1 1 chain   If a manipulator is not always singular, it is still subject to ρ  = Row( j ) nd  y  ( Ρ=MX ) (35) 2 j2 2 2 chain the serial chain singularity conditions identified previously. For a      ρ Row( j ) rd ω  more complete treatment of planar parallel manipulator singularities,  3 j3   3 3 chain  z see Daniali et. al. (1995), Sefriou and Gosselin (1995), and Merlet (1989). where ρ is the actuated joint rate (joint j ) for the ith serial chain and iji i Row( j ) is the jth row of the ith serial chain inverse Jacobian SIMULATION EXAMPLES i ith chain i −1 matrix Ji . (Note this method is equivalent to the rate kinematics in An RPP-RRR-PRR robot is arbitrarily chosen to demonstrate Gosselin et.al. (1996) where ABXΡ+ =0 ; the relationship is thus the inverse position solutions. This robot has 4 solutions (1x2x2), as shown in Fig. 5. The RPP chain has a unique solution, the RRR chain =− −1 MAB.) has elbow-up and elbow-down solutions, and the PRR chain near and For example, consider an RRP-PRP-RRR manipulator. In far. this robot, the third, first, and third joints, respectively, are actuated for serial chains 1, 2, and 3 (j1=3, j2=1, j3=3). The manipulator overall Elbow-Up, Near Solution of RPP,RRR,PRR Robot Elbow-Up, Far Solution of RPP,RRR,PRR Robot 20 20 inverse Jacobian matrix is given in Eq. 36: 18 18 16 16

14 14  +  12 12 cc12// sc 12() LsLsc 223232 / 10 10 L2 11 11 11111x 1  8 8 =−+() 6 6 L1 ss12// 2 cs 12 2 LLcs 2 3 3 / 2 y (36) 4 4 2 22 22 2 222 θ ()+ω 2 2 3 cLssLsLsLsLs1221222232322// /z 0 0 3333333 33 0 5 10 15 20 0 5 10 15 20

Invalid Actuation Schemes Elbow-Down, Near Solution of RPP,RRR,PRR Robot Elbow-Down, Far Solution of RPP,RRR,PRR Robot 20 20 The matrix M maps Cartesian end-effector velocity into 18 18 16 16 actuated joint velocities. If M is invertible, the forward velocity 14 14 12 12 kinematics problem for the overall robot can be solved. 10 10

8 8

6 6 =−1Ρ 4 4 XM (37) 2 2

0 0 0 5 10 15 20 0 5 10 15 20 Figure 5. Four Solutions for RPP-RRR-PRR Manipulator

5 Copyright © 1997 by ASME ACKNOWLEDGMENTS An PRP-PRP-PRP robot is arbitrarily chosen to demonstrate the resolved rate simulation in Fig. 6. The first author would like to acknowledge the reviewers for their helpful comments. 20 18 16 REFERENCES 14 12 Aradyfio D.D. and Qiao D., 1985, "Kinematic Simulation of Novel Robotic 10 Mechanisms Having Closed Chains", ASME Paper 85-DET-81. Cox D.J. and Tesar D., 1981, "The Dynamic Modeling and Command Signal

Y-direction 8 Formulation for Parallel Multi-Parameter Robotic Devices", DOE Report. 6 Daniali H.R.M., Zsomber-Murray P.J.Z., and Angeles J., 1995, "Singularity Analysis of Planar Parallel Manipulators", Mechanism and Machine Theory, Vol. 30, No. 5, 4 pp. 665-678. 2 Gosselin C.M., 1996, "Parallel Computation Algorithms for the Kinematics and Dynamics of Planar and Spatial Parallel Manipulators", Journal of Dynamic Systems, 0 Measurement, and Control, Vol. 118, No. 1, pp. 22-28. 0 5 10 15 20 Gosselin C.M., Lemieux S. and Merlet J.-P., 1996, "A New Architecture of X-direction Planar Three-Degree-of-Freedom Parallel Manipulator", IEEE International Conference on Robotics and Automation, Minneapolis, MN, Vol 4, pp. 3738-3743. Gosselin C.M. and Angeles J., 1988, "The Optimum Kinematic Design of a Figure 6. Resolved-Rate Simulation of PRP-PRP-PRP Planar 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 of CONCLUSION Machinery, 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. This paper has presented the general inverse position and Merlet J.-P., 1996, "Direct Kinematics of Planar Parallel Manipulators", IEEE velocity kinematics solutions for a broad class of planar parallel International Conference on Robotics and Automation, Vol 4, pp. 3744-3749. robots. The class has 3-dof, three serial chains of 3-dof each Merlet J.-P., 1989, "Singular Configurations of Parallel Manipulators and Grassman Geometry", International Journal of Robotics Research, Vol 8, No. 5, pp. 45-56. connecting to the end-effector triangle, one actuator per chain, and R Pennock G.R. and Kassner D.J., 1990, "Kinematic Analysis of a Planar Eight- and P joints. Each serial chain is solved independently given the end- Bar Linkage: Application to a Platform-type Robot", ASME Mechanisms Conference, DE- effector pose and rate. Since the actuation scheme does not affect the 25, pp. 37-43. Sefriou J. and Gosselin C.M., 1995, "On the Quadratic nature of the Singularity inverse solution, there are seven possible serial chains, for a total of Curves of Planar Three-degree-of-freedom Parallel Manipulators", Mechanism and Machine 343 distinct robots (less if the order of chains is unimportant). Theory, Vol. 30, No. 4, pp. 533-551. The primary contributions of this paper are: 1) general Shirkhodaie A.H. and Soni A.H., 1987, "Forward and Inverse Synthesis for a algebraic approach, where one computer program solves all possible Robot with Three Degrees of Freedom", Summer Computer Simulation Conference, Montreal, pp. 851-856. inverse position and velocity problems; 2) all possible independent 3- Sumpter B. and Soni A.H., 1985, "Simulation Algorithm of Oklahoma dof planar serial chains with R and P joints are classified and their Crawdad Robot", 9th Applied Mechanisms Conference, Kansas City, pp. VI.1-VI.3. inverse position and velocity problems solved in a unified and succinct Williams II R.L., 1988, "Planar Robotic Mechanisms: Analysis and manner; 3) overall inverse mapping for actuated joint rates only; and Configuration Comparison", Ph.D. Dissertation, VPI&SU, Blacksburg, VA. Williams II R.L. and Reinholtz C.F., 1988a, "Forward Dynamic Analysis and 4) identification of invalid actuation schemes, which agrees with Power Requirement Comparison of Parallel Robotic Mechanisms", 20th Biennial ASME previously published results. The results of this paper can be used for Mechanisms Conference, Kissimmee FL, DE Vol. 15-3, pp. 71-78. the design, simulation, and control of any 3-dof planar parallel robot. Williams II R.L. and Reinholtz C.F., 1988b, "Closed-Form Workspace Determination and Optimization for Parallel Robotic Mechanisms", 20th Biennial ASME Mechanisms Conference, Kissimmee FL, DE Vol. 15-3, pp. 341-351.