WORKSPACE OPTIMIZATION OF ORIENTATIONAL 3-LEGGED UPS PARALLEL PLATFORMS

Transcription

1 Proceedngs of DETC 02 ASME 2002 Desgn Engneerng Techncal Conferences and Computers and Informaton n Engneerng Conference Montreal, Canada, September 29-October 2, 2002 DETC2002/MECH WORKSPACE OPTIMIZATION OF ORIENTATIONAL 3-LEGGED UPS PARALLEL PLATFORMS Mrcea Badescu, Jeremy Morman and Constantnos Mavrods Robotcs and Mechatroncs Laboratory, Department of Mechancal and Aerospace Engneerng Rutgers Unversty, The State Unversty of New Jersey 98 Brett Rd, Pscataway, NJ, Tel: , Fax: , WEBPAGE: E-mal: badescum@jove.rutgers.edu, mavro@jove.rutgers.edu ABSTRACT In ths paper the workspace optmzaton of an orentatonal 3-legged parallel platform wth unversal (U), prsmatc (P) and sphercal (S) jonts at each leg s performed. The workspace s parameterzed usng four desgn parameters that span a large range of values. In ths paper both the unconstraned and the constraned workspaces (.e. workspace wth jont lmts) are used. For the workspace of each desgn confguraton three performance ndces are calculated usng a Monte Carlo method: a) the workspace volume; b) the average of the nverse of the condton number and c) a Global Condton Index whch s a combnaton of the other ndces. A Genetc Algorthm procedure s used to determne the optmum desgn of the platform usng the Global Condton Index as the objectve functon. Plots of the workspace are presented for the optmal desgn confguraton wth and wthout jont angle lmtatons for the whole workspace and for specfc twst angles. Fnally, for the optmal desgn, t s shown that by ntroducng lmts to the unversal and sphercal jons, the platform's sotropy s mproved. Keywords: parallel platforms, optmal desgn, workspace, Genetc Algorthm, Monte Carlo method. INTRODUCTION In recent years, modular robots were ncreasngly proposed as means to develop reconfgurable and self-reparable robotc systems [1]. To perform mpromptu custom tasks, ncrease the payload to weght rato, and, n cases of emergency, self-repar, future nter-planetary robots and manpulaton systems need to ncorporate modularty and self-reconfguraton capabltes. Modular robots utlze many autonomous unts, or modules, that can be reconfgured nto a vast number of desgns. Ideally, the modules wll be homogeneous, small, and self-contaned. The robot can change from one confguraton to another by manual reassembly, or by tself. Self-reconfgurng robots adapt to a new envronment or functon by changng shape. Modules must nteract wth one another and cooperate n order to realze self-reconfguraton. Also, modular robots can repar themselves by removng and replacng faled modules. Snce one self-reconfgurable modular robot can provde the functonalty of many tradtonal mechansms, they wll be especally suted for space and planetary exploraton, where payload mass must be kept mnmum. Because they promse self-reparablty and vrtually lmtless functonalty, future selfreconfgurable modular robots are expected to be cheaper and more useful than current robot mechansms n space mssons. In ths project we nvestgate the use of 3-legged parallel platforms as jont modules of reconfgurable robots. Parallel platforms are currently beng used n many applcatons as mult-degree of freedom systems wth hgh rgdty, hgh payload to weght rato, hgh precson and low nerta [2, 3]. These propertes are also desred characterstcs for the jont modules of reconfgurable robots. Sx-legged, sx degree-offreedom (DOF) parallel platforms have been used as jont modules for reconfgurable robots n [4]. However, the hgh number of DOF, and thus the hgh number of actvely controlled jonts per module, ncreases complexty and cost. In addton, a purely 3 degree-of freedom translatonal or orentatonal moton would requre actvaton of all sx-module legs, whch means ncrease n energy consumpton. Our approach s to develop reconfgurable robots formed by a seres of 3 DOF 3-legged parallel platform modules wth purely translatonal or orentatonal moton so that they form hybrd knematc chans wth decoupled translaton and orentaton [5]. Fgure 1 shows an example of such a hybrd system. In ths example a 2-arm reconfgurable robotc system s formed from an alternatng sequence of 3 DOF orentatonal and translatonal parallel platform modules. Recently, we studed the workspace optmzaton of a 3 DOF translatonal platform 1 Copyrght 2002 by ASME

2 [6]. In ths paper we study the workspace of a 3 DOF orentatonal platform. Dfferent types of sphercal or orentatonal parallel platforms have been proposed and analyzed. Merlet descrbed several types of 2 and 3 DOF parallel platforms that can be used as wrsts / orentaton mechansms [3]. Tsa classfed the parallel platforms wth rotatonal DOF n sphercal and spatal orentaton mechansms [2, 7]. All the movng ponts of a sphercal parallel platform move on concentrc spheres. Analyss and fabrcaton of a sphercal 3 DOF platform were presented n [8]. Optmzaton of a sphercal fve bar parallel lnkage was done n [9]. Gosseln and Angeles used three desgn crtera, symmetry, workspace volume and sotropy, to optmze a 3 DOF sphercal platform [10]. workspace analyss, desgn optmzaton, or sotropy analyss have been performed yet for ths 3-UPS platform. In order to facltate the development and moton plannng of self-reconfgurable robots wth parallel platforms as modules, the robot modules need to have smlar moton and force capabltes at any one of ther two ends (fxed and movng). Ths means that the platform modules should have smlar type of motons and be able to apply the same forces regardless of whch end s used as the fxed base or the movng component. Of the varous types of sphercal / orentatonal platforms that have been proposed so far, the 3-UPS parallel platform has ths moton and force equvalence of ts two ends. Therefore, ths platform has been selected for further study n ths paper. (a) Overall Vew (b) Close-Up Vew FIGURE 1: Two-Arm Reconfgurable Robot wth 3-DOF Parallel Platform Modules. The 3-legged parallel platform wth unversal (U), prsmatc (P) and sphercal (S) jonts at each leg, wth two tetrahedral movng and fxed platforms connected through a sphercal jont s an orentatonal platform. It s composed of two tetrahedrons wth unequal equlateral trangular bases (Fgure 2). The movng tetrahedron has a smaller trangular base than the base tetrahedron. The two tetrahedrons are connected usng a sphercal jont that determnes the pont around whch the movng tetrahedron rotates. The corners of the two trangular bases are lnked usng three dentcal legs that are connected to the movng platform wth unversal jonts and to the fxed platform wth sphercal jonts. Prsmatc actuators control the length of the legs. Even though the ponts of the movng tetrahedron move on a sphere, due to the dssmlar moton of the legs the platform cannot be called sphercal [2]. The knematcs and the sngularty analyss of ths platform were studed n [2]. To the author's knowledge, no FIGURE 2: The 3-UPS Parallel Platform. In ths paper the desgn optmzaton of orentatonal 3- UPS parallel platforms wth prsmatc, unversal, and sphercal jont constrants s performed. The workspace s parameterzed usng four desgn parameters, whch are the prsmatc jont stroke, the dfference between the dstances of the center to one of the corners of the trangular plate of the base and of the movng platform, the heght of the platform n zero orentaton, and the rato of the heghts of the two tetrahedrons. For a large range of values for these desgn parameters the workspace of the correspondng 3-UPS platforms s calculated. In ths paper both the constraned and unconstraned workspaces are used. The dfference between these two s that for the constraned workspace angular unversal and sphercal jont lmts are mposed. For both workspaces of each desgn confguraton three performance ndces are calculated: a) the workspace volume; b) the average of the nverse of the condton number and c) a Global Condton Index whch s a combnaton of the other two performance ndces. For determnng the optmal desgn a Genetc Algorthm was mplemented. The Global Condton Index was chosen as the objectve functon for optmzaton. The optmzaton was performed wth and wthout mposng jont constrants. Plots of the 3-D workspace of the optmal desgns are generated. Fnally, for the optmal desgn, t s shown that by ntroducng lmts to the angles of the unversal and sphercal jonts, the sotropy of the parallel platform s mproved. MATHEMATICAL TOOLS In ths secton we present the mathematcal and knematcal tools that were used to formulate the workspace optmzaton problem. These tools nclude the drect and nverse knematcs, the platform s Jacoban matrx, ts condton number and the Global Condton Index. For the 2 Copyrght 2002 by ASME

3 drect and nverse knematcs and for the calculaton of the Jacoban matrx, we used the soluton methodology proposed n [2]. For the Global Condton Index we used the defnton ntroduced n [10]. fxed tetrahedron and has center C at the tetrahedron s tp. The pont O s the center of the base trangle. The z a axs s orented along OC. The x a axs s parallel to OA 1 and the y a axs s chosen to obtan a rght hand rectangular coordnate system. The frame B, Cx b y b z b, s rgdly connected to the movng tetrahedron and has the same center as the fxed frame, A. The pont E s the center of the base trangle of the movng tetrahedron. The z b axs s orented along the vector EC. The x b axs s parallel to the EB 1 and y b axs s chosen to obtan a rght hand rectangular coordnate system. As seen n Fgure 5, the poston vectors of ponts A and B wth respect to frames A and B respectvely, can be wrtten as: A a =[a x, a y, -h a ] T, and B b = [b x, b y, -h b ] T (1) FIGURE 3: 3-UPS Parallel Platform. The 3-UPS parallel platform conssts of a fxed tetrahedron, a movng tetrahedron and three dentcal lmbs (see Fgure 3). The tetrahedrons have equlateral trangular bases of dfferent szes (trangles A 1 A 2 A 3 and B 1 B 2 B 3 Fgure 5). The tps of the tetrahedrons are connected usng a sphercal jont. The lmbs are connected to the movng tetrahedron base wth 2- DOF unversal jonts and to the fxed tetrahedron base wth sphercal jonts. A lnear actuator controls the leg length, and forms a prsmatc jont. In Fgure 3 the sphercal jonts are marked wth S, the unversal jonts wth U, and the prsmatc actuated jont wth P. Each unversal jont s treated as two revolute jonts wth axes perpendcular to each other and ntersectng at a pont. In Fgure 4, u 4 s the unt vector along the prsmatc jont axs for leg, u 5 and u 6 are the unt vectors along the axes of the unversal jont of leg. u 7 s the CB edge of the movng tetrahedron. FIGURE 4: Jont Axes n the 3-UPS Platform. A coordnate system s chosen for each tetrahedron as shown n Fgure 5, a) and b). Frames A and B are defned for the base and movng tetrahedrons respectvely. Ther orgn s placed at the common tp of the tetrahedrons and represents the pont around whch the movng tetrahedron rotates. Ther z-axs s perpendcular to each base trangle's plane, and ther x and y- axes are n planes parallel to the base trangle's plane. The x a/b axes are orented parallel to the OA 1 and EB 1 lnes, respectvely. The frame A, Cx a y a z a, s rgdly connected to the a) Base Tetrahedron b) MovngTetrahedron FIGURE 5: Coordnate Systems. The scalar a denotes the dstance between the center, O, of the base trangle of the fxed tetrahedron and any of the corners of the base trangle. The scalar b denotes the dstance between the center, E, of the base trangle of the movng tetrahedron and any of the corners of the base trangle. The parameter c denotes the dfference between a and b. Parameter d denotes the length of leg. Subscrpt denotes one of the legs and can take the values 1, 2 or 3. Letters n bold represent three-dmensonal vectors. Superscrpt A or B on the left of a vector denotes the reference frame where ts coordnates are calculated. Superscrpt T on the rght of a vector denotes the transpose of a vector. Vector d determnes the poston of the leg (A B ) n the base coordnate system: A d =[d,xa, d,ya, d,za ] T (2) The scalar h a denotes the heght of the fxed tetrahedron and s the dstance from pont O to pont C. The scalar h b denotes the heght of the movng tetrahedron and s the dstance from pont E to pont C. The parameter h represents the total heght of the parallel platform n (0,0,0) orentaton and s equal to the sum of h a and h b. The parameter h ab s the rato between 3 Copyrght 2002 by ASME

4 h a and h b. The parameters c and h are taken as ratos to the maxmum length of the leg. The transformaton from the movng frame B to the fxed frame A can be descrbed by a 3x3 rotaton matrx A R B defned by a z-x-z (ϕ-θ-ψ) Euler rotaton. Gven that n the ntal poston (0-0-0 rotaton) the x a and x b axes concde, z a and z b, and y a and y b are n opposte drectons respectvely, the resultng rotaton matrx s gven by: A cφcψ-sφcθsψ cφsψ+sφcθcψ -sφsθ R=sφcψ+cφcθsψ B sφsψ-cφcθcψ cφsθ sθsψ -sθcψ -cθ where c represents the cosne and s the sne of the followng angle. For the drect knematcs the leg lengths d and the trangular bases' geometry are known. The three orentaton angles of the movng platform are determned from expandng the followng three scalar equatons (see [2] for more detals): (3) T d=a+b-2ab, for =1, 2, 3 (4) For the nverse knematcs the orentaton of the movng platform s known and the lmb lengths, d are calculated from the followng three equatons: 2 2 T d =± a +b -2a b, for =1,2,3 (5) The manpulator Jacoban matrx J relates the end-effector veloctes ω to the actuated jont veloctes d : Jxω= Jdd (6) For the 3-UPS platform J x and J d are gven by: ( u 1,7 u1,4 ) ( ) ( u 3,7 u3,4 ) T T Jx = u 2,7 u 2,4 J d = [] I (7) T The condton number of the Jacoban matrx s defned by Equaton (8) 1 k = J J (8) where. denotes the norm 2 of a matrx [11]. Robot manpulator confguratons where k s equal to 1 are called sotropc. In these confguratons the system s able to develop the same amount of forces and veloctes n all endeffector drectons. For hgh values of k, there are end-effector drectons where the manpulator can develop much hgher forces or veloctes than n other drectons. In many applcatons, ths s not a desrable system property because the system loses ts homogenety n force and velocty development. Confguratons n whch k has an nfnte value are sngular confguratons. In these confguratons there are drectons n space where the end-effector can ether not move or not apply forces. The manpulator can also gan extra degrees of freedom n these confguratons. Isotropy s a very local manpulator property because t changes from confguraton to confguraton. A manpulator can have good sotropy n one confguraton but poor sotropy n another confguraton. In many robot applcatons sotropy s an mportant property, and needs to be taken nto account durng the robot's desgn phase. Defnng a global sotropy ndex that wll be able to characterze the system sotropy n the whole workspace s an mportant but dffcult problem to solve. The problem s that a robotc system wll always have some sngular confguratons or confguratons wth poor sotropy. By takng a type of an average of the condton number s only a rough ndcaton of the qualty of the system global sotropy and says nothng about the magntude and number of bad sotropy confguratons. Nevertheless, n ths work we wll use global sotropy ndces based on the average of the condton number, as they are beng used n the current lterature. In ths work, three performance ndces wll be used to characterze a robotc system's workspace: a) The volume of the workspace. Obvously, usng ths performance ndex, optmal desgns correspond to maxmum workspace volume. It s obtaned by multplyng the searchng volume wth the rato of the number of ponts nsde the workspace dvded by the total number of ponts. b) The average of the nverse of the condton number. The nverse of the condton number characterzes the system's sotropy. The average of the nverse of the condton number s calculated by summng the nverse of the condton number n every pont n the workspace and then dvdng t wth the number of ponts consdered n the workspace. Optmal desgns correspond to values of the average condton number that are close to 1. c) The Global Condton Index proposed n [10]. Ths ndex, defned n Equaton (9), s the rato of the ntegral of the nverse condton numbers calculated n the whole workspace, dvded by the volume of the workspace. In realty t s calculated as the product of the sum of the nverse condton number wth the searchng volume dvded by the number of ponts n the workspace. Therefore, t s lke a combnaton of the frst two performance ndces: A 1 η =, where A = dw B k W and B= dw (9) W n whch k s the condton number at a partcular pont n the workspace W. B s the total volume of the workspace. The Global Condton Index s bounded as: 0<η< 1 (10) Isotropc systems correspond to value of η equal to 1 and systems wth bad sotropy correspond to values of η approachng zero. OPTIMIZATION ALGORITHM Based on the platform's drect knematcs, and due to the fact that all legs are equal and that the base and movng trangles are equlateral, the 3-UPS orentatonal platform workspace depends on four geometrc parameters: a) the dfference c between a and b : c =a -b ; b) the total heght of the platform consdered as the sum of the heghts of the two tetrahedra: h=h a +h b (see Fgure 5); c) the rato of the heghts of the two tetrahedra: h ab =h a /h b ; and d) the stroke of the prsmatc jont s. We consdered the followng ranges of possble values for the desgn parameters. The range of values for c s between 0.27 and Platforms wth very small values for c (.e. platforms where the 4 Copyrght 2002 by ASME

5 trangular base and movng plates are almost equal) present extra DOF (self-motons) that could not be controlled from the actuaton moton and such desgns are not acceptable. Platforms wth very large values for c have very small workspace. The total heght of the two tetrahedrons, h, has values between 0.2 and 1.0. Below 0.2 the workspace s very small and has no practcal use. Total heght equal to 1 corresponds to maxmum length. The rato of the heghts of the two tetrahedrons, h ab, was selected to vary between 0.33 and 3.0. The stroke s of the actuator s the maxmum amount of travel for each leg's prsmatc actuator. It s expressed wth a percentage value of the mnmum length of the lmb d mn whch s taken such that the maxmum length s equal to 1. The fact that the leg's maxmum length s consdered to be equal to 1 means that all lengths are normalzed wth respect to the leg's maxmum length as t s specfed from each applcaton's requrements. In ths project the range of values of the stroke of the actuators s consdered to be between 20% and 87.5%. These values were selected based on the techncal specfcatons of the majorty of commercally avalable prsmatc actuators. The mnmum value of each leg's length d mn s equal to 1/(1+s /100). So n each platform's confguraton the value of the leg length d should be between d mn and 1. Due to the fact that the plots of all three performance ndces for any value of the desgn parameters are hghly nonmonotone, a Genetc Algorthm was mplemented to determne the optmal desgn confguraton. It conssts of the followng steps: a) An ntal populaton of 100 members s randomly generated. Each member contans values for the four desgn parameters defned above. Each parameter has an eght dgt resoluton whch means that 256 dfferent values can be chosen for each parameter. b) For each populaton member the platform s geometrc parameters are calculated usng the desgn parameters. c) To determne the values of the performance ndces a Monte Carlo method was mplemented. For each confguraton wth certan values of the desgn parameters, s, c, h, and h ab, 1,728,000 orentatons of the movng tetrahedron are chosen randomly. The ϕ, θ and ψ Euler angles correspondng to a z-x-z rotaton are randomly chosen n the nterval [-180,+180 ]. d) For each orentaton the lengths of the legs are determned usng the nverse knematcs, Equaton (5). If all legs have lengths n the allowed nterval d mn <d<d max the nterference of the legs wth fxed and movng tetrahedrons s checked. Also, for the constraned workspace, the jont angles are determned and verfed f they are between mposed lmts. If the orentaton verfes all constrants, then the Jacoban condton number, k, s determned usng Equaton (8). e) For each confguraton the performance ndces are calculated usng Equatons (9) [Note: In Equaton (9) η s the global condton number, A s the average of the nverse condton number and B s the volume of the workspace]. The populaton member wth the best objectve functon s determned and saved n a fle. It wll be called the kng n further references. As the objectve functon, any one of the descrbed performance ndces can be chosen. f) A matng pool s then created from the ntal populaton selectng the members wth better objectve functon values. The frst member of the matng pool s chosen as the member wth the best value of the objectve functon (the kng). g) A crossover among the members of the matng pool s done wth a 0.45 probablty. At the end, the frst member s replaced wth the member wth the best objectve functon (the kng). h) A mutaton wth 0.05 probablty s then done over the new populaton. The frst member of the populaton s not affected (the kng). ) For the new populaton the cycle starts over. It stops when the value of the objectve functon of the kng s not mproved wth a certan relatve quantty. The unconstraned optmzaton s done wthout mposng the constrants mentoned n step d) above. RESULTS We used a program wrtten n C++ to perform the optmzaton and MATLAB to plot the workspace. We run the optmzaton program for the constraned and unconstraned workspace. FIGURE 6: Workspace of the Optmal Unconstraned Confguraton of the 3-UPS Platform. The optmal desgn for the unconstraned workspace has the followng parameter values: stroke=83.9%, c =0.602, h=0.285, and h ab = Fgure 6 shows the workspace of the optmal unconstraned confguraton. The ponts represent the postons that can be reached by the center, E, of the movng tetrahedron base. E descrbes a sphere wth the center at pont C and a radus h b. To represent the twst angle, ψ, the radus of the sphere descrbed by pont E was changed dependng on the value of the twst angle. For ψ = -180 the radus was reduced by 10% of h b and for ψ = +180 t was ncreased by 10%. A lnear dependence was used for the ntermedary values. The color wth whch the ponts were represented reflects the value of the condton number of the platform s Jacoban at the specfc pont. A value 1 of the condton number was represented wth dark blue color. Because some of the condton numbers had extremely large values (more than 100), n our representaton n MATLAB TM all condton numbers wth a value over 10 were reduced to 10 (Ths was used just for representaton but not for optmzaton). Dark red color s used to plot ponts wth large condton numbers. For the optmzaton of the constraned workspace, we mposed lmts at the varous jonts of the platform. In ths paper we present results from two dfferent schemes of jonts constrants (I and II). For the constraned workspace I, the lmts of the unversal jont angles were set at ±60 for the revolute jont that allows the leg rotaton on CEA planes and ±60 for the other revolute jont. We have to remnd that the 5 Copyrght 2002 by ASME

6 unversal jont can be treated as two revolute jonts. The lmts for the sphercal jont angles were set at ±65. The optmal desgn for the frst constraned workspace was found to have the followng values: stroke=50%, c =0.616, h=0.526, and h ab =2.85. Fgure 7.a) shows the workspace for the optmal platform desgn usng jont constrant scheme I. It can be noted, that due to the nfluence of jont constrants a stroke larger than 50% wll no further ncrease the workspace volume. Fgures 8 and 9 present the top vew of the unconstraned and constraned workspaces for 0 and 30 twst angles, respectvely. Here the effect of the workspace reducton due to the angular constrants s more obvous. a) Unconstraned Workspace a) Frst Scheme of Jont Constrants (I) b) Second Scheme of Jont Constrants (II) FIGURE 7: Workspace of the Optmal Constraned Confguraton of the 3-UPS Platform. We run the optmzaton algorthm for the second scheme of jont constrans. The lmts of the unversal jont angles were set at ±75 for the revolute jont that allows the leg rotaton on CEA planes and ±85 for the other revolute jont. The lmts for the sphercal jont angles were set the same, at ±65. The optmal desgn parameters that were obtaned are: stroke=71%, c =0.62, h=0.44 and h ab =1.75. In ths case the average value of the nverse of the condton number, k, s ncreased and the workspace volume s ncreased relatve to the prevous optmal confguraton. For the constraned workspaces I and II we used the same representaton method as for the unconstraned workspace. From the data provded by the optmzaton programs, t can be seen that the unconstraned workspace volume s less than half the constraned workspace volume. By mposng constrants on the jont angles, one can exclude not only many bad sotropy ponts, but many good sotropy ponts as well. Overall, the average value of the nverse of the condton number s ncreased, whch means that the sotropy of the platform s mproved, and the workspace volume s reduced when jont constrants are mposed. It can also be seen that the jont constrants dd not affect sgnfcantly the optmal value for the desgn parameter c. Ths means that for a better desgn, the dfference n sze between the bases of the tetrahedrons needs to be very large, ndependently of the jont constrants mposed. b) Constraned Workspace (I) c) Constraned Workspace (II) FIGURE 8: Fxed Twst of 0.0. Even though the workspaces shown n Fgures 6-9 are contnuous, not all of the ponts can be reached wth any orentaton. Ths s shown n Fgures 8-b,c) and 9-b,c). Also t can be seen that for small values of the ptch angle, ϕ, the manpulator has a bad sotropy, no matter how small the yaw angle, θ, or the twst angle, ψ, are. From Fgures 7, 8, and 9 we learn that ponts on the workspace can be reached only for certan twst angles. Also we learn that even though some ponts can be reached for a 6 Copyrght 2002 by ASME

7 large range of twst angles for some values of these we get a better sotropy. These results can be used to determne a path between two orentatons mantanng a good sotropy f the twst angle n the ntermedate postons s not crtcal. CONCLUSIONS In ths paper the desgn optmzaton of orentatonal 3- UPS parallel platforms was presented. These platforms can be used as jont modules n reconfgurable robots and the qualty of ther unconstraned and constraned workspace s an mportant desgn feature. The workspace s parameterzed usng four desgn parameters. Three workspace performance ndces are calculated as a functon of the desgn parameters. These performance ndces quantfy the workspace volume, the system sotropy and a combnaton of the prevous two propertes. The optmal desgn s determned for both stuatons wth and wthout jont lmts mposed. Dfferent desgn parameters are obtaned n the two stuatons. The paper presents the total and fxed twst unconstraned and constraned workspaces. ACKNOWLEDGMENTS Ths work was supported by a Rutgers Unversty Research Councl Grant and by NASA's Jet Propulson Laboratory. Mr. Mrcea Badescu was supported by a Teachng Assstantshp from the Department of Mechancal and Aerospace Engneerng. Mr. Jeremy Morman was supported by an NSF REU supplement. REFERENCES 1. McKee, G. and Schenker, P., Edtors, 1999, Sensor Fuson and Decentralzed Control n Robotc Systems II, SPIE Proceedngs Seres, Vol Tsa, L.-W., Robot Analyss The Mechancs of Seral and Parallel Manpulators, John Wley & Sons, New York, Merlet J.-P., Parallel Robots, Kluwer, Dordrecht, Nederland, Hamln, G. and Sanderson, A., "TETROBOT: A Modular Approach to Parallel Robotcs," IEEE Robotcs & Automaton Mag., Vol. 4, No. 1, pp Tsa, L.W. and Josh, S., "Feasblty Study of Hybrd Knematc Machnes," Proceedngs of the 2001 NSF Desgn, Manufacturng and Industral Innovaton Research Conference, Tampa, FL, January 7-10, Badescu M., Morman J. and Mavrods C., "Workspace Optmzaton of 3-UPU Parallel Platforms wth Jont Constrants," Proceedngs of the 2002 IEEE Internatonal Conference of Robotcs and Automaton, Washngton, D.C., May 11-15, Tsa, L. W., 1999, Systematc Enumeraton of Parallel Manpulators, n Parallel Knematc Machnes: Theoretcal Aspects and Industral Requrements, edted by C. R. Boer, L. Molnar-Tosatt, and K. S. Smth, Advanced Manufacturng Seres, publshed by Sprnger- Verlag, ISBN: , pp Gosseln, C., and Hamel, J., The Agle Eye: A Hgh- Performance Three-Degree-of-Freedom Camera-Orentng Devce, Proceedngs IEEE Internatonal Conference on Advanced Robotcs, Psa, Italy, 1994, pp Ouerfell, M., and Kumar, V., Optmzaton of a sphercal fve bar parallel drve lnkage, DE, Advances n Desgn Automaton, Vol. 1, 1991, pp Gosseln, C. and Angeles, J. The Optmum Knematc Desgn of a Sphercal Three-Degree-of-Freedom Parallel Manpulator, ASME J. of Mech., Transmssons, and Automaton n Desgn, Vol. 111, No. 2, 1989, pp Gosseln, C. and Angeles, J. A Global Performance Index for the Knematc Optmzaton of Robotc Manpulators, ASME J. Mech. Des., Vol. 113, No. 3, 1991, pp Chapra, S., Canale, R., Numercal Methods for Engneers: Wth Programmng and Software Applcatons, WCB/McGraw-Hll, a) - Unconstraned Workspace b) - Constraned Workspace (I) c) Constraned Workspace (II) FIGURE 9: Fxed Twst of Copyrght 2002 by ASME