Atlas Based Kinematic Optimum Design of the Stewart Parallel Manipulator

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1 20 CHINESE JOURNAL OF MECHANICAL ENGINEERING Vol. 28,aNo. 1,a2015 DOI: /CJME , avalable onlne at Atlas Based Knematc Optmum Desgn of the Stewart Parallel Manpulator SHAO Zhufeng 1, 2, TANG Xaoqang 1, 2, *, WANG Lpng 1, 2, and SUN Dengfeng 3 1 State Key Laboratory of Trbology, Department of Mechancal Engneerng, Tsnghua Unversty, Bejng , Chna 2 Bejng Key Lab of Precson/Ultra-precson Manufacturng Equpments and Control, Tsnghua Unversty, Bejng , Chna 3 School of Aeronautcs and Astronautcs Engneerng, Purdue Unversty, West Lafayette, IN 47907, USA Receved March 6, 2014; revsed September 23, 2014; accepted September 29, 2014 Abstract: Optmum desgn s a key approach to make full use of potental advantages of a parallel manpulator. The optmum desgn of mult-parameter parallel manpulators(more than three desgn parameters), such as Stewart manpulator, reles on analyss based and algorthm based optmum desgn methods, whch fall to be accurate or ntutve. To solve ths problem and acheve both accurate and ntuton, atlas based optmum desgn of a general Stewart parallel manpulator s establshed, wth ratonal selecton of desgn parameters. Based on the defned sphercal usable workspace(suw), prmary knematc performance ndces of the Stewart manpulator, nvolvng workspace and condton number are ntroduced and analyzed. Then, correspondng performance atlases are drawn wth the establshed non-dmensonal desgn space, and mpact of jont dstrbuton angles on the manpulator performance s analyzed and llustrated. At last, an example on atlas based optmum desgn of the Stewart manpulator s accomplshed to llustrate the optmum desgn process, consderng the end-effector posture. Deduced atlases can be flexbly appled to both quanttatve and qualtatve analyss to get the desred optmal desgn for the Stewart manpulator wth respect to related performance requrements. Besdes, the establshed optmum desgn method can be further appled to other mult-parameter parallel manpulators. Keywords: optmum desgn, performance atlas, parallel mechansm, Stewart 1 Introducton Parallel manpulator s put forward wth respect to the conventonal seral manpulator, whch possesses the sgnfcant structural feature of closed-loop chans. A parallel manpulator s composed of the base and the end effector, whch are connected together by several dentcal lmbs [1]. The base s usually fxed, whle the end effector has multple degrees of freedom. Generally, the number of ndependent actuators equals that of the manpulator s degrees of freedom, wthout consderng actuaton redundancy. Compared wth seral manpulators, parallel manpulators possess evdent advantages, e.g, large load-toweght rato, hgh accuracy, mproved stffness. Based on above merts, parallel manpulators have been wdely appled n the ndustry feld and also attracted great academc attenton [2 3]. However, theores and technologes on parallel manpulators are stll nadequate, whch lmt ther potental n practcal applcatons. As an mportant * Correspondng author. E-mal: tang-xq@mal.tsnghua.edu.cn Supported by Natonal Natural Scence Foundaton of Chna(Grant Nos , ), and Natonal Outstandng Youth Scence Foundaton of Chna(Grant No ) Chnese Mechancal Engneerng Socety and Sprnger-Verlag Berln Hedelberg 2015 ssue of both theoretcal research and engneerng applcaton, optmum desgn s a great approach to mprove the performance of parallel manpulators and wll make parallel manpulators more attractve to the ndustry [4]. Optmum desgn of the parallel manpulator s always recognzed as a challengng ssue,and s manly based on the knematcs now [5]. Many researchers have spent a lot of effort n ths feld that can be summarzed nto three categores: The frst one can be named as the analyss based optmum desgn. Performance ndces are analyzed one by one to accumulate rough desgn prncples, and the dmenson synthess s carred out accordngly [6 7]. The second category s referred as the algorthm based optmum desgn, whch starts wth determnng the exact target, based on sngle or multple performance ndces. Then, the optmum desgn ssue s converted to a mult-objectve optmzaton problem, and solved wth complex nonlnear algorthms, such as genetc algorthms and artfcal neural networks [8 9]. The last one s the atlas based optmum desgn. Based on drawn performance atlases, optmum parameters can be determned accurately and ntutonally [10 12]. The frst method s smple to be carred out, but obtaned desgn prncples are naccurate. Parameters wth better performance can be deduced, whle

2 CHINESE JOURNAL OF MECHANICAL ENGINEERING 21 the optmal result s hard to get, especally when there are contradctons among obtaned prncples. The second and the thrd methods are equvalent on deducng the optmal result. However, algorthm based optmum desgn fals to llustrate the relatonshp between desgn parameters and manpulator performances, and has to restart agan when the target changes. Thus, atlas based optmum desgn s more advantageous. In addton, due to lmtaton on the number of desgn parameters, performance atlas based optmum desgn s only adopted by lower-moblty parallel manpulators wth fewer desgn parameters now. In ths paper, atlas based optmum desgn s carred out on a general 6 degree-of-freedoms(dof) Stewart parallel manpulator for the frst tme, consderng fve desgn parameters, through proper parameter selecton and sensble analyss. In the process of optmal desgn, some typcal knematc ndces are nevtablly adopted to evaluate the performance of the parallel manpulator. Workspace s the basc requrement of practcal applcaton. MERLET, et al [13], carred out the workspace analyss on the planar parallel manpulator, puttng forward the constant orentaton workspace, maxmal workspace, and nclusve workspace. Then, the workspace of spatal parallel manpulators are studed, and workspaces of Stewart manpulators wth dfferent parameters are compared [14]. Further, workspace has been adopted as an mportant evaluaton crteron for the optmum desgn of parallel manpulator [15 17]. The condton number of the knematc Jacoban matrx s recognzed as a comprehensve knematc performance ndex and has been used by some researchers to carry out the optmum desgn. KLEIN, et al [18], dscussed the local condton number and mnmal sngularty value of the knematc Jacoban matrx on redundant manpulators. The condton number of the Jacoban matrx s also known as the dexterty of the parallel manpulator [19]. Later, the global condtonng ndex s ntroduced and used n the optmum desgn of the parallel manpulator [20 21]. The aforementoned ndces are adopted n the followng analyss, and correspondng atlases are deduced, whch reveal the performance trend of the general Stewart manpulator. Our study object, as shown n Fg. 1, s the 6 DOF Stewart parallel manpulator [22 24], consstng of two bodes (the base and the end effector) as well as sx extensble lmbs. Each lmb s drven by a set of servomotor and ball screw. One end of the lmb s connected to the base by a unversal jont, whle the other end s connected to the end effector by a sphercal jont. The remnder of ths paper s organzed as follows. In the next secton, knematcs of the Stewart manpulator s analyzed and the Jacoban matrx s deduced. In secton 3, the non-dmensonal desgn space of the Stewart manpulator s establshed. Sngularty and workspace of the Stewart platform are dscussed n secton 4, and workspace atlases are derved. Global condtonng ndex s dscussed n secton 5, and correspondng atlases are drawn. In secton 6, an example on atlas based optmum desgn of the Stewart manpulator s gven to show the mplementaton procedure. Fnally, conclusons of ths paper are gven. Fg. 1. Confguraton of Stewart parallel manpulator 2 Knematcs Analyss Fg. 2 llustrates the knematc model of the Stewart manpulator. The base s defned by rotaton centers of unversal jonts B 1, B 2, B 3, B 4, B 5, B 6, whle the end effector s defned by rotaton centers of sphercal jonts, marked wth ponts P 1, P 2, P 3, P 4, P 5, P 6. The base frame {B}:O-XYZ s located at the geometrc center of the base, wth the X-axs pontng at the mdpont of secton B 1 B 6, and the Z-axs s perpendcular to the base downward. The end-effector frame {P}:o-xyz s attached to the geometrc center of the end effector, wth the x-axs pontng at the mddle of secton P 1 P 6, and the z-axs perpendcular to the end effector downward. The end-effector poston s descrbed by the translaton vector t = [x, y, z] T, pontng from the orgn of the base frame to the orgn of the endeffector frame. The Euler representaton s adopted to descrbe the endeffector posture, namely, a rotaton of angle about the X-axs, followed by a rotaton of angle around the Y-axs, and fnally followed by a rotaton of angle around the Z-axs. Relatve to the base frame, the rotaton matrx of the end effector equals the product of three basc rotaton matrces R= R3( Z, ) R2( Y, ) R1( X, ) = æcc css sc csc ssö - + sc sss cc ssc cs + -, çè- s cs cc ø (1) where s and c symbols stand for sne and cosne operatons, respectvely. The poston vector of pont P can be descrbed n the base frame by the rotaton matrx and translaton vector as

3 22 Y SHAO Zhufeng, et al: Atlas Based Knematc Optmum Desgn of the Stewart Parallel Manpulator P = Rp + t, = 1, 2,, 6, (2) where p s the poston vector of pont P under the endeffector frame. In order to deduce the Jacoban matrx, Eqs. (6) and (7) are consdered together, whch gve t + ω ( Rp ) = LW s + L s. (8) Take the dot product of Eq. (8) wth the unt lmb vector s at both sdes, and the velocty mappng equaton can be establshed as s t + é( ) ù ë Rp s û ω = L. The above equaton can be organzed n matrx form as JX = L, (9) where J s the Jacoban matrx of the Stewart manpulator, and can be expressed as T ( ) T T ( ) T T J = é ( ) T 1, 1 1 ; 2, 2 2 ; ; 6, ù ês Rp s s Rp s s Rp6 s6 ú. ë û 3 Desgn Space (10) Fg. 2. Knematc model of the Stewart parallel manpulator Next, the lmb vector pont) can be derved as S (pontng from B pont to P S = P -b, (3) and b s the poston vector of pont B expressed n the base frame. Wth the lmb length the unt lmb vector can be obtaned as L = S, (4) s = S L. (5) Takng the dervatve of Eq. (3) wth respect to tme yelds the velocty expresson as S = t + ω ( Rp ), (6) where ω s the angular velocty of the end effector. Accordng to the physcal meanng, S can also be descrbed wth the sldng velocty L and the angular velocty W of the th lmb as S = L s + LW s. (7) The performance of the parallel manpulator depends on the end-effector pose as well as the archtecture. After the telescopc proporton of the lmb s determned, the archtecture of the Stewart manpulator can be descrbed wth fve geometrc parameters(consderng practcal experence, the telescopc range of the extensble lmb s gven as from k to 2k here). Four parameters are llustrated n Fg. 2(b), namely, crcumradus of the end effector r, crcumradus of the base R, dstrbuton angle of the sphercal jont, and dstrbuton angle of the unversal jont. In order to descrbe the workspace center and measure the manpulator volume, the ntal dstance h between the base and the effector s chosen as the ffth geometrc parameter, when each lmb s of the ntal length 1.5k. Thus, three length parameters and two angle parameters are chosen to defne the archtecture of the Stewart manpulator. Let s focus on length parameters frst. Theoretcally, each of these three parameters can take an arbtrary value from zero to nfnty. And, ther combnatons are nfnty. In order to llustrate all possble combnatons wthn a fnte area, length parameters must be normalzed. Defne a dmenson factor as = ( r+ R+ h) 3. (11) Then, we can get three dmensonless parameters as l = r, l = R, l = h, (12) where l 1 +l 2 +l 3 =3. Length parameters can be llustrated n a

4 CHINESE JOURNAL OF MECHANICAL ENGINEERING 23 plane to facltate performance descrpton and analyss. As shown n Fg. 3, l 1 -axs s establshed parallel to the Y-axs, l 2 -axs and l 3 -axs are mutually perpendcular, and the angle between l 1 -axs and l 2 -axs s 3 4. Each combnaton of these three parameters can also be expressed n the Cartesan coordnate system, wth the relatonshp as ìï ïx= 2l3+ l1 2, í ï ïî y= l1. Fg. 3. Desgn space of length parameters The end-effector crcumradus r should not be greater than the base crcumradus R, namely, l 2 l 1. Then, the desgn space of length parameters s determned as shown n Fg. 3(enclosed by red lnes). Theoretcally, range of angle parameters(jont dstrbuton angles) s from zero to sxty degrees. Consderng the jont nstallaton space, the avalable range s narrowed, and s between fve and sxty degrees. Further dscusson on the angle range s carred out n the next secton. 4 Workspace Analyss and Workspace Atlases The reachable workspace of the Stewart manpulator, whch s encrcled by twelve envelope surfaces, has an rregular shape wth the sngular loc nsde. Thus, regular usable workspace(ruw) s defned here as the regularshaped maxmum contnuous workspace wthout a sngularty. Snce the Stewart manpulator s a crcular symmetrc structure, the sphercal usable workspace(suw) s adopted n ths paper, whch s the maxmum contnuous sphercal workspace wth no sngular locus nsde. In most cases, the SUW of the Stewart manpulator s the maxmum nscrbed sphere of the reachable workspace. In the followng analyss, the zero-rotaton SUW s consdered at frst, when rotaton angles of the end effector are zero. 4.1 Sngularty In order to determne the SUW, sngulartes of the Stewart manpulator are dscussed. As shown n Eq. (9), t s obvous that there exsts only the second type sngularty [25] for the Stewart manpulator, when det( J ) = 0. Such sngulartes of the Stewart manpulator can be further dvded nto three categores, such as archtecture sngularty, confguraton sngularty and formulaton sngularty. If the base and the end effector of the Stewart manpulator are smlar polygons, the archtecture sngularty occurs, whch should be completely avoded n the desgn. When the base and the end effector concde, or lmbs are of the same length wth the endeffector rotaton of / 2 around the Z-axs, the confguraton sngularty appears. The formulaton sngularty s usually ntroduced by the adopted descrpton method of the end effector posture. In ths paper, snce the Euler representaton s used, ths type of sngularty happens only when the rotaton angle around the Y-axs s / 2 [26 27]. The archtecture sngularty can be avoded by the ratonal defnton of base and end-effector frames. As shown n Fg. 2(b), base and end-effector frames are establshed wth a devaton angle of / 3 relatve to the jont dstrbuton. And, only when = = / 3, the archtecture sngularty appears. Thus, avalable ranges of angle parameters and should be further narrowed, and s between 5 and 55. At the same tme, the rotaton angle of / 2 s too large to be avalable for the Stewart manpulator. Thus, only the confguraton sngularty, where the base and the end effector concde, needs to be pad attenton to n the follow-up workspace analyss. 4.2 Workspace atlases Wth gven jont dstrbuton angles( = = 10 ), the zero-rotaton SUW atlas can be derved, consderng length parameters, as shown n Fg. 4. The raduses of both maxmum nscrbed sphere and the SUW s obtaned through numercal smulaton wth the MATLAB software. From the results, we can conclude the followng. (1) The desgn space can be dvded nto three regons by two lnes, namely l 2 = l , and l 2 = 1.5. (2) In the left regon, the SUW s dfferent from the maxmum nscrbed sphere of the reachable workspace, and the SUW radus s generally proportonal to h. (3) In the mddle regon, the SUW radus ncrease wth the decrease of r n general. (4) In the rght regon, f l 2 s specfed, the radus of SUW grows wth the ncrease of l 3 and the decrease of l 1. (5) The maxmum SUW radus appears at lower rght and the bottom area of lne l 2 = l Then, let s dscuss the mpact of jont dstrbuton angles on the SUW radus of the Stewart manpulator. Frstly, three groups of l 1, l 2, and l 3 are selected to estmate the relatonshp between radus of the zero-rotaton SUW and jont dstrbuton angles. As shown n Fg. 5, t ndcates that the SUW radus ncreases wth the decrease of the angle sum +, and the radus wll not change for dfferent values of and, when ther sum + s constant.

5 24 Y SHAO Zhufeng, et al: Atlas Based Knematc Optmum Desgn of the Stewart Parallel Manpulator Fg. 4. Workspace atlas under zero-rotaton condton Sold: maxmum nscrbed sphere of the reachable workspace Dashed: SUW In Fg. 6(a), sold lnes show the SUW atlas for = = 30, whle dashed lnes llustrate the workspace atlas for = = 50. In Fg. 6(b), sold curves show the workspace atlas for = 55, = 5, whle dashed curves reveal the workspace atlas for = 5, = 55. From these atlases, we can tell the followng. (1) In the left regon, the SUW radus s manly determned by the sngularty condton. Impact of jont dstrbuton angles cannot be reflected obvously. (2) In the mddle and the rght regons, the SUW radus s affected by the sum of jont dstrbuton angles, and the mpact of jont dstrbuton angles on the radus s more obvous when l 1 s larger. (3) In the mddle and the rght regons, when the sum of jont dstrbuton angles s fxed, specfc values of and have lttle mpact on the SUW radus. Fg. 6. SUW atlases consderng dfferent jont dstrbuton angles Above laws can be explaned by the expresson for the telescopc length of the lmb, whch s Lnt = ( 3k 2) = h + R + r -2Rrcosé 3 ( ) 2 ù ë - + û, (13) Fg. 5. Relatonshp between the SUW radus and jont dstrbuton angles Further, jont dstrbuton angles are analyzed through supermposng on the desgn space of length parameters to confrm ther nfluences on workspace, as shown n Fg. 6. where k s the mnmum length of the lmb, and equals the telescopc length. When the sum of jont dstrbuton angles ( + ) ncreases, the telescopc length s reduced. The reachable workspace of the Stewart manpulator s manly determned by the telescopc length of the lmb. Thus, the reachable workspace and ts nscrbed sphere(suw n the mddle and rght regons) shrnk. In addton, f h s fxed, an ncrease of r wll enlarge the proporton of Rr to h R r n Eq. (13), and makes the mpact of jont dstrbuton angles obvous. In all, n order to obtan a large SUW, sum of jont dstrbuton angles should be reduced, and Stewart manpulators wth large h or large dfference between R and r have advantages.

6 CHINESE JOURNAL OF MECHANICAL ENGINEERING 25 5 Atlases of Global Condtonng Index In mathematcs, the condton number s used to analyze senstvty and uncertanty of the soluton of lnear equatons. When adopted n the parallel manpulator, the condton number of the Jacoban matrx s an mportant local performance ndex, whch ndcates dexterty and sotropy, as well as mples velocty, accuracy and rgd features. The condton number of the Jacoban matrx can be expressed as = max mn, (14) where max and mn are maxmum and mnmum sngular values of the Jacoban matrx, and 1. The local condtonng ndex(lci) s defned as the recprocal of the condton number of the Jacoban matrx. In order to evaluate the performance change of a parallel manpulator wth dfferent desgn parameters, the Global Condtonng Index (GCI) s adopted, whch can be expressed as (3) In the rght area, when l 1 s determned, GCI value grows wth the decrease of l 3 and the ncrease of l 2. Fg. 7. GCI atlas In Fg. 8, the mpact of jont dstrbuton angles and on the GCI value s analyzed for three sets of length parameters. Obvously, the GCI value decreases wth the ncrease of the sum of jont dstrbuton angles. However, when ther sum s fxed, separate jont dstrbuton angles have lttle effect on the GCI value. W 1 d W d W, W = ò ò (15) where W s the regular usable workspace of the parallel manpulator. A larger GCI value promses a better control accuracy n general. The Stewart parallel manpulator possesses sx degrees of freedom n the termnal Cartesan coordnate system, ncludng three rotatonal degrees of freedom and three translatonal degrees of freedom. Consderng the dmensonal homogenety, the Jacoban matrx of the Stewart manpulator s normalzed, usng the end-effector crcumradus r. And, the normalzed Jacoban matrx can be wrtten as é T T Tù T æ ö 1 T æ ö 2 T æ ö 6 N 1, p 1 ; 2, p 2 ; ; 6, p J = s R s 6. çè r ø s R s çè r ø s R s çè r ø êë úû Then, the velocty mappng functon of the Stewart manpulator can be expressed as æ ö t J N ç = N N = r J X L. çè ω ø Based on the Jacoban matrx obtaned above, correspondng GCI atlas can be drawn, as shown n Fg. 7 ( = = 10, zero-rotaton SUW). We can conclude the followng. (1) Larger values are n the center area, whle mnmum values le n left and rght ends. (2) In the left area, for a gven l 1, GCI value grows wth the decrease of l 2 and the ncrease of l 3. Fg. 8. Relatonshp between the GCI value and jont dstrbuton angles

7 26 Y SHAO Zhufeng, et al: Atlas Based Knematc Optmum Desgn of the Stewart Parallel Manpulator The nfluence of jont dstrbuton angles can also be deduced from GCI atlases consderng dfferent jont dstrbuton angles, as shown n Fg. 9. In Fg. 9(a), dashed curves llustrate the GCI atlas for = = 50, whle sold curves correspond to that for = = 30. In Fg. 9(b), dashed curves show the GCI atlas for = 5 and = 55, whle sold curves represent that for = 55 and = 5,.e., the sum of jont dstrbuton angles s a constant value of 60. Comparatve analyss of these curves ndcates the followng. (1) Sum of jont dstrbuton angles has obvous mpact on the GCI atlas. However, specfc values of and have lttle nfluence, f ther sum s nvarant. (2) When sum of jont dstrbuton angles( + ) ncreases, the regon wth large GCI value shrnks, and moves toward the upper left n the desgn space. (3) In most area of the desgn space, the GCI value decreases, when the sum of jont dstrbuton angles + grows. Only n a small area surrounded by lnes l 2 =1.5 and l 2 =l 3, the GCI value may ncrease wth growng angle sum. accordng to engneerng experence. Here, the maxmum length of the lmb s assumed twce the mnmum length. The next step s to establsh the desgn space and draw requred performance atlases. Snce ths part has been completed, we can move on to the thrd step to obtan a canddate regon n the desgn space. If the radus of the zero-rotaton SUW s supposed to be not less than 0.45 and GCI value s larger than or equal to 0.5(performance requrements), the canddate regon can be determned, as shown n Fg. 10. And, canddate ranges of length parameters descrbed n the desgn space can be obtaned approxmately as l1 Ì [ 0.44, 0.71], l2 Ì [ 1.13,1.52], and l3 Ì [ 0.81,1.21 ]. Fg. 10. Canddate regon of the optmum desgn Fg. 9. GCI atlases consderng dfferent jont dstrbuton angles 6 Example of Atlas Based Optmum Desgn The frst step of atlas based optmum desgn for the Stewart manpulator s to determne jont dstrbuton angles and telescopc proporton of the lmb. Based on above analyss, we can conclude that jont dstrbuton angles mpact the manpulator performance manly n the form of ther sum, and smaller sum s ndcatve of a better knematc performance. Thus, the sum of jont dstrbuton angles should be as small as possble, on the condton of meetng the nstallaton requrement of sphercal and unversal jonts. In detaled desgn phase, specfc values of jont dstrbuton angles can be adjusted more precsely wth lttle mpact on manpulator performances. In ths secton, the sum of jont dstrbuton angles s set to 20 ( + = 20 ), n order to make full use of obtaned atlases. The telescopc range of the lmb can be determned In the fourth step, a set of non-dmensonal optmum parameters can be determned by the most desred performance wthn the obtaned canddate regon. If we want to get a manpulator wth maxmum workspace n the canddate regon, accordng to Fgs. 4 and 10, the lowest pont n the canddate regon should be adopted. Thus, the mnmum value of l 1 s adopted, and l 1 =0.44, l 2 =1.36 and l 3 =1.20. Of course, the maxmum GCI value can also be adopted as the target to get another set of optmum parameters. In the ffth step, the dmenson factor s determned, whch converts the non-dmensonal parameters nto practcal dmensonal ones. The Stewart manpulator s usually adopted to mplement the 5-axs machne tool. Thus, the yaw angle, the angle between the end effector and the horzontal plane, s used here to descrbe the end-effector posture. The LCI s adopted to gve more refned descrpton of the manpulator performance. As shown n Fg. 11, f the LCI s supposed to be not less than 0.4, the radus of SUW can be determned, such as 0.29, 0.38, 0.46 and 0.53, dependng on the yaw angle requrement. The SUW radus of the dmensonal manpulator equals the product of dmenson factor and the non-dmensonal SUW radus. For example, f the desred SUW radus s 400mm, the factor can be calculated as

8 CHINESE JOURNAL OF MECHANICAL ENGINEERING 27 1 = 0.4/ 0.29 = 1.38 m, 2 = 0.4/ 0.38 = 1.05 m, 3 = 0.4/ 0.46 = 0.87 m, correspondng to yaw angle requrements of 15, 10 and 5, respectvely. Fg. 11. Radus of SUW consderng end-effector posture The last step s to calculate the dmensonal parameters. The dmensonal parameters of the optmum Stewart manpulator can be determned wth the obtaned dmenson factor and non-dmenson parameters. Takng 2 for example, we can get practcal parameters from Eq. (12) as r = l1 2 = 0.46 m, R= l2 2 = 1.43 m, h= l3 2 = 1.26 m. Together wth the jont dstrbuton angles, locaton of sphercal and unversal jonts can be determned. Then, wth the telescopc proporton, ntal length and telescopc range of the lmb can be fgured out. 7 Conclusons (1) The sphercal usable workspace s defned to descrbe the workspace of the Stewart manpulator, and establshes the bass for global ndex analyss. (2) When the telescopc range of the extensble lmb s gven, the desgn parameters of the general Stewart manpulator s ratonally smplfed nto two angle and three length desgn parameters, namely, ntal dstance, jont dstrbuton angles and raduses of both the base and the end effector. The dmensonless desgn space of length parameters s establshed n the two-dmensonal plane. (3) Prmary knematc ndces such as workspace and condton number have been dscussed, jont dstrbuton angles are analyzed through supermposng on the desgn space of length parameters, and correspondng atlases are deduced. These atlases accurately llustrate varaton trends of the manpulator performance n the fnte nondmensonal desgn space wth respect to parameter changes of the Stewart manpulator, and can be flexbly appled to quanttatve and qualtatve analyss to get the desred optmal desgn. (4) Impact of jont dstrbuton angles on knematc performances s analyzed, whch shows that the sum of jont dstrbuton angles affects manpulator performances sgnfcantly. Generally, the Stewart manpulator wth mnor sum of jont dstrbuton angles promses better knematc performances wth respect to workspace, sotropy and accuracy. At the same tme, t s found that the Stewart manpulator wth a large ntal dstance or large crcumradus dfference between the base and the end effector has an advantage on obtanng a large workspace. (5) Atlas based knematc optmum desgn of the Stewart parallel manpulator s establshed, and the proposed optmum method s well llustrated wth an example, consderng the end-effector posture. The establshed optmum desgn method can be further appled to other mult-parameter parallel manpulators. References [1] TANG X, TANG L, WANG J, et al. Workspace qualty analyss and applcaton for a completely restraned 3-DOF planar cable-drven parallel manpulator[j]. Journal of Mechancal Scence and Technology, 2013, 27(8): [2] HOSTENS I, ANTHONIS J, RAMON H. New desgn for a 6 DOF vbraton smulator wth mproved relablty and performance[j]. Mechancal Systems and Sgnal Processng, 2005, 19(1): [3] LI B, CAO Y, ZHANG Q, et al. Orentaton-sngularty representaton and orentaton-capablty computaton of a specal class of the Gough-Stewart parallel mechansms usng unt quaternon[j]. Chnese Journal of Mechancal Engneerng, 25(6): [4] STAICU S, ZHANG D. Dynamc modelng of a 4-DOF parallel knematc machne wth revolute actuators[j]. Internatonal Journal Manufacturng Research, 2008, 3(2): [5] LIU X J, WANG J, ZHENG H J. Optmum desgn of the 5R symmetrcal parallel manpulator wth a surrounded and good-condton workspace[j]. Robotcs and Autonomous Systems, 2006, 54(3): [6] YAO R, TANG X, WANG J, et al. Dmenson optmzaton desgn of the four-cable drven parallel manpulator n FAST[J]. IEEE/ASME Transactons on Mechatroncs, 2010, 15(6): [7] TANG X, YAO R. Dmensonal desgn on the sx-cable drven parallel manpulator of FAST[J]. Journal of Mechancal Desgn, 2011, 11: [8] CECCARELLI M, LANNI C. A mult-objectve optmum desgn of general 3R manpulators for prescrbed workspace lmts[j]. Mechansm and Machne Theory, 2004, 39(2): [9] GAO Z, ZHANG D, GE Y. Desgn optmzaton of a spatal sx degree-of-freedom parallel manpulator based on artfcal ntellgence approaches[j]. Robotcs and Computer-Integrated Manufacturng, 2010, 26(2): [10] WANG B, HAN W. Performance atlases of global condtonng ndex of planar 3-DOF parallel manpulator wth accuracy redundancy[j]. Advanced Materals Research, 2012, : [11] LIU X J, WANG J, PRITSCHOW G. Performance atlases and optmum desgn of planar 5R symmetrcal parallel mechansms[j]. Mechansm and Machne Theory, 2006, 41(2): [12] GAO F, ZHANG X, ZHAO Y, et al. A physcal model of the soluton space and the atlases of the reachable workspaces for 2-DOF parallel plane wrsts[j]. Mechansm and Machne Theory, 1996, 31(2):

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