Accuracy Analysis of General Parallel Manipulators with Joint Clearance

Transcription

1 007 IEEE Internatonal Conference on Robotcs and Automaton Roma, Italy, Aprl 007 WeC6. Accuracy Analyss of General Parallel Manpulators wth Jont Clearance Jan Meng, ongjun Zhang, Tnghua Zhang, Hong Wang, Zexang L Abstract ue to the jont clearance, parallel manpulators always exhbt some poston and orentaton errors at the moble platform. Ths paper ams to provde a systematc framework for the error analyss problem of general parallel mechansms nfluenced by the jont clearance. A novel and effcent method s proposed to evaluate the maxmal pose errors of general spatal parallel manpulators wth jont clearance. I. INTROUCTION espte the fact that jont clearance smplfes the assembly and manufacturng of parallel mechansms, the generated pose error of lnks, however, can not be gnored when the mechansm requres hgh accuracy. Compared wth other error sources, such as assembly and manufacturng errors and motor actuaton errors, etc, jont clearance has more sgnfcant mpact on the poston accuracy for both seral and parallel manpulators. Therefore, t s qute mportant to provde an accurate model that can predct the effects of jont clearance on the mechansm s postonng performance. As ndcated n [1], contrary to the assembly and manufacturng errors, jont clearance leads to uncertan error motons at an arbtrary pose of the mechansm. Its effects are hghly non-repeatable and can not be rectfed wth any knd of calbraton. Much research has been devoted to the problem of accuracy n parallel mechansms. Some authors appled probablstc analyss to determne the pose error of clearanceaffected jonts and movng platform []. Parent-Castell and Venanz [3] used the vrtual work method to determne the poston that the movng platform reaches when a gven external load s exerted on t. P.Voglewede and I.Ebert-Uphoff [4] amed to predct precsely the moble platform error motons for 3-RRR and some specal parallel mechansms wth jont clearance. In ths paper, we wll provde a deeper nsght nto the accuracy problem of parallel mechansms affected by jont clearance. A general and yet effcent error evaluaton method s also proposed whch can handle any knd of parallel mechansms, no matter planar or spatal, overconstraned or non-overconstraned. Furthermore, ts effcency makes t possble to compute the global maxmum pose errors of a clearance-affected mechansm n a prescrbed workspace, other than just at a gven theoretcal confguraton. Fg. 1. Fg.. Pose Error of A Rgd Body Clearance-affected Revolute Jont II. POSE ERROR ANALYSIS OF PARALLEL MANIPULATORS WITH CLEARANCE-AFFECTE PAIRS A. End-effector and Jont Pose Error Fg.1 shows the end-effector of a parallel mechansm wth some confguraton (ncludng the poston and orentaton) error. Suppose that the nomnal (or deal) confguraton of the end-effector (the dashed rgd body n Fg.1) s g 0 SE(3), whch s the relatve transformaton from the nomnal body frame B to the nertal frame A. Wth some confguraton error, the relatve transformaton from the actual body frame B to the nertal frame A, however, gves rse to the real confguraton g of the end-effector. Let A be another frame attached to the end-effector, whch s chosen such that the relatve transformaton from B to A s g 0 (see Fg.1). Then t s clear that wthout confguraton error, A frame wll co- Ths project s supported by RGC Grant No. HKUST6301/03E, HKUST676/04E, and J.Meng,.Zhang and Z.L are wth epartment of Electrcal and Electronc Engneerng, Hong Kong Unversty of Scence and Technology,Clear Water Bay, Kowloon, Hong Kong eejmeng@ust.hk T.Zhang and H.Wang are wth the vson of Control and Mechatronc Engneerng, HIT Shenzhen Graduate School, P.R. Chna Fg. 3. Clearance-affected Prsmatc Jont /07/$ IEEE. 889

2 WeC6. ncde wth the nertal frame A, and B frame wll concde wth B. Hence the error rgd moton of the end-effector from the nomnal confguraton to the real confguraton can be descrbed ether by the relatve transformaton from B to B, whch s g0 1 g, or the relatve transformaton from A to A, that s g g0 1. Correspondngly, the confguraton error can be defned as (g0 1 g e) and (g g0 1 e), whch are called the confguraton error wth respect to the body frame and the nertal frame respectvely. Normally the error rgd motons g0 1 g and g g 1 0 are restrcted wthn a small neghborhood of e. Hence the two expressons of the confguraton error can be reasonably approxmated by ther frst order terms, that s, g0 1 g e = eê1 xbeê ybeê3 zbeê4 αbeê5 βbeê6 γb e (ê 1 x b + +ê 6 γ b ) where {e = 1,, 6} s the canoncal bass of R 6, and the sx components x b,, γ b (and x a,, γ a ) represent the three translaton and three rotaton errors of the end-effector about the x, y and z axs of the B (and A) frame respectvely. In the above two equatons, the operaton dentfes R 6 wth se(3) (see [5] for more detals). Hence f we let δg b =( x b, y b, z b, α b, β b, γ b ) T and δg a =( x a, y a, z a, α a, β a, γ a ) T, then t s easy to see that g0 1 g e δg ˆb and g g0 1 e δg ˆa.To avod the confuson wth the term confguraton error, δg b wll be called the pose error wth respect to body frame, and δg a the pose error wth respect to the ntal frame. From now on, the pose error of the end-effector other than the confguraton error wll be manly used and studed n ths paper. As the relatve transformaton from B to A frame s g 0, the two expressons of the pose error are related by δg a = Ad g0 δg b, where for g =(p, R), the Adjont map s defned by R ˆpR Ad g = 0 R The pose error of the end-effector s caused by many factors, such as assembly and manufacturng errors (occurrng at the adjacences between the lnks and the jonts), actuaton motor errors, jont clearance, etc. As the major source of the pose error comes from the error motons caused by the jont clearance [6], n ths paper, we wll focus on studyng the mpact of jont clearance on the devatons of the endeffector s confguratons. Hence, we wll assume that jont clearance s the unque error source. In the reman part of ths subsecton, we study the mpact of jont clearance on the pose error of two lnks connected by a clearance-affected jont. Only revolute and prsmatc jonts are consdered n ths paper, but other types of jonts can be analyzed n a smlar manner. A clearance-affected knematc par (.e., a par wth jont clearance) actually has 6-oFs. Thus at a gven nomnal confguraton (the theoretcal confguraton of the par), there are 6 twsts assocated wth a par. Some twsts result n the desred motons of the jont, called deal twsts, whereas others gve rse to the error motons caused by jont clearance. For example, for a clearance-affected revolute par n Fg., f we attach a local coordnate frame C to the bearng wth z-axs along the par s theoretcal axs, then e 6 generates the deal rotatonal motons of the jont, whereas e 1,,e 5 generate error motons of the shaft wth respect to the bearng. The pose error of the relatve confguraton of the two lnks wth respect to the local frame C thus s gven by δυ c = 5 e σ (1) where σ 1, σ, σ 3 represent 3-nfntesmal error translatons along x, y, z axs, and σ 4, σ 5 represent - and g g0 1 e = eê1 xaeê yaeê3 zaeê4 αaeê5 βaeê6 γa enfntesmal error rotatons about x, y axs. Hereafter (ê 1 x a + +ê 6 γ a ) =1 σ,=1,, 5 wll be called the error motons caused by the jont clearance, wth values restrcted by the jont geometry and the magntude of jont clearance. For a partcular desgn of a clearance-affected revolute jont, t s possble to formulate explctly the set of constrants that bound the values of σ. For example, for the journal bearng desgn (Fg.) of a revolute jont, because of the axal symmetry, the x, y axs can be chosen freely. Wth orgn chosen at the jont center, and the geometrcal dmensons L, and the magntude of jont clearance ɛ r, ɛ a gven beforehand, the values of σ are constraned by a set of quadratc and second order cone constrants : ( σ 1 + L σ 5) +( σ L σ 4) ɛ r ( σ 1 L σ 5) +( σ + L σ 4) ɛ r σ 4 + σ5 σ () 3 ɛ a σ 4 + σ5 + σ 3 ɛ a For other types of desgns of clearance-affected revolute jont, though the structure may be more complcated, one can get smlar constrant nequaltes as (). If there s an nertal reference frame A such that the relatve transformaton from C to A s g R ac, then the pose error caused by jont clearance can be wrtten wth respect to the nertal frame A by δυ a = Ad g R ac δυ c = 5 =1 (Ad g R ac e ) σ (3) Smlarly, for a clearance-affected prsmatc par(fg.3), f we attach a local coordnate frame C to the supportng carrage wth x-axs along the par s deal translatonal axs, then the pose error of the slder wth respect to the supportng carrage s gven wth respect to C by δγ c = 6 e j τ j (4) j= where τ j,j =,, 6 are also error motons caused by the jont clearance. If y, z axs are parallel to the two faces of the supportng carrage, and the orgn les on the geometrcal center of the jont (see Fg.3), then values of the 890

3 WeC6. error motons are restrcted by a set of lnear constrants ɛ a + L τ 6 τ 4 + τ ɛ a ɛ a L τ 6 + τ 4 + τ ɛ a ɛ b L τ 5 + τ (5) 4 + τ 3 ɛ b ɛ b + L τ 5 τ 4 + τ 3 ɛ b Supposng that the transformaton from C to the nertal frame A s g P ac, then the pose error s gven wth respect to frame A by δγ a = Ad g P ac δγ c = 6 j= (Ad g P ac e j) τ j (6) Fnally, we remark that the two desgns of Fg. and Fg.3 have ever been ntensvely studed n [7] and [3]. However, the constrants of error motons n those lteratures are formulated as sets of general nonlnear nequaltes. In ths paper, we have reformulated them nto sets of so-called convex constrants, as seen from () and (5). As we wll see later, ths reformulaton wll make the convex optmzaton feasble to the evaluaton of the pose errors. Fg. 4. Tsa s 3-UPU Manpulator B. Gross Pose Error of Jonts The desred motons of an actuated jont are controlled by the motors mounted on the jont. Assumng there s no motor actuaton error, then at a nomnal confguraton, the pose error of an actuated jont s solely rased by ts jont clearance. Hence the expresson of gross pose error of an actuated jont s the same as Eq.(1) and (3). However, for passve jonts n a parallel mechansm, n addton to the errors motons caused by jont clearance, the dle motons about the deal twsts also play a role n the pose errors of ther two lnks. Hence, the gross pose error of a passve revolute jont s gven wth respect to ts local frame by δυ c = 5 e σ + e 6 θ (7) =1 where θ s the dle rotaton error of the shaft about the jont s deal axs. Wth respect to the nertal frame A, the gross pose error s δυ a = Ad g R ac δυ c = 5 =1 (Ad gac R e ) σ +(Ad g R ac e 6 ) θ = 5 =1 (Ad gac R e ) σ + ξ θ where ξ s the deal twst of the jont when expressed n the nertal frame. Smlarly, two gross pose errors of a passve prsmatc jont can be wrtten as Fg. 5. A Parallelogram Jont C. End-effector Pose Error of A Parallel Mechansm In ths subsecton, we study the pose error of the endeffector of a parallel mechansm caused by the clearance of ts consttuent jonts. We use some examples to llustrate how to get the pose error of the end-effector at a general theoretcal confguraton(say, g 0 ), subject to the error motons generated by all the clearance-affected jonts. Example 1: Pose Error Analyss of Tsa s Manpulator Consder a Tsa s 3-UPU manpulator n Fg.4. Each U- jont conssts of two perpendcularly connected clearanceaffected revolute jonts modelled by Fg.. The three prsmatc jonts P 1,,P 3, beng actuated, are also clearanceaffected and modelled by Fg.3. The nertal frame A and the body frame B are located at the center of the base and the moble platform respectvely. And the local frames of R j (denoted Cj R), and that of P (denoted by C P ), =1,, 3, j =1,, 4, are located at the center of each jont, wth x, and 6 δγ c = e j τ j + e 1 µ (8) j= δγ a = Ad g P ac δγ c = 6 j= (Ad gac P e j) τ j + η µ respectvely, where µ s the dle translatonal error along the jont s deal axs, and η s the deal twst representaton of the jont n the nertal frame A. Fg. 6. Orthoglde Manpulator 891

4 WeC6. y axs chosen at one s convenence. The moble platform(or end-effector) deally can undergo three of pure translatons T (3) under some geometrcal condtons [8]. At a nomnal confguraton g 0 T (3) whch s not sngular for the sake of smplcty, the transformaton matrx from Cj R and CP to A, denoted by gj R and gp respectvely, can be obtaned through the nverse knematc analyss of the deal mechansm at g 0. From the last subsecton, the pose error of P s gven wth respect to the nertal frame A by P = 6 l= (Ad g P e l) τ l =1,, 3 and the pose error of R j s gven by R j = 5 l=1 (Ad gj R e l) σ jl + ξ j θ j where =1,, 3,j =1,, 4, and ξ j s the deal twst assocated wth R j. The pose error of th open subchan, =1,, 3, can thus be wrtten by E = P + 4 j=1 R j = 6 l= (Ad g P e l) τ l + 4 j=1 ξ j Θ j + 4 j=1 5 l=1 (Ad g R j e l) σ jl = A dγ + 4 j=1 B j dυ j + J dθ = A dγ + B dυ + J dθ where A = [Ad g P e Ad g P e 6 ], dγ = ( τ,, τ 6 ) T, B j = [Ad g R j e 1 Ad g R j e 5 ], dυ j = ( σ j1,, σ j5 ) T, B = [B 1 B 4 ], dυ = (dυ T 1,,dΥT 4 )T, J = [ξ 1 ξ 4 ], dθ =( θ 1,, θ 4 ) T. Equatng E 1 and E, E and E 3, we can get J dθ 1 dθ dθ 3 = A dγ 1 dγ dγ 3 + B dυ 1 dυ dυ 3 (9) (10) where J1 J J = (11) J J 3 s a nonsngular square matrx, and A1 A A = B1 B, B = A A 3 B B 3 From Eq.(10) and (11), we may see that the dle passve jont motons are unquely determned by the error motons τ l and σ jl caused by the jont clearance. After the dle motons dθ are determned, we can substtute t nto Eq.(9) for any subchan to obtan the pose error of the movng platform the nomnal confguraton g 0. Choosng the frst subchan as example, we can wrte the result n a concse way as follows E g0 = A 1 dγ 1 + B 1 dυ 1 + J 1 dθ 1 = A 1 dγ 1 + B 1 dυ 1 +[J 1 00] dθ 1 dθ dθ 3 = A 1 dγ 1 + B 1 dυ 1 +[J 1 00] J 1 dγ 1 dυ 1 A dγ + B dυ dγ 3 dυ 3 = H (dγ T 1,dΓ T,dΓ T 3,dΥ T 1,dΥ T,dΥ T 3 ) T In the above equaton, we get a lnear transformaton matrx H that maps the error motons of the clearanceaffected jonts, τ l and σ jl, to the pose error of the movng platform E g0. From Eq.(10), τ l and σ jl are free varables of the mechansm, the number of whch reflects the total extra of of the mechansm at a non-sngular nomnal confguraton. However, the values of τ l and σ jl are bounded wthn the sets of convex constrants gven by nequaltes () and (5) for each jont. Therefore, the maxmal pose error of the movng platform at the nomnal confguraton g 0, denoted by Eg m 0, can be obtaned by sx convex optmzatons appled to each components of E g0. If one further want to get the maxmal pose error of the movng platform n a prescrbed workspace W, for example, a cube centered by the home confguraton, he can dscretze W and calculate Eg m at any pont of W. After that, the global maxmal pose error n W can be obtaned by E m = (max g W Em g (1),, max g W Em g (6)) T where E m g () s the th component of E m g. For overconstraned parallel manpulators, however, not all error motons assocated wth jonts clearance are free varables, as seen from the next example. Example : Pose Error Analyss of A Parallelogram Jont In ths example, we study the pose error of a parallelogram (P a ) composng of four clearance-affected revolute jonts. Snce P a usually serves as an extended par n complex parallel mechansms, e.g., the elta robot, we also attach a frame C, called the local frame of the parallelogram jont, to the base lnk of the parallelogram, as shown n Fg.5. Frame C s located at the center of the base lnk, wth z axs parallel to the deal rotaton axs of R j, and x axs pontng to the geometrcal center of R 1. Thus the deal confguraton space of the parallelogram C M sacrclenthex-y plane of C. At each nomnal confguraton g 0 C M, we defne four local frames C j for R j, =1,, j =1,. Asthe choce of x, y axs of C j s free, for our convenence, C j may be chosen wth orgn lyng at the geometrcal center of R j, and x, y, z axs algned wth that of C. Hence, the transformaton from C j to C s gven by I pj g j = 0 1 where p j s the geometrcal center of R j. Assume that R 11 s the actuaton jont, then the pose error of four revolute jonts wth respect to the C frame s gven by R j = 5 (Ad gj e l ) σ jl + ξ j θ j =1,,j =1, l=1 where θ 11 =0. The pose error of th subchan, =1, thus can be easly got as E = R 1 + R = j=1 5 l=1 (Ad g j e l) σ jl + j=1 ξ j Θ j = j=1 A j dυ j + j=1 ξ j Θ j 89

5 WeC6. where A j = [Ad gj e 1 Ad gj e 5 ], dυ j = ( σ j1,, σ j5 ) T, = 1,, j = 1,. As θ 11 = 0, by equatng E 1 and E, we can get J Θ 1 Θ 1 = A Θ dυ 1 dυ 1 dυ (1) where J = [ξ 1 ξ 1 ξ ], and A = [ A 11 A 1 A 1 A ]. Clearly, J s a 6 3 full column rank matrx. J may combne wth three twsts Ad g1 e 3, Ad g1 e 4, and Ad g1 e 5 to form a 6 6 non-sngular square matrx J =[ξ 1 ξ 1 ξ Ad g1 e 3 Ad g1 e 4 Ad g1 e 5 ]. Then Eq.(1) can be re-wrtten as follows J Θ 1 Θ 1 Θ σ 13 σ 14 σ 15 = A dυ 1 σ 11 σ 1 dυ (13) where A =[ A 11 A 1 Ad g1 e 1 Ad g1 e A ]. Therefore, only 17 error motons, dυ 1, dυ and σ 11, σ 1 are free varables of the mechansm. The other three jont error motons σ 13, σ 14, σ 15, plus the dle passve jont motons Θ 1, Θ 1, Θ, are unquely determned by these free varables. enote by B = J 1 A, B() the th row of B, =1,, 6, and g ac the transformaton from the local frame C to the nertal frame A. Then pose error of the end-effector at the nomnal confguraton g 0 s gven wth respect to the nertal frame by E g0 = Ad gac ( R 11 + R ( 1 ) ) dυ11 = Ad gac [A 11 A 1 ] + Ad dυ gac ξ 1 Θ 1 ( 1 ) dυ11 = Ad gac [A 11 A 1 ] dυ 1 dυ 1 +Ad gac ξ 1 B(1) σ 11 σ 1 dυ = H (dυ T 11 dυ T 1 σ 11 σ 1 dυ T ) T subject to the constrants ( σ j1 + L σ j5) +( σ j L σ j4) ɛ r ( σ j1 L σ j5) +( σ j + L σ j4) ɛ r σj4 + σ j5 σ j3 ɛ a σj4 + σ j5 + σ j3 ɛ a σ 13 B(4) dυ 1 σ 14 = B(5) σ 11 σ 15 B(6) σ 1 dυ (15) where =1, and j =1,. Thus the calculaton of maxmal pose error of an overconstraned parallel manpulator at g 0 s a convex optmzaton problem wth lnear equalty constrants. By [9], such a problem can also be quckly solved by very effcent algorthms. P a s often used as an extended passve par n some common parallel mechansms, e.g, the elta robot. In ths case, R 11 s a passve revolute jont nstead of an actuated one, whch mples that θ Assume that the body frame B orgnally concdes wth A frame, then the deal confguraton space of the end-effector s gven by { } I (eˆωθ 11 I)v Q = θ [0, π] where ω s the deal rotaton axs of the parallelogram, and v the vector connectng R 1 and R at the home confguraton. Clearly, at the nomnal confguraton g 0 = Q(θ 11 ), the confguraton error of the end-effector caused by θ 11 s gven wth respect to the nertal frame A by F g0 = dq Q 1 0 ωeˆωθ 11 v θ 11 = θ dθ The total pose error thus s the lnear summaton of F g0 and E g0 (16) T g0 = E g0 + F g0 = H (dυ T 11 dυ T 1 σ 11 σ 1 dυ T ) T +ωeˆωθ11 v θ 11 subject to the constrants of (15). The value of θ 11,as we wll see n the next example, s determned by the loop closure equaton of the parallel mechansm that contans ths P a. Example 3: Pose Error Analyss of Orthoglde The Orthoglde manpulator [10], as shown n Fg.6, conssts of three P RP a R dentcal subchans. The actuated jonts are the three orthogonal prsmatc ones. Ideally, the manpulator s an overconstraned one, wth end-effector moton a subset of T (3). At a nomnal confguraton g 0 T (3), usngthe same approach as before, we may fnd the pose errors of (14) prsmatc jonts P and revolute jonts R j, =1,, 3, j =1,. As the end-effector s connected to the prsmatc jonts through a set of three passve parallelograms, the pose error of the parallelogram s gven by Eq.(16). Thus the pose error of the th subchan, =1,, 3, s shown to be E = P + R 1 + T + R = 6 l= (Ad g P e l) τ l + 5 l=1 (Ad g1 R e l) σ 1l + ξ 1 Θ 1 +H (dυ T 11 dυt 1 σ 11 σ 1 dυ T )T +ω eˆωθ11 v θ l=1 (Ad g R e l) σ l + ξ Θ = A dγ + j=1 B j dω j + H dυ +J dθ = A dγ + B dω + H dυ + J dθ where g P s the transformaton matrx from the local frame of the P to the nertal frame A, and gj R the one from the local frame of R j to A. Furthermore, A = [Ad g P e Ad g P e 6 ], dγ =( τ,, τ 6 ) T, B j = [Ad g R j e 1 Ad g R j e 5 ], dω j = ( σ j1,, σ j5 ) T, 893

6 WeC6. B = [B 1 B ], dω = (dω T 1,dΩT )T, dυ = (dυ T 11 dυ T 1 σ 11 σ 1 dυ T )T, J = [ξ 1 ω eˆωθ11 v ξ ], dθ = ( θ 1, θ 11, θ ) T, =1,, 3, j =1,. Equatng E 1 and E, E and E 3, we can get J dθ 1 dθ dθ 3 = A dγ 1 dγ dγ 3 + B dω 1 dω dω 3 + C dυ 1 dυ dυ 3 where J1 J J = (17) J J 3 s a 1 9 full column rank matrx, and A1 A A = B1 B, B = A A [ 3 ] B B 3 H1 H C = H H 3 Note that the rotaton axs of the twst Ad g P e 4 s along the deal translaton axs of P, =1,, 3, hence the former equaton may be re-wrtten as dθ REFERENCES 1 τ 14 J dθ τ 4 = A dγ 1 dγ + B dω 1 dω + C dυ [1] K.L. Tng, J. Zhu, and. Watkns. The effects of jont clearance 1 on poston and orentaton devaton of lnkages and manpulators. dυ Mechansm and Machne Theory, 35: , 000. dθ 3 dγ 3 dω 3 dυ [] M.Mayouran and J.Rastegar. Stochastc modelng of the mechancal 3 behavor of mechansms n the presence of jont clearances. In τ Proceedngs of 1990 ASME Mechansms Conference, pages , where [3] S. Venanz. Methods for Clearance Influence Analyss n Planar and [ J J1 Ad = g P e ] Spatal Mechansms. Ph thess, Unversty of Bologna, Bologna, 4 J 1 Ad g P e 4 J Ad g P e 4 J 3 Ad g P 3 e 4 s a 1 1 non-sngular matrx, and A A = 1 A A A, 3 A = [Ad g P e Ad g P e 3 Ad g P e 5 Ad g P e 6 ], dγ = ( τ, τ 3, τ 5, τ 6 ) T, =1,, 3. From the above transformed equaton, one may see that only dγ, dω and dυ, = 1,, 3, are free varables of the mechansm. Three jont error motons τ 4, together wth the dle passve jont motons dθ, =1,, 3, are unquely determned by totally =78 free varables. The pose error of the end-effector thus s gven by E g0 = A 1 dγ 1 + B 1 dω 1 + H 1 dυ 1 + J 1 dθ 1 = A 1 dγ 1 + B 1 dω 1 + H 1 dυ 1 +[J 1 0 0] dγ J 1 A 1 dω 1 dυ 1 dγ + B dω + C dυ dγ 3 dω 3 dυ 3 subject to the constrant of () for revolute jonts, (5) for prsmatc jonts, (15) for parallelogram, and the followng lnear equaltes τ 14 τ 4 = (dγ 1T,,dΓ T 3,dΩ T 1,,dΩ T 3,dΥ T 1,,dΥ τ T 3 ) T 34 where the matrx can be derved from Eq.(3). Hence the calculaton of the maxmal pose error of the Orthoglde manpulator at g 0 s a stll a convex optmzaton problem wth lnear equalty constrants. III. CONCLUSION In ths paper, we propose a general method to evaluate the pose(poston and orentaton) error of the end-effectors of parallel manpulators due to the jont clearance. We show that for non-overconstraned parallel manpulators, the error motons caused by jont clearance are free varables subject to some constrants defned by the jont geometry and magntude of clearance. However, for overconstraned parallel mechansms, part of these jont error motons are dependent on the remanng ones. For some partcular desgns of common lower pars, the pose error analyss for both non-overconstraned and overconstraned parallel manpulators can be formulated nto standard convex optmzaton problems, whch makes t possble to compute the maxmum pose errors n a prescrbed workspace other than just at a sngle confguraton. Italy, 004. [4] P. Voglewede and I.E. Uphoff. Applcaton of workspace generaton technques to determne the unconstraned moton of parallel manpulators. ASME Journal of Mechancal esgn, 16:83 90, 004. [5] R. Murray, Z.X. L, and S. Sastry. A Mathematcal Introducton to Robotc Manpulaton. CRC Press, [6] C.H. Han, J.W. Km, J.W. Km, and F.C. Park. Knematc senstvty analyss of the 3-UPU parallel mechansm. Mechansm and Machne Theory, 37: , 00. [7] H.H.S. Wang and B. Roth. Poston errors due to clearance n journal bearng. ASME Journal of mechansms, Transmssons, and Automaton n esgn, 111:315 30, [8] L.W. Tsa. Knematcs of a three-of platform wth three extensble lmbs. n Recent Advances n Robot Knematcs: Analyss and Control (Lenarc J., Husty M.L. Eds.), Kluwer, pages 49 58, [9] S. Boyd and L. Vandenberghe. Convex Optmzaton. Cambrdge Unversty Press, 1st edton, 001. [10].Chablat and P. Wenger. Archtecture optmzaton of a 3-dof translatonal parallel mechansm for machnng applcatons, the orthoglde. IEEE Transacton on Robotcs and Automaton, 19(3): ,