g for optimal rob ot design In this pap er revisit these concepts for parallel rob ots and exhibit some surprising results at least for the author that sho that these concepts ha to manipulated with care for prop er understanding of the kinematics eh ID: 30072 Download Pdf

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g for optimal rob ot design In this pap er revisit these concepts for parallel rob ots and exhibit some surprising results at least for the author that sho that these concepts ha to manipulated with care for prop er understanding of the kinematics eh

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Jacobian, manipulabilit condition um er and accuracy of parallel rob ots J-P Merlet INRIA, BP 93, 06902 Sophia-An tip olis, rance Abstract: Although the concepts of jacobian matrix, manipulabilit and condition um er ha een oating around since the early eginning of rob otics their real signicance is not alw ys ell understo d, although these conditioning indices pla an imp ortan role e.g. for optimal rob ot design. In this pap er re-visit these concepts for parallel rob ots and exhibit some surprising results (at least for the author!) that sho that these

concepts ha to manipulated with care for prop er understanding of the kinematics eha vior of rob ot. In tro duction arallel rob ots are no ada ys lea ving academic lab oratories and are nding their in an increasingly larger um er of application elds suc as telescop es, ne ositioning devices, fast pac aging, mac hine-to ol, medical application. ey issue for suc use is optimal design as erformances of parallel rob ots are ery sensitiv to their dimensioning. Optimal design metho dologies ha to rely on kinetostatic erformance indices and accuracy is clearly ey-issue for man

applications. It has also ey-issue for serial rob ots and consequen tly this problem has een extensiv ely studied and arious accuracy indices ha een dened. The results ha een in general directly transp osed to parallel rob ots. will no review ho ell these indices are appropriate for parallel rob ots. Jacobian and in erse Jacobian matrix Let denotes the generalized co ordinates of the end-eector comp osed of parameters describing the ailable d.o.f. of the end-eector while denotes all the generalized co ordinates of the end-eector. will imp ose no constrain ts on

the hoice of (e.g. for Gough rob ot with planar platform the ose ma represen ted the co ordinates of particular oin ts on the end-eector). The geometry of the rob ot is describ ed its join ts ariables ector The wist of the end eector is comp osed of its translational and angular elo cities and the estricte twist is dened as the restriction of to the ailable d.o.f. of the rob ot. It is ell kno wn that for rob ot ha ving at least rotational d.o.f. is not the time-deriv ativ of as there is no represen tation of the orien tation whose deriv ativ es corresp onds to the

angular elo cities. Ho ev er there exists usually matrix suc that (1)

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In the usual approac the jacobian matrix linearly relates the actuated join elo cities to (2) In this pap er consider only non-redundan rob ots so that matrix is square and will call it the kinematic jac obian feature of parallel rob ots is that it is usually easy to establish an analytical form for while it is often imp ossible to obtain But ma also dene other jacobian matrices rst hanging the parameters in Indeed parallel rob ots dier from their serial coun terpart larger um er of

passiv join ts and it ma th us in teresting to include the passiv join ts ariables If is dened as ma then dene write the in erse kinematics equations as from whic deriv (3) where is )) and is ). This relation allo ws to quan tify the inuence of the measuremen errors on the passiv and actuated join ts ariables on the ositioning errors on the d.o.f. of the end-eector using (1). Although sa that some rob ot ha d.o.f., still the end-eector is d.o.f. rigid dy and ositioning errors on al d.o.f. should exam- ined. It is th us in teresting to determine an in

erse jacobian that in olv es the full wist of the end-eector. In that case write the kinematics equations as 0. If x kno that these kinematics equations ha nite um er of solutions, whic implies that the um er of equations in should By dieren tiation get: (4) where is square matrix while is 6. Pro vided that is square and not singular ma no deriv an in erse jacobian suc that BH (5) where is In most cases ho ev er elo cit analysis allo ws one to obtain simpler in erse jacobian matrix through relation that in olv es only fk (6) where fk is and will called the ful

inverse kinematics jac obian ma further extend this approac to tak in to accoun the design parameters of the rob ot (e.g. the lo cation of the anc hor oin ts of the legs in Gough platform). or that purp ose the kinematics equations will written as and the matrix of the partial deriv ativ es of with resp ect to will allo one to quan tify the inuence of the errors on on the ositioning error of the end-eector.

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As ma seen there is not single in erse jacobian matrix but ulti- plicit of them. Note ho ev er an imp ortan prop ert of the in erse jacobian of (5) with

resp ect to fk the rank of fk is the same than the rank of It is also imp ortan to note that an in erse jacobian in olving the full wist of the end-eector will not homogeneous in terms of units. This will true also for the in erse kinematic jacobian for rob ot in olving oth translation and rotational d.o.f. for the end-eector. Consequen tly man matrix prop erties (suc as the trace, determinan t) will not in arian under hange of units. In this pap er will fo cus on the inuence of on the ositioning errors of the end-eector through fk The necessit of using the full

in erse kinematic jacobian will emphasized on an example. 2.1 Example: the rob ot Tsai [10 has prop osed this rob ot as d.o.f. translation rob ot (gure 1). Eac leg of this rob ot is constituted, starting from the base, join follo ed an extensible leg terminated another join whose axis ar the same than the joint on the ase This constrain allo ws theoretically to obtain only translation for the end-eector. This example will allo Figure 1: The rob ot us to establish metho dology for determining the full in erse kinematic jacobian. But it will also enable to sho the imp ortance of

this matrix. The story is that suc rob ot as designed at Seoul National Univ ersit (SNU) and that is as exhibiting strange eha vior: although the prismatic actuators ere lo ed, the end-eector as exhibiting signican orien tation motion. This phenomena as explained Bonev and Zlatano [1] and later in [2, 11 ]. urthermore motion sensitivit to man ufacturing tolerances has een studied [5, 8] and has sho wn that this rob ot as ery sensitiv e.

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will denote the cen ter of the join ts on the platform and will no calculate the full in erse kinematic jacobian matrix. The

elo cit of the oin ts is BC Let us dene as the unit ector of the leg and compute the dot pro duct of the righ and left terms of the previous equation: BC CB (7) No let us dene the unit ectors of the join axis of the join at These ectors are the same for the base and platform. The angular elo cit of the leg with resp ect to the base and the angular elo cit of the platform with resp ect to the leg are The angular elo cit of the platform is where can obtained from the previous equations. No dene and compute the dot pro duct of the righ and left terms of the previous

equation (8) Com bining equations (7, 8) get the full elo cities equations in olving the wist as fk CB (9) whic establish the full in erse kinematic jacobian. The in erse kinematic jacobian ma extracted from fk as the matrix whose ro ws are the ectors. But an imp ortan oin for accuracy analysis is to consider the lo er part of fk whic sho ws that if the platform ma exhibit orien tation motion that ma innitesimal or nite according to the geometry of the join t. It happ ens that the design of the SNU rob ot as in the later category Manipulabilit It is realistic to assume that the

join errors are ounded and consequen tly so will the ositioning errors. The norm of the ound ma hosen arbitrary as (6) is linear so that simple scaling will allo to determine the ositioning error from the errors obtained for giv en ound. alue of for the ound is usually hosen: jj jj (10)

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min max -1 -1 -1 -1 Figure 2: The mapping et een the join ts errors space and the generalized co ordinates error space induced according to the norm: on top the Euclidean norm and on ottom the innit norm. whic leads to X X (11) classical geometrical in terpretation of

this relation is presen ted for the 2D case in gure 2. If the Euclidean norm is used (10) represen ts circle in the join ts errors space. This circle is mapp ed through matrix in to an ellipse in the generalized co ordinates error space. More generally the mapping transform the yp er-sphere of the join ts errors space in to an ellipsoid, usually called the manipulability el lipsoid In fact the use of the Euclidean norm is not realistic: it implies for exam- ple that if one of the join error is 1, then some ysterious inuence all the other join errors are 0. The appropriate norm

is the innit norm that states that the absolute alue of the join errors are indep enden tly ounded 1. With this norm (10) represen ts -dimensional square in the join ts errors space that is mapp ed in to the kinematics olyhe dr on that includes the manipulabilit ellipsoid, in the generalized co ordinates errors space. Figure illustrates this mapping in the 2D case. It ust noted that, apart of eing more realistic, the previous mapping leads to geometrical ob ject that can more easily manipulated than the ellipsoid. or example assume that one an to determine what are all the ossible

end-eector elo cities that can obtained in dieren oses of the end-eector. or that purp ose will ha to calculate the in tersection of the olyhedra obtained for the oses, ell kno wn problem of computational geometry that can uc more easily solv ed than computing the in tersection of

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ellipsoids. Condition um er large dimension along giv en axis of the kinematics olyhedron indi- cates large amplication error. It is therefore necessary to quan tify this amplication factor. Let us consider the linear system: where is in erse kinematic jacobian

matrix. The error amplica- tion factor in this system expresses ho relativ error in gets ultiplied and leads to relativ error in It haracterize in some sense the dexterit of the rob ot and will used as erformance index. no use norm suc that jj jj jj jjjj jj and obtain jj jj jj jj jj jjjj jj jj jj jj jj The error amplication factor, called the ondition numb er is therefore dened as (J jj jjjj jj The condition um er is th us dep enden on the hoice of the matrix norm. The most used norms are: the 2-norm dened as the square ro ot of the largest eigen alue of matrix

the condition um er of is th us the square ro ot of the ratio et een the largest and the smallest eigen alues of the Euclidean (or rob enius) norm dened for the matrix y: jj jj =1 =1 ij or equiv alen tly as tr(A A): if denotes the eigen alues of then the condition um er is the ratio et een and Note that sometime is also used eigh ted rob enius norm in whic is substituted where is the eigh matrix In these cases, the smallest ossible alue of the condition um er is 1. The in erse of the condition um er, whic has alue in [0,1], is also often used. alue of will indicate that the in erse

jacobian matrix is singular. The condition um er is quite often used as an index to describ rst the accuracy/dexterit of rob ot and, second, the closeness of ose to singularit or the later oin it is in general not ossible to dene mathematical distance to singularit for rob ots whose d.o.f. is mix et een translation and orien tation: hence the use of the condition um er is as alid an index than an other one. But it has the adv an tage of eing single um er for describing the erall kinematic eha vior of rob ot.

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The denition of the condition um er mak es

clear that cannot cal- culate its analytical form as function of the ose parameters except for ery simple rob ot. But robust linear algebra soft are allo ws to calculate it umerically for giv en ose. But for rob ot ha ving oth translation and orien tation d.o.f. there is ma jor dra wbac of the condition um er: the matrix in olv ed in its cal- culation are not homogeneous in terms of units. Hence the alue of the condition um er for giv en rob ot and ose will hange according to the unit hoice, while clearly the kinematic accuracy is constan t. Ma and Ange- les [6] suggested to dene

normalize inverse jac obian matrix dividing the rotational elemen ts of the matrix length suc as the length of the links in nominal osition, or the natur al length dened as that whic min- imizes the condition um er for giv en ose. Still the hoice of the length remains arbitrary as it just allo ws to dene corresp ondence et een rotation and translation and as men tioned ark [9 "this arbitrariness is an una oidable consequence of the geometry of SE(3)". ev aluate the eciency of the condition um er for accuracy ev alua- tion just use our Gough rob ot and ho oses three

reference oses dened the co ordinates of the cen ter and the Euler angles as =53 cm, 0, 0, (roughly the ose obtained for the mid-strok of the actuator), =53 cm, 30 0, (whose orien tation is roughly 1/3 of the ossible rotation around the axis) and 10 =53 cm, 0, 0, 0.(close to the order of the translation orkspace for this orien tation). then computed the absolute alue of the maximal ositioning error at these oses, obtained as the sum of the absolute alue of the elemen ts of the ro ws of the kinematic jacobian, as indicated in the follo wing table. ose 0.1184 0.1268 0.010087 0.1185

0.1184 0.697 0.1189 0.1274 0.01266 0.1333 0.1429 0.808 0.123 0.1309 0.0372 0.15 0.1663 0.7208 It can seen in this table that the ositioning errors are signican tly larger for and compared to As for the errors are usually larger compared to except for the rotation around Hence as far as accuracy is concerned the ordering of the oses from the most to the least accurate is and exp ect to obtain similar ordering for the condition um er. or this rob ot dene the normalized in erse jacobian matrix ob- tained dividing the orien tation comp onen ts of the 53 i.e. roughly the legs

lengths at ose The considered accuracy indices will the determinan of the 2-norm condition um er of the rob enius-norm condition um er of

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the 2-norm and rob enius norm condition um er of the in- erse jacobian matrix obtained when the in erse kinematics equations are based on the co ordinates of oin ts of the end-eector. The ho- sen oin ts will all ossible triplets in the set hence will pro vide ranges for these indices. The results are presen ted in the follo wing table: -29.22 75.14 63.9 152.8 70.2 [9.55,55.47] [258.8,3204.9] -24.64 75.16 73.8 154 80.9 [9.62,43.84]

[218.8,2383.6] -23.93 80.65 68.4 158.3 74.7 [10.06,58.95] [286.5,3618] or it ma seen that the dierence is surprisingly ery small e- een and signican et een The ordering et een is not resp ected for although these indices are coheren when con- sidering or the ordering is resp ected although the hanges in the index are relativ ely small for On the other hand there is sur- prisingly decrease of et een and while there is signican increase et een and Hence none of this condition um ers exhibits completely coheren eha vior with resp ect to the accuracy of this rob ot. This

simple example sho ws clearly that the concept of condition um er has to carefully considered when talking ab out optimal design for rob ot. Isotrop An isotr opic ose of rob ot is dened as ose where is equal to and rob ot whic has only isotropic oses in its orkspace is coined an isotr opic ob ot Designing an isotropic parallel rob ot is often considered as design ob jectiv [3 12 ]. trivial example of isotropic rob ot is serial Cartesian X-Y-Z rob ot whose kinematic jacobian matrix is the iden tit But this is surprising denomination as stricto sensu isotrop indicates that the

erformances of rob ot should the same whatev er is the motion direction. No if assume that all the actuator elo cities of X-Y-Z rob ot are ounded to 1, then the maximal elo cit of the end-eector lie in the range [1 3]: as far as elo cit is considered suc rob ot is far from isotrop Still the concept ma ha some in terest: for example an Cartesian rob ot whose actuator axis are not utually orthogonal will exhibit ratio et een its maximal elo cities er its orkspace that will larger than 3. Hence, instead of using the name "isotropic rob ot" ma consider using the name "maximally regular rob

ot". Global conditioning index The condition um er is lo cal indication for the dexterit of rob ot. ev aluate the dexterit of rob ot er giv en orkspace Gosselin [4 has

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in tro duced the glob al onditioning index (GCI) as: GCI dW dW whic corresp ond to the erage alue of = Clearly this concept mak es sense for the optimal design of rob ot for whic the extremal and erage alue of an erformance are imp ortan design factors. But apart of the alidit of the condition um er that has een discussed in previous sec- tion the problem with the GCI is its calculation. Clearly cannot exp ect

to obtain its closed-form and ust rely on umerical ev aluation. The usual metho is to sample the orkspace using regular grid, compute = at eac no de and appro ximate the GCI as GCI the sum of the = divided the um er of no des. This calculation ma computer in ten- siv as its complexit is exp onen tial with resp ect to the um er of d.o.f. of the rob ot. urthermore this metho do es not allo to get ound on GCI GCI deal with this error problem it is sometimes assumed that if the result with sampling oin ts is close to the result obtained with oin ts, eing signican tly larger than then the

later result is go appro ximation of the index. This assumption will true only if the condition um er is smo oth enough, claim that is dicult to supp ort. Consider for example simple planar serial 2R rob ot: its GCI can com- puted almost exactly as it dep ends only on single parameter. sample this parameter using 10, 20, 10 oin ts and stop the calculation when the relativ error et een GCI GCI is lo er than 0.5% and assumes GCI GCI ). or 50 the relativ error is 0.377% while the relativ error on the GCI is still 1.751%. It ma assumed that suc error will ev en larger for more complex rob

ot. etter ev aluation will probably obtained using Mon te-Carlo in tegration (with an error that decreases as where is the um er of sampling no des) or quasi-Mon te Carlo. In the previous example (whic is not fa orable for Mon te-Carlo metho as there is only one parameter) found out that using the same stop criteria the relativ error on the GCI as reduced to 0.63%. certied ev aluation of the global conditioning index is therefore an op en problem but nev ertheless the calculation of suc index will probably computer in tensiv e. Conclusion Classical dexterit indices suc as the condition

um er are not ery ade- quate for parallel rob ots. In our opinion the most appropriate accuracy in- dices are the determination of the maximal ositioning errors, their erage alues and their ariance. ha presen ted in recen pap er computer in tensiv metho for nding the largest maximal ositioning errors, up to an arbitrary accuracy of d.o.f. rob ot [7 ]. real hallenge is to design algo- rithms for calculating the erage and ariance of the maximal ositioning

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errors er giv en orkspace. An imp ortan oin is that there is no need to calculate these alues exactly as so on as it

is ossible to imp ose ound on the calculation error. Indeed for comparison purp oses an appro ximate alue with guaran teed error will sucien t. References [1] Bonev I.A. and Zlatano D. The ystery of the singular SNU trans- lational parallel rob ot. www.parallemic.org/Reviews/Review004.h tml, June, 12, 2001. [2] Di Gregorio R. and aren ti-Castelli V. Mobilit analysis of the 3-UPU parallel mec hanism assem bled for pure translational motion. ASME J. of Me chanic al Design 124(2):259{264, June 2002. [3] attah A. and Hasan Ghasemi A.M. Isotropic design of spatial parallel manipulators.

Int. J. of ob otics ese ar ch 21(9):811{824, Septem er 2002. [4] Gosselin C. Kinematic analysis optimization and pr gr amming of ar- al lel ob otic manipulators Ph.D. Thesis, McGill Univ ersit Mon tr eal, June, 15, 1988. [5] Han C. and others Kinematic sensitivit analysis of the 3-UPU parallel manipulator. Me chanism and Machine The ory 37(8):787{798, 2002. [6] Ma O. and Angeles J. Optim um arc hitecture design of platform ma- nipulator. In ICAR pages 1131{1135, Pise, June, 19-22, 1991. [7] Merlet J-P and Daney D. Dimensional syn thesis of parallel rob ots with guaran teed giv en accuracy er

sp ecic orkspace. In IEEE Int. Conf. on ob otics and utomation Barcelona, April, 19-22, 2005. [8] aren ti-Castelli V. and Di Gregorio R. Inuence of man ufacturing errors on the kinematic erformance of the 3-UPU parallel mec hanism. In 2nd Chemnitzer Par al lelkinematik Seminar pages 85{99, Chemnitz, April, 12-13, 2000. [9] ark M.K. and Kim J.W. Kinematic manipulabilit of closed hains. In ARK pages 99{108, ortoroz-Bernadin, June, 22-26, 1996. [10] Tsai L-W. Kinematics of three-dof platform with three extensible lim bs. In ARK pages 401{410, ortoroz-Bernadin, June, 22-26, 1996.

[11] olf A., Shoham M., and ark F.C. In estigation of singularities and self-motions of the 3-UPU rob ot. In ARK pages 165{174, Caldes de Mala alla, June 29- July 2, 2002. [12] Zanganeh K.E. and Angeles J. On the isotropic design of general six- degree-of-freedom parallel manipulators. In J-P Merlet B. Ra ani, editor, Computational Kinematics pages 213{220. Klu er, 1995. 10

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