Efficiency Improvement of Measurement Pose Selection Techniques in Robot Calibration

Transcription

1 Effcency Iproveent of Measureent Pose Selecton echnques n Robot Calbraton Y.Wu,2, A. Klchk,2, A. Pashkevch,2, S.Caro 2, B.Furet 3 Ecole des Mnes de Nantes,44307, Nantes, France 2 Insttut de Recherche en Councatons et Cybernétque de Nantes (IRCCyN), 4432, Nantes, France 3 Unversté de Nantes, 44322, Nantes, France (e-al:{ yer.wu, alexandr,klchk, stephane.caro@rccyn.ec-nantes.fr, benot.furet@unv-nantes.fr) Abstract: he paper deals wth the desgn of experents for anpulator geoetrc and elastostatc calbraton based on the test-pose approach. he an attenton s pad to the effcency proveent of nuercal technques eployed n the selecton of optal easureent poses for calbraton experents. he advantages of the developed technque are llustrated by sulaton exaples that deal wth the geoetrc calbraton of the ndustral robot of seral archtecture. Keywords: robot calbraton, easureent pose selecton, test-pose approach. INRODUCION In the usual engneerng practce, the accuracy of a anpulator depends on a nuber of factors. Usually n robotcs, the geoetrc and elastostatc errors are the ost sgnfcant ones. her nfluence on the robot postonng accuracy hghly depends on the anpulator confguraton and essentally dffers throughout the workspace. o acheve good accuracy n all workng ponts, adequate geoetrc and stffness odels are requred. Whle the odel structure s usually well known, the dentfcaton of the odel paraeters (calbraton) s rather te consug and requres essental experental work. For ths reason, optal selecton of easureent poses for robot calbraton s an portant proble, whch s stll n the focus of nuerous research papers (Daney 2002, Sun 2008). At present, the an actvty n ths area s concentrated around the geoetrc calbraton (Khall 2002). On the other hand, the elastostatc calbraton whch s also very portant for any applcatons (such as precse achnng) has attracted less attenton of the researchers (Meggolaro 2005). However, for both of these calbraton procedures, the proble of easureent pose selecton s one of the key ssues allowng to reduce essentally the easureent error pact (Klchk 20). At frst sght, ths proble can be solved usng well known results fro the classcal desgn of experents theory. However, because of the specfcty and nonlnearty of the anpulator geoetrc and elastostatc odels, the proble soluton s not so obvous. he an dffcultes here are n the area of defnton of a reasonable optalty crteron (whch has clear engneerng sense) and also n effcent soluton of the relevant optzaton proble, whch has rather hgh denson. Aong related works, t s worth entonng several papers. he ajorty of the easureent pose selecton technques reles on the optzaton of soe functons dependng on the sngular values of the dentfcaton Jacoban. For exaple, Zhuang used genetc algorth for zaton of the condton nuber of ths atrx (Zhuang 996). In other work (Daney 2005), to decrease the senstvty to local a, Daney developed the local convergence ethod and abu search technque based on the observablty ndex. However, the perforance easures used n these works are rather abstract and are not drectly related to the robot accuracy. Besdes, the related objectve functons are very dffcult for the optzaton due to exstence of a nuber of local a. o fnd the global one, heurstc search s usually used as the nuercal algorths, whch often requre tedous coputatons. All these otvate the research drecton of ths work. In ths paper, the proble of optal desgn of calbraton experents s studed for the case of robot anpulator of seral archtecture. In contrast to other works, the optzaton proble related to easureent pose selecton s forulated usng the proposed perforance easure (test-pose approach), whch has clear physcal eanng and s drectly related to robot accuracy. he an attenton s pad to the effcency proveent of the related nuercal routnes. 2. PROBLEM SAEMEN 2. Geoetrc and Elastostatc Models of Manpulator In ndustral robot controllers, the end-effector poston of the anpulator s usually coputed usng the geoetrc odel. For soe specfc applcatons, such as hgh-speed achnng that generate essental external loadng, the elastostatc odel should be also used. However, n practce, the robot geoetrc paraeters essentally dffer fro the noal values declared n techncal specfcatons and vary fro one robot to another. In addton, elastostatc paraeters of the anpulator are not provded by the robot anufacturers and can be dentfed fro the experents only. So, the anpulator

2 odel paraeter dentfcaton s an portant step n practcal applcaton of ndustral robots. he anpulator geoetrc odel provdes the poston/orentaton of robot end-effector as a functon of the jont varables and ts nherent paraeters. hs odel s usually presented as a product of hoogeneous transforaton atrces, whch after soe transforatons can be presented as the vector functon p g q, П () where vector p denotes the end-effector poston, vector q aggregates all jont angles and П are the vector of unknown paraeters to be dentfed. hese unknowns dffer wth the appled paraeterzaton ethods n robot geoetrc odellng, such as the classcal Denavt and Hartenberg approach and ts odfed verson (Khall 986). In ths paper, there are consdered the ost essental coponents of the vector П, whch are the devatons of the robot lnk lengths l and the offsets q n the actuated jonts. Snce the devatons of geoetrcal paraeters П are usually relatvely sall, calbraton usually reles on the lnearzed odel p g q, П J q, П П (2) 0 g 0 whch ncludes the conventonal geoetrc Jacoban J g q, П0gq, П0/ Π coputed for the noal geoetrc paraeters. П 0 he elastostatc propertes of a seral robotc anpulator represent ts resstance to deforatons caused by external forces/torques and are usually descrbed by the Cartesan stffness atrx, whch s coputed as K C C θ θ θ K J K J (3) where Kθ s a dagonal atrx that aggregates the jont stffness values (that are the unknowns to be dentfed) and Jθ s the correspondng elastostatc Jacoban. hs odel can be derved usng the vrtual jont ethod, whch descrbes all elastostatc propertes of coplant eleents by localzed vrtual sprngs located n the actuated jonts (Salsbury 980). Usng the Cartesan stffness atrx, the elastostatc odel (or force-deflecton relaton) can be expressed as θ θ θ w J K J p (4) where p s the poston deflecton at the robot end-effector caused by the external wrench w, whch ntegrates both the external force and torque. hs lnear relaton can be further used for the calbraton where the desred paraeters to be dentfed are the coponents of atrx. In the frae of ths work, several assuptons concernng calbraton of these odels are accepted: A: For the geoetrc calbraton, each calbraton experent produces two vectors p, q, whch defne the robot end-effector dsplaceents and correspondng jont angles. he lnear relaton between the errors n geoetrc paraeters and the end-effector poston devatons can be wrtten as p = J ( q ) Π (5) g K θ where Jg ( q ) s the Jacoban atrx that depends on anpulator confguraton q and vector Π collects the unknown paraeters to be dentfed. A2: For the elastostatc calbraton, each calbraton experent produces three vectors p, q, w, where w defnes the appled forces and torques. In accordance wth (Pashkevch 20), the correspondng appng fro the external wrench space to the end-effector deflecton space can be expressed as p = θ( ) J q kθjθ( q) w (6) where k θ s a atrx that aggregates the unknown coplance paraeters k,...,k n to be dentfed. Hence, the calbraton experents provde the set of vectors p, q and p, q, w that allow us to estate the devatons n geoetrc paraeters П (copared to the noal values) and absolute values of the elastostatc paraeters ncluded n the dagonal atrx k θ. 2.2 Identfcaton of the Model Paraeters he proble of paraeter dentfcaton of the robot anpulator can be treated as the best fttng of the experental data by correspondng odels. hese data are easured under several assuptons concernng the easureent equpent: A3: he calbraton reles on the easureents of the endeffector poston only (Cartesan coordnates px, py, p z). A4: he easureents errors ε accoodated n each easureent of end-effector poston are treated as ndependent dentcally dstrbuted rando values wth zero expectaton and standard devaton. For coputatonal convenence and takng nto account the nfluence of easureent errors, the geoetrc and elastostatc odels descrbed by separate lnear equatons (5) and (6) can be expressed n the followng ntegrated for p B( q ) Xε (7), where X Π k collects all unknown paraeters (both geoetrc and elastostatc ones), and the atrx B vares dependng on dfferent calbraton cases J 0 B 0 A 33n J A 32n 3n 32n 3n 32n 3n, for geoetrc paraeters, for elastostatc paraeters, for both paraeters where J s the Jacoban atrx that can be obtaned by dfferentatng the anpulator geoetrc odel wth respect to the desred paraeters; and atrx A can be coputed as Aq (, w) ( ) ( ),..., n( ) n ( ) J q J q w J q J q w (9) where Jn ( q ) s the n-th colun vector of the Jacoban atrx for the -th experent. n s the nuber of jonts. Usng usual approach adopted n the dentfcaton theory, the estated unknown paraeters Xˆ Πˆ, kˆ θ can be obtaned usng the least square ethod, whch yelds (8)

3 BB ˆ X B p (0) Usng ths expresson, t can be proved that the covarance atrx for the dentfcaton errors n the paraeters X can be coputed as ˆ 2 cov( X) B B () where s the standard devaton (s.t.d.) of the easureent errors. Hence, the pact of the easureent errors on the paraeter dentfcaton accuracy s defned by the atrx su BB that s also called the nforaton atrx. It s obvous that, fro practcal pont of vew, the covarance atrx should be as sall as possble. However, strct atheatcal defnton of ths noton s not trval and a nuber of dfferent approaches are proposed n lterature. In ost of the related works, the optal easureent poses are obtaned based on zaton of the covarance atrx nor (Atknson 992). hs approach ay provde a soluton, whch does not guarantee the best poston accuracy for typcal anpulator confguratons defned by the anufacturng process. hus, here t s proposed the ndustry-orented perforance easure, 2 0, whch s defned as the ean square error n the end-effector poston after copensaton. o develop ths approach, let us ntroduce several defntons: D: Plan of experents s a set of robot confguratons Q and correspondng external loadngs W that are used for the easureents of the end-effector dsplaceents and further dentfcaton of the desred paraeters. D2. he accuracy of the error copensaton 0 s the dstance between the desred end-effector poston and ts real poston acheved after applcaton of error copensaton technque. D3. he anpulator test-pose s one or set of robot confguratons Q and correspondng external loadngs 0 W 0 for whch t s requred to acheve the best error copensaton 2 (.e. 0 ). In the frae of the adopted notatons, the dstance defnng the error copensaton accuracy can be coputed as p B ˆ 0( XX) (2) where the vectors X and Xˆ are the true paraeters values and ther estates, respectvely. Matrx B 0 corresponds to the test pose (see expressons (8)). Further, takng nto account that p s a functon of the unbased rando varables ε,..., ε, t can be easly proved that the expectaton E( p ) 0. Besdes, the varance can be expressed as 0 0 Var( p) E X B B X (3) where XXˆ X s the dfference between the estated and true values of the paraeters. Expresson. (3) can be rewrtten as trace( B0E( XX ) B0) and after relevant transforatons n accordance wth (0), (), yelds the desred expresson for the copensaton accuracy trace B0 ( ) ( ) 0 B q B q B (4) As follows fro ths expresson, the proposed perforance easure can be treated as the weghted trace of the covarance atrx (), where the weghtng coeffcents are coputed usng the test pose. Hence, the dentfcaton qualty (evaluated va the error copensaton accuracy) s copletely defned by the set of atrces B,..., B that depend on the anpulator confguratons q,..., q. Optal selecton of these confguratons wll be n the focus of next Subsecton. 2.3 Proble of the Measureent Poses Selecton Based on the perforance easure presented n the prevous Subsecton, the correspondng optzaton proble of the easureent pose selecton can be defned as trace B0 ( ) ( ) B q B q B 0, q w (5) subject to C( q, w) 0,, r Here, the atrces C( q, w ) descrbe soe constrants, whch should be taken nto account whle solvng optzaton proble. hese constrants are posed by the work-cell desgn partculartes and usually nclude the anpulator jont lts, the work-cell space lts, easureent equpent ltatons, etc. It should be also entoned that soe constrants are posed to avod collsons between the workcell coponents and the anpulator. Besdes, soe drectons of the appled loadng are preferable for the reason of practcal pleentaton. It should be entoned that the coponent of the atrces C vary wth dfferent calbraton cases and ay nclude soe very specfc constrants. For nstance, for the case of elastostatc calbraton, they can be expressed as q q pz p C z q q p p, C3 r r, C4 C2 F p p F (6) where q and q are the jont lts, F s the robot u payload, p z s the u heght between the end-pont of the calbraton tool and the work-cell floor, r s the u radus to avod collsons between the appled loadng and robot body, s the u angle between the drecton of calbraton tool and z-axs of robot base frae to ensure that the vertcal loadng can be appled, p and p are the boundares of work-cell space. For the case of geoetrc calbraton, the proble of applyng external loadng does not exst. So, C 2, C3 are zero atrces, whle C, C rean the sae as n elastostatc calbraton. 4 he procedure of solvng such an optzaton proble could be very tedous for the case when nuerous easureent confguratons are requred for the calbraton experent. For ths reason, the proble of nterest s to fnd reasonable

4 nuber of dfferent easureent confguratons and to prove the effcency of optzaton routnes eployed n the easureent pose selecton. 3. MEASUREMEN POSE SELECION ECHNIQUES o solve the above defne proble, several technques can be appled. hs secton presents the analyss and propose soe approaches allowng to obtan acceptable soluton n reasonable te. he an dffcultes here are related wth a large nuber of varables and coplex behavour of the objectve functon that has any local a. 3. Usng Conventonal Optzaton echnques he splest way to solve ths proble s to apply conventonal optzaton technques ncorporated n coercal atheatcal software. It s clear that straghtforward search wth regular grd s non-acceptable here because of hgh coplexty and enorous nuber of solutons to be copared. For ths reason, three other algorths have been exaed: () rando search, () gradent search, and () genetc algorth. her coparson study s presented below and suarzed n ables and 2, where two crtera have been used: coputatonal te and the ablty to fnd optal soluton (evaluated va 0, the anpulator accuracy after calbraton). For all coputatonal experents, t was assued that the s.t.d. of the easureent errors s he benchark exaple deals wth the calbraton experents desgn for 6-dof ndustral anpulator KUKA KR270, whose noal paraeters can be found on the anufacturer webste ( he robot has a seral archtecture wth sx actuated revolute jonts, so 24 ndependent geoetrc paraeters should be dentfed n general case. But for ths exaple, to reduce coputatonal efforts and evaluate the algorth capablty before applyng to the proble of real denson, only nne of the ost essental paraeters were dentfed (whch have ajor pact on the postonng accuracy). hs allowed us to obtan realstc assessents of the conventonal optzaton technques capabltes wth respect to the consdered proble where the nuber of desgn varables s hgh enough (72 for 2 confguratons). he frst of the exaed algorth () s based on the straghtforward selecton of the best soluton fro the set of ones generated n a rando way. For ths study, 0,000 solutons were generated for dfferent nubers of easureent confguratons 3, 4, 6,2. As follows fro the obtaned results (see ables and 2), ths algorth s very fast and requres less than 2 utes to fnd the best soluton. However, ths soluton s essentally worse than the optal one (by 5-30%). he second algorth () eploys the gradent search wth bult-n nuercal evaluaton of the dervatves that s avalable n Matlab. he startng ponts were generated randoly and, to avod convergence to the local a, the optzaton search has been repeated 5000 tes (startng fro dfferent ponts). In ths case, t has been obtaned the best result n ters of the desred objectve 0, but coputatonal cost was very hgh (t can overcoe a hundred of hours). So, ths technque s hardly acceptable n practce. It s worth entonng that reducton of the teraton nuber s rather dangerous here, because there are a nuber of local a that the algorth can converge to (see able that ncludes the u, u and average values of 0 obtaned for rando startng ponts). Moreover, as follows fro our experence, 5000 teratons are also not enough here. able. Effcency of conventonal optzaton technques Rando Search Gradent Search Genetc Algorth Algorth Nuber of poses [] ean 0 [] [] [] ean 0 [] [] [] ean 0 [] [] he thrd of the exaed technques () apples genetc algorth (GA) that s based on adaptve heurstc search. he optzaton has been carred out for 00 tes wth populaton sze 50 and 20 generatons (ntal populatons were randoly generated). For llustratve purposes, Fg. presents the effcency of ths algorth for selecton of three optal easureent confguratons. It shows the algorth convergence as well as dvergence of the optal solutons wth respect to coputng te. As follows fro ths fgure, the optzaton results are hghly senstve to the selecton of ntal populaton. In partcular, the dversty of the optal solutons got fro sequental GA runs s about 25%. So, to acheve the global u, the GA should be repeated any tes, whch leads to essental ncrease of the coputatonal efforts (ore than 6 hours of coputatons for the consdered exaple). However, copared to gradent search, GA provdes acceptable accuracy (only 2% worse) whle the coputatonal te s 4 tes less. able 2. Coputatonal te of exaed algorths Algorth Nuber of poses Rando Search 4s 47s.7 Gradent Search 24.2h 37.5h 56.3h 03.6h G enetc Algorth 6.5h 8.3h 0.5h 5.4h As follows fro the ed re the obtan sults, rando search s rather fast but neffcent here, snce t ay produce nonacceptable solutons. In contrast, the gradent search s able to fnd the global u provded that t s repeated any tes wth dfferent startng ponts. As a coprose, the GA

5 provdes nteredate results n ters of accuracy and coputatonal te. However, for probles of the real ndustral sze, the perforances of the GA are also not suffcent. For ths reason, the followng Subsectons are devoted to the proveent of the nuercal optzaton technques eployed n selecton of optal easureent poses. te, [h] 3.2 Applyng Parallel Coputng Fgure. Effcency of GA for selecton of three easure- ent poses (populaton sze 50, 20 generatons) Snce the consdered proble requres nuerous repettons of the optzaton wth dfferent ntal values, applyng parallel coputng looks attractve to speed up the desgn process and to take advantage of ult-core archtecture avalable n odern coputers. o evaluate benefts of the parallel coputng, the sae benchark exaple has been consdered and two algorths have been exaed: ()' parallel gradent search, and ()' parallel GA wth the sae paraeter settngs. he coputatons were carred out on the workstaton wth 2 cores. he obtaned results are presented n able 3, whch gves the coputatonal te for dfferent nuber of easureent poses (the attaned value of the objectve functon 0 s very close to those presented n able ). he obtaned results are qute expected and confr essental reducton of coputatonal efforts. For both optzaton ethods, the consued te has been decreased by the factor of 0-2 (copared to the results n able 2). However, t s not enough yet to solve the proble of real ndustral sze, where several dozen of paraeters should be dentfed (nstead of nne n the benchark exaple). able 3. Coputatonal te of exaed algorths usng parallel coputng Algorth Parallel Gradent Search Parallel Genetc Algorth Nuber of poses h 3.2h 4.9h 8.9h h 3.3 Usng Hybrd Approach o take the advantages of both exaed algorths and effcency of the parallel pleentaton, a hybrd technque has been developed. It should be entoned that soe software packages (Matlab, etc.) already pleent ths dea and use the fnal soluton fro GA as the ntal pont of gradent search. However, snce the randoly generated ntal populatons n GA ay cause hgh dversty of the optal solutons, the selecton of these ntal values s also an portant ssue. For ths reason, the ebedded hybrd opton n GA cannot be drectly used and requres addtonal odfcatons. o prove the effcency of the exstng technque, the startng pont selecton strategy for the gradent search has been odfed. o ensure better convergence to the global u, t has been proposed to use the best half of fnal ponts obtaned fro GA as the startng ponts for the gradent search. Fro our pont of vew, t ensures better dversty of the startng ponts and allows to avod convergence to the local a. Fgure 2. Effcency of the hybrd approach for selecton of three easureent poses he proposed odfcaton has been evaluated usng the sae benchark exaple. For coparson purposes, Fg 2. presents the convergence of the hybrd ethod for the proble of optal selecton of three easureent confguratons studed n the prevous Subsecton (see Fg. ). It shows the ntal ponts (obtaned fro GA), optal solutons as well as the soluton proveent wth respect to te. As follows fro the fgure, the hybrd algorth can converge uch faster, but f the nuber of easureent poses s ncreased up to 2, the coputatonal te s over.6 hour that s stll unacceptable for ndustry. 3.4 New Approach: Reducton of Proble Denson Generally, as follows fro the dentfcaton theory, the only way to prove calbraton accuracy s to ncrease the nuber of easureents (provded that the reducton of the easureent errors s not possble). However, for the anpulator calbraton proble, each easureent s assocated wth a certan robot confguraton that also has nfluence on the fnal accuracy. It s clear that the best result s acheved f all easureent poses are dfferent and have been optzed durng the calbraton experent plannng.

6 On the other hand, as follows fro our experences, the dversty of the easureent poses does not contrbute sgnfcantly to the accuracy proveent f s hgh enough. hs allows us to propose an alternatve whch uses the sae easureent confguratons several tes (allowng to splfy and speed up the easureents). hs approach wll be further referred as "reducton of proble denson". o explan the proposed approach n ore detals, let us assue that the proble of the optal pose selecton has been solved for the nuber of confguratons that s equal to, and the obtaned calbraton plan ensures the postonng accuracy 0. Usng these notatons, let us evaluate the calbra- ton accuracy for two alternatve strateges that eploy larger nuber of experents k : Strategy #2 (proposed): the easureent poses are obtaned by sple repetton the confguratons got fro the low- densonal optzaton proble of sze. Strategy # (conventonal): the easureent poses are found fro the full-scale optzaton of sze k. k It s clear that the calbraton accuracy 0 for strategy # s better than the accuracy correspondng to the strategy #2 that can be expressed as 0 k. However, a s follows fro our study, ths dfference s not hgh f s larger than 3. hs allows us to essentally reduce the sze of optzaton proble eployed n the optal selecto n of easureent poses wthout sgnfcant pact on the postonng accuracy. o deonstrate the valdty of the proposed approach, the benchark exaple has been solved usng strateges # and #2 assug that the total nuber of easureents s equal to 2 (.e. usng dfferent factorzatons such as 2, 6 2, 4 3, 3 4). Relevant results are presented n able 4 (see the last lne). As follows fro the, the factorzato n 2 where all easureent poses are dfferent s only 6% better copared to the factorzaton 3 4 where easureen ts are repeated 4 tes n 3 dfferent confguratons. At the sae te the factorzatons 6 2 and 4 3 gve alost the sae results as the optal factorzatons 2. On the other hand, the coputatonal te of the optal pose generaton for 3 s uch lower than for 2. hs deonstrates the effcency of the proposed approach and justfy ts valdty. able 4. Calbraton accuracy 0 for dfferent factorzatons of the experent nuber k0 Nuber of easureents Nuber of dfferent poses (3 ) (4 ) (3 2) (6) (3 4) (4 3) (6 2) (2 ) Coputng te h Hence, t can be concluded that repeatng experent s wth optal plans obtaned for the lower nu ber of confgura- tons provdes alost the sae perforance as "fulldensonal" optal plan. Obvously, ths reducton of the easureent pose nuber s very attractve for the engneerng practce. 4. CONCLUSIONS he paper presents a new technque for optal selecton of easureent poses n robot calbraton. In contrast to other works, t s proposed to evaluate the qualty of calbraton plans va the anpulator postonng accuracy for a gven test pose, and to reduce nuber of desgn varables n the related optzaton proble by eans of repeatng easureents wth lower nuber of confguratons. hs technque allows us to essentally reduce the coputatonal te for solvng the proble of real ndustral sze. he advantages of the developed technque were confred by a sulaton exaple, where the proposed approach pertted to decrease the coputng te by ore than 0 tes whle losng only 6% of anpulator postonng accuracy. Future work n ths drecton wll deal wth the effcency proveent of the anpulator elastostatc calbraton. ACKNOWLEGMEN he authors would lke to acknowledge the fnancal support of the French Agence Natonale de la Recherche (Project ANR-200-SEGI COROUSSO), France. REFERENCES Atknson, A.C., Donev, A.N., (992). Optal Experent Desgns. Oxford Unversty Press, Oxford. Daney, D., (2002). Optal easureent confguratons for Gough platfor calbraton. ICRA, pp Daney, D., Papegay, Y., Madelne, B., (2005). choosng easureent poses for robot calbraton wth the local convergence ethod and tabu search. Internatonal Journal of Robotcs Research, 24(6), pp Khall W., Klenfnger J.F., (986). A new geoetrc notaton for open and closed-loop robots. Proceedngs of ICRA Conference, pp Khall, W., Besnard, S., (2002). Geoetrc calbraton of robots wth flexble jonts and lnks. Journal of Intellgent and Robotc Systes, 34, pp Klchk, A., Wu,Y., Caro, S., Pashkevch, A., (20). Desgn of experents for calbraton of planar anthropoorphc anpulators. AIM Conference, pp Meggolaro, M.A., Dubowsky, S., Mavrods, C., (2005). Geoetrc and elastc error calbraton of a hgh accu- racy patent postonng syste. Mechans and Machne heory, 40, pp Pashkevch A., Klchk A., Chablat D. (20). Enhanced stffness odelng of anpulators wth assve jonts. Mechans and Machne heory, vol. 46, pp Salsbury, J., (980). Actve stffness control of a anpulator n Cartesan coordnates. Proceedngs of 9th IEEE Conference on Decson and Control, pp Sun, Y., Hollerbach M., (2008). Observablty ndex selecton for robot calbraton. Robotcs&Autoaton, pp Zhuang, H., Wu, J., Huang, W., (996). Optal plannng of robot calbraton experents by genetc algorths. Robotcs and Autoaton, 2, pp