High Efficient Dynamics Calculation Approach for Computed-Force Control of Robots with Parallel Structures

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1 Proceedngs of the 44th IEEE Conference on Decson and Control, and the European Control Conference 25 Sevlle, Span, December 12-15, 25 MoIC19.2 Hgh Effcent Dynamcs Calculaton Approach for Computed-Force Control of Robots wth Parallel Structures Houssem Abdellatf, Martn Grotjahn and Bodo Hemann Abstract Ths paper presents a compact and complete approach for realzng hgh performant control of fully parallel manpulators wth a computed-force control scheme (CFC. The proposed method of dynamcs computaton s based on the prncpal of vrtual power and allows real-tme mplementaton wthout fallng back on naccurate model smplfcatons. The effcency and performance s demonstrated on a 6-dof complex parallel manpulator wthn commercal control hardware setup. Crucal ponts for the enhancement of trackng performance are dscussed n detals. I. INTRODUCTION The majorty of commercal robotc systems or machne tools are controlled wth smple and lnear sngle-jont feedback controller. The trackng performance s however lmted, especally for nonlnear systems n hgh speed range. There s stll a bg gap between the sophstcated control algorthms developed n research and the commercal ones. Researcher have to assume though a part of responsblty on ths ssue, snce sometmes the practcablty of ther developed approaches s neglected. In ths paper a computed-force control algorthm (CFC s carred out for parallel robots from the theory untl applcaton on a commercal control system. It s commonly known that the nverse dynamcs computaton for feedforward control of robotc manpulators s crucal for hgh performance trackng [1], [2], [3]. Ths ssue becomes delcate for parallel knematc manpulators (PKM snce the coupled structure and the hgh nonlnearty make the dynamcs complex to solve [4], [5], [6]. Many approaches were proposed for a computatonal effcent calculaton of the dynamcs. We thnk that there are stll dfferent drawbacks. The proof of practcablty for control was not regarded n some theoretcal works [4], [5], [6], [7]. Besdes, most of common approaches neglect frcton and jont losses. It was demonstrated n [1] that for some systems, especally frcton compensaton yelds sgnfcant mprovement of control performance. Successful mplementaton n robotc systems was shown n [3] and n [8], where frcton was taken nto account. In the other way around, the rgd-lnk dynamcs were strongly smplfed to reduce the computatonal cost and ensures the real-tme ablty of the models. In ths paper we contrbute to remove all these drawbacks by choosng an approprate approach for modellng and computng the nverse dynamcs H. Abdellatf and B. Hemann are wth the Hannover Center of Mechatroncs, Unversty of Hannover, Appelstr. 11, 3167 Hannover, Germany. E-mal: {abdellatf,hemann}@mzh.un-hannover.de M. Grotjahn s wth IAV GmbH, Gfhorn, Germany. E-mal: martn.grotjahn@av.de /5/$2. 25 IEEE 224 of parallel manpulators. The proposed model takes nto account all relevant dynamcs, ncludng nertal nfluences of all bodes and frcton losses n all passve and actve jonts. The presented approach s kept general and remans avalable for the major cases of PKM. To assure the computatonal effcency and real-tme practcablty the JOURDAIN s prncple of vrtual power was regarded for formulatng all effects n a unform way (secton II and III. To make the model more powerful and useful for further applcatons such adaptve control [3] or parameter dentfcaton [1], [9], the equatons of moton are expressed n a parameter-lnear form [6], [1], [11], [12]. The proposed approach s vald and applcable for a wde range of parallel manpulators [2]. The mplementaton n a real test bed wthn a commercal control hardware s proved n terms of computatonal cost and tme (secton IV and n terms of control performance (secton V. The expermental results are carred out wth the nnovatve drect drven parallel robot PaLDA. The machne was developed by the nsttute of producton engneerng of the unversty of Hannover for hgh-speed manpulaton and machnng [2]. In every secton, drawbacks and advantages of alternatve methodologes known from lterature are crtcally dscussed and systematcally compared wth the approach proposed n ths paper. II. KINEMATICS ANALYSIS A general 6-DOF parallel manpulator s consttuted of a movng platform (end-effector platform attached wth sx seral actuated knematc chans to the base platform [2], [4], [5], [6], [7]. Fgure 1 shows a general sketch of such robotc manpulator. The goal of the knematc analyss s the determnaton of the motons of all modelled bodes n respect to the generalzed coordnates. The vector of generalzed coordnates λ s composed of the cartesan coordnates of the end-effector platform ( r E = [ x, y, z ] and the tltng angles (α, β, γ accordng to the cardan or the euler formalsm. Addtonally, the vector of the generalzed veloctes s defned as θ = [ (v T E, (ω ] T T E that ncludes the translatonal and angular veloctes wth reference to a cartesan frame. The algorthm effcency proposed n ths paper s based on treatng each leg as conventonal seral chan. Afterwards the constrants of the closed-loops are taken nto account. The vectors jonng the fxed jonts A j and the platform jonts B j are (r Aj B j =[x j y j z j ] T = ( r A j + ( r E + R E (Er E B j, (1

2 where R E s the orentaton matrx of the end-effector (See the sketch n Fg. 1. The veloctes and acceleratons of B j are gven by v Bj = v E + ω E r E B j, (2 a Bj = a E + ω E r E B j + ω E ω E r E B j. (3 The sngle robot struts can be now consdered as seral chan robots wth respectve end-effector knematcs gven by r Aj B j, v Bj and a Bj. For each body, body-fxed coordnate frames are defned by the modfed DENAVIT- HARTENBERG (MDH notaton [13]. Ths means that the frame s fxed to the lmb. Thez -axs s the axs of jont and the x -axs s the normal of z and z +1.Frame s defned wth respect to frame 1 by the homogenous transformaton matrx T 1 [ ] T 1 R 1 = ( 1r 1 = 1 c ϑ s ϑ a s ϑ c α c ϑ c α s α d s α s ϑ s α c ϑ s α c α d c α 1 whch s a functon of the MDH-parameters ϑ, d, α and a [6], [12]. The abbrevatons s x and c x denote sn(x and cos(x respectvely. The matrx R 1 and the vector defne orentaton and poston of frame wth respect to frame 1. The nverse knematcs of each chan gves an analytc determnaton of the jont varables ϑ (for revolute jonts and d (for prsmatc jonts as well as ther ( 1r 1 tme dervatves. The velocty ( v and angular velocty (ω of each lmb and the correspondng acceleratons can be calculated recursvely by the followng equatons: ( v = ( v 1 + ( ω 1( r 1 ( v = ( v 1 + ( ω 1( r 1 + (4 + e z d (5 ( ω 1( ω 1( r 1 + d e z + 2d ( ω 1e z (6 ( ω = ( ω 1 + e z ϑ (7 ( ω = ( ω 1 + ϑ ( ω 1e z + ϑ e z (8 where e z = [1] T. The tlde-operator ( defnes the crossproduct ãb = a b. The acceleraton a ncludes the gravtatonal acceleraton. Smultaneously the translatonal and rotatonal Jacobans of each lmb can be calculated J T = ( (v = R 1 J T 1 ( 1 r 1 d J R 1 +e z (9 J R = (ω = R 1J R 1 + e ϑ z. (1 wth R 1 =(R 1 T. For the actuated jonts characterzed by the vector of actuated varables q a, the nverse Jacoban of the manpulator J 1 = q a / can be determned n 225 Fg. 1. Scheme of a general 6-DOF parallel manpulator. the same sense. The use of the MDH-notaton s advantageous for the analyss of the forward and nverse knematcs. It allows the applcaton of reducton rules for dynamc parameters and ensures the computatonal effcency wthn standard control and processors [12]. Smlar conclusons were also met recently for parallel manpulators [6]. Furthermore, the real axs drectons of the unversal jonts can be easly taken nto account. The conventonal smplfcaton as ball-and-socket jont s not necessary [4], [5]. III. EFFICIENT DYNAMICS FORMULATION The dynamcs of parallel robots has been formulated n lterature mostly usng the Newton-Euler formalsm, whch s more effcent for such mechansms. To avod the calculaton of constrant forces lke n [5] or [7], the D ALEMBERT prncple of vrtual work was used n [4], [1]. Ths prncple s based on vrtual dsplacements and vrtual work. It s though not approprate for modellng frcton forces that depends however on jont veloctes. To assure the ntegraton of frcton wthn the nverse dynamcs, the equvalent JOURDAIN s prncple of vrtual power s used. It consders vrtual veloctes and vrtual power and allows not only effcent calculaton but also provdes the lnear form of the dynamcs, necessary for accurate model dentfcaton [1], [9]. The power balance equaton s obtaned as ( T δ θ T τ = δ q T qa a Q a τ = Q a, (11 where τ s the vector of the generalzed forces and Q a s the vector of the actuator forces. Equaton (11 means that the vrtual power resultng n the space of generalzed coordnates s equal to the actuaton power. The power balance can be appled for rgd-body forces: ( T qa Q a,rb = τ rb = J T τ rb, (12

3 and for frcton losses n all jonts ( T ( T q q Q a,f = Q q f = J T a Q f. (13 It s mportant to notce that Q a,rb and Q a,f present ALL rgd-body forces and ALL frcton forces transformed nto the actuaton space. The formulaton of parameter-lnear dynamcs s dscussed n the followng subsectons. A. Parameter-lnear Form of Rgd-Body Dynamcs The generalzed rgd-body forces for a manpulator wth N bodes are τ rb = N [J T T (m ( v + ( ω s + ( ω ( ω s =1 ( ( +J T R (I ( ( ω + ( ω (I ( (ω ] + s ( v. (14 wth dynamc parameters of each body : ts mass m, ts frst moment s := [s x s y s z ] T = m ( r C (r C : vector from coordnate frame to centre of mass and ts nerta tensor about the correspondng coordnate frame ( I (. New operators ( and ( are defned: ω I := ( I ( (ω, (15 wth ω := ω x ω y ω z ω x ω y ω z and ω x ω y ω z I =[I xx I xy I xz I yy I yz I zz ] T, (16 whch helps the smplfcaton of the generalzed rgd-body dynamcs [1], [12]: τ rb = N ] [J T T J T R Ω I s =1 }{{} m H }{{} p (17 wth [ Ω = = [H 1 H N ] [ p T 1 p T N ] T, (18 ]. ω ( + ( ω ( ω ( v ( ω + ( ω ( ω ( v (19 Consderng the power balance gven by (11 the actuaton forces resultng from the rgd-body dynamcs can be derved n a lnear form: Q a,rb = [ ] J T H p rb. (2 The dmenson of the parameter vector p rb has to be reduced for an effcent calculaton and to assure the dentfablty of the system [1], [2], [3]. Only few publcatons treated the parameter reducton for PKM systematcally [6], [12], although ths ssue s crucal for dentfcaton or for adaptve control algorthms [1], [3]. Some approaches were 226 presented and mplemented successfully n practce, but the consdered models are sgnfcantly smplfed [3], [8]. In the recently publshed methodology [6], a very good systematc study of parameter reducton of parallel robots s shown, but unfortunately, no practcal results were nvestgated. The proposed algorthm n the followng s based on former works for seral and parallel manpulators [6], [11], [12], [13]. The matrces H n eq. (17-2 can be grouped n sngle seral knematc chans (whch are here the legs or struts, such that a recursve calculaton: H = H 1 L + K (21 can be acheved. The matrces L and K are gven n [12] and derved n [13] for NEWTON-EULER equatons. The frst step consders n elmnatng all parameters p rb,j that correspond to a zero column h j of H, snce they do not contrbute to the dynamcs. The remanng parameters are then regrouped to elmnate all lnear dependences by nvestgatng H. If the contrbuton of a parameter p rb,j depends lnearly on the contrbutons of some other parameters p rb,1j,...,p rb,kj, the followng equaton holds: k h j = a lj h lj. (22 l=1 Then p rb,j can be set to zero and the regrouped parameters can be obtaned by p rb,lj,new p rb,lj,new = p rb,lj + a lj p rb,j. (23 The recursve relatonshp gven n (21 can be used for parameter reducton. If one column or a lnear combnaton of columns of L s constant wth respect to the jont varable and the correspondng columns of K are zero columns, the parameters can be regrouped. Ths leads to the rules whch are formulated n [11], [12] and n [13]. The rules can be drectly appled to the struts, snce they are treated as seral knematc chans and ther coordnate frames are defned wth respect to the MDH-conventon (secton II. For revolute jonts wth varable ϑ, the other MDH-parameters are constant. Ths means that the 9 th,the1 th and the sum of the 1 st and 4 th columns of L and K comply wth the mentoned condtons. Thus, the correspondng parameters I yy, s z and m can be grouped wth the parameters of the antecedent jont 1. For prsmatc jonts however, the moments of nerta can be added to the carryng antecedent jont, because the orentaton between both lnks reman constant. For a detaled nsght, t s recommended to consder [6] and [13]. The end-effector platform closes the knematc loop and further parameter reducton s possble. The veloctes of the platform jont ponts B j and those of the termnal MDH-frames of the respectve leg are the same. It results therefore dependences of energy-functons of the termnal leg body wth those of the platform [11]. Ther masses can be grouped to the nertal parameter of the platform accordng to stener s laws (see secton IV and table I.

4 B. Parameter-lnear Form of Frcton Dynamcs Commonly frcton n robotcs s modelled as force characterstcs dependng on jont veloctes q : Q f =[φ 1 ( q...φ m ( q ] [α 1...α m ] T, (24 where φ k are elemental functons whch can be lnear (e.g. vscous dampng and nonlnear (e.g. coulomb or dry frcton. Regroupng frcton losses n all n jonts yelds [ Q f =[D 1 ( q,...,d m ( q] α T }{{} 1,...,α T T m], (25 }{{} D f p f wth and α T k =[α kj,...,α kn ], (26 D k ( q =dag (φ k ( q 1,φ k ( q 2,...,φ k ( q n. (27 Applyng the JOURDAIN s prncple of vrtual power as gven n eq. (13 leads to the lnear form of the resultng frcton forces n the actuaton space [ Q a,f = J T ( q T D f ] p f. (28 The accurate analyss of frcton for PKM s mostly not regarded n most publcatons, especally n those nterested n the theoretcal dervaton of moton equatons [4], [6], [7]. The practce reveals the necessty of frcton compensaton for control mprovement, whch explans why ts consderaton took mostly place n control applcatons [3], [8], rather than n theoretcal works. However, n such cases frcton s consdered only for the drves, whereas losses n passve jonts are neglected. Ths work proposes the nterface of ths ssue by combnng accurate modellng wth control applcaton. The use of the JOURDAIN s prncple allows a unform dervaton of the ntegral dynamcs of parallel manpulators. The practcal applcaton of the presented theory s dscussed n the followng. IV. APPLICATION ON THE INNOVATIVE HEXAPOD PALIDA. The consdered hexapod PaLDA s equpped wth electromagnetc lnear drect drves as actuators. The struts are varable n length. PaLDA s desgned for hgh-speed handlng and machnng tasks wth low process forces, lke deburng. Drect lnear drves have several advantages compared to conventonal ball screw drves, e.g. reduced mechancal components, no backlash, low nerta wth a mnmzed number of wear parts. Furthermore, hgher control bandwdth and extremely hgh acceleratons can be acheved. The lnear drect drves were orgnally desgned for fast lftng movements. For use n robotc applcaton, they were enhanced by power (coolng, mechancal desgn (reducng backlash and frcton, poston measurng and control. The system was presented at the Hannover ndustral Far n 21 (Fg Fg. 2. The hexapod PaLDA. Left: presentaton n the Hannover ndustral Far, 21. Rght: CAD-Model A. Knematcs and Dynamcs of PaLDA The robot s composed of 6 struts and an end-effector platform. Each strut of the hexapod s composed of three bodes as depcted n Fg. 3. Thus, the whole system s modelled wth 19 bodes: the movable platform (ndex E, 6 dentcal movable cardan rngs (ndex 1, 6 dentcal stators (ndex 2 and 6 dentcal slders (ndex 3. Startng from the robot s nverse knematcs gven by (1, 2, 3, the nverse knematcs of the sngle strut can be solved: l j = x 2 j + y2 j + z2 j (29 ( xj α j = arctan (3 z ( j yj β j = arctan. (31 r j where r j = x 2 j + z2 j. The calculaton of the veloctes and acceleratons as well as the Jacobans of the dfferent bodes s acheved recursvely accordng to (5-1. The necessary defntons of the MDH-Parameters for the struts are gven n Fg. 4. ( (1 (2 Fg. 3. (3 A j z j e z β j e x α j α β j j x j r j Knematcs of sngle strut e y l j y j B j For the calculaton of the dynamcs, mnmal base parameters are necessary. For the rgd-body model the rules

5 Fg. 4. d θ a α 1 α π π β π 2 π 2 3 l π 2 MDH-frames and parameters of the struts Regardng that frcton was not consdered n those approaches, the here dscussed algorthm wth a total of 1987 operatons and ncludng frcton can be consdered as a further mprovement. It s not necessary to parallelze the computaton on several processors lke suggested n [6], [7], snce ths can not be fulflled by commercal and standard control systems. The mplementaton of the computedforce control on PaLDA requred (ncludng path-plannng and jont-control less that.15 ms at a sample rate of.5 ms. Ths excellent real-tme property was acheved on a commercal dspace Power-PC 64e (333 MHz. The dscussed n sectons III-A are appled. It results a model defned by 1 mnmal parameters, whch are gven n table I. A systematc method for the reducton of frcton parameters s not necessary. It s recommended though to examne expermental nvestgatons or system propertes to decde about optmal parametrzaton and modellng. For PaLDA frcton forces n all jonts are modelled as a sum of vscous dampng and dry frcton: Q f = r 1 sgn( q +r 2 q (32 The actuated jonts l j correspond 6 dfferent dry frcton and also 6 dfferent vscous dampng coeffcents. Frcton n the the passve jonts s modelled only as dry frcton wth a common parameter for all α j and another one for all β j - jonts. The frcton model contans therefore 14 dfferent parameters. It s possble to keep the maxmal number of frcton parameters but that would be dsadvantageous n terms of parameter dentfcaton [1]. TABLE I BASIC RIGID-BODY MODEL PARAMETERS. p rb 1 I zz1 + I yy2 + I zz3 2 I xx2 + I xx3 I yy2 I zz3 3 I zz2 + I yy3 4 s y2 5 s z3 6 I xxe + m 6 3 j=1 (r2 By j + rbz 2 j 7 I yye + m 6 3 j=1 (r2 Bx j + rbz 2 j 8 I zze + m 6 3 j=1 (r2 Bx j + rby 2 j 9 s ze + m 6 3 j=1 r Bz j 1 m E +6m 3 B. Computatonal Cost and Model-Parameter Identfcaton The presented approach s mplemented n the computer algebra program MAPLE TM. It allows an automatc generaton of optmzed C-code. The nverse Jacoban gven n (12,13 s nverted by Gaussan elmnaton. The number of operatons of the resultng code s gven n Table II. The total computatonal cost proves the effcency of the approach. As a comparson, the most effcent methodologes known from lterature and presented n [6] and n [7] requre the total of 278 and 215 operatons respectvely. 228 TABLE II COMPUTATIONAL COST FOR THE CALCULATION OF DYNAMICS. +/ / Sngle strut ( End-effector Inverson of J Frcton model Total presentaton of the dynamcs n a mnmal-parameter form s not only computatonal effcent but allows also the use of lnear estmators for parameter dentfcaton [1], [3], [9]. The dentfcaton of rgd-body and frcton model parameters s necessary for accurate parametrzaton of the computed-force control. We presented two dfferent strateges for PKM. In [1] the dentfcaton of rgd-body and frcton models can be acheved separately by usng measurements at dfferent confguratons. In [9] an optmzed harmonc trajectory s used for optmal exctaton and dentfcaton of the model parameters. In both approaches the motor currents are suffcent for the measurement of the actuator forces. The knematcs are obtaned from the actuator lengths measured by nternal hall sensors. V. EXPERIMENTAL RESULTS AND CONTROL IMPROVEMENT In ths secton the results of the proposed dynamcs modellng methodology s llustrated n terms of control mprovement. The concept was mplemented on a commercal control hardware wth a sngle processor. Frst the accuracy of the model predcton n respect to the measured output s nvestgated. Subsequently the nfluence of the computedforce n the control mprovement s demonstrated. For ths purpose a benchmark trajectory s used. It s a crcle n the mddle of the workspace wth an nclnaton of 3 degrees n respect to the cartesan x-axs. The endeffector velocty s 1 ms 1. It was shown n [3] that for PKM, trackng errors already ncrease exponentally above a velocty of.1ms 1. Fgure 5 shows a comparson between the measured and calculated actuator forces whle the benchmark moton for the frst 4 - arbtrarly chosen - actuators. Neglectng frcton yelds mportant devaton of model-predcted dynamcs from the real behavor. Calculaton of rgd-body forces s not suffcent. Frcton s

6 Q a1 Q a Q a2 Q a measured rb model rb and frcton VI. CONCLUSIONS The man dea of ths paper s to present a hgh effcent methodology for the calculaton of complex dynamcs of parallel manpulators. The proposed approach s based on the JOURDAIN s prncple of vrtual power and allows a unform expresson for rgd-body and frcton dynamcs. The resultng computatonal cost s gven and compared to those known from other publcatons. The method enables real-tme calculaton and mplementaton of computed-force control wthout any model smplfcatons nto standard and commercal control systems. The success was substantated wth expermental results that demonstrate the crucal role of frcton compensaton for the sgnfcant enhancement of control accuracy. Fg. 5. Comparson between measured and calculated forces by regardng only rgd-body dynamcs (rb and by addtonally ncludng frcton. e 1 e e 2 e SJ CF (rb CF (rb+frct Fg. 6. Control errors of actuators whle a crcular moton. Comparson between sngle-jont control (SJ, computed force (CF usng only rgdbody dynamcs (rb and wth addtonal frcton compensaton (rb+frct.. crucal for model accuracy. Ths becomes clear by the comparson of trackng errors when usng dfferent models for the computed-force concept. Fgure 6 shows control errors for the same actuators by usng smple sngle-jont control (SJ and computed-force (CF wth only rgd-body dynamcs and wth addtonal frcton compensaton. The expermental results demonstrate prmarly, that SJ-control s not approprate for handlng PKM n the range of hgh speed and hgh dynamcs. The compensaton of rgd-body forces yelds far mprovement of control qualty, mostly n acceleraton phases. Most mportant for the accuracy s the compensaton of frcton whch yelds sgnfcant reducton of the control errors. Ths demonstrates exemplarly the decsve role of frcton dynamcs n control mprovement n practce. It s mportant to menton, that ths ssue s always close-knt wth relable model dentfcaton [1], [9]. 229 REFERENCES [1] M. Grotjahn, B. Hemann, and H. Abdellatf, Identfcaton of frcton and rgd-body dynamcs of parallel knematc structures for model-based control, Multbody System Dynamcs, vol. 11, no. 3, pp , 24. [2] B. Denkena, B. Hemann, H. Abdellatf, and C. Holz, Desgn, modelng and advanced control of the nnovatve parallel manpulator palda, n Proc. of the 25 IEEE/ASME Int. Conference on Advanced Intellgent Mechatroncs, AIM25, Monterry, USA, 25, pp [3] M. Honegger, R. Brega, and G. Schwetzer, Applcaton of a nonlnear adaptve controller to a 6 dof parallel manpulator, n Proc. of the 2 IEEE Int. Conf. on Robotcs and Automaton, San Francsco, 2, pp [4] L.-W. Tsa, Solvng the nverse dynamcs of a stewart-gough manpulator by the prncple of vrtual work, ASME Journal of Mechancal Desgn, vol. 122, no. 5, pp. 3 9, 2. [5] K. Harb and K. Srnvasan, Knematc and dynamcs analyss of stewart pltatform-based machne tool structures, Robotca, vol. 21, pp , 23. [6] W. Khall and S. D. Guegan, Inverse and drect dynamcs modelng of gough-stewart robots, IEEE Transactons on Robotcs, vol. 2, no. 4, pp , 24. [7] C. M. Gosseln, Parallel computatonal algorthms for the knematcs and dynamcs of parallel manpulators, n Proc. of the 1993 IEEE Int. Conf. on Robotcs and Automaton, New York, USA, 1993, pp [8] A. Vvas, P. Pognet, and F. Perrot, Predcton functonal control for a parallel robot, n Proc. of the 23 IEEE/RSJ Int. Conference on Intellgent Robots and Systems, IROS23, Las Vegas, USA, 23, pp [9] H. Abdellatf, B. Hemann, and C. Holz, Tme-effectve drect dynamcs dentfcaton of parallel manpulators for model-based feedforward control, n Proc. of the 25 IEEE/ASME Int. Conference on Advanced Intellgent Mechatroncs, AIM25, Monterry, USA, 25, pp [1] A. Codourey and E. Burdet, A body-orented method for fndng a lnear form of the dynamc equaton of fully parallel robots, n Proc. of the 1997 IEEE Int. Conf. on Robotcs and Automaton, Albuquerque, USA, 1997, pp [11] M. Gauter and W. Khall, Drect calculaton of mnmum set of nertal parameters of seral robots, IEEE Trans. on Robotcs and Automaton, vol. 6, no. 3, pp , 199. [12] M. Grotjahn, J. Kuehn, B. Hemann, and H. Grendel, Dynamc equatons of parallel robots n mnmal dmensonal parameter-lnear form, n Proc. of the 14th CISM-IFToMM Symp. on the Theory and Practce of Robots and Manpulators (RoManSy, Udne, Italy, 22, pp [13] M. Grotjahn and B. Hemann, Determnaton of dynamc parameters of robots by base sensor measurements, n Proc. of the sxth IFAC Symposum on Robot Control (SYROCO, Venna, Austra, 2.