TRAJECTORY PLANNING IN JOINT SPACE FOR FLEXIBLE ROBOTS WITH KINEMATICS REDUNDANCY

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1 The IASTED Internatonal Conference MODELLING, IDENTIFICATION, AND CONTROL Feruary 9-, 00, Innsruck, Austra TRAJECTORY PLANNING IN JOINT SPACE FOR FLEXIBLE ROBOTS WITH KINEMATICS REDUNDANCY S.G. YUE and D. HENRICH Emedded Systems and Rootcs(RESY) Informatcs Faculty Unversty of Kaserslautern D-6765 Kaserslautern, Germany [shgang, W.L. XU Insttute of Technology and Engneerng Massey Unversty Palmerston North New Zealand S.K. TSO Center for Intellgent Desgn, Automaton and Manufacturng Cty Unversty of Hong Kong 8 Tat Chee Avenue Kowloon, Hong Kong mesktao@ctyu.edu.hk Astract: The paper focuses on the prolem of trajectory plannng of flexle redundant root manpulators (FRM) n jont space. Compared to rredundant flexle manpulators, FRMs present addtonal possltes n trajectory plannng due to ther knematcs redundancy. A trajectory plannng method to mnmze vraton of FRMs s presented ased on Genetc Algorthms (GAs). Knematcs redundancy s ntegrated nto the presented method as a plannng varale. Quadrnomal and quntc polynomals are used to descre the segments whch connect the ntal, ntermedate, and fnal ponts n jont space. The trajectory plannng of FRMs s formulated as a prolem of optmzaton wth constrants. A planar FRM wth three flexle lnks s used n smulaton. A case study shows that the method s applcale. Keywords: redundant root, flexle-lnk, trajectory plannng, vraton, genetc algorthms. INTRODUCTION The use of a lght-weght flexle rootc manpulator can ncrease load carryng capacty and operatonal speed. Other potental advantages of flexle roots nclude lower energy consumpton, use of smaller actuators, safer operaton due to reduced nerta, easer transport, and a more complant structure for assemly. Work done on flexle root manpulators n past decades nvolves manly modelng, vraton control and trajectory trackng control, as revewed and dscussed y []. Snce most of the root tasks cannot e conducted wthout certan accuracy, the key prolem of flexle root manpulators s how to reduce the endpont s vratonal error. Ths vraton remans even after the root arm reaches the goal. Several dfferent approaches have already een reported and proven to e feasle on reducng vratons of flexle arms wth one or two lnks. Most of the work s ased on (ether open-loop or closed-loop) control strateges, for example [,]. However, the aove research focuses on flexle root manpulators wth only one or two lnks. It has een found recently that the redundancy, whch s used n root manpulators to acheve addtonal performance whle trackng a gven end-effector trajectory [4,5], has ts specal applcaton n vraton reducton of flexle manpulators. Nguyen and Walker [6] made use of the self-moton to compensate for and damp out flexle deformaton when the degree of redundancy s of the same numer of the deformaton modes to e controlled. Yue [7] presented an optmal method to choose the self-moton for reducton of the vraton of a FRM, n whch jont flexlty s also taken nto account. Km and Park [8] employed the self-moton capalty to solve the trackng control prolem of a FRM, and developed an algorthm n whch the self-moton s evaluated so as to nullfy the domnant modal force of flexural moton nduced y a rgd ody moton. Nevertheless, trajectory plannng of FRMs n jont space was not mentoned n the aove papers. Compared wth one- or two-lnk flexle root manpulators, t s reasonale to expect that knematcs redundancy can also provde addtonal possltes for mnmzng vraton n FRMs through trajectory plannng n jont space. Moreover, the geometrc prolems wth Cartesan paths related to workspace and sngulartes can e avoded f the pont-to-pont task s planned n jont space. Genetc Algorthms (GAs), are populaton-ased, stochastc, and gloal search methods [9]. As dscussed n [4,5], gloal soluton s qute dffcult to reach usng tradtonal methods. The gloal search alty of GAs provdes a posslty for fndng gloal solutons of redundancy. There has een lttle reported work on

2 applyng GAs to trajectory plannng of flexle root manpulators. In the followng sectons, a trajectory plannng method for FRM ased on GAs to mnmze vraton whle movng etween the ntal and fnal ponts s presented. Frst, a fnte element model for descrng dynamcs of FRM s ntroduced (Secton ). Knematcs redundancy s ntegrated nto the plannng method as varales (Secton ). Quadrnomal and quntc polynomal are used to descre the paths whch connect the ntal, ntermedate and fnal ponts n jont space (Secton 4). The trajectory plannng of FRM s formulated as a prolem of optmzaton (Secton 5). Sutale parameters of each polynomal etween two ponts, sutale ntal and fnal confguratons are determned y usng GAs (Secton 6). Fnally, a planar FRM wth three flexle lnks s used n smulaton and a case study s conducted and dscussed (Secton 7).. DYNAMIC MODEL Fnte element method s employed to uld up the dynamc equatons of a FRM wth multple flexle lnks. Wthout takng nto account the jont flexlty, the dynamc equatons of a flexle root manpulator system can then e wrtten as [0]: [ M ]{ Φ & } [ C]{ Φ& } [ K]{ Φ} = { p} () { τ } = ([ D] [ J ]){ q& Z } { h} { e} () where [M] s the n u n u gloal mass matrx, [C] s the n u n u gloal dampng matrx, [K] s the n u n u gloal stffness matrx, {p} s the n u nerta force matrx, { Φ }, { Φ & } and { Φ& & } are the generalzed coordnates vector, generalzed velocty vector and generalzed acceleraton vector respectvely, whch descre the deformaton ehavor of the flexle lnks, n u s the numer of the generalzed coordnates, [D] s the n n nerta mass matrx, { q } and { q& & }are the vectors whch descre the jont angle and angular acceleraton respectvely, {h} s the n centrfugal, gravtatonal, corols of the rgd and flexle couplng terms, {e} s the n lnk flexlty term, [J z ] s the n rotor nertal mass matrx, {t} s the n actuator torque matrx, and n s the numer of jonts n the root system.. REDUNDANCY RESOLUTION of FRM For a redundant root, the numer of degrees of freedom n of a manpulator s greater than the numer of end-effector degrees of freedom m. That s, gven a poston/posture of the end-effector, there are nfnte numer of root confguratons avalale. In ths study, redundancy of a root s used to avod the acute vraton when ts end-effector moves from one pont to another. How can the redundancy of FRM e used to attan addtonal goals, such as avodng acute vraton? Fgure. Dfferent confguratons of a FRM correspond to ntal and fnal pont of trajectory In the prevous methods [6,7], the trajectory of a FRM was planned to reduce vraton when the prescred trajectory was descred n Cartesan space. However, the fnal results ased on prevous methods are local solutons [4,5] and are prone to varous prolems related to workspace and sngulartes, snce the trajectory s gven n Cartesan space []. As mentoned aove, the presented method wll conduct the plannng prolem of FRM n jont space. Ths means that the varous prolems n Cartesan space can e avoded easly. Snce only pont-to-pont trajectory s dscussed n ths paper, there are only two mportant ponts (.e. the ntal pont and the fnal pont of a possle path ) that should e acheved. Correspondng to the two ponts, there are nfnte possle confguratons for each pont due to knematcs redundancy, for example, shown n Fgure. Ths means that the FRM can start wth one sutale pose and end n a dfferent pose, wth the startng and fnal pose determned accordng to the task eng performed. Thus, the redundancy prolem wth respect to ntal and fnal ponts nvolves determnng the ntal and fnal postures. These two postures can e descred y (n-m) parameters n jont space, where (n-m) s the redundant degree of freedom of the FRM. In our approach, as descred n the followng chapters, these (n-m) varales can e determned usng an optmzaton method. Ths means that the redundancy soluton correspondng to ntal and fnal ponts wll e determned usng GAs. The prolems of redundancy correspondng to ntermedate va ponts wll also e solved effcently y plannng n jont space nstead of n Cartesan task space, as descred n the followng sectons.

3 4. TRAJECTORY PLANNING STRATEGY Here, trajectory refers to a tme hstory of poston, velocty, and acceleraton for each degree of freedom. Suppose that the pont-to-pont trajectory conssts of several segments wth contnuous acceleraton at the ntermedate va pont (as shown n Fgure ). The poston of each ntermedate pont s supposed to e unknown n the followng part of the paper. Of course, the ntermedate ponts can also e gven as partcular ponts that should e passed through. Ths s useful especally when there s an ostacle n the workng area. a a θ & a = θ () ( 4θ T 4θ T T T () θ T θ θ T T / 4 T () a = (0) = ) 4 = ( ) The ntermedate pont () s acceleraton can e otaned as: θ & = a 6aT a4t (4) The segment etween the numer m p ntermedate pont and the fnal pont can e descred y a quntc polynomal as: 4 5 θ ( t ) = t t t t t, ( = m ) (5), p where the constrants are gven as Fgure. Intermedate ponts on the pont-to-pont trajectory If we wsh to e ale to specfy the poston, velocty, and acceleraton at the egnnng and end of a path segment, a quadrnomal and a quntc polynomal are requred. Suppose that there are m p ntermedate va ponts etween the ntal and the fnal ponts. Between the ntal pont to m p ntermedate va ponts, a quadrnomal s used to descre these segments as: 4 θ ( t) = a a t a t a t a t, ( = 0, L, m ) (), 0 4 p where the constrans are gven as θ = a (4) 0 4 θ = a a T a T a T a T (5) 0 4 θ & = a (6) θ & = a at at 4a4T (7) θ = a (8) where T s the executng tme from pont to pont. The fve unknowns can e solved as a = θ (9) 0 θ = 0 (6) θ = 4 5 T T T T T (7) θ & = (8) 4 θ & = T T 4 4T 5 5T (9) θ = (0) θ = 6 T 4T 0 5T () and these constrants specfy a lnear set of sx equatons wth sx unknowns whose soluton s 0 = θ () θ & = () = θ & (4) = ( 0θ 0θ (8 ) T (& ) T ) T (5) 4 = ( 0θ 0θ (4 6θ & ) T (θ && ) T ) T (6) 5 = θ θ (6 6 ) T ( & ) T ) T (7) 4 5 ( As formulated aove, the total parameters that should e determned are the jont angles of each ntermedate va pont (n m p parameters), the jont angular veloctes of each ntermedate pont (n m p parameters), the executng tme of each segment (m p parameters) and the ntal and fnal posture of FRM ((n-m) parameters). Therefore, there are a total of (( m p (n-m)) to e determned. It should e ponted out that jont angular acceleraton at each ntermedate pont can e otaned through equaton (4). If all the ntermedate ponts are connected y quntc polynomals, there wll e (( m p (n-m)) parameters to e determned. Ths would e more tme-consumng, so we choose oth quadrnomal and quntc polynomal to generate the segments.

4 5. OPTIMIZING TRAJECTORY USING GAs For a pont-to-pont trajectory plannng prolem for a FRM, the vraton deformaton ampltude s one of the most mportant factors to e consdered n the optmzaton. It s reasonale to assume that the vratonal deformaton ampltude s the ojectve n the path plannng process. Snce the lmtatons of jont angles, jont angular veloctes, jont angular acceleratons and jont torques are consdered n the optmzaton process, the ojectve and constrants are fnally wrtten as: elastc modules Pa, shear modules Pa, lumped mass at each dstal end of lnk 40g, lumped mass at endpont 0g, moment of nerta of ase jont Kgm, second jont Kgm, thrd jont Kgm. The materal for each lnk s alumnum. The case s smulated n the horzontal plane. There s no ostacle n the workng area. The constrants used n the followng cases are: jont angle q [-π,π]rad, angular veloctes ω [-8,8] rad/s, angular acceleratons ε [-40,40]rad/s, computed torques τ [-.5,.5]Nm, τ [-.5,.5]Nm and τ [-.0,.0]Nm, respectvely. s.t. f α mn (8) q ω m = v q q ( =, L, (9), mn, max ω ω ( =, L, (0), mn,max τ ε ε ε ( =, L, (), mn,max τ τ ( =, L, (), mn,max where α v s the largest ampltude of vratonal deformaton durng the pont-to-pont path, q,mn and q,max are the lower and upper permtted angle of th jont, ω,mn and ω,max are the lower and upper permtted angular veloctes of th jont, ε,mn and ε,max are the lower and upper permtted angular acceleratons of th jont, τ,mn and τ,max are the lower and upper computed torque of th jont, respectvely. Snce the asc formulaton of pont-to-pont moton s gven, f the parameters of the moton have een determned, then the optmal trajectory can e easly determned. We use genetc algorthms to search for those parameters. GA programs have een descred n detal [9]. In the procedure of GAs, ntalzaton generates the frst host populaton P 0 randomly. The populaton P N of the N th generaton s formed y survvors of the last generaton and new ndvduals generated through mutaton and crossover. Sngle-pont crossover s used to form the new generaton. The pont-to-pont trajectory s fnally selected when the termnaton condton s satsfed. The termnaton condton of the procedure can e maxmum generatons or a certan value accordng to dfferent demand. 6. CASE STUDY A planar FRM wth three flexle lnks s used for numercal smulaton n case studes, as shown n Fgure. The root has one redundant freedom only n terms of postonng. Its parameters are gven as: length of each lnk 50mm, heght of cross secton mm and wdth 4mm, Fgure. Endpont deformaton ampltude of FRM versus numer of generatons In ths case study, only the vratonal ampltude s assumed to e the ojectve of optmzaton. One ntermedate va pont s assumed. It s also assumed that the executng tme of each segment s one second. The termnaton condton used here s 00 generatons. All the selected trajectores start from the same ntal confguraton. There are seven parameters to e determned n ths case; they are: confguraton at fnal pont, the three jont angles and the three jont angular veloctes at the ntermedate pont. The optmzaton process took aout 0 mnutes wth a Pentum II 500MHz computer. The results are shown n the followng fgures. It s found that the vratonal ampltude s reduced ovously (shown n Fgure ). Vraton at dfferent generatons can e found n detals n Fgure 4. The vratonal ampltude decreased rapdly etween these generatons. Ths reflects the alty of GAs to fnd solutons wth effcency.

5 Fgure 4. Endpont deformaton curves at 0 th,00 th and 00 th generatons n X drecton (left) and Y drecton (rght) The jont angle, angular velocty, angular acceleraton and computed jont torque of each jont at 0 th, 00 th and 00 th generatons are compared n Fgures 5 through 7. It s notceale that frst and second jont angular acceleratons at 00 th or 00 th generaton are qute smooth; however, the thrd jont angular acceleraton at the same generaton has a sharp peak. Ths means that the thrd lnk swngs to avod acute nerta. Snce the computed torque can e nfluenced y vraton, the results at dfferent generatons are also qute dfferent, even at the end of trajectores (as shown n Fgure 7). Fgure 6. The angle, angular velocty, angular acceleraton and Computed jont torque of the second jont at 0 th,00 th and 00 th generaton Fgure 5. The angle, angular velocty, angular acceleraton and computed jont torque of the frst jont at 0 th,00 th and 00 th generaton Fgure 7. The angle, angular velocty, angular acceleraton and Computed jont torque of the thrd jont at 0 th,00 th and 00 th generaton

6 The confguratons at 0 th, 00 th and 00 th generatons are shown n Fgure 8. It s found that the 00 th and 00 th generaton s movng areas are confned to one sde and have shorter trajectores compared wth 0 th generaton. However, the straght lne from the ntal pont to the fnal pont s the shortest one, ut s far from the est one accordng to the GA optmzaton results. descre the segments whch connect the ntal, ntermedate, and fnal ponts n jont space. Sutale parameters of each polynomal etween two ponts and sutale ntal and fnal confguratons were determned usng GAs. Case study of a planar FRM wth three flexle lnks showed that the method s effectve. The geometrc prolems wth Cartesan paths related to workspace and sngulartes were avoded y usng the presented method. It should e noted that jont elastcty was not taken nto account n ths method. ACKNOWLEDGMENT The frst author s a research fellow of the Alexander von Humoldt (AvH) Foundaton. The support of the AvH Foundaton s greatly apprecated y the frst author. Fgure 8. The confguratons of FRM at 0 th generaton (aove), 00 th generaton (mddle) and 00 th generaton (elow) respectvely, the trajectory starts from rght to left 7. CONCLUSIONS In the aove sectons, the prolem of trajectory plannng for FRM was studed n detal. A trajectory plannng method for FRMs ased on GAs to mnmze vraton was presented. Knematcs redundancy was consdered as a plannng varale n the presented method. Quadrnomal and quntc polynomals were used to REFERENCES. W.J.Book, Modelng, desgn, and control of flexle manpulator arms: A tutoral revew, Proc. 9th IEEE Conf. on Decson & Control, 990, A.Mohr, P.K.Sarkar and M.Yamamoto, An effcent moton plannng of flexle manpulator along specfed path, Proc. IEEE Int. Conf. On Rootcs and Automaton, 998, S. Choura, S. Jayasurya and M.A. Medck, On the modelng, and open-loop control of a rotatng thn flexle eam, Transactons of the ASME, J. of Dynamc Systems, Measurement, and Control,, 99, D.N.Nenchev, Redundancy resoluton through local optmzaton: A revew, Journal of Rootc Systems, 6(6), 989, B.Sclano, Knematc control of redundant root manpulators: A tutoral, Journal of Intellgent and Rootc Systems,, 990, L.A. Nguyen, I.D. Walker and R.J.P. Defgueredo, Dynamc control of flexle knematcally redundant root manpulators, IEEE Transactons on Rootcs and Automaton, 8(6), 99, S.G. Yue, Redundant root manpulators wth jonts and lnk flexlty I: dynamc moton plannng for mnmum end-effector deformaton, Mechansm and Machne Theory, (/), 998, S.J. Km and Y.S. Park, Self-moton utlzaton for reducng vraton of a structurally flexle redundant root manpulator system, Rootca, 6, 998, D.E. Goldenerg, Genetc Algorthms n search, optmzaton and machne learnng, (Addson- Wesley, Readng, Mass., 989). 0. S.G. Yue, Y.Q. Yu and S.X. Ba, Flexle rotor eam element for root manpulators wth lnk and jont flexlty, Mechansm and Machne Theory, (), 997, J. J. Crag, Introducton to Rootcs Mechancs and Control (Addson-Wesley Pulshng Company, USA, 986).