Dynamic Model of a 7-DOF Whole Arm Manipulator and Validation from Experimental Data

Transcription

1 Dynami Model of a -DOF Whole Arm Manipulaor and Validaion from Experimenal Daa Zaira Pineda Rio, Andrea Lehini-Visinini and Rodrigo Quian Quiroga Deparmen of Engineering, Universiy of Leieser, Universiy Road, Leieser, U.K. Keywords: Absra: Whole Arm Manipulaor Model, Friion Model, Friion Idenifiaion. The presen paper desribes he design of he dynami model of a degrees of freedom whole arm manipulaor implemened in SimMehanis. The friion phenomena of he manipulaor is idenified, represened hrough a fied model and inluded in he sysem model wih he aim of inremen he auray of he model wih respe o he real sysem. The haraerisis of he model make i suiable o es and design onrol sraegies for moion and friion ompensaion in MATLAB/Simulink. INTRODUCTION The ompuaion of he dynami model of a robo manipulaor plays an imporan role in simulaion of moion, analysis of he manipulaor s sruure and design of opimal onrol algorihms. The inlusion of he effes of friion in a mehanial sysem model when simulaing onrol sraegies, helps o improve he performane of he onroller o be implemened in he real sysem (Kosi e al., ; Indri, ; Bompos e al., ). Mos of he whole arm manipulaors mahemaial models are based in he ompuaion of muli-links serial robo s mahemaial equaions. These equaions are obained using he Newon-Euler reursive mehod o alulae he Coriolis, enrifugal and inerial fores observed when he end-effeor is in moion. Moreover, a Jaobian approah may be implemened in parallel for mapping beween Caresian and join spae, in order o minimize singulariy ondiions ha inrease he ompuaion load of he onrol algorihm (Lau and Wai, ; Sousa e al., 9). This mehodology involves boh operaional and join fores. In mos ases hese mahemaial models are implemened using high level ompuing languages as MATLAB (Corke, 99), C/C or Forran. Neverheless, in order o avoid signifian ompuaion ime, some auhors have found in MAT- LAB/SimMehanis a omforable ool o design mehanial sysems used for experimenal verifiaion. The apabiliy of his ool yields appropriae resuls when working wih joins wih DOF and when all he manipulaor s inerial parameers are known (MahWorks, ). Seion offers a brief desripion of he real manipulaor sysem. Seion gives insighs on he design of he model and shows he imporane of experimenal daa in he developmen of he friion model. In Seion he response of he real sysem is ompared o ha of he model when simulaing some experimens. Finally, seion presens some onlusions relaed o his work. THE REAL SYSTEM The real sysem is a -degrees-of-freedom whole arm manipulaor (WAM) from Barre Tehnology In. I is a join orque onrolled manipulaor equipped wih onfigurable PD/PID onrol and graviy ompensaion. The informaion relaed o he joins onfiguraion, join moor drives and he body par masses, enre of graviy and ineria marix is provided by he manufaurer in he WAM ARM User s Manual (Barre Tehnology In, ). Figure shows he onfiguraion and aahed frames of he -DOF sysem wih a grasper. All he joins of he manipulaor are DOF revolue joins. An image of he real sysem is shown in Figure, onsising in a Barre -DOF Whole Arm Manipulaor wih an aahed BH-series BarreHand. For he developmen of his proje he -DOF manipulaor is onfigured wih join PD onrol and graviy ompensaion given by Equaion (), where he join orque τ is expressed as he sum of he differene Pineda Rio Z., Lehini-Visinini A. and Quian Quiroga R.. Dynami Model of a -DOF Whole Arm Manipulaor and Validaion from Experimenal Daa. DOI:./ In Proeedings of he 9h Inernaional Conferene on Informais in Conrol, Auomaion and Robois (ICINCO-), pages - ISBN: Copyrigh SCITEPRESS (Siene and Tehnology Publiaions, Lda.)

2 ICINCO - 9h Inernaional Conferene on Informais in Conrol, Auomaion and Robois Ig(y()) I Zool ie() lir() _ J,J Z,Z KP Xool lu() P liy() K Ds X X,X J Figure : Configuraion of he join PD onrol wih graviy ompensaion. r() is he referene rajeory, e() is he response error, P represens he manipulaor, g(y()) is he graviy ompensaion, KD and KP are he derivaive and proporional gains, respeively. Z X,X Z Z J J,J Z,Z X,X,X Z Zbase J THE MODEL The -DOF WAM model is buil by four bloks: Seven modules ha ompue he join referene rajeories. Xbase Figure : WAM -DOF Denavi-Harenberg arhieure wih aahed frames as shown in (Barre Tehnology In, ). The dynami model of he sysem. Fied friion models for eah join of he manipulaor. Join PD onrollers onfigured and uned as hose implemened in he real sysem. The dynami model of he manipulaor is based on he onfiguraion and physial haraerisis of he real sysem, he parameers of he friion model are deermined hrough he idenifiaion of he join friions whils he join rajeory and join onroller are emulaions of hose exisen on he real sysem.. Trajeory Generaion Figure : The Barre -DOF Whole Arm Manipulaor wih he BH-series BarreHand. beween he referene and measured posiion, namely posiion error q, muliplied by a onsan proporional gain KP, he derivaive of he error q muliplied by a respeive derivaive gain KD and he ompensaion for graviy g, whih is a funion of he join posiion. τ = KP q KD q g () The WAM join PD onrol blok diagram is shown in Figure. The onrol loop uses he referene signal r(), namely a join rajeory, and he measured posiion y() o ompue he presen error. Then he derivaive of he error is alulaed, and he graviy ompensaion is added o onform he join orque u() o be applied o he manipulaor s join. The referene signal used o perform every join roaion is a linear segmen paraboli blend (LSPB) rajeory defined as follows. Up o a ime he rajeory is paraboli wih linear veloiy, a he rajeory hanes o linear wih onsan veloiy and zero aeleraion, finally afer a ime ( f ) he rajeory hanges o paraboli again and he veloiy dereases linearly unil reahing zero. The moion produed when applying his veloiy profile o any join is ranslaed in a roaion from he iniial join posiion qi of he manipulaor o q f radians. Equaion () defines he desribed rajeory, where qi is he iniial posiion, q f is he final posiion reahed in a ime f, q = q is he value of he onsan veloiy exhibied from ime o ime ( f ) and q is he value of he desired onsan aeleraion. qi q q() = () (q q q ) q < i f f f q f q ( f ) f < f

3 DynamiModelofa-DOFWholeArmManipulaorandValidaionfromExperimenalDaa [rad] [ lrad/s] [rad/s ] Posiion [s] Veloiy.. [s] Aeleraion. -. [s] Figure : Example of a rapezoidal veloiy profile when he values of he variables are se as follows q i =., q f =. rads, q =. rads/s, f = s and =. An example of he rajeory given by Equaion () an be seen in Figure where a roaion of. rads is performed. In order o generae he LSPB join rajeory, he real sysem ses he value of he ime of hange as, and he veloiy q and aeleraion q are alulaed using Equaions () and (), aking ino aoun he measured values of he iniial and he final join posiion q i and q f. In simulaions, he join rajeory is emulaed by he model onsidering he experimenal values of he iniial join posiion q i, he final join posiion q f and he exeuion ime f. The value of is se as and he veloiy and aeleraion are also alulaed using Equaions () and (). Finally, aking all he parameers previously esimaed, he join rajeory is generaed using Equaion (). q = q (q f q i ) f ( (. f )) () = q (). The Dynami Model The dynami model equaion for a robo manipulaor is generally wrien in he form (Siiliano e al., 9): B(q) qc(q, q) qg(q)f = τ () where q, q, q are he posiion, veloiy and aeleraion veors of all he joins ha shape he manipulaor, B(q) is he inerial marix, C(q, q) q is he Coriolis-Cenrifugal marix, G(q) is he graviy veor and F is he veor of friion, whih is usually obained using a fied friion model. The dynami model of he WAM Arm was designed and simulaed using MATLAB/SimMehanis, and mos of he inerial daa provided by he manufaurer had o be adaped aording o he inerial referene sysem. The MATLAB/SimMehanis oolbox is a blok diagram modelling environmen like Simulink, reaed by MATLAB, o design and simulae mehanial sysems. The oolbox onains several modules ha represen pariular bodies and whih inerial properies an be speified by he user. A grea advanage of using SimMehanis in modelling mehanial sysems is ha he oolbox is promp o be used wih Simulink so ha onrol rouines an be added wih ease in order o analyse he behaviour of he sysem s dynamis under moion onsrains. Posiion [rads] Torque [Nm].. -. Daa for friion modelling J.Id_.id - Samples (T=.s) Samples (T=.s) Veloiy [rad/s] 9 Samples (T=.s) Torque [Nm] Samples (T=.s) Figure : Regiser of posiion and orque in he Barre WAM when roaing join by. rads.. The Friion Model Friion phenomena in robo manipulaors may affes he auray of he sysem in posiion onrol and when moving he manipulaor a very low veloiies. Mos of he friion models employed when modelling posiion onrolled mehanisms, inlude several omponens as he sliding friion (or Coulomb friion), he breakdown friion (or siion) and he visous friion (Bona and Indri, ). In mos ases if he manipulaor is expeed o displae a medium or medium-high veloiies, he friion an be expressed mahemaially as a funion of he join veloiy onsidering he effes of he visous friion and he Coulomb friion only (Guran e al., ; Lewis e al., ; Kelly e al., ; Siiliano e al., 9). This simplified friion model is known as lassi friion model: F = F sign(v)σ v () where F in he parameer for Coulomb friion, σ is he visous damping oeffiien and v is he veloiy of he sysem. The value of he parameers for he lassi friion model an be esimaed from experimenal daa 9

4 Friion [Nm] Friion [Nm] Friion [Nm] Friion [Nm] Friion [Nm] Friion [Nm] Friion [Nm] ICINCO-9hInernaionalConfereneonInformaisinConrol,AuomaionandRobois Friion-Veloiy Map of Join Veloiy [rad/s] Friion-Veloiy Map of Join Veloiy [rad/s] Friion-Veloiy Map of Join Friion-Veloiy Map of Join Veloiy [rad/s] Veloiy [rad/s] - Friion-Veloiy Map of Join... Friion-Veloiy Map of Join. Friion-Veloiy Map of Join Measured daa Esimaion Veloiy [rad/s] Veloiy [rad/s] Veloiy [rad/s] Figure : Friion veloiy maps orresponding o eah join of he DOF manipulaor. The measured daa is obained afer roaing eah join of he manipulaor a differen onsan veloiies whils he esimaed value is ompued using a fied lassi friion model. hrough he exeuion of join roaions a onsan veloiy, his proess in no sraigh forward and is explained in deail in seion..... Friion Idenifiaion The proedure for he idenifiaion of he friion model parameers for eah join of he -DOF whole arm manipulaor follows several seps. Firs a losed loop PD onrol wih graviy ompensaion is used o move he link following a rapezoidal veloiy profile a differen posiive and negaive veloiies, he measuremens of posiion in addiion wih he orresponding sampling ime are employed o ompue he veloiy of he link in eah experimen. Aferwards, he values of orque and veloiy during he onsan veloiy sage of he rajeory are averaged and used o onsru he friion-veloiy map, whih is basially a orque versus veloiy plo. Finally, he parameers for he friion model are esimaed o fi he friion-veloiy map by applying a leas-squares minimizaion mehod (Canudas de Wi and Lishinsky, 99; Canudas de Wi e al., 99; Olsson e al., 99; Johnson and Lorenz, 99) using n i= [F(v i ) ˆF(v i )] () where F(v i ) is he measured orque a erain veloiy v i (i.e he friion fore), and ˆF(v i ) is he value ompued by he friion model represened in Equaion (). Figure shows an example of he measured posiion and orque in he manipulaor when roaing join. The daa segmen in he inse of he figure orresponds o he linear hange in he posiion whih is used o ompue he join veloiy. The averages of he ompued veloiy v i and measured orque F(v i ), respeively, yield o ordered pairs [F(v i ),v i ] ha shape he friion veloiy map. The obained friion model parameers for eah Table : Friion model parameers for he seven joins of he robo manipulaor. V /V sands for posiive and negaive veloiies respeively. Join Coulomb friion Visous friion F σ (V /V ) (V /V )./.9./../../-.9.99/.9.9/../. -./-.99.9/. -.9/../..9/../..9/.9 join of he -DOF whole arm manipulaor are lised in Table. There an be noed ha he firs join is he mos affeed by he Coulomb friion F whils he DOF ha shape he wris of he manipulaor are he leas influened. These parameers were used o esimae he join friion veloiy maps ploed in Figure, where he veloiy maps obained using real daa are also shown. The measured friion suied perfely he shape of he lassi friion model, whih reinfores he iniial hoie of he friion model for he developmen of his proje. In general, he esimaion of friion is very aurae for all he joins, and i is suiable enough o be onsidered when ompensaing he friion phenomena manifesed in he real manipulaor. SIMULATIONS Several join roaional moion experimens were performed wih he aim of evaluae he auray of model wih respe o he real manipulaor. Firs, a LSPB join

5 Posiion Error [rads] Posiion Error [rads] Posiion Error [rads] Posiion Error [rads] Posiion Error [rads] Posiion Error [rads] Posiion Error [rads] DynamiModelofa-DOFWholeArmManipulaorandValidaionfromExperimenalDaa ix - Join ix - Join ix - Join ix - Join Time [se] Time [se] Time [se] Time [se] ix - Join ix - Join ix - Join Time [se] Time [se] Time [se] Manipulaor Model Figure : Real and simulaed join error of he DOF manipulaor when exeuing a speifi rajeory. Every link is roaed one a a ime and he error is alulaed using he referene posiion minus he aual posiion. rajeory wih a roaion of. rad a a veloiy of. rads/se was used as a referene in he real sysem, and he join posiion was measured while he manipulaor was in moion. Subsequenly, he join posiion error was alulaed o be used as a performane measuremen. Laer, in simulaion, he same join rajeory was generaed using experimenal daa as desribed in Seion.. The response of he model was reorded and he model join posiion error was alulaed. The posiion error was hosen as a performane measuremen in order o have a loser omparison of he response of boh he real manipulaor and he model when exeuing he exa same rajeory. As menioned earlier, he real sysem uses a PD onrol wih graviy ompensaion, and herefore, hroughou he simulaions a PD onrol was used and zero graviy was assumed. During he firs se of simulaions every link was roaed one a a ime. The seond se onsised in one simulaion only, where all he joins were roaed a he same ime. This gave a oal of eigh simulaions performed by he model, o be ompared o he behaviour of he real sysem when exeuing he exa same rajeory. The graphs in Figure show a omparison of he posiion error obained in he firs se of simulaions. This praie helped o prove he veraiy of he model wih friion join per join and showed ha, regardless of he real manipulaor is exposed o ransduer noise and sensor errors, he response of he model is aurae. The seond se of simulaions was more an evaluaion of reliabiliy, due o in real exerise he manipulaor migh operae in Caresian spae and all he joins would roae a he same ime. The response of he model during his simulaion was good as expeed despie of he lear differene beween he simulaed and he real posiion error of he hird join. Figure shows he resuls of he seond se of simulaions. The differene in errors beween model and real sysem are expeed due o he disurbanes a whih he real sysem is exposed, suh as nonlineariies aused by he join moors and he ineria of he sysem. Oher soures of disurbane o be onsidered are he join auaors and he sheme used for graviy ompensaion. CONCLUSIONS The idenifiaion of friion phenomena in he manipulaor is helpful o observe he effes ha friion has in he overall performane of he sysem, so a ompensaion ehnique, if neessary aording o he appliaion ask of he manipulaor, may be implemened on he urren join PD/PID onrol. On he oher hand, he inlusion of a friion model in he dynami model of he mehanial sysem esablish a good plaform for simulaions in order o observe and analyse he response of he manipulaor o differen onrol sraegies in a more realisi senario. The presened dynami model wih friion is promp o be used as a referene in he design of join onrol sraegies implemened in Simulink. By aking advanage of he fa ha real join rajeory is ompleely emulaed in our simulaions by seing up he desired roaion angle and exeuion ime, a lose omparison of he simulaion resuls wih respe o experimenal daa is also possible. The developmen of a dynami model wih friion is exremely useful for observing and improving he performane of he manipulaor by analysing he effeiveness of he on sysem onrol. In erain ases, some ypes of simuli uilised in onroller uning ehniques are no feasible o be implemened in experimens whils he model an be easily manipulaed in simulaions.

6 Posiion Error [rads] Posiion Error [rads] Posiion Error [rads] Posiion Error [rads] Posiion Error [rads] Posiion Error [rads] Posiion Error [rads] ICINCO-9hInernaionalConfereneonInformaisinConrol,AuomaionandRobois ix - Join ix - Join ix - Join ix - Join Time [se] Time [se] Time [se] Time [se] ix - 9 Join ix - Join ix Join Time [se] Time [se] Time [se] Manipulaor Model Figure : Real and simulaed join error of he DOF manipulaor when exeuing a speifi rajeory. All he joins are roaing a he same ime and he error is alulaed using he referene posiion minus he aual posiion. REFERENCES Barre Tehnology In (). WAM Arm User's Manual. Bompos, N., Aremiadis, P., Oikonomopoulos, A., and Kyriakopoulos, K. (). Modeling, full idenifiaion and onrol of he misubishi pa- robo arm. In IEEE/ASME Inernaional Conferene on Advaned Inelligen Meharonis,. Bona, B. and Indri, M. (). Friion ompensaion in robois: an overview. In h IEEE Conferene on Deision and Conrol. Canudas de Wi, C. and Lishinsky, P. (99). Adapive friion ompensaion wih parially known dynami friion model. In Inernaional Journal of Adapive Conrol and Signal Proessing. Canudas de Wi, C., Olsson, H., Asrom, K., and Lishinsky, P. (99). A new model for onrol of sysems wih friion. In IEEE Transaions On Auomai Conrol. Corke, P. (99). A robois oolbox for malab. In IEEE Robois and Auomaion Magazine. Guran, A., Pfeiffer, F., and Popp, K. (). Dynamis wih friion. In Series on Sabiliy, Vibraion and Conrol of sysems. Indri, M. (). Conrol of manipulaors subje o unknown friion. In h IEEE Conferene on Deision and Conrol. Johnson, C. and Lorenz, R. (99). Experimenal idenifiaion of friion and is ompensaion in preise, posiion onrolled mehanisms. In IEEE Transaions on Indusry and Appliaions. Kelly, R., Sanibanez, V., and Loria, A. (). Conrol of Robo Manipulaors in Join Spae. Springer. Kosi, D., de Jager, B., Seinbuh, M., and Hensen, R. (). Modeling and idenifiaion for high performane robo onrol: An rrr-roboi arm ase sudy. In IEEE Transaions on Conrol Sysem Tehnology. Lau, H. and Wai, L. (). A jaobian-based redundan onrol sraegy for he -DOF WAM. In Sevenh Inernaional Conferene on Conrol, Auomaion, Robois and Vision. Lewis, F., Dawson, D., and Abdallah, C. (). Robo Manipulaor Conrol: Theory and Praie. Marel Dekker, seond ediion. MahWorks (). MATLAB/SimMehanis. Olsson, H., Asrom, K., Canudas de Wi, C., Gafver, M., and Lishinsky, P. (99). Friion models and friion ompensaion. In European Journal of Conrol. Siiliano, B., Siavio, L., Villani, L., and Oriolo, G. (9). Robois: Modelling, Planning and Conrol. Springer, London. Sousa, C., Coresao, R., and Queiros, P. (9). Complian o-manipulaion onrol for medial robois. In Proeedings of he nd Conferene on Human Sysem Ineraions.