Robust Tracking Control for Constrained Robots

Transcription

1 Aailabl oli at Procia Egirig 4 ( 9 97 Itratioal Symosium o Robotics a Itlligt Ssors (IRIS Robust rackig Cotrol for Costrai Robots Haifa Mhi, Olfa Boubakr* Natioal Istitut of Ali Scics a chology, Ctr Urbai Nor, BP uis, uisia Abstract I this ar, a ol robust trackig cotrol law is roos for costrai robots ur ukow stiffss iromt. h stability a th robustss of th cotrollr ar ro usig a Lyauo-bas aroach whr th rlatioshi btw th rror yamics of th robotic systm a its rgy is istigat. Fially, a 3DOF costrai robotic arm is us to ro th stability, th robustss a th safty of th roos aroach. h Authors. Publish by Elsir Lt. Slctio a/or r-riw ur rsosibility of th Ctr of Humaoi Robots a Bio-Ssor (HuRoBs, Faculty of Mchaical Egirig, Uirsiti kologi MARA. ywors: Positio/forc cotrol; Robust cotrol; Ukow stiffss iromt; Costrai robot.. Itrouctio Robust trackig cotrol ur ukow costrai iromt is o of th mai imortat issus i robotic fil. A umbr of sigificat ars ar roos i this ara. hy iclu fuzzy logic bas cotrollrs [, ], ural tworks bas cotrollrs [3, 4] a sliig mo cotrollrs [5, 6]. I th most cass, ths rsarch ars ar ot cocr by th comromis btw robustss a safty for th trackig cotrol roblm. Oly fw ars ar ot to such subjct [7, 8]. his ar is lac i this cotxt. Furthrmor, th cotrollr roos i this ar is simlr tha thos roos i [7] a [8]. h stability a th robustss of th roos aroach ar show, by simulatio rsults, o a robotic maiulator costrai to a circular trajctory.. Problm formulatio Cosir a costrai robotic systm with grs of from scrib by th yamical mol [9]: M ( θθ & + H (θ,θ & + G(θ = U J (θ F ( whr θ, θ&,& θ x R ar joit ositio, joit locity a joit acclratio ctors, rsctily. M ( θ R is th irtia matrix, H ( θ, θ & R is th ctor of ctrifugal a Coriolis forcs a G( θ R is th ctor of graity trms. U R is th graliz joit forc ctor, F R is th ctor of cotact graliz forcs xrt by th maiulator o th iromt a is th task sac imsio. Lt X, X & R th Cartsia ositio ctor a th Cartsia locity ctors of th robotic systm fi, rsctily, i th task sac by: * Corrsoig author. l.: ; fax:.: arss: olfa.boubakr@isat.ru.t.

2 Haifa Mhi a Olfa Boubakr / Procia Egirig 4 ( 9 97 X = h(θ ( X & = h( θ θ = J (θθ& (3 whr h( θ : R R is a ctor of oliar fuctios scribig th forwar kimatic mol a J ( θ R is th Jacobia matrix assum to b full rak. h tmoral riati of th imatic mol (3 is gi by: X & = J& ( θθ& + J (θθ & (4 x whr X & R is Cartsia -ffctor acclratio. Gi a sir Cartsia ositio of th -ffctor X R th cotrol roblm aims to sur: lim X = t t f X (5 Notatio: I th followig, w will aot th followig otatios: M ( θ = M, H (θ, θ & = H, G(θ = G, J (θ = J 3. Robust ositio/forc cotrollr horm: h costrai robotic systm scrib by th yamical mol: M & θ + H + G = U J F (6 is asymtotically stabl ur th ucrtai forc mol: whr a R x cotrol law is scrib by: F = ( + ( X X (7 ar th iromt stiffss a th stiffss ukow ucrtaity, rsctily, a th robust U = J [ ( X X + ( X& X& ] + J ( X& X& σ sig(σ + G (8 whr σ is a oliar fuctio fi by: [( X X + Λ( X X ] σ = C & & (9 x for th costat ctor C R a th ositi costat matrics a x, R satisfyig th followig coitios: x Λ R, if thr xist iagoal gai matrics > + < (

3 Haifa Mhi a Olfa Boubakr / Procia Egirig 4 ( 9 97 Rmark: o satisfy safty of th robotic systm a th iromt, cotrollr (8 is bas o th ucrtai forc mol (7. h oliar fuctio sig (σ is us to sur robustss by cotrollig at th sam tim costrai robot motio a costrait forc. O th othr ha, th oliar fuctio is also abl to limit th graatio of trackig rformac occurrig urig saturatio. Stability Proof: Cosir for th costrai robot systm (6 th rror ctor fi i th joit sac by: Φ = θ θ ( a th rror ctor fi i th task sac by: Y ( Φ = X ( θ X ( Ur th ukow forc mol (7 a th robust cotrol law (8 w ca writ that: M( Φ Φ & + H( Φ, + J ( Φ Y( Φ + J ( Φ Y( & Φ + J ( Φ Y( & Φ σ sig( σ = (3 whr : = = Followig th som mtho rst i [, ] w ca ro that: Φi M ( Φ H( Φ, = (4 i t = Φi D( Φ, = J P( Φ = J ( Φ Y( Φ Φ ( Φ Y( & Φ + J ( Φ Y( & Φ σ sig( σ (5 (6 Imos, ow, to th systm (3 to ha a Lyauo Hamiltoia fuctio fi by: V ( Φ, Φ & = (Φ,Φ & + P(Φ P( (7 h rror systm (3 is asymtotically stabl if V ( Φ, satisfis th thr coitios imos by Lyauo thorm []. For roig th first a th sco coitios, w ri th sam lomts as thos rst i []. h coitios + ar th wll obtai. o ro th thir Lyauo coitio, th riati of th xrssio (7 gis: < V (Φ (Φ, Φ & P(Φ = + t t t (8 From quatios (5 a (6 w ca writ:

4 Haifa Mhi a Olfa Boubakr / Procia Egirig 4 ( 9 97 ( Φ, ( Φ = ( Φ Φ && + M = ( Φ Φ && + M M H( Φ, t t (9 Furthrmor, P t ( Φ = P( Φ = Φ J ( Φ Y ( Φ ( So, w ca ro that: ( Φ, = M ( Φ Φ && + H ( Φ, + Φ & J ( Φ Y( Φ V t ( From (3 w ca writ: Substitutig th sco mmbr of ( i ( gis: M( Φ Φ & + H( Φ, + J ( Φ Y( Φ = J ( Φ Y( & Φ J ( Φ Y( & Φkσ sig( σ ( ( Φ, = J ( Φ Y( & Φ J ( Φ Y( & Φ σ sig( σ V t (3 Usig rlatios (3, ( a ( gis: ( Φ, = Y& ( Φ Y( & Φ Y& ( Φ Y( & Φ σ sig( σ V t (4 h thir Lyauo coitio is th rifi if is ositi fiit. 4. Simulatio rsults Simulatio rsults ar carri out usig a 3DOF robotic systm usig th hysical aramtr ata gi i [3, 4] for th costrai circular motio scrib by: for θ [ ] = [ π π π] chos as: i i x( t =.76 cos( 3πt y( t =.76 si( 3πt 3 πt 3 πt θ =, t a t s f =. For sufficit coitios (, th umrical aramtrs ar = Λ = iag[ ] = 3, C = [ ] = iag[ ], = [ 5 ], = iag[ 3 5], iag[ 35] iag =. Fig. a Fig. show th olutio of th robot i th Cartsia sac with rsct to th costrai circular trajctory a th smooth rofil of th robust cotrol laws (8, rsctily.

5 Haifa Mhi a Olfa Boubakr / Procia Egirig 4 ( Positio y (m Dsir ositio Actual ositio Positio x (m Fig.. E ffctor trajctory 3 U U U3 Cotrol iut (N.m tim (s Fig.. Cotrol laws Discussio: It is clar that th currt cotrol algorithm ca b imlmt i may ral alicatios. For xaml, th roos aroach ca b rlat for rhabilitatio ic alicatios, s for xaml [-], a for bial a humaoi robots urig th imact a oubl suort hass, s for xaml [5-6]. I futur istigatios, a comarati aalysis with th rlat works [7-8] ar also la.

6 Haifa Mhi a Olfa Boubakr / Procia Egirig 4 ( Coclusio his ar rooss a siml robust cotrollr for motio trackig of costrai robots ur ukow stiffss iromt. h roos aroach taks car o th comromis btw robustss a safty for th trackig cotrol roblm. h stability a th robustss of th cotrollr ar ro usig a Lyauo-bas aroach. Futur istigatios will cocr th alicatio of th roos aroach for humaoi a rhabilitatio robotic ics. Rfrcs [] iguchi,., Fukua,.,. Positio/Forc Cotrol of Robot Maiulators for Gomtrically Ukow Objcts Usig Fuzzy Nural Ntworks, IEEE rasactios o Iustrial Elctroics 47,. 64. [] Yuksl, H., Yuus, Z.A., Nurka, Y.,. MIMO Fuzzy Sliig Mo Cotroll Dual Arm Robot i Loa rasortatio, Joural of th Frakli Istitut 348,.886. [3] atic, D., Vukobratoic, M., 998. A Nural Ntwork Bas Classificatio of Eiromt Dyamics Mols for Comliat Cotrol of Maiulatio Robots, IEEE rasactios o Systms, Ma a Cybrtics 8,. 58. [4] Jug, S., Hsia,.C.,. Robust Nural Forc Cotrol Schm ur ucrtaitis i robot yamics a ukow iromt, IEEE rasactios o Iustrial Elctroics 47,. 43. [5] Bi, Y., Cha, S.P., Dawi, W., 994. Variabl Structur Aati Motio a Forc Cotrol of Robot Maiulators, Automatica 3, [6] Ch, H., Xia, J.,. Robust Comliat Assmbly Automatio Usig a Iustrial Robot, Procigs of th 6th IEEE Cofrc o Iustrial Elctroics a Alicatios (ICIEA, [7] Jalai, J., ha, S. G., Hrrma, G., Mlhuish, C.,. A Nol Aroach of Robust Acti Comliac for Robot Figrs. I: Li,.H., u,.y., sai, C.C., Hsu, C.C., sg, C.C., Vaakkat, Balts, P.J., Arso, J., Wog, C.C., Jss, N., uo, C.H., Yag, H.C. (Es., Nxt Wa i Robotics. Commuicatios i Comutr a Iformatio Scic,. 5-57, Srigr-Hilbrg. [8] Mhi, H., Boubakr, O.,. Nw Robust rackig Cotrol for Saf Costrai Robots Ur Ukow Imac Eiromt, Lctur Nots i Comutr Scic 749. [9] Chiarii, S., Siciliao, B., Villai, L A sury of robot itractio cotrol schms with xrimtal comariso, IEEE/ASME rasactios o Mchatroics 4,. 73. [] Mhi, H., Boubakr, O., Stiffss a imac cotrol usig Lyauo thory for robot-ai rhabilitatio. Itratioal Joural of Social Robotics, DOI:.7/s [] Mhi H., Boubakr O.,. Rhabilitatio of a huma arm suort by a robotic maiulator: A ositio/forc coorati cotrol. Joural of Comutr Scic 6,. 9. [] Sloti, J.J.E., Li, W.,99. Ali Noliar Cotrol, Prtic Hall. [3] Mhi, H., Boubakr, O.,. Imac Cotrollr u by Particl Swarm Otimizatio for Robotic Arms. Itratioal Joural of Aac Robotic Systms 8,. 93. [4] Mhi, H., Boubakr,O.,. Positio/forc Cotrol Otimiz by Particl Swarm Itlligc for Costrai Robotic Maiulators, Procigs of th th IEEE Itratioal Cofrc o Itlligt Systms Dsig a Alicatios (ISDA, [5] Aloulou, A.., Boubakr, O.,. Miimum jrk-bas cotrol for a thr imsioal bial robot. Lctur Nots i Comutr Scic 7,. 5. [6] Aloulou, A., Boubakr, O.,. Cotrol of a St Walkig Combi to Arms Swigig for a hr Dimsioal Humaoi Prototy. Joural of Comutr Scic 6,. 886.