Sliding Mode Controller with RBF Neural Network for Manipulator Trajectory Tracking

Transcription

1 IAENG Internatonal Journal of Apple Mathematcs, 5:, IJAM_5 Slng Moe Controller wth RBF Neural Network for Manpulator rajectory rackng Hatao Zhang, Mengmeng Du an Wenshao Bu Abstract In orer to ecrease the chatterng generate by the slng moe controller (SMC) n the manpulator trajectory trackng, we present a new aaptve slng moe controller base on raal bass functon (RBF) neural network. he slng moe varable structure control s use for resstng sturbance an guaranteeng the system stablty, an the RBF neural network s ntrouce to reuce the swtchng gan through self-learnng ablty. he nput of RBF neural network s the slng moe functon, an ts output s the swtchng gan whch can be ajuste aaptvely. Uner the conton of exstng moel errors an external sturbances, the smulaton stues on the mult-jont rg manpulator show that the propose algorthm can obtan goo performance both n trackng the trajectory an reucng the chatterng. Inex erms manpulator, RBF neural network, slng moe varable structure control, trajectory trackng I. INRODUCION he manpulator system s a tme-varyng, strong couplng an nonlnear system. he manpulator trajectory trackng have attracte conserable attentons of scholars []-[3]. Slng-moe control s a robust esgn methoology usng a systematc scheme base on a slng surface an Lyapunov s stablty theorem. he man avantage of SMC s that the system uncertantes an external sturbances can be accommoate because of the nvarance characterstcs of system slng contons. However, the swtchng control n SMC results n control gan chatterng. When the moel of the manpulator s precsely known, Zha et al. [] propose a new mprove ual power reachng law base on the tratonal slng moe control. he aaptve tems were ae base on the ual power reachng law, the varables an k were ajuste to effectvely mprove the approachng spee. he smulaton result shows that the metho has less chatterng for the manpulator trajectory trackng. However, manpulator system s a complex nonlnear system, whose ynamc parameters are ffcult to be forecaste precsely. In fact, t s almost mpossble to obtan exact ynamc Manuscrpt receve Aprl 7, 5; revse July 5, 5. hs work was supporte n part by the key scentfc research project of unverstes an colleges of Henan Provnce uner Grant 5A33. Hatao Zhang s wth the Informaton Engneerng College, Henan Unversty of Scence an echnology, Luoyang, CO 73 Chna (corresponng author: e-mal: zhang_hatao@63.com). Mengmeng Du s wth the Informaton Engneerng College, Henan Unversty of Scence an echnology, Luoyang, CO 73 Chna. Wenshao Bu s wth the Informaton Engneerng College, Henan Unversty of Scence an echnology, Luoyang, CO 73 Chna. moels because of such uncertantes as nonlnear frctons an flexbltes of the jonts an lnks of manpulator. In the lterature [5], the genetc algorthm was use to optmze the swtchng functon, the sze of the chatterng as the nex of ftness functon was optmze, an a swtchng functon wth relatve mnmum chatterng was constructe. Fey et al. [6] presente a slng moe control base on sturbance observer for robot, the couplng system was ecouple by the feeback jont angle; t acheve hgh precson, but the smple control structure emane that the control parameters must be known. In lterature [7], the slng moe control was apple to eal wth the robust control of space robot n capturng operaton of the target an controllng the spacecraft moton uner unknown parameters. he saturaton functon was ntrouce n orer to avo the chatterng phenomena, the smulaton results prove the feasblty of the algorthm. he self-learnng characterstcs an hgh parallel computng characterstc of neural network are very powerful, t can approxmate the nonlnear systems wth arbtrary accuracy, an t also has a strong robustness. In aton, amng at the approxmaton errors of the neural network, most of the researches get the result that the trackng errors can be unformly ultmately boune or can be kept arbtrarly small f some gan parameters are suffcently large. Ln et al. [8] combne slng moe control an RBF neural network to esgn the controller. he output of RBF neural network was regare as the nput of slng moe controller. It acheve a certan effect on elmnatng the chatterng, but t use the objectve functon to estmate the network weghts, so t can not realze the aaptve weght upate onlne. he controller of lterature [9] was compose of SMC, RBF neural network, an fuzzy control. A Lyapunov functon was selecte for the esgn of the SMC, an RBF neural network was propose to compute the equvalent control. he weghts of the RBF neural network were ajuste accorng to an aaptve algorthm. Fuzzy logc was use to ajust the gan of the correctve control of the SMC. he real tme mplementatons ncate that the propose metho can be apple to manpulator trajectory control. In ths paper, the RBF neural network an slng moe controller s esgne serally, RBF neural network approxmate the swtchng gan, an the robust term s use to elmnate the neural network error. he smulaton results emonstrate that the chatterng an the steay state errors are elmnate an satsfactory trajectory trackng s acheve. he paper s organze as follows. In secton, the relate knowlege of the manpulator s ntrouce. In secton 3, the sle moel control metho s escrbe, an ts savantages are scusse; the slng moe controller wth (Avance onlne publcaton: November 5)

2 IAENG Internatonal Journal of Apple Mathematcs, 5:, IJAM_5 RBF neural network s esgne an the stablty proof s presente. Secton gves an example of a two egrees of freeom robot arm to show the effect of the propose metho. In Secton 5 the conclung remarks are scusse. x x h h j m f ( x) II. PRELIMINARIES x n h m A. Moel of Manpulator A stanar metho for ervng the ynamcs equatons of a mechancal system s va the Euler-Lagrange equatons. Usng ths metho, the ynamcs equatons of a n egree of freeom rg manpulator can be escrbe n the followng general form []: Dqq ( ) Cqqq (, ) Gq ( ) () Where D( q ) s an n n nerta matrx, whch s a postve efnte matrx. Cqq (, ) s an n n matrx contanng the centrfugal an Corols forces. Gq ( ) s an n vector contanng gravty torques. q s jont poston, q s jont velocty, q s jont acceleraton. s jont rve torque. enotes the external sturbance an t s boune. he ynamc characterstcs of manpulator system are as follows: Property : Inerta matrx D( q ) s a symmetrc postve efnte matrx boune by x x D( q) x x, where an are known postve constants. Property : he matrx D( q) C( q, q ) s skew-symmetrc, so t satsfes x Dqx ( ) xcqqx (, ) n x R. Property 3: he unknown sturbance satsfes b where b s a postve constant. Our objectve s to esgn a trajectory trackng controller whose output s the control torque, whch can make the poston of jonts track the esre trajectory accurately. B. RBF Neural Network RBF neural network was propose by J. Mooy an C. Darken n the 98s, an the corresponng theory was evelope by Powell. It s a powerful fee forwar neural network archtecture []. hs type of network was apple to the real multvarable nterpolaton problem an was frst formulate as neural network by Broomhea an Lowe. In the control engneerng, the RBF neural network s usually use as a tool for moelng nonlnear functon up to a small error tolerance because of ts goo capabltes n functon approxmaton. he structure of typcal RBF neural network s mae up of a collecton of parallel processng unts calle noes as shown n Fg. : Fg.. he structure of RBF neural network he RBF neural network has a fee forwar archtecture wth an nput layer, a hen layer, an an output layer. he hen layer s responsble for nonlnear transformaton from the nput space to the hen space. he output layer s lnear, whch s esgne to prove the response to the nput sgnal. here s a layer of processng unts calle hen unts between the nputs an outputs. Each of them s mplemente by a raal bass functon. he nput layer of the network has n unts for an n mensonal nput vector. he nput unts are fully connecte to the hen layer unts, whch are n turn fully connecte to the output layer unts. In ths paper, the number of nput layer s, the number of hen layer s 5, the number of output layer s. he nput-output mappng relatonshp s: m f ( x) h( x) () j Where x s the nput sgnal of neural network, f ( x) s the output sgnal, hx ( ) s the Gaussan bass functon, an enotes neural network weghts. he actvaton functon of the hen layer s generally a Gaussan functon whch s expresse as: ( ) exp( j / j ) h x x c b (3) Where c j are the RBF centers n the nput vector space. b j enote the wth of the noe, j,,, m. Uner the conton of the followng assumptons, RBF neural network can approxmate contnuous functons wth any egree of accuracy n a compact set [].. he neural network output f ˆ ( x, ˆ ) s contnuous.. he eal approxmaton of the neural network output s f ˆ( x, ), for a very small postve number, whch has: f x fˆ x () max ( ) (, ) Where s an n n matrx, an enotes the best approxmaton of neural network weghts. Defne the approxmaton error of eal neural network, namely: ˆ f( x) f( x, ) (5) (Avance onlne publcaton: November 5)

3 IAENG Internatonal Journal of Apple Mathematcs, 5:, IJAM_5 By the approxmaton capablty of RBF neural network, the moelng error s boune, we assume that t s. ˆ sup f( x) f( x, ) (6) Where: ˆ(, ) ( ) f x h x (7) III. HE DESIGN OF HE CONROLLER A. he Desgn of the Slng Moe Controller Slng moe varable structure s a class of nonlnear control. he characterstc of ths kn of control s that t has no fxe structure, an can change movement way accorng to the specfc crcumstances so as to acheve the presupposton state. Because the gven trajectory has no assocaton wth the system state an the external sturbance, so the control system has strong robustness, an operaton metho of ths control s smple. he control objectve s to rve the jont poston q to the esre poston q. Defne the trackng error: eq q (8) Defne the slng surface: s e e (9) Where ag[ n],. Defne the reference state [3]: qr qsq e () q q s q e () r Assumng that there s no external nterference, we choose the control nput : ˆ Ksgn s () Dq ˆ Cq ˆ Gˆ As (3) ˆ r r Where ˆD, Ĉ, Ĝ are the estmate values of D, C, G respectvely, an K ag[ KK Knn ] s a agonal postve efnte matrx n whch K s a postve constant an A ag[ a aan] s also a agonal postve efnte matrx n whchs a a postve constant. sgn( ) s a sgn functon, t s gven as follows:, s sgn( s), s, s Substtutng Eq. () an (3) nto Eq. (), we can get: () Ds ( C A) sf Ksgns (5) Where f Dqr Cq r G, D Dˆ D, C Cˆ C, G Gˆ G. Assumng that f f, where f boun s the boun bounary of f, we choose K such that: K f (6) boun Defne the Lyapunov functon canate: L s D( q) s/ (7) Dfferentatng (7) an conserng Eq. (), (3) an (5), we get [3]: Ls Ds s Ds / s [( CA) sf Ksgn() s Cs] s [ Asf Ksgn( s)] s [ f Ksgn( s) s As] n [ s( f K sgn( s))] s As sas (8) he control law of () an (3) may be senstve to uncertantes n the process of trackng an may lea to chatterng. In aton, there s a ffculty to etermne the swtchng gan of a slng moe controller to acheve esre performance. At present, a tral an error proceure s commonly use to tune the parameters of the slng moe controller. hus the problem consere n our work s to propose a metho to tune aaptvely the swtchng gan K of the SMC n orer to acheve accurate an robust trackng of the manpulator wth mnmum chatterng phenomena. B. he Desgn of the RBF Neural Network Slng Moe Controller (RBFNNSMC) he esgn of the structure s shown as Fg. : (Avance onlne publcaton: November 5)

4 IAENG Internatonal Journal of Apple Mathematcs, 5:, IJAM_5 e e s s s e ce ˆD q K v Cˆ e q e q Fg.. he structure of trajectory trackng control wth RBFNNSMC Manpulator equaton s shown as Eq. (), the new control law s as follows: Dq Cq ˆ Gˆ AsK v (9) ˆ r r Where K [ k, k] s the output of RBF neural network of two jont manpulator; v s the robust term, v( b )sgn( s), t s use for overcomng the neural network approxmaton error an external nterference. Substtutng Eq. (9) nto Equaton (), yel: Ds ( CA) sf K v () he aaptve law s esgne as: Ps h( s ) () Where, represents the jont an jont of manpulator. P s a symmetrc postve efnte matrx, an ts nverse matrx s exstng. Let us ntrouce the canate Lyapunov functon: L s Ds/ ( P )/ () ˆ G q q L( s Ds s Ds s Ds)/ ( P P )/ ( sds sds ) / ( P ) ( ) ( ) s DsCs P [ ( ) ] ( ) s CA sf K vcs P ( ) ( ) s Ass f K P s v ( ) ( ) s As s f k P s v Because of k h s h s ( ) ( ), we obtan: ( ) ( ) L s As s f h h P s v ( ) ( ) s As s f h s h P s v ( ) ( ) s As s f h s h P s v Substtutng Eq. () nto Eq. (5), yel: [ ( )] () (5) L s As s f h s s v (6) he exstence of a very small postve real number, whch makes the Eq. (6) satsfy: f hs ( ) s, (7) hen : ( ) s f h s s s (8) herefore: Where ˆ (3) Dfferentatng (), we get: L s As s s v as s s ( b ) (9) Where ag[, ], a,, b. Accorng to Eq. (9), only when s, L, the aaptve law asymptotc convergence. Fnally, we may get the followng concluson: lm s lm( e e) (3) t t (Avance onlne publcaton: November 5)

5 IAENG Internatonal Journal of Apple Mathematcs, 5:, IJAM_5 Namely: lm q q, lm q q (3) t t From the above analyss t can be seen that the RBFNNSMC metho can guarantee that the trackng errors converge arbtrarly close to zero. he followng case stuy base on smulaton wll emonstrate ths concluson. IV. SIMULAION RESULS A. he Smulaton of wo DOF manpulator In ths secton, a smulaton stuy s conucte to emonstrate the performance of our algorthm. A smple two egrees of freeom (DOF) manpulator s shown n Fg. 3. he ynamc equaton s gven as follows: Dqq ( ) Cqqq (, ) Gq ( ) (3) Where: p p p3cosq p p3cosq Dq ( ) p p3cos q p p3qsn q p3( q q)sn q Cqq (, ) pq 3 sn q pgcos q p5gcos( q q) Gq ( ) pg 5 cos( q q) (33) (3) (35) [.sn( ).sn( )] t t (36) term s. he parameters of the Gaussan bass functon are 5 5 c 5 5, b 5. he control parameters ag{,}, A ag{7,7}, P ag{,,,,}. In the robust term,.3, b.. SIMULINK an S functon s use to esgn the control system. In orer to compare the avantages of the propose metho, uner the same gven parameters, we smulate the slng moe control metho an the propose metho. he smulaton results are shown as Fg. -Fg.. Jont (ra) Jont (ra) Jont (ra) SMC poston trackng eal poston sgnal me (s) Fg.. Poston trackng of SMC SMC poston trackng eal poston sgnal RBFNNSMC poston trackng eal poston sgnal r r m q m Jont (ra) RBFNNSMC poston trackng eal poston sgnal m q Fg. 3. Structure of the manpulator he mass of lnk s m.kg, the mass of lnk s kg, the length of lnk s r m, an the length of lnk s r.87m. he p g p ( m m ) r, p mr, mrr, p ( m m) r, p5 mr, an m/ s. he esre trajectory of the manpulator s q sn( t) an q sn( t), an the ntal states of the manpulator are q() q(), q() q (). he RBF neural network nput s slng moe functon s an ts fferental me (s) Fg. 5. Poston trackng of RBFNNSMC It can be seen from Fg. an Fg. 5, the poston trackng curves of jont an of manpulator are both eal uner the control of the two kn of algorthms. (Avance onlne publcaton: November 5)

6 IAENG Internatonal Journal of Apple Mathematcs, 5:, IJAM_5 Jont (ra) Jont (ra) Jont (ra) Jont (ra) tme(s) tme(s) Fg. 6. Poston trackng error of SMC x tme(s) Fg.7. Poston trackng error of RBFNNSMC Fg. 6 an Fg. 7 show the poston trackng error of the two jonts. Fg. 6 shows that the poston trackng error uner the control of SMC, the seay error of jont fluctuates between.5 to -.5, an the seay error of jont fluctuates between. to -.. Fg. 7 shows that the poston trackng error uner the control of RBFNNSMC, the steay state error of jont ranges from. to -.6, the steay state error of jont ranges from. to -.. From the error comparson of jont, we can know that the control precson of RBFNNSMC s better than SMC. As to the steay state error of jont, t s har to compare the RBFNNSMC quanttatvely wth SMC, but the ntal error uner the control of RBFNNSMC s smaller than SMC. From Fg. 6 an 7 we can prove, the global performance of the system s mprove usng the RBF neural network, an because of usng of the robust control term, the averse effects whch cause by the neural network approxmaton error s compensate effectvely. Jont (ra/s) Jont (ra/s) Jont (ra/s) Jont (ra/s) - - SMC velocty trackng eal velocty sgnal SMC velocty trackng eal velocty sgnal me (s) Fg. 8. Velocty trackng of SMC RBFNNSMC velocty trackng eal velocty sgnal RBFNNSMC velocty trackng eal velocty sgnal me (s) Fg. 9. Velocty trackng of RBFNNSMC From Fg. 8 an Fg. 9, we can conclue that the velocty trackng errors have bg fluctuaton uner the control of SMC. However, the velocty curve uner the control of RBFNNSMC can trace the gven velocty sgnal smoothly. Control nput (Nm) Control nput (Nm) tme(s) tme(s) Fg.. Control torque of SMC (Avance onlne publcaton: November 5)

7 IAENG Internatonal Journal of Apple Mathematcs, 5:, IJAM_5 6 Control nput (Nm) tme(s) Control nput (Nm) tme(s) Fg.. Control torque of RBFNNSMC Fg. an Fg. show the output control torque of manpulator, Fg. shows the chatterng of SMC s very bg, so ths control metho s very ffcult to apply n practce. After jonng the RBF neural network controller, the system can aaptvely ajust the swtchng gan, whch reuces the system chatterng cause by slng moe control. K K tme(s) tme(s) Fg.. he aaptve change of gan uner the RBFNNSMC Fg. shows the changng process of swtchng gan uner the ajustment of the RBF neural network. B. he Smulaton of hree DOF manpulator In orer to verfy the effectveness of the algorthm, we use the assembly manpulator n the lterature [] to o the smulaton. It s a space manpulator wth three egrees of freeom. he agram s shown n Fg. 3, the frst jont o the translatonal moton, the secon jont an the thr jont o the revolvng moton. Fg.3. he agram of the three DOF manpulator We can know the parameters from the lterature [], the mass of lnk s m kg, the mass of lnk s m kg, the mass of lnk 3 s m3 kg, the length of lnk s l.m, an the length of lnk 3 s l.m. Dynamc matrces are shown below: a Dq ( ) a a3 (37) a3 a 33 Cqq (, ) b b3 (38) b3 ( mm m3) g Gq ( ) (39) Where a m m m3 () aml /3 m3l3 /3m3l ll3m3cos( q3) () a3 a3 m3l3 /3 ll3 cos( q3 ) m3 / () b ll3 sn( q3) m3q 3 / (3) b3 m33 l l sn( q3) q3 / m33 l l sn( q3) q / () b m l l sn( q ) q / (5) he esre trajectory of the manpulator s q sn( t), q sn( t), q3 sn( t), an the ntal states of the manpulator are q() q() q3(), q() q() q 3(). External sturbance s [.sn( ).sn( ).sn( )] t t t. he RBF neural network parameters are c an b he nputs are the slng moe functon of the three jonts respectvely, namely, s (), s() an s (3). he structure of the RBF neural network s he control parameters ag{8,8}, A ag{3,3} an (Avance onlne publcaton: November 5)

8 IAENG Internatonal Journal of Apple Mathematcs, 5:, IJAM_5 P ag{,,,,}. In the robust term,.3, b.. he smulaton results are shown as Fg. -Fg. 8. Jont (ra) Jont (ra) Jont 3 (ra) Poston Error (ra) RBFNNSMC poston trackng eal sgnal RBFNNSMC poston trackng eal sgnal RBFNNSMC poston trackng eal sgnal Fg.. Poston trackng of RBFNNSMC Jont Poston Error (ra) Poston Error (ra) x -3 Jont x -3 Jont me (s) Fg.5. Poston trackng error of RBFNNSMC We can conclue from the Fg. that trackng curves of the three jonts are very eal. Fg. 5 are the poston error curves of three jonts. he trackng error of jont s.; the trackng error of the jont an jont 3 s also very small, t s -3 orer of magntue.. Fg. 6 s the changng process of swtchng gan uner the ajustment of the RBF neural network. Jont orque (Nm) Jont orque (Nm) Jont orque (Nm) Jont orque (Nm) Jont3 orque (Nm) Jont3 orque (Nm) me (s) Fg. 7. Control torque of RBFNNSMC me (s) Fg. 8. Control torque of SMC Fg. 7 s the output torque uner the control of RBFNNSMC. Fg. 8 s the output torque uner the control of tratonal SMC. Because of the gravty term of jont an jont 3 are zero, so the output torques are very small n the Fg. 7 an Fg. 8. We can see that the chatterng s very bg uner the control of tratonal SMC. However, there s almost no chatterng uner the control of RBFNNSMC. K.5 K K x x me (s) Fg. 6. he aaptve change of gan uner the RBFNNSMC V. CONCLUSION hs paper manly stues the mult-jont manpulator system wth uncertantes for the trajectory trackng control. hs paper combnes the slng moe control wth RBF neural network. he slng moe control s use to resst nterference an ensure the stablty of the system, RBF neural network s use to ajust the slng moe gan onlne, so as to reuce the chatterng of output torque. he RBF neural network approxmaton error s overcome by ang the robust term. he esgn guarantees the close-loop stablty by usng Lyapunov metho. he smulaton results show that the propose control metho s approprate for esgn of manpulator wth uncertanty an the external nterference. (Avance onlne publcaton: November 5)

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