Tracking Control for an Underactuated Two-Dimensional Overhead Crane

Transcription

1 Trackng Control for an Uneractuate Two-Dmenonal Overhea Crane D.T. Lu *, W.P. Guo Inttute of Computer Scence an Technology, Yanta Unverty, Shanong Provnce, P.R. Chna ABSTRACT In th paper, the trackng control problem conere for the payloa tranportaton wth an uneractuate twomenonal overhea crane. Two lng moe controller are egne to perform the trajectory trackng. One propoe to control hotng an lowerng the upene payloa, an the other one propoe to control both trolley potonng an payloa wayng. Conerng the econ lng moe controller ue to control two egree of freeom (DOF), a fuzzy nference algorthm propoe to ynamcally ajut the couplng factor between the two DOF. The two controller make the payloa track a preefne trajectory an be afely tranporte a fat an accurately a poble wth a mall wng angle, an then place the payloa at the ere poton. Smulaton are performe wth the propoe controller an the reult how ther effectvene. Keywor: Slng moe, fuzzy nference, trajectory trackng, uneractuate ytem, crane.. Introucton Overhea crane are wely ue n workhop an harbor to tranport all kn of mave goo. It ere for the overhea crane to tranport payloa to the requre poton a fat an a accurately a poble wthout collon wth equpment whle placng payloa at an approprate poton. In aton to thee requrement, the payloa wng angle houl be kept a mall a poble an t ere that the payloa can track a preefne trajectory. Much work ha been one n controllng the overhea crane. Garro [] aopte the nput hapng control metho; however, nput hapng mut be precalculate accurately accorng to the ytem moel an the approach lack robutne to eternal turbance. Lee [] ue fuzzy logc only for ant-wng control an poton ervo control for poton an wng ampng. Mchael [] aopte fuzzy logc to control both potonng an wng ampng, but t wa ffcult to et both the fuzzy rule an the parameter of the controller only accorng to eperence. Moreover, mot of the above metho o not take payloa hotng an lowerng nto account. Ant-way trajectory trackng of overhea crane ha been tue n the followng lterature. To mprove the performance of wng uppreon, Lee [] propoe a nonlnear wtchng control law for -D overhea crane wth the feeback lnearzaton metho an the Lyapunov tablty theorem. Lee [5] propoe an ant-wng trajectory trackng control law wth a couplng control by efnng a couple varable of the trolley travelng moton an the loa way moton. However, the rawback of that work le n that they have conere le practcal ue uch a robutne agant parameter uncertanty, unmoele ynamc, an actuator nonlnearty []. Slng moe control (SMC) a robut egn methoology ung a ytematc cheme bae on a lng moe urface an the Lyapunov tablty theorem [7]. The man avantage of SMC that the ytem uncertante an eternal turbance can be hanle uner the nvarance charactertc of the ytem lng moe tate wth guarantee ytem tablty [7]. The SMC wa ue by Er [8] an Manour [9] for potonng control an hotng control, but another tate feeback control cheme wa neee to control the payloa wng. A reference moel wa efne by Haanul [0] to track through a lnearze ytem moel. All thoe metho have ffculte n automatcally tunng the relatonhp between poton control an way control. Park [] propoe aaptve fuzzy lng- Journal of Apple Reearch an Technology 597

2 Trackng Control for an Uneractuate Two Dmenonal Overhea Crane, D.T. Lu / moe ant-way trackng control an Yang [] propoe aaptve trackng control for overhea crane, but the payloa hotng an lowerng were not conere. Th paper preent a oluton to control the overhea crane, where the payloa wng, crane moton, an payloa hotng (an lowerng) are conere altogether. It well known that SMC capable of tacklng non-lnear ytem wth parameter uncertante an eternal turbance hence the eternal turbance an ytem parameter varyng are not taken nto coneraton n moelng. Th paper organze a follow: Secton gve the ynamcal moel of the overhea crane. In Secton, two lng moe controller are egne for the overhea crane an the fuzzy tunng algorthm egne to ajut the control of two DOF. In Secton, mulaton are performe an the reult are gven. In Secton 5, the concluon are preente. of the overhea crane ytem can be obtane: ( m m ) m l co ml co ml n m l n f m l m n m l m gco f ml ml co m l l m gln 0 (). Dynamcal moel of the two-menonal overhea crane Fgure how the overhea crane ytem, where m, m,, l,, f an f, repectvely, are the trolley ma, payloa ma, trolley poton, rope length, wng angle, rvng force of the trolley an rvng force of hotng (lowerng) along the rope. There are three DOF (.e.,, l an ) to be controlle but only two control acton (.e., f an f ) are avalable thu the overhea crane an uneractuate ytem. Accorng to the Lagrangan equaton: t L ) L ( T q q (,, ) () where L K U the Lagrangan functon, K the ytem knetc energy, U the ytem potental energy, q the generalze coornate (here, l or ), an T the eternal force (here f or f ). Wthout conerng frcton, the moton equaton Fgure. Two-menonal overhea crane. From the ytem moton equaton (), the followng equaton can be obtane: ( f f n) / m l f / m l g co ( f f n)n / m (( f f n)co / m l g n )) / l () Aume the trolley poton, the rope length l, an the wng angle ; moreover, l, 5,. The tate equaton of the overhea crane ytem can be rewrtten a 598 Vol. 0, Augut 0

3 Trackng Control for an Uneractuate Two Dmenonal Overhea Crane, D.T. Lu / ( f f / m (( f f n ( f f n ) / m f n )co )n / m / m 5 g co g n () ) /. Degn of the trackng controller for the crane The control objectve of the overhea crane to tranport the payloa along a preefne trajectory from the ntal poton to the fnal poton. In the control proce, the payloa houl be tranporte a fat a poble an the payloa wayng houl be a mall a poble. The control problem are comple becaue the overhea crane uner-actuate. The trackng error vector efne a e (,,,) (5) where upercrpt ncate the correponng ere value. To apply the SMC trategy, three tme-varyng lng moe functon are efne for each DOF a follow: Durng the tranportaton proce, the payloa houl be tranporte along the preefne trajectory; moreover, the payloa often mave. Therefore, the wng angle houl not be controlle by ytem nput f n orer to prevent tuaton uch a lowerng n the hotng proce an hotng n the lowerng proce, but t can be controlle by ytem nput f. Conequently, f control two DOF, namely, the crane poton an the payloa wng angle. Another lng moe functon efne by k (0) where k et a a real negatve varable to be egne below an t can be regare a a couplng factor between lng moe functon an. The couplng factor can not only connect lng moe urface 0 an 0 but alo ajut ther mportance n the ytem control; therefore, the couplng factor wll play an mportant role n the nteractve control between lng moe urface 0 an 0. In SMC, f the followng contnuou ynamc are mpoe on the correponng (=,): gn( ) () e e () gn( ) () e5 e (7) e e (8) where (=,,) are utable potve contant. The preefne trajectory planne accorng to the tranportaton tance an the obtacle tanng n the tranportng envronment. Wthout conerng the wng angle, the preefne trajectory can be epree a f ( ) (9) the lng moe conton ( ) are atfe, n whch an are utable potve contant. That to ay, ytem tate wll acheve the correponng lng moe urface 0 n a fnte amount of tme. Where gn() a gn functon: gn( ) 0 for 0 for 0 for 0 The atfe lng moe conton an table lng moe urface can enure the tablty of Journal of Apple Reearch an Technology 599

4 Trackng Control for an Uneractuate Two Dmenonal Overhea Crane, D.T. Lu / the overhea crane control ytem. A well known, ung the gn functon n SMC often caue ytem chatterng n practce that not epecte. One oluton to ntrouce a bounary layer aroun the lng moe urface [7]; therefore, aturaton functon are aopte to replace the gn functon n () an (): at( / ) Φ at( / ) Φ Φ () () where potve contant factor (=,) are the wth of the bounary layer. Saturaton functon at( / Φ ) are efne a at( / Φ / Φ ) gn( / Φ ) for for From (), (7), (8) an (0), we have / Φ / Φ 5 5 ( ) at( / Φ ) 0 (5) ( ) k ( ) k ( ) at ( / Φ ) 0 () Generally, the ere wng angle zero thu we have 0 an 0. From (), (5) an (), we can obtan the followng control law: f m n ( b an g co ) ma where f a ( k( m (7) ( b a n g co ) g n at ( / Φ ))/( k co ) 5 ) (8) (9) b ( 5 ) at( / ) 5 Φ (0) Conerng that the couplng factor play an mportant role n the coornate control between the crane poton an the payloa wng, k houl be approprately choen to acheve atfactory ecouplng performance for fferent ytem tate. Here, an aaptve fuzzy tunng metho propoe to generate the couplng factor. A well known, f the wng angle large, the payloa wayng control houl be renforce n orer to amp the payloa wng, an f the wng angle mall, the crane poton control houl be b k gven to enure trengthene. A bae value a bac performance, a ynamc varable k ynamcally tune n real tme, an a breath K ue to lmt the range of the couplng factor. Then the ynamc couplng factor efne by b k k Kk () Therefore, the couplng factor can be automatcally b ajute wthn [ k, k b K ] through tunng the ynamc varable accorng to the ytem tate. The fuzzy rule et can be epree a F R : IF IS THEN k IS K where R the th rule of n rule, F an K are the correponng memberhp functon of an k n the th rule. The mplfe nference metho ue to erve the nference reult a k ( n F ( ) K ) / F ( ) n () ( ) where F the frng egree of the th rule. By the way, the motor pee uually lmte n practce, an the trolley rate pee ve tranforme from the motor rate pee the optmal tranportaton pee. For long-tance tranportaton, n orer to prevent the acceleraton 00 Vol. 0, Augut 0

5 Trackng Control for an Uneractuate Two Dmenonal Overhea Crane, D.T. Lu / from becomng too great urng the acceleraton proce an to make the trolley tranport wth v e after the acceleraton proce, poton error e can be lmte to e at(( ) / ) where coeffcent can be calculate by ve / Ung () an () n (), we get e ve at( e / ) () () When 0, tranport velocty wll approach trolley rate pee v e an keep t untl the eceleraton proce take place becaue lng moe functon approache zero uner the control acton of rvng force f n (7). The eceleraton proce begn when poton error e become le than. Remark : The tablty of the cloe-loop ytem cont of the acceblty of the lng moe urface an the tablty on the lng moe urface. The frt one apparently guarantee through the egn proce of the lng moe controller, an the econ one ece by couplng factor k n Equaton (0) an the efnton of the wng angle an rvng force f. Aume a potve angle an potve f recton are efne a Fgure how. For lng moe functon 0, a negatve rvng force requre uch that lng functon 0 approache lng moe urface ; however, for lng moe functon 0, a potve rvng force requre uch that lng functon approache lng moe urface 0. That to ay, the role of the two lng moe functon n control are ncontent. Therefore, couplng factor k houl be negatve for the tablty on the lng moe urface. Remark : In practce, rope length alway larger than zero. Swng angle wthn ( /, / ) to enure afe operaton; moreover, varable k negatve. Hence, the term ( k co ) n Equaton (9) greatly larger than zero. Therefore, there are no ngularte of the controller of Equaton (7) an (8). Remark : Wth the propoe control cheme, the overhea crane can be tablze to a fe poton or a trajectory. The aturaton functon n Equaton () oe not affect the acceblty of the lng moe urface an the tablty on the lng moe urface. Therefore, the aturaton functon n Equaton () wll not affect the tablty of the cloe-loop ytem.. Control Smulaton Smulaton are performe wth the ytem gven by () an the controller gven by Equaton (7) an (8) wth (), (7), (8), (0), (9), (0), () an (). In the mulaton, the parameter of the overhea crane ytem are choen a follow: trolley ma m 0kg, payloa ma m 0kg, g 9.8m/ gravtatonal acceleraton. The hotng an lowerng moton wll track the reference trajectory, along whch the upene payloa wll not colle wth other object an the energy conume n tranportaton the lowet. Aumng that the obtacle n the tranportng envronment look lke thoe n Fgure, a parabolc reference trajectory calculate offlne for tranportng. Hence, Equaton (9) can be epree a l l l ( P ( ) P ) for for P for P P P (5) Journal of Apple Reearch an Technology 0

6 Trackng Control for an Uneractuate Two Dmenonal Overhea Crane, D.T. Lu / In mulaton, the above parameter are choen: l 0 5, 0.05, P 0, P 0. The reference trajectory alo hown n Fgure. Sytem performance entve to the parameter of the lng urface [, ]. For the lng urface lope, f a large value avalable, the ytem wll be more table but the trackng accuracy may be egrae becaue of a longer reachng tme of the repreentatve pont to the urface. Converely, f a mall value choen, the convergence pee on the lng urface telf wll be low, leang to longer trackng tme. For the lng moe wtch control value, a large value wll lea to a large chatterng an a mall value wll lea to a long reachng tme. Moreover, the wtch value lmte by the ytem mamum output. The ntroucton of a bounary layer reuce chatterng at the cot of eterorate the trackng precon,.e., a large value wll lea to a mall chatterng but a low trackng precon. Accorng to the above prncple, the parameter of the lng moe urface are choen a 0.5, 0.8, 8, 8, 0, 0, Φ 0.5, Φ 0.5 ; the parameter of couplng b factor are k, K 0. ; the trolley rate pee v e m/ ;an the mamal acceleraton (or eceleraton) of the trolley m/. The elf-tunng fuzzy nference controller take the abolute value of the wng angle a anteceent varable whoe fuzzy ubet are contructe a mall (S), meum (M) an bg (B) an are hown n Fgure. S M Fgure. Memberhp functon of ynamc varable. B Fgure. Reference trajectory. 0 Vol. 0, Augut 0

7 Trackng Control for an Uneractuate Two Dmenonal Overhea Crane, D.T. Lu / The ynamc varable of the couplng factor can be nferre by the fuzzy rule n Table. Anteceent varable Conequent varable k S 0.0 M 0.5 B.0 Table. Rule for ynamc varable. Ung the propoe control law wth the above parameter an zero ntal conton, the frtmulaton performe wth MATLAB. The tranportaton tak to move the payloa that locate at (0, 0) along the preefne trajectory to (0, 0). The mulaton reult are hown n Fgure. Then, the mulaton reult howe the followng: poton overhoot lttle, an collon wth other objectve prevente. The velocty of the trolley quckly approache the rate pee hence the tranportng tme harply hortene. The wng angle can be ampe n all the tranportaton proce, an the payloa can trctly track the preefne trajectory. In orer to valate the robutne of the propoe control algorthm wth the ame tranportaton tak, the econ mulaton wth nonzero ntal conton, eternal turbance an ytem parameter varyng performe. The payloa ntal wng angle 0.05ra, a man-mae wng angel (0.05ra) at 0 ae to the overhea crane ytem, an ytem parameter m change from 0kg to 8kg. The econ mulaton reult are hown n Fgure 5. The reult how the propoe control algorthm can effectvely ret eternal turbance an ytem parameter varaton, an the ntal wng angle can be ampe quckly. A epecte, n the contant velocty zone, the trolley tranport wth the rate pee an the payloa wng angel almot zero. Moreover, the propoe elf-tunng fuzzy nference algorthm can automatcally mprove the control performance through ajutng the mportance of thoe two lng moe urface n control. (a) Poton repone (b) Velocty repone Journal of Apple Reearch an Technology 0

8 Trackng Control for an Uneractuate Two Dmenonal Overhea Crane, D.T. Lu / (c) Sway angle repone () Trolley rvng force repone (e) Rope length repone (f) Rope rvng force repone Fgure. Smulaton reult wthout nonzero ntal conton, etern turbance an ytem parameter varyng. 5. Concluon The ynamc moel of overhea crane obtane accorng to the Lagrangan equaton, an two lng moe controller are propoe for an overhea crane to tranport the payloa along a preefne trajectory. One lng moe controller egne to hot an lower the payloa to track the preefne trajectory, an another one ue to control the trolley poton an payloa wng angle. Becaue the overhea crane an uneractuate ytem, a elf-tunng fuzzy nference algorthm ha been propoe to ajut the relatonhp between trolley potonng control an payloa wayng control. The mulaton reult have upporte the theoretcal reult an have hown the effectvene of the propoe control law. The performance of the propoe algorthm le affecte by the ntal conton, eternal turbance an parameter uncertanty. Conequently, the propoe algorthm ha a great potental for hghperformance control of overhea crane. 0 Vol. 0, Augut 0

9 Trackng Control for an Uneractuate Two Dmenonal Overhea Crane, D.T. Lu / (a) Poton repone (b) Velocty repone (c) Sway angle repone () Trolley rvng force repone (e) Rope length repone (f) Rope rvng force repone Fgure 5. Smulaton reult wth nonzero ntal conton, etern turbance an ytem parameter varyng. Journal of Apple Reearch an Technology 05

10 Trackng Control for an Uneractuate Two Dmenonal Overhea Crane, D.T. Lu / Acknowlegement Th work upporte by the Natonal Natural Scence Founaton of Chna (No.7508) an the Reearch Awar Fun for Outtanng Mle-age an Young Scentt of Shanong Provnce (No.BS009DX0). Reference [] Garro S., Aberrahm M., Gménez A., Dez R., an Balaguer C. Ant-wngng nput hapng control of an automatc contructon crane. IEEE Tranacton on Automaton Scence an Engneerng, 5(): , 008. [0] Haanul B.A.M. Swng-free tranport ung varable tructure moel reference control. Proceeng of IEEE Southeat conference, pp.85-9, 00. [] Yang J.H., Shen S.H. Novel approach for aaptve trackng control of a -D overhea crane ytem. Journal of Intellgent &Robotc Sytem, (): 59-80, 0. [] Chang C.Y., Hu K.C., Chang K.H. Ant-way an moton plannng control of overhea crane. Internatonal Journal of Innovatve Computng Informaton an Control, (7): -8, 00. [] Palm R. Robut control by fuzzy lng moe. Automatca, 0(9): 9-7, 99. [] Lee H.H. an Cho S.K. A new fuzzy-logc ant-wng control for nutral three- menonal overhea crane. In Proceeng of IEEE nternatonal Conference on Robotc & Automaton, pp.95-9, 00. [] Mchael J.N. an Mohame B.T. Control of overhea crane ung a fuzzy logc controller. Journal of Intellgent an Fuzzy Sytem, No.8, pp.-8, 000. [] Lee H. H. An ant-wng trajectory control of overhea crane wth hghpee hotng. Proceeng of the Amercan Control Conference, Denver, CO, Jun. pp. 0 5, 00. [5] Lee H., Lang Y., an Segura D. A new approach for the antwng control of overhea crane wth hghpee loa hotng, Internatonal Journal of Control, 7(5): 9 99, 00. [] Park M.S., Chwa D., an Hong S.K. Antway trackng control of overhea crane wth ytem uncertanty an actuator nonlnearty ung an aaptve fuzzy lng-moe control. IEEE Tranacton on Inutral Electronc, 55(): 97-98, 008. [7] Ewar C., Collet E. F. an Frman L., Avance n Varable Structure an Slng Moe Control, Sprnger- Verlag, 00. [8] Er M.J., Zrb M. an Lee K. L. Varable Structure Control of an Overhea Crane. Proceeng of IEEE Internatonal Conference on Control Applcaton, pp.98-0, 998. [9] Manour, A.K. an Mohame Z. Robut Control Scheme for an Overhea Crane. Journal of Vbraton an Control, 7(): 95-, Vol. 0, Augut 0