Application of Linear Model Predictive Control and Input-Output Linearization to Constrained Control of 3D Cable Robots

Transcription

1 Moern Mechancal Engneerng, 2011, 1, o: /mme Publshe Onlne November 2011 ( Applcaton of Lnear Moel Prectve Control an Input-Output Lnearzaton to Constrane Control of 3D Cable Robots Abstract Al Ghasem Member of Young Researchers Club, Scence an Research Branch, Islamc Aza Unversty, ehran, Iran E-mal: Receve September 3, 2011; revse October 13, 2011; accepte October 25, 2011 Cable robots are structurally the same as parallel robots but wth the basc fference that cables can only pull the platform an cannot push t. hs feature makes control of cable robots a lot more challengng compare to parallel robots. hs paper ntrouces a controller for cable robots uner force constrant. he controller s base on nput-output lnearzaton an lnear moel prectve control. Performance of nput-output lnearzng (IOL) controllers suffers ue to constrants on nput an output varables. hs problem s successfully tackle by augmentng IOL controllers wth lnear moel prectve controller (LMPC). he effecttveness of the propose metho s llustrate by numercal smulaton. Keywors: Cable Robots, Input-Output Lnearzaton, Lnear Moel Prectve Control 1. Introucton After a moton smulator wth parallel knematc chans was ntrouce n 1965 by D. Stewart [1], parallel manpulators receve more an more attenton because of ther hgh stffness, hgh spee, hgh accuracy, compact an hgh carryng capablty [2]. hey have been use wely n the fels of moton smulators, force/torque sensors, complance evces, mecal evces an machne tools [3,4]. A parallel robot s mae up of an en-effector, wth n egrees of freeom, an a fxe base lnke together by at least two nepenent knematc chans [5]. Actuaton takes place through m smple actuators. Parallel robots rawbacks are ther relatvely small workspace an knematcs complexty. Cable robots are a class of parallel robots n whch the lnks are replace by cables. hey are relatvely smple n form, wth multple cables attache to a moble platform or an en-effector. Cable robots posses a number of esrable characterstcs, nclung: 1) statonary heavy components an few movng parts, resultng n low nertal propertes an hgh payloa-to-weght ratos; 2) ncomparable moton range, much hgher than that obtane by conventonal seral or parallel robots; 3) cables have neglgble nerta an are sutable for hgh acceleraton applcatons; 4) transportablty an ease of sassembly/- reassembly; 5) reconfgurablty by smply relocatng the motors an upatng the control system accorngly; an, 6) economcal constructon an mantenance ue to few movng parts an relatvely smple components [6,7]. Consequently, cable robots are exceptonally well sute for many applcatons such as hanlng of heavy materals, nspecton an repar n shpyars an arplane hangars, hgh-spee manpulaton, raply eployable rescue robots, cleanup of saster areas, an access to remote locatons an nteracton wth hazarous envronments [6-12]. For these applcatons conventonal seral or parallel robots are mpractcal ue to ther lmte workspace. However, cables have the unque property they cannot prove compresson force on an en-effector. Some research has been prevously conucte to guarantee postve tenson n the cables whle the en-effector s movng. he ea of reunancy was utlze n cable system control [13,14]. hs paper ntrouces a controller for cable robots uner force constrant. By conserng, lnear moel prectve covers fferent constrant such as nput constrants. he goal s to apply the lnear moel prectve control to the nput-output lnearze system to account for the constrants. A varety of nonlnear control esgn strategy has been propose n the past two ecaes. Input-output lnearzaton (IOL) an nonlnear moel base control are the most wely stue esgn technques n nonlnear control. he

2 70 A. GHASEMI central ea of the nput output lnearzaton approach s to algebracally transform the nonlnear system nto lnear one an apply a sutable lnear control esgn technque [15,16]. LMPC s prmarly evelope for process control. herefore ts applcaton n robot control has less been reporte. he ncpent nterest n the applcatons of MPC ates back to the late 1970s. In 1978, Rchalet et al. [17], presente the Moel Prectve Heurstc Control (MPHC) metho n whch an mpulse response moel was use to prect the effect at the output of the future control actons. Lnear moel prectve control refers to a class of control algorthms that compute a manpulate varable profle by utlzng a lnear process moel to optmze a lnear or quaratc open-loop performance objectve subject to lnear constrants over a future tme horzon. he frst move of ths open loop optmal manpulate varable profle s then mplemente. hs proceure s repeate at each control nterval wth the process measurements use to upate the optmzaton problem. Durng 1980s, MPC quckly became popular partcularly n chemcal process nustry ue to the smplcty of the algorthm an to the use of the mpulse or step response moel, whch s preferre, as beng more ntutve an requrng no prevous nformaton for ts entfcaton [18]. A cable-suspene robot s actuate by servo motors that control the tensons n the cables. A major savantage of cable robots s that each cable can only exert tenson. hs constrant leas to performance eteroraton an even nstablty, f not properly accounte for n the control esgn proceure. Due to ths feature, well known results n robotcs for trajectory plannng an control are not rectly applcable to them. Several approaches nclung a lyapunov base controller wth varable gans an a feeback lnearzng controller wth varable gans [19], feeback lnearzaton together wth metho of reference sgnal management [20], lyapunov base slng controller wth metho of sgnal management [21] have been suggeste to satsfy the postve tenson n the cables whle the platform s movng. In ths paper a lnear moel prectve control s apple to lnearze moel. Moel prectve control, a computer control algorthm that utlzes an explct moel to prect the future response of a system s an effectve tool for hanlng constrane control problems. cable connecton n the platform frame. herefore, = a R b c s the vector representng the length of each cable an l s the recton of tenson force along each cable, where c s the poston vector of mass center of platform parameterze. R s the rotaton matrx between the two frames, the base an the movng, Fgure System Dynamcs he nerta of each lnk of cable robots s neglgble compare to that of the platform because the so-calle lnk s just a cable or wre. herefore, the ynamcs of the lnks can be gnore whch wll sgnfcantly smplfy the ynamc moel of the manpulator. One can erve the Newton-Euler equatons of moton of the manpulator wth respect to the center of mass on the platform as follows [10] F mamc f mg ( n s number of cables) M I I n ext 1 ext n 1 Rb f l I I where m an I are the mass an nerta tensor of the platform nclung any attache payloa; g s the gravty acceleraton vector; an f ext an τ ext are external force an moment vectors apple to the platform. c an α are the lnear an angular acceleraton vectors of the platform; an f are the force vector an force value of the th cable. Equaton (1) can be rewrtten nto a compact form as: M x Cx D Ju (2) where mi M,, f D C 033 I 033 I ext mg ext (1) 2. Knematcs Moelng of the Cable Robots he knematc notaton of a spatal cable-rven manpulator s presente n Fgure 1, where P an B are two attachng ponts of the th cable to the platform an the base, respectvely. a represents the poston vector of B n the base frame an b shows the poston vector of the Fgure 1. General knematcs of a cable robot.

3 A. GHASEMI 71 l1 ln J, J jacopan matrx Rb1l1 Rbnln f1 u f n I 3*3 s a 3*3 entty matrx an 0 I I z 0 z I I 0 y I I I x y x Equaton (2) can be wrtten nto a steay state form as: where y I x X F Gu y E X h X x x 06* n X, F, G 1 1 x M Cx D M J wth constrants 0 u u max 4. Input-Output Lnearzaton he ffculty of the trackng control esgn can be reuce f we can fn a rect an smple relaton between the system output y an the control nput u. Inee, ths ea consttutes the ntutve bass for the so-calle nput -output lnearzaton approach to nonlnear control esgn. hs artcle s ame to use the LMPC whch covers fferent constrants such as nput constrants. Because of LMPC s usually use for lnear screte systems. In the begnnng we wll lnearze the ynamcs equatons base on n Input-Output Lnearzaton. o generate a rect relatonshp between the output y an the nput u, Dfferentate the output y wth respect to tme t n Equaton (3), we have y L h X L h u t F gj j j1 where L F h (X) an L gj h (X) are the Le ervatves of h (X) wth respect to F(X) an g j (X), respectvely. If L gj h (X) = 0, y = L F h (X) then the r th ervatve of y can be represent n the followng form. n r r r1 F gj F j j1 In ths way, we can wrte all plant s nput-output equatons as n y L h X L L h u t (3) r1 r r 1 11 r11 y 1 LF h L 1 g L 1 F h1 Lg L h n F 1 u r q r q rq1 rq1 yq LF hq Lg L h 1 F qlg L h n F q where q an r are the number of egree of freeom of the robot an the relatve orer of the plant, respectvely. Equaton (4) can be represente n the followng compact form: v P Wu Now u can be obtane as: u W W W vp where 1 W s the pseuo nverse of W. 1 W W WW For a cable robotc system, t can be easly shown that the system has no zero ynamcs. herefore, the ecouplng matrx W s full rank an the control law s well X efne an a sutable change of coornates X y y y y yq y q where yel s a close- n the normal form [15]. loop system y xr 2 1,,q r1 rq n A Bv y H 5. Lnear Moel Prectve Control he basc structure of MPC to mplement s shown n Fgure 2 A moel s use to prect the future plant outputs, base on past an current values an on propose future control actons. hese actons are calculate by optmzer takng nto account the cost functon as well as constrants. he optmzer s another funamental part of the strategy as t proves the control acton. each component of ths structure s escrbe n more etal In the followng of ths artcle. Fgure 2. Basc structure of MPC. (4)

4 72 A. GHASEMI he goal s to apply the lnear moel prectve control to the nput-output lnearze system to account for the constrants. Snce the lnear moel prectve s more naturally formulate n screte tme, the lnear subsystem n (5) s scretze wth a samplng pero to yel k1 A kb vk (6) yk H k where A,B an H are obtane rectly from the contnuous-tme matrces [22]. Also, the state-epenent relaton between u(k) an v( k) s obtane as r vk L h r F X k LGLFhX k uk hs mappng can be rewrtten n the followng form v k PX k W X k u k 5.1. Constrant Mappng When lnear moel prectve control s apple to the system, t s necessary to map the constrants from the orgnal nput space to lnearze system. By conserng v s a new nput to be etermne, o obtan constrants on the new nput, he nput constrant mappng s performe usng nput output lnearzaton law an the current state measurement x(k). he transforme constrants can be etermne on each samplng pero by solvng the followng optmzaton problem: mn mn k vmn k j k P X k j max ukjk W X k j k u k j k 0 j N 1 uk ( jk ) u u k j j u max vmax k j k P X k j k W X k j k u k j k 0 j N 1 (7) where Z (k + j k) s the precte value of the system state (Z) at tme k + j base on the nformaton avalable at tme k. Note that the varable X (k + j k) cannot be calculate prove that the nput sequence s calculate, whch s not possble untl the constrants are specfe. herefore, at the begnnng (k = 0), the nput constrant over the entre control horzon can be presente by: v k j k v k k mn mn j1,, N1 max v k j k v k k max j1,, N1 hen, we use nputs calculate at last samplng tme to etermne future constrants at the current samplng tme. (8) herefore, Equaton (7) wll be change nto Equaton (9). mn max mn uk ( jk ) 1 max 1 uk ( jk ) max v k j k P X k j k W X k j k1 ] u k j k 0 jn1 v k j k P X k j k W X k j k1 u k j k 0 jn1 umn u k j j u (9) Now Equaton (9) can be solve to obtan v mn (k+ j k) an v max (k + j k). If W(,j) s postve, the control u(j) must be the smallest value for v mn an the largest value for v max an f W(,j) s negatve, then t must be the largest value for v mn an the smallest value for v max Lnear Moel Prectve Control Desgn he goal s to apply LMPC to the lnearze system to account for these constrants. Now the moel (6) s use n the nfnte horzon lnear moel prectve strategy propose by Muske an Rawlngs [23]. herefore, the openloop optmal control problem that the nput control foun by mnmzng the nfnte horzon crteron, can be expresse as V( kk ) j1 k j k H QH k j k mn ( ) ( ) vk j k vk j 1 k R vk j k vk j 1 k v k j k v S v k j k v (10) where ξ an v are target values for ξ an v, respectvely, an Q,S, an R, are postve sem efnte matrces. In orer to obtan value v(k + j k) t s necessary to mnmze the functonal of Equaton (10) to o ths value of the precte output are calculate as functon of pas values of nputs an outputs an future control sgnals obtan an expresson whose mnmzaton leas to the looke for values. he ecson vector s efne as V(k k) = [v(k k) v( k+1 k) v(k+n-1 k)], where N s the control horzon. All future moves beyon the control horzon are set equal to the target value v. As scusse n [23], the matrx A s unstable an n orer for the lnear moel prectve problem to have a feasble soluton t s necessary to mpose the equalty constrant ξ(k + N k) = ξ. o obtan a fnte set of ecson varables, nputs beyon the control horzon are set equal to the esre value: v(k + j k) = v, j N. herefore, the nfnte horzon lnear

5 A. GHASEMI 73 moel prectve problem Equaton (10) can be wrtten as a fnte horzon problem. V( kk ) vk N kv 1 mn 1 N 1 k j k H QH k j k j1 vk j k vk j 1 k 1 S v kn k v v k jk v S v k jk v R v k j k v k j k (11) hs optmzaton problem must be solve subjecte to the followng constrants: k N k (12) v k j k v k j k v k j k mn mn N 1 N 2 EE A B A B B N K N A HQH A 0 he soluton of Equaton (13) belongs to the regular system. o fn the solutons of the trackng one, that of the regular system shoul be shfte nto the orgn of the system to the steay state escrbe by ξ, an v. he esre values must le wthn the feasble regon efne by nput constrants for lnear moel prectve control an mnmze the control effort (Equaton (15)). Straghtforwar algebrac manpulaton of quaratc subject to : objectve functon of the corresponng regular system presente n Equaton (11) results n the followng stan- I A B 0 U ar quaratc program form: H 0 y mn V HHV 2V GG kffvk 1 (13) 0 DD U C Vkk where S B KN 1 A 0 GG, FF B K0 A 0 Equaton (13) shoul be solve subjecte to the followng constrants: DDV CC N (14) EE V A k where by conserng vmax k k I vmax k N 1 k 6* N DD, CC, I 6* N vmn k k vmn kn 1 k 2 mn ( 0 U 0m2m Imm U U U v m m Im m U U Qs 2 m*2m (15) U an Qs are esre value of the nput an a postve efnte matrx, respectvely. herefore control law s the summaton of the answers of Equaton (13) an Equaton (15) whch can be shown n the followng form. ( W X k k v k kp( X k k u k W X k k v k k P X k k (16) Fgure 3 shows schematc of the propose structure control. Fgure 3. Schematc of control structure. N 1 B K N1B R2S B AKN2 B S B A K0 B N 2 B KN 2 AB S B KN 2 B R 2S B A K0 B HH N1 N2 B K0 A B B K0 A B B K0 B R2S

6 74 A. GHASEMI 6. Smulaton In ths secton, smulaton results of applyng moel prectve control on a 3D cable robot wll be presente. able 1. shows the mensons of the cable robot. We conser a efnte movement of the platform from X 0 = [ 0.1, 0.1,1.5,0.002,.001, 0.001] to X = [0.2sn(t), 0.4 sn(t), t t 3,0,0,0], on a esre trajectory shown n otte lnes by Fgures 4(a) to 4(f), for poston/orentaton of the platform. Also, we conser the parameters of the moel precttve controller as: the control horzon N = 25, S, Q, an R of Equaqon (10), R = I, Q = I, S = 0.01I, an Q s of Equaton (15), Q s = I. Snce the controller nees platform s poston/orenttaton, at frst, we must solve forwar knematcs of the robot. hs has been carre out by the authors usng neural network algorthm [24]. Fgure 4 shows the moel prectve controller wth nput-out lnearzng worke qute well an a trajectory trackng are one. Fgure 5 shows the sx tensons n the cables vs. tme. As t can be seen, all of them reman postve urng the moton. able 1. Dmensons of the cable robot. Poston vector X(m) Y(m) Z(m) Poston vector x(m) y(m) z(m) a b a b a b a b a b a b 6 (a) (b) (c) () (e) Fgure 4. Plots of esre an actual poston an orentaton of the platform, (a)-(c) poston n X-Y-Z rectons, respectvely, ()-(f) orentaton aroun X-Y-Z recton, respectvely. (f)

7 A. GHASEMI 75 (a) (b) (c) () (e) (f) Fgure 5. Plots of tenson trajectores. 7. Conclusons In ths paper, a lnear moel prectve controller together wth an nput-output lnearzng control strategy for a constrane robotc system, a 3D cable robot, was evelope an evaluate. he control system s comprse of: 1) an nput-output lnearzng controller that accounts for cable robot nonlneartes; 2) a constrant mappng scheme that transforms the actual nput constrants nto nput constrants on the feeback lnearze system; an 3) a lnear moel prectve controller that proves explct compensaton for nput constrants. he smulaton re- of the propose metho. It sults showe the effectveness s worth nothng that ths approach can be extene for the reunant cable robots. 8. References [1] D. Stewart, A Platform wth Sx Degrees of Freeom, Proceengs of the Insttuton of Mechancal Engneer, Vol. 180, No. 15, 1965, pp o: /pime_proc_1965_180_029_02 [2] J. -P. Merlet, Parallel Robots, Sol Mechancs an Its Applcatons, Kluwer, Norwell, [3] J. -P. Merlet, Stll a Long Way to Go on the Roa for Parallel Mechansms, A Keynote Speech at Desgn Engneerng echncal Conferences, Montreal, 29 September-2 October [4] J. -P. Merlet, Parallel Robots, Open Problems, INRIA Sopha-Antpols, France. [5] L. -W. sa, Robot Analyss, he Mechancs of Seral an Parallel Manpulators, Wley, New York, [6] P. Bosscher, A.. Rechel an I. Ebert-Uphoff, Wrench- Feasble Workspace Generaton for Cablerven Robots, Journal of Intellgent an Robotc Systems, Vol. 22, No. 4, 2006, pp [7] A.. Rechel an I. Ebert-Uphoff, Force-Feasble Workspace Analyss for Unerconstrane Pontmass Cable Robots, Proceengs of IEEE Internatonal Conference on Robotcs an Automaton, New Orleans, 26 Aprl-1 May 2004, pp [8] S. Kawamura, W. Choe, S. anaka an S. R. Panan, Development of an Ultrahgh Spee FALCON Usng Wre Drve System, In Proceengs of the 1995 IEEE Internatonal Conference on Robotcs an Automaton, May 2003, pp [9] P. Lafourcae, M. Llbre an C. Reboulet, Desgn of a Par-

8 76 A. GHASEMI [10] X. Dao, O. Ma an R. Paz, Stuy of 6-DOF Cable Robots for Potental Applcaton of HIL Mcrogravty Con- tact-dynamcs Smulaton, In Proceengs of the AIAA Moelng an Smulaton echnologes Conference an Exhbt, Keystone, August 2006, pp [11] P. Gallna, G. Rosat an A. Ross, 3-D.O.F. Wre Drven Planar Haptc Interface, Journal of Intellgent an Robotc Systems,Vol. 32, No. 1, 2001, pp o: /a: [12] J. Albus, R. Bostelman an N. Dagalaks, he NIS Robocrane, Journal of Robotc Systems, Vol. 10, No. 5, 1993, pp o: /rob [13] Y. Q. Zheng, Workspace Analyss of a Sx DOF Wre- Drven Parallel Manpulator, Proceengs of the WORK- SHOP on Funamental Issues an Future Research Derecton for Parallel Mechansms an Manpulators, Quebec, 3-4 October 2002, pp [14] W. J. Shang, D. Cannon an J. Gorman, Dynamc Analyss of the Cable Array Robotc Crane, Proceengs of the IEEE Internatonal Conference on Robotcs an Automaton, Detrot, May 1999, pp [15] J. J. Slotne an W. Lepng, Apple Nonlnear Control, Prentce Hall, Englewoo Clffs, [16] H. Khall, Nonlnear Systems, 3r Eton, Prentce-Hall, Upper Sale Rver, [17] J. Rchalet, A. Raault, J. L. estu an J. Papon, Moel allel Wre-Drven Manpulator for Wn unnels, In Proceengs of the Workshop on Funa mental Issues an Future Research Drectons for Parallel Mechansms an Manpulators, Quebec, 3-4 October 2002, pp Prectve Heurstc Control: A pplcaton to Inustry Processes, Automatca, Vol. 14. No. 2, 1978, pp o: / (78) [18] C. E. Garca, D. M. Prett an Morar, Moel Prectve Control: heory an Practce a Survey, Automatca, Vol. 25, No , pp o: / (89) [19] A. B. Alp an A. K. Agrawal, Cable Suspene Robots: Desgn, Plannng an Control, Internatonal Conference on Robotcs Robotcs & Automaton, Washngton, DC, 9-13 May 2002, pp [20] S. R. Oh an A. K. Agrawal, Controller Desgn for a Non-reunant Cable Robot Uner Input Constrant, ASME Internatonal Mechancal Engneerng Congress & Exposton, November 2003, Washngton, DC. [21] S. R. Oh an A. K. Agrawal, A Control Lyapunov Approach for Feeback Control of Cable-Suspene Robots, IEEE Internatonal Conference on Robotcs an Automaton, Aprl 2007, pp [22] F. Frankln, J. Powell an L. Workman Dgtal Control of Dynamc Systems 2n Eton, Ason Wesley, Boston, 1994, pp [23] K. R. Muske an J. B. Rawlngs, Moel prectve control wth lnear moels, AIChE Journal, Vol. 39, No. 2, 1993, pp o: /ac [24] A. Ghasem, M. Eghtesa an M. Far Neural Network Soluton for Forwar Knematcs Problem of Cable Robot, Journal of Intellgent an Robotc Systems, Vol. 60, No. 2, 2010, pp o: /s z