Active Vibration Control Based on a 3-DOF Dual Compliant Parallel Robot Using LQR Algorithm

Transcription

1 The 009 IEEE/RSJ Internatonal Conference on Intellgent Robots an Systems October -5, 009 St. Lous, USA Actve Vbraton Control Base on a 3-DOF Dual Complant Parallel Robot Usng LQR Algorthm Yuan Yun an Yangmn L Abstract In recent years, many applcatons n precson engneerng requre a carefusolaton of the nstrument from the vbraton sources by aoptng actve vbraton solaton system to acheve a very low remanng vbraton level especally for the very low frequency uner 0Hz vbraton sgnals. In ths paper, base on the prevous research experences n the systematcal moelng an stuy of parallel robots, a hybr robot s escrbe an the vbraton moes gven by usng Lagrange s Equatons. Then the present stuy aresses the ssues relate to the actve vbraton control schemes for the MIMO system usng LQR algorthm. Fnally, numercal smulatons on the effect of actve vbraton control are presente. I. INTRODUCTION There are three funamental control strateges to regulate or control the response of a system: passve control, semactve control, an actve control. Desgn an mplementatons of passve solaton systems have been stue for many years. Passve solaton systems generally consst of one or several stages of mass-sprng-amper systems ntrouce n the propagaton path, whose parameters are ajuste to acheve the esre corner frequency an a reasonable compromse between the amplfcaton at resonance an the hgh-frequency attenuaton. The passve ampng s necessary to lmt the amplfcaton at resonance, but t tens to reuce the hgh-frequency attenuaton of the solaton system. Sem-actve control has been evelope as a compromse between passve an actve control. A sem-actve control system can acheve favorable results through selectve energy sspaton, but s ncapable of njectng energy nto a system. Actve solaton has been ntrouce to allow to acheve smultaneously a low amplfcaton at resonance an a large attenuaton at hgh-frequency. Actve control can change the propertes of the system base on the change n the nstantaneous operatng contons as measure by sensors. To counteract the vbraton on precson nstruments, actve solaton systems are best sute, snce these unts acheve a very low remanng vbraton level, especally for low frequency sturbances wthout the resonance behavor of a passve solaton system. Actve vbraton control technology conssts of a mxture of mechancal engneerng, structural mechancs, control engneerng, materal scences an computer scence. Nowaays, a carefusolaton of the nstrument from the Ths work s supporte by the Macao Scence an Technology Development Fun uner Grant No.:06/008/A an research commttee of Unversty of Macau uner Grant No.: UL06/08-Y/EME/LYM0/FST. Yuan Yun an Yangmn L(corresponng author) are wth the Department of Electromechancal Engneerng, Faculty of Scence an Technology, Unversty of Macau, Av. Pare Toms Perera, Tapa, Macau SAR, Chna, ya77406@umac.mo, yml@umac.mo vbraton sources s requre n many precson engneerng applcatons, whch s realze by aoptng actve vbraton solaton system to acheve a very low remanng vbraton level especally for the very low frequency uner 0Hz vbratons. Actve vbraton solaton project base on parallel manpulator s on-gong amng to cross the brge between the structural ynamcs an control communtes, whle provng an overvew of the potental of smart materals for sensng an actuatng purposes n actve vbraton control. Parallel manpulators can offer the avantages of hgh stffness, low nerta, an hgh spee capablty whch have been ntensvely researche an evaluate by nustry an nsttutons n recent years. An some esgners aopt the flexure hnges nstea of conventonal mechansm jonts snce the backlash an frcton n the conventonal jonts nfluence the performances of parallel mechansms remarkably. Mcro/nano postonng manpulators an actve vbraton control evces are ncreasngly beng mae of parallel manpulators ue to ther characterstcs of hgh precson an hgh spee capablty. Therefore, a lot of esgners focus ther attentons on mult-egree of freeom hybr manpulators or evelopng we range flexure hnges. A spatal complant 3-DOF parallel robot wth SMA pseuo-elastc flexure hnges was presente n, whch has a workspace larger than mm 3 an resoluton s better than µm. A parallel structure for macro-mcro systems was propose n. In ths new esgn, the macro-moton (DC motor) an mcro moton are connecte by a parallel structure, the two motons are couple uner one complant mechansm framework. At the same tme, a kn of ual parallel mechansm was evelope 3, calle a 6-PSS parallel mechansm an a 6-SPS one, whch s ntegrate wth we-range flexure hnges as passve jonts to ensure the large workspace of the whole system an hgh precson moton. A XYZ-flexure parallel mechansm was propose wth large splacement an ecouple knematcs structure 4, whch has a large moton range beyon of mm. The pezo actuator s a well-known commercally avalable evce for managng small splacements, whch has the avantages of hgh precson, large force generaton, submllsecon response, no magnetc fels et al. Nowaays, pezo actuators n hgh precson postonng systems an actve vbraton control systems have been nvestgate wely. However there are lmte research works on usng pezo actuators n parallel manpulator for actve vbraton control. Ths research s concerne wth the evelopment of a /09/$ IEEE 775

2 system that can acheve three hgh accurate translatonal postonng an a 3-DOF actve vbraton solaton, whch attenuates the vbraton transmsson above some corner frequences, to protect the payloa from the jtters nuce by the varous sturbance sources. In ths paper, base on the prevous research experences n the systematcal moelng an stuy of parallel robots, a novel ual 3-DOF parallel robot wth flexure hnges wll be escrbe. A vbraton moel of the parallel manpulator wll be presente by Lagrangan s equatons. Then the present stuy aresses the ssues relate to the actve vbraton control schemes for the MIMO system usng LQR algorthm wth the pezo actuators as the actuators, laser sensors for splacement an accelerometers as the sensors. Fnally, numercal smulatons about the ynamc characterstcs an control effects wll be presente. II. SYSTEM DESCRIPTION The esgne 3-DOF ual parallel platform s a 3-DOF ual parallel mechansm combnng a 3-PUU parallel mechansm wth another spatal 3-UPU one. In the 3-PUU parallel structure, the prsmatc actuators prove three translatonal macro motons wth mcron level accuracy an cubc centmeter workspace. At the same tme, the mcro moton s prove by a spatal 3-UPU structure whch can ncrease the accuracy of the whole system to the nanometer level or prove a strctly acceleraton level for payloa place on the movng platform. 3-PUU an 3-UPU knematcal structures wth conventonal mechancal jonts can be arrange to acheve only translatonal motons wth some certan geometrc contons satsfe. Usng flexure hnges at all jonts, the parallel platform conssts of a moble platform, a fxe base, an three lmbs wth entcal knematc structure. Each lmb connects the moble platform to the fxe base by one prsmatc actuator, one flexure unversal (U) hnge an a pezo actuator followe by another U jont n sequence as shown n Fg., where the frst U jont s fxe at the prsmatc actuator an the secon one connects to the movng platform actuate by a pezo actuator to offer the mcro motons wth merts of nvolvng smooth moton, hgh accuracy, an fast response, etc. The flexure unversal hnge s a slener shaft confguraton wth very hgh torsonal stffness whch s aopte as passve jont to ensure the large workspace of the whole system an hgh precson moton. Hgh precson ceramc motors are aopte as the prsmatc actuators to prove the macro moton for the mechansm. III. DYNAMIC MODEL Snce the macro moton s aopte for the rough postonng, the knematc analyss s necessary for the 3-PUU structure. The etals have been publshe n 5. In ths secton, the vbraton moel usng Lagrange s equatons wll be escrbe frst for the 3-UPU structure, an then Kane s ynamcs moelng wll be bult up to verfy the results. Fg.. A. Knematc Analyss The 3-DOF 3-PUU/3-UPU ual parallel manpulator. In ths case, the prsmatc actuators for macro moton are self-locke when the mcro moton s avalable. ) Coornate System: Let L=l l l 3 T be the vector of the three PZT actuate length varables an the vector r oo =x y z T of the reference pont o be the poston of the movng platform. As shown n Fg., let b be the vector ob an m be the vector o M. The mass of movng platform s M. The mass of each strut s m l. C (=,,3) s the center of mass for the th lmb. Let a lmb fxe rght-hane coornate system wth orgn C (=,,3) locate at the center of mass for the th lmb, wth axs rectons etermne by an orthonormal set of unt vectors ĉ j (j=,,3). The hat ncates unt length, the nex correspons to the th lmb, an the nex j stngushes the three vectors. ĉ 3 s along the th lmb, towar the th flexure hnge whch connects the movng platform. ĉ s perpencular to the vector ob when the movng platform s n ts home poston, an ĉ s n the recton ĉ ĉ 3. Let a reference frame ŝ j attach to the fxe platform at the center o wth the ŝ towar the pont B an ŝ 3 vertcal the fxe platform. Fx a coornate system f j to the movng platform at the center o wth f towar the pont M an f3 vertcal the movng platform. R s a three-orer entty transformaton matrx from f j coornate system to ŝ j. R c s the transformaton matrx from ĉ j coornate system to ŝ j. In orer to etermne the angles of flexure hnges rectly, an ntal coornate system s set on orgn C wth axs rectons etermne by an orthonormal set of unt vectors ĉ j0 when the movng platform s n the home poston. The axs rectons are locate concent wth the corresponng lmb fxe coornate system ĉ j when the movng platform s n the ntal poston. Let the orentaton of the ĉ j coornate system, relatve to the ĉ j0, be escrbe by consecutve postve rotatons q about the ĉ 0 an q about the move two-axs. The rotaton matrx R c0 s: R c0 = ĉ 0 ĉ 0 ĉ 30 ĉ ĉ ĉ 3. () It s assume that the small-angle approxmatons hol for angles qk, the rotaton matrx s: 0 q R c0 = 0 q. () q q 776

3 Fg.. Coornate system of the 3-UPU parallel platform B. Vbraton Moel In ths paper, the vbratonal moes establshe by usng Lagrange s equaton whch s a well known tool for establshng equatons of moton of screte systems. The key pont of Lagrange s equatons s knetc energy, whch can be formulate favorably wth respect to a movng coornate system as well. ) Knetc Energy of the System: For the movng platform, the knetc energy can be gven by ) Generalze Spees for the System: Defne generalze spees u for the system as the tme rate of change of the generalze coornates of q=x y z T n the nertal reference frame: u = ẋ ẏ ż T. (3) An ntermeate reference frame s ntrouce prevously to permt escrbng the angular velocty of each lmb to the ntal lmb poston. Desgnate the unt vectors of these ntermeate reference frames by m j. The expressons for the angular veloctes of the reference frame of each upper arm wth respect to ntal reference frame of upper arm are: The poston vectors are: ω C = q ĉ 0 q m. (4) r oo = ĉ 3 + b m, r m = ĉ 3 + b, r C = ĉ 3 + b. (5) Dfferentatng Eq. (5) wth respect to tme: v m = v o, v C = v o = ĉ 3 + ω C ĉ 3 (6) ĉ 3 + ω C ĉ 3 Dot-multplyng both ses of Eq. (6) by ĉ 3: = v o. (ĉ 3) T v o = ( =,,3) (7) whch can be assemble nto a matrx form: l l l3 T = A vo (8) where A = (ĉ 3) (ĉ 3) (ĉ 3 3) T. Cross-multplyng both ses of Eq. (6) by ĉ 3: ω C = ĉ 3 v o (9) The lnearze acceleratons are: a o = ( ω C + ε C ) ĉ 3 + ω C (ω C ĉ 3) + ĉ 3 a C = a o (0) Cross-multplyng both ses of Eq. (0) by ĉ 3, the angular acceleraton of each lmb can be obtane by: ε C = ĉ 3 a o ω C (ω C ĉ 3)(ω C ĉ 3) () E km = M(ẋ + ẏ + ż ) () For each lmb, assume rotatons about a fxe axs, the knetc energy can be gven by where E kl = m l v C + I C ω C I C = 3 m ll (3) an accorng to Eq. (4), the knetc energy of the whole system can be erve by E k = E km + E kl = M(ẋ +ẏ +ż )+ ( m l v C = = + I C (( q ) + ( q ) )) = ( M m l)(ẋ + ẏ + ż ) + = 6 m ll (( q ) + ( q ) )) (4) ) Potental Energy of the System: The potental energy of movng platform s gven by E pm = Mgz. (5) Snce the system s nvestgate n tny vbraton envronment, accorng to the Eq. (5), we can obtan ĉ 3 = x x b + x m y y b + y m (6) z z b + z m Let ĉ 0 ĉ 0 ĉ 30 = r r r3 r r r3 r3 r3 r33 (7) Accorng to Eq. () an (), the consecutve postve rotatons q an q about the move two-axs are: r r r3 x x b + x m q r r r3 y y b + y m = q r3 r3 r33 z z b + z m q = H r x r y r 3 z q = H + r x + r y + r 3 z (8) 777

4 where H = r (x b x m ) + r (y b y m ) + r 3 (z b z m ) H = r (x m x b ) + r (y m y b ) + r 3 (z m z b ) For each lmb, the potental energy can be wrtten as: E pl = m l g z z b + z m + k (q) + k(q ) (9) Hence, the potental energy of the whole system can be gven by E p = Mgz + (m l g z z b + z m + k (q ) + k(q ) ) = (0) 3) Lagrange s Equatons: Let F Dj an F a j be the sturbance force an the actuate force of each lmb assocate to the q j respectvely. The Lagrange s equatons are: where u j ) E k q j + E p q j = F Dj + ẋ ) = (M m l)ẍ + = = F a j () 3 m l(((r ) + (r ) )ẍ +(r r + r r )ÿ + (r 3r + r 3r ) z) ẏ ) = (M m l)ÿ + = 3 m l((r r + r r )ẍ +((r ) + (r ) )ÿ + (r 3r + r 3r ) z) ż ) = (M m l) z + = 3 m l((r r 3 + r r 3)ẍ +(r r 3 + r r 3)ÿ + ((r 3) + (r 3) ) z) E k x = 0, E k y = 0, E k z = 0 E p x = (k( r )(H r x r y r 3 z) = +k ( r )(H + r x + r y + r 3 z)) E p y = (k( r )(H r x r y r 3 z) = +k ( r )(H + r x + r y + r 3 z)) E p z = Mg + 3m lg + = (k ( r 3 )(H r x r y r 3 z) + k ( r 3 )(H + r x + r y + r 3 z)) () The fnal forwar ynamc equaton of ths parallel multboy system can be wrtten as: where M = (M m l)i + 3 m l M = M u + K q = Q F + F D (3) M, = (r ) + (r ) r r + r r r 3r + r 3r rr + rr (r) + (r) r3r + r3r rr 3 + rr 3 rr 3 + rr 3 (r3) + (r3) F D s the sturbance force actng on the mass center of movng platform. I s the entty matrx. In ths case, the pertnent flexure hnge stffness has the relatonshp of k = k j = k ( =,,3,j =,,3) Hence, the stffness matrx of the whole system s: r +r r l r +r r l = = = K = k r r +r r r l +r l = = = = r r 3 +r r 3 l = r r 3 +r r 3 l = r r 3 +r r 3 l r r 3 +r r 3 l r 3 +r 3 l Let F a be the vector of force exerte by the th pezo actuator on each lmb at the mass center of the strut n the global coornate system. The generalze external force matrx s: k 3 ( r H r H ) = Q F = F a + k 3 ( r l = H r H ) = k 3 ( r3 H r3 H ) Mg 3m lg = Let the F a be the matrx of scalars of rvng forces gven by: F a = F F F 3 T (4) where F a = F ĉ 3. IV. CONTROL STRATEGY The smple form of loop shapng n scalar systems oes not exten rectly to mult-varable (MIMO) plants, whch are characterze by transfer matrces nstea of transfer functons. The noton of optmalty s closely te to MIMO control system esgn 6. In ths secton, the control strategy wll be analyze by usng the lnear quaratc regulator (LQR) metho, whch s a well-known esgn technque that proves practcal feeback gans. A process of the actve vbraton contros shown n Fg. 3. The sensors on the movng platform etect the sturbances actng on the system frst. Then the sgnals are converte by A/D sgnal converter an fee back to the controller to calculate the rvng forces whch wll be sent to three pezo actuators, an fnally realze the actve vbraton solaton. The block agram for ths feeback actve vbraton contros shown n Fg. 4, where s the 778

5 Fg. 3. Process of the actve vbraton control. where α an β s the unetermne coeffcent. The full state feeback controller gan matrx G can be solve by LQR metho, an u = GX (9) Substtute Eq. (9) to Eq. (7), the close loop system ynamcs s gven by: Ẋ = à X + D (30) where à = A BG. Fnally, the tme response of the close loop control moel can be obtane by usng fferental equaton solver. Fg. 4. Block agram for feeback actve vbraton control. sturbance vector, u s the control vector an Y s the measurement vector. As shown n Fg. 5, let X b an X m be the 3 egree of freeom splacements of base platform an movng platform. Hence, the Eq. (3) can be wrtten as: M Ẍ m + K (X m X b ) = Q F + F D (5) Accorng to the Eq. (5), the state vector conssts of X m wth the 3 coornates: Ẋm 0 I Xm 0 0 = Ẍm M + K 0 Ẋm M M K FD 0 + M Q F (6) Ẍm X b Let the state vector, sturbance vector an control vector be Ẋm FD X =, =, u = Q F respectvely. Hence, the generalze plant of the control problem s gven by: where X b Ẋ = A X + B u + D, Y = X (7) A = 0 I M K 0 D =, B = 0 0 M M K. 0 M Let the weghte matrx Q an R be: K 0 Q = α, R = βi. (8) 0 M Fg. 5. Actve vbraton solaton system., V. NUMERICAL SIMULATIONS The parameters of the parallel mechansm are shown n Table I. As the well elastc nature of the selecte materal of we-range flexure hnges, the workspace of mcro moton only epens on the lmts of pezo actuator. The maxmal usable nscrbe workspace of mcro moton, when the nputs of ceramc motors are zero an self-locke at the ntal poston, s a column wth a raus of 66.6µm an 38.6µm heght accorng to 5 of our prevous work. TABLE I GEOMETRIC AND MATERIAL PARAMETERS Item Raus of movng platform r Raus of fxe platform R Raus of flexure hnge r f Length of flexure hnge l f Raus of strut r s Length of strut l s Moulus of elastcty of flexure hnge E f Moulus of elastcty of strut E s A. Influence of Unetermne Coeffcent Value 0mm 60mm 0.9mm 0mm 6.35mm 0mm 30GPa 70GPa Frst, n orer to etermne the weghte matrx Q an R, the nfluence of the unetermne coeffcents α an β shoul be consere. Snce the response an rvng forces are relate to the rato of α an β, the value of β s set to 00 n ths paper accorng to the emprcal value. The tren of the z-axs response of movng platform an the maxmal rvng force s shown n Fg. 6. In ths state, t s obvously that the maxmal rvng force can be ecrease wth ncrease the value of β. But the effect of control gets worse. In ths case, n orer to control the splacement of movng platform uner nm, α an β are set to 00 an for takng conseraton of the response of movng platform an maxmal rvng force. B. Actve Vbraton Control As shown n Fg. 7, the re lnes represent to the open loop response of the movng platform an the blue lnes represent to the close loop response by usng LQR metho. It s obvously that the sturbance coul be elmnate about 50-70% especally the sturbance actng along the z-axs. 779

6 Fg. 6. Relatonshp between splacement/rvng force an β (α=00). (a) Actuator (b) Actuator (c) Actuator 3 (a) X-axs splacement (b) Y-axs splacement Fg. 8. Drvng forces of the three actuators. (c) Z-axs splacement () X-axs acceleraton contrbutons to the research on actve vbraton solaton base on parallel manpulators. After ths work, some control moelng for actve vbraton control wll be gven an optmal control algorthm wll be scusse. Fnally, a prototype of the parallel robot wll be evelope, an the expermental nvestgaton wll be carre out to verfy the vbraton control performance of the prototype. (e) Y-axs acceleraton Fg. 7. (f) Z-axs acceleraton Postons an acceleratons of the movng platform The change of the rvng forces of three pezo actuators are shown n Fg. 8. VI. CONCLUSION In ths paper, a novel ual 3-DOF parallel robot wth flexure hnges has been presente. Ths system can acheve three hgh accurate translatonal postonng wth a 3-DOF actve vbraton solaton an exctaton functon to the payloa place on the movng platform. Base on the prevous research experences n the systematcal moelng an stuy of parallel robots, a vbratonal moel of the parallel manpulator has been presente by usng Lagrange s ynamcs. Then the actve vbraton control scheme for the MIMO system usng LQR algorthm has been ntrouce. Fnally, some smulaton results by MATLAB are shown to verfy the valty of the control strategy. The nvestgatons of ths paper are expecte to make REFERENCES J. Hesselbach, A. Raatz, J. Wrege, an S.Soeteber, Desgn an analyss of a macro parallel robot wth flexure hnges for mcro assembly tasks, n Proc. of 35th Internatonal Symposum on Robotcs, Pars, France, 004, No.TU4-04fp. P. R. Ouyang, Hybr ntellgent machne systems: Desgn, Moelng an Control, Ph.D. Thess, Unversty of Saskatchewan, Canaa, W. Dong, L. N. Sun, an Z. J. Du, Desgn of a precson complant parallel postoner rven by ual pezoelectrc actuators, Sensors an Actuators A: Physcal, Vol. 35, pp , X. Tang an I. -M. Chen, A large-splacement 3-DOF flexure parallel mechansm wth ecouple knematcs structure, n IEEE/RSJ Internatonal Conference on Intellgent Robots an Systems, Bejng, Chna, 006, pp Y. Yun an Y. L, Performance Analyss an Optmzaton of a Novel Large Dsplacement 3-DOF Parallel Manpulator, n Proc. of IEEE Internatonal Conference on Robotcs an Bommetcs, Bangkok, Thalan, pp MIT, Maneuverng an Control of Surface an Unerwater Vehcles, Fall Y. Yun, Y. L an Q. Xu, Actve vbraton control on A 3-DOF parallel platform base on Kanes ynamc metho, n Proc. of SICE Annual Conference, Chofu, Tokyo, Japan, 008, pp R. D. Hampton an G. S. Beech, A Kane s Dynamcs Moel for the Actve Rack Isolaton System, NASA/TMł00C0063, Marshall Space Flght Center, AL, Y. L an Q. Xu, Knematc analyss an ynamc control of a 3-PUU parallel manpulator for caropulmonary resusctaton, Proc. of th Int. Conf. on Avance Robotcs, pp. 344C35, M. S. Whorton, H. Buschek an A. J. Calse, Homotopy Algorthm for Fxe Orer Mxe H /H Desgn, Journal of Guance, Control, an Dynamcs, Vol. 9, No. 6, pp. 6C69, 996. M. Whorton, Robust Control for Mcrogravty Vbraton Isolaton Usng Fxe-orer, Mxe H /µ Desgn, AIAA Guance, Navgaton, an Control Conference an Exhbt, Austn, Texas, Aug. -4,