Reference Input Shaping to Reduce Move Time of a Very. Flexible One-Link Manipulator

Transcription

1 Reference Input Shaping to Reuce Move Time of a Very Flexible One-Link Manipulator Withit Chatlatanagulchai 1*, Tanapon Srivonga 1 an Peter H. Meckl 1 Control of Robot an Vibration Laboratory (CRV Lab), Department of Mechanical Engineering, Faculty of Engineering, Kaetart Univerity, Bangkok 19, Thailan School of Mechanical Engineering, Purue Univerity, Wet Lafayette, IN , USA * Correponing author, fengwtc@ku.ac.th 1/5

2 ABSTRACT When the robot manipulator i require to be light, long, or to move fat, it link ten to flex. When the link flexe, it en poition i ifficult to control. In fact, the flexibility bring about uneire attribute uch a nonminimum phae, tip vibration, an complicate ynamic. A a reult, unlike that of the rigi-link robot, control of the flexible-link robot i more challenging an till an open reearch problem. We propoe eigning a twoegree-of-freeom control ytem from a imple moel. The controller enure goo tracking an proper iturbance rejection. The cloe-loop natural frequency i then compute, an the reference input i hape uch that it energy content i low aroun the natural frequency. The experiment how that thi propoe metho reuce tip vibration ignificantly reulting in fater move time. Key wor: vibration reuction, flexible link, comman haping, hape reference input, quantitative feeback /5

3 INTRODUCTION The pace tation activitie in the 8 purre the interet in control of the flexible-link robot. The long boom in pace mut be light, long, an accurately poitione; it link i flexible. Since the well-unertoo control of the rigi-link robot cannot apply, reearcher ha to fin new an effective control an moeling metho. Over the pat three ecae, having alo been encourage by an emerging inutrial tren for lighter an fater robot with higher payloa capacity, tremenou effort have been pent in moeling an control a well a other area uch a tructural optimization an enor an actuator electronic. Neverthele, an effective control that can be applie to all ituation ha not been foun. Moeling of the flexible-link robot can be ivie into approximate an exact metho. Hating an Book (1987) are among the firt to propoe an approximate moel from aume moe approach. Intereting comment about the limitation of thi approach i given by Wang an Viyaagar (1989). Jen et al. (1996) propoe a low-orer tranfer function from the Rayleigh- Ritz metho, an Krihnan an Viyaagar (1998) propoe a reuce-orer moel bae on the Hankel-norm minimization. The exact metho, although i not uitable for control eign, ecribe the behavior of the robot more accurately. The moel from the Euler-Bernoulli beam theory wa obtaine in the work of Bellezza et al. (199), Low (1997), 3/5

4 an Iemamai an Chatlatanagulchai (8). The lat work compare the exact moel with an approximate moel. Majority of the flexible-link robot control ue active metho, where external energy i provie to the ytem to control it repone. Example that ue fixe moel are that of Chapnik et al. (1993), whoe controller i an open-loop compute torque, Jnifene an Fahim (1997), in which elaye eflection i ae to counteract the non-minimum phae effect, an Baicu et al. (1998), where a nonlinear controller bae on the backtepping approach wa applie. Example that ue aaptive cheme are given by Caruone et al. (1993), Konno an Uchiyama (1995), Mougal et al. (1994), Yang et al. (1997), an Cawara an Unbehauen (), where metho uch a gain cheuling, fuzzy logic, aaptive control, an neuro-fuzzy control were applie. Paive metho inclue traitional ajutment of pring an amping contant an reference input haping, where the input i altere o le energy i injecte aroun the natural frequency. The latter can be ivie into two cheme. Firt i bae on the convolution of the reference input with a equence of impule. Thi cheme wa propoe by Singer an Seering (199). It improvement can be een in the work by Tze an Yurkovich (1993), Khorrami et al. (1995), an Mohame an Tokhi (4). Zuo an Wang (199) pointe out the limitation of the firt cheme, regaring banwith an frequency reprouction. 4/5

5 The econ cheme ue optimization etting. Prove by Meckl (1988), the amplitue of the reiual vibration i irectly proportional to the input pectrum magnitue aroun the natural frequency. Therefore, a erie repreentation can be forme to imitate the bang-bang acceleration profile an to reuce the pectrum energy aroun any eire frequency. Chatlatanagulchai et al. (6) applie thi technique to a two-link flexiblejoint robot. In thi paper, we eign a control ytem from a imple moel obtaine from a ytem ientification of an actual robot. Being imple, thi thir-orer moel ha uncertainty. We then eign a robut controller an a prefilter uing the quantitative feeback theory, whoe etail can be foun in the work by Chatlatanagulchai et al. (7) for the SISO cae an Chatlatanagulchai et al. (8) for the MIMO cae. Thi type of control ytem can hanle uncertainty by incorporating it in the loop haping proce. The natural frequency i then foun from the cloe-loop ytem. The comman haping propoe by Meckl (1988) i then applie to obtain a reference trajectory whoe energy i reuce aroun the cloe-loop natural frequency, reulting le excitation, hence le vibration an ignificantly fater move time. The paper i organize a follow. In the metho ection, we introuce a one-link flexible-link robot an how we obtain a imple robot moel. Control ytem eign an imulation reult are ubequently icue, followe by etermination of the cloe-loop natural frequency an haping of the reference input. We then apply the reference input to an actual 5/5

6 robot in the CRV laboratory. The experimental et-up an reult are given in the reult ection. Then, ummary, unolve part of the problem, an new reearch irection are preente in the concluion ection. METHODS Obtaining a imple robot moel J, m p p P y w( t, x ) y x x x θ ( t ) α ( x, t ) x Figure 1 Diagram of a one-link flexible-link robot. A flexible-link robot normally ha etting in Figure 1. In our cae, we are interete in controlling it tip poition, o point P i at the payloa. The output variable to be controlle i α which conit of the bae angle θ, obtaine from the bae optical encoer, an the link angle, obtaine from train gauge attache along the flexible link. Our goal i to move α o the link i move from point to point with leat ettling time poible. 6/5

7 α v α 1. m ( ra ) Time ( ) Figure Robot moel valiation. The oli line i the moel output; the otte line i the actual output. Several frequency-varying ine ignal were given a input voltage v to the robot. The output α wa recore. The input an output pair were fitte together uing Matlab Sytem Ientification toolbox to prouce everal tranfer function. The experiment wa repeate everal time with new et of input ignal. At the en, we obtaine a tranfer function with goo valiation reult a α 1.18 = P = v , (1) whoe valiation reult i hown in Figure. Care wa taken not to over-fit the moel with any particular et of ata. 7/5

8 Control ytem eign an imulation Having obtaine the moel, we then eigne the control ytem, whoe iagram i in Figure 3. α i the reference input. O i a iturbance at plant output, uch a groun vibration, to be attenuate. G an F are the controller an prefilter. O α F + e v + G P α Figure 3 Diagram of a fee-forwar/ feeback ytem. The plant P i given in (1), of which each parameter wa allowe to eviate by ± 1%, which i a number obtaine from performing everal ytem ientification. On the Nichol chart, the o-calle plant template { P } were plotte at pertaining frequencie. Figure 4 how the plant template at five frequencie, where tar mark nominal poition. 8/5

9 Open-Loop Gain (B) ra / 1 ra / 1 ra /.5 ra /.1 ra / Open-Loop Phae (eg) Figure 4 Plant template at variou frequencie. We eigne the control ytem to achieve goo tracking, tability, an iturbance attenuation. A a reult, two pecification were et, in the frequency omain, a 1 1 GP < δ + () an δ LB FGP < < δ 1+ GP UB, (3) where δ = 4 B an δ LB an δ UB are two tranfer function 3.9 δ LB = /5

10 an δ UB = 16.3, each having uitable tep repone. Specification (3) i for tracking, an it i eay to how that pecification () enure goo tability an plant-output iturbance attenuation. 3.1 ra / 1.5 ra / Open-Loop Gain (B) -1-1 ra / 1 ra / ra /.1 ra /.5 ra / 1 ra / 1 ra / 35 ra / Open-Loop Phae (eg) Figure 5 QFT wort-cae boun an open-loop hape. Specification were converte into boun on the Nichol chart. When each boun wa rawn, all point in the correponing plant template were 1/5

11 incorporate to enure that the reulting controller, eigne in the next tep, will be robut for all poible plant variation in the template. Two boun, for () an (3), were obtaine for each frequency. Then the two boun were combine an the tricter portion wa electe to create the wort-cae boun. Figure 5 contain five wort-cae boun for five elective frequencie. In the loop haping tep, the controller G wa eigne uch that the reulting open-loop hape L = GP lie in an acceptable region. Figure 5 how the open-loop hape after the loop haping. The controller i in the form G = ( + ) 1. 11/5

12 1 1+ GP ( B) FGP 1+ GP ( B ) -1 δ S - -4 δ UB - -6 δ LB α ( ra ) frequency ( ra / ) frequency ( ra / ) ( a ) α ( ra ) ( b) UB α LB time ( ) time ( ) ( c ) ( ) α Figure 6 Simulation reult. (a) Stability. (b) Tracking. (c) Diturbance rejection. () Tracking in the time omain. The prefilter i trivial an i left a F = 1. Figure 6 how the imulation reult. Each oli line in the pack repreent the imulation of a one-point plant in the template. In Figure 6(a), the pecification () i the otte line wherea the tar mark the eigne frequencie. It i een that the pecification () i met for all poible plant variation. 1/5

13 Figure 6(b) how the tracking reult, which fall out of the boun only at high frequencie, the region not inclue in the eign to avoi noie. Figure 6(c) plot the output α when the tep iturbance O wa applie. We ee that the iturbance i appropriately attenuate. Figure 6() contain the tracking reult, where LB an UB are two time-omain boun obtaine from paing a quare wave α into the tranfer function δ LB an δ UB. The imulate output α wa able to follow it eire α within the boun mot of the time for all plant variation. 1 ( / ) α ra B time ( ) α ( ra / ) ( a) ( ).5 8 ra / 1 1 ( / ) ( b) frequency ra α ra ( ) time ( ) ( c ) ( ) time Figure 7 Square-wave reference input. (a) Reference 13/5

14 acceleration α. (b) Frequency pectrum of α. (c) Reference velocity α. () Reference poition α. Shaping of the reference input The natural frequency of the cloe-loop ytem can be compute. For the nominal cloe-loop ytem, it i 8 ra /. The reference input for the robot to follow can be obtaine from integrating the bang-bang acceleration profile. Figure 7 how the bang-bang acceleration profile a well a it reulting reference velocity an poition. It can be een from Figure 7(b) that the reference input pectrum amplitue i high over the natural frequency 8 ra /. Mentione by Meckl (1988), there i a irect relationhip between the magnitue of the input pectrum at the natural frequency an the amplitue of the reiual acceleration. We are certain that inputting thi reference input woul reult in high-level of reiual vibration. 14/5

15 ( / ) α ra B time ( ) α ( ra / ) ( a) ( ).5 8 ra / ( / ) ( b) 1 1 frequency ra α ra ( ) time ( ) ( c ) ( ) time Figure 8 Shape reference input. (a) Reference acceleration α. (b) Frequency pectrum of α. (c) Reference velocity () Reference poition α. α. To improve the ituation, we followe the work by Meckl (1988) by haping the reference input o that it energy content i lower aroun 8 ra /, uing the fact that any input function can be repreente by the erie L B l, (4) α l ( ) = Φ ( t) f t l= 1 l 15/5

16 where B l i the coefficient for each harmonic an 1 t t αl t Φ l ( t ) = αl + inαl coα l T f Tf Tf i a rampe inuoial function with α l a a characteritic number aociate with each harmonic an T f a move time. B l can be choen to minimize the cot function Tf / T 11 1 f J = f t t + f t t + T F T Tf T / i= 1 f * [ 1 ( )] [ 1 ( )] ρ ( ωi ) ( ωi f ), (5) where ρ i a relative weighting between two objective the firt term to mimic the bang-bang profile for fat move, the econ term to lower the energy content aroun ω n an T i the move time to cover a itance when the input i a ingle cycle of a quare wave of amplitue y f F max. The cot function contain 11 frequencie urrouning reonance; thi number can be change arbitrarily. In our cae, ± 1% eviation from ω n = 8 ra / wa ue; the boun on the frequency ω are.9ω < ω < 1.1 ω. The percent i n i n eviation wa compute from each plant in the plant template. We contructe α uing the erie (4). The cot function (5) wa minimize uing the eign parameter: L =, ρ = 1, ω n = 8 ra /, an F = The eire move itance i y = ra. Figure 8(a) how the max. f hape α together with it frequency pectrum in Figure 8(b). The pectrum 16/5

17 magnitue aroun the natural frequency i greatly reuce. Time plot of the correponing α an α are given in Figure 8(c) an (). Note that the eire move time increae from.67 in the quarewave cae (Figure 8()) to 1.4 in the rampe inuoial cae (Figure 8(c)). Thi i becaue the energy that wa pulle out from the pectrum aroun the natural frequency reult in a lower eire poition profile. However, a will be een in the experimental reult, the rampe inuoial profile greatly reuce the robot move time becaue it inuce a lot le reiual vibration than the quare-wave profile. RESULTS Experimental et-up an reult Figure 9 i a photograph of our flexible-link robot. An accelerometer i mounte with the payloa whoe location i at the tip. Three train gauge are ue to meaure tranveral eflection, an an optical encoer i ue to meaure motor an hub angle. A teel ruler i ue a the flexible link with an effective length of.54 m. Figure 1 i a block iagram of the experimental arrangement. A hot computer, with neceary oftware, i ue to communicate with uer an a target computer. The target computer contain a ata acquiition car whoe function are to acquire enor ignal an to en out actuator comman from the control algorithm. The hot an target computer are connecte to each 17/5

18 other via a LAN line. Control ignal i ent a voltage to a motor amplifier boar to amplify to a level that can rive the DC motor. An IC chip accelerometer i mounte at the tip to meaure linear acceleration. There i an in-houe-eign op-amp circuit to amplify an filter train gauge ignal. A DC power upply upplie require current to the motor amplifier boar. DC Motor Strain Gauge Optical Encoer Payloa an Accelerometer Figure 9 Photograph of the flexible-link robot in our laboratory. 18/5

19 Accelerometer Payloa Hot Computer Target Computer Strain Gauge Motor θ Encoer On-Boar LAN Software Labview Labview Realtime Labview Control Labview Simulation Matlab LAN Car Intel Pro1S Data Acquiition Car NI-61 AI1 AI AI3 Op-Amp Circuit Motor Amplifier Boar CTR AI4 AO V DC Power Supply Software Labview Realtime OS Labview ETS Licene Figure 1 Block iagram of the experimental et-up an aociate harware. The ampling time of 1 m wa ue for the harware an for icretizing the controller. A witch wa mae in the program to alternate between upplying the hape reference input an the unhape (bang-bang) reference input. Figure 11(a) plot the robot tip poition α veru it eire quare-wave trajectory α. By alternating between the hape an unhape reference input, we clearly ee that, with hape input, the robot 19/5

20 wa able to move fater with ignificantly le ettling time than with the unhape cae, which uffer from evere reiual vibration. Figure 11(b) how the ignal from the accelerometer at the tip, with ignificantly higher acceleration output een uring the unhape perio. 3 α, α ( ra ).5 α α hape unhape hape unhape hape ( / ) x m ( a) ( ) time hape unhape hape unhape hape time ( b ) ( ) /5

21 Figure 11 Experimental reult for hape an unhape cae. (a) Angular poition output α an it eire value α. (b) Tip acceleration ẋ. CONCLUSIONS With haping of the reference input, the flexible-link robot achieve fater point-to-point motion ue to le reiual vibration. Not only that the robot can be operate fater, but alo that le vibration reuce amage of both the robot an it payloa. The algorithm preente in thi paper i eay to implement, requiring only the information on the natural frequencie of the ytem. The unolve part of the algorithm i that the hape input can only be obtaine off-line. Therefore, the robot path mut be pre-pecifie. Although thi work well for repetitive robot like thoe in inutrie, the algorithm will not work with the robot with unknown or aaptive path. More reearch houl be on an on-line algorithm. ACKNOWLEDGEMENTS The firt author woul like to thank Craig Borgheani an Teraoft, Inc for their evaluation copy of the QFT toolbox. Thi work i performe at the Control of Robot an Vibration Laboratory, which i ituate at an partially 1/5

22 upporte by the Reearch an Development Intitute of Prouction Technology (RDiPT) of Kaetart Univerity, Thailan. Thi work i alo ponore by the grauate-level reearch funing from the Grauate School of Kaetart Univerity, Thailan. LITERATURE CITED Baicu, C.F., C.D. Rahn an D.M. Dawon Backtepping bounary control of flexible-link electrically riven gantry robot. IEEE/ASME Tran. on Mechatronic. 3(1): Bellezza, F., L. Lanari an G. Ulivi Exact moeling of the flexible lewing link. Proceeing of the IEEE International Conference on Robotic an Automation. Cincinnati. Ohio Caruone, J., K.S. Buchan an G.M.T. D Eleuterio Experiment in eneffector tracking control for tructurally flexible pace manipulator. IEEE Tran. on Robotic an Automation. 9(5): Cawara, F.M. an H. Unbehauen.. A neurofuzzy approach to the control of a flexible-link manipulator. IEEE Tran. on Robotic an Automation. 18(6): Chapnik, B.V., G.R. Heppler an J.D. Aplevich Controlling the impact repone of a one-link flexible robotic arm. IEEE Tran. on Robotic an Automation. 9(3): /5

23 Chatlatanagulchai, W., V.M. Beazel an P.H. Meckl. 6. Comman haping applie to a flexible robot with configuration-epenent reonance. Proc. of the 6 American Control Conference. Minneapoli. MN Chatlatanagulchai, W., B. Ineemeeak an W. Siwakoit. 7. Quantitative feeback control of a penulum with uncertain payloa. Journal of Reearch in Engineering an Technology. 4(4): Chatlatanagulchai, W., C. Srinangyam an W. Siwakoit. 8. Trajectory control of a two-link robot manipulator carrying uncertain payloa uing quantitative feeback theory. Journal of Reearch in Engineering an Technology. 5(1): Hating, G.G. an W.J. Book A linear ynamic moel for flexible robotic manipulator. Control Sytem Magazine Iemamai, K. an W. Chatlatanagulchai. 8. Exact mathematical moel of the flexible-link robot. Proceeing of the n Conference of Mechanical Engineering Network of Thailan. Patumthani Jen, C.W., D.A. Johnon an R. Gorez A reuce-orer ynamic moel for en-effector poition control of a flexible robot arm. Mathematic an Computer in Simulation. 41: /5

24 Jnifene, A. an A. Fahim A compute torque/time elay approach to the en-point control of a one-link flexible manipulator. Dynamic an Control. 7: Khorrami, F., S. Jain an A. Tze Experimental reult on aaptive nonlinear control an input prehaping for multi-link flexible manipulator. Automatica. 31(1): Konno, A. an M. Uchiyama Vibration uppreion control of patial flexible manipulator. Control Eng. Practice. 3(9): Krihnan, H. an M. Viyaagar Control of ingle-link flexible beam uing Hankel-norm-bae reuce-orer moel. IEE Proc. Control Theory Appl. 145(): Low, K.H A note on the effect of hub inertia an payloa on the vibration of a flexible lewing link. Journal of Soun an Vibration. 4(5): Meckl, P.H Control of vibration in mechanical ytem uing hape reference input. PhD Thei. Department of Mechanical Engineering. Maachuett Intitute of Technology. 16 p. Mohame, Z. an M.O. Tokhi. 4. Comman haping technique for vibration control of a flexible robot manipulator. Mechatronic. 14: /5

25 Mougal, V.G., K.M. Paino an S. Yurkovich Rule-bae control for a flexible-link robot. IEEE Tranaction on Cont. Sy. Tech. (4): Singer, N.C. an W.P. Seering Prehaping comman input to reuce ytem vibration. ASME Tran. J. Dynam., Mea., Contr. 11(1): Tze, A. an S. Yurkovich An aaptive input haping control cheme for vibration uppreion in lewing flexible tructure. IEEE Tran. on Control Sytem Technology. 1(): Wang, D. an M. Viyaagar Tranfer function for a ingle flexible link. IEEE Int. Conf. on Robotic an Automation Yang, J.H., F.L. Lian an L.C. Fu Nonlinear aaptive control for flexible-link manipulator. IEEE Tran. on Robotic an Automation. 13(1): Zuo, K. an D. Wang Cloe loop hape-input control of a cla of manipulator with a ingle flexible link. Proc. of the 199 IEEE Int. Conf. on Robotic an Automation. Nice. France /5