Stability Improvements of Robot Control by Periodic Variation of the Gain Parameters

Transcription

1 Proceedings of the th World Congress in Mechanism and Machine Science ril ~4, 4, ianin, China China Machinery Press, edited by ian Huang Stability Imrovements of Robot Control by Periodic Variation of the Gain Parameters amás Inserger Budaest University of echnology and Economics, Deartment of lied Mechanics, 5, Budaest, Hungary Gábor Stéán Budaest University of echnology and Economics, Deartment of lied Mechanics, 5, Budaest, Hungary bstract: In this aer, a PD osition control of a one degree of freedom system is investigated for the case when the gain arameters are varying eriodically. Due to the samling effect of the control rocess, the mathematical model is a time eriodic delay-differential equation. he investigation of this equation shows that the stability roerties and the settling time of the control rocess can be imroved by roer gain variation. wide range of stability charts are determined for constant and eriodically varying control gains and imrovements in stability are shown. he settling time is also investigated, and the gains resulting the fastest signal decay are also determined. Keywords: Samling effect, Periodic gains, Stability Classical -DOF model of robot control Consider the DOF mechanical model of osition control shown in Figure. he osition of mass m is sensed and a control force Q is alied to ush the mass into the desired osition. PD Q m x Introduction Position control is a frequent mechanical controlling roblem in robotics. he aim is to drive the robot arm into a desired osition. o achieve a clear icture about the behaviour of the control, digital effects, like samling should also include into the mechanical model. Samling is a kind of delay in information transmission that often leads to unstable oscillations [~3]. nalytical investigation of simle models with degree of freedom (DOF) lay a central role in understanding technical henomena and forming the common sense in design work. Proerly choosen gain arameters and samling time does not always rovide stable motion and fast settling time, since this arameters are often hedged by other technical conditions. For these cases, eriodic variation of the gains may stabilizes or seed the control. he idea of stabilizing by arametric excitation comes from the classical examle of the endulum: the uer osition of a endulum can be stabilized by vibrating its ivot oint [4]. Parametric excitation is often used to stabilize otherwise instable systems, like instable machining oerations [5,6]. In this aer, the effect of gain variation on the stability and robustness of osition controlled machines is investigated. Caused by the samling effect, the governing equation is a delay-differential equation (DDE). DDE s usually have infinite dimensional hase sace, and the stability conditions often have no analytical form [7~9]. In the case of digital samling, the governing DDE has a finite dimensional discrete reresentation, so the roblem is reduced to a finite eigenvalue roblem, and stability criteria can be given in closed form [~]. Stability henomena are observed by constructing stability charts in the lane of the control arameters. Figure DOF mechanical model of osition control he system is governed by the simle differential equation m & x Q. () Let the samling time be denoted by τ, and the th samling interval be t [ t,t + τ ), where t τ, Z. In the th samling interval, the control force Q is comuted from the data measured at the revious samling at time t, as it is demonstrated in Figure. If the control has roortional and differential gains, then the control force reads Q ( t) Px( t ) Dx& ( t ), t [ t,t + τ ). () hus, the governing equation is m & x ( t) Px( t ) Dx& ( t ), t [ t,t + τ ). (3) x Q τ t x - x x + - t t t t t t + t - Figure Samling effect

2 By introducing dimensionless time t / τ, Eq. (3) obtain the form x ( ) x( ) dx ( ), [ + ), (4) where Pτ / m, d Dτ / m are dimensionless arameters and rime denotes the derivatives with resect to the dimensionless time. lthough Eq. (4) is a delayed differential equation, it still has the finite dimensional discrete reresentation x( + ) x ( + ) a d + / x( x, (5) a where a x dx is the acceleration of the mass during the th interval (details will be exlained later). If all the eigenvalues of the coefficient (transition) matrix are in modulus less than one, than the system is asymtotically stable [3,4]. he stability roerties and the settling time is shown in Figure 3. he charts shows the stable domain in the -d arameter lane. he curve denoted by µ is the boundary curve of asymtotic stability actually, it is a combination of a arabola and a vertical line. he curves denoted by µ.9,.8 and.7 are the curves, where the largest eigenvalue in modulus is ust equal to.9,.8 and.7, resectively. Inside the curve denoted by.7, the amlitude of the oscillation decays at least to 7% in each samling. his gives an orientation where to find the gain arameters resulting the fastest settling time. It can be easily shown by analytical calculations that the best settling time is achieved for the gains /7, d 7/54, when µ /3. his oint is denoted by P in Figure 3. In the next section, the secial case will be investigated, where the control gains are varying ste by ste in each samling interval. 3 -DOF model of robot control with eriodic gain variation Consider the case, when the gains has eriodically varying discrete values in each discretization interval: P P π τ P t + cos + τ P D D π τ D t + cos + τ D, (6a), (6b) t [ t,t + τ ),,, K, k. (6c) Here, P and D are the mean values, and P and D are the relative amlitude of the gain variations. he gains are alternating between the discrete values ( P ) P ( P ), or D ( + D ) and ( D ) P + and D, resectively (see Figure 4). he time eriod of this variation is k τ, that is P + k P and D + k D. his kind of variation of the gains is ust a theoretical aroach. In fact, the control force is continuous in time, and it does not have any brakes. lthough, the gains determined by Eqs. (6a) and (6b) can be used as a good aroximation of the real forces. he dimensionless control gains are also iece-wise constant functions: + cos π +, (7a) d d + d cos π + d, (7b) [ + ),,, K, k, (7c) where, P, τ / m and d. D,τ / m. he time eriod is now equal to k. For k and, the dimensionless roortional as a function of dimensionless time gain can be seen in Figure 4. Finally, the dimensionless equation of motion reads x ( ) x( ) d x ( ), [ + ), (8) Figure 3 Stability chart for Eq. (4) where + k and k d + d are eriodic arameters. k 3 k 3 (+ ) (- ) (+ ) (- ) Figure 4 Control gain variations

3 4 Construction of the transition matrix he solution of Eq. (8) can be determined for each samling interval. he exression in the right hand side is the acceleration a of the mass in the th interval, consequently, the motion can be integrated as x ( ) x + a, [ + ), (9) x( ) x( + x ( + a, [ + ), () where a x( ) d x ( ) is constant for each samling interval. Now, substitute + into Eqs. (9) and (): x ( + ) x +, () x( + ) x( + x + a. () t this oint, a the connection between the states at times instant and + can be given by the ma where y a + y, (3) x( y x and a /. d he transition matrix over the time eriod k of the system (8) can be given by couling the solution over the forthcoming samling intervals: Φ k K. (4) k he stability roerties and the oscillation decay ratios are determined by the eigenvalues of the transition matrix Φ. he modulus of the eigenvalues gives the ratio of the oscillation decay over a samling interval. If all the eigenvalues are in modulus less than one, than the control rocess is stable [3,4]. 4 Stability charts Stability charts are determined in the lane of main gain arameters and d. he transition matrices and the critical eigenvalues were comuted in oints of the charts, and a contour lot was used to obtain boundary lines. In Figures 5 and 6, stability charts can be seen for k and, when one gain variation takes and times of samling time, resectively. Six different values of relative Figure 5 Stability charts for k

4 Figure 6 Stability charts for k gain amlitudes and d were investigated resulting a 6 6 matrix of stability charts. he left uer charts of Figures 5 and 6, when and d, is ust the same as the chart of Figure 3. he variation of the relative gain amlitudes can be followed as they change from to 5. In the case, when the relative gain amlitudes are greater than one, the resultant gains may become also negative. he resented boundary lines denote the arameter oints, where the eigenvalue greatest in modulus is equal to,.9,,.,.. Outside these boundary curves, the control rocess is unstable, and vibrations arise. hese unstable domains are denoted by U in the figures. he first thing that can be observed in Figures 5 and 6 is that the 6 6 matrices of stability charts are not symmetric, that is, the variation of the roortional gain has different effect on stability as the variation of the differential gain. nother oint is that the smallest the eriod k is the better the imrovement in stability is, as it can be seen by the comarison of the charts in Figures 5 and 6. his means that faster variation in the gain arameters results more stable control. First, consider the case k in Figure 5. For the cases d.5 and, no relevant differences instability can be observed, the stability chart are almost the same. For the case d, the largest imrovement in stability is shown u for the arameter 5. his corresonds to a very strongly variation, since is changing between ( + ) 6 and ( ) 4. In this case, the there is no relevant imrovement in the settling time correlating to the control with constant gains. For the case, the largest imrovement in stability is shown u for the arameter d. In this case, d is alternating between d ( + d) d and d ( d). his corresonds to the case, when in every second interval, the control is only roortionally controlled. he best imrovement in settling time is achieved by the case and d, when both gains are zero for every second samling interval. In this case, /, d and / (5) d 3/ Φ d. (6) s it is can be seen in Figure 7, the maximum eigenvalue in modulus goes also under.. Furthermore, if / 4

5 and d 7 / 8, all the eigenvalues of the transition matrix Φ is. his oint is denoted by P in Figure 7. For d >, the stability roerties get worse until the stable region disaear, like in the charts d 5,,.,.5,. show the effect of various gain variations. hese matrices of stability charts are not symmetric matrices, that is the variation of the roortional gain is not identical to the variation of the differential gain. For some gain variation, significant imrovement in stability and settling time was exerienced. It was found that the faster the variation of the gains are, the better the stability roerties are. secial combination of arameters was found (k,, d ), where all the eigenvalues of the transition matrix was. However, this case needs further investigations. In site of the simle model and the fact that the gain variations alied in this study are very intermittent and are not always realizable, the results give an orientation for real structures how to choose control tye and gain values. cknowledgements his research was suorted by the Hungarian National Science Foundation under grant no. OK 376/99. Figure 7 Stability chart for k, and d Now, consider the case k in Figure 6. Here the stability imrovements are not so good as in the case k, but the stability boundaries are more comlex. he best stability roerties are achieved for gain amlitudes, d. his chart is shown in Figure 8. If the variation amlitude is increased then stability roerties get worse. Figure 8 Stability chart for k, and d 5 Conclusions he PD osition control of a one degree of freedom system was investigated for the case when the gain arameters are varying eriodically. he delay effect of samling was built in the mathematical model. he stability of the control was resented via stability charts. 6 6 matrices of stability charts was constructed to References. Raibert, M.H., Craig, J.J., Hybrid osition force control for comuter controlled maniulators, SME Journal of Dynamic Systems, Measurement and Control, : 5-33, 98.. Craig, J.J., Introduction to Robot Mechanics and Control, ddison-willey, Reading, Stéán, G., Haller, G., Quasieriodic oscillations in robot dynamics, Nonlinear Dynamics, 8: 53-58, Inserger,., Horváth, R., Pendulum with harmonic variation of the susension oint, Periodica Polytechnica, 44(): 39-46,. 5. Sexton, J.S., Stone, B.J., he stability of machining with continuously varying sindle seed, nnals of the CIRP, 7(): 3-36, Inserger,., Stéán, G., Chatter suression of turning rocess via eriodic modulation of the sindle seed - a DOF analysis, Proceedings of 3rd Conference on Mechanical Engineering GÉPÉSZE, Budaest, (), Hale, J.K., Lunel, S.M.V., Introduction to Functional Differential Equations, Sringer-Verlag, New York, Stéán, G., Retarded Dynamical Systems, Longman, Harlow, Inserger., Stability analysis of eriodic delay-differential equations modeling machine tool chatter, PhD dissertation, Budaest University of echnology and Economics Budaest,.. Stéán, G., Varying delay and stability in dynamical systems, Zeitschrift für angewandte Mathematik und Mechanik, 7(4): 54-56, 99.. Kollár, L.E., Stéán, G., Hogan, J.S., Samling Delay and Backlash in Balancing Systems, Periodica Polytechnica, 44(): 77-84,.. Kovács, L.L., Stéán, G., Dynamics of digital force control alied in rehabilitation robotics, Meccanica, in ress,. 3. Kuo, B.C., Digital Control Systems, SRL Publishing Comany, Chamaign, Illinois Lakshmikantham, V., rigiante, D., heory of Difference Equations, Numerical Methods and lications, cademic Press, London, 988.