31055 Toulouse Cedex

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1 Rocsedingafh19921EEE atomrtiaul Caafmwm 011 abotia uy1 AutanutiOa ~pmoq-may1992 DYNAMC MODELLNG AND FRCTON COMPENSATED CONTROL OF A ROBOT MANPULATOR JONT Gomes, S. C. P. CMtien, J. P. CERT/DERA - 2, Avenue Edouard Belin (BP 4025) Toulouse Cedex Abstract nternal disturbances of a robot joint drive constitute important sources of non-linearity, that make djfficult the design of control laws with a high precision level. The most important of these disturbances is the friction torque and there is no consensus about the mathematical friction model that better approximate the reality. This fact motivated the realization of this work, that &scribes a dynamic modelling of a robot joint drive (motor and gear), based on experimental results. The friction model is a finction of angular position and velocity. This model was used for the design of compensation mechanisms which experimental results were good in a general manner. The velocity- &pendent friction torque component was written in a dfferent form. This made possible an analysis of closed loop poles behavior that can be useful in the elaboration of gain scheduling control laws. 1 ntroduction Most of the physical phenomenons have a non-linear nature. Some of then may have a linear dynamic model approximation, but most of them are extremely non-linear. One can stand out the phenomenon which involves relative movement of masses in presence of friction like, for instance, the motor-driven joint of a robot. Friction constitutes the main some of non-linearity, that makes the time response of the system extremely dependent on the reference input amplitude level. The major part of the published works that deal with control systems of mechanical manipulators, consider the dynamics of each motor-driven joint as linear as far as the control law synthesis is considered. The proposed models contain only viscous friction, as can be seen in the Figure (la). Many authors who take into account the non-linear behavior of a joint, make use of models based on the Coulomb's model (Figure (16)) [l], j131. Generally, Coulomb friction (dry friction) is considered together with static and viscous friction (Figures (C) and (16)) [31,[41, [71, 181, [lo], [14]. The transition from static to dynamic state (stiction) is supposed instantaneous in the models of Figures (lb) and (lc), while the model of Figure (16) includes friction reduction for small velocities. Experimental works that adopt the model of Figure (C) [9]. consider only dry friction once it has shown a value close to that of the static friction. We can also mention Da"s model [ 111, which is very different of the models based on the Coulomb's model. n spite of having a complex physical nature, friction has been shown to be very repeatable [4]. One important goal in this research domain is to eliminate the non-linear friction through compensation mechanisms (some authors suggest adaptive techniques [5], 161, [14]). n 161, an analysis of stability when control laws with friction compensation are used is given. With the objective of linearize the dynamic of a standard motor-driven joint of a robot, an initial modelling work was based on experimental results in open loop, leading to a friction model function of the angular position and velocity. The latter is indsed the most important. 'No friction compensation mechanisms were tested experjmentally: one for the position dependent part and another for the velocity dependent part. The objective of testing the dynamic model was always present during development of the work, and for this reason, all experimental results were compared with the results obtained in simulations. The experimental data was never filtered. This was important for the modelling of the position-dependent friction torque component. 2 Equipment Description The facility used to obtain the experimental results is a two degree of freedom robot manipulator [12], located at the Center of Studies and Research of Toulouse (CERTDERA). The two joints are driven by DC servomotors through harmonic drive gears [2]. The modelling work was based on experiments with one of the joints. The gear output axis is connected to a quasi-rigid arm. At the end of this arm there is a "payload" supported on air bearings (Figure 2). There are four sensors that provide the rotor angular velocity (tachometer), the load angular position (optical ender), the torque through the gear output axis (torquemeter) and the load angular velocity (gyrometer). State variables values as well as identified parameters may be provided on the motor side or on the /92 $ EEE 1429

2 load side. For better understanding, acronyms (ms) (motor side) or (S) (load side) will be added to each variable or parameter. 3 Dynamic Modelling Considering Newton's formalism, the differential equations describing the drive joint dynamics are: where: eq = 5.87 Kg m2 (ms)(motor side inertia),. = 0.9 Kg m2 (s)(load inertia), Keq = (s) (joint stiffness), n = 80 (reduction factor), 8, is the rotor angular position, Os is the load angular position, T, is the electromagnetic motor torque and Tf is the friction torque. Tm is controlled with precision by the drive electronics, which are tuned to work in rotor current control mode. 3.1 The friction torque model The experimental results of the friction torque modelling motivated the choice of the model shown on Figure (C), but with some modifications. The minimum torque needed to achieve a steady state with a non zero velocity is Nm(ms). Above this minimum torque, excitation of the joint by step inputs of increasing value were applied, and for each torque step, the averaged velocity was measured in steady state. n this way, the friction torque was obtained as a function of angular velocity, in a torque region called here dynamic equilibrium region. The complementary torque region (Tm < NM(ms)) is called here the static equilibrium region. The friction torque points were interpolated, and second order polynomial approximation achieved a better matching of the little curvature observed on the experiments then the linear approximation. Figure (3) shows the friction curve obtained from 18 measurement points, 9 for each rotation sense, and the polynomial approximation curve, which was extrapolated to the null velocity. The polynomial equations of the velocity dependent friction model read: f +.fvpie + coefpiz if i, > o ppifn)sign(tm) if 8, = 0 (2) fn + funee + Coefnbz if ie < o where: fp = NM(ms); fn = NM(ms) fup = (ms); fun= (ms) Coefp = w(ms) ; Coef,, = ' w(ms) ; The fiiction parameters are clearly different for each rotation sense. The differences between the dry friction Up or fn) and the minimum torque necessary to obtain the dynamic equilibrium (supposed static friction) are not very significant. This fact did not encourage the use of the friction model shown on Figure (16) as a basic model (i.e. the static friction will be taken equal to the dry friction in the sequel). The signals of velocity, position and torque measured experimentally, showed oscillations which frequency vary as a linear function of the average rotor rotation frequency, in the fonn: foac = 2 frotor. The disturbing torque driving these oscillations is probably due to small relative decentralization between the wave generator and the circular spline (harmonic drive components). This originates a periodic variation in the normal pressure between the teeth in contact, creating a periodic variation in the friction torque. This friction torque was approximated by a sinusoidal function of rotor angular position, in the form: Tfp = -A sin(28, + 4) (3) where Tfp is the position dependent friction torque. To identify parameters A and 4, a compensating sinusoidal torque was injected together with a step torque and the parameters values which minimize the oscillation were identified. The amplitude and phase vary with the load angular position and with the rotation sense, hence making a performing rejection of T difficult. Figure (&) shows the rotor velocity as a funchon f9 of the load position for three different angular positions regions and for the two rotation senses, without the friction compensation. Figure (46) is similar to Figure (44, but friction compensation was implemented (i.e. open loop cancellation of Tfp as a function of 0,). The pafameter values are A = NM(ms), q5c = 0.6~ if 8, > 0 and 4c = 0.8~ if 8, < 0, where dc is the compensation torque phase. Figures (44 and (46) show that there are situations in a which the compensation is very performing, but there are others for which the performance is less significant. Armstrong [4] identified the presence of a position-dependent friction torque component (work made with an harmonic drive gear). He suggested a compensation with the help of correction tables function of the angular position. This compensation showed a good performance. The system of differential equations including the complete friction torque model is thus given by: Keq eqie + X(6.s - nes) + j j + fujie + Coefji:+ -A sin(2ee + 4~ - T, = o (4) ai. - K,~ ($- e.) =

3 where i may be p or n, according to positive or negative velocity. Figure (5) shows experimental and simulations results in open loop for one alternating step level of input torque. Table (1) shows the maximum position errors obtained with 8 different input torque levels, similar to that which generated Figure (5). The total average maximum error amounts to 2.7%. 4 Arm Position Control with Rejection of the Velocity-Dependent Friction Component A simple proportional control was used to test the possibility of lineatizing the joint dynamics by friction rejection, according to the block diagram of Figure (6), where fcome = NM(ms), fo = NM(ms) and 81 = 1.4 y(ms) = 1 %(is). The compensation block is based on a similar scheme proposed in [9]. t was implemented on this way to avoid vibrations due to input toque commutations in the neighborhood of null velocity. Experiments made with torques under the minimum limit to obtain stationary non zero velocity, allowed to determine torque regions shown on Figure (7). t was verified that if an input torque with absolute magnitude between f' and fj is applied, it will produce a little movement of the gear load axis, but after a short transient, the velocity becomes null again and the static equilibrium is established. The commutation of this torque between positive and negative values creates vibrations on the load side that may reach important amplitude levels, especially when the torque is near fi. The compensation term fcomp is nearly 95% of the dry friction (fi = fp or fn). This value gave satisfactory results without the risk of vibrations. The constant fo was obtained experimentally. Figure (10) shows experimental results of step responses, using Kp = 0.178%(ms) and Kd = 0. This Figure shows clearly the non-linearity of the dynamics, as the response shapes are extremely dependent in the level of the reference input. Figure (8) was obtained under the same conditions as Fi,gure (lo), but with the compensation block of equation (5) in the control, and it shows that the dynamic becomes approximately linear. Figure (11) shows responses for four reference input levels using a proportional gain Kd Merent of zero: Kp = y(ms) and Kd = %(ms). The steady state errors are shown on table (2). These errors can be eliminated with the help of an integral control component activated around the steady state. This was tested in practical and a zero error was obtained for all amplitude levels reference input. This technique may be an interesting alternative to avoid instability problems and responses with overshoot generally obtained with the integral control. To test the model efficiency, all the experimental results in closed loop were compared with the respective simulations. Figure (14) exhibits an experimental result (one of the curves of Figure (8)) and the corresponding simulation. The compensation block (Comp Se, U ) was (* 1 simulated in the same conditions as in the experiment. n this Fip the angular position of the load Os, the rotor velocity 8, (experimental result is truncated due ADC saturation), the output torque and the control input torque are compared. Considering all the simulations in closed loop, the averaged maximum position error is 2.17%. The results in closed and open loop indicate that the model is satisfactory. 5 Linearized Model Closed Loop Poles Behavior The velocity-dependent friction torque component (equation (2)) has a strong discontinuity at Se = 0. This is a source of problems not only for a digital implementation of a friction compensation block (see Figure (7)). as discussed in section 4, but also in the simulations runs with slow changes in the signal of angular velocity. Oscillations appear in simulations which do not exist experimentally. This proves that, for the digital implementations, models of the Coulomb family are not realistic in situations which involve angular movements with small accelerations in the vicinity of null velocity. To run simulations as those of Figure (14). the concept of a variable viscous friction coefficient was introduced: where fj, Coe fj and f,,i are parameters previously defined (equation (4)). The set of differential equations may be written in the state space form as: '0, \ f,' and TfP are updated at each intlegration step. Obviously, equation (6) is singular at 8, = 0. As a consequence, numerical integration methods diverge for 8, 1431

4 values very close to zero. A velocity limit (eint) was thus introduced: ejnr = 2$(ms) = 1.43?(Zs) provide good results. This value is important because ijnt very close to zero may let the dynamic be more damped then it is really in this region of small velocities. The variable viscous friction coefficient may also be useful in the analyze of poles behavior of the non-linear systems. Assuming an hypothetical linear approximation through a constant viscous friction coefficient (fm), the transfer function in closed loop is given by: Several velocity levels ware considered that span the available velocities. For each velocity level j, the correspondent f:j was obtained. The fjj values were substituted in place of fm in the transfer function above and so, several linearized dynamics were considered. n this way, for a given value of Kp, there is a root locus due to the changes in velocity. For instance, the root locus due to the velocity evolution fop Kp = 1.5 Y!m) and K, = 0 is seen on Figure (12). For small velociues, the low frequency poles are real and the system is more damped, while for higher velocities, the poles become complex and the systems is consequently less damped. For each gain value Kp, there will be one mt locus specified due to the changes in velocity. Analysis through the poles behavior may be useful in the elaboration of gain scheduling control laws, as is the case of the gain scheduling proportional function of the velocity Kp = Kp (be) to obtain a constant damping rate (t)(see Figure (13)). Once there is a linearized dynamic for each velocity level, it is possible to obtain the corresponding gain values Kp that produce a constant damping rate. The points (KP, ie) were approximated by a curve that has an hyperbolic form. Figure (9) shows a simulation of a tracking with this control law. This simulation was repeated for two different situations: for a load inertia 50% greater than the nominal value; for the dry friction 25% greater than the nominal value. n both cases, the response stayed almost the same of Figure (9). This indicate that this control has a good robustness to the changes in parameters. Due to problems with time real implementation (it is nbt possible to implement a simple function Kp = Kp ), there are no experimental results. For the step reference input, it is necessary to saturate the input torque given by this control law, to avoid overshoot problems due to a great value of the torque at the beginning of the angular movement. Another application is the non-linear friction compensation from the gain scheduling K,, on the form: Ku = fm - f: (10) where fm is the viscous friction coefficient of the desirable linear dynamic. At this time, the experimental results are not available for the same reason of the gain scheduling proportional, but the friction compensation from equation (10) may have a good performance, depending mainly upon the noise level of the velocity signal. 6 Conclusions The work presented had threemain objectives: to obtain a motor-driven joint dynamic model and test it against experimental results; use the friction model to show the possibility of building up compensation mechanisms; exhibit the non-linear dynamics through the analysis of the poles behavior, that make the design of gain scheduling control laws possible. These goals were achieved in the following way: i- a good linearization level was obtained with the compensation of the velocity-dependent fiiction torque component, but m m efficient compensation mechanisms can be obtained, and this is a subject for further research. The approaches that may provide better practical results are: use of gain scheduling control (equation lo), or a research of a friction model more appropriate to digital implementations. ii- from the control view point, the position-dependent friction torque component does not cause problems, especially when rigid loads are used. The velocitydependent component is really more important, because it is the main some of dynamic non-linearity. iii- the proportional control law used with the fiiction compensation might not be satisfactory for applications in a which a great precision level are required, because a steady state error remains (table 2). This error can be eliminated with the help of an integral control component activated around the steady state. iv- the gain scheduling proportional function of the velocity has a good robustness to the changes in parameters, and may be one interesting alternative to the integral control. The question of the flexible mode remaining on the simulations as well as on the experimental results after compensation of the dry friction (Figure (14)), which is due to the in-joint elasticity, has not been considered in the study. but the joint model becomes linear after compensation and the use of classical control design techniques should allow to derive a satisfactory solution for this problem. 1432

5 Figures and lsbles la- V~scourfriclion lb- Covlomb or dry fricfwn a '. 0' model rdls(ms) meaaurcmcnt Figure 3 Modelled and measured fxiction torque witboil friction compent:tion [rd/t(mt)] b. with friction rompentation [rd/t(mt)l Figure 1 The principals friction torque models. 6,.--. ~ _ load potition dcg(1t) 8.O load potition deg(lt) 8. Figure 4 Rotor velocity & a function of load angular position, with and without friction compensation loid position deg(1s) t Table 1 Maximum position simulation errors obtained from experimental results. Rcrarmccinput stcrdyrtltcamr a Table 2 Steady state errors obtained using the compensatron block. time: 6.0 experimental result - experimental rc~ult aimulation result rimulation rcrult Figure 5 Simulation and experimental results (open loop). Drive joint f Tenninallwd Rigid ann Figure 2 The one degree of freedom manipulator of use for experimental tests. Figure 6 Block diagram of the proportional control with friction compensation. 1433

6 40.0 load position and reference input deg(1s) \ Figure 7 The different torque regions load position and reference input deg(1s) time a 8.b Figure 8 Experimental step responses with compensation block. ) time a CO Figure 11 Experimental ste responses with compensation block and K, Jifferent from zero osition control deg(1s) K-X-CK*W***N J Fi ure 12 The root locus due to vefocity variation load position reference input Figure 9 Simulation of a tracking with the gain scheduling Kp = Kp, for the case = time (s) Figure 13 The root locus due to velocity variation with K~ = K~ (i,). 1434

7 250.0 rotor velocity rd/s(ms) input motor torque NM(ls) b position control deg(1s) experimental reiult iimulatian rciult 8.b time (s) baiile experimental rciult fcrcncc input time ( roult baiile Figure 14 Simulation and experimental results. Bibliography [l] Gougoussis A. and Donath M. Coulomb friction joint and drive effects in robot mechanisms. EEE Trans. Robotics and Automation. 2:82&836, [2] Sabot J., Bouchareb A. and Briere Ph. Pertubations dynamiques induites par les dducteurs utilisb en mbotique. n Theory of Machines and Mechanisms Proceedings of the 7th World Congress, volume 2, pages , September [3] Armstrong-Helouvry B. Stick-slip arising from stribeck friction. EEE Trans. Robotics and Automation, pages , May [4] Armstrong B.S.R. Dynamics for robot control: fiiction modeling and ensuring excitation during parameter idenz@ation. PhD thesis, Stantford University - California, [5] Canudas de Wit C. and Seront V. Robust adaptive friction compensation. EEE Trans. Robotics and Automation, pages , May [6] Canudas de Wit C., Noel P., Aubin A., Brogliato B. and Drevet P. Adaptative friction compensation in mbot manipulators: low velocities. EEE Trans. Robotics and Automation, pages , May [7] Dupont P. E. Friction modeling in dynamic robot simulation. EEE Trans. Robotics and Automation, pages , May [8] Dupont P. E. Avoiding stick-slip in position and force control through feedback. EEE Trans. Robotics and Automation, pages , April [9] Kubo T., Anwar G. and Tomizuka M. Application of nonlinear friction compensation to robot arm control. EEE Trans. Robotics and Automation, , [lo] Canudas C., Astrom K. J. and Braun K. Adaptative friction compensation in dc-motors drives. EEE Trans. Robotics and Automation, RA-3, December [ll] Steenman G. J. J. Modelling and simulation of a manipulator joint. National Aerospace Labratory, NLR TR U, Amsterdam, [12] Chrt5tien J. P. SECAEUX : an experimental set-up for the study of active control of flexible structures. n American Control Conference, volume 2, pages , Pittsburgh, Pa, June [13] Townsend W. T. The effect of the coulomb friction and stiction on the force control. EEE Trans. Robotics and Automatwn, Gilbert J. W. and Winston G. C. Adaptative compensation for an optical tracking telescope. Automatica, ,