Carlos Canudas de Wit. ENSIEG-INPG, BP 46, 38402, ST. Martin d'heres, France. ultrasonic motor. more sophisticated friction model of the form:

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1 CONTROL OF FRICTION-DRIVEN SYSTEMS Carlos Canudas de Wit Laboratoire d'automatique de Grenoble, UMR CNRS 558 ENSIEG-INPG, BP 46, 384, ST. Martin d'heres, France Abstract There exist many examples of friction-driven systems, where the transfer of power between the actuators (master) and the system to be controlled (slave) is done via a friction interface. Examples of such a systems are piezoelectric actuators, and vehicles. This paper is devoted to study the control design for such a systems. The paper rst presents the \kinematic" control where the velocity of the master system is seen as the control input. We then extend the control design to the case where the slave system dynamics is also considered. Finally we also study some robustness issues, concerning the disturbance rejection properties of the slave system. Some simulations are presented. Friction-driven systems, nonlinear sys- Keywords: tems. 1. Introduction There exist many examples of friction-driven systems. Those are systems where the transfer of power between the actuators (master) and the system to be controlled (slave) is done via a friction interface. Contact friction in ties is one well known example. The power is generated by the combustion motor driving the vehicle wheel axis. The tie/road contact friction transfers part of the motor power to the vehicle body producing the requested motion. Other example is the piezoelectric actuators. They can be linear or rotational motors for which the motion is produced via the coupling frictional forces between rotor and stator. Ultrasonic motors are made with piezoceramics whose eciency is size-insensitive. They consist in high-frequency supply (signal input), a vibrator (master), and a slider (slave). The vibrator is composed of a piezoelectric component, that deform under the application of a high-frequency voltage signal. Under certain conditions, this deformation creates a traveling wave on the stator, inducing a rotor motion due to the friction interface, see Fig. 1. In the same way that a surfer will slide on a sea wave. Their area of applications is large. They are used in automatic focusing mechanism, in motor for watch, micro-walking machines, etc. See [8] for a comprehensive treatment on this topic. ROTOR PIEZOELECTRIC VIBRATOR Friction interface Traveling wave Orbital motion of contact surface Figure 1: An example of a friction-driven system: the ultrasonic motor. The majority of the previous works in this area uses classical Coulomb and viscous models to describe the friction interface between the master and the slave system. These models are described as a one-to-one map between the relative (or the sliding) velocity and the contact friction force. In the area of piezoelectric motors, [1] studies the inuence of the contact surface and contact pressure on classical Coulomb model. In [9] the authors propose a more sophisticated friction model of the form: F = = arctan( x) where is the friction coecient and x is the relative displacement of the surface in contact. Similar models are used also in the automotive area to describe tie/road contact friction. These type of models capture part of the spring-like behaviour observed during the small displacement during the sticktion phase (elasto-plastic deformation of the bristles in contact), but it disregard most of the dynamic eects of the friction behaviour (hysteresis). They also fails at capturing the stick-slip behaviour. In this paper we use one of the recently proposed friction models that include most of the desired friction characteristics (stick-slip, Stribeck, Coulomb, viscous and other hysteresis friction eects). The LuGre friction model [4] can be written in a compact form and allows for analytic studies (stability and existence of solution), generally not possible with discontinuous friction models. The paper is devoted to study the control design for such a systems. We rst presents the \kinematic" control where the velocity of the master system is seen as the control input. This design has two purposes. From one hand, it provides a very good insight inhowtode-

2 u J Master - F ω 1 ω F r Slave I u Master ω v~ r ω Friction F - r Slave Figure : The two-wheels friction-driven system. sign the control law for the complete master slave system. From another, it is directly useful in application such as the piezoelectric actuators, where the tangent velocity of the traveling wave (the master's system velocity) can be controlled freely. We then extend this study to the case where the slave system dynamics is also considered. Finally we also look at some robustness issues concerning the disturbance rejection properties of the slave system. Some simulations are presented at the end of the paper.. Models In this paper we consider systems of the form: I _! = ;r F (1) J _! 1 = F + u () _z = ~v ; j~vj z (3) system (1) describes the so called slave system, i.e. the system to be controlled. System () is the the master system, or the driving system, which provides power to the slave system, via the friction interface (3). The system above describes, in particular, two wheels of: inertia J and I, radius and r, and rotational velocity! 1, and!, respectively. The control input is described by u, and the friction torque induced by the punctual contact is given by r i F, see (Fig ). F is the tangent contact friction force modeled by the LuGre model 1 proposed in [4], and having the tangent sliding velocity ~v as its input. F,~v, and are given by: F = z + 1 _z (4) ~v = r! ;! 1 (5) = F C +(F S ; F C )e ;(~v=v) (6) where F C and F S, are the Coulomb, and the break-away friction, satisfying F C <F S. These two parameters depend on the contact pressure (or force) between the two systems. v is the Stribeck velocity. =1=, and 1 are the spring-like, and viscous friction coecient. 1 The LuGre model has been rewritten in its \singular perturbed form". Note that in [4], the variable z is replaced here by z= = z, with small positive. This is to easily understood the motivation of the control design presented here. Figure 3: Block schema of a friction-driven system Remark: Model (3), has the following two notable property: (i) if jz()j <F C,thus jz(t)j <F C 8t, and (ii) 1 >F S g() F C >. In particular Property (i) ensures that the internal friction states are bounded and that its upper bound is given by the Coulomb friction parameter. The gure 3 shows the block schema of the system under consideration. An analogy can also be maid with an electrical circuit by considering torques as a currents, and velocities as a voltages. Each of the three systems, can thus be visualized as an electrical (passive) system. The intermediate system (the friction interface) inject energy from the master to the slave system during the acceleration phases. Inversely, energy is extracted back from the slave to the master system during the breaking phases. The problem under consideration is thus to design a control law for u such that the slave system's velocity! tends to the desired velocity! d. Only the regulation problem will be here considered, i.e. _! d =. 3. Kinematic control The purpose of this analysis is not only to provide a rst insight in how to design a control low for the complete system, but it is also directly applicable for systems such as the piezoelectric actuators, where the master system dynamics is strongly decoupled from the slave one, and the tangent velocity of the traveling wave (master system velocity) has a much more fast dynamics than the angular rotor velocity (angular velocity of the slave system). The relation between the applied voltage (real input) and the tangent velocity of the traveling wave of the stator at the contact point (! 1 ), can be approximate by aninvertible (static) map between these two variables, see [6]. In consequence, the variable (! 1 ), can be seen as the control system input. Consider thus, that! 1 = u is the control input, the system (1)-() simplies to: I _! = ;r F (7) _z = (r! ; u) ; jr! ; uj z (8) with F, and g dened as before.

3 Let e =! ;! d,_e =_!, and dene as = r! ; u then, the error equation (in open-loop) writes as: I _e = ;r (z + 1 _z) (9) _z = ; jj z (1) Based on the observation that the friction dynamics (1) is much more faster than the driven system dynamics (9) because > is small, the \slow dynamics" of system above, can be approximated as, I _e ;r sign() which suggest to dene as, yielding, = ke (11) I _e ;r sign(ke ) From property (ii), this equation, if true, will imply \convergence " in nite-time of e to zero. The parameter r and the function weights the converge-time needed to reach zero. Although this analysis is approximated, the control law obtained in this way can be prove to be globally stable, without any additional approximation. This result is shown next. Theorem 1 Consider the system (7)-(8), with the control law u given as: u = 1 [r! ; ke ] then the error e, and the state z are bounded, and tend asymptotically to zero. Proof: Consider the exact closed-loop error equation resulting from this controller, and the function I _e = ;r (z + 1 _z) (1) _z = ke ; jke j z (13) V = I e + r k z The sense of convergence is here not rigorous. In the best case will mean to reach a neighborhood O() of zero, for some t t 1 >. However, to prove this we require further technicalities since the error equations (1)-(9) are not in the standard singular perturbed form required by the Tikinov's conditions (i.e. analicity of the right hand side of the dierential equations, and exponential stability of the fast sub-system). We will not pursue further this analysis here since an alternative prove is possible without any approximation. from which we get, _V = ;e (z + 1 _z)+ z k (ke ; jke j z) (14) = ;e 1 _z ; je jz = ; 1 e (ke ; jke jz ) ; je jz = ; 1 ke (1 ; sign(e )z ) ; je jz ; 1 ke (1 ; jzj ) ; je jz (15) (16) (17) (18) From properties (i) and (ii), we have that the terms in the parenthesis in Equation (18) will be always positive. Hence this gives: _V ; je jz The LuGre model (13) is locally Lipschitz, and hence the signal e and z are uniformly continuous (see [3]). Implying that V _!. Hence, from the above inequality and property (ii), we have that je jz!. Since both e,and z are also bounded, only two cases may be possible: If e!. Thus from the boundedness and the continuity of the signals, we have that _e!. Therefore from the error equations we have that z + 1 _z!, and that _z!. from which we conclude that z!. If z!, and from similar arguments as before, we also have that _z!. Which from (13), indicates that the only possible value for e is zero. We see thus that in these two cases, both signal e and z tend asymptotically to zero. 4. Master-slave control We consider in this section the dynamics of the full system (1)-(3). The idea (as before) is to make (asymptotically)! 1 to converge to a value such that r! ;! 1 tends to r! ; r! d = r e. The desire value for! 1 is thus! 1 d = r r1!d. For this we introduce: e =! ;! d _e =_! (19) e 1 =! 1 ; r! d _e 1 =_! 1 () Consider then the following simple proportional controller, u = ;ke 1 (1) then, the closed-loop error equations are: I _e = ;r F () _z = (r e ; e 1 ) ; j(r e ; e 1 )j z (3) J _e 1 = F ; ke 1 (4)

4 We consider this time V = I e + J e 1 + z which, after some calculations, gives: _V = ;ke 1 ; j~ejz ; 1~e ~e ; j~ej z ;ke 1 ; j~ejz ; 1~e ;ke 1 ; j~ejz 1 ; jzj (5) (6) (7) with ~e = r e ; e 1. Last inequality holds from the fact that jz(t)j g(~v(t)) 8t. Following the same arguments that in the proof of the previous theorem, we conclude that: e 1!, and that ~ez!. Or, equivalently, that e 1 and e z tends to zero. From the analysis of the closed-loop equations (using continuity of the signal and the convergence of the signals to its equilibria manifolds, we can conclude that all the signals are bounded and that e 1, e, and z converge asymptotically to zero. The following has thus been proved. Theorem Consider the system (1)-(3), with the control law u given as: u = ;k(! 1 ; r! d ) then the errors e 1, e, and the state z are bounded, and tend asymptotically to zero. 5. Robustness issues The previous control law is based on the assumption that no disturbances are present. It is thus possible to set the master system's velocity toavalue such that the slave system reaches the desired set point. If torque disturbances are present, thus the equilibria points will be shifted away from its desired values. It is thus important to redesign the controller such a to cope with this diculty. Torque disturbances occurring on the master system may be easily rejected because they will satisfy the matching condition. We consider thus the more complex case of disturbances appearing only on the slave system. In particular we consider: I _! = ;r F + (8) J _! 1 = F + u (9) _z = ~v ; j~vj z (3) with constant. The idea is to redene (dynamically) the desired value of! 1 as a (slow-varying) function of the errors in the slave system, in order to compensate for the shifting of the equilibria point due to. To this aim, we dene! d 1 as! 1 d = r! d r ; k P 1 k e ; k Z I k and consider the following control law: t e u = ;k(! 1 ;! 1) d (31) = ;k(! 1 ; r Z t! ) d ; k P e ; k I e (3) = ;ke 1 ; k P e ; k I Z t e (33) where e 1 =! 1 ; r r1!d and e =! ;! d, follows the same denition as before. With this controller, the closed-loop equations are: I _e = ;r F + (34) _z = (r e ; e 1 ) ; j(r e ; e 1 )j z (35) J _e 1 = F ; ke 1 ; k P e ; k I^ (36) _^ = e (37) If e 1 tends to some constant c 1,thus F! k 1 c 1 + k P e + k I R e. Therefore the system (34) will tends to: or, equivalent to: I _e + r Z (k P e + k I e )= ; r k c Ie + r (k P _e + k I e )= Due to the integral action, we have that e!. However, this is possible only if the magnitude of disturbance is smaller than the steady-state value of F times r. This condition appears from the equilibrium manifolds of the error equation (34)-(37) given by all the points e 1 e z = F ^, satisfying: ; e 1 ; je 1j g(~v ) z = r (38) r = (39) k I ^ = r ; ke 1 (4) e = (41) From (38) we can see that the stationary friction value should be equal to. Since z(t) (;F C F C ), a necessary r condition for this equilibria to exist is that be smaller r than the level of Coulomb friction, i.e. jj <r F C (4)

5 Figure 4: Time evolution of the slave velocity! (dotted), and its desired reference w d (bold), for the ideal case. Figure 5: Time evolution of the friction force F (t). Ideal case. At this point is important to underline that the solutions of dynamics of the LuGre friction model, allows for stationary values of z in the set (;F C F C ). It is also worth to notice that since g(~v(t)) is lowerbounded by F C, the condition (4) ensures also that the only possible solution for e 1, in Equation (39) is zero. Therefore from (4), we have that^ = Since r1. k I r ~v = r! ;! 1 = r e ; e 1 the stationary value for sliding velocity will be thus ~v =, with a non-zero steady-state friction value. This is not contradictory, it means that the friction model will be operating in the sticktion domain, where the spring-like behaviour dominates. The value of F C is proportional to the normal forces at the contact point (contact pressure) and the friction material coecient <<1. In applications like the ultrasonic motors, the contact pressure can be set arbitrarily. Hence disturbances with relative high values can be rejected. In other applications like invehicles, the normal force is mainly given by product of the vehicle's mass and the gravity vector. Its value, and hence its capacity to reject important disturbances, is thus limited. A formal analysis of the above robust controller will be presented in the nal version of this paper is space is available. Some simulations are next presented. 6. Simulations We present in this section some simulations of the master-slave control, under ideal and non-ideal conditions. Parameters used for simulations are: Master-slave parameters: J =: [Kg=m ], I = :4 [Kg=m ], =:3[m], r =:5 [m]. Friction parameters: F C =3[N], F S =4[N], v = :1[m=sec], =5: [N=m], 1 =:797 [N=m]. Control parameters: k =:1, k P =:5, k I =:5 Disturbance is = 1 for the non-ideal case. The rst set of simulations concern the ideal case with the control law given in Theorem 1, is used. We can observe from Figures 4, and 5 that the slave velocity converge to its desired value, and that the friction forces are settle to zero. The non-ideal case with the control 31 is shown in Figures 6, and 7. We see that the! tends rst to a high desired velocity value. Then the integral action of the extended controller brings back the slave velocity to its desired value. Note also that F tends to N, which corresponds to =r =1=:5. 7. Conclusions We have presented several control designs for frictiondriven systems. Although these systems are widely spread, there has not been previous attempt to study the control aspect related to them. We have introduced the LuGre dynamic friction model in this study, and shown that simple controllers can be used to solve the velocity set-point tracking problem. 8. Acknowledgments Thanks are due to Prof. this problem to the author. Tomizuka for introducing References [1] Bliman, P.A. (199), \Mathematical study of the Dahl's friction model" European Journal of Mechanics, A./Solid., Vol. 11, No6, 199.

6 Figure 6: Time evolution of the slave velocity! (dotted), and its desired reference w d (bold), for the non-ideal case: =1. 6 [] Bliman, P.A., and M. Sorine (1997), \Friction modeling by hysteresis operator. Application to Dahl, sticktion and Stribeck eects" private correspondence. [3] Canudas de Wit (1998), \Comments on: A New Model for Control of Systems with Friction", to appear as in IEEE TAC, [4] Canudas de Wit, C., H. Olsson, K.J. Astrom, and P. Lischinsky, (March, 1995), \A New Model for Control of Systems with Friction", IEEE TAC, Vol. 4, No. 3, pp [5] Canudas de Wit, and P. Lischinsky, (1997), \Adaptive friction compensation with partially known dynamic friction model", International Journal of Adaptive Control and Signal Processing, Vol. 11, pp (1997). [6] Canudas de Wit, and S. Shakhesi, (1998), \Control design for ultrasonic motors with dynamic friction interface", Submitted to the IFAC 1999 word congress, as an invited session on Mechatronic Systems. [7] Hagood, McFarlard, IEEE Trans. on Ultrasonic, ferro-electronics and frequency control, Vol 4, No. pp. 1-3, 199. [8] K. Uchino, \Piezoelectric actuators and ultrasonic motors",kluwer Academy Publishers. [9] Glenn, T.S., Hagood, N.W., \Developmentofatwoside piezoelectric rotary ultrasonic motor for high torque", SPIE, Vol 341, pp , [1] Zharii, O.Y. \Frictional contact between the surface wave and a rigid strip", J. Of Applied Mechanics, Trans of ASME, Vol 63, pp.15-, March Figure 7: Time evolution of the friction force F (t). Nonideal case: =1