Modification of the Leuven Integrated Friction Model Structure

Transcription

1 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 4, APRIL Modification of the Leuven Integrated Friction Model Structure Vincent Lampaert, Jan Swevers, and Farid Al-Bender Abstract This note presents a modification of the integrated friction model structure proposed by Swevers et al., called the Leuven model. The Leuven model structure allows accurate modeling both in the presliding and the sliding regimes without the use of a switching function. The model incorporates a hysteresis function with nonlocal memory and arbitrary transition curves. This note presents two modifications of the Leuven model. A first modification overcomes a recently detected shortcoming of the original Leuven model: a discontinuity in the friction force which occurs during certain transitions in presliding. A second modification, using the general Maxwell slip model to implement the hysteresis force, eliminates the problem of stack overflow, which can occur with the implementation of the hysteresis force described in [10]. Index Terms Friction, mechatronics, motion control, nonlinear systems. I. INTRODUCTION Friction in mechanical systems is a nonlinear phenomenon which can cause control problems such as static errors, limit cycles and stickslip. In order to design controllers for highly accurate machines friction has to be taken into account. Friction is a result of extremely complex interactions between the surface and the near surface regions of the two interacting materials and other substances present such as lubricants. Detailed analysis of friction experiments reveals two friction regimes: the presliding regime and the sliding regime. In the presliding regime the adhesive forces (at asperity contacts) are dominant such that the friction force appears to be a function of displacement rather than velocity (e.g., the Dahl model [4] describes the friction behavior as a function of the position). This is so because the asperity junctions deform elasto-plastically, thus behaving as nonlinear springs. As the displacement increases more and more junctions will break resulting eventually in working in the sliding regime. In the sliding regime all the asperity junctions are broken apart such that the friction force is a function of the velocity (as described in classical friction models [1]). Accurate modeling and control of mechanical systems with friction requires a model which includes both regimes. Armstrong et al. [2] derive a general model structure which includes several experimentally observed friction properties. The model uses a switching function between the two regimes. Such a switching function is physically not justified and may result in implementation problems. In order to overcome this problem, Canudas de Wit et al. [3] reformulated the model into a integrated friction model, known as LuGre model, i.e., a set of equations integrating the sliding and presliding regime without the need of a switching function. Swevers et al. propose a more elaborated model, called the Leuven model, which includes the friction properties of the LuGre model and a more accurate Manuscript received December 6, 2000; revised May 24, 2001 and November 5, Recommended by Associate Editor M. Reyhanoglu. This work presents research results of the Belgian Programme on Interuniversity Poles of Attraction by the Belgian State, Prime Minister s Office, Science Policy Programming. It was supported by Katholieke Universiteit Leuven s Concerted Research Action under Grant GOA/99/04. The scientific responsibility is assumed by the authors. The authors are with the Mechanical Engineering Department, Katholieke Universiteit Leuven, B3001 Heverlee, Belgium ( Vincent.Lampaert@mech.kuleuven.ac.be). Publisher Item Identifier S (02) modeling of the presliding regime using a hysteresis function with nonlocal memory. This paper gives some modifications on the Leuven model: i) it tackles a recently detected shortcoming, viz. a discontinuity in the friction force upon closing a hysteresis loop and ii) proposes a more appropriate implementation of the hysteresis force based on the General Maxwell Slip model which eliminates the problem of stack overflow. Section II describes in more detail the Leuven model. Section III describes the first modification of the new integrated friction model to overcome the discontinuity in the friction force and Section IV discusses the implementation of the hysteresis force. II. THE LEUVEN MODEL The integrated friction model by Swevers et al. [10] is a more elaborated friction model than the LuGre model proposed by Canudas de Wit et al. [3]. It consists of two equations: a friction force equation (1) and a nonlinear state equation (2). The model uses a state variable z which can be envisaged as the average deflection of the asperity junctions dt =v 1 0 sgn F d (z) s(v) 0 F b F d (z) s(v) 0 F b F f =F h (z) v: (2) dt v is the current velocity, n is a coefficient used to shape the transition curves and s(v) is a function that models the constant velocity behavior. s(v) is given by s(v) =sgn(v) F c +(F s 0 F c ) e 0(jvj=V ) : (3) F h (z) is the hysteresis force, i.e., the part of the friction force exhibiting hysteresis behavior with state variable z as input. The hysteresis force is a static nonlinearity with a nonlocal memory. This means that: a new transition curve (i.e., a branch of the hysteresis curve) is initiated at velocity reversal; the shape of the curves is determined by the past extremum values of F h, i.e., the shape is independent of the particular manner of variation of z between the extremum points; the value of F h after time t 0 depends not only on the value of F h at time t 0 and the values of z after t 0, but also on the past extremum values of F h. The implementation of the hysteresis force consists of two parts F h (z) =F b + F d (z): The value of the F h (z) at the beginning of a transition curve (i.e., at velocity reversal) is represented by F b. The transition curve which is active at a certain time is represented by F d (z). F d (z) is a point-symmetrical strictly increasing function of z. The implementation of F h (z), as is presented in [10], requires two memory stacks: one for the minima of F h in ascending order (stack min) and one for the maxima of F h in descending order (stack max). The stacks grow at velocity reversals and shrink when internal loops are closed. The value of F b, F d (z) and z are calculated according to the following rules. At velocity reversal a new transition curve is started; the state variable z and F d (z) are reset, the former value of F h (z) is placed on the stack max = min in the case of going from positive negative to negative positive velocity and becomes the new value of F b. n (1) /02$ IEEE

2 684 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 4, APRIL 2002 Fig. 1. (Top left) Position of a mass as a function of time. (Top right) Friction force as a function of time. (Bottom left) Friction force as a function of the position of the mass. (Bottom right) Microviscous term ( (=dt)) as a function of time. Closing an internal loop, called wiping out; the last values on the stacks min and max associated with this internal loop are removed from the stack. The new value of F b is the top value of the stack min = max for positive/negative velocity. The value of z and F d (z) are recalculated such that a transition curve starts at the new value of F b and the value of F h (z) equals the former value. Working on a transition curve; (1) yields the new value of z and F d (z). The value of F b remains the same. This model can account accurately for experimentally obtained friction characteristics: Stribeck effect in sliding, friction lag, varying break-away, stick-slip behavior and hysteretic behavior in presliding. This last property cannot be modeled with the LuGre model. III. FIRST MODIFICATION OF THE LEUVEN MODEL A. Adaptation of the State Equation Fig. 1 shows a shortcoming in the Leuven model. The figure gives the simulation results when a force is applied on a mass and on the mass acts a friction force simulated with the Leuven model. The top left figure shows the position of a mass as a function of time and the top right figure shows the friction force as a function of the time. The bottom left figure shows the friction force as a function of the position of the mass. The bottom right figure shows the microviscous term ( 1(=dt)) as a function of time. At point a the friction force is discontinuous, due to the discontinuity of the microviscous term 1(=dt). The problem lies in closing an internal loop of the hysteresis function. Closing an internal loop resets the value of z and F d (z) to zero and sets the value of F b equal to the value of F h (z) such that F h (z) =F b + F d (z) does not change. This however yields discontinuities of F d (z), F b, the argument of the nonlinear state equation F d (z)=(s(v) 0 F b ) and consequently also of =dt. Since the friction force F f depends on =dt it will also be discontinuous, which is physically not possible. Starting a new internal loop (at point b in Fig. 1) results also in a discontinuity in F d (z)=(s(v)0 F b ). However, =dt and F f will be continuous due to the fact that the velocity v is zero at that moment (see (1)). To overcome the discontinuity in the friction force, the argument of the nonlinear state equation F d (z)=(s(v) 0 F b (z)) is changed to F h (z)=s(v). With this argument, the new nonlinear state equation becomes dt = v 1 0 n sgn F h (z) F h (z) : (4) s(v) s(v) The variables =dt and F f will now be continuous functions. This modification does not change the two steady-state behaviors of the Leuven model discussed in [10]. In the case of constant velocity different from zero and in steady state (=dt =0), the state equation becomes: yielding dt =0 s(v) =F h (z): The friction equation reduces to F f = s(v)+ 2 v: The function s(v) determines the constant velocity characteristics in the sliding regime at low velocity while 2v becomes significant at high velocity.

3 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 4, APRIL In the case of zero velocity behavior, which corresponds to steady state in presliding, the friction force reduces to dt =0 F f =F h (z): These conclusions are the same as with the nonlinear state equation in [5] and [10]. B. Relation With the LuGre Model The nonlinear state equation is after the modification of the same form as the state equation of the LuGre model. Setting F d (z) equal to 0z and n equal to 1, simplifies the Leuven model to the LuGre model. The modified model adds more flexibility to simulate the behavior in presliding regime, which results in more accurate results. IV. SECOND MODIFICATION OF THE MODEL A. The Maxwell Slip Model of the Hysteresis Force In [10], the hysteresis force F h (z) is implemented using two stacks min and max. Another well known method, related to the method used by Swevers, et al. for describing hysteresis functions is the Preisach model [9]. Both methods are cumbersome to implement in real-time systems because the size of the stack must be chosen in advance, which can cause stack overflow if many loops are initiated. Another implementation of the hysteresis function is used in [6], [11]. The idea is to put N elastoslide elements in parallel. Each element i has one common input z and one output F i and each element is characterized by its own maximum force W i, linear spring-constant k i and state variable i (see Fig. 2). The state variable i describes the position of element i. The elements have no mass, yielding a static relationship between the force F i and the relative displacement (z 0 i) for each element. The relationship is shown in Fig. 3 and can be described by if jz 0 ij < W i k i else then F i = k i (z 0 i ) i = const. F i = sgn (z 0 i) Wi i = z 0 sgn (z W 0 i) k : The hysteresis force is equal to the sum of hysteresis forces (F i )of each element N F h = F i : (5) i=1 To understand the behavior of the Maxwell slip model, consider the following simulation. Suppose that the model is in its initial state (the values of i equal the value of z) which corresponds to zero deflection of the asperity junctions. When a small force is applied to the model all the elements are sticking and the initial stiffness will be the sum of all the element stiffnesses 6k i. When the force F i on element i reaches the saturation level W i, then element i starts to slip and a new value for i must be calculated. The total stiffness of the whole system decreases with the stiffness of the spring of element i. The whole system is in stick when at least one element is still in stick, the whole system is in slip when all the elements slip. This construction of a superposition of many elasto-slide elements was initially formulated by J. C. Maxwell [8] and in the limit as the number of elasto-slide elements become infinite, the model is referred to as Generalized Maxwell slip which is equivalent to the two other representations. Fig. 2. Fig. 3. Maxwell Slip model for N elements. The characteristic of an element. B. Considerations of the Maxwell Slip Implementation Fig. 4 shows the deflection of the asperities z as a function of time and Fig. 5 shows the hysteretic force F h as a function of the deflection z. The top figure shows the implementation described by Swevers, et al., and the bottom figure shows the Maxwell slip implementation, where 10 elasto-slide elements are used to approximate the real transition curve. The figures show an identical hysteresis behavior and the presence of nonlocal memory in the hysteresis function for both implementations. Point a can be reached by several different ways, which is typical for systems with nonlocal memory. The only disadvantage of using the Maxwell slip implementation is its piecewise approximation of the hysteresis function. This disadvantage must however be put into perspective: in the real-time implementation of Swevers et al. the transition curve F d, which depends only on the value of z, is chosen as a connection of piecewise linear functions resulting just like the Maxwell slip model in a piecewise approximation. As a consequence of working with a piecewise approximation, it may happen that the working range stays within the range of the first linear piece for very small displacements resulting in no energy dissipation, as in a real hysteresis function. However, the microviscous term in the friction force equation imposes dissipation of energy and prevents limit cycle problems due to the approximation. Choosing the appropriate values W i and k i for each element in order to approximate the real hysteresis is a nonlinear problem. A better method is choosing the appropriate values W i and 1 i for each element, with 1 i equal to W i =k i. 1 i corresponds to the maximum deflection of the spring before the element begins to slip. By using these new parameters the model equations can be rewritten in the following form: N F h (k) = W i8i (z(k);i(k); 1i(k)) i=1

4 686 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 4, APRIL 2002 Fig. 4. The deflection variable z as a function of time. Fig. 5. The hysteresis force as a function of the deflection variable z. Top figure shows implementation by Swevers et al. Bottom figure shows Maxwell slip implementation using ten elements.

5 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 4, APRIL with 8 i(1) a nonlinear function. Pre-assigning the values of 1 i results in a nonlinear equation which is linear in the unknown parameters W i. These parameters can be determined using a least-squares method [7]. The chosen values for 1 i are equally spaced within the range of deflection of the asperity (the maximum value of z). The more elements used, the more accurate the approximation will be, but on the other hand the computational complexity is proportional to the number of elements used. The mimimum 1 i -value is limited by the noise on the measurement results. If the 1 i -value is smaller than the noise level, the model will create inner loops due to the noise and not to the change in deflection. The advantage of the Maxwell slip implementation is the elimination of the stack overflow problem. Looking at the free response of a mass-spring system with limited friction from an initial state which does not correspond to an equilibrium, the position and the state variable z will have a lightly damped oscillating behavior, resulting in several velocity reversals without closing of inner loops, causing the addition of maximum and minimum values of F h (z) on the stacks min and max. When the maximum lengths of the stacks are limited this can lead to the problem of stack overflow. The Maxwell slip method uses only a fixed number of memory places equal to the number of elements used in the implementation. In the Maxwell slip implementation, the initial curve of the hysteresis behavior is implicitly taken into account in the equations. For the implementation described in [10], working on the initial curve and reentering the initial curve needs an extra implementation for those two cases. [10] J. Swevers, F. Al-Bender, C. Ganseman, and T. Prajogo, An integrated friction model structure with improved presliding behavior for accurate friction compensation, IEEE Trans. Automat. Contr., vol. 45, pp , Apr [11] M. Versteyhe, Development of an Ultra-Stiff Piezostepper With Nanometer Resolution, Ph.D. dissertation, Division PMA, Dept. of Mechanical Engineering, Katholieke Universiteit Leuven, Leuven, Belgium, A Limit to the Capability of Feedback Yanxia Zhang and Lei Guo Abstract Feedback is ubiquitous and is a basic concept in the area of control, where it is used primarily for reducing internal or external uncertainties, or both. In this note, we will study the capability of feedback in dealing with both internal and external uncertainties for a class of th order nonlinear autoregressive control systems. The size of the uncertainty is described by the Lipschitz constant (say ) of the uncertain nonlinear function in consideration. It is shown that if and satisfy a certain inequality, then there exists no globally stabilizing feedback for the corresponding class of uncertain systems, and thus finding a quantitative limit to the capability of the feedback mechanism in dealing with structural uncertainties. Index Terms Feedback, nonlinear, stability, uncertainity. V. CONCLUSION This paper briefly discusses the integrated friction model structure, called the Leuven model, and proposes two improvements to this model. The first modification reformulates the nonlinear state equation in order to obtain always a continuous friction force. The second modification solves the problem of stack overflow, which may occur with the implementation method of the hysteresis force proposed in [10]. The General Maxwell slip model is a better way to implement the hysteresis force. Even with a limited number of elements, it is possible to approximate the hysteresis force accurately. REFERENCES [1] B. Armstrong-Hélouvry, Control of Machines With Friction. Boston, MA: Kluwer, [2] B. Armstrong-Hélouvry, P. Dupont, and C. Canudas de Wit, A survey of models, analysis tools and compensation methods for the control of machines with friction, Automatica, vol. 30, no. 7, pp , [3] C. Canudas de Wit, H. Olsson, K. Aström, and P. Lischinsky, A new model for control of systems with friction, IEEE Trans. Automat. Contr., vol. 40, pp , May [4] P. R. Dahl, A Solid Friction Model, The Aerospace Corporation, El Segundo, CA, Tech. Rep. TOR-158( ), [5] C. Ganseman, Dynamic Modeling and Identification of Mechanisms With Application to Industrial Robots, Ph.D. dissertation, Division PMA, Dept. of Mechanical Engineering, K.U. Leuven, Leuven, Belgium, [6] M. Goldfarb and N. Celanovic, A lumped parameter electromechanical model for describing the nonlinear behavior of piezoelectric actuators, Trans. ASME, J. Dyna. Syst., Measure. Control, vol. 119, pp , Sept [7] V. Lampaert and J. Swevers, On-line identification of hysteresis functions with nonlocal memory, in Proc. Int. Conf. Advanced Intelligent Mechatronics, July 2001, pp [8] B. J. Lazan, Damping of Materials and Members in Structural Mechanics. London, U.K.: Pergamom, [9] I. D. Mayergoyz, Mathematical Models of Hysteresis. New York: Springer-Verlag, I. INTRODUCTION Feedback is ubiquitous and is a basic concept in automatic control. Its primary objective is to reduce the effects of the plant uncertainty on the desired control performance. The uncertainty of a plant includes both internal (structure) uncertainty and external (disturbance) uncertainty and, in general, the former is harder to cope with than the latter. How to design efficient feedback laws to cope with various plant uncertainties has been a key issue in the development of automatic control [1] [3]. However, from a philosophical point of view, there must be some limits to the capability of feedback in dealing with uncertainties. Finding such limits is of fundamental importance, since control scientists may not waste their time on constructing control laws for systems with uncertainties which are already beyond the capability of control, while control engineers may be more cautious (or confident) when applying their new control methods (robust, adaptive, intelligent, etc.) bravely to complex systems which practically contain many uncertainties. Unfortunately, the question about the limits of feedback is a conundrum and on which only a few existing areas of control theory can shed some light. Robust control and adaptive control are two such areas where structural uncertainty of the plant is the main concern in the controller design. Robust control usually requires that the true plant lies in a (small) ball centered at a known nominal model and often assumes that the controllers are selected from certain given classes of systems [4]. The need of a nominal model, with reliable model error bounds in robust Manuscript received June 1, 2001; revised December 6, Recommended by Associate Editor L. Y. Wang. This work was supported by the National Natural Science Foundation of China. The authors are with the Institute of Systems Science, Chinese Academy of Sciences, Beijing , P.R. China ( Lguo@control.iss.ac.cn). Publisher Item Identifier S (02) /02$ IEEE