Adaptive Compensator of Single State Elastoplastic Friction Model

A nonlinear friction is an unavoidable phenomenon frequently experienced in mechanical system between two contact surfaces. An adaptive compensator is designed to achieve tracking of a desired velocity trajectory in the presence of friction force described by a single state elastoplastic friction model. The adaptive compensator includes an adaptive observer and a computed force controller. The closed loop system is shown to be stable using Lyapunov second method. Simulation results show the effectiveness of the proposed compensator.


Introduction
Friction occurs in all mechanical systems, eg.bearings, transmission, hydraulic and pneumatic cylinders, valves, brakes and wheels.In many engineering applications, frictional contact occurs between machine parts and the characterization of contact behavior becomes an important subject in solving tribological problems such as friction induced vibration wear.Issues related to mechanical sealing, performance and life of machine elements, and thermal are few examples.The pioneering work of Greenwood and Williamson (1966) has been utilized by many researchers (Ibrahim and Rivin 1994;Mulhearn and Samuels 1962;Abdo andFarhang 2005, Seabra andBerthe 1987) as a basic for further extension to obtain contact models for general or specific contact problems for mainly elastic contact.On the other hand, the work of Pullan and Williamson (1972) utilized as a basic model for pure plastic contact.In an attempt to bridge the gap between the pure elastic and pure _______________________________________ *Corresponding author's e-mail: jdabdo@squ.edu.omplastic contact, researchers Halling and Nuri (1972), Greenwood andTripp (1971), andAbdo andAl-Yahmadi (2004) developed a wide intermediate range of interest where elastic-plastic contact triumph.
In many cases the classical friction model cannot capture the characteristics such as Stribeck effect, stick-slip, pre-sliding hysteretic motion, break-away force, which play a significant role in application on high precision positioning control (Abdo et al 2010, Abdo andAl-Yahmadi 2009).The role of friction modeling can be categorized according to whether or not the friction compensation is model-based.Examples of non-model-based compensators include high-gain feedback, and impulsive control.
This paper contains four sections including the introduction.Section 2 reviews the single state elastoplastic friction model given by Dupont et al. (2002).Section 3 presents an adaptive compensator to achieve velocity tracking in the presence of the friction force described by the single state elastoplastic friction model and a proof of it is also given in this section.
Since the single state elastoplastic friction model contains a bounded function to describe different friction phases and depends on the immeasurable elastic state, the proposed compensator uses an adaptive observer and a computed force controller.Stability analysis of the proposed adaptive compensator is carried out using Lyapunov second method.Simulation results are given in Section 4 and conclusion is drawn in Section 5.

Single State Elastoplastic Friction Model
In this section we review the single state elastoplastic friction model given by Dupont et al. (2002), based on which we design an adaptive velocity compensator to follow a desired velocity trajectory.The rigid body displacement x is composed of elastic (z) and plastic (w) components as, Friction models define the elastic dynamics explicitly, while the plastic displacement w is defined implicitly.Following Dupont et al. (2002) model, the friction force is given by: (2) (3) where z ss is defined as (4) and f ss (x) is the steady state friction force also called the Stribeck function which is shown in Fig. 1.The Stribeck function f ss (x) is bounded from below and above as ( 5) and (6) (7) Figure 2 shows a typical shape of m (z, m) as given in (Dupont et al. 2002).

Adaptive Compensator
In this section an adaptive velocity compensator is designed to achieve tracking of a desired velocity trajectory v d (t) in the presence of friction force that was described by Eqs.(2 and 3).A similar approach was proposed by Canudas et al. (1995) but with a simple friction model and constant m (z, x).Consider the equation of motion of a single mass m subject to driving force u and friction force fl  and in matrix form Eq. ( 9) can be given as ( 10) The only available signal for measurement is the velocity v of the rigid body, hence, expressing y = v in terms of the state x leads to

Adaptive Observer
The adaptive observer is given as ( 14) Depending on Kaman-Yakubovich (KY) Lemma there exist a positive definite symmetric matrices P = P T > 0 and Q = Q T > 0, such that

Controller
The controller u is chosen as > 0 is an adaptation gain.

Stability Analysis
The close loop system given by Eqs. ( 9), ( 14),( 16) and ( 17) is analyzed in this section utilizing the Lyapunov Second Method.Consider hence (16) From the selected gain L discussed in section 3.1 to make the transfer function c(sI -A + Lc) -1 b SPR there exist a positive definite symmetric P and Q which satisfies Eq. ( 13).
Substituting for the control u into (3.1)we obtain

Simulation Results
The desired velocity function is chosen as v d (t) = 10 sin (10t) this is a standard test signal.The elastoplastic friction model parameters are given in Table 1.
The Stribeck friction function is given by: The gains of the observer and controller are given in Table 2.
It is obvious from Figures 4-7 that the proposed adaptive compensator provides an accurate tracking capability to the desired trajectory while keeping all other signals bounded.

Conclusions
In this paper a single state elastoplastic friction model is reviewed and an adaptive compensator is proposed to track a desired velocity trajectory in the presence of friction force model.The adaptive compensator includes an adaptive observer which consists of a state estimator and an adaptation law to one of the unknown functions of the friction model.The controller cancels the nonlinearities in the friction model using the estimated state and parameter.The model stability is determined by using Lyapunov analysis.Simulation of the proposed adaptive compensator to track both sinusoidal and square wave signals show almost perfect tracking and signals are bounded in the closed loop system which indicate the proposed compensator is effective.
the velocity of the mass (v = x).

Figure 1 .
Figure 1.Stribeck curve of steady state friction force f ss (x) versus rigid body velocity Real (PR) function A strictly proper rational function G(s) is positive real (PR) if G(s) is analytic for Re [s] 0 and Re [G(j ) > 0 for al -< .Definition 2 Strictly Positive Real (PR) function A strictly proper rational function G(s) is strictly positive real (SPR) if G (s-) is positive real for some real > 0. The Kaman-Yakubovich (KY) Lemma (Narendera and Annaswamy 1989): A strictly proper rational function G(s) with state space realization (A,b,c) G(s) = c (SI -A) -1 is strictly positive real (SPR) if there exists positive definite symmetric matrices P=P T >0 and Q=Q T >0, such that PA + A T P = -Q (13)

Figure 6 .
Figure 6.Rigid body velocity v and desired square wave velocity v d