Adaptive Friction Compensation of Servo Mechanisms with Nonlinear Viscous Friction

In this paper, a SDOF system with nonlinear viscous friction under harmonic excitation is studied. A nonlinear viscous friction model is built as an exponential expansion of the relative slip velocity. According to the adaptive control theory, an adaptive controller for the friction compensation of the SDOF system with nonlinear viscous friction is put forward by using the model-based parameterization techniques. The good performance of the proposed control scheme is illustrated through the computer simulation. INTRODUCTION Friction compensation is an important research topic in the field of motion control of mechanical systems with friction. There have been many works on model-based friction compensation [1,2]. The accuracy of the fixed friction compensation is always affected by the parameter estimation of friction model. So the application of fixed friction compensation is limited by its shortcomings. The adaptive friction compensation has been an important branch among the many model based friction compensation methods [3,4]. The principal advantage of adaptive friction compensation relative to the fixed compensation lies in its ability to track changes in friction. Viscous friction is the most popular ingredient in friction model utilized for control of mechanical systems. From the research works on friction compensation, the adaptive friction compensation methods are all concentrated on the linear viscous friction, in which the viscous friction is considered as a linear function of velocity. While the friction compensation about nonlinear viscous friction in mechanical systems is seldom considered at present. In this paper the adaptive friction compensation theory is introduced to the mechanical system with nonlinear viscous friction based on the present research works on the nonlinear viscous friction. The adaptive friction compensation controller of mechanical system with nonlinear viscous friction is presented. This paper is organized as follows: In Section 1 the nonlinear viscous friction model is introduced. In Section 2 the adaptive friction compensation for nonlinear viscous friction is put forward. Simulation example is used to demonstrate the performance of the addressed method. Conclusions are drawn in the last. NOMENCLATURE Adaptation error = ) 3 , 2 , 1 ( ~ = i i θ Adaptation gain =γ Adaptive parameter = ) 3 , 2 , 1 ( ˆ = i i θ Applied force input = f Arbitrary large numbers = q n, Coefficients = μ c cv , Coulomb friction = h Desired position = d x Differential gain = d k Elastic element = ) (x Fs Exciting signal = ) (t p Mass = m Nonlinear viscous friction = ) (x Fd & Observer gain =κ Position = x Position error = e Proportional gain = p k Stiffness = k METHODS 1 Nonlinear viscous friction model 1 Copyright ©2005 by ASME The nonlinear viscous friction model has been put forward in paper [5,6]. The nonlinear viscous friction model is a function of exponential expansion of velocity. The viscous friction coefficients are identified using the energy balance method. The nonlinear viscous friction model in paper [5] is introduced in this paper. The differential equation of motion for the SDOF system with nonlinear viscous friction can be written ) ( ) , ( t p x x F x m = + & & & (1) where, , = v 1 , , is the nonlinear viscous friction model, is purely elastic element., is the exiting signal,The Eq.(1) can be rewritten ) ( ) ( ) , ( x F x F x x F s d + = & ∑ + = n v v d x c x h x F ) sgn( ) ( & & &