422 Asian Journal of Control, Vol. 9, No. 4, pp. 422-425, December 2007 Manuscript received July 5, 2005; revised July 19, 2006; accepted January 2, 2007. R. Guerra and L. Acho are with the CITEDI-IPN, Otay Mesa, San Diego, CA 92154, USA (e-mail: rguerra{leonardo} @citedi.mx). －Brief Paper－ ADAPTIVE FRICTION COMPENSATION FOR TRACKING CONTROL OF MECHANISMS R. Guerra and L. Acho ABSTRACT Because friction is a phenomenon that is present in the vast ma- jority of mechanical systems producing some unwanted effects such as tracking errors, limit cycles, and stick-slip motion, friction model based compensation has been previously proposed. We present a sim- ple adaptive friction compensator, developed from a simple friction model, that achieves the control objective (friction compensation). This simple model was effectively used to obtain a friction compen- sator with smooth terms avoiding the use of signum and absolute functions presented in previously reported works on friction compen- sation. Considering that the velocity is bound away from zero and us- ing Lyapunov stability analysis, exponential stability of the closed loop system is shown; i.e., the tracking errors and the parameter esti- mation error converge exponentially to zero. Because our friction compensator is based on a simple friction model, numerical experi- ments using a more representative friction model are given to support our theoretical findings. KeyWords: Friction, adaptive compensation, non-linear systems. I. INTRODUCTION Friction is highly nonlinear and poses a considerable challenge for compensation, this phenomenon is present, in some degree, in almost all mechanical systems. This is the reason for it being extensively studied (see [1-7], among many others). Friction modeling has also been a topic studied by many authors ([6-14], just to name a few). For the most part it causes unwanted effects such as tracking errors, limit cycles, and stick-slip motion, that degrade the performance of the controlled system. Therefore, to achieve trajectory and velocity tracking more closely, it is necessary to consider it when designing a control scheme. Compensator design based on friction model has been pre- sented in [3,4,6,7,15]. From the compensator point of view, a simple friction model is preferred (see [3] and [4]). At present, there is no universal friction model, one of the most popular models, the LuGre model [6], is widely used because it recreates the main effects of friction, but from a compensator design point of view, it is too complex be- cause it requires six parameters to recreate a single force. For this reason, and inspired by the works presented by [3] and [4], we propose a simple friction model to obtain a simple compensator which has smooth terms avoiding the use of signum and absolute functions traditionally present in friction compensation (see, for instance, [3] and [4]). With the assumption as in [3], that the velocity is bound away from zero (this is a restriction that can be seen as a compromise between analytical proof and structural sim- plicity) and using Lyapunov stability analysis, exponential stability of the closed loop system is shown; i.e., the track- ing errors and the parameter estimation error converge ex- ponentially to zero. Also, simulation experiments using the LuGre friction model in the system and our compensator developed using a simple friction model are presented to support our main result. The rest of this paper is distributed as follows: Section 2 presents a brief review of the compensator design pre- sented by [4]. Section 3 presents our compensator design.