In this paper a new approach for the formulation of the friction forces velocity function is introduced. The scope of this formulation is to facilitate the implementation of control laws for systems where friction forces appear. The friction model includes the exponential decay part, the Coulomb and viscous friction. The introduced formulation is based on the observation that the friction coefficient function of velocity can be presented as the solution of a linear differential equation. Due to this linearity, the parameters of the derived differential equation can be estimated easily by an adaptive system. The estimation of these parameters is equivalent to the estimation of the friction coefficient in the full range of operational velocities. This knowledge gives to the designed control systems the potential to avoid successfully the stick-slip phenomenon.

A control law for one D.O.F. system, where friction appears, is designed in order to prove the applicability of the proposed formulation of the friction model in control systems. A MRAC adaptive algorithm estimates the differential friction model parameters, using the measured friction force, while a sliding controller adjusts the motion of the mechanical system. The proposed friction model can be used in any control system where friction forces have to be compensated. The linear form of the model is suitable for common adaptive estimators. Therefore, the proposed structure is suitable for robotic applications, such as assembly, deburring, etc.