Analytical describing function of LuGre friction model

LuGre friction model is a comprehensive model for friction, which can simulate important nonlinear phenomena such as stiction, Stribeck effect, variable break-away force, pre-sliding displacement, and frictional lag. In this paper, an analytical describing function (DF) for this friction model is derived. The theoretical analysis is examined by comparing its result with some existing experimental results. The determined DF can approximate the LuGre friction force with constant viscous damping in a sinusoidal motion. Afterwards, as an application, its effect on the stability of a haptic device (HD) is studied. For this purpose, some simulations in MATLAB software are performed by considering the HD as a one-degree of freedom system with an effective mass that moves with the LuGre friction model, and stability analysis is performed theoretically (using the DF of the LuGre) and compared with the simulation result.