Necessary and sufficient conditions for passivity of the LuGre friction model

Friction is a nonlinear phenomenon difficult to describe analytically. To capture its effect in mechanical systems, a bristle-based dynamical model, known as the LuGre model, was proposed in the literature. It is difficult to assess whether this (or any other) mathematical model constitutes a bona fide friction model. It should, however, reflect the dissipative nature of friction, which mathematically translates into the requirement of defining a passive operator from velocity to friction force. We give necessary and sufficient conditions for this property to hold for the LuGre model. The conditions are expressed in terms of a simple algebraic inequality involving the parameters of the model. If this inequality does not hold, we construct an input signal that generates a periodic orbit along which the passivity inequality is violated.