Quasi-steady state approximation of the dynamics of a planar motorcycle

Modelling the dynamics of a vehicle is the starting point for any control and optimization strategy. Generally speaking, as the complexity of a model grows, possible instability and nontrivial dynamic coupling make the exploration of vehicle trajectories a rather challenging task. The aim of this paper is to present a way to simplify a detailed model into a simpler one, the trajectories of which being close to those of the original system. For the sake of presentation, we describe a planar motorcycle model including front and rear suspensions and chain drive. The motorcycle model is composed of six rigid bodies and its equations of motion are derived using a Lagrangian approach. A suitable chain model is described. In particular, we provide a method that avoids explicit inclusion of the dynamics of the drive sprocket. This motorcycle model is then simplified allowing us to compute an estimate of the tire forces and the fast suspension dynamics. Dynamic simplification is achieved by imposing holonomic constraints at the wheels and quasi-steady state conditions on the suspensions. A simple example, a mass-wheel-suspension system, is discussed, comparing the dynamic simplification to that obtained using singular perturbation theory. A comparison between the complete dynamics and the simplified quasi-steady state model is reported by means of simulation.