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This paper introduces a rigid motorcycle model that captures many important aspects of real motorcycle dynamics including sliding and load transfer. The model is used to demonstrate a dynamic inversion procedure which maps a desired flatland trajectory into a corresponding (state-input) trajectory for the rigid motorcycle model. This inverse trajectory is the solution of an optimal control problem that is computed using the projection operator approach for the optimization of trajectory functionals, a recently developed optimization technique. The effectiveness of the proposed strategy is illustrated using a trajectory computation for a realistic path that is traversed with a demanding speed profile. The rigid motorcycle model detailed in this paper is also of interest as a nontrivial example of a mechanical system with nonideal holonomic constraints.