An efficient Newton method for general motorcycle kinematics

Abstract

This paper presents a detailed study of the kinematics of single-track vehicles, with a special emphasis on motorcycles. We consider a general class of tyre profiles as well as general vehicle geometry. We show that the kinematic problem may be reduced to the problem of finding the zero of a (single) nonlinear equation in the pitch angle which may then be solved using a safeguarded Newton method, providing rapid convergence. Special care, enabled by the systematic use of rotation matrices, is taken to understand the range of pitch angles for which all quantities in the equation are well defined. The paper provides a fast and numerically reliable algorithm that can be used within analysis tools such as those involving numerical integration of system dynamics.

Keywords:

• kinematics
• motorcycle
• bicycle
• single-track vehicles
• Newton method

Acknowledgements

We thank Professor Ruggero Frezza for interesting discussions on motorcycle kinematics. This research was supported in part by Ducati Corse and MSC Software.