Symbolic derivation of nonlinear benchmark bicycle dynamics with holonomic and nonholonomic constraints

We present a symbolic method for modeling nonlinear multibody underactuated systems with holonomic and nonholonomic constraints. Using MAPLE software, we are able to solve the quartic holonomic constraint analytically. We then use the constraints and extra Lagrange-Euler equations to systematically eliminate all the auxiliary coordinates and Lagrange multipliers, thereby obtaining a minimum set of unconstrained nonlinear analytic ordinary differential equations corresponding to the degrees of freedom of the system. The method is applied to a benchmark bicycle, in which all the six ground contact constraint equations are eliminated, leaving analytic coupled ordinary differential equations corresponding to the bicycle rear body roll, steer angle, and rear wheel rotation degrees of freedom without any approximation. This reduced analytic model offers insights in understanding complex nonlinear bicycle dynamic behaviors and enables the development of an efficient model suitable for real time control outside of the linear regime.