Symbolic derivation of nonlinear benchmark bicycle dynamics with toroidal wheels

In this paper we present a nonlinear benchmark bicycle dynamics with toroidal wheels. Using a symbolic mathematic tool, we obtain two holonomic and four nonholonomic constraint equations due to front and rear wheels of the bicycle. We show that the two holonomic constraints cannot be expressed in quartic form, hence no analytic solution for the body pitch angle, for given steer and roll angles unless the minor (crown) radius of the torus are the same for both the front and rear wheels. Applying a standard procedure for developing dynamics in robotics, we construct constrained Euler-Lagrange equations for the 4 connected rigid bodies in the Whipple-Cornell-Delft benchmark bicycle. We eliminate Lagrange multipliers, auxiliary coordinates and obtain a set of three coupled nonlinear ordinary differential equations, corresponding to the rear body roll, steer and front wheel rotation motions. The coefficients of the equations are shown to depend only on rear body roll and steer angles. Finally, we prove the dynamic equation for the benchmark bicycle is the same as that of an underactuated manipulator, which has been studied extensively in robotics.