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In this paper, we propose the reduced model for the full dynamics of a bicycle and analyze its nonlinear behavior under a proportional control law for steering. Based on the Gibbs-Appell equations for the Whipple bicycle, we obtain a second-order nonlinear ordinary differential equation (ODE) that governs the bicycle&#x2019;s controlled motion. Two types of equilibrium points for the governing equation are found, which correspond to the bicycle&#x2019;s uniform straight forward and circular motions, respectively. By applying the Hurwitz criterion to the linearized equation, we find that the steer coefficient must be negative, consistent with the human&#x2019;s intuition of turning toward a fall. Under this condition, a critical angular velocity of the rear wheel exists, above which the uniform straight forward motion is stable, and slightly below which a pair of symmetrical stable uniform circular motions will occur. These theoretical findings are verified by both numerical simulations and experiments performed on a powered autonomous bicycle.

Reduced Dynamics and Control for an Autonomous Bicycle