  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's controlled motion. Two types of equilibrium points for
the governing equation are found, which correspond to the bicycle'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'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.
