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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'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.

Stability of rider-controlled bicycles is related to the appropriate coupling between leaning and steering. Inspired by this driving experience, we propose a linear control law between steer and lean angles to balance motion of an unmanned bicycle. The selection of the control parameters is based on stability analysis for the controlled bicycle dynamics, in which the control law and the constant rear-wheel velocity ω0 serve as servo-constraints imposed to the bicycle system. To facilitate the stability analysis of the controlled motion, we use tools from geometric mechanics to reduce the bicycle's dynamics, leading to a two-dimensional dynamic system. We thereafter study its relative equilibria and related stability. Theoretical results show that there is a critical value of ωc, above which the bicycle can move in a stable uniform straight forward motion, and below which a pair of relative equilibria related to stable uniform circular motion exist. The driving rule of steering toward a fall (STF) and counter-steering (CST) can also be explained theoretically. Finally, we fabricate a powered autonomous bicycle and propose an error correction algorithm to eliminate the drift error of the gyroscope sensor. Our experimental results are in good agreement with the theoretical predictions. Generally, this paper presents the mathematical basis for designing the more advanced and intelligent autonomous bicycles, and may find potential applications in other vehicle systems.

This paper presents experimental results to verify the reduced dynamics modeling and stability analysis of an autonomous bicycle, following the theoretical framework presented in Part I. A self-fabricated autonomous bicycle is introduced, and experiments are conducted to verify its stability in uniform straight motion (USM) and uniform circular motion (UCM). While the experimental results are qualitatively consistent with the theoretical findings, they also reveal that the stability of the bicycle is affected by uncertainties in the real system, primarily stemming from the drift error of the gyroscope sensor used to measure the bicycle's lean angle. Parameter uncertainty analysis confirms the gyroscope drift as the main source of uncertainty. To compensate for this drift error, a modified linear control law with a time-varying intercept term is proposed. This results in a three-dimensional reduced dynamic system governing the bicycle's controlled motion. Theoretical analysis and experiments demonstrate that the bicycle under the modified linear control law achieves stable USM and UCM with a desired steer angle, and exhibits an interesting phenomenon of a limit cycle motion when the control parameters are set such that the trivial relative equilibrium becomes unstable through a supercritical Hopf bifurcation.