Reduced Dynamics and Control for an Autonomous Bicycle

TL;DR

This paper develops a simplified nonlinear model of an autonomous bicycle's dynamics, analyzes its stability under proportional steering control, and validates findings through simulations and experiments.

Contribution

It introduces a reduced nonlinear model based on Gibbs-Appell equations and provides stability analysis with experimental verification.

Findings

Stable straight and circular motions depend on steer coefficient and angular velocity.

A critical angular velocity exists for stability of straight motion.

Experimental results confirm theoretical stability conditions.

Abstract

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.