Control of a Bicycle Using Virtual Holonomic Constraints

TL;DR

This paper presents a method using virtual holonomic constraints to control a bicycle model, enabling it to traverse convex curves with bounded roll angle and speed, and analyzes the resulting dynamics.

Contribution

It introduces a systematic way to generate virtual holonomic constraints for bicycle control and proves properties of the constrained dynamics.

Findings

Bicycle can traverse convex curves with steady-state periodic speed.

VHC can be generated as a solution to a scalar periodic differential equation.

Constrained dynamics are generally not Lagrangian.

Abstract

The paper studies the problem of making Getz's bicycle model traverse a strictly convex Jordan curve with bounded roll angle and bounded speed. The approach to solving this problem is based on the virtual holonomic constraint (VHC) method. Specifically, a VHC is enforced making the roll angle of the bicycle become a function of the bicycle's position along the curve. It is shown that the VHC can be automatically generated as a periodic solution of a scalar periodic differential equation, which we call virtual constraint generator. Finally, it is shown that if the curve is sufficiently long as compared to the height of the bicycle's centre of mass and its wheel base, then the enforcement of a suitable VHC makes the bicycle traverse the curve with a steady-state speed profile which is periodic and independent of initial conditions. An outcome of this work is a proof that the constrained dynamics of a Lagrangian control system subject to a VHC are generally not Lagrangian.