Symmetries in the wheeled inverted pendulum mechanism

TL;DR

This paper explores the role of symmetries and connections in the dynamics of the Wheeled Inverted Pendulum, simplifying equations for control design by analyzing nonholonomic constraints and conserved quantities.

Contribution

It introduces a geometric framework using Ehresmann connections and symmetry groups to derive simplified dynamical equations and identify conserved quantities in the WIP system.

Findings

Derived reduced Euler-Lagrange equations for WIP

Identified nonholonomic momenta and their relation to symmetries

Compared two symmetry groups: SE(2) and SE(2) x S^1

Abstract

The purpose of this article is to illustrate the role of connections and symmetries in the Wheeled Inverted Pendulum (WIP) mechanism - an underactuated system with rolling constraints - popularized commercially as the Segway, and thereby arrive at a set of simpler dynamical equations that could serve as the starting point for more complex feedback control designs. The first part of the article views the nonholonomic constraints enforced by the rolling assumption as defining an Ehresmann connection on a fiber bundle. The resulting equations are the reduced Euler-Lagrange equations, which are identical to the Lagrange d'Alembert equations of motion. In the second part we explore conserved quantities, in particular, nonholonomic momenta. To do so, we first introduce the notion of a symmetry group, whose action leaves both the Lagrangian and distribution invariant. We examine two symmetry groups - $SE (2)$ and $SE(2) \times \mathbb{S}^{1}$. The first group leads to the purely kinematic case while the second gives rise to nonholonomic momentum equations.