Symmetries in the wheeled inverted pendulum mechanism
Provides a theoretical framework for simplifying the dynamics of an underactuated nonholonomic system, which may aid in control design, but the results are theoretical and not validated experimentally.
This paper uses geometric mechanics (connections and symmetries) to derive simplified dynamical equations for the Wheeled Inverted Pendulum (Segway), arriving at reduced Euler-Lagrange equations and nonholonomic momentum equations for two symmetry groups.
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.