A well-known problem in the application of the Interconnection and Damping Assignment technique for the stabilization of underactuated mechanical systems is dissipation in unactuated coordinates, since it may impede the definiteness requirements for the closed-loop system. Recently, the expansion of the closed-loop Hamiltonian function by a cross term between coordinates and momenta has been explored showing promising results. However, the large number of free parameters is an issue for the tuning of the closed-loop system, and the solution of the matching partial differential equations (PDEs) remains a difficult task. In this work, we aim at giving the closed-loop augmented Hamiltonian more structure in order to simplify the controller parametrization. The result is desired behavior at the equilibrium avoiding the solution of PDEs. Simulations and experiments demonstrate the applicability of the method.

This paper develops the equations of motion in the reduced space for the wheeled inverted pendulum, which is an underactuated mechanical system subject to nonholonomic constraints. The equations are derived from the Lagrange-d'Alembert principle using variations consistent with the constraints. The equations are first derived in the shape space, and then, a coordinate transformation is performed to get the equations of motion in more suitable coordinates for the purpose of control.

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) × S 1 . The first group leads to the purely kinematic case while the second gives rise to nonholonomic momentum equations.

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