Variational integrators for underactuated mechanical control systems with symmetries

TL;DR

This paper develops variational integrators for underactuated mechanical systems with symmetries, preserving geometric properties and handling higher-order derivatives, advancing numerical methods for optimal control in complex mechanical systems.

Contribution

It introduces a variational formalism and constructs geometric integrators specifically for higher-order underactuated systems on trivial principal bundles, including constrained cases.

Findings

Preservation of symplectic structure and momentum in integrators

Effective handling of higher-order derivatives in control systems

Extension to constrained and underactuated systems

Abstract

Optimal control problems for underactuated mechanical systems can be seen as a higher-order variational problem subject to higher-order constraints (that is, when the Lagrangian function and the constraints depend on higher-order derivatives such as the acceleration, jerk or jounces). In this paper we discuss the variational formalism for the class of underactuated mechanical control systems when the configuration space is a trivial principal bundle and the construction of variational integrators for such mechanical control systems. An interesting family of geometric integrators can be defined using discretizations of the Hamilton's principle of critical action. This family of geometric integrators is called variational integrators, being one of their main properties the preservation of geometric features as the symplecticity, momentum preservation and good behavior of the energy. We construct variational integrators for higher-order mechanical systems on trivial principal bundles and their extension for higher-order constrained systems and we devote special attention to the particular case of underactuated mechanical systems