The variational discretization of the constrained higher-order Lagrange-Poincar\'e equations

TL;DR

This paper develops a variational discretization method for higher-order constrained Lagrange-Poincaré equations in mechanical systems with symmetries, enabling numerical integration and optimal control applications.

Contribution

It introduces a discrete connection approach to derive discrete constrained higher-order equations, extending variational integrators to complex symmetric systems.

Findings

Derivation of discrete higher-order Lagrange-Poincaré equations

Establishment of a well-defined local flow for numerical integration

Application to optimal control problems in underactuated systems

Abstract

In this paper we investigate a variational discretization for the class of mechanical systems in presence of symmetries described by the action of a Lie group which reduces the phase space to a (non-trivial) principal bundle. By introducing a discrete connection we are able to obtain the discrete constrained higher-order Lagrange-Poincar\'e equations. These equations describe the dynamics of a constrained Lagrangian system when the Lagrangian function and the constraints depend on higher-order derivatives such as the acceleration, jerk or jounces. The equations, under some mild regularity conditions, determine a well defined (local) flow which can be used to define a numerical scheme to integrate the constrained higher-order Lagrange-Poincar\'e equations. Optimal control problems for underactuated mechanical systems can be viewed as higher-order constrained variational problems. We study how a variational discretization can be used in the construction of variational integrators for optimal control of underactuated mechanical systems where control inputs act soley on the base manifold of a principal bundle (the shape space). Examples include the energy minimum control of an electron in a magnetic field and two coupled rigid bodies attached at a common center of mass.