Lagrange-Poincar\'e reduction for optimal control of underactuated mechanical systems

TL;DR

This paper extends Lagrangian reduction techniques to higher-order constrained systems with symmetries, deriving equations and applying them to optimal control problems like underactuated robots.

Contribution

It introduces higher-order Lagrange-Poincaré equations and operators, enabling analysis of complex mechanical systems with symmetry and constraints.

Findings

Derived higher-order Lagrange-Poincaré equations from classical reduction.

Introduced higher-order Lagrange-Poincaré operator for system characterization.

Applied framework to optimal control of underactuated systems like Elroy's Beanie and snakeboard.

Abstract

We deal with regular Lagrangian constrained systems which are invariant under the action of a symmetry group. Fixing a connection on the higher-order principal bundle where the Lagrangian and the (independent) constraints are defined, the higher-order Lagrange-Poincar\'e equations of classical mechanical systems with higher-order constraints are obtained from classical Lagrangian reduction. Higher-order Lagrange-Poincar\'e operator is introduced to characterize higher-order Lagrange-Poincar\'e equations. Interesting applications are derived as, for instance, the optimal control of an underactuated Elroy's Beanie and a snakeboard seens as an optimization problem with higher-order constraints.