Inexact trajectory planning and inverse problems in the Hamilton--Pontryagin framework

TL;DR

This paper develops a geometric framework for higher-order trajectory planning problems involving Lie group actions, providing new insights into Euler-Lagrange equations, conservation laws, and numerical schemes with applications in computational anatomy and quantum control.

Contribution

It introduces a Hamilton-Pontryagin variational formulation for higher-order problems, offering a new interpretation of node equations and a structure-preserving numerical integration scheme.

Findings

Derived a geometrically insightful form of Euler-Lagrange equations.

Reinterpreted node equations as Legendre-Ostrogradsky momenta with conservation properties.

Presented a numerical scheme that preserves geometric features of the continuous problem.

Abstract

We study a trajectory-planning problem whose solution path evolves by means of a Lie group action and passes near a designated set of target positions at particular times. This is a higher-order variational problem in optimal control, motivated by potential applications in computational anatomy and quantum control. Reduction by symmetry in such problems naturally summons methods from Lie group theory and Riemannian geometry. A geometrically illuminating form of the Euler-Lagrange equations is obtained from a higher-order Hamilton-Pontryagin variational formulation. In this context, the previously known node equations are recovered with a new interpretation as Legendre-Ostrogradsky momenta possessing certain conservation properties. Three example applications are discussed as well as a numerical integration scheme that follows naturally from the Hamilton-Pontryagin principle and preserves the geometric properties of the continuous-time solution.