Differential Dynamic Programming on Lie Groups: Derivation, Convergence Analysis and Numerical Results

TL;DR

This paper extends Differential Dynamic Programming to systems on Lie groups using a coordinate-free, Lie-theoretic approach, providing convergence analysis and numerical validation on T SO(3).

Contribution

It introduces a Lie-theoretic, coordinate-free DDP framework for systems on Lie groups, with convergence analysis and practical implementation details.

Findings

Algorithm converges under certain conditions

Effective on systems on T SO(3)

Provides a coordinate-free derivation

Abstract

We develop a discrete-time optimal control framework for systems evolving on Lie groups. Our work generalizes the original Differential Dynamic Programming method, by employing a coordinate-free, Lie-theoretic approach for its derivation. A key element lies, specifically, in the use of quadratic expansion schemes for cost functions and dynamics defined on manifolds. The obtained algorithm iteratively optimizes local approximations of the control problem, until reaching a (sub)optimal solution. On the theoretical side, we also study the conditions under which convergence is attained. Details about the behavior and implementation of our method are provided through a simulated example on T SO(3).