Optimal Regulators in Geometric Robotics

TL;DR

This paper develops an intrinsic geometric framework for optimal control in robotic systems on Riemannian manifolds, extending classical control methods and applying them to rigid body attitude regulation.

Contribution

It introduces existence results for optimal control using Riemannian geometry and extends the linear quadratic regulator to systems on manifolds.

Findings

Derived a Riemannian PMP formulation.

Extended LQR to Riemannian manifolds with Riccati equations.

Applied theory to rigid body attitude regulation.

Abstract

The aim of this paper is to give some existence results of optimal control of robotic systems with a Riemannian geometric view, and derive a formulation of the PMP using the intrinsic geometry of the configuration space. Applying this result to some special cases will give the results of avoidance problems on Riemannian manifolds developed by A. Bloch et al. We derive a formulation of the dynamic programming approach and apply it to the quadratic costs and extend the linear quadratic regulator to robotic systems on Riemannian manifolds and giving an equivalent Riccati equation. We give an optimisation aspect of the Riemannian tracking regulator of F. Bullo and R.M. Murray. Finally, wee apply the theoretical developments to the regulation and tracking of a rigid body attitude.