Pontryagin calculus in Riemannian geometry

TL;DR

This paper applies Pontryagin's calculus to Riemannian geometry in robotics, deriving optimal control equations that incorporate the Riemann curvature tensor to improve understanding of mechanical system control.

Contribution

It introduces a novel application of Pontryagin's framework to Riemannian configuration spaces in robotics, explicitly involving the Riemann curvature tensor.

Findings

Derived covariant form of optimal control equations

Explicit use of Riemann curvature tensor in control laws

Enhanced understanding of control in curved configuration spaces

Abstract

In this contribution, we study systems with a finite number of degrees of freedom as in robotics. A key idea is to consider the mass tensor associated to the kinetic energy as a metric in a Riemannian configuration space. We apply Pontryagin's framework to derive an optimal evolution of the control forces and torques applied to the mechanical system. This equation under covariant form uses explicitly the Riemann curvature tensor. This contribution is dedicated to the memory of Claude Vall{\'e}e (1945-2014).