Optimal control of robotic systems and biased Riemannian splines

¹ Instituto de Matemática, Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil
² Robotics and Mechanical Engineering, Oregon State University, Corvallis, OR, USA

^* Corresponding author: alejandro@matematica.ufrj.br; Ross.Hatton@oregonstate.edu

Received: 17 November 2023
Accepted: 16 March 2024

Abstract

In this paper, we study mechanical optimal control problems on a given Riemannian manifold (Q, g) in which the cost is defined by a general cometric g̃. This investigation is motivated by our studies in robotics, in which we observed that the mathematically natural choice of cometric g̃ = g^* – the dual of g – does not always capture the true cost of the motion. We then, first, discuss how to encode the system’s torque-based actuators configuration into a cometric g̃. Second, we provide and prove our main theorem, which characterizes the optimal solutions of the problem associated to general triples (Q, g, g̃) in terms of a 4th order differential equation. We also identify a tensor appearing in this equation as the geometric source of “biasing” of the solutions away from ordinary Riemannian splines and geodesics for (Q, g). Finally, we provide illustrative examples and practical demonstration of the biased splines as providing the true optimizers in a concrete robotics system.