Analysis of a Dry Friction Force Law for the Covariant Optimal Control of Mechanical Systems with Revolute Joints

Juan Antonio Rojas-Quintero (), François Dubois, Hedy César Ramírez- de-Ávila, Eusebio Bugarin, Bruno Sánchez-García and Nohe R. Cazarez-Castro
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Juan Antonio Rojas-Quintero: CONAHCYT—Tecnológico Nacional de México, I. T. Ensenada, Ensenada 22780, BC, Mexico
François Dubois: Laboratoire de Mathématiques d’Orsay, Université Paris-Saclay, 91400 Orsay, France
Hedy César Ramírez- de-Ávila: Tecnológico Nacional de México, I. T. Tijuana, Tijuana 22414, BC, Mexico
Eusebio Bugarin: Tecnológico Nacional de México, I. T. Ensenada, Ensenada 22780, BC, Mexico
Bruno Sánchez-García: Tecnológico Nacional de México, I. T. Tijuana, Tijuana 22414, BC, Mexico
Nohe R. Cazarez-Castro: Tecnológico Nacional de México, I. T. Tijuana, Tijuana 22414, BC, Mexico

Mathematics, 2024, vol. 12, issue 20, 1-26

Abstract: This contribution shows a geometric optimal control procedure to solve the trajectory generation problem for the navigation (generic motion) of mechanical systems with revolute joints. The mechanical system is analyzed as a nonlinear Lagrangian system affected by dry friction at the joint level. Rayleigh’s dissipation function is used to model this dissipative effect of joint-level friction, and regarded as a potential. Rayleigh’s potential is an invariant scalar quantity from which friction forces derive and are represented by a smooth model that approaches the traditional Coulomb’s law in our proposal. For the optimal control procedure, an invariant cost function is formed with the motion equations and a Riemannian metric. The goal is to minimize the consumed energy per unit time of the system. Covariant control equations are obtained by applying Pontryagin’s principle, and time-integrated using a Finite Elements Method-based solver. The obtained solution is an optimal trajectory that is then applied to a mechanical system using a proportional–derivative plus feedforward controller to guarantee the trajectory tracking control problem. Simulations and experiments confirm that including joint-level friction forces at the modeling stage of the optimal control procedure increases performance, compared with scenarios where the friction is not taken into account, or when friction compensation is performed at the feedback level during motion control.

Keywords: Lagrangian systems; optimal control; Rayleigh’s dissipation function; friction force laws; nonlinear mechanical systems; Riemannian geometry; Pontryagin’s maximum principle (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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