The contact problem in Lagrangian systems subject to bilateral and unilateral constraints, with or without sliding Coulomb's friction: A tutorial

Alejandro Blumentals (1) , Bernard Brogliato (1) , Florence Bertails-Descoubes (1)

1 BIPOP - Modelling, Simulation, Control and Optimization of Non-Smooth Dynamical Systems

This work deals with the existence and uniqueness of the acceleration and contact forces for Lagrangian systems subject to bilateral and/or unilateral constraints with or without sliding Coulomb’s friction. Sliding friction is known to yield singularities in the system, such as Painlevé’s paradox. Our work aims at providing sufficient conditions on the parameters of the system so that singularities are avoided (i.e., the contact problem is at least solvable). To this end, the frictional problem is treated as a perturbation of the frictionless case. We provide explicit criteria, in the form of calculable upper bounds on the friction coefficients, under which the frictional contact problem is guaranteed to remain well-posed. Complementarity problems, variational inequalities, quadratic programs and inclusions in normal cones are central tools.

Mots clés: KKT conditions; Lagrangian systems; Unilateral constraint; Bilateral constraint; Complementarity problem; Coulomb’s friction; Gauss’ principle

Domaines: Mécanique [physics.med-ph]; Mécanique des solides [physics.class-ph]; Optimisation et contrôle [math.OC]

Alejandro Blumentals, Bernard Brogliato, Florence Bertails-Descoubes. The contact problem in Lagrangian systems subject to bilateral and unilateral constraints, with or without sliding Coulomb's friction: A tutorial. Multibody System Dynamics, 2016, 38 (1), pp.43-76. ⟨10.1007/s11044-016-9527-6⟩. ⟨hal-01096394v2⟩