Multibody dynamics with unilateral constraints and friction

Abstract

In the view of robotics or granular media mechanics, we ask for the statement of the dynamical evolution problem for multibody systems with contacts and friction. The "event driven" approach does not allow to state an evolution problem in a systematic way: it gives a peculiar status to the instants of impact and to the contact points although they are unknowns of the problem. This matter of fact has given rise to a new approach in the frictionless case [4,6] we generalize to the case including friction. Following to the point of view of Lagrange on the equation of the dynamics, we emphasize the systematic use of virtual powers and duality. This bias suggests to put emphasis on generalized reaction forces in the statement of the evolution problem and not on local reaction forces of the real world. This point of view allows to escape from pathologies e.g. Painlevé paradox [5] and Kane Paradox [3]. The proposed evolution problem accounts for impacts and friction in case of multiple unilateral constraints. We prove the existence and uniqueness of the solution to the associated rate problem.Until the late 70s, the point of view on the dynamics of multibody system with unilateral constraints was "event driven" and gave a special status to the events of impacts within the formulation. This point of view is neither practical for theoretical investigations nor for numerical computation because the instants of impacts are unknowns of the problem.Schatzman and Moreau [4,6] proposed a global in time formulation of the evolution problem (through the application of the measure theory) which had important important by-products, such as existence and uniqueness results for the solution to the problem, e.g. [1], or the rise of an efficient numerical strategy commonly referred to as "time stepping scheme".In case of dry friction, the archetypal force law is the Coulomb law. Since it is formulated in terms of local forceselements of the Euclidian 3-D space in contrast with generalized forces in R n -such a formulation gives a special status to the contact points. According to multibody system theory, constraints are described through constraint functions which are functions of the configuration of the system and the material points which realize the contact are unknown of the problem. Hence, this point of view requires a contact point survey at every single instant time and is anew an "event driven" formulation.In the paper, we extend the evolution problem point of view from the frictionless case to the case with friction. We expect as future by-products computational methods and mathematical results: today the only system which can be mathematically adressed is the collection of points which for an existence and uniqueness results is proven in [2] whereas counter-examples known as Painlevé paradox show that the use of Coulomb law leads to ill-posed problems.2 A dynamical contact problem with friction formulated in the configuration space.Advocating local reaction forces in the equation of motion requires a preliminary contact poin...