Singular mass matrix and redundant constraints in unilaterally constrained Lagrangian and Hamiltonian systems

Abstract

This article deals with the analysis of the contact complementarity problem for Lagrangian systems subjected to unilateral constraints, and with a singular mass matrix and redundant constraints. Previous results by the authors on existence and uniqueness of solutions of some classes of variational inequalities are used to characterize the well-posedness of the contact problem. Criteria involving conditions on the tangent cone and the constraints gradient are given. It is shown that the proposed criteria easily extend to the case where the system is also subjected to a set of bilateral holonomic constraints, in addition to the unilateral ones. In the second part, it is shown how basic convex analysis may be used to show the equivalence between the Lagrangian and the Hamiltonian formalisms when the mass matrix is singular.

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

“…Further quadratic programs obtained for the unilaterally constrained case may be found in [9, §4.4.1], using Dorn's duality. They summarize various results (including (11) and (15)), some of which were obtained by Moreau [43,44] and Lötstedt [38]. It is worth noting that (14) is written in a more general setting in [22,Eq.…”

“…The next proposition gathers results from Moreau [43,44], Lötstedt [38], Pang and Trinkle [49], and Brogliato and Goeleven [9,11]. [19].…”

“…This in fact was proved by Moreau [43,44] using another reasoning. The case where M(q) has low rank r < n may be tackled via the tools in [3]; however, this is outside the scope of this article; see [11]. It is nevertheless worth noting that the formulations in (14) or (4) do not a priori rely on the assumption that M(q) 0, and this will be the case of all their extensions in the sequel.…”

“…Surprizingly enough, it is only recently that the analysis of this case has received attention [9,11], though it may represent the most common case in practice [50]. Let us write the bilateral constraints on the acceleration level.…”

“…The contact problem with mixed (bilateral and unilateral) constraints has received surprisingly less attention. Its analysis in the frictionless case with redundant constraints and a singular mass matrix is given in [11], whereas in the frictional sliding case, it is established in [27] that it becomes a mixed linear complementarity problem but no explicit condition for its solvability is given.…”