Kinetic quasi-velocities in unilaterally constrained Lagrangian mechanics with impacts and friction

Abstract

Quasi-velocities computed with the kinetic metric of a Lagrangian system are introduced, and the quasi-Lagrange equations are derived with and without friction. This is shown to be very well suited to systems subject to unilateral constraints (hence varying topology) and impacts. Energetical consistency of a generalized kinematic impact law is carefully studied, both in the frictionless and the frictional cases. Some results concerning the existence and uniqueness of solutions to the so-called contact linear complementarity problem, when friction is present, are provided.

The contact problem in Lagrangian systems with redundant frictional bilateral and unilateral constraints and singular mass matrix. The all-sticking contacts problem

“…This theorem can be restated equivalently as: v ∈ Q M ⇒ q ∈ Q ⋆ M . The next corollary is proved in [18], and is a consequence of results in [23].…”

Feedback control of multibody systems with joint clearance and dynamic backlash: a tutorial

“…where proj M (q(t)) is the projection in the kinetic matrix metric. This law has limited prediction capabilities for certain multiple (simultaneous) impacts [61,Chapter 3], and can be improved while remaining in a kinematic setting, to allow for different coefficients of restitution at different impact points, and also tangential effects [17]. The advantage of Moreau's impact law is that it is kinematically, kinetically and energetically consistent for all e n ∈ [0, 1] [36].…”

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

“…Let us also notice that a necessary condition for the MLCP in (16) to possess at least one solution is that Φ is nonempty. This is guaranteed by Assumption 1.…”

“…The Lagrange multipliers associated with the bilateral constraints are denoted as μ ∈ R p . Therefore, the right-hand side of the dynamics in (1) becomes equal to ∇h(q)λ + ∇f (q)μ and the MLCP in (16) …”

“…Transformations applied to systems with holonomic constraints usually aim at decoupling the dynamics into a "tangent" (independent of the contact force) and "normal" (contact force dependent) dynamics, using various annihilators of ∇h(q) [15,16,27,29]. Systems with a singular mass matrix are descriptor variable systems, or implicit systems in the quasi-linear form G(ẋ, x, t, λ) = E(x)ẋ + H (x, t, λ), where…”