Piecewise-differentiable trajectory outcomes in mechanical systems subject to unilateral constraints

TL;DR

This paper establishes conditions for piecewise-differentiable dependence of trajectory outcomes on initial conditions in mechanical systems with unilateral constraints, extending previous differentiability results to variable activation sequences.

Contribution

It introduces new conditions ensuring piecewise-differentiability of trajectories despite changing constraint activation sequences, advancing the understanding of system stability and controllability.

Findings

Provides conditions for piecewise-differentiability of trajectories

Extends differentiability results to variable constraint sequences

Discusses implications for stability and controllability

Abstract

We provide conditions under which trajectory outcomes in mechanical systems subject to unilateral constraints depend piecewise-differentiably on initial conditions, even as the sequence of constraint activations and deactivations varies. This builds on prior work that provided conditions ensuring existence, uniqueness, and continuity of trajectory outcomes, and extends previous differentiability results that applied only to fixed constraint (de)activation sequences. We discuss extensions of our result and implications for assessing stability and controllability.