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

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.