  Sensitivity analysis of multibody systems computes the derivatives of general
cost functions that depend on the system solution with respect to parameters or
initial conditions. This work develops adjoint sensitivity analysis for hybrid
multibody dynamic systems. Hybrid systems are characterized by trajectories
that are piecewise continuous in time, with finitely-many discontinuities being
caused by events such as elastic/inelastic impacts or sudden changes in
constraints. The corresponding direct and adjoint sensitivity variables are
also discontinuous at the time of events. The framework discussed herein uses a
jump sensitivity matrix to relate the jump conditions for the direct and
adjoint sensitivities before and after the time event, and provides analytical
jump equations for the adjoint variables. The theoretical framework for
sensitivities for hybrid systems is validated on a five-bar mechanism with
non-smooth contacts.
