Sensitivity analysis for hybrid systems with state-triggered jumps is experiencing renewed attention for the control of robots with intermittent contacts. The basic assumption that enables this type of analysis is that jumps are triggered when the state reaches, transversally, a sufficiently smooth switching surface. In many scenarios of practical relevance, however, this switching surface is just piecewise smooth and, moreover, a perturbation of the initial conditions or the input leads to a different number of jumps than the nominal trajectory's. This work extends the sensitivity analysis in this context, under the assumptions that (i) at least locally, the intermediate perturbation-dependent jumps lead the system to reach always the nominal post-impact mode and (ii) once a switching and corresponding intermediate jump has occurred, its corresponding constraint remains active until reaching the nominal post-impact mode. Numerical simulations complement and validate the theoretical findings.

This work explores the stabilization of desired dynamic motion tasks involving hard impacts at non-negligible speed for humanoid robots. To this end, a so-called reference spreading hybrid control law is designed showing promising results in simulation. The simulations are performed employing a dynamical model of an existing humanoid robot and impacts are assumed to be inelastic. The desired motion task consists of having the robot balancing on one foot while repeatedly making and breaking contact with a wall by means of one hand. The simulation results illustrate that the considered controller is suited to control humanoid robot motions with impacts.