Parameter to state stability of control Lyapunov functions for hybrid system models of robots

Model based controllers, by virtue of their dependence on a specific model, are highly sensitive to imperfections in model parameter estimation leading to undesirable behaviors, especially in robots that undergo impacts. With the goal of quantifying the effect of model imperfection on the resulting output behavior from a control Lyapunov function (CLF) based controller, we formally derive a measure for model parameter mismatch and show that a bounded measure leads to an ultimate bound on the CLF. This is also extended to the discrete map by introducing an impact measure. The measure is controller and path dependent, and not just parameter dependent, thereby differentiating it from existing methods. More specifically, if traditional methods yield ultimate boundedness for a bounded parameter uncertainty, the proposed “measure” uses the notion of input to state stability (ISS) criterion to establish stability of model based controllers. The main result of this paper establishes that the proposed CLF based controller is parameter to state stable (PSS) for a class of robotic hybrid systems-systems with impulsive effects. These formal results motivate the construction of a robust controller-combining a computed torque term with a traditional PD term-that yields stricter convergence rates and bounds on the errors. This is demonstrated on the bipedal robot AMBER with a modeling error 30%, wherein the stability of the proposed controller is verified in simulation.