SUMMARY

Controllers that are functions of the states and the model of the system, are highly sensitive to imperfections in real world implementation leading to undesirable behaviors, especially in robots that undergo impacts. These imperfections are due to factors like inadequate sensing, inaccurate parameter estimation, saturations in control input, unmodeled disturbances and so on. The main focal point of this proposal is the stabilization of these imperfect nonlinear hybrid systems via Control Lyapunov Functions (CLFs); specifically applying to bipedal walking robots. We are going to consider two factors in the proposal, that are very frequently observed in bipedal walking robots---model parameter uncertainty and imperfect sensing, and realize a set of robust (CLF) based controllers to stabilize these imperfect systems via the notion of Input to State Stability (ISS). For parameter uncertainty, we first derive a nominal CLF based controller that stabilizes a known model, and then by using the principles of ISS derive a robust CLF based controller---combining a computed torque term with a traditional PD term---that stabilizes the uncertain model. The goal is to show that the robust CLF based controller renders the system Input to State Stable (ISS) w.r.t. a ``measure" input, where the measure to evaluate the parameter uncertainty is obtained formally. If traditional methods yield ultimate boundedness for a bounded parameter uncertainty, the proposed measure establishes Parameter to State Stability (PSS) of the hybrid system. This is demonstrated on the bipedal robot AMBER with a modeling error 30%, wherein the stability of the proposed controller is verified in simulation. In a similar manner, to address the problem of implementing state-based parameterized trajectories on complex robotic systems, we first derive a state based CLF that renders the ideal system stable, and then derive a robust controller---time dependent CLF---and show that the resulting hybrid system is Time to State Stable (TSS). These results are verified on the bipedal humanoid robot DURUS both in simulation and experimentally; it is demonstrated that a close match between time based tracking and state based tracking can be achieved as long as there is a close match between the time and phase based desired output trajectories.