Energy Shaping of Mechanical Systems via Control Lyapunov Functions with Applications to Bipedal Locomotion

摘要

This dissertation presents a method which attempts to improve the stability properties of periodic orbits in hybrid dynamical systems by shaping the energy. By taking advantage of conservation of energy and the existence of invariant level sets of a conserved quantity of energy corresponding to periodic orbits, energy shaping drives a system to a desired level set. This energy shaping method is similar to existing methods but improves upon them by utilizing control Lyapunov functions, allowing for formal results on stability. The main theoretical result, Theorem 1, states that, given an exponentially-stable limit cycle in a hybrid dynamical system, application of the presented energy shaping controller results in a closed-loop system which is exponentially stable.\n\nThe method can be applied to a wide class of problems including bipedal locomotion; because the optimization problem can be formulated as a quadratic program operating on a convex set, existing methods can be used to rapidly obtain the optimal solution. As illustrated through numerical simulations, this method turns out to be useful in practice, taking an existing behavior which corresponds to a periodic orbit of a hybrid system, such as steady state locomotion, and providing an improvement in convergence properties and robustness with respect to perturbations in initial conditions without destabilizing the behavior. The method is even shown to work on complex multi-domain hybrid systems; an example is provided of bipedal locomotion for a robot with non-trivial foot contact which results in a multi-phase gait.