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

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. The 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

A Foot Placement Strategy for Robust Bipedal Gait Control

This thesis introduces a new measure of balance for bipedal robotics called the foot placement estimator (FPE). To develop this measure, stability first is defined for a simple biped. A proof of the stability of a simple biped in a controls sense is shown to exist using classical methods for nonlinear systems. With the addition of a contact model, an analytical solution is provided to define the bounds of the region of stability. This provides the basis for the FPE which estimates where the biped must step in order to be stable. By using the FPE in combination with a state machine, complete gait cycles are created without any precalculated trajectories. This includes gait initiation and termination. The bipedal model is then advanced to include more realistic mechanical and environmental models and the FPE approach is verified in a dynamic simulation. From these results, a 5-link, point-foot robot is designed and constructed to provide the final validation that the FPE can be used to provide closed-loop gait control. In addition, this approach is shown to demonstrate significant robustness to external disturbances. Finally, the FPE is shown in experimental results to be an unprecedented estimate of where humans place their feet for walking and jumping, and for stepping in response to an external disturbance