For general hybrid systems, we develop new stabilization results that can be used to achieve asymptotically stable locomotion for bipedal robots with series compliant actuation. The stabilization contributions build upon previous results involving partially rapidly exponentially decaying control Lyapunov functions. Such functions are useful when the dynamics that remain when the function is constrained to zero exhibits an asymptotically stable set and the solutions starting in this set have time domains that satisfy a uniform average dwell-time constraint. In a new result of independent interest, we establish that such an average dwell-time condition is robust; in particular, it degrades gracefully under perturbations and as the initial conditions move away from the asymptotically stable set. From this robustness result and the existence of a partially rapidly exponentially decaying control Lyapunov function, we establish local asymptotic stabilization. The result is then applied to robot locomotion. We conclude by showing that, because of the high-gain nature of the feedback, it is possible in some situations for the basin of attraction to become arbitrarily small as the gain becomes arbitrarily large. Future simulation studies will investigate whether this phenomenon occurs for the robot application.