Phase Uncertainty to State Stability of Continuous Periodic Orbits

The paper shows sufficiency conditions for stability of continuous periodic orbits under phase uncertainty. Phase based uncertainty is a trait of bipedal walking robots, where the desired trajectories are parameterized by a monotonous function. This monotonous function, called the phase variable, is often affected by intermittent perturbations due to noisy sensors. We will mainly focus on continuous periodic orbits obtained via parameterized trajectories, and then analyze their stability properties under a noisy phase estimation. In other words, our focus is on examples where phase variables are difficult to compute, and therefore are imperfect. We will show that stable periodic orbits subject to phase based uncertainty are input to state stable.