  This study investigates formal-method-based trajectory optimization (TO) for
bipedal locomotion, focusing on scenarios where the robot encounters external
perturbations at unforeseen times. Our key research question centers around the
assurance of task specification correctness and the maximization of
specification robustness for a bipedal robot in the presence of external
perturbations.
  Our contribution includes the design of an optimization-based task and motion
planning framework that generates optimal control sequences with formal
guarantees of external perturbation recovery. As a core component of the
framework, a model predictive controller (MPC) encodes signal temporal logic
(STL)-based task specifications as a cost function. In particular, we
investigate challenging scenarios where the robot is subjected to lateral
perturbations that increase the risk of failure due to leg self-collision. To
address this, we synthesize agile and safe crossed-leg maneuvers to enhance
locomotion stability.
  This work marks the first study to incorporate formal guarantees offered by
STL into a TO for perturbation recovery of bipedal locomotion. We demonstrate
the efficacy of the framework via perturbation experiments in simulations.
