  Safety guarantees are valuable in the control of walking robots, as falling
can be both dangerous and costly. Unfortunately, set-based tools for generating
safety guarantees (such as sums-of-squares optimization) are typically
restricted to simplified, low-dimensional models of walking robots. For more
complex models, methods based on hybrid zero dynamics can ensure the local
stability of a pre-specified limit cycle, but provide limited guarantees. This
paper combines the benefits of both approaches by using sums-of-squares
optimization on a hybrid zero dynamics manifold to generate a guaranteed safe
set for a 10-dimensional walking robot model. Along with this set, this paper
describes how to generate a controller that maintains safety by modifying the
manifold parameters when on the edge of the safe set. The proposed approach,
which is applied to a bipedal Rabbit model, provides a roadmap for applying
sums-of-squares verification techniques to high dimensional systems. This opens
the door for a broad set of tools that can generate safety guarantees and
regulating controllers for complex walking robot models.
