  Guaranteeing safe behavior on complex autonomous systems -- from cars to
walking robots -- is challenging due to the inherently high dimensional nature
of these systems and the corresponding complex models that may be difficult to
determine in practice. With this as motivation, this paper presents a
safety-critical control framework that leverages reduced order models to ensure
safety on the full order dynamics -- even when these models are subject to
disturbances and bounded inputs (e.g., actuation limits). To handle input
constraints, the backup set method is reformulated in the context of reduced
order models, and conditions for the provably safe behavior of the full order
system are derived. Then, the input-to-state safe backup set method is
introduced to provide robustness against discrepancies between the reduced
order model and the actual system. Finally, the proposed framework is
demonstrated in high-fidelity simulation, where a quadrupedal robot is safely
navigated around an obstacle with legged locomotion by the help of the unicycle
model.
