  The key innovation of our analytical method, CaRT, lies in establishing a new
hierarchical, distributed architecture to guarantee the safety and robustness
of a given learning-based motion planning policy. First, in a nominal setting,
the analytical form of our CaRT safety filter formally ensures safe maneuvers
of nonlinear multi-agent systems, optimally with minimal deviation from the
learning-based policy. Second, in off-nominal settings, the analytical form of
our CaRT robust filter optimally tracks the certified safe trajectory,
generated by the previous layer in the hierarchy, the CaRT safety filter. We
show using contraction theory that CaRT guarantees safety and the exponential
boundedness of the trajectory tracking error, even under the presence of
deterministic and stochastic disturbance. Also, the hierarchical nature of CaRT
enables enhancing its robustness for safety just by its superior tracking to
the certified safe trajectory, thereby making it suitable for off-nominal
scenarios with large disturbances. This is a major distinction from
conventional safety function-driven approaches, where the robustness originates
from the stability of a safe set, which could pull the system
over-conservatively to the interior of the safe set. Our log-barrier
formulation in CaRT allows for its distributed implementation in multi-agent
settings. We demonstrate the effectiveness of CaRT in several examples of
nonlinear motion planning and control problems, including optimal,
multi-spacecraft reconfiguration.
