  Generating receding-horizon motion trajectories for autonomous vehicles in
real-time while also providing safety guarantees is challenging. This is
because a future trajectory needs to be planned before the previously computed
trajectory is completely executed. This becomes even more difficult if the
trajectory is required to satisfy continuous-time collision-avoidance
constraints while accounting for a large number of obstacles. To address these
challenges, this paper proposes a novel real-time, receding-horizon motion
planning algorithm named REachability-based trajectory Design via Exact
Formulation of Implicit NEural signed Distance functions (REDEFINED). REDEFINED
first applies offline reachability analysis to compute zonotope-based reachable
sets that overapproximate the motion of the ego vehicle. During online
planning, REDEFINED leverages zonotope arithmetic to construct a neural
implicit representation that computes the exact signed distance between a
parameterized swept volume of the ego vehicle and obstacle vehicles. REDEFINED
then implements a novel, real-time optimization framework that utilizes the
neural network to construct a collision avoidance constraint. REDEFINED is
compared to a variety of state-of-the-art techniques and is demonstrated to
successfully enable the vehicle to safely navigate through complex
environments. Code, data, and video demonstrations can be found at
https://roahmlab.github.io/redefined/.
