Reachability-based Trajectory Design via Exact Formulation of Implicit Neural Signed Distance Functions

TL;DR

The paper tackles real-time, safety-guaranteed motion planning for autonomous vehicles in dense, dynamic environments by marrying offline zonotope reachability with an online neural implicit signed distance function. REDEFINED constructs exact distance measures between the ego vehicle's zonotope forward reachable sets and obstacles using a fixed-weight ReLU network, enabling a batched, differentiable collision-avoidance constraint within a receding-horizon optimization. Key contributions include (i) an exact RDF representation via a neural network, (ii) a real-time online optimization framework that leverages batched constraint evaluations, and (iii) comprehensive experiments showing improved planning efficiency and robustness over state-of-the-art baselines like REFINE, SOS-RTD, and NMPC. The approach offers practical impact by delivering collision-free trajectories with real-time performance, while highlighting limitations related to zonotope over-approximation and scalability to higher dimensions.

Abstract

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/.

Figures (5)

• Figure 1: REDEFINED constructs safe trajectories in a receding-horizon fashion from an initial state (yellow circle) to a goal state (green star) while avoiding obstacles (white cars). REDEFINED first performs offline reachability analysis using a closed-loop full-order vehicle dynamics model to construct control-parameterized, zonotope reachable sets (purple boxes) that over-approximate all possible states of the vehicle over a finite planning horizon. During online planning, REDEFINED computes a parameterized controller by solving an optimization problem that selects subsets of pre-computed zonotope reachable sets. Typically this can be done by either requiring that the distance between the obstacles and reachable set are greater than zero (QP-based formulation) or by incorporating a constraint that ensures that the two sets do not intersect with one another (non-intersection constraint); however, solving these formulations of the trajectory design problem can be computationally prohibitive. Instead, REDEFINED uses a novel neural implicit representation to compute the exact signed distance between the vehicle's zonotope reachable sets and obstacles. By ensuring the signed distance between the reachable sets and obstacles is greater than zero, REDEFINED is able to guarantee the ego vehicle does not collide with any obstacles while executing the trajectory.
• Figure 2: Overview of REDEFINED. During each planning iteration, a family of reachable sets are constructed that correspond to different maneuvers. In this figure, subsets of purple zonotope reachable sets corresponding to the control parameter shown in purple ensures a collision-free path while the other control parameter shown magenta might lead to collisions with white obstacles.
• Figure 3: Graphical illustration of signed distance between zonotopes. The top row illustrates the positive distance function between two zonotopes. First, the gray zonotope is buffered by the generators of the purple zonotope. Then, the distance is computed between the purple zonotope's center and the closest line segment of the buffered zonotope. The bottom row graphically illustrates the negative distance between two zonotopes. First, the pink zonotope's generators buffer the gray zonotope. Then, the negative distance is computed by projecting the pink zonotope's centers on each of the buffered zonotope's half-spaces.
• Figure 4: Mean number of constraint and constraint gradient evaluations of REDEFINED vs. REFINE within one optimization problem.
• Figure 5: An illustration of REDEFINED, REFINE, SOS-RTD, and NMPC running on the same simulated scenario. (a) REDEFINED is able to find a solution at $t = 27.0$[s] to perform a lane change and finally succeeds. (b) REFINE fails to find a solution at $t = 27.0$[s] and therefore executes fail-safe maneuver to stop the ego vehicle. (c) SOS-RTD stops the ego vehicle at $t = 23.0$[s] because of its conservative reachable sets. (d) NMPC leads the ego vehicle to collision at $t = 27.2$[s]. In each set of images, the ego vehicle and its executed trajectory are colored in black. Zonotope reachable sets for REDEFINED, REFINE and polynomial reachable sets for SOS-RTD are colored in purple. Other vehicles are obstacles and are depicted in white. Waypoints are plotted as green stars.

Theorems & Definitions (17)

• Definition 1
• Definition 2
• Definition 3
• Definition 5
• Definition 6
• Definition 8
• Definition 11
• Theorem 12
• Theorem 13
• proof