Contingency-Aware Planning via Certified Neural Hamilton-Jacobi Reachability

Kasidit Muenprasitivej; Derya Aksaray

cs.ROMar 17, 2026

Contingency-Aware Planning via Certified Neural Hamilton-Jacobi Reachability

Authors: Kasidit Muenprasitivej, Derya Aksaray

Organizations: Department of Electrical and Computer Engineering, Northeastern University, Boston, MA, USA, 02115

Abstract

Hamilton-Jacobi (HJ) reachability provides formal safety guarantees for dynamical systems, but solving high-dimensional HJ partial differential equations limits its use in real-time planning. This paper presents a contingency-aware multi-goal navigation framework that integrates learning-based reachability with sampling-based planning in unknown environments. We use Fourier Neural Operator (FNO) to approximate the solution operator of the Hamilton-Jacobi-Isaacs variational inequality under varying obstacle configurations. We first provide a theoretical under-approximation guarantee on the safe backward reach-avoid set, which enables formal safety certification of the learned reachable sets. Then, we integrate the certified reachable sets with an incremental multi-goal planner, which enforces reachable-set constraints and a recovery policy that guarantees finite-time return to a safe region. Overall, we demonstrate that the proposed framework achieves asymptotically optimal navigation with provable contingency behavior, and validate its performance through real-time deployment on KUKA's youBot in Webots simulation.

Figures & tables

Fig. 1: Preview of finite-time guaranteed contingency planning using Fourier Neural Operator–based Hamilton–Jacobi reachability. The robot state lies within a safe under-approximation (blue) of the true reachable set (green). The final trajectory (red, solid) is obtained by following a gradient-based policy toward the safe set (light green) while avoiding obstacles (grey), which may be unknown (red, dashed) and detected by onboard sensors (cyan). (Rightmost) Realization of the contingency control policy on KUKA’s youBot.

Fig. 2: Overview of the planning framework in Alg. 1 with $K=1$ goal (red star) and initially unknown obstacles (red, dashed). The $\varepsilon$ -sublevel set of the FNO-predicted reachable set (dark blue) is guaranteed to under-approximate the true reachable set (green). The FNO model is trained over a local domain $D^{\rm loc}$ (blue) and integrated into planning within the global environment $\Omega$ (grey). Prior to planning, the heading-agnostic reachable sets (cyan) must sufficiently overlap to contain a closed Euclidean ball (yellow), ensuring feasibility of the nominal planner with step length $\delta$ across adjacent sets. The final trajectory toward the goal is shown in red.

Training Epochs	$\varepsilon$ ( $L^{\infty}$ )	$\eta_{\rm include}$ $(\mathcal{RA}_{\phi}^{\varepsilon}\ /\ \mathcal{RA}_{\phi})$	$\varepsilon_{0}$ ( $\mathbb{W}^{1,2}$ )	$\mathbf{meas}(D_{\rm violate})$ over $\mathbf{meas}(\mathcal{X})$	MSE $[\times 10^{-4}]$
1500	0.483	1.0 / 0.9927	0.118	2.465 %	9.62
2000	0.455	1.0 / 0.9928	0.114	2.281 %	8.60
2500	0.442	1.0 / 0.9937	0.110	2.138 %	7.84
3000	0.431	1.0 / 0.9935	0.107	2.011 %	7.06

TABLE I: FNO Training and Approximation Metrics

Scenario	Success	Avg. $T_{\rm reach}$	Avg. $V(\mathbf{x},0)$ (failure)	Avg. Dist. to $\mathcal{L}$ (failure)
1 Obs	100%	4.564	—	—
3 Obs	98.00%	4.600	0.054	1.544
5 Obs	96.80%	4.769	0.111	1.582
7 Obs	95.60%	4.707	0.019	2.046

TABLE II: Contingency Policy Case Studies (500 runs each)

Panel of Figure 5 — Fig. 3: Comparison of two cases in the multi-goal planning and routing framework: (a) an environment with known obstacles (grey) and no reachable-set constraint for contingency planning; (b) an environment in which all obstacles are a priori unknown (red, dashed) and later detected (grey), with a guaranteed finite-time reachability strategy to a safe set (green, dashed) enforced at every state along the trajectory. The robot starts at the red star, and trajectory colors denotes the sequence of the goals visited.

	Unknown Map		Priori-Known Map
RRT X Constraint	Dist.	Time	Dist.	Time
$\mathbf{p}\in\Omega_{\rm feasible}$	105.614	89.494 s	105.428	64.741 s
$\mathbf{p}\in\Omega$	103.879	47.604 s	102.568	42.067 s

TABLE III: Multi-Goals Planning Case Studies (10 runs each)

Fig. 4: Contingency-aware multi-goal planning on KUKA’s youBot simulation. On the plot, the robot initializes at the red star, and the trajectory color encodes the active goal index. While navigating toward the green star, an adversarial event is triggered and activates the contingency mechanism, prompting the robot to execute a certified fallback policy (black, squared) toward the safe set (green, dashed). Upon reaching the safe set, the robot resumes the mission and visits the remaining goals in a locally optimal sequences.

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Learning Reachability-Based Safety Representations for Real-Time Robot Navigation in Unknown and Dynamic Environments

Bojan Derajić

Faculty IV – Electrical Engineering and Computer Science of Technische Universität Berlin

Safe Robot Navigation Dynamic Obstacle Avoidance Reachability Analysis Implicit Neural Representations Control Barrier Functions

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Safe robot navigation in unknown, dynamic environments requires more than geometric collision checking: a robot still far from an obstacle may be unable to avoid it because of its speed, orientation, or the obstacle's motion. Hamilton--Jacobi (HJ) reachability analysis captures this in a value function whose sign separates states that admit a safe continuation from those where collision is unavoidable, but computing it online for every newly observed environment is too expensive. This dissertation trains neural implicit representations offline on reachability solutions to give optimization-based controllers this information in real time. Since the value function never exceeds the constraint, it equals a directly computed constraint, such as a signed distance, minus a nonnegative dynamics correction. Learning only this correction keeps represented obstacles outside the predicted safe set despite approximation errors. The first two of five methods use the learned function as a terminal constraint in model predictive control, for static local environments and predicted obstacle motion. The next two use it as a barrier candidate in control barrier function safety filters, for static environments and, through pairwise barriers in relative coordinates, for a varying number of moving obstacles. Simulations and hardware experiments on ground and aerial robots show better collision avoidance than geometric and hand-designed baselines, especially for short horizons and moving obstacles. The fifth method replaces the grid-based offline solver, which limits the first four to low-dimensional models. For control-affine dynamics with polytopic inputs, the HJ Hamiltonian is maximized at a vertex, so a parallelizable tree search over vertex inputs can generate training labels. With a loss from the HJ variational inequality, these labels yield barrier candidates for systems with up to 13 states.

Abstract

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