Hamilton-Jacobi Reachability Filters

• HJ Reachability Filters are safety-critical control mechanisms that compute backward reachable sets via HJ PDE to guarantee state constraint satisfaction.
• They integrate methodologies like grid-based dynamic programming, neural PDE solvers, and latent-space models to handle nonlinear and high-dimensional dynamics.
• Recent advancements extend these filters to model-free, data-driven frameworks, enabling seamless integration with MPC, RL, and planning for autonomous safety.

Hamilton-Jacobi (HJ) Reachability Filters are a class of safety-critical control methodologies that employ the mathematical framework of Hamilton-Jacobi reachability analysis to monitor system states and enforce state-constraint satisfaction. These filters operate by computing and leveraging the value function associated with the backward reachable set of unsafe or failure states, providing formal guarantees for safe operation—even under nonlinear dynamics, nonconvex constraints, disturbances, and high-dimensional or partially observed systems. Recent work extends traditional state-space HJ filters to latent and data-driven settings, dramatically broadening the scope of safe control in robotics and autonomous systems (Nakamura et al., 2 Feb 2025, Borquez et al., 2023, Ganai et al., 2024).

1. Mathematical Formulation and Core Principles

The foundational principle of HJ reachability filtering is the backward reachable set characterization for nonlinear control systems,

$$\dot{x} = f(x, u), \quad x \in \mathbb{R}^n,\; u \in U$$

given a failure (unsafe) set $\mathcal{F} = \{x : \ell(x) < 0\}$ for some margin $\ell$. The core object is the value function $V(x, t)$, which, under nominal discrete or continuous-time dynamics, evolves according to the (terminal- or infinite-horizon) Hamilton-Jacobi partial differential equation (PDE):

$$\frac{\partial V}{\partial t}(x, t) + \min_{u \in U} \nabla_x V(x, t) \cdot f(x, u) = 0,$$

with terminal condition $V(x, T) = \ell(x)$. The zero- or subzero-level sets of $V$ define the backward reachable sets: states that will inevitably reach $\mathcal{F}$ within time $T$, regardless of the admissible control policy (Nakamura et al., 2 Feb 2025, Tabbara et al., 19 Nov 2025).

Safety filtering is realized by interpreting $V$ as a certificate: if $\mathcal{F} = \{x : \ell(x) < 0\}$0 there exists a control policy to remain safe; if $\mathcal{F} = \{x : \ell(x) < 0\}$1, no control can avert failure (Nakamura et al., 2 Feb 2025).

For stochastic or high-dimensional systems, value iteration and RL-style Bellman backups yield discounted or time-relaxed surrogates:

$\mathcal{F} = \{x : \ell(x) < 0\}$2

with $\mathcal{F} = \{x : \ell(x) < 0\}$3 ensuring contraction (Nakamura et al., 2 Feb 2025).

2. Classical State-Space HJ Filters

Traditional HJ reachability filters operate by:

• Precomputing the value function $\mathcal{F} = \{x : \ell(x) < 0\}$4 on a discrete grid or via neural approximators (Bansal et al., 2020, Tabbara et al., 19 Nov 2025),
• At runtime, querying $\mathcal{F} = \{x : \ell(x) < 0\}$5 to determine whether the current state is safe,
• Modifying or projecting a base/nominal controller's action to ensure future state trajectories never enter the unsafe set $\mathcal{F} = \{x : \ell(x) < 0\}$6.

Three principal families of such filters, as detailed in (Borquez et al., 2023), are:

Filter	Core Mechanism	Properties
Least-Restrictive (LR)	Directly switches to the optimal safety override if $\mathcal{F} = \{x : \ell(x) < 0\}$7	Minimal intervention, can induce bang-bang control
Smooth LR (SLR)	Solves a QP to minimally modify nominal action on boundary	Smoother, low jerk
Smooth Blending (SB)	Blends nominal and safe control with parameter $\mathcal{F} = \{x : \ell(x) < 0\}$8	Continuous, tunable conservativeness

These filters guarantee forward invariance of the safe set by projecting the base policy onto the set of actions that preserve (or do not decrease) $\mathcal{F} = \{x : \ell(x) < 0\}$9 (Borquez et al., 2023, Kumar et al., 2023). In high-dimensional settings, neural approximators or trajectory-optimization solvers may be used in place of grid-based methods (Bansal et al., 2020, Kirchner et al., 2020).

3. Learning-Based and Model-Free Extensions

The classical paradigm of HJ filters is limited by the "curse of dimensionality". To address this, multiple methodologies have been developed:

• Deep Neural PDE Solvers: Methods such as DeepReach parameterize $\ell$0 by a neural network with periodic or sinusoidal activations, trained using the unsupervised PDE residual loss to approximate the viscosity solution of the HJI variational inequality (Bansal et al., 2020).
• Classification-Based Filters: For control-affine systems, the boundary of the optimal safety action is learned as a collection of binary classifiers, each selecting the bang-bang extremum per control dimension (Rubies-Royo et al., 2018).
• Model-Free Q-Filters: Approaches such as Deep QP Safety Filter and verifiable multiplicative Q-networks learn Q-functions or derivatives directly from data, using contraction-based Bellman operators to ensure convergence to the viscosity solution, and enforce safety online via quadratic program (QP) or mixed-integer QCQP checks (Kim et al., 29 Jan 2026, Li et al., 27 May 2025).
• HJ-Patch and Barrier Value Function Refinement: Existing approximate or neural CBFs are "patched" near the safety boundary using local HJ updates, yielding certifiably invariant sets with drastically reduced computational overhead compared to global reachability DP (Tonkens et al., 2023).

4. Latent-Space and High-Dimensional Generalizations

Latent HJ reachability filters generalize the state-space HJ formalism to systems where the true state is not directly observable—e.g., robotics with raw image inputs:

• The system is mapped into a learned latent space by a generative world model (e.g., RSSM, transformer-based) trained on multimodal trajectory data (Nakamura et al., 2 Feb 2025).
• Safety/failure sets are specified as classifiers in the latent space: a margin function $\ell$1 trained on labeled observation-embeddings.
• The HJ-Bellman equation is applied in the latent space, propagating the safe set and synthesizing safety-preserving actions without explicit recovery demonstrations.
• At runtime, the process is: encode observation $\ell$2 to latent $\ell$3, assess proposed action by imagining its effect via the model, and override with safety-preserving action only if necessary.

Empirical results show that such filters not only generalize safety monitoring to rich observation modalities (vision, proprioception) but also guard against complex hazards not easily hand-coded, such as bag-spill or dexterous manipulation failures. Latent HJ reachability achieved near-parity with privileged state-access filters and outperformed conventional CMDP baselines in both simulated and real hardware (Nakamura et al., 2 Feb 2025).

5. Integration with Planning and Control Architectures

HJ reachability filters are compatible with a wide variety of control frameworks and planners:

• Model Predictive Control (MPC): The safety value function $\ell$4 is incorporated as a constraint, ensuring the MPC horizon always terminates in the backward reachable set—yielding recursive feasibility and closed-loop safety (Wang et al., 29 Jun 2025).
• Sampling-Based Planning: Certified neural approximations to the HJ value function (e.g., Fourier Neural Operators) enable contingency-aware pruning of unsafe samples during RRT or incremental multi-goal planning (Muenprasitivej et al., 17 Mar 2026). Under-approximation guarantees ensure that only truly recoverable states are considered, with recovery policies invoked upon dangerous contingencies.
• Reinforcement Learning (RL): HJ filters act as safety shields in RL, overriding agent actions that would violate the safe set as predicted by the reachability critic. Both "shielded exploration" and safe online adaptation are supported (Ganai et al., 2024, Kim et al., 29 Jan 2026).

6. Algorithmic Implementation and Scalability

Key implementation steps for HJ reachability filters include:

1. Collection (or simulation) of observation-action trajectories, with safe and unsafe labels for constraint learning,
2. Solving the HJ PDE via grid-based DP, neural PDE solver, trajectory-optimization, or latent-space Bellman backup, depending on system dimensionality and observability (Kirchner et al., 2020, Bansal et al., 2020, Nakamura et al., 2 Feb 2025),
3. Runtime evaluation of safety conditions and policy override with efficient lookup (grid or NN), QP/MI-QCQP or learned classifier composition,
4. For latent-space filters, embedding constraints and world dynamics enable operation directly from raw observations (Nakamura et al., 2 Feb 2025).

Scalability strategies include decomposition, local patching, neural architectures attuned to PDE structure (sinusoidal, multiplicative), and layered verification pipelines. Latent and NN-based methods have demonstrated tractable real-time performance on systems with dimension $\ell$5–$\ell$6 (Bansal et al., 2020, Wang et al., 29 Jun 2025).

7. Empirical Validation and Performance

Table: Example Empirical Performance of HJ Reachability Filters

System/Scenario	Baseline Violation	HJ Filter Violation	Safety Success
Vision-based Dubins Car (Nakamura et al., 2 Feb 2025)	4%	2%	96–97.6%
Block Manipulation (vision)	36%	20%	80%
Franka Skittles Bag (hardware)	73.4%	26.4%	>95% (spill<5%)
Rocket Landing (SLR filter) (Borquez et al., 2023)	–	0%	100%

In all cases, HJ-based safety filtering reduces failure rates relative to unfiltered or CBF-only baselines, and closes the gap between robust theoretical safety and practical, tractable deployment.

8. Key Limitations and Research Directions

Notwithstanding their rigor, HJ reachability filters face challenges:

• State-space grid-based solvers are limited to low-dimensional ($\ell$7) systems without decomposition or neural parameterization (Tabbara et al., 19 Nov 2025, Bansal et al., 2020).
• Guaranteeing invariance for neural-approximate value functions, especially under nonconvex dynamics or partial observability, requires additional local verification, as in HJ-Patch or formal verification with MIQCQP (Tonkens et al., 2023, Li et al., 27 May 2025).
• Real-time, high-fidelity deployment on complex hardware is practical with NN-based surrogates or latent-space models, but their approximation-induced conservatism must be explicitly quantified.
• Integration with RL, MPC, and planning algorithms is an active area, particularly for systems operating in uncertain, dynamic, or unstructured environments (Ganai et al., 2024, Muenprasitivej et al., 17 Mar 2026).

Recent advances in model-free, verifiable safety Q-filters, hybrid patching algorithms, and latent-space generalizations continue to expand the application and formal robustness of HJ reachability filters in safety-critical autonomous control (Nakamura et al., 2 Feb 2025, Li et al., 27 May 2025, Kim et al., 29 Jan 2026).