⚡ electrical engineering

Hamilton-Jacobi Reachability for Hybrid Systems: Unified Goal-Driven Control with Safety Guarantees

This paper extends Hamilton-Jacobi reachability analysis to hybrid dynamical systems by introducing a unified framework that computes safe sets and integrates performance objectives through least-restrictive safety filters and backward reach-avoid tubes, thereby guaranteeing both safety and task completion for robotic systems in contact-rich environments.

Original authors: Javier Borquez, Shuang Peng, Somil Bansal

Published 2026-09-16

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Javier Borquez, Shuang Peng, Somil Bansal

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Robots that move through the real world face a fundamental challenge: their actions are rarely smooth and continuous. A dog running through a park, a spacecraft docking with a station, or a drone landing on a ship all experience sudden shifts in how they move. They might be sliding on wheels, then suddenly jump into the air, or switch from a gentle glide to a hard impact. In engineering terms, these systems are "hybrid," meaning they combine smooth, flowing motion with sudden, discrete jumps from one mode of operation to another. For a robot to be useful and safe, it must not only navigate these shifts but also guarantee that it never falls into a trap from which it cannot escape. This requires a way to look ahead and know, with absolute certainty, which paths are safe and which lead to failure, even when the robot's environment is unpredictable.

For decades, engineers have used a powerful mathematical tool called Hamilton-Jacobi reachability to solve this problem for smooth systems. This method allows a computer to calculate a "safe zone," a map of every possible starting position from which a robot can avoid a crash, no matter how the wind blows or how the ground shifts. However, this tool broke down when applied to hybrid systems. The sudden jumps between modes, like a robot's foot hitting the ground, created a complexity that the old math could not handle. Without a way to compute these safe zones for hybrid systems, engineers were forced to guess or rely on trial and error, leaving robots vulnerable to unexpected failures in complex environments.

A team of researchers has now bridged this gap by extending the reachability method to handle these hybrid systems. They developed a new framework that can calculate safe zones for robots that switch between different behaviors, such as walking, jumping, or flying. Their work does two distinct things. First, they created a safety filter that acts like a vigilant supervisor. This filter watches a robot's planned path and only steps in when a planned move would lead to a crash. If the robot tries to jump into a wall, the filter gently nudges the plan to a safe alternative, but if the plan is already safe, it lets the robot proceed exactly as intended. This ensures the robot stays safe without forcing it to move clumsily or unnecessarily. Second, they developed a method to plan not just for safety, but for success. This new approach calculates a path that guarantees the robot will reach its goal, like a docking port or a finish line, while simultaneously avoiding all obstacles. It finds a route that is both safe and effective, even when the robot must switch between different modes of movement to get there.

To prove their ideas work, the researchers tested their system on a four-legged robot in a challenging obstacle course inspired by dog agility competitions. The course included tight weaving poles, steep slopes, low overhangs requiring the robot to crawl, and slippery terrain. The robot had to navigate this entire course without hitting anything. The researchers ran the robot using two different low-level control systems: one based on traditional physics equations and another based on artificial intelligence learning. In both cases, the new high-level planner successfully guided the robot through the course. When the robot approached a low overhang, the planner automatically switched its mode to a crawling gait to avoid hitting its head. When it reached a slippery patch, it switched to a faster gait to maintain traction. The system handled these switches seamlessly, ensuring the robot never entered a dangerous state.

The experiments showed that the new method is robust even when the robot is pushed or pulled by external forces. In one test, a human physically dragged the robot toward an obstacle. Instead of crashing, the robot's planner immediately recognized the danger and switched its behavior to a safer mode, allowing it to recover and continue the task. This ability to recover from unsafe situations is a significant improvement over previous methods, which often left robots stuck once they entered a dangerous zone. The researchers also compared their approach to other advanced planning techniques in computer simulations. While other methods could sometimes reach the goal, they frequently crashed into obstacles when the environment became unpredictable. The new method maintained a high success rate, reaching the goal safely in nearly every trial, even when the robot was subjected to strong, random pushes.

The work relies on a pre-calculated map of safety that the robot consults in real time. This map is generated offline, a process that took about seven hours on a standard computer for the specific task. Once the map is ready, the robot can use it instantly, making decisions in about one millisecond. This speed is crucial for real-world applications where a robot must react to changes in the environment faster than a human can blink. The researchers validated their system on a real quadruped robot, a Unitree Go1, which was tracked by a high-speed motion capture system to ensure precise measurements of its position and movement. The robot successfully completed the entire course multiple times, demonstrating that the theoretical safety guarantees hold up in the messy, unpredictable reality of the physical world.

While the method is powerful, the researchers acknowledge its current limitations. The calculations required to generate the safety map become extremely difficult as the number of variables increases, a problem known as the "curse of dimensionality." This means that for now, the system works best for robots with a manageable number of moving parts, like the four-legged robot used in the study. For more complex machines with many degrees of freedom, the computational cost is currently too high. The team suggests that future work will need to combine their rigorous safety guarantees with new learning-based techniques to handle these more complex systems. They also note that their current approach assumes the robot knows the layout of the obstacles beforehand. In a dynamic world where obstacles move or appear unexpectedly, the system would need to update its safety map on the fly, a challenge they plan to address in future research.

Despite these limitations, the achievement represents a significant step forward in making autonomous robots truly reliable. By providing a mathematical guarantee that a robot can reach its goal without crashing, even when it must switch between different ways of moving, the researchers have removed a major barrier to deploying robots in complex, unstructured environments. The ability to plan for both safety and success simultaneously means that robots can be trusted to operate in places where failure is not an option, from search and rescue missions to delicate manufacturing tasks. The framework turns the chaotic nature of hybrid motion into a manageable problem, ensuring that as robots become more agile and versatile, they remain fundamentally safe.

Technical Summary: Hamilton-Jacobi Reachability for Hybrid Systems

1. Problem Statement

Hybrid dynamical systems, which combine continuous dynamics with discrete mode transitions, are essential for modeling robotic systems in contact-rich environments (e.g., jumping robots, legged locomotion). However, ensuring safety and performance in these systems is challenging due to the intricate coupling between continuous state evolution and discrete events (such as mode switches, state resets, and forced transitions).

Existing Hamilton-Jacobi (HJ) reachability analysis provides a rigorous framework for continuous-time nonlinear systems to compute Backward Reachable Tubes (BRTs) and synthesize safety controllers. However, applying these methods to hybrid systems has been limited by:

Structural Assumptions: Previous hybrid methods often restricted analysis to linear dynamics or excluded controlled inputs.
Objective Integration: Existing frameworks focused primarily on verification (reachability or avoidance) but failed to integrate these safety guarantees with goal-reaching objectives. Real-world applications require controllers that not only avoid failure but also actively achieve tasks.
Complexity: Handling controlled transitions, forced transitions, and state resets simultaneously within a unified reachability framework remains computationally and theoretically demanding.

2. Methodology

The paper extends classical HJ reachability to hybrid systems by defining a generalized value function over both discrete and continuous states. The approach is built upon three core components:

A. Hybrid System Model

The system is modeled as a tuple $H = ((Q \times X), (U \times D), f, \text{Inv}, \Sigma, R)$ , where:

$Q$ is a finite set of discrete modes.
$X, U, D$ represent continuous states, controls, and disturbances.
$f$ defines continuous dynamics $\dot{x} = f_i(x, u, d)$ within an invariant set $S_i$ .
$\Sigma$ includes discrete actions for controlled transitions (when $x \in S_i$ ) and forced transitions (when $x \notin S_i$ ).
$R$ defines reset maps $x^+ = R_{ij}(x)$ upon transition.

B. Hybrid Backward Reachable Tubes (hBRT)

Building on prior work (Borquez et al., 2024b), the authors utilize a constrained Hamilton-Jacobi-Isaacs Variational Inequality (HJI-VI) to compute the hBRT. The value function $V(x, q_i, t)$ is updated by selecting the optimal outcome between:

Discrete Transitions: Taking a controlled switch $\sigma_{ij}$ or a forced switch $\varsigma_{ik}$ with associated reset maps.
Continuous Evolution: Staying in the current mode and evolving according to the continuous dynamics.
The solution characterizes the set of states from which the system can reach a goal (liveness) or avoid a failure set (safety) under worst-case disturbances.

C. Two Control Mechanisms

The paper proposes two distinct mechanisms to utilize these reachable sets:

Hybrid Least Restrictive Filter (hLRF):
- Function: A safety filter that wraps any nominal hybrid policy $\pi_{nom}$ .
- Mechanism: It intervenes only when necessary. For discrete transitions, it checks if the nominal switch leads to a safe state (value function $V \geq 0$ ). If not, it overrides with the optimal safe switch. For continuous control, it acts as a boundary controller, modifying inputs only when the system is on the safety boundary.
- Goal: Guarantee forward invariance of the safe set while minimally altering the nominal behavior.
Hybrid Backward Reach-Avoid Tubes (hBRAT):
- Function: A unified formulation that simultaneously enforces safety (avoiding failure set $L$ ) and task completion (reaching goal set $G$ ).
- Mechanism: The authors derive a new constrained HJI-VI where the terminal condition is $V(x, q, T) = \max(g(x), l(x))$ . The value function evolution combines the cost of reaching the goal and avoiding the failure set.
- Goal: Synthesize control policies that guarantee the system reaches the target within a time horizon without ever entering the failure set.

D. Numerical Implementation

The value functions are computed using a grid-based level-set method (Algorithm 1 for hBRT, Algorithm 2 for hBRAT). The algorithm propagates the value function backward in time, updating it at each step to account for continuous dynamics and discrete transitions across all modes.

3. Key Contributions

Unified Hybrid Safety Filter (hLRF): A real-time mechanism that guarantees safety for hybrid systems with state resets and forced transitions, wrapping any nominal policy. It is proven to ensure forward invariance of the safe set.
Hybrid Reach-Avoid Formulation (hBRAT): A novel mathematical framework that jointly encodes safety and goal-reaching objectives in hybrid systems. It derives a new HJI-VI that characterizes the set of states guaranteeing both task completion and safety under worst-case disturbances.
Validation on Real Hardware: Successful deployment on a quadrupedal robot (Unitree Go1) navigating a complex Barkour-style obstacle course, demonstrating the framework's ability to handle mode switching (walking, climbing, crawling) and recover from external disturbances.

4. Results

Simulation Studies

Planar Jumping Robot: The hLRF successfully modified nominal policies (straight-line movement and low hops) to clear obstacles by triggering optimal jumps, ensuring safety with minimal intervention.
Low Earth Orbit Spacecraft: The filter prevented collisions in a cluttered orbital environment, managing controlled thrusting and forced cooldown transitions.
Quadruped Simulation (Barkour): Compared against Model Predictive Path Integral (MPPI), Deterministic MPC, and MPC with Control Barrier Functions (CBF-MPC).
- Safety: The hBRAT planner achieved the highest "Safe Reach" rates across all disturbance levels (0.81 at $\sigma=0.4$ m/s), significantly outperforming baselines (MPC: 0.48, MPC-CBF: 0.44).
- Robustness: While baselines reached the goal, they frequently collided with obstacles under high disturbance. The hBRAT planner maintained safety by recovering from disturbances and adhering to the reach-avoid tube.

Real-World Experiments

Hardware: A Unitree Go1 quadruped navigated a course with weave poles, slopes, overhangs, and slippery terrain.
Performance: The system successfully completed the task with both a model-based (MPC) and a learned (RL) low-level controller.
- Safety: 100% success rate and 100% safe reach rate over 10 trials with both controllers.
- Recovery: The hBRAT-based planner demonstrated the ability to recover from external physical disturbances (e.g., being pushed into an overhang) by autonomously switching modes (e.g., from walking to crawling) to regain safety, a capability lacking in previous frameworks.
- Efficiency: Online inference took approximately 1 ms per query, enabling real-time operation.

5. Significance and Claims

The paper claims to bridge the critical gap between safety verification and goal-directed control in hybrid systems. By extending HJ reachability to handle hybrid dynamics with a unified reach-avoid formulation, the authors provide a method that:

Guarantees Safety and Task Completion: Unlike previous methods that treated safety and goals separately, this framework synthesizes policies that are provably safe and goal-complete.
Enables Minimal Intervention: The hLRF allows the use of high-performance nominal controllers while providing a rigorous safety net that intervenes only when absolutely necessary.
Handles Complex Hybrid Dynamics: The framework explicitly accounts for controlled transitions, forced transitions, and state resets, which are prevalent in legged robotics but often ignored in continuous-only safety filters.

The authors acknowledge limitations, specifically the computational complexity scaling exponentially with state dimension (curse of dimensionality) and the reliance on a priori knowledge of the environment for offline computation. They position this work as a foundational step toward safer, more capable robotic systems in dynamic environments, with future work targeting learning-based approximations for higher dimensions and online reachability updates.