Hybrid dynamical systems are a popular and versatile tool to model robotic systems exhibiting continuous and discrete dynamics. They are “hybrid” because they contain continuous and discrete state variables, wherein continuous flow is interleaved with discrete transitions. For instance, a jumping robot exhibits continuous dynamics as it pushes against the ground or jumps through the air, but undergoes discrete transitions when it launches or lands. However, their complexity and richness present challenges, particularly when it comes to ensuring the safety and performance of such systems.
Controllers for such systems demand simultaneous reasoning about the continuous evolution of states and discrete mode transitions, a task that is conceptually and computationally demanding. To safely leap over an obstacle, for example, a robot must decide not only whether to jump, but precisely when to engage the jump and at what velocity, such that the resulting trajectory clears the obstacle without midair collision. This requires careful coordination between continuous control and discrete decisions.
Hamilton-Jacobi (HJ) reachability analysis is a powerful tool for understanding and controlling hybrid dynamical systems. It characterizes the backward reachable tube (BRT), the set of all initial states from which the system is guaranteed to enter a failure set, regardless of the control strategy. This BRT defines the system’s unsafe set, while its complement corresponds to the safe region, from which the system can avoid failure under appropriate control. Beyond identifying this safe set, HJ reachability also synthesizes a corresponding safety controller that ensures the system remains within the safe region over time.
The utility of HJ reachability lies in its ability to handle general nonlinear dynamics, control bounds, and dynamic uncertainty during BRT computation. Accordingly, several methods have been developed to theoretically characterize and numerically compute BRTs for continuous-time dynamical systems. However, analogous frameworks for hybrid dynamical systems remain limited. These limitations—ranging from structural assumptions to restricted applicability—are discussed in detail below.
Contributions and Paper Organization
The two contributions of this paper are as follows:
Together, these contributions bridge the gap between safety verification and goal-directed hybrid control, enabling both online safety enforcement and offline synthesis of provably safe, task-complete control policies.
Safety Filtering for Hybrid Systems
While the computation of the hybrid backward reachable tube (hBRT) provides a rigorous characterization of the unsafe region—that is, the set of states from which failure is inevitable—it is not sufficient on its own. In practice, safety must be maintained while the system pursues operational goals. To address this, safety guarantees are integrated into the execution of a nominal policy, the nominal control policy, which is typically designed to optimize performance metrics such as efficiency, speed, or task success, but often without formal consideration of the system’s safety constraints.
Hybrid Reach-Avoid Case Studies
The proposed hybrid backward reach-avoid tube (hBRAT) framework is validated through two hybrid system case studies designed to test its ability to synthesize guaranteed goal-reaching behavior under safety constraints. These examples showcase how the approach ensures that the system reaches a specified goal set within a finite time horizon while avoiding entry into defined failure regions. Each scenario introduces distinct hybrid control challenges—including mode transitions, unactuated dynamics, and dwell-time constraints—demonstrating the versatility and robustness of the hybrid reach-avoid formulation across different system configurations.
To illustrate the hybrid reach-avoid approach, the planar jumping robot example introduced previously is considered. The hybrid dynamical system formulation and the same simulation parameters are used.
Robustness to External Disturbances
To further evaluate closed-loop robustness, random environmental disturbances were introduced by placing slippery material before the overhang obstacle in the form of green plastic film. These changes were not encoded in the failure set when computing the hBRAT. Notably, the framework remained robust to these dynamic uncertainties with both low-level locomotion controllers. This robustness was guaranteed by the hBRAT under the assumption that the overall disturbances lie within the bounds of the adversarial disturbance considered during its calculation.
The proposed framework integrates the high-level reach-avoid planner with low-level locomotion controllers, enabling safe and efficient real-world navigation in the Barkour environment setup. The hierarchical structure allows seamless integration of different low-level controllers, including both model-based and learned policies, as demonstrated in the experiments.
Nevertheless, the presented framework provides only the top level of a hierarchical planning architecture. Additional modules such as environmental perception, real-time state estimation, and disturbance estimation play a vital role in achieving safe and agile locomotion in real-world environments. Integrating the framework within such modules remains future work.
Discussion and Future Work
This work extends the classical Hamilton-Jacobi reachability framework to hybrid dynamical systems, introducing a generalized methodology for computing hybrid backward reachable tubes and hybrid backward reach-avoid tubes. By leveraging these reachable sets, a hybrid least restrictive safety filter is developed, which minimally modifies a nominal hybrid policy to prevent unsafe transitions. A hybrid reach-avoid control framework is also developed, which simultaneously guarantees safety and task completion.
The effectiveness of these methods is demonstrated through extensive simulations and real-world hardware experiments on quadrupedal robots, showcasing their applicability in contact-rich and dynamic environments.
A key challenge in hybrid reachability remains scalability to higher-dimensional systems. The framework currently relies on grid-based numerical methods, whose computational complexity scales exponentially with the number of continuous states, limiting direct applicability to high-dimensional hybrid systems. Future work will explore recent learning-based approaches for solving high-dimensional Hamilton-Jacobi-Isaacs variational inequalities, as well as structure-exploiting techniques such as sum-of-squares programming for polynomial dynamics and zonotope methods for affine dynamics. These methods offer a promising direction for improving computational efficiency while maintaining rigorous safety guarantees.
Another limitation of the current approach is its reliance on a priori knowledge of the full environment to compute the hybrid BRT or BRAT offline before deployment. In many real-world scenarios, robotic systems operate in partially known or dynamically changing environments, where precomputing the full reachable set is impractical.
Future work could explore incremental or online reachability methods, where reachable sets are progressively refined as new information about the environment becomes available. Another direction is extending the framework to black-box hybrid systems, where the dynamics, mode domains, or transition surfaces are not explicitly defined but instead learned from data. By leveraging model-learning techniques or data-driven approximations of reachable sets, it may be possible to apply reachability-based safety filtering and reach-avoid control when an analytical representation of the hybrid system is unavailable.
Finally, applying the proposed framework to a broader range of robotic systems presents an opportunity for further investigation. While the approach is validated on quadrupedal locomotion, hybrid reachability and safety filtering have applications in legged manipulation, multi-contact locomotion, and aerial-ground hybrid vehicles. Extending the framework to these domains would further demonstrate the versatility and impact of reachability-based hybrid safety guarantees, enabling robots to operate in increasingly complex and uncertain environments.
Statements and Declarations
Hardware experiments video: DOI 10.6084/m9.figshare.28950725.v2
Frequently Asked Questions
What is a hybrid dynamical system?
A hybrid dynamical system combines continuous state evolution with discrete transitions between modes. A jumping robot, for example, flows continuously while pushing against the ground or moving through the air and transitions discretely when launching or landing.
What does the hybrid backward reachable tube identify?
The hybrid backward reachable tube identifies states from which entering a failure set is inevitable, regardless of the control strategy. It therefore characterizes the unsafe region of the system.
How does the safety filter work?
The hybrid least restrictive safety filter minimally modifies a nominal hybrid policy to prevent unsafe transitions. This allows the robot to pursue operational goals while maintaining safety guarantees.
What limits the scalability of the proposed framework?
The framework uses grid-based numerical methods whose complexity grows exponentially with the number of continuous states. This limits direct application to high-dimensional hybrid systems.