Manifold-constrained Hamilton-Jacobi Reachability Learning for Decentralized Multi-Agent Motion Planning

TL;DR

HaMMAR introduces manifold-constrained Hamilton-Jacobi reachability learning to enable decentralized multi-agent motion planning under task-induced constraints. By projecting dynamics onto the tangent space of constraint manifolds and extending neural HJR solvers like DeepReach, it produces a safety-valued function $V_\theta$ on the constraint manifold and integrates it into a receding-horizon trajectory optimizer that avoids collisions without assuming other agents' policies. The approach is demonstrated on 2D circle-constrained dynamics and high-DOF UR5 manipulation tasks (object-carrying, cup-holding, doorway-crossing), where it surpasses baselines in safety and task-feasibility while maintaining real-time performance. While effective, the method faces challenges such as potential drift off the manifold, increased conservatism due to worst-case safety analysis, and the lack of formal safety guarantees for neural approximations, guiding future work on tighter manifold enforcement and uncertainty quantification.

Abstract

Safe multi-agent motion planning (MAMP) under task-induced constraints is a critical challenge in robotics. Many real-world scenarios require robots to navigate dynamic environments while adhering to manifold constraints imposed by tasks. For example, service robots must carry cups upright while avoiding collisions with humans or other robots. Despite recent advances in decentralized MAMP for high-dimensional systems, incorporating manifold constraints remains difficult. To address this, we propose a manifold-constrained Hamilton-Jacobi reachability (HJR) learning framework for decentralized MAMP. Our method solves HJR problems under manifold constraints to capture task-aware safety conditions, which are then integrated into a decentralized trajectory optimization planner. This enables robots to generate motion plans that are both safe and task-feasible without requiring assumptions about other agents' policies. Our approach generalizes across diverse manifold-constrained tasks and scales effectively to high-dimensional multi-agent manipulation problems. Experiments show that our method outperforms existing constrained motion planners and operates at speeds suitable for real-world applications. Video demonstrations are available at https://youtu.be/RYcEHMnPTH8 .

Figures (6)

• Figure 1: Our method, HaMMAR, successfully solving a real-world cup-holding task with two UR5 manipulators. Transparent models indicate the goal configurations. The sequence illustrates how the manipulators move from their initial states and avoid potential collisions to reach their goals.
• Figure 2: A comparison of the BRS obtained from the ground truth (left), the manifold-constrained HJR value function (middle), and the unconstrained HJR value function (right), each shown at three time slices. Backward reachable regions appear in white, backward unreachable regions in blue, and off-manifold regions in orange. The BRS obtained with the constrained formulation closely aligns the ground truth, while the unconstrained one over-approximates the BRS.
• Figure 3: An example of HaMMAR successfully solving an object-carrying task. The robots’ current configurations are shown in solid gray, while each object is shown in red or blue, with transparent counterparts indicating their respective goal positions. The sub-figures illustrate a single trial at different time steps, where the manipulators start from their initial configurations and get close to each other (a-b), adjust their configurations to avoid a potential collision (c-d), and then safely move towards their goals (e). The corresponding video demonstration is available in the supplementary materials.
• Figure 4: An example of HaMMAR successfully solving a cup-holding task. The robots’ current configurations are shown in solid gray with their goals shown in transparent green. The sub-figures illustrate a single trial at different time steps, where the manipulators start from their initial configurations and get close (a-b), avoid a potential collision (c-d), and then safely move towards their goals (e). The corresponding video demonstration is available in the supplementary materials.
• Figure 5: An example of the five-UR5 cup-holding task. Start configurations are shown in solid gray while goals are shown in transparent green.