Hybrid zonotopes with ADMM MIP heuristic enable efficient motion planning for hybrid systems.
Hybrid zonotopes, an advanced set representation, are used to model non-convex constraints in motion planning for hybrid systems, enabling more efficient optimization by reducing memory complexity and providing tighter convex relaxations compared to traditional methods . When paired with a mixed-integer programming (MIP) framework, hybrid zonotopes allow for compact modeling of obstacle-free spaces and environment-specific constraints such as noise restrictions or energy limitations .
To address the computational challenges of mixed-integer formulations, a novel alternating direction method of multipliers (ADMM) heuristic has been proposed specifically to exploit the structure of hybrid zonotopes . This heuristic improves convergence rates in solving mixed-integer quadratic programs (MIQPs) for model predictive control (MPC)-based planning, outperforming state-of-the-art solvers in terms of speed and robustness . The ADMM-based approach leverages the decomposable nature of hybrid zonotope constraints, making it particularly suitable for embedded optimization in real-time robotic applications.
The integration of hybrid zonotopes with structure-exploiting solvers has been applied to autonomous driving and uncrewed aerial systems (UAS), where both motion and energy dynamics must be jointly optimized . For instance, in energy-aware planning, constrained zonotopes represent state and input constraints while hybrid zonotopes encode spatial constraints like obstacles or restricted zones . By exploiting these representations within a custom MIQP solver, real-time performance is achieved, with optimization times under one second demonstrated in simulation and processor-in-the-loop tests .
Furthermore, the convex relaxation of certain hybrid zonotopes is provably tight—equal to their convex hull—allowing for stronger bounds in branch-and-bound algorithms and reducing the number of iterations needed for convergence . This property, combined with reachability-based logical constraints, enhances planning efficiency in complex environments .
These advances demonstrate that hybrid system planning using hybrid zonotopes and tailored MIP heuristics, such as the ADMM-based method, offers a powerful framework for AI-driven motion and behavior planning in robotics and control systems, particularly where real-time performance and non-convex constraints are critical .
This research addresses the computational complexities inherent in motion planning for hybrid systems, which are characterized by the interplay between continuous dynamics and discrete mode switches (e.g., a robot making contact with a surface). The authors utilize hybrid zonotopes, a specific type of set representation known for its computational efficiency in affine transformations and Minkowski sums, to model the system's reachable sets. By formulating the planning problem as a Mixed-Integer Program (MIP), the framework seeks optimal trajectories that satisfy both safety constraints and logical switching conditions.
The key contribution of this work is a novel heuristic based on the Alternating Direction Method of Multipliers (ADMM) designed to solve the resulting MIP more efficiently than standard commercial solvers. Since MIPs are NP-hard and often suffer from poor scalability in real-time applications, the proposed ADMM heuristic decomposes the problem to accelerate the search for feasible integer solutions. This approach leverages the structure of hybrid zonotopes to tighten the formulation, effectively balancing the need for rigorous verification with the computational demands of online planning.
This material is significant for the robotics and control communities because it provides a scalable pathway to verify and synthesize complex motions that involve hybrid dynamics. It bridges the gap between formal methods (which offer guarantees but are often slow) and heuristic planning (which is fast but lacks formal guarantees). By enabling efficient motion planning for systems with discrete logic, this research has direct implications for advancing autonomy in legged locomotion, manipulation, and other safety-critical applications where contact and mode switching are prevalent.
This paper introduces a novel approach for motion planning in hybrid systems by combining hybrid zonotopes (a set-based representation for uncertain hybrid dynamics) with a mixed-integer heuristic based on the Alternating Direction Method of Multipliers (ADMM). The method leverages ADMM’s decomposition capabilities to efficiently handle the nonlinear constraints and mixed-integer nature of hybrid system planning, which often arise in robotics and control applications. By reformulating the problem as a convex relaxation with integer cuts, the proposed heuristic enables scalable computation while preserving solution quality.
The key contributions include: 1. Hybrid Zonotope Representation: A compact, set-based encoding of hybrid system trajectories that accounts for both continuous dynamics and discrete transitions, improving robustness to uncertainty. 2. ADMM-MIP Hybrid Heuristic: A tailored optimization approach that decouples the problem into subproblems, accelerating convergence while maintaining feasibility. 3. Scalability: Demonstrates practical performance on high-dimensional systems, making it suitable for real-time applications in robotics (e.g., motion planning under uncertainty).
This work is significant for AI planning and optimization in robotics/control, where hybrid systems—combining discrete logic (e.g., mode switches) and continuous dynamics (e.g., kinematics)—pose challenges for traditional methods. The proposed framework bridges gaps between set-theoretic approaches and discrete optimization, offering a promising direction for robust, efficient hybrid system planning.
Source: [arXiv:2602.17574](https://arxiv.org/abs/2602.17574)