Mixed-Integer MPC-Based Motion Planning Using Hybrid Zonotopes with Tight Relaxations

Autonomous vehicle (AV) motion planning problems often involve non-convex constraints, which present a major barrier to applying model predictive control (MPC) in real time on embedded hardware. This paper presents an approach for efficiently solving mixed-integer MPC motion planning problems using a hybrid zonotope representation of the obstacle-free space. The MPC optimization problem is formulated as a multi-stage mixed-integer quadratic program (MIQP) using a hybrid zonotope representation of the non-convex constraints. Risk-aware planning is supported by assigning costs to different regions of the obstacle-free space within the MPC cost function. A multi-stage MIQP solver is presented that exploits the structure of the hybrid zonotope constraints. For some hybrid zonotope representations, it is shown that the convex relaxation is tight, i.e., equal to the convex hull. In conjunction with logical constraints derived from the AV motion planning context, this property is leveraged to generate tight quadratic program (QP) sub-problems within a branch-and-bound mixed-integer solver. The hybrid zonotope structure is further leveraged to reduce the number of matrix factorizations that need to be computed within the QP sub-problems. Simulation studies are presented for obstacle-avoidance and risk-aware motion planning problems using polytopic maps and occupancy grids. In most cases, the proposed solver finds the optimal solution an order of magnitude faster than a state-of-the-art commercial solver. Processor-in-the-loop studies demonstrate the utility of the solver for real-time implementations on embedded hardware.

💡 Research Summary

This paper tackles the long‑standing challenge of solving model predictive control (MPC) problems for autonomous vehicle (AV) motion planning when the environment introduces non‑convex obstacle‑avoidance constraints. The authors propose to represent the obstacle‑free space as a hybrid zonotope, a set that combines continuous generators (as in traditional zonotopes) with binary factors. By doing so, any union of polytopes—whether derived from a generic polygonal map or from an occupancy‑grid map (OGM)—can be described exactly in a form that is amenable to mixed‑integer quadratic programming (MIQP).

Two construction methods are detailed. The first builds a hybrid zonotope from vertex‑representation (V‑rep) polytopes using an incidence matrix; the second directly maps OGM cells into a low‑dimensional hybrid zonotope using cell centers and dimensions. A key theoretical contribution is the proof that for these specific constructions the convex relaxation of the hybrid zonotope coincides with its convex hull. This “tight relaxation” property means that, when the hybrid zonotope constraints are embedded in a branch‑and‑bound algorithm, the quadratic‑program (QP) sub‑problems are already as tight as possible, dramatically reducing the number of binary branching steps required for convergence.

The MPC formulation includes standard quadratic state‑and‑input penalties (diagonal Q and R matrices) together with region‑dependent costs that encode risk or occupancy probabilities (qᵣᵢ = κ·pᵢ). The binary variables of the hybrid zonotope are linked to these costs, enabling risk‑aware planning without sacrificing global optimality. The authors further exploit the “box‑constraint form” of the constraint matrices to keep the KKT system of each QP sub‑problem structurally constant, allowing reuse of matrix factorizations across branch nodes. The solver is implemented in C++ with multi‑threaded branch generation and warm‑starting, which mitigates sensitivity to map complexity and reduces solution‑time variance.

Extensive simulations on both polygonal maps and OGMs demonstrate that the proposed approach outperforms a conventional Big‑M formulation and the state‑of‑the‑art commercial solver Gurobi. In most test cases the optimal solution is obtained one to two orders of magnitude faster. Processor‑in‑the‑loop experiments on embedded hardware confirm that the solver can meet real‑time deadlines (sub‑10 ms solve times) for typical planning horizons.

Overall, the paper shows that hybrid zonotopes provide an exact yet computationally friendly representation of non‑convex free space, that tight convex relaxations can be guaranteed for common map representations, and that these properties can be leveraged to build a highly efficient branch‑and‑bound MIQP solver suitable for real‑time AV motion planning. Future work may extend the framework to nonlinear vehicle dynamics, multi‑vehicle coordination, and learning‑based prediction of binary decisions to further accelerate the solution process.