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

Autonomous vehicle (AV) motion planning problems often involve nonconvex constraints, which present a major barrier to applying model predictive control (MPC) in real time on embedded hardware. This article 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 multistage mixed-integer quadratic program (MIQP) using a hybrid zonotope representation of the nonconvex constraints. Risk-aware planning is supported by assigning costs to different regions of the obstacle-free space within the MPC cost function. A multistage 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) subproblems within a branch-and-bound mixed-integer solver. Simulation and processor-in-the-loop (PIL) 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. PIL studies demonstrate the utility of the solver for real-time implementations on embedded hardware.