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

TL;DR

An approach for efficiently solving mixed-integer MPC motion planning problems using a hybrid zonotope representation of the obstacle-free space using a multi-stage MIQP solver that exploits the structure of the hybrid zonotope constraints.

Abstract

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.

Figures (5)

• Figure 1: Problem overview for risk-aware motion planning.
• Figure 2: Obstacle avoidance map with convex regions displayed around white obstacles. The current AV position is displayed using a green diamond and the blue dots show point-to-region reachability for regions 2, 3, and 5. The red dots show region-to-region reachability for regions 3, 5, 6, and 12 from region 9. Referencing Definition \ref{['def:reachability']}, the red and blue dots are spaced at an interval of $d_{max}$ for time steps less than $k_n$. Given an assumed direction of travel, the plotted spacing between the final two dots is reduced as needed to prevent the final dot from jumping a region or entering its interior.
• Figure 3: Simulation results for double integrator model using: (a) a polytopic map with convex obstacles, (b) a polytopic map with non-convex obstacles, (c) a binary OGM, (d) an OGM with both obstacles and cell-dependent costs, and (e) a binary OGM with three spatial dimensions.
• Figure 4: Average and maximum solution times for the double integrator example using an MPC horizon of $N=15$. The obstacle maps correspond to those shown in Fig. \ref{['fig:trajectories']}. "HZ" denotes a hybrid zonotope representation of the obstacle free space, while "H-rep" denotes a union of H-rep polytopes using the Big-M method. "WS" indicates that the binary variables were warm started.
• Figure 5: Processor-in-the-loop results for double integrator model compared to optimal trajectories from Fig. \ref{['fig:trajectories']}.

Theorems & Definitions (16)

• Proposition 1
• proof
• Definition 1
• Definition 2
• Definition 3
• Theorem 1
• proof
• Lemma 1
• proof
• Lemma 2