Hybrid System Planning using a Mixed-Integer ADMM Heuristic and Hybrid Zonotopes

TL;DR

The paper addresses the challenge of planning for hybrid (piecewise affine) systems with mixed-integer constraints on embedded hardware by developing a general hybrid zonotope reachability framework and a new ADMM-FP mixed-integer programming heuristic. It introduces lifted planning formulations and graphs of functions to efficiently represent feasible trajectories while achieving tighter convex relaxations than traditional MLD-based methods. The ADMM-FP heuristic leverages the hybrid zonotope structure, uses perturbations and restarts, and benefits from warm-starts based on convex relaxations, yielding improved feasibility rates and practical runtimes in numerical experiments and a laboratory autonomous-vehicle demonstration. Practically, the approach enables real-time embedded planning for complex hybrid dynamics and lays groundwork for safety verification and extension to broader planning tasks, albeit without formal convergence guarantees for the heuristic.

Abstract

Embedded optimization-based planning for hybrid systems is challenging due to the use of mixed-integer programming, which is computationally intensive and often sensitive to the specific numerical formulation. To address that challenge, this article proposes a framework for motion planning of hybrid systems that pairs hybrid zonotopes - an advanced set representation - with a new alternating direction method of multipliers (ADMM) mixed-integer programming heuristic. A general treatment of piecewise affine (PWA) system reachability analysis using hybrid zonotopes is presented and extended to formulate optimal planning problems. Sets produced using the proposed identities have lower memory complexity and tighter convex relaxations than equivalent sets produced from preexisting techniques. The proposed ADMM heuristic makes efficient use of the hybrid zonotope structure. For planning problems formulated as hybrid zonotopes, the proposed heuristic achieves improved convergence rates as compared to state-of-the-art mixed-integer programming heuristics. The proposed methods for hybrid system planning on embedded hardware are experimentally applied in a combined behavior and motion planning scenario for autonomous driving.

Figures (11)

• Figure 1: Hybrid zonotope reachability analysis is used to construct planning problems for PWA systems. Then an ADMM-based mixed integer programming heuristic efficiently finds feasible, and often nearly optimal, solutions.
• Figure 2: Reachable sets of two-equilibrium system computed using Proposition \ref{['prop:union_of_gofs']} and \ref{['eq:pwa_single_mode_GOF']}, \ref{['eq:gof_recursion_orig']}.
• Figure 3: Convex relaxations for the 5-step reachable set $\mathcal{X}_5$ of the two-equilibrium system. Note that $\mathit{CR}(\mathcal{X}_5) \neq \mathit{CH}(\mathcal{X}_5)$ for the PWA -- Sharp Union case because the generalized intersection in \ref{['eq:gof_recursion_orig']} does not preserve sharpness.
• Figure 4: Comparison of solution methods for an MILP feasibility problem for which the feasible space is a random, non-empty hybrid zonotope. In these box and whisker plots, the box encloses the middle 50% of data points, and the whiskers enclose all other data points. The median is depicted as a black line.
• Figure 5: Trajectories generated by ADMM-FP compared to the globally optimal trajectory for a 40-step motion planning problem. The blue set is $\mathcal{P}$ and the green set shows the first two dimensions of $\mathcal{F}_N$.

Theorems & Definitions (11)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Proposition 1
• Remark 1
• Lemma 1
• Theorem 1
• Proposition 2
• Proposition 3