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.
Dynamic systems that exhibit both continuous and discrete behavior are called hybrid systems. These are often used to model systems with multiple dynamic modes. In robotics, hybrid systems often arise when there are contact dynamics, such as in cases of walking robots or manipulators [1]. Hybrid systems can also appear in combined task and motion planning [2] or, analogously, integrated behavior and motion planning for autonomous driving [3]- [5].
This paper is concerned with the formulation and solution of motion planning problems for linear systems with disjoint constraints and hybrid systems where each dynamic mode is an affine system. This class of hybrid systems is equivalently described as piecewise affine (PWA) systems, mixed-logical dynamical (MLD) systems, or discrete hybrid automata (DHA) [6]. PWA systems in particular are widely used in robotics, with examples including legged robots [7], skidding UGVs [8], and robotic manipulation systems [9], [10]. Planning problems for PWA systems are challenging to solve in general, with computational efficiency very sensitive to the specific numerical formulation [11].
Reachability analysis-i.e., rigorously quantifying the states that a dynamic system can achieve-is a powerful tool for All authors are with the Department of Mechanical Engineering, The Pennsylvania State University, University Park, PA 16802 USA (e-mail: jrobbins@psu.edu, thompson@psu.edu, jglunt@psu.edu, hcpangborn@psu.edu). formulating planning problems. Advantages can include the ability to account for disturbances [12], [13], and favorable numerical structure and properties [14]. Zonotopes [15] and constrained zonotopes [16] are advanced set representations that are widely used to compute reachable sets for linear systems. Hybrid zonotopes [17] are an extension of constrained zonotopes that can be used to compute reachable sets for hybrid systems and represent non-convex constraints.
A general treatment of MLD system reachability analysis using hybrid zonotopes was provided in [17]. Modeling with MLD systems generally requires a modeling language like HYSDEL [6], however, which limits applicability to systems where the model is not known a priori. This can occur, for example, in combined behavior and motion planning, where the allowed behaviors are environment-dependent, and when using data-based models as in [7]. Hybrid zonotope-based reachability analysis for some specific DHA systems was also considered in [18]. A comprehensive treatment of hybrid zonotope-based reachability analysis for PWA systems has not yet been provided.
Hybrid zonotopes are a mixed-integer set representation, and as such, planning problems formulated using hybrid zonotope reachability analysis are mixed-integer programs (MIPs). The use of MIPs in motion planning is an active area of research. While MIPs can model non-convex and disjoint constraints, solving them is NP-hard in general. Typically, solution approaches are based on branch-and-bound or branchand-cut methods, often using powerful commercial solvers such as Gurobi or CPLEX [19]. Problem-tailored solvers have been proposed as well [3], [20]. See [19] for a review of MIPs in motion planning.
For mixed-integer convex programs, branch-and-bound and branch-and-cut methods are guaranteed to converge to the optimal solution but can be computationally intensive, limiting their utility for embedded applications. In particular, memory requirements can be prohibitive for complex problems due to the need for a queue data structure to store sub-problems [21]. These challenges have led some authors to consider mixedinteger programming heuristics for embedded applications. The most widely-used mixed-integer heuristic is the feasibility pump [22]. The feasibility pump is effective at finding feasible solutions in practice, and is used as a primal heuristic in most high-performance MIP solvers [23]. In contrast with branching-based methods, the feasibility pump is purely iterative and does not require any sub-problem queue, alleviating memory challenges. Because the method is heuristic, there are no convergence guarantees. A disadvantage of the feasibility pump is that each iteration requires the solution of a linear program (LP).
For embedded applications, MIP heuristics based on accelerated dual gradient projection [24] and the alternating direction method of multipliers (ADMM) [25]- [29] have been proposed as well. Like the feasibility pump, ADMM is an iterative method and does not require a sub-problem queue. ADMM-based heuristics are simple to implement, and iterations typically have much lower computational complexity than the feasibility pump. Feasible MIP solutions with low objective function values are often found, motivating the use of ADMM heuristics as a substitute for traditional branchingbased solution methods [25]. Performance can be inconsistent however, and most implementations require running the heuristic several times with different randomly ge
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