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

Joshua A. Robbins; Andrew F. Thompson; Jonah J. Glunt; Herschel C. Pangborn

cs.ROFeb 19, 2026

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

Authors: Joshua A. Robbins, Andrew F. Thompson, Jonah J. Glunt, Herschel C. Pangborn

Organizations: Department of Mechanical Engineering, The Pennsylvania State University, University Park, PA 16802 USA

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 & tables

Panel of Figure 1 — Fig. 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.

method	$n_{Gc}$	$n_{Gb}$	$n_{C}$	$\nnz{G}$	$\nnz{A}$
PWA	$Nn_{Gc,\psi}+n_{G,0}$	$Nn_{Gb,\psi}$	$N(n_{C,\psi}+n_{x})$	$\mathrm{nnz}(G_{\psi,+})$	$N(2\mathrm{nnz}(G_{\psi,+})+\mathrm{nnz}(A_{\psi}))+n_{x}n_{G,0}$
MLD	$N(n_{G,u}+n_{Gc,w}+n_{e})$	$Nn_{Gb,W}$	$N(n_{C,w}+n_{e})$	$Nn_{x}n_{G,\mathrm{uw}}$	$N^{2}\left(\frac{n_{e}n_{G,\mathrm{uw}}}{2}\right)+N(n_{C,w}(n_{Gc,w}+n_{Gb,w})$
	$+n_{G,0}$			$+n_{x}n_{G,0}$	$+2n_{e}+n_{G,\mathrm{uw}}/2+n_{G,0})$

TABLE I: Memory complexity upper bounds for

N

-step reachable sets

\mathcal{X}_{N}

using MLD system reachability calculations and the proposed PWA system reachability calculations.

Panel of Figure 3 — Fig. 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 ( 26 ) does not preserve sharpness.

set	method	$n$	$n_{Gc}$	$n_{Gb}$	$n_{C}$	$\mathrm{nnz}(G)$	$\mathrm{nnz}(A)$
$\mathcal{X}_{15}$	MLD	2	182	15	150	2	692
$\mathcal{X}_{15}$	PWA cond.	2	92	30	75	12	442
$\mathcal{X}_{15}$	PWA sharp	2	122	30	105	12	502
$\mathcal{Z}_{15}$	PWA cond.	32	152	30	135	34	722
$\mathcal{Z}_{15}$	PWA sharp	32	182	30	165	34	782

TABLE II: Memory complexity of unconstrained reachable sets

\mathcal{X}_{15}

(Def. 2 ) and lifted, constrained reachable sets

\mathcal{Z}_{15}

(Def. 3 ).

parameter	$\rho$	$\epsilon_{p}$	$k_{\mathrm{restart}}$	$k_{ph1}$	$k_{ph2}$	$l_{\mathrm{buf}}$	$\epsilon_{\mathrm{buf}}$
value	10	0.001	5000	10000	90000	20	0.001

TABLE III: Default parameter values for ADMM-FP.

parameter	$\alpha_{0}$	$\phi$	$t$	$\delta_{\alpha}$	$l_{\mathrm{buf}}$
value	1	0.9	20	0.005	10

TABLE IV: Default parameter values for OFP [ 33 ] .

Panel of Figure 7 — Fig. 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.

Panel of Figure 8 — Fig. 6: Trajectories produced by ADMM-FP prior to convergence at the specified numerical tolerance $\epsilon_{p}=0.001$ . Colors indicate the primal residual norm $r_{p}$ of a given trajectory, with convergence corresponding to $r_{p}<\epsilon_{p}$ .

Panel of Figure 9 — Fig. 7: Times to find a feasible solution, peak memory usage, and percentage of cases where a feasible solution was found for 100 randomly generated motion planning problem. Whiskers in (a) and bands in (b) enclose all data points.

statistic	solve time (sec)	sub-optimality (%)
median	0.23	19.1
minimum	0.06	3.51
maximum	1.29	31.6

TABLE V: ADMM-FP solution statistics over 25 trials for a representative reach-avoid problem.

statistic	run time (sec)	iteration count
median	0.99	14,915
minimum	0.03	377
maximum	7.8	121,561

TABLE VI: ADMM-FP solution statistics over 50 trials with

\epsilon_{p}=0.01

for the ball and paddle system.

Panel of Figure 12 — Fig. 8: Example ball and paddle system solution with $\epsilon_{p}=0.01$ . The black bar is the ceiling and the red bar is the paddle. The opaque ball corresponds to the planned trajectory while the translucent ball corresponds to the open-loop simulation using the planned control inputs.

Panel of Figure 13 — Fig. 9: Velocity constraints for combined behavior and motion planning system.

Panel of Figure 14 — Fig. 11: Comparison of convex relaxation (C.R.) initialization and warm-start (W.S.) initialization for randomly generated combined behavior and motion planning scenarios. Warm-start times include the time to solve ( 46 ), and convex relaxation times include the time to solve the relaxation of ( 19 ). Whiskers enclose all data points.

Panel of Figure 15 — Fig. 12: Overhead images from the experimental demonstration of combined behavior and motion planning. Imperfect correction of the fisheye effect in the overhead camera results in some minor misalignment of the behavior and motion plan with the camera images.

	unit	median	min	max
ADMM-FP time	ms	5.2	3.4	103.0
ADMM-FP iterations	count	114	21	3888
Warm start time	ms	15.4	4.1	34.5
Eq. ( 25 ) and Eq. ( 24b ) time	ms	1.9	1.3	1.9
Alg. 2 time	ms	4.4	2.9	6.4

TABLE VII: Problem formulation and solution time statistics for combined behavior and motion planning experiment.

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