arXiv subjects

Andrew F. Thompson

Publications and source records attributed to Andrew F. Thompson.

9 recordsLinked to original sources

arXiv2026

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

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.

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arXiv2026

Motion Planning with Metric Temporal Logic Using Reachability Analysis and Hybrid Zonotopes

Metric temporal logic (MTL) provides a formal framework for defining time-dependent mission requirements on autonomous vehicles. However, optimizing control decisions subject to these constraints is often computationally expensive. This article presents a method that uses reachability analysis to implicitly express the set of states satisfying an MTL specification and then optimizes to find a motion plan. The hybrid zonotope set representation is used to efficiently and conveniently encode MTL specifications into reachable sets. A numerical benchmark highlights the proposed method's computational advantages as compared to existing methods in the literature. Further numerical examples and an experimental application demonstrate the ability to address time-varying environments, region-dependent disturbances, and multi-agent coordination.

eess.SY↗

arXiv2025

Integration of a Graph-Based Path Planner and Mixed-Integer MPC for Robot Navigation in Cluttered Environments

The ability to update a path plan is a required capability for autonomous mobile robots navigating through uncertain environments. This paper proposes a re-planning strategy using a multilayer planning and control framework for cases where the robot's environment is partially known. A medial axis graph-based planner defines a global path plan based on known obstacles, where each edge in the graph corresponds to a unique corridor. A mixed-integer model predictive control (MPC) method detects if a terminal constraint derived from the global plan is infeasible, subject to a non-convex description of the local environment. Infeasibility detection is used to trigger efficient global re-planning via medial axis graph edge deletion. The proposed re-planning strategy is demonstrated experimentally.

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arXiv2025

Automated Functional Decomposition for Hybrid Zonotope Over-approximations with Application to LSTM Networks

Functional decomposition is a powerful tool for systems analysis because it can reduce a function of arbitrary input dimensions to the sum and superposition of functions of a single variable, thereby mitigating (or potentially avoiding) the exponential scaling often associated with analyses over high-dimensional spaces. This paper presents automated methods for constructing functional decompositions used to form set-based over-approximations of nonlinear functions, with particular focus on the hybrid zonotope set representation. To demonstrate these methods, we construct a hybrid zonotope set that over-approximates the input-output graph of a long short-term memory neural network, and use functional decomposition to represent a discrete hybrid automaton via a hybrid zonotope.

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arXiv2024

Energy-Aware Predictive Motion Planning for Autonomous Vehicles Using a Hybrid Zonotope Constraint Representation

Uncrewed aerial systems have tightly coupled energy and motion dynamics which must be accounted for by onboard planning algorithms. This work proposes a strategy for coupled motion and energy planning using model predictive control (MPC). A reduced-order linear time-invariant model of coupled energy and motion dynamics is presented. Constrained zonotopes are used to represent state and input constraints, and hybrid zonotopes are used to represent non-convex constraints tied to a map of the environment. The structures of these constraint representations are exploited within a mixed-integer quadratic program solver tailored to MPC motion planning problems. Results apply the proposed methodology to coupled motion and energy utilization planning problems for 1) a hybrid-electric vehicle that must restrict engine usage when flying over regions with noise restrictions, and 2) an electric package delivery drone that must track waysets with both position and battery state of charge requirements. By leveraging the structure-exploiting solver, the proposed mixed-integer MPC formulations can be implemented in real time.

eess.SY↗

arXiv2024

Error Bounds for Compositions of Piecewise Affine Approximations

Nonlinear expressions are often approximated by piecewise affine (PWA) functions to simplify analysis or reduce computational costs. To reduce computational complexity, multivariate functions can be represented as compositions of functions with one or two inputs, which can be approximated individually. This paper provides efficient methods to generate PWA approximations of nonlinear functions via functional decomposition. The key contributions focus on intelligent placement of breakpoints for PWA approximations without requiring optimization, and on bounding the error of PWA compositions as a function of the error tolerance for each component of that composition. The proposed methods are used to systematically construct a PWA approximation for a complicated function, either to within a desired error tolerance or to a given level of complexity.

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arXiv2024

Reachability Analysis Using Hybrid Zonotopes and Functional Decomposition

This paper proposes methods for reachability analysis of nonlinear systems in both open loop and closed loop with advanced controllers. The methods combine hybrid zonotopes, a construct called a state-update set, functional decomposition, and special ordered set approximations to enable linear growth in reachable set memory complexity with time and linear scaling in computational complexity with the system dimension. Facilitating this combination are new identities for constructing nonconvex sets that contain nonlinear functions and for efficiently converting a collection of polytopes from vertex representation to hybrid zonotope representation. Benchmark numerical examples from the literature demonstrate the proposed methods and provide comparison to state-of-the-art techniques.

eess.SY↗

arXiv2023

Set-valued State Estimation for Nonlinear Systems Using Hybrid Zonotopes

This paper proposes a method for set-valued state estimation of nonlinear, discrete-time systems. This is achieved by combining graphs of functions representing system dynamics and measurements with the hybrid zonotope set representation that can efficiently represent nonconvex and disjoint sets. Tight over-approximations of complex nonlinear functions are efficiently produced by leveraging special ordered sets and neural networks, which enable computation of set-valued state estimates that grow linearly in memory complexity with time. A numerical example demonstrates significant reduction of conservatism in the set-valued state estimates using the proposed method as compared to an idealized convex approach.

eess.SY↗

arXiv2020

A pole-to-equator ocean overturning circulation on Enceladus

Enceladus is believed to have a saltwater global ocean with a mean depth of at least 30~km, heated from below at the ocean-core interface and cooled at the top, where the ocean loses heat to the icy lithosphere above. This scenario suggests an important role for vertical convection to influence the interior properties and circulation of Enceladus' ocean. Additionally, the ice shell that encompasses the ocean has dramatic meridional thickness variations that, in steady state, must be sustained against processes acting to remove these ice thickness anomalies. One mechanism that would maintain variations in the ice shell thickness involves spatially-separated regions of freezing and melting at the ocean-ice interface. Here, we use an idealized, dynamical ocean model forced by an observationally-guided density forcing at the ocean-ice interface to argue that Enceladus' interior ocean should support a meridional overturning circulation. This circulation establishes an interior density structure that is more complex than in studies that have focused only on convection, including a shallow freshwater lens in the polar regions. Spatially-separated sites of ice formation and melt enable Enceladus to sustain a significant vertical and horizontal stratification, which impacts interior heat transport, and is critical for understanding the relationship between a global ocean and the planetary energy budget. The presence of low salinity layers near the polar ocean-ice interface also influences whether samples measured from the plumes are representative of the global ocean.

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