A Hamilton–Jacobi Formulation for Optimal Coordination of Heterogeneous Multiple Vehicle Systems

Kirchner, Matthew R.; Debord, Mark; Hespanha, João P.

doi:10.1109/iros45743.2020.9340864

Cited by 5 publications

(4 citation statements)

References 25 publications

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“…Under a set of mild Lipschitz continuity assumptions, there exists a unique value function (12) that satisfies the following Hamilton-Jacobi (HJ) equation [15] with V (t, χ) being the viscosity solution of the partial differential equation (PDE)…”

Section: Hamilton-jacobi Formulationmentioning

confidence: 99%

“…We note here that under mild assumptions, the viscosity solution of (17 such that ū optimizes (12). By Pontryagin's theorem [18] there exists adjoint trajectories p (s) := p (s; χ) and λ (s) := λ (s; χ) such that the function…”

Section: Hamilton-jacobi Formulationmentioning

confidence: 99%

“…New research has emerged in an attempt to bridge this technological gap, including trajectory optimization approaches [5], [6], [7], machine learning techniques [8], [9], [10], and sub-problem decomposition [11], [12]. The structure of the sensor placement problem lends itself well to the later strategy.…”

Section: Introductionmentioning

confidence: 99%

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Trajectories for the Optimal Collection of Information

Kirchner¹,

Grimsman²,

Hespanha³

et al. 2023

Preprint

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We study a scenario where an aircraft has multiple heterogeneous sensors collecting measurements to track a target vehicle of unknown location. The measurements are sampled along the flight path and our goals to optimize sensor placement to minimize estimation error. We select as a metric the Fisher Information Matrix (FIM), as "minimizing" the inverse of the FIM is required to achieve small estimation error. We propose to generate the optimal path from the Hamilton-Jacobi (HJ) partial differential equation (PDE) as it is the necessary and sufficient condition for optimality. A traditional method of lines (MOL) approach, based on a spatial grid, lends itself well to the highly non-linear and non-convex structure of the problem induced by the FIM matrix. However, the sensor placement problem results in a state space dimension that renders a naive MOL approach intractable. We present a new hybrid approach, whereby we decompose the state space into two parts: a smaller subspace that still uses a grid and takes advantage of the robustness to non-linearities and non-convexities, and the remaining state space that can be found efficiently from a system of ODEs, avoiding formation of a spatial grid.

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Section: Hamilton-jacobi Formulationmentioning

confidence: 99%

Section: Hamilton-jacobi Formulationmentioning

confidence: 99%

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Trajectories for the Optimal Collection of Information

Kirchner¹,

Grimsman²,

Hespanha³

et al. 2023

Preprint

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“…In literature, there are some algorithms for solving high-dimensional optimal control problems (or the corresponding Hamilton-Jacobi PDEs), which include optimization methods [18,24,21,23,15,14,87,60,55], max-plus methods [1,2,29,35,39,67,66,68,69], tensor decomposition techniques [28,44,86], sparse grids [10,37,53], polynomial approximation [51,52], model order reduction [4,57], optimistic planning [9], dynamic programming and reinforcement learning [13,11,3,8,89], as well as methods based on neural networks [5,6,27,47,42,45,46,58,73,78,81,84,62,20,…”

mentioning

confidence: 99%

SympOCnet: Solving optimal control problems with applications to high-dimensional multi-agent path planning problems

Meng¹,

Zhang²,

Darbon³

et al. 2022

Preprint

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Solving high-dimensional optimal control problems in real-time is an important but challenging problem, with applications to multi-agent path planning problems, which have drawn increased attention given the growing popularity of drones in recent years. In this paper, we propose a novel neural network method called SympOCnet that applies the Symplectic network to solve high-dimensional optimal control problems with state constraints. We present several numerical results on path planning problems in two-dimensional and three-dimensional spaces. Specifically, we demonstrate that our SympOCnet can solve a problem with more than 500 dimensions in 1.5 hours on a single GPU, which shows the effectiveness and efficiency of SympOCnet. The proposed method is scalable and has the potential to solve truly high-dimensional path planning problems in real-time.

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