Trajectories for the Optimal Collection of Information

TL;DR

This paper develops an optimal path planning method for sensor-equipped aircraft to improve target tracking by minimizing estimation error, using a hybrid PDE-ODE approach to handle complex non-linearities efficiently.

Contribution

It introduces a novel hybrid PDE-ODE method for optimal sensor placement and path planning, overcoming computational challenges in high-dimensional, non-convex problems.

Findings

Hybrid approach reduces computational complexity.

Effective sensor path optimization improves tracking accuracy.

Method handles non-linear, non-convex problem structures.

Abstract

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 by found efficiently from a system of ODEs, avoiding formation of a spatial grid.