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$\rm{A}^{\rm{SAR}}$: $\varepsilon$-Optimal Graph Search for Minimum Expected-Detection-Time Paths with Path Budget Constraints for Search and Rescue

Eric Mugford, Jonathan D. Gammell

TL;DR

The paper addresses minimum expected-detection-time trajectory planning for search and rescue under a path-budget and stochastic drift. It introduces A$^{ m{SAR}}$, an $oldsymbol{ extvarepsilon}$-optimal, A*-style graph-search algorithm that provides formal guarantees by using an admissible heuristic $ ilde h$ and a budget $T$, optimizing the finite-horizon objective $J(\sigma)= extstyle\sum_{t=1}^{T} extstyle\sum_{v} b(v,t|\sigma)$. The method relies on a belief-state representation that evolves with target motion via $P(v_j|v_i,t)$ and observation updates with glimpse probability $q(v)$, yielding a tractable bound on the true $ ext{MTTD}$ objective. Simulations across multiple maritime scenarios show A$^{ m{SAR}}$ achieving better solution quality faster than prior ACO and parallel-track baselines, and a Lake Ontario field trial demonstrates practical applicability with a 150 s detection time under real drift conditions.

Abstract

Searches are conducted to find missing persons and/or objects given uncertain information, imperfect observers and large search areas in Search and Rescue (SAR). In many scenarios, such as Maritime SAR, expected survival times are short and optimal search could increase the likelihood of success. This optimization problem is complex for nontrivial problems given its probabilistic nature. Stochastic optimization methods search large problems by nondeterministically sampling the space to reduce the effective size of the problem. This has been used in SAR planning to search otherwise intractably large problems but the stochastic nature provides no formal guarantees on the quality of solutions found in finite time. This paper instead presents $\rm{A}^{\rm{SAR}}$, an $\varepsilon$-optimal search algorithm for SAR planning. It calculates a heuristic to bound the search space and uses graph-search methods to find solutions that are formally guaranteed to be within a user-specified factor, $\varepsilon$, of the optimal solution. It finds better solutions faster than existing optimization approaches in operational simulations. It is also demonstrated with a real-world field trial on Lake Ontario, Canada, where it was used to locate a drifting manikin in only 150s.

$\rm{A}^{\rm{SAR}}$: $\varepsilon$-Optimal Graph Search for Minimum Expected-Detection-Time Paths with Path Budget Constraints for Search and Rescue

TL;DR

The paper addresses minimum expected-detection-time trajectory planning for search and rescue under a path-budget and stochastic drift. It introduces A, an -optimal, A*-style graph-search algorithm that provides formal guarantees by using an admissible heuristic and a budget , optimizing the finite-horizon objective . The method relies on a belief-state representation that evolves with target motion via and observation updates with glimpse probability , yielding a tractable bound on the true objective. Simulations across multiple maritime scenarios show A achieving better solution quality faster than prior ACO and parallel-track baselines, and a Lake Ontario field trial demonstrates practical applicability with a 150 s detection time under real drift conditions.

Abstract

Searches are conducted to find missing persons and/or objects given uncertain information, imperfect observers and large search areas in Search and Rescue (SAR). In many scenarios, such as Maritime SAR, expected survival times are short and optimal search could increase the likelihood of success. This optimization problem is complex for nontrivial problems given its probabilistic nature. Stochastic optimization methods search large problems by nondeterministically sampling the space to reduce the effective size of the problem. This has been used in SAR planning to search otherwise intractably large problems but the stochastic nature provides no formal guarantees on the quality of solutions found in finite time. This paper instead presents , an -optimal search algorithm for SAR planning. It calculates a heuristic to bound the search space and uses graph-search methods to find solutions that are formally guaranteed to be within a user-specified factor, , of the optimal solution. It finds better solutions faster than existing optimization approaches in operational simulations. It is also demonstrated with a real-world field trial on Lake Ontario, Canada, where it was used to locate a drifting manikin in only 150s.

Paper Structure

This paper contains 13 sections, 1 theorem, 13 equations, 4 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. A* For Search and Rescue (A$^{\mathrm{SAR}}$)
  4. Notation
  5. Problem Preliminaries
  6. Problem Formulation
  7. Path Budgeted Search (Algorithm \ref{['alg:asar']})
  8. Heuristic Calculation (Algorithm 2)
  9. Experiments
  10. Drift Model
  11. Simulated Experiments
  12. Lake Ontario Field Experiment
  13. Conclusion

Key Result

Theorem 1

The estimate of the objective-to-go, $\hat{h}(\cdot)$, never overestimates the optimal objective-to-go, $h^*(\cdot)$, and is admissible, i.e.,

Figures (4)

  • Figure 1: Operational field experiment on Lake Ontario, Canada. A manikin (top right) drifted freely for one hour and was modelled with OpenDrift open_drift, with initial particle positions in light grey and the predicted position at search time in dark grey. A$^{\mathrm{SAR}}$ computed the minimum MTTD path, which was flown by a fixed-wing UAV (bottom right). Planned (blue) and flown (magenta) paths are shown up to their intersection with the manikin at 150s. The UAV image of the manikin is shown top left and its true trajectory (black) was calculated after the experiment from an attached GPS tracker.
  • Figure 2: The optimal searcher path (orange) calculated by A$^{\mathrm{SAR}}$ on a toy target distribution for a path length budget of 30, a perfect sensor and fixed starting position. The searcher maintains a target belief state that is updated and plotted with the searcher path. A$^{\mathrm{SAR}}$ plans a path to optimize the MTTD of a given object probability distribution. The belief state probability is reduced according to the sensor model when A$^{\mathrm{SAR}}$ searches a vertex.
  • Figure 3: Experimental results for Bay of Fundy (\ref{['fig:1']}), Lake Ontario (\ref{['fig:2']}), Salish Sea (\ref{['fig:3']}) and Arctic Ocean (\ref{['fig:4']}) scenarios comparing an approach from Pérez-Carabaza_2018 and A$^{\mathrm{SAR}}$. The results are expressed as a percentage of the optimal solution. The median values of the ACO planner after 100 trials are plotted along the ant generations axis with nonparametric 99% confidence intervals. The results of A$^{\mathrm{SAR}}$ are plotted along the suboptimality factor, $\varepsilon$, axis. The parallel track (PT) baseline results for each scenario are included in the figure labels. A$^{\mathrm{SAR}}$ outperforms ACO and parallel track in all tested scenarios for all suboptimality factors while providing optimality guarantees.
  • Figure 4: The OpenDrift model and the paths found by A$^{\mathrm{SAR}}$, an ACO approach Pérez-Carabaza_2018 and parallel track on the Lake Ontario scenario. Fig \ref{['fig:icra_sub1']} shows the OpenDrift Monte Carlo particle drift used to model object location. Figs. \ref{['fig:icra_sub2']}, \ref{['fig:icra_sub3']}, and \ref{['fig:icra_sub4']} show the A$^{\mathrm{SAR}}$, parallel track, and ACO searcher paths after 19 steps, respectively. A$^{\mathrm{SAR}}$ outperforms parallel track and ACO by prioritizing areas with high probability first and conducting the search efficiently, avoiding unnecessary revisits to recently explored vertices when moving toward other high-probability regions.

Theorems & Definitions (1)

  • Theorem 1: Admissibility