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Drone Air Traffic Control: Tracking a Set of Moving Objects with Minimal Power

Chek-Manh Loi, Michael Perk, Malte Hoffmann, Sándor Fekete

TL;DR

An algorithm based on geometric insights that is able to find optimal solutions for the $\min \max$ variant of the problem, which aims at minimizing peak power consumption.

Abstract

A common sensing problem is to use a set of stationary tracking locations to monitor a collection of moving devices: Given $n$ objects that need to be tracked, each following its own trajectory, and $m$ stationary traffic control stations, each with a sensing region of adjustable range; how should we adjust the individual sensor ranges in order to optimize energy consumption? We provide both negative theoretical and positive practical results for this important and natural challenge. On the theoretical side, we show that even if all objects move at constant speed along straight lines, no polynomial-time algorithm can guarantee optimal coverage for a given starting solution. On the practical side, we present an algorithm based on geometric insights that is able to find optimal solutions for the $\min \max$ variant of the problem, which aims at minimizing peak power consumption. Runtimes for instances with 500 moving objects and 25 stations are in the order of seconds for scenarios that take minutes to play out in the real world, demonstrating real-time capability of our methods.

Drone Air Traffic Control: Tracking a Set of Moving Objects with Minimal Power

TL;DR

An algorithm based on geometric insights that is able to find optimal solutions for the variant of the problem, which aims at minimizing peak power consumption.

Abstract

A common sensing problem is to use a set of stationary tracking locations to monitor a collection of moving devices: Given objects that need to be tracked, each following its own trajectory, and stationary traffic control stations, each with a sensing region of adjustable range; how should we adjust the individual sensor ranges in order to optimize energy consumption? We provide both negative theoretical and positive practical results for this important and natural challenge. On the theoretical side, we show that even if all objects move at constant speed along straight lines, no polynomial-time algorithm can guarantee optimal coverage for a given starting solution. On the practical side, we present an algorithm based on geometric insights that is able to find optimal solutions for the variant of the problem, which aims at minimizing peak power consumption. Runtimes for instances with 500 moving objects and 25 stations are in the order of seconds for scenarios that take minutes to play out in the real world, demonstrating real-time capability of our methods.
Paper Structure (28 sections, 4 theorems, 3 equations, 11 figures)

This paper contains 28 sections, 4 theorems, 3 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Related Work
  4. Hardness
  5. Stationary Disk Cover
  6. Nearest-Neighbor Heuristic
  7. Integer Programming
  8. Computing the Candidate Set
  9. IP Formulation
  10. Algorithmic Approaches
  11. Ensuring Feasibility
  12. Improving Extended Solutions
  13. Iterative Algorithm
  14. Initial Solution
  15. Extending the Solution
  16. ...and 13 more sections

Key Result

Theorem 1

Given a KDC instance $I=(\mathcal{P}, \mathcal{Y})$, a cost bound $c$, and an optimal solution for the static instance at time $t_0$, determining whether there exists an assignment of radii at a later time $t_1 > t_0$ that covers all points with total cost at most $c$ is NP-hard.

Figures (11)

  • Figure 1: Solution of the KDC problem for $n=10$ moving points and $m=5$ stations. The points move along the dotted lines. The stations are shown as green triangles and are opaque, if the radius is zero. We start with an IP solution at time $t=0$. Our algorithm can find a solution over time, that uses at most 8022.23□m of area over the whole time interval. This solution is matched by a lower-bound computed by solving the static DC problem at this time step.
  • Figure 2: The supporting point of station $y$ changes from $p_1$ to $p_2$, to ensure coverage of the objects at all times.
  • Figure 3: Degenerate cases after which the assignment of supporting point might change. In both cases we choose $p_2$ as the new supporting point.
  • Figure 4: (Left) Station $y_2$ takes over a point $p_2$. The objective values of the red and blue disks are equal at this point in time. We change the supporting point of both $y_1$ and $y_2$ at the same time. (Right) Generalization of the case depicted on the left to more than two stations.
  • Figure 5: Multiple stations and objects in the same event point. Either the blue or the red disk cover the objects, but not both.
  • ...and 6 more figures

Theorems & Definitions (8)

  • Theorem 1
  • proof
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof