UAV Routing and Coordination in Stochastic, Dynamic Environments

Abstract

Recent years have witnessed great advancements in the science and technology for unmanned aerial vehicles (UAVs), for example, in terms of autonomy, sensing, and networking capabilities. This chapter surveys algorithms on task assignment and scheduling for one or multiple UAVs in a dynamic environment, in which targets arrive at random locations at random times, and remain active until one of the UAVs flies to the target’s location and performs an on-site task. The objective is to minimize some measure of the targets’ activity, for example, the average amount of time during which a target remains active. The chapter focuses on a technical approach that relies upon methods from queueing theory, combinatorial optimization, and stochastic geometry. The main advantage of this approach is its ability to provide analytical estimates of the performance of the UAV system on a given problem, thus providing insight into how performance is affected by design and environmental parameters, such as the number of UAVs and the target distribution. In addition, the approach provides provable guarantees on the system’s performance with respect to an ideal optimum. To illustrate this approach, a variety of scenarios are considered, ranging from the simplest case where one UAV moves along continuous paths and has unlimited sensing capabilities, to the case where the motion of the UAV is subject to curvature constraints, and finally to the case where the UAV has a finite sensor footprint. Finally, the problem of cooperative routing algorithms for multiple UAVs is considered, within the same queueing-theoretical framework, and with a focus on control decentralization.

Appendices

Appendix A

1.1 A.1 The Continuous Multi-median Problem

Given a set $\mathcal Q$ ⊂ ℝ^d and a vector P = (p ₁,…, p _m ) of m distinct points in $\mathcal Q$, the expected distance between a random point q, generated according to a probability density function φ, and the closest point in P is given by

$$\matrix{{H_m (P,{\cal Q}):} \hfill & { = {\rm{E}}\left[ {\mathop {\min }\limits_{i \in \left\{ {1,...,m} \right\}} \parallel p_i - q\parallel } \right]} \hfill \cr {} \hfill & { = \sum\limits_{i = 1}^m {\int_{{\cal V}_i \left( {P,{\cal Q}} \right)} {\parallel p_i - q\parallel \varphi (q)dq,} } } \hfill \cr }$$

where $\mathcal V$(P, $\mathcal Q$) = ($\mathcal V$ ₁(P, $\mathcal Q$),…, $\mathcal V$ _m (P, $\mathcal Q$) is the Voronoi partition of the set $\mathcal Q$ generated by the points P. In other words, q ∈ $\mathcal V$ _i (P, $\mathcal Q$) if ∥ q − p _i ∥ ≤ ∥ q − p _k ∥, for all k ∈ {1,…, m}. The set $\mathcal V$ _i is referred to as the Voronoi cell of the generator p _i . The function H _m is known in the locational optimization literature as the continuous Weber function or the continuous multi-median function; see (Agarwal and Sharir 1998; Drezner 1995) and references therein.

The m-median of the set $\mathcal Q$, with respect to the measure induced by φ, is the global minimize

$$\matrix{{P_m^* ({\cal Q}) = {\rm{argmin }}H_m \left( {P,{\cal Q}} \right).} \cr {P \in {\cal Q}^m } \cr }$$

Let $H_m^* ({\cal Q}) = H_m (P_m^* ({\cal Q}),{\cal Q})$ be the global minimum of H _m . It is straightforward to show that the map P ↦ H ₁(P, $\mathcal Q$) is differentiable and strictly convex on $\mathcal Q$. Therefore, it is a simple computational task to compute $P_1^* ({\cal Q})$. It is convenient to refer to $P_1^* ({\cal Q})$ as the median of $\mathcal Q$. On the other hand, the map P ↦ H _m (P, $\mathcal Q$) is differentiable (whenever (p ₁,…, p _m ) are distinct) but not convex, thus making the solution of the continuous m-median problem hard in the general case. It is known (Agarwal and Sharir 1998; Megiddo and Supowit 1984) that the discrete version of the m-median problem is NP-hard for d ≥ 2. Gradient algorithms for the continuous m-median problems can be designed by means of the equality

$${{\partial H_m \left( {P,{\cal Q}} \right)} \over {\partial p_i }} = \int_{{\cal V}_i \left( {P,{\cal Q}} \right)} {{{p_i - q} \over {\parallel p_i - q}}\varphi (q)dq.}$$

The set of critical points of H _m contains all configurations (p ₁,…, p _m ) with the property that each p _i is the generator of the Voronoi cell $\mathcal V$ _i (P, $\mathcal Q$) as well as the median of $\mathcal V$ _i (P, $\mathcal Q$). We refer to such Voronoi diagrams as median Voronoi diagrams. It is possible to show that a median Voronoi diagram always exists for any bounded convex domain $\mathcal Q$ and density φ.

The dependence of $H_m^* ({\cal Q})$ on m plays a crucial role in the design and analysis of algorithms relying on geometric optimization. However, finding the exact relationship for the general case is difficult; hence, it is of great interest to provide bounds on $H_m^* ({\cal Q})$. This problem is studied thoroughly in Papadimitriou (1981) for square regions and in Zemel (1984) for more general compact regions. It is known that, in the asymptotic case

$$\left( {m \to + \infty } \right),\,H_m^* ({\cal Q}) = c_{{\rm{hex}}} \sqrt {A/m}$$

almost surely, where c _hex ≈ 0.377 is the first moment of a hexagon of unit area about its center. This optimal asymptotic value is achieved by placing the m points at the centers of the hexagons in a regular hexagonal lattice within $\mathcal Q$ (the honeycomb heuristic). Working towards the above result, it is also known that for any m ∈ ℕ,

$${2 \over 3}\sqrt {{A \over {\pi m}}} \le H_m^* ({\cal Q}) \le c({\cal Q})\sqrt {{A \over m},}$$

where c($\mathcal Q$) is a constant depending on the shape of $\mathcal Q$.

Appendix B

1.1 B.1 The Euclidean Traveling Salesman Problem

The Euclidean TSP is formulated as follows: given a finite set D of n points in ℝ^d, find the minimum-length closed curve through all points in D. In graph theoretical language, a tour of the point set D is a spanning cycle of the complete graph with vertex set D; the length of a tour is the sum of all Euclidean distances between points in the tour.

The asymptotic behavior of stochastic TSP problems for large n exhibits the following interesting property. Let ETSP(n) be a random variable returning the length of the Euclidean TSP tour through n points, independently and uniformly sampled from a compact set $\mathcal Q$ of unit area; in Beardwood et al. (1959), it is shown that there exists a constant β ₂ such that, almost surely,

$$\mathop {\lim }\limits_{n \to + \infty } {{{\rm{ETSP(}}n{\rm{)}}} \over {\sqrt n }} = \beta _2.$$

In other words, the optimal cost of stochastic TSP tours approaches a deterministic limit and grows as the square root of the number of points to be visited; the current best estimate of the constant appearing in the limit is β ₂ = 0.7120 ± 0.0002; see Percus and Martin (1996) and Johnson et al. (1996). Similar results hold in higher dimensions, and for nonuniform point distributions from Steele (1990), the above limit takes the general form

$$\matrix{{\mathop {\lim }\limits_{n \to + \infty } {{{\rm{ETSP(}}n{\rm{)}}} \over {n^{1 - 1/d} }} = \beta _d \int_{\cal Q} {\bar \varphi (q)^{1 - 1/d} dq} } & {{\rm{almlst surely,}}} \cr }$$

where $\bar \varphi$ is the density of the absolutely continuous part of the distribution φ from which the n points are independently sampled. Notice that the bound holds for all compact sets: the shape of the set only affects the convergence rate to the limit. According to Larson and Odoni (1981), if Q is a “fairly compact and fairly convex” set in the plane, the estimate ETSP(n) ≈ $\beta _2 \sqrt n$ for values of n as low as 15. Remarkably, the asymptotic cost of the stochastic TSP for uniform point distributions is an upper bound on the asymptotic cost for general point distributions, that is,

$$\mathop {\lim }\limits_{n \to + \infty } {{{\rm{ETSP(}}n{\rm{)}}} \over {n^{1 - 1/d} }} \le \beta _d.$$

This follows directly from an application of Jensen’s inequality, that is,

$$\int_{\cal Q} {\bar \varphi (q)^{1 - {1 \over d}} dq, \le \left( {\int_{\cal Q} {\bar \varphi (q)dq} } \right)^{1 - {1 \over d}} \le \varphi ({\cal Q})^{1 - {1 \over d}} = 1.}$$

The TSP is known to be NP-hard, which suggests that there is no general algorithm capable of finding the optimum tour in an amount of time polynomial in the size of the input. Even though the exact optimal solutions of a large TSP can be very hard to compute, several exact and heuristic algorithms and software tools are available for the numerical solution of Euclidean TSPs.

The most advanced TSP solver to date is arguably concorde (Applegate et al. 1998). Heuristic polynomial-time algorithms are available for constant-factor approximations of TSP solutions, such as Christofides’ algorithm, providing a 3/2 approximation factor (Christofides 1972). On a more theoretical side, Arora (1997) proved the existence of polynomial-time approximation schemes, providing a (1+ε) constant-factor approximation for any ε > 0.

A modified version of the Lin-Kernighan heuristic (Lin and Kernighan 1973) is implemented in linkern; this powerful solver yields approximations in the order of 5 % of the optimal tour cost very quickly for many instances. For example, in numerical experiments on a 2.4 GHz Pentium machine, approximations of random TSPs with 1,000 points typically required about 2 s of CPU time. Both concorde and linkern are written in ANSI C and, at the time of writing, are freely available for academic research use at http://www.tsp.gatech.edu/concorde/index.html.

In this chapter, several routing policies were presented requiring on-line solutions of large TSPs. Practical implementations of the algorithms will rely on heuristics, such as Lin-Kernighan’s or Christofides’. If a constant-factor approximation algorithm is used, the effect on the asymptotic performance guarantees of our algorithms can be simply modeled as a scaling of the constant β _d .