Stochastic and Dynamic Routing Problems for Multiple Uninhabited Aerial Vehicles

Abstract

Consider a routing problem for a team of vehicles in the plane: target points appear randomly over time in a bounded environment and must be visited by one of the vehicles. It is desired to minimize the expected system time for the targets, i.e., the expected time elapsed between the appearance of a target point, and the instant it is visited. In this paper, such a routing problem is considered for a team of Uninhabited Aerial Vehicles (UAVs), modeled as vehicles moving with constant forward speed along paths of bounded curvature. Three algorithms are presented, each designed for a distinct set of operating conditions. Each is proven to provide a system time within a constant factor of the optimal when operating under the appropriate conditions. It is shown that the optimal routing policy depends on problem parameters such as the workload per vehicle and the vehicle density in the environment. Finally, there is discussion of a phase transition between two of the policies as the problem parameters are varied. In particular, for the case in which targets appear sporadically, a dimensionless parameter is identified which completely captures this phase transition and an estimate of the critical value of the parameter is provided. Special Euclidean group in two dimensions (i.e., the group of rigid planar motions) t time, s T the steady-state system time, s v vehicle speed, ms −1

Dynamic Vehicle Routing for Robotic Systems

“…One can observe that in heavy-load the optimal system time for the m-Dubins DTRP is of the order λ 2 /(mv) 3 , whereas for the m-DTRP it is of the order λ/(mv) 2 . Therefore, our analysis [18] rigorously establishes the following intuitive fact: boundedcurvature constraints make the optimal system much more sensitive to increases in the demand generation rate. Perhaps less intuitive is the fact that the optimal system time is also more sensitive with respect to the number of vehicles and the vehicle speed in the m-Dubins DTRP as compared to the m-DTRP.…”

“…In this section, we consider the m-DTRP described in the earlier sections with the addition of differential constraints on the vehicle's motion [18]. In particular, we concentrate on vehicles that are constrained to move on the plane at constant speed v > 0 along paths with a minimum radius of curvature ρ > 0.…”

“…In [18], we report results from illustrative numerical experiments on the performance of MC, SL and mBT policies. A close observation of the system time in the light-load case shows that the territorial MC policy is optimal as d ρ → 0 + and the gregarious SL policy is constant-factor optimal as d ρ → +∞.…”

“…We refer to this approach as "algorithmic queueing theory" for dynamic vehicle routing. The power of algorithmic queueing theory stems from the wide spectrum of aspects, critical to the routing of robotic networks, for which it enables a rigorous study; specific examples taken from our work in the past few years include complex models for the demands such as time constraints [11], [12], service priorities [13], and translating demands [14], problems concerning robotic implementation such as adaptive and decentralized algorithms [15], [16], complex vehicle dynamics [17], [18], limited sensing range [19], and team forming [20], and even integration of humans in the design space [21].…”

UAV Routing and Coordination in Stochastic, Dynamic Environments

“…This suggests the existence of a critical value of d ⇢ below which the partitioning policy using Median Voronoi partitions is efficient, and above which the partitioning policy that uses dynamic partitions is efficient. The details of such phase transitions can be found in Enright et al (2009).…”