Dubins Vehicle Tracking of a Target with Unpredictable Trajectory

Motivated by a fixed-speed, fixed-altitude Unmanned Aerial Vehicle (UAV), we seek to control the turning rate of a planar Dubins vehicle that tracks an unpredictable target at a nominal standoff distance. To account for all realizations of the uncertain target kinematics, we model the target motion as a planar random walk. A Bellman equation and an approximating Markov chain that is consistent with the stochastic kinematics is used to compute an optimal control policy that minimizes the expected value of a cost function based on the nominal distance. Our results illustrate that the control can further be applied to a class of continuous, smooth trajectories with no need for further computation. NOMENCLATURE r Dubins-target standoff distance d Nominal distance φ Viewing angle σ Target noise intensity β(r) Distance-dependent discount factor u ∈ U Admissible turning-rate control dw Increment of unit intensity Wiener process V (r,φ) Value function INTRODUCTION The use of Unmanned Aerial Vehicles (UAVs) to track, protect, or provide surveillance of a ground-based target has recently been the focus of much attention and research. The UAV is assumed to fly at a constant altitude and with a bounded turning ∗Address all correspondence to this author. radius. This behavior is modeled by a planar Dubins vehicle [1], which gives a good approximation for feasible UAV trajectories, even for realistic UAV kinematics [2]. Our goal in this work is to develop a control policy that allows the Dubins vehicle to maintain a nominal standoff distance from the target without the knowledge of the future target trajectory. Path planning and shortest path problems in time and space for Dubins vehicles have been studied previously, in [3–6], among others. This type of vehicle has further been studied for the tracking, geolocation, and coverage time of one or more targets by one or more UAVs [7–12]. Ding et. al. [7, 8] employ Pontryagin’s principle to plan optimal paths that maximize the Dubins vehicle coverage time of a ground target that is stationary or moving in a straight line. Along similar lines, Quintero et. al. [11] use dynamic programming to develop a control policy that minimizes the geolocation error covariance of the observations of a target that is again assumed to move in a straight line. In both of these works, the control is computed with respect to the current distance between the Dubins vehicle and the target. Although this distance is a relative coordinate, the associated cost function provides a control that is optimal over a planning horizon for that specific target trajectory only. Tracking and surveillance of an unknown trajectory would require a separate computation for each possible trajectory, which we avoid with the work presented herein. Instead, the future motion of the target from its current position is assumed to be a planar random walk [13]. The Dubins vehicle tracking control based on this assumption accounts for wild target kinematics, perhaps that of a target avoiding pursuit. Moreover, any continuous, smooth target trajectory can be considered as a realization of the random walk, although the probability of a particular realization may be very small. Therefore, although strictly speaking our tracking control is optimal in the expected value sense for the random walk, it can be applied to a wider class of continuous and smooth target trajectories. The target speed is included in the form of the intensity of the noise characterizing the random walk. To clarify results we assume a constant noise intensity, although the target speed is not necessarily required to be constant, as described below. The optimal control feedback policy minimizes the expected value of a cost function which depends on the nominal distance between the Dubins vehicle and the target. The policy is computed off-line using a Bellman equation and an approximating Markov chain that is consistent with the stochastic kinematics [14]. Stochastic problems in the control of Dubins vehicles typically concentrate on variants of the Traveling Salesperson Problem and other routing problems, in which the target location is unknown or randomly-generated [15–17]. Other works examine control methods that direct Dubins vehicle motion toward the maximum of a scalar field [18, 19]. To the best of our knowledge, our work is the first use of stochastic optimal control for the type of problem at hand. In what follows, we first formulate our problem for the case of a Brownian target and then derive the corresponding relative kinematics model for this problem. Next, the dynamic programming methodologies to compute the optimal control for this problem are provided, and the effectiveness of this approach is demonstrated for Brownian targets and targets with unknown trajectory. We conclude with a discussion and direction for future research. PROBLEM FORMULATION We consider a UAV flying at a constant altitude in the vicinity of a ground-based target, tasked with maintaining a nominal distance from the target. The target is located at position ~rT (t) = [xT (t), yT (t)] T at the time point t (see Fig. 1), and since we do not account for the possibility of antagonistic target trajectories, no knowledge of the UAV kinematics or state is assumed. The UAV, located at position ~rA(t) = [xA(t), yA(t)] T , moves in the direction of its heading angle θ at a constant speed vA. The turning rate is determined by a non-anticipative [14], bounded control u(t)∈ U ≡{u : |u| ≤ umax}, which has to be found. Note that by considering a model with more constraints (e.g., fixed vA and altitude, single integrator) tracking is more difficult for the UAV. In our problem formulation, the target motion is unknown. Drawing from the field of estimation, the simplest signal that can be used to describe an unknown model suggests that the motion of the target should be described by a 2D Brownian particle: dxT (t) = σdwx, dyT (t) = σdwy (1) where dwx and dwy are increments of unit intensity Wiener processes along the x and y axes, respectively, which are mutually x̂ ŷ