The Dubins Traveling Salesperson Problem with Stochastic Dynamics

Motivated by applications in which a nonholonomic robotic vehicle should sequentially hit a series of waypoints in the presence of stochastic drift, we formulate a new version of the Dubins vehicle traveling salesperson problem. In our approach, we first compute the minimum expected time feedback control to hit one waypoint based on the Hamilton-Jacobi-Bellman equation. Next, minimum expected times associated with the control are used to construct a traveling salesperson problem based on a waypoint hitting angle discretization. We provide numerical results illustrating our solution and analyze how the stochastic drift affects the solution. NOMENCLATURE v Speed of Dubins vehicle ∆xi x component of distance from Dubins vehicle to waypoint i ∆yi y component of distance from Dubins vehicle to waypoint i θ Heading angle of Dubins vehicle u Feedback-controlled turning rate N Number of waypoints K Number of possible final heading angles θ f k Discrete final heading angle at which the Dubins vehicle should hit the waypoint, k = 1, ...K T (·) Minimum expected time to hit the waypoint μ(n) Index in 1, . . . ,N of the nth waypoint to be visited c(·) Edge length between two nodes ∗Address correspondence to this author. INTRODUCTION The problem of navigating a nonholonomic robotic vehicle or small aerial vehicle through a series of waypoints in minimum time has recently received much attention. Applications include target surveillance by mobile sensors and small aerial vehicles [1,2], vehicle routing [3], and environmental sampling [4], for example. The kinematics of the robotic vehicles is often approximated by that of a Dubins vehicle [5], which is constrained to move at fixed speed in the direction of its heading angle along paths of bounded curvature. The problem of finding the shortest path for the Dubins vehicle through a series of points has been extensively analyzed [6, 7] and has become known as the Dubins Traveling Salesman Problem (Dubins TSP [or DTSP]). Variations on this problem have considered worst-case bounds [8] for the DTSP and include extensions for multiple vehicles [9] and for other dynamical systems [10, 11]. Recent interest has paid particular attention to the stochastic case, in which the waypoint locations arise from a spatial or spatio-temporal stochastic process, and a DTSP is formulated minimize the expected time to hit or service all waypoints [6,7,12,13]. However, in each of these stochastic DTSPs, the stochasticity arises in the locations of the stationary waypoints, while the motion of the vehicle is deterministic. The scenario in which the relative motion between the Dubins vehicle and the waypoints is stochastic has alluded attention in literature thus far. Therefore, in this paper, we provide a method for a Dubins vehicle under the influence of stochasticity to plan and execute a closed path (tour) through a series of waypoints before returning to the initial location in minimum (expected) time. In reality, it may be difficult for the Dubins vehicle to follow a prescribed sequence of points due to stochastic drifts. Because the stochastic drift can influence the tour taken by the Dubins vehicle in practice, our offline approach based on the Hamilton-JacobiBellman equation takes into account the possibility for the vehicle to be “blown” off course by the stochastic effects when approaching a waypoint, and we further include the possibility for a waypoint to be hit at a heading angle that was not intended. Previous work for a Dubins vehicle to hit a single waypoint in minimum expected time in the presence of stochastic drifts can be found in [14], and see [4] for the case of an a priori fixed tour. This paper is organized as follows. In the following section, we formulate the problem and describe how it may be decoupled in terms of a stochastic optimal control problem and a subsequent traveling salesperson optimization problem. Next, we describe how these two stages are linked through an appropriate selection of the state space variables. We then present in more detail the stochastic optimal control problem and the TSP problem. We provide simulations based on our proposed method, and finally, we conclude with a discussion our findings and provide directions for future research. PROBLEM FORMULATION Here we formulate the problem of navigating a Dubins vehicle in the presence of stochastic drift to visit a series of waypoints and then return to its initial position (e.g., a runway or dock) in minimum expected time. The Dubins vehicle has position (x(t),y(t)) and moves in the direction of its heading angle θ(t) at fixed speed v relative to the stochastic drift so that its kinematics is described by the equations dx(t) = vcos(θ)dt + σdwx (1) dy(t) = vsin(θ)dt + σdwy (2) dθ(t) = udt, |u| ≤ 1 (3) where σ is a known noise intensity of the stochastic drift, and where dwx and dwy are mutually independent increments of a standard Wiener process. This type of stochasticity could arise in the model of an uncertain drift field in which the Cartesian components of the drift are assumed to be independent, with zero mean, and with known variance, for example. The feedbackcontrolled turning rate is u, which is bounded as u ∈ [−1,1]. The positions of N waypoints are fixed at (xi,yi), i = 1, . . . ,N. The evolution equations for the Cartesian components of the distance to a waypoint i, ∆xi(t) = xi − x(t) and ∆yi(t) = yi− y(t), are d∆xi(t) =−vcos(θ)dt + σdwx (4) d∆yi(t) =−vsin(θ)dt + σdwy (5) dθ(t) = udt, |u| ≤ 1, (6) where we have chosen to use the symmetry of the Wiener process to revert the sign of the diffusion factor from (1)-(2). ∆yi ∆xi