ON SAMPLING BASED METHODS FOR THE DUBINS TRAVELING SALESMAN PROBLEM WITH NEIGHBORHOODS

Acta Polytechnica CTU Proceedings 2:57–61, 2015 doi:10.14311/APP.2015.1.0057 © Czech Technical University in Prague, 2015 available online at http://ojs.cvut.cz/ojs/index.php/app ON SAMPLING BASED METHODS FOR THE DUBINS TRAVELING SALESMAN PROBLEM WITH NEIGHBORHOODS Petr Váňa∗ , Jan Faigl Dept. of Computer Science, Czech Technical University in Prague, Technická 2, 166 27 Prague, Czech Republic ∗ corresponding author: Abstract. In this paper, we address the problem of path planning to visit a set of regions by Dubins vehicle, which is also known as the Dubins Traveling Salesman Problem Neighborhoods (DTSPN). We propose a modification of the existing sampling-based approach to determine increasing number of samples per goal region and thus improve the solution quality if a more computational time is available. The proposed modification of the sampling-based algorithm has been compared with performance of existing approaches for the DTSPN and results of the quality of the found solutions and the required computational time are presented in the paper. Keywords: dubins vehicle, dubins maneuver, DTSP, DTSPN. 1. Introduction Path planning for a non-holonomic vehicle is a fundamental problem of surveillance mission where an unmanned aerial vehicle (such as a fixed-wing) is requested to visit a given set of locations. The basic model of such a vehicle with the limited turning radius is called the Dubins vehicle [1] for which the combinatorial problem of finding optimal sequence of visits to the locations is known as the Dubins Traveling Salesman Problem (DTSP) [2]. In this paper, we consider a generalized problem of the DTSP where the particular waypoints to be visited can be selected from a set of possible locations. Due to the similarity of this problem with the so-called Traveling Salesman Problem with Neighborhoods [3], the problem is called as the Dubins Traveling Salesman Problem with Neighborhoods (DTSPN) [4]. The DTSPN is a suitable problem formulation to address surveillance missions with unmanned aerial vehicles, where it is required to take a camera snapshot (or other type of measurement) of each target location. Such a snapshot can be acquired from some distance of the target location and thus it is not necessary to visit the location exactly. It is rather a more suitable to select the waypoints in such a way that all locations are covered while the total cost of the final path is minimal. There are several approaches to address the DTSP and also the DTSPN in the literature [2, 4–6] including our work [7]. Therefore, in this paper, we provide an overview of the existing approaches to address the DTSPN and compare their performance according to the trade-off between the solution quality and computational requirements. In particular, we focus on the sampling-based algorithm [4] which is able to provide high quality solutions for very high number of samples. However, more samples increase the computational burden, and therefore, the algorithm is not directly suitable for real-time applications. On the other hand, we can get a quick estimation of the solution quality using only few samples. This has motivated us to modify the original algorithm [4] to provide a first solution quickly that is then improved if a computational time is left. The paper is organized as follows. A brief introduction to the problem background is presented in the next section. The addressed problem is formally introduced in Section 3. A brief overview of the existing approaches to solve the DTSPN is provided in Section 4. The proposed modification of the samplingbased approach is presented in Section 5 together with the analysis of its computational complexity. Evaluation results of the comparison of the existing approaches with the proposed modification are reported in Section 6. Finally concluding remarks are denoted to Section 7. 2. Related Work The problem of curvature-constrained path planning has been studied years ago and the fundamental work has been published in 1957 by Dubins. In [1], he proved that the optimal path between two configurations q1 , q2 ∈ SE(2) consists of only straight line segments and segments with the minimum turning radius. He also showed that only 6 maneuvers can be optimal, which can be further divided into two main types: • CCC type: LRL, RLR; • CSC type: LSL, LSR, RSL, RSR; where C stands for a circle turn (R – right, L – left) and S for a straight segment. Even though the optimal path for Dubins vehicle between two configurations is known and it is straightforward to compute, this is not sufficient to directly solve the DTSP. It is because the orientation of the vehicle at the waypoints is not known and thus it must be determined together with the optimal sequence of 57 Petr Váňa, Jan Faigl visits to the waypoints. Moreover, the optimal orientation depends on the sequence and vice-versa, which make the problem difficult to address. Probably the simplest approach to address the DTSP is the Alternating Algorithm (AA) proposed in [2]. In this approach, headings are established in the way that even edges are connected by straight line segments and odd edges are connected by Dubins maneuvers. In addition, the authors show that the length of the optimal solution of the DTSP can be bounded by LT SP κdn/2eπρ, where ρ is minimum turning radius, LT SP is the length of the optimal solution of the Euclidean TSP, n is the number of the goals, and κ < 2.658. Based on the similar idea, authors of [5] proposed a receding horizon algorithm called the look-ahead (LA) approach. The heading is determined with respect to the next point in the sequence. This algorithm investigates each point only once, and therefore, the LA approach is very fast and suitable for real-time application while the authors reported better results than the AA. The optimal solution of the Dubins planning to visit a given sequence of waypoints that are at the distance longer than 4ρ is presented in [8]. The approach is based on the convex optimization; however, the optimization needs to be solved several times because of possible alternation of the maneuvers directions. The authors bound the number of possible combinations to 2n−2 for n waypoints. The DTSPN is a generalization of the DTSP, where each goal is extended by a neighborhood (goal region). As the DTSP is known to be NP-hard [6], also its generalization the DTSPN is NP-hard. 3. Problem Definition The addressed problem is motivated by surveillance missions with fixed-wing aerial vehicles, which are nonholonomic vehicles due to their kinodynamic constraints. These vehicles are often modeled as the Dubins vehicle [1], which can go only forward at a constant speed and has a minimum turning radius ρ. The configuration space of such a vehicle can be represented as SE(2), where each configuration q ∈ SE(2) is a triplet (x, y, θ), where (x, y) is the vehicle position in a plane and θ ∈ S1 is the orientation of the vehicle. The mathematical mode (...truncated)