THE DUBINS TRAVELING SALESMAN PROBLEM WITH CONSTRAINED COLLECTING MANEUVERS

Acta Polytechnica CTU Proceedings 6:34–39, 2016 doi:10.14311/APP.2016.6.0034 © Czech Technical University in Prague, 2016 available online at http://ojs.cvut.cz/ojs/index.php/app THE DUBINS TRAVELING SALESMAN PROBLEM WITH CONSTRAINED COLLECTING MANEUVERS 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 introduce a variant of the Dubins traveling salesman problem (DTSP) that is called the Dubins traveling salesman problem with constrained collecting maneuvers (DTSP-CM). In contrast to the ordinary formulation of the DTSP, in the proposed DTSP-CM, the vehicle is requested to visit each target by specified collecting maneuver to accomplish the mission. The proposed problem formulation is motivated by scenarios with unmanned aerial vehicles where particular maneuvers are necessary for accomplishing the mission, such as object dropping or data collection with sensor sensitive to changes in vehicle heading. We consider existing methods for the DTSP and propose its modifications to use these methods to address a variant of the introduced DTSP-CM, where the collecting maneuvers are constrained to straight line segments. Keywords: Dubins vehicle, DTSP, non-holonomic vehicle, path planning, collecting maneuver. 1. Introduction The Dubins traveling salesman problem (DTSP) [1] has been subject of studies for many years. The problem is to find a shortest tour visiting a given set of target locations by the Dubins vehicle [2] that models a fix-wing aircraft (or car-like vehicle) with a limited minimal turning radius and a constant forward velocity. The herein studied path planning problem is motivated by surveillance and rescue missions to visit a set of target locations by a fix-wing aerial vehicle. Such a problem can be formulated as the variant of the combinatorial optimization the Traveling salesman problem (TSP) where the target locations have to be connected by Dubins maneuvers. The problem is to determine a sequence of visits to the targets together with the most suitable heading of the vehicle at each target location such that the total tour length is minimized. Dubins proved that the optimal path between two configurations of the Dubins vehicle can be found analytically by enumerating six possible maneuvers [2]. However, it is necessary to determine the optimal headings to find the optimal maneuver analytically, which in fact depend on the optimal sequence of visits to the targets. Therefore, this challenging problem is called the Dubins traveling salesman problem (DTSP) rather than just the TSP to distinguish computational challenges arising from the constrained turning radius. The DTSP is NP-hard [3] as for zero minimal turning radius, the problem becomes the TSP. Even though an ordinary formulation of the DTSP provides a solution of the problem to visit a set of target locations by an aircraft, the problem considered in this paper requires to perform specific collecting maneuvers to reliably accomplish the mission. Such a requirement is arising from the need to get a visual contact with the drop-off location. Then, the vehicle 34 Figure 1. An example of the DTSP-CM instance with straight collecting maneuvers in a cargo airdrop mission. Target locations are depicted as small green disks, maneuvers are the red straight line segments, and particular trajectories for a reliable cargo drop-off are in green. The blue curves stand for the Dubins maneuvers which connect the determined collecting maneuvers into the closed tour. is required to move straight ahead during the dropping phase itself. An example solution of such an object dropping mission is depicted in Fig. 1. In this paper, we propose a novel extension of the ordinary DTSP to address the requirement of specific maneuvers for target accomplishment. The introduced problem is called the Dubins traveling salesman problem with collecting maneuvers (DTSP-CM). Further, we study a special case of the DTSP-CM where collecting maneuvers are straight line segments (denoted as the DTSP-SCM) and we propose modifications of the existing approaches to the DTSP to solve this constrained variant of the DTSP-CM. The paper is organized as follows. An overview of the related work with various DTSP approaches vol. 6/2016 is provided in Section 2. The proposed problem is formally introduced in Section 3 and modifications of the existing DTSP approaches are proposed in Section 4. Evaluation of the proposed modifications of the existing approaches is presented in Section 5. Finally, conclusion remarks are in Section 6. 2. Related work Various approaches have been proposed to address the DTSP that can be broadly divided into three main classes: 1) decoupled approaches, 2) sampling-based approaches, 3) and evolutionary-based approaches. In this section, a brief overview of representative approaches of each particular class is provided to clarify the context for the introduced DTSP-CM and proposed modifications of the particular approaches to solve the constrained variant of the DTSP-CM with straight line collecting maneuvers. Probably the first approach to the DTSP is the Alternating algorithm (AA) introduced in [1] which can be categorized as a decoupled approach, since it addresses the DTSP in two consecutive steps. First, a sequence of visits to the targets is determined, e.g., by a solution of the underlying Euclidean TSP without considering the vehicle motion constraint. In the second step, headings at the target locations are established by applying straight line segments for even segments followed by closing the tour with Dubins maneuvers for odd segments. A similar decoupled strategy has been utilized also in the receding horizon based approach [4] with the horizont of two or three target locations ahead which significantly outperforms the original AA in terms of the solution quality. Sampling-based approaches [5, 6] represent the second class of existing solutions for the DTSP. These approaches are based on sampling of possible headings into a finite discrete set and the DTSP is transformed into the Generalized Asymmetric TSP (GATSP) and further into the ATSP by Noon-Bean transformation [7]. The resulting ATSP can be then solved by existing solvers such as Concorde [8] or LKH [9]. The third class of the DTSP approaches are evolutionary based algorithms [10]. They encoded solutions into individuals in the population and used similar crossover and mutation operators like for the standard TSP. Although evolutionary based approaches can be computationally demanding, their main advantage is the ability to search for a global optima. In [11], authors proposed a novel memetic algorithm to improve performance of the evolutionary based approach by utilizing a local optimization method. Recently, the evolutionary multi-objective algorithm NSGA-II [12] has been deployed in [13] to address path plannin (...truncated)