New heuristic algorithms for the Dubins traveling salesman problem

Journal of Heuristics https://doi.org/10.1007/s10732-020-09440-2 New heuristic algorithms for the Dubins traveling salesman problem Luitpold Babel1 Received: 3 November 2018 / Revised: 23 July 2019 / Accepted: 8 February 2020 © The Author(s) 2020 Abstract The problem of finding a shortest curvature-constrained closed path through a set of targets in the plane is known as Dubins traveling salesman problem (DTSP). Applications of the DTSP include motion planning for kinematically constrained mobile robots and unmanned fixed-wing aerial vehicles. The difficulty of the DTSP is to simultaneously find an order of the targets and suitable headings (orientation angles) of the vehicle when passing the targets. Since the DTSP is known to be NP-hard there is a need for heuristic algorithms providing good quality solutions in reasonable time. Inspired by standard methods for the TSP we present a collection of such heuristics adapted to the DTSP. The algorithms are based on a technique that optimizes the headings of the targets of an open or closed subtour with given order. This is done by discretizing the headings, constructing an auxiliary network and computing a shortest path in the network. The first algorithm for the DTSP uses the order of the targets obtained from the solution of the Euclidean TSP. A second class of algorithms extends an open subtour by successively adding a new target and closes the tour if all targets have been added. A third class of algorithms starts with a closed subtour consisting of few targets and successively inserts a new target into the tour. The individual algorithms differ by the number of headings to be optimized in each iteration. Extensive simulation results show that the proposed methods are competitive with state-of-the-art methods for the DTSP concerning performance and superior concerning running time, which makes them applicable also to large-scale scenarios. Keywords Dubins traveling salesman problem · Curvature-constrained traveling salesman problem · Dubins vehicle · Heuristic methods B Luitpold Babel 1 Fakultät Betriebswirtschaft, Institut für Mathematik und Informatik, Universität der Bundeswehr, München, Werner-Heisenberg-Weg 39, 85579 Neubiberg, Germany 123 L. Babel 1 Introduction The Traveling Salesman Problem (TSP) is one of the most intensely studied problems in combinatorial optimization. Given a set of targets, the task is to determine a shortest tour that visits each target precisely once and returns to the start. If the distance between any two targets is equal to the Euclidean distance, the problem is called the Euclidean Traveling Salesman Problem (ETSP). The Asymmetric Traveling Salesman Problem (ATSP) is a problem where the distance between two targets is not symmetric but depends on the direction of the traversal. Further variations of the TSP include the Traveling Salesman Problem with Neighborhoods (TSPN) where each target of the tour is allowed to move in a given region, the Bottleneck Traveling Salesman Problem (BTSP) where the largest distance between two targets in the tour should be minimized, and others. For more details see e.g. the reviews of Lawler et al. (1985), Laporte (1992) and Gutin and Punnen (2007). The route planning problems listed above are aimed at finding best possible tours without taking into account the characteristics of the vehicle. However, when working with real-world vehicles, one has to consider kinematic constraints such as the minimum turning radius. Vehicles with motion constraints imposed by the steering mechanism satisfy a non-holonomic constraint. Such vehicles are not able to follow paths obtained from solutions of the classical TSP problems. Car-like mobile robots or fixed-wing aerial vehicles that move forward at a constant speed and turn with upper bounded curvature can be modeled as a Dubins vehicle (see e.g. Tang and Özgüner 2005; Otto et al. 2018). The traveling salesman problem for a Dubins vehicle is usually called Dubins Traveling Salesman Problem (DTSP) or Curvature-constrained TSP (see LaValle 2006). The DTSP has attracted considerable attention due to many civil and military applications. A typical setup is monitoring a collection of spatially distributed targets by an unmanned aerial vehicle (UAV). This might concern traffic control over specific locations, intelligence gathering and reconnaissance of suspicious targets for antiterrorism operations, security missions and monitoring of critical infrastructure and other point of interests, support of combat missions by intelligence, surveillance and reconnaissance (ISR) operations, battle damage assessment (confirming a target and verifying its destruction), and others. For further applications see e.g. Epstein et al. (2014) and Otto et al. (2018). It is well known (see Dubins 1957) that a shortest path between two points in the plane with prescribed initial and terminal tangents and a constraint on the curvature consists of a concatenation of straight lines and circle segments with maximal curvature. More precisely, each path is of the form CSC or CCC where C stands for a concave or convex circle segment with maximal curvature and S for a line segment. The solutions are commonly called Dubins paths. Alternative proofs have been obtained by Reeds and Shepp (1990) using advanced calculus, and by Boissonnat et al. (1994) from the standpoint of optimal control using Pontryagin’s maximum principle. The original idea of Dubins has been refined by Shkel and Lumelsky (2001) in order to speed up the computation of the paths. The particular challenge associated with the DTSP is not only to find an optimal order of the targets but also suitable headings (orientation angles) of the vehicle when 123 New heuristic algorithms for the Dubins traveling salesman… passing the targets. Since the target order is discrete but the heading angles are continuous, we obtain an optimization problem with both discrete and continuous decision variables. Just like the TSP and its variations mentioned above, the DTSP turns out to be NP-hard (see Le Ny et al. 2012). Therefore, unless P = NP, there are no efficient algorithms to solve any of these problems to optimality. Hence there is a need for methods providing approximate solutions in reasonable time. Numerous heuristic methods have been developed for the DTSP. One popular approach is to use the order obtained from the corresponding ETSP and determine suitable headings of the targets. Representatives of such methods include the Alternating Algorithm of Savla et al. (2008) that replaces the even-numbered edges of an ETSP tour by a Dubins path. The algorithms of Rathinam et al. (2007) and Ma and Castanon (2006) are look-ahead algorithms that consider a short sequence of targets in each iteration. While the former restricts to two targets, the second algorithm searches for the minimal path through three targets at a time. Another method proposed by Macharet and Campos (2014) assigns or (...truncated)