Computational Intelligence and Neuroscience

Volume 2016, Article ID 2720630, 15 pages

http://dx.doi.org/10.1155/2016/2720630

## An Application of Self-Organizing Map for Multirobot Multigoal Path Planning with Minmax Objective

Department of Computer Science, Faculty of Electrical Engineering, Czech Technical University in Prague, Technická 2, 166 27 Prague 6, Czech Republic

Received 30 November 2015; Accepted 5 May 2016

Academic Editor: Silvia Conforto

Copyright © 2016 Jan Faigl. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In this paper, Self-Organizing Map (SOM) for the Multiple Traveling Salesman Problem (MTSP) with minmax objective is applied to the robotic problem of multigoal path planning in the polygonal domain. The main difficulty of such SOM deployment is determination of collision-free paths among obstacles that is required to evaluate the neuron-city distances in the winner selection phase of unsupervised learning. Moreover, a collision-free path is also needed in the adaptation phase, where neurons are adapted towards the presented input signal (city) to the network. Simple approximations of the shortest path are utilized to address this issue and solve the robotic MTSP by SOM. Suitability of the proposed approximations is verified in the context of cooperative inspection, where cities represent sensing locations that guarantee to “see” the whole robots’ workspace. The inspection task formulated as the MTSP-Minmax is solved by the proposed SOM approach and compared with the combinatorial heuristic GENIUS. The results indicate that the proposed approach provides competitive results to GENIUS and support applicability of SOM for robotic multigoal path planning with a group of cooperating mobile robots. The proposed combination of approximate shortest paths with unsupervised learning opens further applications of SOM in the field of robotic planning.

#### 1. Introduction

Self-Organizing Map (SOM) is an unsupervised neural network proposed by Kohonen in 1982 as a technique to map a high-dimensional input space into a lower dimensional (usually 2D) output space. Although SOM has been originally proposed for data visualization, it has been applied to many other problems including a solution of the Traveling Salesman Problem (TSP) [1]. The TSP stands to find a closed shortest tour to visit a given set of cities (locations) such that each city is visited exactly once and the tour returns to the starting city. It is known that the TSP is NP-hard and it is a well studied problem in operational research [2], where efficient heuristics have been proposed [3, 4].

On the other hand, the earliest application of SOM to the TSP was proposed independently by Angéniol et al. [5] and Fort [6] in 1988. Since that, several approaches have been developed to improve performance of the unsupervised learning of SOM for the TSP, for example, by a combination with -opt heuristic [7], using inhibition mechanism [8], considering geometric properties of the associated solution [9], and so forth; see extensive overviews in [10–12]. However, most of the approaches consider the Euclidean variant of the TSP in which cities are locations in a plane. Although few works on SOM for other routing problems have been published [13–16], non-Euclidean TSP is relatively unnoticed by the research community. It is probably because of the main difficulty of SOM for the non-Euclidean TSP that is a determination of the best matching neuron to the input signal presented to the network. It can be nontrivial to evaluate a suitable distance function and thus it can decrease performance of any algorithm based on elastic net principles [13].

For the unsupervised learning of SOM for the TSP, the best matching neuron is determined as the distance between the neuron weights and cities, which can be easily computed as the Euclidean distance. In robotic planning, the problem is to find a shortest path to visit a given set of cities and such a distance corresponds to the length of the shortest path among obstacles, which is more computationally demanding than computation of the Euclidean distance. The requirement for collision-free paths connecting the particular cities in the tour is the main reason why the problem is called the multigoal path planning (MTP) rather than the TSP to emphasize this difficulty [17]. Therefore, we aim to extend existing SOM approaches for the TSP to address more challenging MTP problems.

A simple and fast approximation of the shortest path in the polygonal domain has been proposed in [18] that enables to deploy the SOM for the TSP [8] to the robotic MTP. In this paper, the approximation is further developed to address the multirobot variant of the MTP, where shortest paths (for robots) among obstacles are requested to visit the given set of locations in the polygonal domain. The addressed problem is considered as a variant of the Multiple Traveling Salesman Problem (MTSP) with minmax objective [19] in which we aim to minimize the longest tour. This variant has a suitable objective function for motivational inspection planning or search and rescue scenarios, where it is desired to search the given environment as quickly as possible, and the total mission time corresponds to the length of the longest path a robot has to travel [20].

This presented work reports on an extension of the SOM for the MTSP with minmax objective proposed in [16] to a more general approach for the multirobot multigoal path planning problem to visit a given set of locations in the polygonal domain . The proposed approach is based on our previous work on approximation of the shortest path in for SOM-based solution of routing problems [18, 20, 21]. Therefore, we focus on an evaluation of the proposed extensions and an alternative inexpensive procedure for the competitive rule for the SOM-based MTSP-Minmax. The performance of the proposed approach is compared with a combinatorial heuristic algorithm for the MTSP-Minmax called GENIUS [22]. The presented results indicate that the proposed extensions make SOM competitive with the combinatorial approach from the solution quality and required computational time points of view. Furthermore, solutions found by SOM provide interesting features in relation to the robotic motivational problem, where SOM tends to provide mutually noncrossing tours for the robots.

The rest of the paper is organized as follows. An overview of the related work is presented in the next section. The problem statement, used notation, and terminology are introduced in Section 3. A detailed description of the selected reference combinatorial algorithm [22], the considered SOM for the MTSP [16], and utilized approximation of the shortest path in are presented in Section 4 to provide a better understanding of the proposed extensions and the evaluated algorithms’ variants. The proposed extensions of the SOM for the MTSP [16] to address the multirobot MTP problem are presented in Section 5. Evaluation results and comparisons of the algorithms in several problems motivated by the inspection planning are reported in Section 6. Conclusion and remarks about further work are discussed in Section 7.

#### 2. Related Work

Various problems can be formulated as the TSP or MTSP, but in our case, the considered problem is motivated by path planning problems in inspection and search missions, where single or a group of mobile robots is requested to visit a given set of locations as quickly as possible. The problem is called the multigoal path planning (MTP) problem [17, 23] in robotics, and the additional problem to the standard formulation of the MTSP is the necessity to consider paths among obstacles to avoid possible collision of the robots with obstacles in the workspace.

For a simple case, when paths between two locations are given, the multigoal path planning problem can be directly formulated as the TSP [24]. In general, a determination of such a collision-free path for a mobile robot can be computationally very demanding [25]. However, if a point robot can be assumed and the robot workspace can be represented by the polygonal domain, the shortest path roadmap approach can be used [26]. Thus, a solution of the TSP in a form of the found tour, for example, using visibility graph, can be considered as the requested collision-free path (solution) of the MTP for a single mobile robot.

In robotic planning, cities can represent sensing locations at which the robot gathers information about its surrounding environment to “see” the whole workspace [27]. The problem of searching the workspace is called the inspection task, and one of the feasible approaches is based on a formulation as a problem of finding the set of sensing locations and consecutive solution of the TSP [28]. Suitable sensing locations can be found by a sensor placement algorithm, for example, [29–32]. Then, a group of cooperating mobile robots can be used to decrease the required time to inspect the environment and thus the inspection task can be formulated as the MTSP [20].

Several methods for the MTSP have been proposed in literature which is also the case for a very closed problem formulation known as the Vehicle Routing Problem (VRP) where capacity of each vehicle is considered [19]. Beside auction-based techniques [33] and multiagent solutions [34], soft-computing techniques such as genetic algorithms have been proposed for these problems [35]. Note that the MTSP can be transformed into the TSP using transformation proposed in [36]; however, such a solution can be highly degenerated for the MTSP with minmax objective. It is because, in the TSP, the total tour length is minimized, that is, a tour with zero length can be provided while a sum of the lengths of the all tours for individual salesmen can be minimal. Therefore, it is necessary to address the minmax objective directly [37].

The MTSP-Minmax has been addressed by the combinatorial heuristic in [22], where authors propose to find optimal solution of the MTSP-Minmax using the distance constraint VRP formulation. A solution of the MTSP is used as the distance constraint that is gradually decreasing and if the VRP does not have a solution, the previous solution of the MTSP is considered as the optimal solution.

Soft-computing techniques have been also applied to the MTSP-Minmax, such as ant colony optimization [38], genetic algorithms [39], and also SOM in [16]. Particular soft-computing approaches for Euclidean instances of the MTSP-Minmax have been evaluated in [40] and our early results on robotic problems with obstacles in [41]. Therefore, in the presented work, we are focused on the evaluation of the SOM-based solution of the multirobot multigoal path planning and its comparison with the GENIUS algorithm [22]. The used SOM is directly based on [16] combined with the ideas proposed in [41] that have been accompanied by the approximation of the shortest path originally proposed in [18, 21], which significantly decreases the required computational time for SOM adaptation. Therefore, GENIUS [22], the considered SOM [16], and the approximations of the shortest path are described in detail in Section 4.

#### 3. Problem Statement

The studied problem of the multirobot multigoal path planning with minmax objective is motivated by inspection missions, where a group of mobile robots is requested to visit a given set of sensing locations, where sensor measurements are taken. In particular, the mission is to inspect all reachable areas of the environment as quickly as possible. The environment is represented by a polygonal map and the given sensing locations are determined in such a way that the whole environment is covered by visiting them [32]. It is assumed that a map of the environment is available; each robot has a differential drive and its shape can be bounded by a disk with a limited radius. For simplicity and without loss of generality, a point robot is considered in the polygonal domain created by enlarging the original map by the radius of the disk, and all free space is reachable by the robot. Then, the multigoal path planning problem is formulated as the MTSP-Minmax that can be defined as follows:* for a given polygon with holes **, a set of cities (sensing locations) ** lying inside ** and ** salesmen (robots) find ** closed tours starting at the selected city ** such that each city ** is visited by one salesman and the length of the longest tour is minimized.* The city is called the depot in the rest of this paper.

##### 3.1. Used Notation

The SOM adaptation schema is considered in the polygonal domain ; therefore, few terminology notes are presented here to clarify the used terms and symbols for underlying geometrical structures utilized in the approximation of the shortest path in .

The robot workspace is represented by the polygonal map consisting of vertices and thus is a closed, multiply connected region, whose boundary is a union of line segments, forming closed polygonal cycles (polygons), where is the number of holes (obstacles). A distance between two points inside is a length of a path among obstacles that can be a straight line segment or consists of the map vertices. Thus, a path between two points and consists of a finite number of straight line segments joining the points and vertices of .

can be divided into a set of nonoverlapping convex polygons that are formed from vertices. Such convex polygons are called cells and represent* convex polygon partition* of ; that is, each cell forms a closed polygonal cycle of line segments joining vertices. A line segment is called* diagonal* if it connects two nonadjacent vertices and it is entirely contained in . A point inside is always inside some cell and a path between two points and can be constructed from the shortest path between vertices of and .

Regarding SOM for the TSP, weights of a particular neuron represent a point (called node) that lies in and therefore is always inside some cell. Such a cell of the node is denoted as . An overview of the used symbols is in Symbols section at the end of the paper.

##### 3.2. Quality of Solution

The motivational problem of the multigoal path planning for a group of cooperating robots is formulated as the MTSP. The minmax variant of the MTSP leads to minimizing the longest tour and therefore we consider the maximal length of the individual tours as one of the solution quality indicators. However, SOM and also GENIUS are randomized algorithms and therefore the performance indicators should be computed from several trials. For the TSP, the usual indicators are the percentage deviation of the mean solution to the optimum tour (denoted as the PDM) and the percentage deviation from the optimum of the best solution value, denoted as the PDB. Finding an optimal solution for the considered instances of the MTSP-Minmax is computationally very demanding and therefore the best found solution for the particular problem instance (found by the evaluated algorithms) is considered as the reference solution. The longest tour of this reference solution is denoted as and it is used to compute the PDM and PDB as follows:(i)Reference solution length of the particular problem instance found as the longest tour of the best solution of several solutions found by particular selected algorithm(s).(ii)Maximal length of the individual tours in a solution of the MTSP. Consider (iii)Percentage deviation to the reference solution value of the mean solution value . Consider (iv)Percentage deviation to the reference solution value of the best solution value . Consider where is the best (the shortest the longest tour) solution from several solutions of the particular problem instance found by a particular algorithm variant.

The advantage of the percentage deviations of the tour lengths is that they provide a scale independent metric for particular instances of the MTSP and thus it can be used to aggregate results for various problems and many trials. However, it does not provide any indication to how the workload is divided into the particular robots; that is, what are the differences in the lengths of the individual tours? We propose two quality indicators to measure the quality of cooperation. The first is a percentage deviation of the lengths in a tour. This indicator is called a* Cooperative Quotient (CQ)* and its zero value means an* ideal cooperation*. The second indicator considers the total travelled distance by all robots and it is called* Collaborative Effort (CE)*. These indicators are computed as follows:(i)Cooperative Quotient is compouted as where is the root of the sample variance, , and is the average value of the tour lengths.(ii)Collaborative Effort is computed as

#### 4. Use Approaches

##### 4.1. GENIUS

The GENIUS algorithm has been used to find a solution of the MTSP-Minmax in [22]. It is a combinatorial method representing a general approach for the TSP that is based on two heuristics: GENI (Generalized Insertion) and US (Unstringing and Stringing) [42]. The first heuristic is a construction method while the second heuristic is an optimization method. Tours are initially constructed by GENI. After that, the tabu search technique is used to exchange cities from one tour to another, while GENI is utilized for vertices inserting/removing. Finally, the US optimization procedure is used. It removes a vertex from the tour and inserts the vertex into the same tour by GENI. The procedure is repeated until a vertex reinsertion improves the quality of solution. The parameter of the GENI algorithm defines the size of the neighborhood that is used to select the best possible vertex insertion. Performance of the tabu search can be controlled by three additional parameters: , and . The parameter determines the size of the global neighborhood to select an appropriate tour for a vertex exchange and controls the number of iterations for which a move of vertex according to the particular tour is declared tabu. The maximum allowed number of iterations without improvement is defined by the parameter.

Recommended values of parameters have been suggested by authors [22]. Two sets of parameters can be considered. The first set can be called* fast*, because it provides a compromise between computational requirements of the algorithm and the quality of solution. The second set provides high quality solutions, but it is computationally demanding. That is why the algorithm with this set of parameters is denoted as GENIUS-*quality* in this paper. Note that for each operation stored in the tabu list, the value of is selected randomly from the interval .

GENIUS is a combinatorial approach; therefore, only distances between cities are need. In the case of the Euclidean TSP, distances can be computed as the Euclidean distance, while for the motivation problem of multigoal path planning, shortest paths between cities have to be found. The shortest paths can be determined from the full visibility graph that can be constructed in [43], where is the number of vertices of and is the number of cities. All shortest paths between cities can be found by Dijkstra’s algorithm in , where is the number of edges of the visibility graph. All distances of the shortest paths can be precomputed and stored in the distance matrix.

##### 4.2. SOM Adaptation Schema for the MTSP-Minmax

The SOM for the MTSP-Minmax [16] uses two-layered competitive learning networks, where each network contains two-dimensional input vector and an array of output units. An association between the learning network and its geometrical representation of one TSP tour is shown in Figure 1. An input vector represents coordinates of the city and weights and can be interpreted as coordinates of the node . Nodes are connected to a ring representing the tour; thus, an individual ring of nodes is created for each salesman. The network is initialized with small random connection weights and cities are then sequentially applied to the network in a random order to avoid local minima. The output nodes compete to be the winner for a given city according to the following competitive rule:where denotes the Euclidean distance between the city and the node , is the length of the ring, into which the node belongs, and is the average length of the rings. Basically, the rule prefers nodes from shorter rings and thus it aims to minimize the longest ring (tour).