Robot-Assisted Sensor Network Deployment and Data Collection

Robot-Assisted Sensor Network Deployment and Data Collection Yu Wang Abstract— Wireless sensor networks have been widely used in many applications such as environment monitoring, surveillance systems and unmanned space explorations. However, poor deployment of sensor devices leads (1) bad network connectivity which makes data communication or data collection very hard; or (2) redundancy of coverage which wastes energy of sensors and causes redundant data in the network. Thus, in this paper, we propose using a mobile robot to assist the sensor deployment and data collection for unmanned explorations or monitoring. We assume that the robot can carry and deploy the sensor devices, and also have certain communication capacity to collect the data from the sensor devices. Given a set of interest points in an area, we study the following interesting problems: (1) how to decide minimum number of sensor devices to cover all the interest points; (2) how to schedule the robot to place these sensor devices in certain position so that the path of the robot is minimum; and (3) after the deployment of sensors, how to schedule the robot to visit and communicate with these sensor devices to collect data so that the path of the robot is minimum. We propose a complete set of heuristics for all these problems and verify the performances via simulation. I. I NTRODUCTION Wireless sensor networks [1] have tremendous prospects due to their relatively lower cost and capability of obtaining valuable information from locations that are beyond human reach. A sensor network consists of a set of sensor nodes1 that spread over a geographical area. These sensors are able to perform processing as well as sensing and are additionally capable of communicating with each other. Due to its widerange potential applications such as battlefield, emergency relief, environment monitoring, surveillance system, space explorations, and so on, wireless sensor network has recently emerged as a premier research topic. Most of current research on wireless sensor networks assume the cost of each sensor is cheap thus the number of sensors in a network could be sufficient large (hundreds or thousands) to cover the target area and maintain the network connectivity. However, in many real applications (such as space exploration), certain kind of sensor devices could be very expensive, and it is impossible to have thousands of them to deploy. In addition, since the sensors would have relatively weak radios, internode separation is very common in sensor networks. On the other hand, even if the number of sensor is sufficient and the radio is strong enough, poor deployment of sensor Yu Wang is with Department of Computer Science, University of North Carolina at Charlotte, USA. Email: [email protected]. The work of Wang is supported, in part, by funds provided by Oak Ridge Associated Universities and the University of North Carolina at Charlotte. Changhua Wu is with Department of Science and Mathematics, Kettering University, USA. Email: [email protected] 1 In this paper the term node often represents a sensing device or called a sensor. We often interchange them here. Changhua Wu devices could also lead to (1) bad network connectivity which makes data communication or data collection very hard; or (2) redundancy of coverage which wastes energy of sensors and causes redundant data in the network. Thus, in this paper, we propose using a mobile robot to assist the sensor deployment and data collection for unmanned explorations or monitoring. We assume that the robot can carry and deploy sensor devices, and also have certain communication capacity to collect the data from these sensor devices. Recent years have seen the growing interest in mobile sensor networks [2]–[7] or robot-assisted sensor networks [8]–[11]. In mobile sensor networks, all or partial of the sensor nodes have motion capability endowed by robotic platforms. Mobile sensor networks have more flexibility, adaptively and even intelligence compared with stationary wireless sensor networks. Mobile sensors can dynamically reposition themselves to satisfy certain requirements on monitoring coverage, network connectivity, or fault tolerance. However, to make every sensor have motion capability increases the cost of each sensor and maybe not feasible in most applications. On the other hand, robots are large complex systems with powerful resources and can interact with sensor nodes. The new paradigm of robot assisted sensor networks is of ubiquitous sensors embedded in the environment with which the robot interacts: to deploy them, to harvest data from them, and to task them. In turn, the sensors can provide the robot with models that are highly adaptive to changes in the environment and can re-task the robots with feedback from sensors. Therefore, we believe that robotics will have a profound effect on sensor networks. Most previous research on robot-assisted sensor networks [8]–[11] study using the robot to achieve coverage, localization and navigation. In this paper, we focus on coverage and path planning. Given a set of interest points in an area, we study the following interesting problems: (1) how to decide minimum number of sensor devices to cover all the interest points; (2) how to schedule the robot to place these sensor devices in certain position so that the path of the robot is minimum; and (3) after the deployment of sensors, how to schedule the robot to visit and communication with these sensors to collect data so that the path of the robot is minimum. An illustration of this scenario is depicted in Figure 1 where the rover using one path to deploy the sensors and the other path to collect data from deployed sensors. We propose a complete set of heuristics for all these problems and verify the performances via simulations. A potential application of our proposed robot-assisted sensor network design is for unmanned space explorations. Unmanned space explorations have tremendous prospects well-known graph theoretic problem, the traveling salesman problem [15]. We assume that the 2D space does not have any obstacles and the robot can move towards any direction freely. The objective of our path planning is to minimize the total length of the path which the robot travels. We study how to deploy sensors and schedule the robot path such that the total travel distance is minimized and the coverage is guaranteed. III. M ODELS AND P ROBLEMS A. Models Fig. 1. Illustration of the scenario where a robot (a rover) deploys sensor nodes around interest points and assists to collect data from the unconnected sensor network. Here, the red path is the deployment path, while the blue one is the data collection path. due to their relatively lower cost and capability of obtaining valuable information from locations that are beyond human reach. The impact of unmanned missions and the use of automated remote monitoring stations and robotic platforms in space have been observed from several successful ventures in the past. Examples include the NASA Mars rovers that are designed to negotiate unpredictable surface conditions and provide valuable data, video samples as well as physical samples through remote control. Our proposed approach can allow the robot (rover) efficiently deploy and maintain the sensor networks which enable data collection over large areas over extended periods of time. The proposed coordinated remote data deployment and collection approach can extend the reach and lifetime of both space rovers and smart sensors. II. R ELATED W ORK Sensor Coverage: Since each sensor covers a limited area, adequate coverage of a large area requires appropriate placement of sensors based on collective coverage and cost constraints. The previous research on sensor coverage mainly focuses on studying how to determine the minimum set of sensors for covering every location or certain objects (interest points) in the target field. Different coverage models and methods are surveyed by Cardei and Wu [12]. Robot-Assisted Sensor Networks: Mobile or robotassisted sensor networks have been studied recently. Most previous research concentrate on using the robot or mobile sensors to help sensor network to achieve coverage [2], [5], localization [6]–[9], [11], target detection [3], fault-tolerance [4], [10], and navigation [9]. In this paper, we study how to use a robot assisting the sensor deployment and data collection, with a focus on efficient path planning. Path Planning: One of the most important problems in robotics is path planning (or called motion planning) [13], [14], which is aimed at providing robots with the capability of deciding automatically which motions to execute in order to achieve certain specific goals. It arises in a variety of forms. The common form requires finding a short geometric collision-free path for a single robot in a known static workspace. In this paper, we do not focus on such kind of path planning. The problem we concentrate on is similar to a We assume that a set of m interest points (or called targets), denoted by P = {p1 , p2 , · · · , pm }, are distributed in a 2-dimensional plane. The objective of our mission is to deploy a set of sensor devices, denoted by S = {s1 , s2 , · · · , sn } to form a sensor network to monitor or track these interest points. Each sensor node si is equipped with a sensor which can monitor a disk region centered at si with radius rS , i.e., if the distance between pl and si is less than rS then sensor si can monitor the interest point pl . We assume that single sensor can monitor multiple points inside its sensing region. Each sensor node si has an omnidirectional antenna so that it can talk to all sensor nodes or the robot within a disk region centered at si with radius rT . Hereafter, we call rS and rT the sensing range and the transmission range respectively. We assume all sensor nodes are equipped with same hardware devices, thus, they have the same2 fixed rS and rT . We assume the robot R has a larger transmission range than the sensor node, i.e., it can talk with sensor node si if it is inside the transmission range of si . The robot parks at point v0 initially and need to return v0 after all operations. It can travel to any point in the 2-dimensional plane during the operations. B. The Problem The problem we study is how to efficiently schedule a robot to (1) deploy a set of sensor nodes S to guarantee the coverage of all interest points P and (2) collect data from these sensor nodes. Here, the efficiency of the path scheduling means the scheduled path for the robot to travel is shortest. We treat this problem as two sub-problems separately: deployment problem and data collection problem. For the deployment problem, given the set of interest points P , we study how to find the positions V = {v1 , v2 , · · · , vn } of sensor nodes S where they will be deployed by the robot, such that (1) the sensor network guarantees the full coverage of all interest points P and uses the minimum number n of sensor nodes S; and (2) the path ΠD = v0 v1 v2 · · · vn v0 which the robot will travel to deploy sensors at those positions has the minimum total length. For the data collection problem, given the set of deployed sensors V , we study how to find the turning positions (or called pause points) U = {u1 , u2 , · · · , uk } where the robot pauses and collects data from sensors, such that (1) the robot 2 However, our proposed methods can be easily extended to the case with heterogeneous sensing and transmission ranges. can communicate with every sensor during the round trip and make the minimum number k of stops; and (2) the path ΠC = v0 u1 u2 · · · uk v0 which the robot will travel to collect data on those pause points has the minimum total length. Notice that the deployment problem and the data collection are essentially the same except the range of coverage is different (one uses the sensing range, the other uses the transmission range), thus we use the same set of heuristics to solve these two problems. p1 exists many heuristics for it. The simplest and most classical method is a greedy method, in which you always greedily select the subset which can cover the maximum number of uncovered elements. This greedy algorithm can achieve an approximation ratio of O(ln s) where s is the size of the largest subset. Inapproximability results [16], [17] show that the greedy algorithm is essentially the best-possible polynomial time approximation algorithm for set cover under plausible complexity assumptions. Algorithm 1 shows our greedy algorithm and Figure 3 illustrates the results from Algorithm 1 on the example shown in Figure 2. p3 p6 p7 p9 p4 p5 p2 p8 Interest Point Robot v0 Fig. 2. The set of interest points P (black nodes) and the initial position of the robot (red triangle). Here, 15 areas can be defined as A by the 9 sensing disks and their intersections. as1 as3 as2 as5 as4 Algorithm 1 Greedy algorithm to select the minimum number of areas where to deploy sensors Input: A set of areas A = {a1 , a2 , · · · , al } and a set of interest points P = {p1 , p2 , · · · , pm }. Output: A subset of areas AS = {as1 , as2 , · · · , asn } to place the n sensors S. 1: Initially, set all interest points uncovered and the uncovered counter k = m. Let the potential coverage ci of each area ai equal to the number of disks DjS intersecting with this area. Here, each interest points pj could be covered by a sensor placed in area ai . We call pj can be covered by ai . 2: while k! = 0 do 3: Select area aj with the largest potential coverage cj (using IDs to break a tie) and add it into the selected subset AS ; 4: Mark all interest points covered by aj covered; 5: k = k − cj ; 6: Update the ck for all adjacent areas ak . 7: end while Interest Point Robot v0 Fig. 3. Grey areas are the subareas selected by Algorithm 1 where a sensor needs to be deployed. IV. ROBOT-A SSISTED S ENSOR D EPLOYMENT In this section, we describe our algorithm for how to deploy the sensors with assistance from the mobile robot. As shown in Figure 2, we first use the sensing range rS to draw a disk DiS for each interest point pi . To guarantee all interest points are covered by sensors, we need at least one sensor node inside each disk DiS to monitor pi . However, one sensor can sit in the intersection of multiple disks to monitor multiple targets. Thus, we define the areas formed by the disks and their intersections, denoted by A = {a1 , a2 , · · · , al }, putting a sensor in an area ai covers one or multiple interest points. The first optimization problem is how to select the minimum number of areas to deploy sensor nodes to guarantee the coverage. This problem is actually the minimum set cover problem which aims to find the minimum number of subsets to cover the whole space. The minimum set cover problem is a NP-hard problem [15]. However, there After selecting the area to place the sensors, we need to decide their exact positions. Since the positions can affect the total length of the path that the robot needs to visit, we consider the position problem joint with the path schedule problem. In other words, we propose an algorithm to schedule the robot to deploy the sensors in each selected area asi , so that the total length of the path travelled by the robot is minimum. This problem is actually the traveling salesperson problem with neighborhoods (TSPN) which is also a NP-hard problem [18]. The classical TSP studies what is the shortest round-trip route that visits each point exactly once and then returns to the starting point, given a set of point in a plane. TSPN studies what is the shortest roundtrip route that visits each area exactly once and then returns to the starting area, given a set of areas. There are several approximation algorithm exists for TSPN, however most of them are very complex and not practical at all. Our algorithm is an iterative algorithm in which each step we add a new turn point inside one of the unvisited areas such that the distance added to the robot path is minimum. Assume, we have n areas needed to be visited (deploying the sensor) and initially all areas are unvisited, the algorithm will terminate after n rounds, since each round it adds a new turn point in the path and covers an unvisited area. Algorithm 2 shows the detailed algorithm. Algorithm 2 Path schedule and sensor placement algorithm: to select the turn points of the robot to deploy sensors Input: A set of areas AS = {as1 , as2 , · · · , asn }. Output: A path ΠD = v0 v1 v2 · · · vn v0 which the robot use for sensor deployment. 1: Initially, set all selected areas asi unvisited and the unvisited counter k = n. Let the path ΠD = v0 v0 . 2: while k! = 0 do 3: For each edge on vi vi+1 in path ΠD and every unvisited area aj , we draw an ellipse which uses vi and vi+1 as its foci and is tangent to aj . Let vj be the tangent point. See 4(a) for illustration. If select aj to visit between ai and ai+1 , the distance added to the path ΠD will be ||vi vj || + ||vj vi+1 || − ||vi vi+1 ||. 4: It is obvious that we want to select the unvisited area which adds the least distance to path ΠD . For example, in Figure 4(b), ap , hence vp , is a better choice than aj . Assume we select ap which is the best for all edges in ΠD and all unvisited areas, we mark ap visited, and insert vp between vi and vi+1 in ΠD . Thus the number of edges in the path increases by one. k = k − 1. 5: end while v3 v2 v4 v5 v1 Interest Point Sensor Robot v0 Fig. 5. Path ΠD (red line) generated by Algorithm 2. Here, green squares are the positions to place the sensors (also the turn points of the robot). v3 v2 v4 v5 v1 Interest Point Sensor Robot v0 Fig. 6. The deployed sensors (green squares) and their sensor ranges (solid circles) after the sensor deployment phase. s2 aj aj u3 u2 ap vp vi s4 vj vj vi+1 s3 s5 u4 s1 vi u1 vi+1 Interest Point Sensor Robot v0 v0 D ! (a) ! D v0 (b) Fig. 4. (a) For each edge on vi vi+1 in path ΠD and every unvisited area aj , we draw the ellipse which uses vi and vi+1 as its foci and is tangent to aj . The distance added to the path ΠD by visiting aj is ||vi vj || + ||vj vi+1 || − ||vi vi+1 ||. (b) We select the unvisited area which adds the least distance to path ΠD . In this example, ap is a better choice than aj . Notice that in Step 3 of Algorithm 2 we need to draw an ellipse which is tangent to aj . This can be done by two ways. We can start with a small ellipse and increase its size until it reaches aj . However, how to decide the initial size of the ellipse and what size to increase at each step are difficult to answer. The second way to do is using binary search. We first randomly select a point b inside aj . We use ||vi b|| + ||vi+1 b|| as the major axis to draw the ellipse which guarantees to intersect with aj . Then we reduce the major axis by half, if the ellipse does not intersect with aj , we increase the major Pause Point Fig. 7. The robot-assisted data collection: the robot travels via the blue path to collect data from each sensor. Here, the green dash circle is the communication range of the sensor. axis, otherwise further reduce it. By recursively doing this, we can find the ellipse which is tangent to aj efficiently. In practice, if the sensing range is small compared with the distance between all areas, we can just use the ellipse via b to estimate the optimal ellipse. Figure 5 shows the path ΠD generated by Algorithm 2. Path ΠD represented by red line is the path that the robot will follow to place the n sensors, while the green squares are the positions to place the sensors (also the turn points of the robot). Figure 6 shows the deployed sensors and their sensing ranges after the deployment phase. It is clear that every interest point is covered at least by one sensor. V. ROBOT-A SSISTED DATA C OLLECTION After the robot has deployed the sensors, all sensors begin to collect information about the interest points. All Path by the greedy method. P=415.6257 Path by the proposed method. P=336.8159 120 120 Path by a genetic travel salesperson method. P=352.5129 120 100 100 100 80 80 80 60 40 40 40 20 20 20 0 −20 −20 60 60 0 20 40 60 80 100 120 greedy method 0 0 −20 −20 −20 −20 0 20 40 60 80 100 120 generic method for TSP Fig. 8. 0 20 40 60 80 100 120 proposed method Sample paths found by the three methods. information will be sent to a centralized control center. However, due to the fact that the communication range of sensor is limited, the sensor network may be partitioned to components far away from each other. Adding more sensor nodes can improve the connectivity, however, it is not feasible in many applications, such as space exploration with expensive sensor devices. In such scenario, the mobile robot can help. We assume that the robot is also equipped with communication devices and can collect data from the deployed sensors. The path planning for the robot is again an optimization problem where we try to minimize the total distance traveled by the robot. Here, given the set of deployed sensors S = {s1 , s2 , · · · , sn } and their positions v1 , v2 , · · · , vn , we study how to schedule the robot to visit certain pause points U = {u1 , u2 , · · · , uk } where the robot can collect data from sensors, such that (1) the robot can communicate with every sensor during the round trip and make the minimum number k of stops; and (2) the path ΠC = v0 u1 u2 · · · uk v0 which the robot will travel to collect data on those pause points has the minimum total length. For the first half problem, we first use transmission range rT of each sensor si to draw the areas to be covered, and then run the greedy algorithm (Algorithm 1) to select minimum number of pause points to cover all sensor nodes. The problem is essentially the same as the one in the deployment phase except that the range of coverage is transmission range rT instead of sensing range rS . For the second half problem, we need to schedule the robot to visit these selected areas using shortest round trip. By using the same heuristic (Algorithm 2), we can find a solution ΠC and return the turn points u1 u2 · · · uk of the robot, shown as the blue path in Figure 7. Notice that if some sensors can communicate and transfer data with each other, then it will suffice for the robot to visit only one of these sensors to pick up data. For this situation, we can merge these sensors’ transmission ranges to a union area and use it as a single area in the input of Algorithm 1 instead of several individual areas. By asking the sensors to increase their transmission ranges, the connectivity of the sensor network can increase, which will lead to less areas the robot needs to visit. This is a tradeoff between the communication cost plus power consumption at sensors and the power consumption at the robot. For example, if the transmission range of each sensor is infinitely large, then the robot does not need to move to collect the data. If the transmission range is infinitely small, the robot needs to visit each sensor at its position to collect the data. VI. S IMULATION S TUDIES We carried out several simulation experiments to evaluate the proposed method. As we have discussed earlier, the sensor deployment and data collection are actually one problem. Therefore, we only simulate in the context of sensor deployment. Conclusions made from the simulation in sensor deployment can be applied to the data collection problem. In the simulation, all sensors have the same sensing range. For simplification, the visiting point vi of each area ai is chosen to be the center of ai . However, this simplification does not undermine the virtual of the proposed approach. In the simulation, we compared the travel distance by the proposed approach with two traditional methods: greedy method and near-optimal solution from traveling salesman problem. In the greedy method, the robot starts from the current interest point and goes to the next point which is closest to the current one until all interest points have been visited and returns to the original position. The near optimal solution from traveling salesman problem is obtained by a genetic algorithm [19]. In the simulation, we want to know how the proposed approach performs with regard to the number of interest points and the sensing range compared with the two traditional methods. Figure 8 shows a case of the simulation, in which 20 interest points, shown as black square dots, are randomly generated within a 100 × 100 field. In this case, the sensing range is 8. The total travel distances found by the greedy method, the generic TSP method, and the proposed method are 415.6257, 352.5129 and 336.8159. As we can see, when there is overlapping between the disks, the travel distance found by the proposed approach can be considerably smaller than the distances found by the genetic TSP method and the the greedy method on the interest points. In the next two simulation experiments, we will compare the three methods with regard to the number of interest points and the sensing ranges. Figure 9 shows the experiment in evaluating the proposed method with regard to the number of The length of travel path 600 greedy method 550 proposed method generic tsp method 500 450 400 350 300 250 200 150 5 10 15 20 Number of interest points 25 30 Fig. 9. Travel distance comparison with regard to number of interest points. The length of travel path 500 450 400 350 300 greedy method 250 proposed method generic tsp method 200 5 10 15 20 Sensing range Fig. 10. Travel distance comparison with regard to sensing range interest points and a fixed sensing range. The sensing range in this simulation is 8, and the number of interest points varies from 5 to 29. We can see that the total travel distance increases almost proportionally to the number of interest points. Among the three methods, our proposed approach can always achieve the smallest travel distance. Figure 10 shows travel distances found by the three methods with regard to a fixed number (20) of random interest points and various sensing ranges (from 5 to 19). It shows that for the greedy method and the genetic TSP method, the total travel distance do not vary much with regard to the sensing range, which is caused by the fact that they always go to the interest points instead of the overlapping areas. The travel distance found by the proposed approach, however, steadily decreases with increasing sensing range. This demonstrates that when the sensing range is large, there is high probability of overlapping, therefore traveling only to the overlapping regions will save large amount of time and energy cost. Since this paper does not intend to propose a method for finding near optimal solution for the traveling salesman problem, we did not compare the path finding algorithm, described in Algorithm 2, in the proposed method with the greedy method and the genetic method. VII. C ONCLUSION In this paper, we studied how to use a mobile robot to assist the sensor deployment and data collection for unmanned explorations or monitoring using sensor networks. Given a set of interest points, we proposed a set of heuristics to (1) decide minimum number of sensor devices to cover all the interest points; (2) schedule the robot to place the sensor devices in certain positions so that the path of the robot is minimum; and (3) schedule the robot to visit and communicate with these sensor devices to collect data so that the path of the robot is minimum. We also verified the performances via simulations, and the results demonstrated a nice performance of our method compared with the greedy algorithm and a genetic algorithm for TSP. R EFERENCES [1] I. F. Akyildiz, W. Su, Y. Sankarasubramaniam, and E. Cayirci, “Wireless sensor networks: a survey,” Computer Networks, vol. 38, no. 4, pp. 393–422, 2002. [2] A. Howard, M. J. Mataric, and G. S. Sukhatme, “Mobile sensor network deployment using potential fields: A distributed, scalable solution to the area coverage problem,” in Proc. of 6th International Symp. on Distributed Autonomous Robotics Systems (DARS), 2002. [3] T.-L. Chin, P. Ramanathan, K. K. Saluja, and K.-C. Wang, “Exposure for collaborative detection using mobile sensor networks,” in Proc. of IEEE International Conf. on Mobile Adhoc and Sensor Systems, 2005. [4] G. Wang, G. Cao, T. L. Porta, and W. Zhang, “Sensor relocation in mobile sensor networks,” in Proceedings of IEEE INFOCOM, 2005. [5] Y. C. H. Yongguo Mei, Yung-Hsiang Lu and C. S. G. Lee, “Reducing the number of mobile sensors for coverage tasks,” in Proceedings of International Conference on Intelligent Robots and Systems, 2005. [6] C. Wu, W. Sheng, and Y. Zhang, “Mobile self-localization using multidimensional scaling in robotic sensor networks,” Int’l J. on Intelligent Control and Systems, pp. 163-175, vol. 11, no. 3, 2006. [7] W. Sheng, G. Tewolde, and Y. Guo, “Distributed robot-assisted node localization in active sensor networks,” in Proc. of the 2006 IEEE International Conference on Mechatronics and Automation, 2006. [8] H. Lee, H. Dong, and H. Aghajan, “Robot-assisted localization techniques for wireless image sensor networks,” in Proc. of the Annual IEEE Communications Society Conference on Sensor, Mesh and Ad Hoc Communications and Networks, 2006. [9] P. Corke, R. Peterson, and D. Rus, “Localization and navigation assisted by networked cooperating sensors and robots,” International Journal of Robotics Research, vol. 24, no. 9, pp. 771–786, 2005. [10] Y. Mei, C. Xian, S. Das, Y. C. Hu, and Y.-H. Lu, “Replacing failed sensor nodes by mobile robots,” in International Conference on Distributed Computing Systems, Workshop on Wireless Ad hoc and Sensor Networks (WWASN), 2006. [11] A. Panousopoulou, E. Kolyvas, Y. Koveos, A. Tzes, and J. Lygeros, “Robot-assisted target localization using a cooperative scheme relying on wireless sensor networks,” in Proceedings of IEEE Control and Decision Conference, (CDC), 2006. [12] M. Cardei and J. Wu, “Energy-efficient coverage problems in wireless ad hoc sensor networks,” Computer Communications Journal (Elsevier), vol. 29, no. 4, pp. 413–420, 2006. [13] J.-C. Latombe, Robot Motion Planning. Springer, 1991. [14] P. C.-Y. Sheu and Q. Xue, Intelligent Robotic Planning Systems. World Scientific Pub Co Inc, 1991. [15] T. H. Cormen, C. E. Leiserson, R. L. Rivest, and C. Stein, Introduction to Algorithms. MIT Press, Cambridge, Massachusetts, 2001. [16] C. Lund and M. Yannakakis, “On the hardness of approximating minimization problems,” Journal of the ACM (JACM), vol. 41, no. 5, pp. 960–981, 1994. [17] U. Feige, “A threshold of ln n for approximating set cover,” J. ACM, vol. 45, no. 4, pp. 634–652, 1998. [18] A. Dumitrescu and J.S.B. Mitchell, “Approximation algorithms for TSP with neighborhoods in the plane,” in Proc. of ACM Symposium on Discrete Algorithms, 2001. [19] J. Kirk, “Traveling Salesman Problem - Genetic Algorithm,” http://www.mathworks.fr/matlabcentral/fileexchange/loadFile.do ?objectId=13680&objectType=file