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Distributed RangeFree Localization Algorithm Based on SelfOrganizing Maps
EURASIP Journal on Wireless Communications and Networking volumeÂ 2010, ArticleÂ number:Â 692513 (2009)
Abstract
In Mobile Ad Hoc Networks (MANETs), determining the physical location of nodes (localization) is very important for many network services and protocols. This paper proposes a new Distributed RangeFree Localization Algorithm Based on SelfOrganizing Maps (SOMs) to deal with this issue. Our proposed algorithm utilizes only connectivity information to determine the location of nodes. By utilizing the intersection areas between radio coverage of neighboring nodes, the algorithm has maximized the correlation between neighboring nodes in distributed implementation of SOM and reduced the SOM learning time. An implementation of the algorithm on Network Simulator 2 (NS2) was done with the mobility consideration to verify the performance of the proposed algorithm. From our intensive simulations, the results show that the proposed scheme achieves very good accuracy in most cases.
1. Introduction
Recently, mobile adhoc network localization has received attention from many researchers [1]. Many algorithms and solutions have been presented so far. These algorithms are ranging from simple to complicated schemes, but they can be categorized as rangebased and rangefree algorithms. Rangefree algorithms utilize only connectivity information and the number of hops between nodes. The others utilize the distance measured between nodes by either using the TimeOfArrival (TOA) [2], TimeDifferentialOfArrival (TDOA) [3], AngleOfArrival (AOA) [4], or ReceivedSignalStrengthIndicator (RSSI) [5] technologies. However, they usually need extra hardware to achieve such measurement. When calculating the absolute location, most schemes need at least three anchors (nodes that are equipped with Global Positioning System or know their location in advance).
DVHOP is a typical rangefree algorithm. It was proposed by Niculescu and Nath [6] as an Adhoc Positioning System (APS). DVHOP uses distancevector forwarding technique to get the minimum hop count from a node to heard anchors. By using corrections calculated by anchors (average hopdistance between anchors), nodes estimate their location by using lateration (triangulation) method. Besides DVHOP, some other algorithms seem to be more complicated, but have better accuracy. The Multidimensional Scaling Map (MDSMAP) proposed by Shang et al. [7] is an example. MDSMAP is originated from a data analytical technique by displaying distancelike data in geometrical visualization. It computes the shortest paths between all pairs of nodes to build a distance matrix and then applies the classical Multidimensional Scaling (MDS) to this matrix to retain the first two largest eigenvalue and eigenvector to a 2D relative map. After that, with three given anchors, it transforms the relative map into an absolute map based on anchors' absolute location. There are some variances of MDSMAP such as centralized method: MDSMAP(C), and distributed one: MDSMAP(P). But, in the distributed method, to get the absolute location, nodes need global information about the subnetwork's map that contains at least three anchors. Tran and Nguyen [8] proposed a new localization scheme based on Support Vector Machine (SVM). The authors have contributed another machine learning method to the localization problem, and proved the upper bound error of this method.
Regarding the localization based on SelfOrganizing Maps, some researchers have employed SOM directly or with some modification. The method presented by Giorgetti [9] employed the classical SOM to the localization. This method uses centralized implementation and requires thousands of learning steps in convergence of network topology. The authors also realize that this method is good for small and medium size networks of up to 100 nodes. S. Asakura et al. proposed a distributed localization scheme [10] based on SOM. Hu and Lee [11] also proposed another version of distributed localization based on SOM. In this work, the authors employed a deduced SOM version [12]. But, this method still needs too many iterations (at least 4000) to make the topology to be converged with a relatively low accuracy. In another work [13], the authors use SOM to track a mobile robot with the utilization of surrounding environments from readings of sensor data. In the work presented by Ertin and Priddy [14], another version of SOM was used to implement the localization in wireless sensor networks. This paper extends one of our previous work [15] to improve and adapt it with mobility scenarios. The main contribution of this paper is the utilization of intersection between radio coverage of neighboring nodes in our modified SOM, and the adaptation of the algorithm to the mobility scenarios. It is also noted that our method was verified in both MATLAB and NS2 environments.
2. Motivation for Distributed SOMBased Localization
2.1. SelfOrganizing Maps
The SelfOrganizing Maps (SOMs) were invented by Kohonen [16]. SOM provides a technique for representation of multidimensional data into much lowerdimensional spaces (usually one or two dimensions). It uses a process known as vector quantization. The nature of SOM is a neural network working in unsupervised learning manner. The SOM learning process can be summarized as follows.

(1)
Initialization: assign the initial weight, , to each neuron in the SOM network.

(2)
Finding the BMU: determine the Best Matching Unit (BMU) or winning neuron at the iteration by using Euclidean minimumdistance criterion:
(1)where represents the input pattern, is the total number of SOM neurons, and the input pattern has dimensions.

(3)
Weight adjusting: Adjust the weights of the BMU and its neighbors using the following rule
(2)where is the learning rate at time step and is the function for topological neighborhood of neuron at time step
Steps and are repeated until the convergent criterion is satisfied.
2.2. Motivation for Distributed SOMBased Localization
Suppose that we have a mobile adhoc network of connected nodes, in which only a small number of nodes know their location in advance (anchor nodes). Now we have to determine the location of the remaining nodes that do not know their location, especially in distributed manner. In our proposed scheme, one can think that a mobile adhoc network itself is an SOM network, in which each neuron is a node in that network, and these neurons are connected to their 1hop neighboring nodes (nodes have direct radio links). The topological position and the weight of each neuron are associated with its estimated location. The learning process takes place locally at each node, where the input pattern is estimated location of the node (this input is dynamically changed over time except that the anchors use their known location). The neighborhood neurons of a node are determined by its 1hop neighboring nodes. It is obvious that each node becomes the BestMatching Unit (BMU) at its local region. So when updating weights at the BMU, only its 1hop neighbors' weights are updated. The BMU node also receives updates from other nodes when it becomes 1hop neighbor of other nodes. Anchors do not update their known positions during the learning process, so if the network has some nodes know their location in advance (anchors), then each node will utilize the information from these anchors by adjusting its location towards the estimated absolute location based on the information from these heard anchors. At the end of the learning process, the weight at each node (SOM neuron) is its estimated location.
3. Proposed Distributed Localization Algorithm Based on SOM
In this section, we will introduce about our proposed Distributed Rangefree Localization Algorithm (LSSOM). The first two sections describe about initialization and learning stages of the main algorithm. The mobility consideration is presented in the third section.
3.1. Initialization Stage
In the initialization stage, each anchor in the network broadcasts a packet to its neighboring nodes. This packet contains the anchor's location and a hop count initialized to one. When a node receives a packet that contains anchor information, node then decides to discard or forward the packet to its neighboring nodes or not with the following rules.

(1)
If the packet is already in the cache, the node then compares the hop count of the packet with that of the cached packet. If the hop count of the arrival packet is less than that of the cached packet, then the cached packet is replaced with a new arrival packet, and forwarded to its neighboring nodes with hop count modified to add one hop. If the hop count of the arrival packet is greater than or equal to that of cached packet, then it is dropped.

(2)
If the packet is not in the cache, then it is added to the cache and forwarded to its neighboring nodes with hop count modified to add one hop.
Having information from some anchors, the nodes now initialize their location ready for SOM learning process. In our proposed method, the initial location of a node is calculated based on either randomized value (if node does not receive enough information from three anchors) or a value calculated using a trilateral method. In this initialization stage, nodes also exchange information (using short "HELLO" message broadcast) so that each node has information about its neighboring nodes (1hop neighbors). Each node also exchanges information about 1hop neighbors (just the IDs of 1hop neighbors) with its neighboring nodes, so that all nodes in the network have information about both 1hop and 2hop neighboring nodes.
3.2. Learning Stage
Before going into our algorithm details, let us formulate the mathematical notations which will be used in this paper. We represent a wireless adhoc network as an undirected connected graph. The vertices are nodes' locations, and edges are the connectivity information (direct connection between neighboring nodes). The target wireless adhoc network is formed by anchors with known locations and N nodes with unknown locations. The unknown nodes have actual locations denoted as and estimated locations denoted as .
() Estimated location exchange: at this step, each node forwards its estimated location to all of its neighbors, so that it also knows the estimated location of its neighbors as with is the number of nodes within its communication range.
() Local update of relative location: we will now shape the topology at each region formed by the node with location together with all of its neighboring nodes. The node with location plays as the input vector and becomes the winning neuron for that region. Consequently, the neighboring nodes of the node with location will receive the updating vector from node with location . Suppose that the node with the estimated location has neighbors. The locations of these neighbors are denoted as . Based on classical SOM, neighboring nodes of the node with location will update their weight with the following formula:
where is calculated using
in which is the learning rate exponential decay function at iteration defined in (5).
where denotes the time step of the total learning steps. But, updating by using (3) means that the neighboring nodes will move toward the location determined by . This will lead to the problem as showed in Figure 1. From Figure 1, the nodes with location and are the neighbors of the node with location , but is not the neighbor of . In the worst case, the estimated location of the node with location falls into the radio range of the node with location , then the node with location may not escape from that wrong location throughout the learning process (dead location) as illustrated by position .
In this paper, we propose an algorithm to solve this problem as follows. Suppose that at the node with location , we have to update location for the neighbor node with location . First, we find out other neighboring node of the node with location that are not the neighbor of the node with location (this is done easily because each node knows its neighbors' neighbors) and find their estimated location falls into radio range of the node with location . Now we calculate the vector that has the direction towards the intersection area as illustrated by the dashed area in Figure 2. As illustrated in Figure 3, this vector is calculated using
where denotes the maximum communicable range between node with location and node with location . We use vector as a guidance to update the location of the node with location by changing (3) to (7),
The update by (7) makes each node move toward the intersection area as showed in Figure 2. This update also maximizes the correlation between the neighboring nodes that is the key problem for the speed and accuracy of topological convergence using SOM. In (7), is a learning bias parameter calculated using
with is a learning threshold. This threshold determines the step to apply this modification. Basically, we can apply this modification after several steps of SOM learning when nodes are in relative order to ensure the convergence of the learning process. At the end of this step, the node with location transmits its neighbor location updates based on (7) to all of its neighbors. As a result, it also receives the similar updates from its neighboring nodes as . Node with location now calculates its newly estimated location by averaging its current location and the updates from the neighboring nodes using
The learning process is repeated times.
3.3. Mobility Consideration
In MANETs, nodes may move in arbitrarily manner, so the movement of nodes will affect the performance of the algorithm. To adapt LSSOM with MANETs, we proposed a repeated learning algorithm as follows.
() First Time Initialization. Anchors participate in localization will flood the network just one time, so that nodes can calculate the initial location for fast topology convergence.
() Repeated Learning. At specified interval, nodes perform neighboring detection by exchanging short "HELLO" messages. Having neighboring information, nodes now proceed with the learning process.
The algorithm is illustrated by the flowchart in Figure 4.
4. Simulation Evaluations
To evaluate the performance of our proposed method, we use the average error ratio in comparison with the radio range of the nodes presented in
4.1. Simulation Parameters
To ease the comparison, we call the method in the existing work [10] as SOM, and our proposed method as LSSOM. We conducted the simulation for static and mobile scenarios by using MATLAB (we integrated SOM, DVHOP, and LSSOM into the program received from [7]) and NS2, respectively. For static scenarios, each experiment is done on thousands of randomly generated topologies that are deployed by 100 nodes on an area of 10 by 10. For mobile scenarios, we simulated on networks with 25 randomly distributed nodes on an area of 300 by 300 square meters. The propagation model is TwoRayGround and transmission range of each node is 100 meters. The common parameters used in simulation are as follows. Number of SOM learning steps is 15, and Learning bias threshold is 1.
4.2. Static Networks
With static networks, we study how the accuracy is influenced by the connectivity level (the average number of neighboring nodes that a node has direct communication with), and the number of anchor nodes deployed. Figure 5 shows the average error with different connectivity levels. The result indicates that LSSOM achieves very good accuracy over the SOM, DVHOP, MDSMAP(C), and even MDSMAP(P) from sparse to dense networks. Especially with very sparse networks, LSSOM still performs better than the others. The performance with the variance of anchors is showed in Figure 6. We find that LSSOM increases accuracy when the number of anchors increases. When the number of anchors increases, LSSOM improves accuracy much better than the others. We have tested and realized that on the grid deployment with 50% position error, LSSOM gets better accuracy than the random deployment. Figure 7 shows the average error through each SOM learning step. LSSOM needs only 15 to 30 learning steps to achieve a stable result. Comparing to thousands of learning steps in the traditional SOM, LSSOM decreases network overhead and computational cost. Figure 8(a) shows one of the actual topology that is generated during the simulation. Figures 8(b), 8(c), and 8(d) show the topologies estimated with DVHOP, SOM and LSSOM, respectively. In these figures, the rectangles and the circles denote the anchor nodes and the unknown nodes, respectively. From the figures, one can realize that LSSOM outperforms the topology regeneration. Especially, it is resistant to the perimeter effect.
Figure 9 shows another actual topology in random deployment experiment, and the topology estimation result is reported in Figure 10. The lines in Figure 10 show the differences between the actual location and estimated location of each method used. The shorter the line, the better the accuracy is.
Figure 11 shows the distribution of nodes localized for 1000 randomly generated networks with 100 nodes, number of anchors ranging from 4 to 15, and connectivity is selected randomly from 7 to 15. From Figure 11, 80% of nodes localized with the error around 30% for LSSOM.
4.3. Mobile Networks
We have implemented LSSOM in NS2 environment, in which LSSOM is designed as an agent installed on each node and runs completely in parallel manner. In our simulation scenarios, 4 anchor nodes are deployed and they are also moving. The movement of nodes is simulated with the RandomWayPoint mobility model. The mobility scenarios are generated with maximum speed of 10 meters per second. Figure 12 shows the performance of LSSOM by simulation time. From Figure 12, we can see that LSSOM will give a stable estimation accuracy after the time period for initialization and initial learning. The delay period depends on the network configuration. In our simulation on NS2, the difficulty is that the transmission delay and packet collision at MAC layer. We just simply solve the packet collision by using randomized packet exchange scheduling. Figure 13 shows the throughput for generated packets by simulation time. We see a burst of traffic at the beginning because of the anchor flooding in the initialization stage. From this observation, we can easily find that the cost for network flooding is very expensive. Figure 14 shows the throughput of dropping packets due to collision. We realize that during the learning process, about 30% of exchanging messages were dropped. Figure 15 shows the distribution of dropping packets at each node. Number of dropped packets of nodes near the center of topology is greater than that of nodes near the perimeters. It is to infer that the number of packet dropped will increase with the connectivity level, and we should consider this problem when designing a practical localization system.
5. Conclusions
We have presented our proposed Distributed Rangefree Localization Algorithm Based on SelfOrganizing Maps (LSSOMs) in this paper. By introducing the utilization of intersection areas between radio coverage of neighboring nodes, the algorithm maximizes the correlation between neighboring nodes in distributed SOM implementation. With this correlation maximization, our method increases the quality of the topology estimation and reduces the time of the topological convergence. With our proposed solution for mobility management, LSSOM is capable of working with networks having high mobility. From intensive simulations, the results show that LSSOM has achieved good accuracy over the original SOM and other algorithms. LSSOM has reduced the SOM learning steps to just around 15 to 30 steps. Besides that, LSSOM is capable of working not only with static networks, but also with mobile networks. Future work will investigate in a more precise distance measurement method to make LSSOM to be more flexible.
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Tinh, P., Kawai, M. Distributed RangeFree Localization Algorithm Based on SelfOrganizing Maps. J Wireless Com Network 2010, 692513 (2009). https://doi.org/10.1155/2010/692513
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DOI: https://doi.org/10.1155/2010/692513