## 2 Sensor Network Localization
### Positioning Using Angle of Arrival  
In many cases, practical situations require one to be aware of a sensor's physical coordinates. Installing GPS receivers in every sensor is not always optimal from the cost-related and other perspectives. Typically, only a few nodes (seed nodes, landmarks, etc) of the network are equipped with GPS and know their physical location. The rest of the nodes can only communicate with other nodes and determine relative location characteristics such as distance, angles, etc. Various localization techniques are used to obtain location of all the sensors in the network. The following method assumes that the network consists of two types of nodes: usual and more capable nodes - which knows its position. [PERSON] and [PERSON] propose a method by which nodes in an ad hoc network collaborate in finding their position and orientation, assuming that a small part of the network has a position capability. Also, every node in the network has a capability to determine the angle of the arriving signal (AOA).  
Each node in the network has one fixed main axis (which may not be the same for different nodes) and the node is able to measure all angles against this axis (Figure 1).  
Every node in an ad hoc network can only communicate with its immediate neighbors within the radio range, and its neighbors may not always be landmarks, i.e., the nodes that know their position. [PERSON] propose in [17] a method to forward orientation in such way, that the nodes which are not in direct contact with landmarks can determine its orientation with respect to the landmarks. Orientation means bearing, radial, or both. Bearing is an angle measurement with respect to another object. A radial is the angle under which the object is seen from another point. The authors examine two algorithms: Distance Vector Bearing (DV-Bearing), which allows each node to get a bearing to landmark, and DV-Radial, which allows a node to get a bearing and radial to a landmark. The propagation in both algorithms works the following way: nodes adjacent to landmarks determine their bearing/radial directly from landmark and send to the network the information about their position. The method of computing node's bearing and radial at each step is shown in Figure 2.  
DV-Bearing algorithm works the following way. Nodes \(A\), \(B\), and \(C\) are neighbors and they can communicate with each other. Suppose that the node \(A\) needs to find its bearing to node \(L\), which is not within radio range of node \(A\) but within radio range of nodes \(B\) and \(C\). Since \(A\), \(B\), and \(C\) can locate each other than the node \(A\) can determine all the angles in triangles \(ABC\) and \(BCL\). But that would allow to calculate the angle \(LAC\) and consequently the bearing of \(A\) to \(L\), which is equal to \(c+LAC\). Once node \(A\) knows three bearings to landmarks, which are not at the same line, then it can calculate its own location by triangulation.  
The DV-Radial algorithm works the same way but with only one difference that node \(A\) needs to know not only bearings of nodes \(B\) and \(C\) to node \(L\), but also the radials of \(B\) and \(C\) from \(L\). The knowledge of radials improves accuracy of the algorithm. When all angles are measured against the same direction (for example, when compass is available) then these two methods become identical.  
Figure 1: Nodes with AOA capability.