# Underwater Electromagnetic Sensor Networks, Part II: Localization and Network Simulations
## 3 Algorithms for Self-Localization and Navigation-Aid Applications
### Aid to Underwater Navigation  
In this section, we consider a different localization algorithm for underwater navigation aid. Underwater navigation is useful in applications where robotic devices or UUVs require orientation to perform some task, such as collecting data from a sink node or exploring the seabed. There are many general localization algorithms in the literature, both distributed and centralized, such as [17; 19; 20; 21]. For the specific application of underwater navigation, the UUV needs to estimate its own position by measuring distances to every node within reach. Since we assume the medium is isotropic, triangulation methods for outdoor localization are sufficient. We assume that the nodes know their position from a previous calibration, such as the one described in Section 3.1. Then, these nodes transmit a signal to the UUV at a fixed power level together with their local position (if unknown by the UUV). We assume the UUV knows the prefixed transmit power level. Then, the UUV can estimate the distance \(\hat{d}\) to each node using the channel form that was described in Section 2. In particular, it needs to solve the following implicit equation:  
\[P_{tx}-P_{rx}-L(d_{0})=\eta_{1}\frac{\hat{d}}{d_{0}}+10\eta_{2}\log\big{(} \frac{\hat{d}}{d_{0}}\big{)}, \tag{5}\]  
where \(P_{tx}\) corresponds to the prefixed transmit power level (in dBm); \(P_{rx}\) the received power at the UUV (in dBm); and \(\eta_{1}\), \(\eta_{2}\), \(d_{0}\) and \(L(d_{0})\) are the channel parameters derived in Section 2. Since the UUV measures all distance estimates directly, it is preferable that it performs all of the computations (i.e., in a centralized manner), so that we avoid the nodes transmitting other data to each other. Regarding the localization problem, the following optimization problem is frequently proposed:  
\[\min_{u}\sum_{i=1}^{n}(\hat{d}_{i}^{2}-\|u-x_{i}\|^{2})^{2} \tag{6}\]  
where \(x_{i}\) represent the network node's coordinates; \(u\) the unknown UUV target coordinates; \(\hat{d}_{i}\) the estimated distances from the sensors to the UUV; and \(n\) represents the number of measurements available (nodes that can reach the UUV). The localization problem proposed in (6) is static, i.e., it does not consider the speed of a UUV navigating in the network. We remark that in order to consider a dynamic UUV, the model should also take into account the time delay inherent to the underwater communication network. Problem (6) is non-convex and is also suboptimal in the maximum-likelihood (ML) sense. Nonetheless, since the noise variance is relatively small compared to the channel attenuation (as discussed in Section 2), the solution to the problem yields a good estimate that is close to the theoretical bound of the ML estimator (so that it can be considered accurate enough for many applications). A common formulation that is an alternative to Problem (6) is to obtain the exact ML estimator (see, e.g., [22] Section II-A). Nevertheless, this is a more difficult problem and requires the use of suboptimal methods. It turns out that the algorithms that aim to suboptimally solve the exact ML may yield poorer estimates than the solution of (6), so we do not consider these methods in our simulations. A more thorough discussion can be found in [22]. There are several algorithms in the literature that try to solve (6). For instance, the best linear unbiased estimator proposed in [19] yields a suboptimal result since it solves a relaxed approximation of (6). Other linear estimators, such as the one proposed in [20], minimize an error function derived from (6) rather than the problem itself. Additionally, majorization methods, such as the one proposed in [17], solve the problem by successive convex approximations that converge to local minimization points. An optimal solution of (6) is however available as presented in [22]. Its solution can be found by reformulating the objective to a non-convex quadratic problem with a single quadratic constraint, which presents strong duality and can be solved optimally [23].