## 2 Navigation and Localization
### Acoustic Navigation
#### Acoustic Ranging
The underwater target position (\(P_{t}\)) is calculated using the moving vehicle positions (\(P_{i}\)) and the ranges measured between the moving vehicle and the target (\(\overline{r_{i}}\)) expressed as:  
\[\overline{\mathrm{r}_{i}}=\|\mathrm{P_{t}-P_{i}}\|+\mathrm{w_{i}} \tag{7}\]  
where \(w_{i}\) is a zero mean Gaussian measurement error. Different methods can be used to solve the system and find the target position through ranges: linearize the function and find a closed-form least squares solution; or use an iterative minimization algorithm to minimize a cost function related to the maximum likelihood estimate. [PERSON] et al. [51] presented an AUV localization system that relied on the computation of the ranges between the vehicle and one or more underwater beacons, the location of which may be unknown. The aim of the system was to compute in real time an estimate of the position of the AUV and simultaneously construct a map composed by the estimations of the locations of the beacons. Experiments were performed with three autonomous marine vehicles following three different trajectories. Minimum-energy estimation, projection filters, and multiple-model estimation techniques were used as observers to compare the results. A combination of those estimators produced the best results in terms of error in the trajectory followed by the AUV, which was reduced from tens of meters up to some meters in the first three minutes of the test. [PERSON] et al. [52] presented a solution to range-only localization using a Sum of Gaussian (SoG) filter. Two variations of the SoG filter were proposed and tested in real experiments, where an AUV performed an autonomous localization and homing maneuver. The results in all experiments showed that the AUV was able to home with an error smaller than 4 m. Results were corroborated by a vision-based algorithm. [PERSON] et al. [50] developed a range-only underwater target localization system. A wave glider performed as a moving LBL in simulations and real sea tests. The aim of the study was to determine the best path and its characteristics, such as number of points, radius and offset, to obtain the desired target localization performance. Results showed that with a minimum number of 12 points, radius greater than 400 m and offset as low as possible, the Root-Mean-Square Error (RMSE) can be of less than 4 m.  
[PERSON] et al. [53] presented a new method to solve problems of LBL systems such as communication synchronization among hydrophones. The system considers a Strapdown Inertial Navigation System (SINS) and the formation of a matrix of several virtual hydrophones. A single sound source is placed at the bottom of the sea and sends periodic signals meanwhile a single hydrophone is installed on the AUV. In the AUV navigation trajectory, four selected recent positions of the AUV are regarded as four virtual hydrophones of the LBL matrix, which constitute a virtual LBL matrix window. Simulation results indicate that the proposed method can effectively compensate for the position error of SINS. Thus, the positioning accuracy can be confined to 2 m.