## 2 Navigation and Localization
### Dead-Reckoning and Inertial Navigation  
The simplest method to obtain a position for a moving vehicle is by integrating its velocity in time. This method is known as dead-reckoning [8]. DR requires to know the velocity and direction of the vehicle, which is usually accomplished with a compass and a water speed sensor. The principal problem is related to the presence of an ocean current, as illustrated in Figure 2, because it will add a velocity component to the vehicle which is not detected by the speed sensor. Then, the accuracy of the method will be strongly affected especially when the vehicle navigates at a low velocity. Figure 1: Autonomous Underwater Vehicle (AUV) technologies for localization and navigation. Inertial sensors can be used to improve the navigation accuracy and reliability of DR methods. The INS consist of three mutually-orthogonal accelerometers aligned to a gyroscopic reference frame. Measured accelerations are integrated to obtain the desired velocity, position, and attitude information of the vehicle. The fact that inertial navigation is self-contained--it neither emits nor receives any external signal--is one of its most significant strengths, making it a stealthy navigation solution, immune to interference or jamming [26]. However, the error on the pose estimations is known to increase over time, and depends on the accuracy of the sensors used. Mathematically, the total acceleration, denoted as \(\tilde{r}\), can be expressed as follows [27]:  
\[\tilde{r}=a+g, \tag{1}\]  
where \(a\) is the acceleration calculated by the INS and \(g\) is the gravitational acceleration. Since the accelerometers do not sense the gravity, the position of the vehicle obtained by integrating the acceleration measurements will result with an error. Gyroscopic drifts are also a source of error that can result in significant misalignments between the sensor frame and the earth-fixed reference frame, causing navigation errors that also grows over time. Using a Global Positioning System (GPS) is a common method used to correct these errors. However, to correct the error accumulated by the INS, the vehicle must go to the surface to obtain a new GPS location at regular intervals, which can result in a waste of time and resources. Integration of an INS and a GPS data can also be a complex process, since those systems are based in completely different principles. Even if the instruments were perfect, the estimations of an INS would result with an error [9]. The gyroscopic reference frame is aligned to a reference ellipsoid model of the earth. The reference ellipsoid conforms roughly to the shape of the earth, and in particular to mean sea level. If the mass of the earth were homogeneously distributed within the ellipsoid, the gravity vector would be normal to the reference ellipsoid surface. However, due to the inhomogeneous distribution of the earth mass, the gravity vector can have significant components tangential to the reference ellipsoid surface (known as vertical deflections) as shown in Figure 3. Since an INS cannot distinguish between the tangential components of earth gravity and the horizontal acceleration of the vehicle, these gravity disturbances cause errors in the INS velocity and position estimations. Figure 2: Dead-Reckoning drift effect. The latest advances in MEMS inertial sensors are having profound effects on the recent availability of MEMS-Inertial Measurement Units (IMUs), that has become clearly attractive for a wide range of applications where size, weight, power, and cost are key considerations [28]. This set of sensors can be used to implement an Attitude and Heading Reference System (AHRS) or an INS. Some MEMS-based systems commercially available are showed in Table 1. Nevertheless, despite technological developments in inertial sensors, INS are underway to reduce the accuracy drift at the level of a few meters uncertainty over one hour of unaided inertial navigation [29].