# Applying Neural Networks in Aerial Vehicle Guidance to Simplify Navigation Systems
## 2 Vehicle Modeling
### Navigation  
Navigation refers to the determination during the flight, of the position and attitude of the vehicle, and target position.  
Determining the position of the vehicle may consist of integrating accelerometers' measurements to be hybridized with GNSS sensor information. The details of these calculations are out of the scope of this paper, see [1; 2; 28; 29] for details on this approach.  
As it was mentioned before, determining attitude involves knowing two or more different vectors in two different reference systems. The velocity vector and the line of sight vectors can be the two needed vectors. If a GNSS sensor device is equipped on the aircraft, velocity vector can be directly obtained from sensor measurements in the NED triad, which can be expressed as shown in (7), where \(v_{x_{NED}},v_{y_{NED}}\) and \(v_{z_{NED}}\) are the components of this velocity in NED axes.  
Figure 2: Scheme of the GNC process.  
\[\overrightarrow{v_{NED}}=[v_{x_{NED}},v_{y_{NED}},v_{z_{NED}}]^{T} \tag{7}\]  
In parallel, the same velocity vector can also be calculated in body triad from a set of accelerometers, one on each of the axes. Integrating each of their measures along time, the velocity vector is obtained as shown in (8). Here, \(a_{x_{B}},a_{y_{B}}\) and \(a_{z_{B}}\) are the components of the acceleration in body axes as measured by the accelerometers and \(\overrightarrow{\omega_{b}}\) is the estimated angular speed expressed in body axes. Note that, at this point, \(\overrightarrow{\omega_{b}}\) is unknown, and the algorithm for estimating it will be shown in the following sections.  
\[\overrightarrow{v_{B}}=\int\Big{\{}[a_{x_{B}},a_{y_{B}},a_{z_{B}}]^{T}+ \overrightarrow{\omega_{b}}\times\overrightarrow{v_{B}}\Big{\}}dt \tag{8}\]  
Similarly, the line of sight vector must be obtained in NED and body reference systems, \(\overrightarrow{LOS_{NED}}\) and \(\overrightarrow{LOS_{B}}\), respectively. \(\overrightarrow{LOS_{NED}}\) can be easily obtained from GNSS sensor information. However, the semiactive laser kit is needed to derive \(\overrightarrow{LOS_{B}}\), and this sensor signal is not available until the vehicle is close enough to the target. This means another vector is needed to successfully estimate the attitude of the vehicle during all the phases of flight.  
The gravity vector poses as a natural candidate for such a challenge. Notice that determining the gravity vector in NED triad is straightforward. It is always parallel to \(\overrightarrow{z_{NED}}\). Its expression is shown in (9), where \(g\) is the gravity acceleration, which is a fixed constant in this model (9.81 m/s\({}^{2}\)). Note that precision may be increased using more sophisticated models, i.e., it can be made variable with longitude, latitude, and altitude.  
\[\overrightarrow{gNED}=g[0,0,1]^{T} \tag{9}\]  
However, the gravity vector expressed in another reference system, i.e, body axes, is also needed. However, although there are multiple available approaches to obtain it, none of them is simple and/or require additional sensors. For example, it can be estimated determining the constant component of the measured acceleration employing a low pass filter, where Jerk in body axes is calculated by derivation of the acceleration; then, it is integrated to obtain the nonconstant component of acceleration and, by subtracting this nonconstant component from the measured acceleration, gravity vector may be estimated. However, this method is not valid when the aircraft rotates. Another method to obtain the gravity vector is to integrate the mechanization equations [30] to obtain it from the resulting expressions. However, gyros are needed to implement this method. Therefore, the keystone of the presented attitude determination method is determining gravity vector in body axes.  
An estimation method for the gravity vector, which is valid for nonrotating and rotating aircraft and which is only based on accelerometers, is presented in the following subsection.