# Complete Systematic Error Model of SSR for Sensor Registration in ATC Surveillance Networks
## 3 Azimuth Bias
### Antenna Deviation  
The first factor that affects the azimuth bias is the deviation between the vertical axis of the antenna reference and the reference of the rotation axis. This deviation will be defined with three parameters describing the rotation in the three axes of the Cartesian coordinate system. In order to determine the antenna deviation, the coordinated system is defined as: \(y\)-axis, parallel to the ideal antenna boresight; \(z\)-axis, orthogonal to the \(y\)-axis, resulting that both axes compose the ideal plane of the 2D antenna pattern (\(z\)-axis is the ideal antenna rotation axis); the \(x\)-axis, orthogonal to the \(y\)-axis and \(z\)-axis positive to the right. In order to derive the model, the biased antenna coordinated system (\(x^{\prime},\ y^{\prime},\ z^{\prime}\)) is calculated. The axes of the coordinated system (\(x,y,z\)) are rotated to get the new system (\(x^{\prime},\ y^{\prime},\ z^{\prime}\)) and the measured azimuth (\(\theta_{m}\)) is calculated in the new system. Under the assumption that the deviation angles of the antenna axes are small (below \(1^{\circ}\)), the order of the rotation around each axis has a negligible influence in the model. The three rotation angles (Figure 1) correspond to the following bias parameters components:  
* Azimuth offset (\(\theta_{0}\)): it is generated with the rotation of the antenna in the horizontal plane. This offset is constant for all the targets of the sensor. This rotation is defined positive in counter-clockwise direction. * Antenna squint (\(s_{ant}\)): it is the bias produced due to the rotation in the orthogonal plane to the antenna boresight. This component produces an azimuth bias dependent on the target elevation (angle of line of sight respect to horizontal plane). This rotation is defined positive in counter-clockwise direction. * Antenna tilt (\(t_{ant}\)): produced due to the rotation of the plane of the 2D antenna pattern around the \(x\)-axis. As radars used in ATC applications do not measure the elevation, this bias has not consequences in the first order terms of the azimuth bias. This rotation is defined positive in clockwise direction. The rotation matrix for each one of the axes are:  
\[R_{z}(\theta_{0})=\left( \begin{array}{ccc}\cos(\theta_{0})&\sin(\theta_{0})& 0\\ -\sin(\theta_{0})&\cos(\theta_{0})&0\\ 0&0&1\end{array} \right) \tag{4}\]  
\[R_{x}(t_{ant})=\left( \begin{array}{ccc}1&0&0\\ 0&\cos(t_{ant})&-\sin(t_{ant})\\ 0&\sin(t_{ant})&\cos(t_{ant})\end{array} \right) \tag{5}\]  
Figure 1: Representation of: (**a**) antenna azimuth offset, (**b**) antenna squint (**c**) and antenna tilt. \[R_{y}(s_{antt})=\left( \begin{array}{ccc}\cos(s_{antt})&0&-\sin(s_{antt})\\ 0&1&0\\ \sin(s_{antt})&0&\cos(s_{antt})\end{array} \right) \tag{6}\]  
Applying these rotations, the resultant coordinate system is:  
\[\left( \begin{array}{c}x^{\prime}\\ y^{\prime}\\ z^{\prime}\end{array} \right)=R_{z}(\theta_{0})\cdot R_{x}(t_{ant})\cdot R_{y}(s _{ant})\cdot\left( \begin{array}{c}x\\ y\\ z\end{array} \right) \tag{7}\]  
Due to the convention in geographical systems of the azimuth with the origin in the \(y\)-axis, positive in clockwise, the measured azimuth \((\theta_{m,ant})\) is:  
\[\theta_{m,ant}=arctan\bigg{(}\frac{x^{\prime}}{y^{\prime}}\bigg{)} \tag{8}\]  
These calculations are made under two assumptions: the error is only caused by the deviations of the antenna and this antenna can only acquire the targets that are in the 2D antenna main lobe. A multidimensional Taylor series on the bias variables \(\theta_{0},s_{ant}\) and \(t_{ant}\) is made to derive the following linearized approximation:  
\[