# Complete Systematic Error Model of SSR for Sensor Registration in ATC Surveillance Networks
## 3 Azimuth Bias
### Axis Skew  
The axis skew is the inclination of the rotation axis. It causes an effect similar to the antenna deviation but in this case the bias also depends on the azimuth. For this model we use the local Cartesian coordinated system in the radar position: the \(y\)-axis pointing to the north and the \(x\)-axis pointing to the east both in the Earth horizontal plane. As in the previous subsection the biases of the azimuth can be calculated with rotations of the system axis as is represented in Figure 2.  
In order to be coherent with the names of the rotations, the rotation in \(y\)-axis will be named axis squint and the rotation in \(x\)-axis will be named axis tilt. In this case, the tilt gives errors on the measures because it is the rotation in the \(x\)-axis and not in the orthogonal plane to the antenna boresight.  
In this case the rotation matrix are:  
\[R_{x}(t_{axis})=\left(\begin{array}{ccc}1&0&0\\ 0&\cos(t_{axis})&-\sin(t_{axis})\\ 0&\sin(t_{axis})&\cos(t_{axis})\end{array}\right) \tag{12}\]\[R_{y}(s_{axis})=\left(\begin{array}{ccc}\cos(s_{axis})&0&-\sin(s_{axis})\\ 0&1&0\\ \sin(s_{axis})&0&\cos(s_{axis})\end{array}\right) \tag{13}\]  
The new coordinates system (\(x^{\prime},\ y^{\prime},\ z^{\prime}\)) is:  
\[\left(\begin{array}{c}x^{\prime}\\ y^{\prime}\\ z^{\prime}\end{array}\right)=R_{x}(t_{axis})\cdot R_{y}(s_{axis})\cdot\left( \begin{array}{c}x\\ y\\ z\end{array}\right) \tag{14}\]  
The measured azimuth is calculated again (\(\theta_{m,axis}=\arctan(\frac{x^{\prime}}{y^{\prime}})\)) and the result is approximated linearizing with a multidimensional Taylor series on the bias variables \(s_{ant}\) and \(t_{ant}\). In this case the azimuth is not particularized to zero (\(\theta\approx 0\)) because the reference axes are fixed to north and the ideal measurement can have any value of azimuth. The linear approximation, rejecting higher order terms in Taylor series is:  
\[\theta_{m,axis}\approx\theta+t_{axis}\tan(\varphi)\sin(\theta)-s_{axis} \tan(\varphi)\cos(\theta) \tag{15}\]  
Therefore, the azimuth bias due to the axis skew is:  
\[\Delta\theta_{axis}=\theta_{m,axis}-\theta\approx t_{axis}\tan(\varphi) \sin(\theta)-s_{axis}\tan(\varphi)\cos(\theta) \tag{16}\]  
An alternative equivalent model to the one in (16) can be derived as follows. Under the assumption that the inclination of the axis is small, the spherical triangles with components t\({}_{axis}\) and \(s_{axis}\) can be approximated by the right triangle of Figure 3, expressed in terms of \(\beta_{axis}\) and \(\alpha_{axis}\) where \(\beta_{axis}\) represents the total skew and \(\alpha_{axis}\) the direction where the axis is skewed. The new relation is:  
\[s_{axis}=\beta_{axis}\sin(\alpha_{axis}) \tag{17}\] \[t_{axis}=\beta_{axis}\cos(\alpha_{axis})\]  
Substituting (17) in (16) the final model could be written, after some minor algebra, as:  
\[\Delta\theta_{axis}=\beta_{axis}\sin(\theta-\alpha_{axis})\tan(\varphi) \tag{18}\]  
Figure 2: Skew of the rotation axis of the radar.  
The azimuth offset due to this effect at the direction of the skew is zero, and it follows a sinusoidal law with azimuth. The model with \(s_{axis}\) and \(t_{axis}\) parameters in (16) is equivalent to the model with \(\beta_{axis}\) and \(\alpha_{axis}\) parameters in (18). The former model (16) results generally in better stability when used for the development for bias estimation algorithms.