# Complete Systematic Error Model of SSR for Sensor Registration in ATC Surveillance Networks
## 3 Azimuth Bias
### Optical Encoder  
In modern radars the azimuth of the antenna boresight is determined with an optical encoder that is allocated in the rotating radar shaft. In the installation of this encoder, small calibration errors can be made and these are transformed in azimuth biases. One of the errors is the misalignment between azimuth reference of the encoder and the axis, this is an azimuth offset. This error is the same kind that the azimuth offset of the antenna and the azimuth offset of the rotating axis, described in Section 3.1. Other error to be considered is the eccentricity bias. Usually this is the encoder error with the biggest impact in the measures [19]. In the installation, the rotation axis (O) is not exactly placed in the geometric center (O') of the encoder. The resultant geometry is shown in the Figure 4, supposing that the difference of the centers is in the azimuth \(\theta=0\). Resolving the triangle shown in Figure 5 and making a deduction similar to [20; 21] we have the following relation:  
\[\left. \begin{array}{c}A=\Delta R\ \sin(\theta)\\ A=R\ \sin(\alpha)\end{array} \right\}=>\sin(\alpha)=\frac{\Delta R}{R}\sin(\theta) \tag{19}\]  
where, \(\Delta R\) is the eccentricity offset, \(R\) is radius of the encoder and \(\theta\) and \(\alpha\) are angles of the triangle shown in Figure 5.  
Using the relation between sine and cosine for \(\alpha\) (\(\cos(\alpha)=\sqrt{1-\sin^{2}(\alpha)}\)) we also have the following relation:  
\[\cos(\alpha)=\sqrt{1-\left(\frac{\Delta R}{R}\right)^{2}\sin^{2}(\theta)}= \frac{1}{R}\sqrt{1-\Delta R^{2}\sin^{2}(\theta)} \tag{20}\]  
Figure 3: (**a**) Right triangle of the axis skew and (**b**) relation between the rotation of the axis and the skew of the rotation axis and its direction. On the other hand, from Figure 5 we can also see:  
\[r=R\;\cos(\alpha)-\Delta R\cos(\theta)r=-\Delta R\cos\theta+\sqrt{R^{2}-\Delta R^{2 }\sin^{2}(\theta)} \tag{21}\]  
Finally:  
\[\left. \begin{array}{c}h=R\;\sin(\theta_{m})\\ h=r\;\sin(\theta)\end{array} \right\}=>\theta_{m}=arcsin\Big{(}\frac{r}{R}\sin( \theta)\Big{)} \tag{22}\]  
The measured azimuth is also linearized with a Taylor series on the bias variable \(\Delta R\) in order to get an approximation to model. Assuming that \(\frac{\Delta R}{R}\ll 1\) the approximation of \(\Big{(}\frac{\Delta R}{R}\Big{)}^{2}=0\) can be made. With these simplifications, the azimuth measurement can be approximated as:  
\[\theta_{m}\approx\theta-\frac{\Delta R}{R}\sin(\theta) \tag{23}\]  
The second error produced by the optical encoder is due to the swash between the encoder and the horizontal plane. A small swash of the encoder produces that the projection of the circumference is an ellipse and the determination of the azimuth has errors when this azimuth does not point to the ellipse axis. In Figure 6a a schematic model of the optical encoder is represented. The circle line on the \(x\)-\(y^{\prime}\) plane represents the mechanical encoder and the dotted line on the \(x\)-\(y\) plane represents the projection of the rotated mechanical encoder circumference on the horizontal plane. The ideal azimuth  
Figure 4: Optical encoder with eccentricity bias. Figure 5: Resolution of the resultant triangle for eccentricity bias. is represented on the horizontal plane but the measured azimuth is obtained by the rotated encoder. In Figure 6b both mechanical encoder circumference and its projection on the horizontal plane are represented in the same plane in order to represent the 3D projection in a 2D figure to see the difference between the real azimuth and the measured azimuth. Following the methods used in [18] to convert geocentric latitude in reduced latitude we can obtain the relation between \(\theta_{m}\) and \(\theta\). Using the next variables:  
\[ \begin{array}{l}x=r\,\sin(\theta_{m})\\ y_{m}=r\cos(\theta_{m})\\ y=y_{m}\cos(s_{enc})\end{array}