## 3 Method to Obtain the Rolling Information
### Model Verification Based on 6 DoF Trajectory  
From the analysis above, the design and optimization of the tracking system depends on the frequency of input \(B_{\mathrm{Z}}\) to a large extent. In practice, the rotational speed of a spinning projectile is determined by the type of projectile and launching platform. Meanwhile, information of geomagnetic field can be obtained from WMM and IGRF mentioned in chapter 2 and yaw and pitch are given by GPS receivers (or other sensors) in this proposed technology. In this simulation, a 6-DoF trajectory [31] would provide information needed to verify the designed tracking loop. Based on a 6-DoF trajectory, as shown in Figure 9(a), of a 155 mm artillery projectile, the properties of which are listed in Table 2, the designed tracking loop is applied to obtain the roll angle of spinning projectile based on the information of yaw and pitch (obtained from GPS in practice) is shown in Figure 9(b). Table 2 also provides the details of geomagnetic field (assuming that declination and inclination changes slightly in the range of howitzer). Figure 10(a) shows the change of amplitude of during flight, while Figure 10(b) shows the amplitude changing within 0.03 second. From Figure 9(b), yaw varies (from 0\({}^{\circ}\) to 4\({}^{\circ}\)) slighter than pitch (from 50\({}^{\circ}\) to \(-\)67\({}^{\circ}\)), and the rotational speed drops from 300 Hz to 134 Hz in the form of acceleration function. Figure 11 indicates that in flight, amplitude of \(B_{\mathrm{Z}}\) changes in the form of sinewave along with the change of pitch and yaw. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Properties of Projectile** & **Specifications** & **Properties of Geomagnetic Field** & **Specifications** \\ \hline Mass (kg) & 46.88 & Strength (mGs) & 500 \\ Width (m) & 0.866 & Declination (\({}^{\circ}\)) & \(-\)6.8285 \\ Axial Inertial (kg\(\cdot\)m\({}^{2}\)) & 0.1658 & Inclination (\({}^{\circ}\)) & 59.263 \\ Initial Attitude (\({}^{\circ}\)) & 0 (Yaw); 51 (Pitch) & Sampling Frequency (Hz) & 1000 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Properties of projectile and field. Figure 10: Model of 6-degrees of freedom (DoF) trajectory: (**a**) Trajectory of projectile and (**b**) information of the speed, pitch and yaw from 6-DoF. Based on (14), (15), (20), and \(\xi=0.707\), noise bandwidth--\(B_{\rm PLL}\) and \(B_{\rm FLL}\)--are two parameters to be determined to optimize the 2-order FLL-assisted 3-order PLL tracking loop. From Figures 8 and 9 it can be seen that the frequency at 3 dB attenuation increases with the raising of noise bandwidth. For the tracking accuracy, the cut-off frequency has to be not less than the lowest spinning speed (3 Hz). However, with a higher range of rotational speed, the cut-off frequency can be improved slightly at the aim of a quick transient response especially for FLL. Thus, in the circumstance of spinning speed dropping from 300 Hz to 134 Hz, the \(B_{\rm PLL}\) is set as 0.65 and \(B_{\rm FLL}\) is set as 0.7, which also indicates that transient performance is prior to steady-state performance in FLL. Meanwhile, the period of input signal is from 3.33 ms to 7.46 ms. Based on the analysis, integration time \(T_{\rm roll}\) should be five times more than \(T\) to ensure tracking accuracy, it is set as 40 ms. Therefore, based on the 6-DoF trajectory above, the optimized parameters are listed at Table 3.  
After optimization of the designed system, the tacking results are given as follows. Figure 12 shows the tracking results. There are amplitude of input \(B_{Z}\) and output of tracking system, the phase of tracking signals, phase discriminator output, as well as error of phase tracking. (a) indicates that the output of tracking system tracks the input well.