## 6 Conclusions  
Real-time estimation of roll angle of spinning projectile based on phase-lock on signals from single-axis magnetometer is proposed and verified. This technology is based on information, position, velocity, yaw, and pitch from GPS receiver (or other sensors). Unlike the conventional measurement by inverse trigonometric operation based on amplitude of multiple outputs, single-axis magnetic sensor enables this system to measure rolling information without compensation based on a second-order frequency-locked loop (FLL) assisted third-order phase-locked loop (PLL). A mathematical model based on coordinate transformation of the local North-East-Down (NED) coordinate system and body coordinate system was set up to analyze the relationship between roll angle and magnetic signal. There is a difference (compensation angle) between roll angle and phase information of magnetic signal detected. And obtaining the phase information and compensation angle is the key to figure out roll angle. According to the practice, the spinning projectile rotates in the form of acceleration function, a second-order FLL-assisted third-order PLL is selected to track the phase information. Parameters of tracking loop depends on the frequency of input and are discussed to optimize system to keep  
Figure 23: Results of group 3: (**a**) Description of digital amplitude of _BZ;_ (**b**) Comparison of pulses produced by hall sensor and controller; (**d**) Description of speed figured out from controller; (**e**) Description of speed error and phase error. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Group**} & **Nominal** & **Average** & **SD** & **SD** \\  & **Speed** & **(Phase Error)** & **(Speed Error)** & **(Phase Error)** \\ \hline
1 & 5 r/s & 1.487\({}^{\circ}\) & 0.047\({}^{\circ}\) & 1.929\({}^{\circ}\) \\
2 & 5 r/s & 1.827\({}^{\circ}\) & 0.045\({}^{\circ}\) & 2.906\({}^{\circ}\) \\
3 & 5 r/s & 2.320\({}^{\circ}\) & 0.050\({}^{\circ}\) & 2.684\({}^{\circ}\) \\ \hline \multicolumn{2}{c}{Average} & 1.878\({}^{\circ}\) & 0.047\({}^{\circ}\) & 2.506\({}^{\circ}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Results of experiments. excellent transient performance as well as accurate in different speed situation. A 6-DoF trajectory is given to verify the tracking system designed and results show an outstanding tracking performance in both phase (error less than 3\({}^{\circ}\)) and speed (error less than 0.1 Hz). Generally, error of roll angle measurement is required by guidance and control module. To eliminate the measurement error from tracking loop, the selection of parameters of noise bandwidth and integration time should meet the requirement that 3-sigma jitter is no more than one fourth of the pull-in range of discriminator. In the range of howitzer (~30 km), the change of declination and inclination has little effect on the compensation angle (deviation mostly keeps within 0.15\({}^{\circ}\)) and if necessary, the geomagnetic information should be stored in advance and exported to the tracking system to figure out compensation angle timely. Some filtering measurements should be taken to deal with attitude information from GPS receivers and error of yaw and pitch should be less than 5\({}^{\circ}\) so that SD of deviation of compensation angle mostly keeps within 6\({}^{\circ}\). A magnetic sensor with a high resolution helps to eliminate error from blind area and the period of blind area would not last too long for attitude keeps changing in flight. Experiments on rotary table are carried out to verify the proposed method. In the circumstance of nominal speed 5 Hz and attitude changed, the tracking loop is able to track both phase and speed accurately and stably. The mean of phase error is within 2.4\({}^{\circ}\) and SD of phase error is no more than 3\({}^{\circ}\). Profound research, such as improving accuracy and experiments with higher speed and changing attitude, will be conducted. Conceptualization, Q.S. and Z.D. (Zhaowei Deng); Methodology, Q.S., Z.D. (Zhaowei Deng), Z.D.