## 3 Method to Obtain the Rolling Information
### Design of Tracking Loop  
The phase-locked loop (PLL) is a kind of closed-loop control system. It obtains the information about the phase and frequency of the input through a numerically controlled oscillator (NCO) creating synchronous output [28]. In chapter 2, measurement model indicates that with the rotation of spinning  
Figure 4: Process of obtaining roll angle. projectile, the geomagnetic vector projecting on cross-section \((B_{\mathrm{Z}})\) changes in the form of sinewave, which is the input of PLL. PLL achieves the phase information of input by obtaining the frequency and phase of local signal from NCO. As shown in Figure 5, PLL consists of three parts, discriminator, loop filter, and NCO. It shows the output of NCO is fed back to the front, forming a closed loop control system. It also can be presented as the block diagram of FLL for the only difference between FLL and PLL is the discriminator, where arctangent discriminator [29] is applied in PLL to obtain the difference of phase, and cross-product frequency tracking [29] is applied in FLL to obtain the difference of frequency. \(u_{\mathrm{i}}(s)\) and \(u_{\mathrm{o}}(s)\) are the Laplace transformations of input \(B_{\mathrm{Z}}\) and the output, respectively; \(u_{\mathrm{e}}(s)\) obtained by discriminator is the difference between \(u_{\mathrm{i}}(s)\) and \(u_{\mathrm{o}}(s)\); \(K_{\mathrm{d}}\) is the gain of discriminator; \(u_{\mathrm{d}}(s)\) is the output of discriminator; \(F(s)\) is a loop filter, the output of which is \(u_{\mathrm{f}}(s)\);\(\frac{K_{\mathrm{o}}}{s}\) is the Laplace transformation of NCO. Thus, transfer function of above system can be expressed as  
\[H(s)=\frac{u_{\mathrm{o}}(s)}{u_{\mathrm{i}}(s)}=\frac{K_{\mathrm{o}}K_{d}F(s) }{s+K_{\mathrm{o}}K_{d}F(s)}=\frac{KF(s)}{s+KF(s)} \tag{10}\]  
where, \(F(s)\) is the transfer function of loop filter, \(K_{d}\) and \(K_{\mathrm{o}}\) are the gain of discriminator and NCO, respectively. The gain of loop filter can be expressed as  
\[K=K_{\mathrm{o}}K_{d} \tag{11}\]  
PLL is able to track the target signals with low noise precisely, but the narrow bandwidth restricts its accuracy in high-dynamic situations. Furthermore, it does not work properly when tracking signals are with much noise. In contrasts, the FLL owns a comparatively wide bandwidth and good dynamic performance. However, FLL is applied to track frequency of signal and has less efficiency in obtaining phase information. To satisfy both accuracy and dynamic performance, a FLL-assisted PLL loop combining advantages of PLL and PLL was designed to track the target signals quickly and accurately. What is shown in Figure 6 is the discrete time system of second-order FLL-assisted third-order PLL. The combined loop filter is the Z transform of \(F(s)\) in Figure 5, which consists of \(F_{\mathrm{p}}(s)\), a transfer function of third-order PLL, as well as \(F_{\mathrm{f}}(s)\), transfer function of second-order FLL. Both frequency discriminator and phase discriminator in Figure 6 make up discriminator in Figure 5. \(p_{\mathrm{e}}\) and \(f_{\mathrm{e}}\) are the outputs of discriminators input to second-order FLL and third-order PLL, respectively. There are parameters to be determined to optimize system to keep good performance in different situations, which are \(T_{\mathrm{roll}}\), unit delay (it is also the integration time of integrate and dump process in Figure 7 and discussed in chapter 4), damping ration \(\xi\) (\(a_{2}=2\xi\)), natural frequency \(\omega_{\mathrm{nf}}\) of second-order FLL as well as natural frequency \(\omega_{\mathrm{np}}\) of third-order PLL. Details of the selection of parameters \(a_{3}\) and \(b_{3}\) can be found in \"Understanding GPS: principles and ap pl  
Figure 5: Block diagram of phase-locked loop in frequency domain (Laplace transform).