## 3 Parameter Self-Calibration Method for Loops in Deeply Coupled Systems Based on Norm Analysis
### Parameter Self-Calibration Method for the Tracking Loop in a Deeply Coupled System  
A method based on the norm property was proposed to analyze the relationship between the IMU errors and the system velocity error above; the next step is to make the judgement on the stability of the tracking loop in deeply coupled systems in highly dynamic environments.  
For the deeply coupled system, the total tracking error of carrier loop could be given as follows:  
\[\sigma_{PLL}=\sqrt{{\sigma_{IPLL}}^{2}+{\sigma_{v}}^{2}+{\sigma_{A}}^{2}}+ \frac{\theta_{e}}{3}, \tag{22}\]  
\[\sigma_{IPLL}=\frac{180}{\pi}\sqrt{\frac{B_{L}}{C/N_{0}}}\bigg{[}1+\frac{1}{2 T_{coh}\cdot C/N_{0}}\bigg{]}, \tag{23}\]  
\[\sigma_{A}=360\frac{c}{\lambda}T_{coh}\sigma_{A}(\tau), \tag{24}\]  
\[\theta_{e}=360\frac{{a_{n}}^{n}R^{(n)}}{\lambda{B_{L}}^{n}}, \tag{25}\]  
where \(\sigma_{IPLL}\) is the standard deviation of phase error caused by the thermal noise, \(\sigma_{v}\) is the standard deviation of phase error caused by the jitter of the reference oscillator frequency, \(\sigma_{A}\) is the standard deviation of phase error caused by the frequency drift of the Allan crystal oscillator, and \(\theta_{e}\) is the dynamic stress error caused by the relative movement.  
Further, \(B_{L}\) is the equivalent bandwidth of the loop, \(C/N_{0}\) is the carrier-to-noise ratio, \(T_{coh}\) is the coherent integration time, \(c\) is the speed of light, \(\lambda\) is the carrier wavelength, \(\sigma_{A}(\tau)\) is the Allan standard deviation of the frequency, and \(a_{n}\) is filter parameters of the loop; \(n\) is the loop order, \(a_{1}=0.25,a_{2}=0.53,a_{3}=0.7845\)[8].  
It needs to be emphasized that \(R^{(n)}=\frac{d^{n}R}{d^{n}}\), where \(R^{(n)}\) is the \(n\)th-order derivative of the distance between the vehicle and the satellite.  
Because the second-order locked loop is sensitive to the accelerated movement and the third-order loop is sensitive to the jerk movement, the velocity error should be transformed to the equivalent relative movement. Assuming that the INS-aided velocity error in one loop-aided cycle is \(\delta v^{e}\), the \((n-1)\)th-order derivative is as follows:  
\[(\delta V^{e})^{(n-1)}=R^{(n)}, \tag{26}\]  
where \(R^{(n)}\) represents the \(n\)th-order derivative of the position error vectors.  
In order to keep the tracking loop stable, 3 times the variance of the phase measurement error should be less than 45\({}^{\circ}\)[14]. According to Equations (22), (25) and (26), the following inequation could be inferred:  
\[\sqrt{{\sigma_{IPLL}}^{2}+{\sigma_{v}}^{2}+{\sigma_{A}}^{2}}+120\frac{{a_{n}} ^{n}{(\delta V^{e})}^{(n-1)}}{\lambda{B_{L}}^{n}}\leq 15. \tag{27}\]  
Furthermore, according to Equations (19) and (21), the \((n-1)\)th-order derivative of the INS-aided velocity error is  
\[(\delta V^{e})^{(n-1)}=\left\{\sqrt{3}\bigg{[}\Delta T+\sqrt{2}\omega_{e} \Delta T^{2}+\frac{\sqrt{6}\omega\Delta T^{3}}{2^{2}}+\frac{\mu^{2}\Delta T^{ 3}}{\mu^{4}}\bigg{]}\|\delta A^{b}\|_{2}+\sqrt{6a^{2}}(\Delta T^{2}+\frac{ \sqrt{2}\omega_{e}\Delta T^{3}}{2})\|\delta\omega^{b}\|_{2}\right\}^{(n-1)} \tag{28}\]where \(\sigma_{v}\) is about \(2^{\circ}\), \(n\) is the loop order, \(a\) is the sum acceleration of the vehicle, and \(\Delta T\) is the time interval between two INS-aided moments.  
So far, all the key parameters of the receiver are involved in Equations (27) and (28), and the whole deeply coupled system can be configured to satisfy the engineering requirement in consideration of the vehicle maneuvering. If the IMU precision related to \(\delta A^{b}/\delta\omega^{b}\) and the dynamic requirement of the deeply coupled system related to \(a\) are defined, it can be directly determined whether the system can guarantee the stability of the tracking loop or not. The entire parameter self-calibration method is proposed above; the correctness and advantages will be proved next.