## 4 Tracking Loops Error in the Ultra-Tightly Integrated GNSS/INS Receiver  
In the ultra-tightly integrated GNSS/INS receiver, the dynamic stress noise is reduced by using an additional IMU to measure and compensate the dynamics experienced by the receiver antenna. After the dynamics are compensated, the dynamic stress noise for second-order PLL is mainly affected by the acceleration measurement error caused by the IMU errors, not by the acceleration of the motions that the receiver antenna experiences in LOS. Therefore, the dynamic stress noise (1-sigma) for second-order PLL in the ultra-tightly integrated GNSS/INS receiver is expressed according to [27]:  
\[\theta_{UT}=\frac{1}{3}\cdot K_{2}\,\frac{\delta\overline{a}}{\left(B_{PLL} \right)^{2}}\cdot\frac{360\cdot L_{c}}{c} \tag{11}\]  
In the equation, \(\delta\overline{a}\) is the measurement error vector of the acceleration in LOS. Meanwhile, the acceleration error vector caused by the IMU can be modeled by:  
\[\delta\overline{f}_{IMU}=\overline{\
abla}+\overline{g}\cdot\overline{\alpha }\left(0\right)+\overline{g}\cdot\overline{\varepsilon}\cdot t_{d} \tag{12}\]  
where \(\overline{\
abla}\) is the accelerometer bias. \(\overline{\varepsilon}\) is the gyro bias. \(\overline{g}\) is the acceleration vector of gravity. \(\overline{\alpha}\left(0\right)\) is the initial misalignment error. \(t_{d}\) is the drift time if the IMU is unaided. Since the acceleration of the pseudorange is measured by the IMU in LOS, the \(\delta a\) in Equation (11) is substituted by the \(\delta f_{IMU}\) in Equation (12). Therefore, the dynamic stress noise of the PLL in the ultra-tightly integrated GNSS/INS receiver is dependent on the IMU errors and can be expressed for second-order loop:  
\[\theta_{UT}=\frac{1}{3}\cdot K_{2}\frac{\delta\overline{f}_{{}_{\mathit{ IMU}}}}{\left(B_{PLL}\right)^{2}}\cdot\frac{360\cdot L_{c}}{c} \tag{13}\]  
Similar to Equation (10), the tracking error of the PLL in the ultra-tightly integrated GNSS/INS receiver can be expressed:  
\[\sigma_{PLL}=\sigma_{T}+\theta_{UT}+\sigma_{A}+\sigma_{\phi}\leq 15^{\circ} \tag{14}\]  
Tracking Performance Comparisons between Standard Receivers and Ultra-Tightly Integrated GNSS/INS Receivers  
In the ultra-tightly integrated GNSS/INS receiver, the tracking performances can be improved as the dynamic stress noise is reduced. According to the models analyzed in this paper, the PLL performances in the standard receiver and the ultra-tightly integrated GNSS/INS receiver are compared to verify the improvement induced by the ultra-tight integration. In the comparisons, an IMU is selected to estimate the acceleration of the pseudorange. The gyro bias and accelerometer bias of the IMU are respectively 50 \(\mathrm{\char 37}\)h and 1 mg. The integration time length of the correlators is 1 ms. Moreover, a temperature compensated crystal oscillator (TCXO) is selected and the oscillator's g-sensitivity is 5 parts/g. Figures 3-6 depict the tracking performances in the standard receiver for second-order PLL. The tracking threshold which keeps the PLL locked is 15 \(\mathrm{\SIUnitSymbolMicro}\) in the simulations. The minimum loop error can be obtained when the loop bandwidth is optimal. Figure 3: 2 nd loop phase error _vs_. BPLL with different C/N\({}_{0}\) in the standard receiver (0.5 g LOS acceleration). Figures 3 and 4 together illustrate the influence of C/N0 on the phase error of the PLL. Figure 3 shows the change of the overall phase error in different C/N0 scenarios, and then Figure 4 further analyzes which error source causes this change. As shown in Figure 3, when the LOS acceleration is assumed to be 0.5 g, the C/N0 of the GNSS signal for the unaided second-order PLL in the standard receiver should be no less than 30 dB-Hz to keep the loop locked. Hence, the GNSS signal whose C/N0 is less than 30 dB-Hz is defined as a challenge to the standard receiver in this  
Figure 4: 2 nd loop error distributions with different C/N\({}_{0}\) in the standard receiver (0.5 g LOS acceleration, 20 Hz PLL bandwidth).