## 3 Steady-State Error Modeling and Analysis
### Steady-State Error Modeling of INS-Aided Second-Order PLLs  
Under vehicle dynamic conditions, \(\Delta\!f_{MU}(s)\) are the main error sources in the INS branch. Hence, while \(\Delta\!f_{MU}(s)\) is modeled in detail, which refers to our preliminary study in the literature [22], \(K_{a}(s)\) and \(e^{-n_{b}}\) are simply modeled as random constants in the INS-aided PLL error transfer function (Equation (1)). For the horizontal movement of vehicles, the lower the satellite elevation is, the larger the PLL mapping error from the INS is. Meanwhile, the north error and the east error of the INS are symmetrical in the north-east-down (NED) navigation frame. Here, we use the north direction as an example. If the vehicle is aligned with the NED navigation frame and moves toward the north, the PLL tracking a satellite at the north direction of the vehicle will map the largest error from the INS. The largest error of the aiding information should be considered for the parameter design of the PLLs. Therefore, without loss of generality, assume that the vehicle is aligned with the NED navigation frame and a satellite is at the north direction of the vehicle; \(\Delta\!f_{MU}(s)\) can be expressed by:  
\[\Delta\!f_{MU}(s)=\frac{2\pi}{\lambda}\left\{\frac{1}{s}\cdot \left[\frac{b_{{}_{\alpha x\_c}}}{s}+\frac{GM_{\alpha x}(0)+w_{{}_{G\alpha x} }(s)}{s+\frac{1}{T_{a}}}+w_{\alpha x}(s)\right]\right. \tag{9}\] \[\left.+\frac{g}{s^{2}}\cdot\left[\frac{b_{{}_{\alpha x\_c}}}{s}+ \frac{GM_{{}_{\beta x\_c}}(0)+w_{{}_{G\alpha x}}(s)}{s+\frac{1}{T_{g}}}+w_{{} _{\beta x\_c}}(s)\right]+\frac{1}{s}\cdot\delta\!V_{{}_{N}}(0)+\frac{g}{s^{2}} \cdot\phi_{{}_{E}}(0)\right\}\]  
where \(b_{{}_{\alpha x\_c}}\) is the constant bias of the \(X\)-axis accelerometer (_i.e._, forward direction), \(b_{{}_{\alpha x\_c}}\) is the constant bias of the \(Y\)-axis gyro (_i.e._, pitch gyro), \(GM_{\alpha x}(0)+w_{{}_{G\alpha x}}(s)\) is a first-order Gauss-Markov process representing the bias instability of the \(X\)-axis accelerometer, \(GM_{{}_{\beta x\_c}}(0)+w_{{}_{G\alpha x}}(s)\) is a first-order Gauss-Markov process representing the bias instability of the \(Y\)-axis gyro, \(w_{\alpha x}(s)\) is the white noise of the \(X\)-axis accelerometer, \(w_{{}_{\beta x\_c}}(s)\) is the white noise of the \(Y\)-axis gyro, \(\delta\!V_{{}_{N}}(0)\) is the north initial velocity error after correction of the GNSS/INS integration algorithm and \(\phi_{{}_{E}}(0)\) is the east initial attitude error (_i.e._, pitch error) after coupled navigation correction. Next, this detailed model of the INS bias errors \(\Delta\!f_{{}_{\mathit{h}\mathit{U}}}(s)\) is analyzed for the error modeling of the INS-aided PLL. Since the error sources, including the INS bias-type errors and the INS scale factor-type errors, are corrected when the coupled navigation results are updated, the relationship between each error source and the tracking error in time domain should be analyzed in the update interval of coupled navigation. While the tracking errors caused by the error sources with a random constant feature are analyzed by transforming their transfer functions to the time domain, the tracking errors caused by the error sources with the noise feature are analyzed by Monte Carlo simulation. Only considering the INS bias-type errors in Equation (1), taking Equation (9) into Equation (1), we get the concrete model reflecting the relationship between all of the INS bias-type errors and the phase tracking error. The influence of each INS bias-type error on the tracking error can be quantitatively analyzed from the concrete model. Analysis results in the literature [23] showed that, compared with the tracking error caused by other INS bias-type errors, the tracking error caused by the north initial velocity error after correction of the GNSS/INS integration algorithm \(\delta\!V_{{}_{N}}(0)\) was dominant.