# An Adaptive INS-Aided PLL Tracking Method for GNSS Receivers in Harsh Environments
## 3 Tracking Error Analysis of Delta Doppler Aided PLL
### Dynamic Stress Error
When the LOS dynamic is the first-order or second-order change of phase, as \(\ddot{\theta}\left(k\right)\) is a constant value, the aiding delay will not affect the dynamic stress error, so the dynamic stress of the second-order loop caused by the acceleration can be removed. When there is a jerk in the LOS dynamic, we have:  
\[\theta\left(k\right)=\frac{1}{6}pk^{3}+\frac{1}{2}ak^{2}+vk+\Theta,k\geq 0 \tag{13}\]  
where \(\Theta\) is the initial phase, \(v\) is the phase change rate, \(a\) and \(p\) are the acceleration and jerk of the phase change, respectively. With the existence of the jerk \(p\), the aiding Doppler rate will have the error caused by the aiding delay, so we have:  
\[df_{dopp}\left(k-1\right)=\ddot{\theta}\left(k-1\right)-p\Delta t \tag{14}\]  
In addition, because of the calculation error of satellite acceleration \(\Delta a_{kSat}\) and the LOS acceleration error \(\Delta a_{kINS}\) caused by the error of INS, the aiding Doppler rate can be expressed as:  
\[ \begin{array}{c}df_{dopp}\left(k-1\right)=\ddot{\theta}\left(k-1\right)- \left(p\Delta t+\Delta a_{kSat}+\Delta a_{kINS}\right)\\ =\left\{\theta\left(k\right)-2\theta\left(k-1\right)+\theta\left(k-2\right) \right\}/T^{2}-\left(p\Delta t+\Delta a_{kSat}+\Delta a_{kINS}\right)\end{array}  \tag{15}\]  
Then the aiding information \(A_{dopp}(k-1)\) can be obtained as:  
\[A_{dopp}\left(k-1\right)=pT^{2}(k-1)+aT-\left(p\Delta t+\Delta a_{kSat}+ \Delta a_{kINS}\right)T \tag{16}\]  
Substitute the Equations (16) and (13) into Equation (12), and according to Equation (3), the dynamic stress error of delta Doppler aided PLL can be derived as:  
\[E[\Psi_{ss}]=\frac{\Delta a_{kSat}+\Delta a_{kINS}+p\Delta t}{\omega_{n}^{2}} (1+\zeta\omega_{n}T+\frac{1}{4}(\omega_{n}T)^{2}) \tag{17}\]  
\(\Delta a_{kSat}\) is usually small enough to be ignored [14], and \(\Delta a_{kINS}\) can be expressed as [19]:  
\[\Delta a_{kINS}=\delta f_{ib}^{b}\cos\left(\omega_{s}t_{c}\right)+\delta\omega _{ib}^{b}\delta\frac{\sin\left(\omega_{s}t_{c}\right)}{\omega_{s}} \tag{18}\]  
where \(\delta\omega_{ib}^{b}\) and \(\delta f_{ib}^{b}\) are the total angular rate and force error in the body frame, \(g\) is the acceleration due to gravity, \(\omega_{s}\) is the [PERSON] frequency and \(t_{c}\) is time with no INS error corrections. The aiding delay \(\Delta t\) can be analyzed as follows: the aiding delay error of Doppler with a jerk dynamic is illustrated in Figure 3, which gives a way to determine the value of \(\Delta t\) as well. When the integration time is equal to the aiding information update time (\(T=T_{f}\)), we usually use the acceleration at the time \(t_{0}\) to aid the loop over the interval from \(t_{0}\) to \(t_{0}+T\). Then we will obtain an average acceleration over the integration interval as (note that herein the update of NCO includes the update of the loop and the update of the aiding information):  
\[a_{M}=a_{0}+\frac{T}{2}p \tag{19}\]  
where \(a_{0}\) is the acceleration at the time \(t_{0}\). With the aiding acceleration \(a_{f}=a_{0}\), the equivalent aiding delay can be derived as:  
\[\Delta t=\frac{a_{M}-a_{f}}{p}=\frac{T}{2} \tag{20}\]  
The equation above is obtained under the condition of \(T=T_{f}\), let's assume that \(\Delta t\) is a weighted result of integration time and aiding information update timewith weights \(k_{1}\) and \(k_{2}\), and then let:  
\[\Delta t=k_{1}\cdot T_{f}+k_{2}\cdot T \tag{21}\]  
Decreasing the integration time (loop update time) or the aiding information update time can improve the dynamic performance of INS-aided PLL. According to the selection of the loop update time and aiding information update time, the weights \(k_{1}\) and \(k_{2}\) should satisfy following conditions:  
\[\left\{ \begin{array}{cc}k_{1}+k_{2}=\frac{1}{2}&T_{f}=T\\ k_{1}\cdot T_{f}+k_{2}\cdot T<\frac{T}{2}&T_{f}<T\\ k_{1}\cdot T_{f}+k_{2}\cdot T<\frac{T_{f}}{2}&T<T_{f}\end{array} \right. \tag{22}\]  
From Equation (22), we can find symmetry between integration time and update time of aiding information.