# Sample-Wise Aiding in GPS/INS Ultra-Tight Integration for High-Dynamic, High-Precision Tracking
## 2 Performance of Traditional Doppler Aiding  
Both the scalar tracking-based ultra-tight integration and the vector tracking-based deep integration use Doppler-aided tracking loops, which allows the INS or the integrated navigation filter to account for the platform dynamics such that the receiver can narrow its loop bandwidths. To simplify the analysis of the effects of the aiding data update rate, this section will focus on the dynamic process of the scalar tracking based PLL. According to the fundamental theory of PLL [27], the linear phase model of a PLL can be extended to describe the dynamic process of a PLL aided by external Doppler data, as shown in Figure 1.  
In Figure 1, \(\theta_{e}\left(t\right)=\theta_{i}\left(t\right)-\theta_{o}\left(t\right)\) is the phase error between the input phase \(\theta_{i}\left(t\right)\) and the PLL output phase \(\theta_{o}\left(t\right)\); \(f_{a}\left(t\right)\) is the aiding Doppler provided by the INS estimate; \(F\left(p\right)\) denotes the transfer operator of the loop filter in the time-domain analysis, whose counterpart is \(F\left(j\omega\right)\) in the frequency domain, and \(F\left(s\right)\) in the Laplace domain. For an unaided PLL, the closed-loop phase transfer function is \(H\left(s\right)=\frac{F\left(s\right)}{s+F\left(s\right)}\), and the phase error transfer function is \(H_{\varepsilon}\left(s\right)=1-H\left(s\right)=\frac{s}{s+F\left(s\right)}\). The phase error \(\theta_{\varepsilon}\left(s\right)\) is determined by the input signal phase \(\theta_{i}\left(s\right)\) and the phase noise \(n\left(s\right)\) as follows:  
\[\theta_{\varepsilon}\left(s\right)=H_{\varepsilon}\left(s\right)\theta_{i} \left(s\right)-H\left(s\right)n\left(s\right) \tag{1}\]  
Generally the term \"high dynamics\" is embodied in two aspects: high frequency and large amplitude of the range/velocity/acceleration/jerk variations. For the case study of the behavior of the loop under high dynamics, a typical sinusoidal trajectory in the LOS (line of sight) direction is selected in this paper due to its following characteristics: (1) any complicated LOS dynamics can be decomposed into a number of sinusoidal components with different amplitudes and frequencies; and (2) it can provide severe high-order dynamics with continuous derivatives and limited range of motion, which is easy to control and convenient for testing. Therefore the LOS Doppler is simply modeled as:  
\[f_{d}\left(t\right)=A\sin\left(\omega t\right) \tag{2}\]  
The carrier phase of the incoming signal is:  
\[\theta_{i}\left(t\right)=-\frac{A}{\omega}\text{cos}\left(\omega t\right)+ \theta_{0} \tag{3}\]  
Since second-order and higher-order PLLs are not sensitive to frequency/phase steps, the steady state carrier phase tracking error is only caused by the first item in Equation (3):  
\[\theta_{\varepsilon}(t)=-\frac{A}{\omega}\left|H_{\varepsilon}\left(\omega \right)\right|\text{cos}\left[\omega t+\Phi_{\varepsilon}\left(\omega\right)\right] \tag{4}\]  
It is seen that the sinusoidal dynamic stress error cannot be completely eliminated, and the amplitude of the error transfer function \(\left|H_{\varepsilon}\left(\omega\right)\right|\) represents the ratio of the amplitude of the phase error to that of the incoming phase. For a Doppler aided PLL, it can be easily derived from Figure 1 such that:  
\[\theta_{\varepsilon}\left(s\right)=H_{\varepsilon}\left(s\right)\theta_{ia} \left(s\right)-H\left(s\right)n\left(s\right) \tag{5}\]  
\[\theta_{ia}\left(s\right)=\theta_{i}\left(s\right)-\theta_{a}\left(s\right) \tag{6}\]  
\[\theta_{a}\left(s\right)=\frac{1}{s}f_{a}\left(s\right) \tag{7}\]  
where \(\theta_{ia}\left(s\right)\) is defined as the equivalent input phase of the aided PLL, and \(\theta_{a}\left(s\right)\) is the feedback phase introduced by Doppler aiding.