## 1 Introduction  
The tracking loop design of the traditional global navigation satellite system (GNSS) receiver is suffering a dilemma. In order to reduce the thermal noise to improve the signal receiving accuracy in the GNSS receiver, the bandwidth of its tracking loops should be narrow enough. However, the narrow bandwidth would increase dynamic stress error [1]. Therefore, receiver measurement errors during dynamic periods are systematically larger than those during static periods [2]. The inertial navigation system (INS) has a superior dynamic characteristic, which is highly complementary to GNSS. GNSS/INS integration can be briefly classified as loosely-coupled, tightly-coupled and deeply-coupled, listed in order of complexity. While GNSS navigation results are fused with INS information in the loosely-coupled integration, GNSS observation results are fused with INS information in the tightly-coupled integration. The GNSS receiver's baseband structure need not be adjusted in loosely-coupled integration and tightly-coupled integration, but satellite signal receiving performance cannot be improved. The deeply-coupled integration is the fusion of GNSS and INS information in the signal processing level, which could further take advantage of the INS dynamic characteristic to improve satellite signal acquisition and tracking performances under dynamics [3]. Based on the type of tracking loops used in receivers, deeply-coupled integration can be implemented in two different ways, which are shown in Figure 1, respectively named: (1) scalar-based architecture; and (2) vector-based architecture [4]. The term \"scalar-based architecture\" refers to aiding the individual tracking loops by inertial measurements, proposed by Stanford University [5]. By contrast, the term \"vector-based architecture\" is considered as a vector-based receiver integrated with an inertial measurement unit (IMU), in which the traditional code and carrier tracking loops are eliminated, proposed by MIT [6]. Although the vector-based architecture can make fuller use of the available information and get better signal sensitivity, the architecture is more complex and challenging for realizing a real-time system. Meanwhile, since the deeply-coupled Kalman filter output accuracy is insufficient for carrier phase tracking, individual tracking loops still need be used in the vector-based architecture [7]. As a simpler approach, the scalar-based architecture, whose tracking loops are individual, is valuable for carrier phase tracking. Figure 1: Two different architectures of GNSS/INS deeply-coupled integration. (**a**) Scalar-based architecture; (**b**) vector-based architecture. NCO means numerically controlled oscillator. Most previous research on the tracking loops of deeply-coupled integration was mainly based on relatively simple modeling and simulation. There was no literature available on the complete modeling for the INS branch in the INS-aided tracking loops, so there was no mature model supporting the design and evaluation of INS-aided tracking loops, such as the design of the tracking loop's bandwidth, selection of inertial sensors, implementation optimization of aiding information delay and quantitative analysis of tracking performance [6, 7, 8, 9, 10, 11, 12, 13, 14, 15]. [PERSON] _et al._ proposed a mathematical structure of the INS-aided delay lock loop (DLL) in 1998, shown in Figure 2a[9]: the INS branch is simply modeled as \(\left(\mathrm{l}\!-\!b\right)/\left(\tau s\!+\!1\right)\), where \(b\) is the error of the IMU scalar factor and \(\tau\) is delay time in the filter. However, the error of the IMU bias and the navigation error were not considered in the model. [PERSON] proposed a mathematical structure of the INS-aided phase lock loop (PLL) in 2003, shown in Figure 2b[10]: the IMU is simply modeled as a low pass filter, which could not correctly reflect the error transformation of IMU; the INS information delay to tracking loops was not considered.