## 1 Introduction  
The global navigation satellite system (GNSS) can only work in those environments where the number of GNSS satellites in view is no less than four. This leads to inconveniences and difficulties for GNSS applications in high-dynamic or weak signal scenarios where it is not easy to acquire navigation satellites. Hence, integrated GNSS/INS navigation systems have been developed for these GNSS-challenged environments. Inertial navigation systems (INSs) are capable of high update rates, while GNSS has high navigation accuracy. By fusing them together, the strengths and weaknesses of GNSS receivers and INS uniquely complement each other. Generally, the three architectures of integrated navigation systems, listed in order of complexity, are loose integration, tight integration, and ultra-tight integration [1, 2]. In the loose integration and tight integration, the GNSS needs stable and strong signals for navigation applications. However, these signals are difficult to receive in the environments where line of sight (LOS) to satellites is not readily available, e.g., urban areas, indoors and dense forest areas. Such environments either completely block the GNSS signals or attenuate them to a power level which is 10-30 dB lower than nominal signal power [3]. Therefore, the ultra-tight integration was developed for GNSS receivers to realize high-dynamic navigation under weak signal conditions, by aiding the tracking loops of receivers. The acquisition and tracking capabilities of receivers are improved by this ultra-tight integration method [4, 5]. [PERSON] at the Charles Stark Draper Laboratory proposed in 2000 an ultra-tightly integrated navigator with extended range code tracking [6]. This led many researchers to pay attention to the ultra-tight integration since it has better tracking and navigation performance than the standard receiver and the tight integration [7, 8]. Nowadays, some successes have been achieved in the various investigations on ultra-tight integration, such as the demonstration of the anti-interference capacity [9, 10], the design and implementation of the dual-mode GNSS/INS ultra-tight integration [11] and the (Micro Electro Mechanical System) MEMS ultra-tight integration [12]. Nevertheless, most previous research about ultra-tight integration was still mainly on the architecture and filter design of the ultra-tightly integrated GNSS/INS navigation system [13, 14, 15]. The performance comparisons of the standard receiver and the ultra-tightly integrated GNSS/INS receiver were usually achieved by some experimental analyses [16]. No research used models of tracking loop errors to analyze the performance improvements brought by the ultra-tight integration, although the error models can demonstrate the essence of any performance improvements. Compared to the previous research, this paper presents the sources and compositions of tracking loop errors in both the standard receiver and the ultra-tightly integrated GNSS/INS receiver, and establishes the mathematical formulas of every error as well. Based on the tracking loop error analysis and comparisons of the two receivers, the advantages of the ultra-tightly integrated GNSS/INS receiver are starkly evident, especially in the high dynamics scenario. Moreover, the error distributions illustrate the proportions of the major noise sources in the two receivers and the noise reduction brought about by the ultra-tight integration. In the ultra-tightly integrated GNSS/INS receiver, the level of the performance improvement is impacted by the quality of the inertial measurement unit (IMU) used. Some researchers have compared the tracking and navigation performances of ultra-tight integrations with different grade IMUs by simulation experiments [7, 15], but there has been no research to derive the mathematical relationshipbetween the level of the performance improvement and the IMU quality.