## 1 Introduction  
Navigation is becoming more and more significant in fields such as aviation, aerospace, marine applications and traffic guidance [1, 2]. In an urban region, the popularity of the global navigation satellite system (GNSS) provides precise location results and becomes an indispensable part of citizen's daily life [3, 4]. However, in GNSS-dentunnels, underwater or signal-shielding environments, the absence of GNSS signals sets up an obstacle to obtain navigation information and brings inconvenience to people's life [5; 6]. Fortunately, the inertial navigation system (INS) is an ideal alternative solution to resolve position, velocity and attitude regarding angular and acceleration signals acquired using the inertial measurement unit (IMU) in a GNSS-denied situation [7; 8]. Generally, a typical IMU composes of three gyroscopes and accelerometers, consistently measuring angular and acceleration information of the carrier in inertial coordinate [9; 10]. With known angular and acceleration information, the carrier's motion state can be estimated through a navigation algorithm as long as its previous motion state is given. However, due to the measuring error of inertial sensors and the imperfection of the navigation algorithm, navigation errors are consistently generated in the navigation process. To make matters worse, there is no external information to correct this mistake; navigation inaccuracy accumulates with time and degrades its reliability in a long-term navigation situation [11; 12]. In order to solve this problem, we propose a calibration and compensation method to eliminate the influence of gyro bias and improve the performance of navigation accuracy. Theoretically, the accuracy of INS is determined by two factors. One is the measuring accuracy of inertial sensors. Since the attitude, position and velocity is estimated based on the angular and acceleration signals obtained from inertial sensors. The measuring errors of individual sensors would result in an inaccuracy in velocity and attitude estimation, thereby introducing errors in the navigation algorithm and degrading the calculation performance of the position algorithm [13; 14]. Another factor is the reliability and robustness of the navigation algorithm [15; 16]. The purpose of the navigation algorithm is to estimate the initial carrier's state and update the motion state of the carrier [17]. In the navigation-resolving process, a carrier's motion state is determined by its previous motion state, with a measured angular and linear acceleration increment during one calculation period. Thus, the accuracy of initial alignment determines the navigation accuracy. In addition, the motion state updating process would also integrate movement information from several individual coordinates, and the limitation of measuring and calculation speed would also introduce non-commutative error [18]. The navigation algorithm's resolving process should also reduce the impact of measurement or calculation errors to improve its navigation accuracy. Regarding improving the navigation accuracy, current studies largely concentrate on optimizing the navigation algorithm. For instance, in the initial alignment process, Kalman filtering (KF) is a widely used method to improve attitude estimation accuracy after the raw alignment method [19; 20; 21; 22; 23; 24]. In order to improve the reliability and speed of the initial measurement, the optimization-delayed marginalization (RODM) alignment method is proposed [25]. In the navigation-updating process, the asynchronous time misalignments within different sensors were also analyzed and compensated through a 40-D KF method [26]. Although the improvements in the algorithm ensure a better navigation performance, the sensor's measurement error will also degrade the performance of the navigation algorithm and is an obstacle in terms of improving the performance of the navigation algorithm [27; 28].