In this paper, the quaternion of ${b}$ -frame based nonlinear error model is proposed for the INS. According to the definition of the attitude error, the quaternion of ${b}$ -frame based nonlinear error model has been derived by additive and multiplicative relationship. Then, a discussion and analysis is made to compare the proposed model in additive and multiplicative relationship with the quaternion of ${n}$ -frame based nonlinear error model. In addition, an analysis and comparsion of ${b}$ -frame and ${n}$ -frame based nonlinear error model is made to have a theoretical justification to describe why ${b}$ -frame is superior to ${n}$ -frame based nonlinear error model. Finally, field tests are carried out to access the performance of the quaternion of ${b}$ -frame and ${n}$ -frame based nonlinear error model in the additive and multiplicative relationship for the INS.

In this paper, an error model based on Special Euclidean group SE (3) of inertial navigation system (INS) is proposed using the quaternion. The attitude and velocity are formulated together as element of the SE (3). The quaternion based error model has a better performance than the Euler Angles based error model, especially for the large initial attitude error. The conclusion can be also applied to the error model based on Lie group due to the superiority of the quaternion. Further, two field tests for the INS are carried out to investigate the benefits and drawbacks of the quaternion and Euler Angles based error model based on SE (3) or Special orthogonal group SO (3) for the INS.

In this paper, a new Euler angle model without approximation based on SE(3) is proposed to solve the problem that the low common frame performance of conventional Euler angle error model based on SE(3) for inertial navigation system (INS) when the misalignment angle error is large. Firstly, the Euler angle error is represented by the relationship between Euler angle and the attitude matrix. Then, the left/right error model based on SE(3) without approximation is rederived, extending the conventional Euler angle error model based on SE(3) to the application of large misalignment angle error. Finally, the validity of the proposed model is verified by theoretical analysis and field experiments. The results show that the proposed model is more applicable to INS/GPS and INS/DVL, respectively. Especially in the case of large misalignment angle, the convergence and accuracy of the proposed model are higher than the conventional model.

In response to the lack of specific demonstration and analysis of the research on the necessity of the Lie group strapdown inertial integrated navigation error model based on the Euler angle, two common integrated navigation systems, strapdown inertial navigation system/global navigation satellite system (SINS/GNSS) and strapdown inertial navigation system/doppler velocity log (SINS/DVL), are used as subjects, and the piecewise constant system (PWCS) matrix, based on the Lie group error model, is established. From three aspects of variance estimation, the observability and performance of the system with large misalignment angles for low, medium, and high accuracy levels, traditional error model, Lie group left error model, and right error model are compared. The necessity of research on Lie group error model is analyzed quantitatively and qualitatively. The experimental results show that Lie group error model has better stability of variance estimation, estimation accuracy, and observability than traditional error model, as well as higher practical value.

Vertical information (vertical velocity and altitude) is important in three-dimensional navigation. In order to improve the vertical accuracy of strapdown inertial navigation system/global navigation satellite system integrated navigation (SINS/GNSS) while reducing costs, firstly, the propagation laws of vertical error sources in two types of SINS/GNSS (vertical velocity set to 0 and not set to 0) are systematically analyzed. Furthermore, a vertical accuracy improvement method considering the gravitational anomaly is proposed. In this method, the gravitational anomaly is considered as one of the vertical error sources. Then, the processing method of error sources in integrated navigation is referenced, and two processing modes of gravitational anomaly are designed. The first method is to represent the gravitational anomaly as one of the system states of SINS/GNSS. The number of dimensions for the system is expanded from 15 to 16. The corresponding mathematical model is derived to ‘absorb’ the vertical errors caused by the gravitational anomaly. The second is to represent the gravitational anomaly as one of the vertical system noises. Thereby, the Kalman filter is adjusted in real time using the above adaptive method to improve the accuracy of the estimated state. The corresponding errors are then suppressed. Field experiments show that both modes of the proposed method can effectively improve the vertical accuracy.

Gravity matching is a key technology in gravity-aided inertial navigation. The traditional Sandia inertial matching algorithm introduces linearization errors using the linear error model, which can diminish navigation accuracy. To address this issue, we propose a novel gravity-matching algorithm based on modified adaptive transformed cubature quaternion estimation (MA-TCQUE), designed for a nonlinear error model to enhance accuracy in gravity-aided navigation. Additionally, the proposed algorithm can estimate the measurement noise matrix demonstrating improved filtering stability in complex and dynamic environments. Finally, simulation and experimental results validate the advantages of the proposed matching algorithm compared to existing state-of-the-art methods.