# A General Euler Angle Error Model of Strapdown Inertial Navigation Systems
## 5 Conclusions  
The existing misalignment angle error model cannot directly describe the attitude error propagation, especially in large-angle attitude movements. The misalignment angle is a small angle, which is characterized by the calculation of the three-axis deviation angle between the computational frame and the navigation frame. Compared with the proposed general Euler angle error model, it does not reflect and optimize the body's attitude movement directly. Euler angles are the most intuitive measurements of attitude movements. The Euler angle errors of SINS, different from the misalignment of platform inertial navigation systems, comprise two parts: relative Euler angle errors and convected Euler angle errors. The proposed convected Euler angle error model and the general Euler angle error model, as well as their conclusions, have been validated by the simulation experiments. The general Euler angle error model can accurately describe attitude error propagation rules, which contribute to the design and operation of SINS. The attitude can be measured in real-time, then attitude movements can be optimized in large-angle attitude movements.  
**Acknowledgments:** The work described in the paper was supported by National Natural Science Foundation of China under grant 61571030, grant 61421063, and grant 61722103; the National High Technology Research and Development Program of China (863 program 2015 AA124001 and 2015 AA124002); the International (Regional) Cooperation and Communication Project under grant 61661136007; in part by Basic Scientific Research under grant YWF-17-BJ-Y-71; and in part by the Open Foundation under grant EX166840042.  
**Author Contributions:** [PERSON] and [PERSON] devised the algorithm and wrote the manuscript; and [PERSON] and [PERSON] contributed to the design of the experiments and the discussion of the method. All the authors contributed in the English corrections and approved the final manuscript.  
**Conflicts of Interest:** The authors declare no conflicts of interest.  
**Nomenclature**  
_Notation_  
\(H\)Height, m  
\(L\)Latitude, degree  
\(\omega\)Angular velocity vector, rad/s  
\(\Phi\)Misalignment, degree  
\(\delta\omega_{intx}^{b}\)The angular velocity error of the navigation frame with respect to the inertial Frame denoted in the body frame along \(X\) axis, rad/s  
\(\delta\omega_{inty}^{b}\)The angular velocity error of the navigation frame with respect to the inertial Frame denoted in the body frame along \(Y\) axis, rad/s  
\(\delta\omega_{intz}^{b}\)The angular velocity error of the navigation frame with respect to the inertial Frame denoted in the body frame along \(Z\) axis, rad/s  
\(\left[\begin{array}{cc}\delta\theta^{\otimes}&\delta\gamma^{\otimes}& \delta\varphi^{\otimes}\end{array}\right]^{T}\)The relative Euler angle error, rad  
\(\left[\begin{array}{cc}\delta\theta^{*}&\delta\hat{\gamma}^{*}&\delta \hat{\varphi}^{*}\end{array}\right]^{T}\)The convected Euler angle error, rad  
_Subscripts_  
\(i\)The inertial frame  
\(b\)The body frame  
\(n\)The navigation frame  
\(e\)The Earth frame  
\(c\)The computational frame