## 2 Introduction of Tri-Axis RINS
### Error Propagation Equations of RINS  
According to the basic principle of INS, the error propagation equations of INS are shown in Equations (4) and (5). \(\overset{\rightarrow}{\phi}^{n}=\left[\phi_{E},\phi_{N},\phi_{U}\right]^{T}\) is the misalignment angle of the n-frame and \(\overset{\rightarrow}{\phi}^{n}\) is the differential of \(\overset{\rightarrow}{\phi}^{n}\).\(\overset{\rightarrow}{v}^{n}=\left[v_{E},v_{N},v_{U}\right]^{T}\) is the velocity of RINS, \(\Delta\overset{\rightarrow}{v}^{n}=\left[\Delta v_{E},\Delta v_{N},\Delta v_{ U}\right]^{T}\) is the velocity error of RINS, and \(\Delta\overset{\rightarrow}{v}^{n}\) is the differential of the velocity error. \(\overset{\rightarrow}{\omega}_{en}\) is the angular velocity of the n-frame relative to the Earth and \(\Delta\overset{\rightarrow}{\omega}_{en}^{n}\) is the error of \(\overset{\rightarrow}{\omega}_{en}^{n}\cdot\overset{\rightarrow}{g}\) is the acceleration of gravity, and \(\Delta\overset{\rightarrow}{g}\) is the computational error of the acceleration of gravity.  
\[\overset{\rightarrow}{\phi}^{n}=\overset{\rightarrow}{\phi}^{n}\times \left(\overset{\rightarrow}{\omega}_{ie}^{n}+\overset{\rightarrow}{\omega} _{en}^{n}\right)+\left(\Delta\overset{\rightarrow}{\omega}_{ie}^{n}+\Delta \overset{\rightarrow}{\omega}_{en}^{n}\right)-C_{b}^{n}C_{s}^{b}\Delta \overset{\rightarrow}{\omega}_{is}^{s} \tag{4}\]  
\[\Delta\overset{\rightarrow}{v}^{n}=\overset{\rightarrow}{f}_{is}^{n}\times \overset{\rightarrow}{\phi}^{n}+\overset{\rightarrow}{v}^{n}\times\left(2 \Delta\overset{\rightarrow}{\omega}_{ie}^{n}+\Delta\overset{\rightarrow}{ \omega}_{en}^{n}\right)-\left(2\overset{\rightarrow}{\omega}_{ie}^{n}+ \overset{\rightarrow}{\omega}_{en}^{n}\right)\times\Delta\overset{ \rightarrow}{v}^{n}+C_{b}^{n}C_{s}^{b}\Delta\overset{\rightarrow}{f}_{is}^{s }+\Delta\overset{\rightarrow}{g} \tag{5}\]  
It is clear to see that the errors of the IMU are introduced into the error equation through \(C_{b}^{n}C_{s}^{b}\Delta\overset{\rightarrow}{\omega}_{is}^{s}\) and \(C_{b}^{n}C_{s}^{b}\Delta\overset{\rightarrow}{f}_{is}^{s}\). By integration, the errors in angular velocity and acceleration are converted to attitude and velocity errors, and the velocity errors are again integrated into position errors. The expressions of \(\Delta\overset{\rightarrow}{\omega}_{is}^{s}\) and \(\Delta\overset{\rightarrow}{f}_{is}^{s}\) are as follows, which is the theoretical basis for the error analysis below.  
\[\Delta\overset{\rightarrow}{\omega}_{is}^{s}=\begin{bmatrix}\Delta K_{gx}&0& -\delta_{xy}^{G}\\ -\delta_{yz}^{G}&\Delta K_{gy}&\delta_{gx}^{G}\\ \delta_{zy}^{G}&-\delta_{zx}^{G}&\Delta K_{gz}\end{bmatrix}\cdot\overset{ \rightarrow}{\omega}_{is}^{s}+\begin{bmatrix}\Delta e_{x}\\ \Delta e_{y}\\ \Delta e_{z}\end{bmatrix} \tag{6}\]  
\[\Delta\overset{\rightarrow}{f}_{is}^{s}=\begin{bmatrix}\Delta K_{xx}&-\delta _{xz}^{A}&0\\ \delta_{yz}^{A}&\Delta K_{ay}&0\\ -\delta_{zy}^{A}&\delta_{xx}^{A}&\Delta K_{az}\end{bmatrix}\cdot\overset{ \rightarrow}{f}_{is}^{s}+\begin{bmatrix}\Delta a_{x}\\ \Delta a_{y}\\ \Delta a_{z}\end{bmatrix} \tag{7}\]  
From the analysis above, when the configuration of the RINS is determined, the key factor affecting the projection of IMU-related errors under the geographic coordinate system is \(C_{s}^{b}\), which is the main reason why studies related to the rotation schemes of the RINS are so popular.  
Figure 3: Definitions of gyros’ and accelerometers’ installation errors. (**a**) Installation errors of the gyros. (**b**) Installation errors of the accelerometers.