## 2 Introduction of Tri-Axis RINS
### Definitions of Coordinate Systems and Symbols  
For clarity of calculation and expression, we need to define several coordinate systems related to the tri-axis RINS. They are the carrier coordinate frame (noted as b), the navigation coordinate frame representing the local-level geographic coordinate frame (noted as n), the sensor coordinate frame (noted as s), and the gyro installation frame (noted as g). Besides, some symbols also need to be defined; all of these are summarized in Table 2. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Error Components** & \multicolumn{2}{c}{**Error Values**} \\  & **Horizontal Gyro** & **Vertical Gyro** \\ \hline Gyro zero bias stability & 0.002\({}^{\circ}\)/h & 0.0035\({}^{\circ}\)/h \\ (1\(\sigma\), 100 s) & (1\(\sigma\), 100 s) \\ Gyro angle random walk coefficient & 0.0003\({}^{\circ}\)/\(\sqrt{h}\) & 0.0005\({}^{\circ}\)/\(\sqrt{h}\) \\ Gyro scale factor stability & 10 ppm & 20 ppm \\ \hline Accelerometer zero bias stability & 20 ug & \\ Accelerometer noise & 10 ug & \\ Accelerometer scale factor stability & 25 ppm & \\ \hline Sampling frequency & 200 Hz & \\ Angle encoder accuracy & \textless{}3\({}^{\prime\prime}\) & \\ Angle encoder resolution & 0.019\({}^{\prime\prime}\) & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of the sensors. Figure 1: Structure of Tri-axis RINS. \begin{table}
\begin{tabular}{l l l} \hline \hline
**Name** & **Symbol** & **Name** & **Symbol** \\ \hline The geographic coordinate frame & & The carrier coordinate frame (b-frame) & \(O-X_{g}Y_{g}Z_{b}\) \\ (n-frame) & & The gyro installation & \\ The sensor coordinate frame & & frame & \(O-X_{g}Y_{g}Z_{g}\) \\ (s-frame) & & (g-frame) & \\  & & The accelerometer installation & \\ Ture angular velocity and acceleration of n-frame & \(\overset{\text{\scalebox{0.8}{$\rightarrow$}}}{\omega}^{\text{\scalebox{0.8}{$ \rightarrow$}}}{\overset{\text{\scalebox{0.8}{$\rightarrow$}}}{\overset{ \text{\scalebox{0.8}{$\rightarrow$}}}{\overset{\text{\scalebox{0.8}{$ \rightarrow$}}}{\overset{\text{\scalebox{0.8}{$\rightarrow$}}}{\overset{ \text{  
The definition of \"s-frame\" requires the following clarification: Since the gimbal axis inevitably changes during rotation, we choose the x and y accelerometer measuring axes to determine the \(X_{s}OY_{s}\) plane of the s-frame, which \(OX_{s}\) is the projection of the x gyro measurement axis in the \(X_{s}OY_{s}\) plane. \(OY_{s}\) is perpendicular to \(OX_{s}\) and close to the measurement axis direction of the y-accelerometer \(OY_{s}\), \(OX_{s}\) and \(OY_{s}\) fits the right-hand rule. After the output angles of the three encoders return to zero, the s-frame and the b-frame overlap. Figure 2 shows the relationship between the s-frame and the b-frame. When the rotational angles of the three gimbals are zero, the inner, middle, and outer gimbals point in the \(Z_{s}\), \(X_{s}\), \(Y_{s}\) direction respectively. According to the rotation structure of tri-axis RINS, we can define the direction cosine matrix \(C_{b}^{s}\) according to Equation (1). \[ \begin{split}& C_{b}^{s}=R_{z}(\phi_{1})\cdot R_{x}(\phi_{2}) \cdot R_{y}(\phi_{3})\\ &=\left[\begin{array}{ccc}\cos\phi_{1}&\sin\phi_{1}&0\\ -\sin\phi_{1}&\cos\phi_{1}&0\\ 0&0&1\end{array}\right]\left[\begin{array}{ccc}1&0&0\\ 0&\cos\phi_{2}&\sin\phi_{2}\\ 0&-\sin\phi_{2}&\cos\phi_{2}\end{array}\right]\left[\begin{array}{ccc}\cos \phi_{3}&0&-\sin\phi_{3}\\ 0&1&0\\ \sin\phi_{3}&0&\cos\phi_{3}\end{array}\right]\end{split}  \tag{1}\]  
The installation errors of gyros and accelerometers are defined in Figure 3. The g-frame and the a-frame are non-orthogonal coordinate systems consisting of the actual pointing of the gyros and accelerometers, and the transformation matrices from the g-frame and a-frame to the s-frame are shown in Equations (2) and (3). Take \(\delta_{xy}^{G}\) as an example to illustrate how installation errors are defined. The superscript \(G\) represents the gyroscope, and the superscript \(xy\) stands for the non-orthogonal angle between \(OX_{g}\) and \(OZ_{s}\) along \(OY_{s}\).