## 2 Single-Axis Rotating Inertial Navigation Systems
### Error Model of the SRINS  
In the p-frame, the actual angular velocity and specific force sensed by the IMU are represented by \(\mathbf{\omega}_{ip}^{p}\) and \(\mathbf{f}_{ip}^{p}\), respectively, and they satisfy the following conditions:  
\[\left\{\begin{array}{l}\mathbf{\omega}_{ip}^{p}=\mathbf{C}_{b}^{p}\mathbf{\omega}_{ib}^{b}+ \mathbf{\omega}_{bp}^{p}\\ \mathbf{f}_{ip}^{p}=\mathbf{C}_{b}^{p}\mathbf{f}_{ib}^{b}+\mathbf{f}_{bp}^{p}\end{array}\right. \tag{6}\]  
In the p-frame, the platform only has an angular velocity in the z-direction relative to the b-frame, and there are no linear accelerations, so:  
\[\mathbf{\omega}_{bp}^{p}=\left[\begin{array}{ccc}0&0&\omega\end{array}\right]^ {T}\]  
\[\mathbf{f}_{bp}^{p}=\mathbf{0}\]  
If the measurement errors are represented by \(\delta\mathbf{\omega}_{ip}^{p}\) and \(\delta\mathbf{f}_{ip}^{p}\), then the angular velocity and specific force measured by the IMU can be expressed as:  
\[\left\{\begin{array}{l}\mathbf{\tilde{\omega}}_{ip}^{p}=\mathbf{\omega}_{ip}^{p}+ \delta\mathbf{\omega}_{ip}^{p}\\ \mathbf{\tilde{f}}_{ip}^{p}=\mathbf{f}_{ip}^{p}+\delta\mathbf{f}_{ip}^{p}\end{array}\right. \tag{7}\]  
If the control error of the indexing mechanism is ignored, then:  
\[\delta\mathbf{\omega}_{ip}^{p}=\mathbf{C}_{b}^{p}\delta\mathbf{\omega}_{ib}^{b}+\delta \mathbf{\omega}_{bp}^{p}=\mathbf{C}_{b}^{p}\delta\mathbf{\omega}_{ib}^{b}\]  
\[\delta\mathbf{f}_{ip}^{p}=\mathbf{C}_{b}^{p}\delta\mathbf{f}_{ib}^{p}+\delta\mathbf{f}_{bp}^{p} =\mathbf{C}_{b}^{p}\delta\mathbf{f}_{ib}^{p}\]  
According to the error equations of the strapdown inertial navigation system (SINS) provided in [24], the error equation of the SRINS can be expressed as:  
\[\begin{array}{l}\dot{\mathbf{\phi}}=-\mathbf{\omega}_{in}^{n}\times\dot{\mathbf{\phi}}+ \delta\mathbf{\omega}_{in}^{n}-\mathbf{C}_{b}^{n}\mathbf{C}_{b}^{p}\delta\mathbf{\omega}_{ip}^ {p}\\ \delta\dot{\mathbf{\sigma}}=\mathbf{f}^{n}\times\dot{\mathbf{\phi}}-(2\delta\mathbf{\omega}_{ ie}^{n}+\mathbf{\omega}_{en}^{n})\times\mathbf{\upsilon}^{n}-(2\mathbf{\omega}_{ie}^{n}+\mathbf{ \omega}_{en}^{n})\times\delta\mathbf{\upsilon}^{n}+\mathbf{C}_{b}^{n}\mathbf{C}_{p}^{b} \delta\mathbf{f}_{ip}^{p}+\delta\mathbf{g}^{n}\\ \delta\dot{L}=\frac{1}{R_{\rm M}+\hbar}\delta\
u_{N}-\frac{\
u_{N}}{(R_{\rm M} +\hbar)^{2}}\delta\hbar\\ \delta\dot{\lambda}=\frac{\sec L}{R_{N}+\hbar}\delta\
u_{E}+\frac{\sec L\tan L}{ R_{N}+\hbar}\delta L-\frac{\
u_{E}\sec L}{(R_{N}+\hbar)^{2}}\delta\hbar\\ \delta\dot{h}=\delta\
u_{U}\end{array} \tag{8}\]  
In the equation above, \(\dot{\mathbf{\phi}}\) represents the misalignment angle error from the n-frame to the computational coordinate system (n\({}^{\prime}\)-frame), so the actual direction cosine matrix is:  
\[\mathbf{C}_{b}^{n}=\mathbf{C}_{n}^{n}\mathbf{C}_{b}^{n^{\prime}}=(\mathbf{I}+\dot{\mathbf{\phi}} \times)\mathbf{C}_{b}^{n^{\prime}}\]  
\(\mathbf{\upsilon}^{n}\) represents the velocity in the n-frame; \(\delta\mathbf{\upsilon}^{n}\) represents the velocity error, and \(\delta\mathbf{\upsilon}^{n}=\left[\begin{array}{cc}\delta\
u_{E}&\delta\
u_{N} &\delta\
u_{U}\end{array}\right]\) ; \(L\), \(\lambda\), \(h\) represent the latitude, longitude, and altitude in the n-frame, respectively; \(\delta L\), \(\delta\lambda\), \(\delta h\) represent the latitude, longitude, and altitude errors, respectively; \(R_{\rm M}\) and \(R_{N}\) represent the meridian radius of curvature and the prime vertical radius of curvature; and \(\mathbf{\omega}_{ie}^{n}\) and \(\mathbf{g}^{n}\) represent the Earth's rotation angular velocity and the Earth's gravitational acceleration.