## 2 Single-Axis Rotating Inertial Navigation Systems  
### Definition of an SRINS Coordinate System  
Before introducing the error equations of the SRINS, it is necessary to define the relevant coordinate systems. The coordinate systems used in this paper mainly include the navigation coordinate system (n-frame, the body coordinate system (b-frame), the platform coordinate system (p-frame), and the IMU measurement coordinate system (t-frame). The n-frame is represented by \(o_{n}x_{y}y_{n}z_{n}\), with \(o_{n}x_{n}\) pointing to the geographic east, \(o_{n}y_{n}\) pointing to the geographic north, and \(o_{n}z_{n}\) pointing upward, perpendicular to the local rotating ellipsoid surface. The b-frame is represented by \(o_{b}x_{b}y_{b}z_{b}\), with \(o_{b}x_{b}\) pointing to the right along the transverse axis of the body, \(o_{b}y_{b}\) pointing forward along the longitudinal axis of the body, and \(o_{b}z_{b}\) pointing upward, perpendicular to the body. The attitude matrix of the b-frame relative to the n-frame is represented by \(C_{b}^{n}\).  
Since the IMU of the INS is fixed to the platform of the rotation mechanism, the t-frame can be considered to coincide with the p-frame. Generally, an SRINS uses the \(z\) axis as its rotation axis. Assuming that at the initial moment, the b-frame coincides with the p-frame, when rotating counterclockwise around the \(z\) axis at an angular velocity \(\omega\) (as shown in Figure 1), the attitude matrix between the b-frame and the p-frame can be expressed as:  
\[C_{b}^{p}=\left[\begin{array}{ccc}\cos\omega t&\sin\omega t&0\\ -\sin\omega t&\cos\omega t&0\\ 0&0&1\end{array}\right] \tag{1}\]  
The rotation path of the SRINS used in this paper is shown in Figure 2. The IMU rotates back and forth at an angular velocity of 6\({}^{\circ}\)/s. During one modulation cycle, the rotation mechanism controls the IMU to rotate clockwise for one full turn, and then immediately rotate counterclockwise for one full turn. There is no time interval between the two modulation cycles.  
Figure 1: Rotation Diagram.  
During the rotational modulation, the constant drift of the gyroscope and the zero bias of the accelerometer in the b-frame for an SRINS can be expressed as follows:  
\[\mathbf{\varepsilon}^{b}=\mathbf{C}^{b}_{p}(t)\mathbf{\varepsilon}^{p}=\left[\begin{array}{ ccc}\cos\omega t&-\sin\omega t&0\\ \sin\omega t&\cos\omega t&0\\ 0&0&1\end{array}\right]\left[\begin{array}{c}\varepsilon^{p}_{x}\\ \varepsilon^{p}_{y}\\ \varepsilon^{p}_{z}\end{array}\right]=\left[\begin{array}{c}\varepsilon^{ p}_{x}\cos\omega t-\varepsilon^{p}_{y}\sin\omega t\\ \varepsilon^{p}_{x}\sin\omega t+\varepsilon^{p}_{y}\cos\omega t\\ \varepsilon^{p}_{z}\end{array}\right] \tag{2}\]  
\[\
abla^{b}=\mathbf{C}^{b}_{p}(t)\
abla^{p}=\left[\begin{array}{ccc}\cos\omega t&- \sin\omega t&0\\ \sin\omega t&\cos\omega t&0\\ 0&0&1\end{array}\right]\left[\begin{array}{c}\
abla^{p}_{x}\cos\omega t- \
abla^{p}_{y}\sin\omega t\\ \
abla^{p}_{y}\\ \
abla^{p}_{z}\end{array}\right]=\left[\begin{array}{c}\
abla^{p}_{x}\cos \omega t-\
abla^{p}_{y}\sin\omega t\\ \
abla^{p}_{x}\sin\omega t+\
abla^{p}_{y}\cos\omega t\\ \
abla^{p}_{z}\end{array}\right] \tag{3}\]  
From the equations above, it can be seen that the constant drift of the gyroscope and the zero bias of the accelerometer perpendicular to the axis are modulated into sinusoidal signals. Therefore, using the modulation method shown in Figure 2, the equivalent constant drift of the gyroscope and the equivalent zero bias of the accelerometer in the b-frame within one modulation cycle can be expressed as follows:  
\[\begin{array}{l}\frac{1}{7}\int_{0}^{T}\xi^{b}dt=\frac{1}{7}(\int_{0}^{T/2} \xi^{b}dt+\int_{T/2}^{T}\xi^{b}dt)\\ =\frac{1}{7}\left(\int_{0}^{T/2}\left[\begin{array}{c}\varepsilon^{p}_{x