## 2 Single-Axis Rotating Inertial Navigation Systems
### State Equations of the SRINS  
It is known that the single-axis rotation modulation technique can modulate the constant drift of the gyroscope and the accelerometer bias to zero in the roll and pitch axes in the b-frame. However, as indicated in references [22; 23], the gyroscope is affected by the gravitational field and the geomagnetic field, resulting in an equivalent bias in the geographic frame (g-frame, which is consistent with n-frame). This unmodulatable equivalent bias in the g-frame can cause significant errors in navigation results. Therefore, in this paper,zero biases \(\varepsilon_{x}^{b}\), \(\varepsilon_{y}^{b}\) modulated by rotating machine are replaced by the equivalent biases \(\varepsilon_{x}^{n}\), \(\varepsilon_{y}^{n}\) in the geographic frame. Due to the accelerometer's sensitivity to the gravitational field and the geomagnetic field, its zero bias is still represented by \(\
abla_{x}^{b}\) and \(\
abla_{y}^{b}\).  
The state equation for the SRINS can be written as:  
\[\dot{\mathbf{X}}_{SRINS}=\mathbf{F}_{SRINS}\mathbf{X}_{SRINS}+\mathbf{G}\mathbf{u} \tag{9}\]  
where \(\mathbf{F}_{SRINS}\) is the state transition matrix of the SRINS, \(\mathbf{u}\) is the state noise matrix, and \(\mathbf{G}\) is the state noise transition matrix.  
The elements in the state vector are:  
\[\mathbf{X}_{SRINS}=\left[\begin{array}{cccc}\phi&\delta\
u^{n}&\delta\mathbf{p}^{n} &\left[\begin{array}{cccc}\varepsilon_{x}^{n}&\varepsilon_{y}^{n}&\varepsilon _{z}^{b}\end{array}\right]&\
abla^{b}\end{array}\right]^{T} \tag{10}\]  
For the SRINS, \(\varepsilon_{x}^{b}\) and \(\varepsilon_{y}^{b}\) can be modulated. \(\
abla^{b}\) represents the accelerometer bias. Therefore, \(\mathbf{F}_{SRINS}\) can be expressed as:  
\[\mathbf{F}_{SRINS}=\left[\begin{array}{cccc}\mathbf{F}_{11}&\mathbf{F}_{12}&\mathbf{F}_{13 }&\mathbf{F}_{14}&\mathbf{O}_{3\times 3}\\ \mathbf{F}_{21}&\mathbf{F}_{22}&\mathbf{F}_{23}&\mathbf{O}_{3\times 3}&\mathbf{F}_{25}\\ \mathbf{O}_{3\times 3}&\mathbf{F}_{32}&\mathbf{F}_{33}&\mathbf{O}_{3\times 3}&\mathbf{O}_{3\times 3} \end{array}\right] \tag{11}\]  
where:  
\[\mathbf{F}_{11}=-(\mathbf{\omega}_{ln}^{n}\times)\]  
\[\mathbf{F}_{12}=\left[\begin{array}{cccc}0&-\frac{1}{R_{M}+\hbar}&0\\ \frac{1}{R_{N}+\hbar}&0&0\\ \frac{\hbar\omega_{ln}}{R_{N}+\hbar}&0&0\end{array}\right]\]  
\[\mathbf{F}_{13}=\left[\begin{array}{cccc}0&0&\frac{\upsilon_{N}}{(R_{M}+\hbar)^ {2}}\\ -\omega_{ie}\sin L&0&-\frac{\upsilon_{E}}{(R_{N}+\hbar)^{2}}\\ \omega_{ie}\cos L+\frac{\upsilon_{E}\sec^{2}L}{R_{N}+\hbar}&0&-\frac{ \upsilon_{E}\tan L}{(R_{N}+\hbar)^{2}}\end{array}\right]\]  
\[\mathbf{F}_{14}\approx\left[\begin{array}{cccc}-1&0&0\\ 0&-1&0\\ 0&0&-\mathbf{C}_{b}^{n}(3,3)\end{array}\right]\]  
\[\mathbf{F}_{21}=(\mathbf{f}^{n}\times)\]  
\[\mathbf{F}_{22}=(\mathbf{v}^{n}\times)\left[\begin{array}{cccc}0&-\frac{1}{R_{M}+ \hbar}&0\\ \frac{1}{R_{N}+\hbar}&0&0\\ \frac{\tan L}{R_{N}+\hbar}&0&0\end{array}\right]-[(2\mathbf{\omega}_{ie}^{n}+\mathbf{ \omega}_{en}^{n})\times]\]  
\[\left\{\begin{array}{cccc}\beta=0.00530240\\ \beta_{1}=0.00000582\\ \beta_{2}=3.08\times 10^{-6}\text{s}^{-2}\\ \beta_{3}=8.08\times 10^{-9}\text{s}^{-2}\end{array}\right.\]  
\[\mathbf{F}_{23}=(\mathbf{v}^{n}\times)\left[\begin{array}{cccc}0&0&\frac{\upsilon_ {N}}{(R_{M}+\hbar)^{2}}\\ -2\omega_{ie}\sin L&0&-\frac{\upsilon_{E}}{(R_{N}+\hbar)^{2}}\\ 2\omega_{ie}\cos L+\frac{\upsilon_{E}\sec^{2}L}{R_{N}+\hbar}&0&-\frac{ \upsilon_{E}\tan L}{(R_{N}+\hbar)^{2}}\end{array}\right]\]  
\[+\left[\begin{array}{cccc}0&0&0\\ -2\beta_{3}\cos 2L&0&-\beta_{3}\sin 2L\\ -g_{e}(\beta-4\beta_{1}\cos 2L)sin2L&0&\beta_{2}\end{array}\right]\]\[F_{32}=\left[\begin{array}{ccc}0&\frac{1}{R_{M}+\hbar}&0\\ \frac{\sec L}{R_{N}+\hbar}&0&0\\ 0&0&1\end{array}\right]\]  
\[F_{33}=\left[\begin{array}{ccc}0&0&-\frac{v_{N}}{(R_{M}+\hbar)^{2}}\\ \frac{v_{E}\sec L\tan L}{R_{N}+\hbar}&0&-\frac{v_{E}\sec L}{(R_{N}+\hbar)^{2}} \\ 0&0&0\end{array}\right]\]