# A Novel SINS/SRS/CNS Multi-Information Fusion Global Autonomous Navigation Method
## 3 The Multi-Information Fusion Autonomous Navigation System Based on SINS/SRS/CNS
### The Principle of Spectral Redshift Navigation
#### 3.2.1 State Equations  
In the middle-low latitudes, the \(g\) frame was chosen as the navigation coordinate frame. The SINS error equations in the \(g\) frame can be obtained as follows [21].  
The mathematical platform error angle equation can be denoted as:  
\[\dot{\mathbf{\Phi}}^{g}=\mathbf{\Phi}^{g}\times\left(\mathbf{\omega}_{ie}^{g}+\mathbf{\omega} _{cg}^{g}\right)+\left(\delta\mathbf{\omega}_{ie}^{g}+\delta\mathbf{\omega}_{cg}^{g} \right)-\mathbf{C}_{b}^{g}\mathbf{e}^{b} \tag{9}\]  
where \(\mathbf{\Phi}^{g}\) is the mathematical platform error angle in the \(g\) frame; \(\mathbf{\omega}_{ie}^{g}\) is the projection of the Earth's rotation angular velocity, \(\mathbf{\omega}_{ie}\), in the \(g\) frame; \(\mathbf{\omega}_{cg}^{g}\) is the angular velocity of the \(g\) frame to the \(e\) frame, projected in the \(g\) frame; \(\delta\mathbf{\omega}_{ie}^{g}\) and \(\delta\mathbf{\omega}_{cg}^{g}\) are the errors of \(\mathbf{\omega}_{ie}^{g}\) and \(\mathbf{\omega}_{cg}^{g}\); \(\mathbf{C}_{b}^{g}\) is the direction cosine matrix from the \(b\) frame to the \(g\) frame; and \(\mathbf{\varepsilon}^{b}\) is the gyroscope drift in the \(b\) frame.  
The velocity error equation can be denoted as:  
\[\delta\dot{\mathbf{v}}^{g}=\mathbf{f}^{g}\times\mathbf{\Phi}^{g}-\left(2\delta\mathbf{\omega} _{ie}^{g}+\delta\mathbf{\omega}_{cg}^{g}\right)\times\mathbf{v}^{g}-\left(2\mathbf{\omega} _{ie}^{g}+\mathbf{\omega}_{cg}^{g}\right)\times\delta\mathbf{v}^{g}+\mathbf{C}_{b}^{g}\mathbf{ \
abla}^{b} \tag{10}\]  
where \(\delta\mathbf{v}^{g}\) is the velocity error in the \(g\) frame; \(f^{g}\) represents the specific force in the \(g\) frame; and \(\mathbf{\
abla}^{b}\) is the accelerator drift in the \(b\) frame.  
The position error, \(\delta\mathbf{p}^{g}\), comprises the latitude error, \(\delta L\); the longitude error, \(\delta\lambda\); and the altitude error, \(\delta h\). The equations of position error are as follows:  
\[\left\{\begin{array}{ll}\delta L=\frac{\delta\mathbf{v}_{N}}{R_{M}+h}\\ \delta\lambda=\frac{\delta\mathbf{v}_{E}}{R_{N}+h}\sec L+\frac{\mathbf{v}_{E}}{R_{N}+h} \sec LgL\delta L\\ \delta h=\delta\mathbf{v}_{U}\end{array}\right. \tag{11}\]  
where \(R_{M}\) and \(R_{N}\) are the radii of curvatures in meridian and prime vertical, respectively, and \(V_{E}\) and \(V_{N}\) are the east velocity and north velocity, respectively.  
The gyroscope and accelerator drift error equations can be denoted as  
\[\left\{\begin{aligned} \dot{\mathbf{\varepsilon}}^{b}&=0\\ \dot{\
abla}^{b}&=0\end{aligned}\right. \tag{12}\]  
According to Equations (9)-(12), select \(\mathbf{\Phi}^{g}\), \(\delta\varpi^{g}\), \(\delta\mathbf{p}^{g}\), \(\mathbf{\varepsilon}^{b}\), and \(\
abla^{b}\) as the state vector:  
\[X^{g}(t)=\begin{bmatrix}\mathbf{\Phi}^{g}&\delta\mathbf{v}^{g}&\delta\mathbf{p}^{g}& \mathbf{\varepsilon}^{b}&\
abla^{b}\end{bmatrix}^{\mathsf{T}} \tag{13}\]  
Then, the state equation of the SINS/SRS/CNS multi-information fusion autonomous navigation system in the middle-low latitudes can be established:  
\[\dot{\mathbf{X}}^{g}(t)=\mathbf{A}^{g}(t)\mathbf{X}^{g}(t)+\mathbf{G}^{g}(t)\mathbf{W}^{g}(t) \tag{14}\]  
where \(\mathbf{X}^{g}(t)\) is the state vector; \(\mathbf{A}^{g}(t)\) is the state transition matrix; \(\mathbf{G}^{g}(t)\) is the system noise coefficient matrix; and \(\mathbf{W}^{g}(t)\) is the system noise vector.