# 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  
When the photon frequency emitted by a celestial body produces a color change in the spectrum, the spectrum emitted by the celestial body differs from the spectrum the carrier receives. The redshift is an essential parameter of the spectrum, including the relative velocity. The redshift, \(z\), is defined as:  
\[z=\frac{\lambda-\lambda_{0}}{\lambda_{0}}=\frac{f_{0}-f_{m}}{f_{m}} \tag{1}\]  
where \(\lambda_{0}\) is the initial wavelength of the spectrum lines, \(\lambda\) is the measured wavelength; \(f_{0}\) is the initial frequency of the spectral lines; and \(f\) is the measured frequency. The redshift equation can be obtained from the equation:  
\[1+z=\frac{f_{0}}{f_{m}}=\frac{1+\left|\mathbf{v}\right|\text{cos}\theta/c}{\sqrt{1- \left|\mathbf{v}\right|^{2}/c^{2}}} \tag{2}\]  
where \(\mathbf{v}\) is the velocity of the carrier relative to the celestial body in the \(i\) frame; \(\theta\) is the angle between the astro-carrier vector and \(\mathbf{v}\); and \(c\) is the light velocity. According to the ephemeris of the celestial bodies and the carrier attitude, combined with the measurement of spectral redshift, the flight velocity of the carrier can be obtained [17; 18]. The basic principle of spectral redshift autonomous navigation is as follows. The schematic diagram of spectral redshift navigation is shown in Figure 3:  
During flight, the carrier can detect the light signal emitted by the celestial body. From the Doppler principle, it can be known that the spectral frequency received by the carrier is not equal to the frequency the celestial body emitted, so the relative velocity between the carrier and the celestial body can be calculated by measuring the spectral redshift. When the number of non-collinear celestial bodies detected by the carrier is not less than three, the carrier velocity can be obtained by geometrical relations. The solution process is as follows [16; 19; 20]. Figure 2: The relation between the frames. According to Equation (2), when the carrier moves relative to the celestial body, it can be obtained that:  
\[|\mathbf{v}|\cos\theta=(1+z)\sqrt{c^{2}-\left|\mathbf{v}\right|^{2}}-c \tag{3}\]  
The above equation is also the radial velocity of the carrier relative to the celestial body:  
\[v_{r}=|\mathbf{v}|\cos\theta=(1+z)\sqrt{c^{2}-\left|\mathbf{v}\right|^{2}}-c \tag{4}\]  
Equation (4) is applied to the first reference celestial body:  
\[v_{r1}=(1+z_{1})\sqrt{c^{2}-\left|\mathbf{v}_{p}-\mathbf{v}_{1}\right|^{2}}-c \tag{5}\]  
where \(v_{r1}\) is the radial velocity of the carrier relative to the first reference celestial body; \(z_{1}\) is the redshift of the carrier relative to the first reference celestial body; \(\mathbf{v}_{p}\) is the velocity of the carrier in the \(i\) frame; and \(\mathbf{v}_{1}\) is the velocity of the first reference celestial body in the \(i\) frame. The equation group can be listed by selecting three reference celestial bodies:  
\[ \begin{cases}v_{r1}=(1+z_{1})\sqrt{c^{2}-\left|\mathbf{v}_{p}-\mathbf{v}_{1}\right|^{ 2}}-c\\ v_{r2}=(1+z_{2})\sqrt{c^{2}-\left|\mathbf{v}_{p}-\mathbf{v}_{2}\right|^{2}}-c\\ v_{r3}=(1+z_{3})\sqrt{c^{2}-\left|\mathbf{v}_{p}-\mathbf{v}_{3}\right|^{2}}-c\end{cases}  \tag{6}\]  
Figure 3: The schematic diagram of spectral redshift navigation, where \(\mathbf{r}_{1},\mathbf{r}_{2},\mathbf{r}_{3},\mathbf{r}_{p}\) represent the position vectors of celestial body 1, celestial body 2, celestial body 3, and carrier \(\mathbf{P}\) in the inertial coordinate system, respectively; \(\mathbf{r}_{1p},\mathbf{r}_{2p},\mathbf{r}_{3p}\) represent the position vectors from three celestial bodies to the carrier; \(\mathbf{v}_{r1},\mathbf{v}_{r2},\mathbf{v}_{r3}\) are the radial velocities of the carrier relative to three reference celestial bodies; and \(\mathbf{v}_{p}\) is the velocity vector of the carrier in the \(i\) frame. According to the geometrical relation, \(\mathbf{v}_{p}\) and \(v_{r1},v_{r2},v_{r3}\) satisfy:  
\[