## 2 Mathematical Models
### Adaptive Interpolation Algorithm  
As seen in Figure 2, where C01 represents a BDS GEO satellite, both the rise and set functions of the satellite-to-Moon visibility and the period function of the Earth occultation between the Earth and the Moon are multi-hump functions, which are irregular and do not have specific analytical expressions. It is known from previous work [5; 10] that the solution to a similar function can be approximated using the idea of segment interpolation.  
For the satellite-to-Moon visibility, this paper utilizes the idea of segmented Hermite interpolation to construct a cubic polynomial between the subintervals \(t_{0}\) to \(t_{1}\) and calculates the maximum step from the second-order derivatives of the corresponding visibility functions, as well as the visibility function values in satisfying a certain error limit. According to [17], the function of segmented cubic Hermite interpolation can be written as:  
\[\begin{array}{l}S(t)=f(t_{0})+f^{\prime}(t_{0})(t-t_{0})+a(t-t_{0})^{2}+b(t-t _{0})^{3}\\ a=(3\frac{f(t_{1})-f(t_{0})}{k^{2}}-\frac{f^{\prime}(t_{1})+2f^{\prime}(t_{0}) }{h})\\ b=(\frac{f^{\prime}(t_{1})+f^{\prime}(t_{0})}{k^{2}}-2\frac{f(t_{1})-f(t_{0})} {k^{3}})\end{array} \tag{3}\]  
Figure 1: Geometric illustration of satellite-to-lunar visibility.  
where \(t_{0}\) denotes the starting point for constructing the cubic polynomial, \(t_{1}\) denotes the end point, and \(h=t_{1}-t_{0}\). When the fourth-order derivative exists, the approximation error can be described by:  
\[R(t)=V(t)-S(t)=\frac{V^{(4)}(\eta)}{4!}((t-t_{0})(t-t_{1}))^{2} \tag{4}\]  
where \(t,\eta\in[t_{0},t_{1}]\), \(R(t)\), denotes the interpolation error residual term, \(V(t)\) denotes the true function, \(V^{(4)}(\eta)\) is the fourth-order derivative of \(V(t)\) and can be obtained using higher-order Hermite interpolation, and \(S(t)\) denotes the constructed cubic polynomial. The quadratic polynomial of the function can be approximated by the following equation:  
\[\begin{array}{l}V^{(4)}(t)\approx 120\kappa_{5}t+24\kappa_{4}\\ \kappa_{4}=\frac{4}{h^{2}}\alpha-\frac{4}{h^{2}}\delta-\frac{24}{h^{3}}\chi\\ \kappa_{5}=\frac{24}{h^{2}}[V(t_{0})-V(t_{1})]+\frac{4}{h^{4}}\left[\dot{V}(t_ {0})-\dot{V}(t_{0}+\frac{h}{2})+\dot{V}(t_{1})\right]\\ \alpha=\left[V(t_{0})+4V(t_{0}+\frac{h}{2})+V(t_{1})\right]\\ \beta=\left[\dot{V}(t_{0})(2t_{0}+3t_{1})+10\dot{V}(t_{0}+\frac{h}{2})(t_{0}+t_ {1})+3\dot{V}(t_{1})(3t_{0}+2t_{1})\right]\\ \chi=\left[V(t_{0})(2t_{0}+3t_{1})-V(t_{1})(3t_{0}+2t_{1})\right]\end{array} \tag{5}\]  
Using \(V^{(4)}(t)\) to place limits on the errors, the following is obtained:  
\[|R(t)|_{\text{max}}\leq|\frac{V^{(4)}(n)_{\text{max}}}{4!}|\left[(t-t_{0})(t-t _{1})^{2}\right]_{\text{max}}\leq|5a_{5}n+a_{4}|_{\text{max}}(\frac{h^{2}}{4} )^{2} \tag{6}\]  
Then, the step is:  
\[\hat{h}=(\frac{16\epsilon}{|5\kappa_{5}\eta+\kappa_{4}|_{\text{max}}})^{1/4} \tag{7}\]  
The solution for the step is an iterative process, and the termination condition of the iterative solution is:  
\[\frac{|\hat{h}_{k}-\hat{h}_{k-1}|}{\hat{h}_{k-1}}\leq u \tag{8}\]  
Figure 2: Visibility image of the C01 GEO satellite.  
From the above equation, it is clear that the first-order derivative of the original function is required to perform the segmented Hermite interpolation, where the first-order derivative of the satellite elevation angle is:  
\[\dot{V}(t)=\frac{1}{\|\Delta r\|}(\Delta\dot{r}\cdot r_{s}+\Delta r\cdot\dot{r} _{s})-\frac{1}{\|\Delta r\|}(\Delta r\cdot\Delta\dot{r})\Delta r\cdot r_{s} \tag{9}\]  
The first-order derivative of the Earth occultation model is:  
\[\dot{L}(t)=\frac{(r_{s}\times\Delta r)\cdot(\dot{r}_{s}\times\Delta r+r_{s} \times\Delta\dot{r})}{\|r_{s}\times\Delta r\|\cdot\|\Delta\|}-\frac{\Delta \cdot\dot{\Delta}\cdot\|r_{s}\times\Delta r\|}{\|\Delta\|^{3}} \tag{10}\]  
where \(\Delta\dot{r}=v_{m}-v_{s}\).