# Autonomous and Earth-Independent Orbit Determination for a Lunar Navigation Satellite System
## 3 Proposed Sensors
### Crater Navigation
This expression might be rearranged to matrix form for the left-hand side as  
\[ \begin{bmatrix}S^{T}T_{E_{i}P}B_{i}T_{PE_{i}}S&S^{T}T_{E_{i}P}B_{i}(\mathbf{p}_{ P_{i}}-\mathbf{r}_{P})\\ (\mathbf{p}_{P_{i}}-\mathbf{r}_{P})^{T}B_{i}^{T}T_{PE_{i}}S&(\mathbf{p}_{P_{i}}-\mathbf{r}_{P} )^{T}B_{i}(\mathbf{p}_{P_{i}}-\mathbf{r}_{P})\end{bmatrix} =s_{i}C_{i}. \tag{16}\]  
Figure 7: Lunar craters projected on a spacecraft’s camera lens. The independence of the matrix upper-left hand corner to the spacecraft position vector can be used to find the scalar \(s_{i}\) as the least squares-like solution,  
\[\hat{s}_{i}=\frac{\text{Vec}(S^{T}C_{i}S)^{T}\text{Vec}(S^{T}T_{E_{i}P}B_{i}T_{PE _{i}}S)}{\text{Vec}(S^{T}C_{i}S)^{T}\text{Vec}(S^{T}C_{i}S)}, \tag{17}\]  
where the operator \(\text{Vec}()\) converts a matrix into a vector, with each column of the matrix'stacked' on top of the other. The upper-right-hand corner of the matrix in Equation (16) can then be used to create a linear system in terms of the spacecraft position vector \(r_{P}\) for \(m\) craters,  
\[ \begin{bmatrix}S^{T}T_{E_{1}P}B_{1}\\ \vdots\\ S^{T}T_{E_{m}P}B_{m}\end{bmatrix} \mathbf{r}_{P}= \begin{bmatrix}S^{T}T_{E_{1}P}B_{ 1}\mathbf{p}_{P_{1}}-\hat{s}_{1}S^{T}C_{1}\mathbf{k}\\ \vdots\\ S^{T}T_{E_{m}P}B_{1}\mathbf{p}_{P_{m}}-\hat{s}_{m}S^{T}C_{m}\mathbf{k}.\end{bmatrix}  \tag{18}\]  
The visible craters on a standard lunar projection are highlighted in Figure 7, where each corresponding projected ellipse matrix \(A_{i}\) to a corresponding crater ellipse \(C_{i}\) is plotted. The associated covariance of each measurement may be calculated by the ellipse coefficient covariance \(P_{\mathbf{a}}\). The associated covariance measurement model covariance matrix is calculated by  
\[R_{\mathbf{a}}=\frac{d\mathbf{h}}{d\mathbf{a}}P_{\mathbf{a}}\frac{d\mathbf{h}^{\top}}{d\mathbf{a}}. \tag{19}\]  
The position covariance estimated is then  
\[P_{\mathbf{r}}=\left(\frac{d\mathbf{h}}{d\mathbf{r}}R_{\mathbf{a}}^{-1}\frac{d\mathbf{h}^{\top}}{d \mathbf{r}}\right)^{-1}. \tag{20}\]  
These derivatives can be simply solved by performing a vector derivative on Equation (18). The position estimation performances of crater navigation at varying altitudes are illustrated in Figure 8. A sensor similar to the horizon sensor is treated, with a \(40^{\circ}\) field of view and a resolution of 1024 px. Each performance in Figure 8 takes an average of 100 position estimates at randomised lunar rotations. These results are juxtaposed by the average number of craters identified, which can vary by not only altitude but also the side of the lunar surface visible. The lunar far or dark side, being the side opposite to the tidally locked face, has a higher number of impact craters given that it is less protected by the Earth. Performance of the proposed crater technique increases as the spacecraft nears the lunar surface, as more craters become identifiable. It then plateaus at 1000 km above the surface, at a performance of approximately 200 m. This however might be caused by the elliptical fit approach of this work. AI techniques, which can provide more accurate identification fitting, might significantly improve the result. The camera optics are also constrained. Within the ELFO operating range, performances of approximately 1 km to 12 km are witnessed in Figure 8. This variation is not appropriate for autonomous positioning, and may also present challenges to the EKF since the estimated covariance will change significantly. Craters are only visible when the sun is overhead, down to an angle of elevation of approximately \(20^{\circ}\). This constraint may be maintained by taking the dot product of the sun \(\hat{\mathbf{p}}_{s}\) and crater \(\hat{\mathbf{p}}_{i}\) unit position vector in the lunar fixed frame, \(\cos 20^{\circ}<\hat{\mathbf{p}}_{s}\cdot\hat{\mathbf{p}}_{i}\), where the hat indicates it is a unit vector. In this study, 70\({}^{\circ}\) is taken as the angular limit between the sun direction and the crater normal; otherwise, it is treated as too dark to be detectable.