# Autonomous and Earth-Independent Orbit Determination for a Lunar Navigation Satellite System
## 3 Proposed Sensors
### Crater Navigation  
Using craters for navigation around the Moon, alongside other planetary bodies, was initially considered in the context of asteroid rendezvous [45; 46]. Likening them to ellipses, similar approaches to horizon ellipse fitting have been developed for location pinpointing and crater identification. Crater estimation techniques have received significant renewed interest in the context of lunar exploration [47; 48]. This technique has been especially considered for the case of lunar descent, where decimetre performances are usually targeted [16; 49]. The work of [PERSON] provides a linear least-square estimate for the general crater positioning estimation problem and will be used here [47]. Traditional crater identification algorithms have been studied [50], but machine learning and Artificial Intelligence (AI) present higher performances, popularly adopted by [16; 47; 51]. Many databases have been generated by various recent lunar missions, including the NASA Lunar Reconnaissance Orbiter and Chinese Chang'e Lunar surveyor, as well as by ground-based astronomy. The simple database created by the Lunar and Planetary Institute is used in this analysis. Even though craters are described as circular, the projections on the camera frame provide appropriate ellipse shapes for fitting to be evaluated, and position error estimated [52]. In crater-based navigation, three different references are treated, the optical sensor body frame \(B\), the lunar planetary frame \(P\), and the local East-North-Up (ENU) frame of  
Figure 6: Position error of a spacecraft at a position in the solar system plane with origin at the Sun, using triangulation from the Sun, Earth, and Mars with an object localisation accuracy of 0.01\({}^{\circ}\). the crater \(E\). The transformation between the body and planetary frame \(T_{PB}\) is known by the star tracker, as introduced in Section 3.1. The crater ellipse projection is developed in a similar fashion to the horizon, and is illustrated in Figure 7. The profile can be expressed by Equation (3), where the crater \(i\) measured across the image plane can be denoted by a matrix \(A_{i}\). This is after a set of image points is fit to an equivalent ellipse by a routine first proposed by [PERSON] [53], providing a set of conic coefficients, \(\mathbf{a}^{\tau}=\left\{A\quad B\quad C\quad D\quad F\right\}\). The approach also provides a covariance with the associated non-linear least-square estimate, which is expressed as a \(5\times 5\) matrix \(P_{\mathbf{a}}\). Similarly, the shape of the crater is known within the crater database by the shape matrix \(C_{i}\), defined as in Equation (4). The ellipse is defined in the crater's local ENU frame \(E_{i}\), centred at the ellipse origin \(\mathbf{p}_{P_{i}}\) in the Moon's planetary frame. The ENU frame is highlighted on the projected equator in Figure 7. The ellipse relation between \(A_{i}\) and \(C_{i}\), after appropriate identification by a machine learning algorithm [16; 51], can be related through a series of projection transformations for the spacecraft located at a position \(\mathbf{r}_{P}\)[47]. Defining first the homography \(H_{P_{i}}\) for the \(i\)th crater,  
\[H_{P_{i}}=\left[T_{PE_{i}}S\quad\mathbf{p}_{P_{i}}\right], \tag{12}\]  
where \(T_{PE_{i}}\) is the transformation matrix between the fixed lunar planetary and crater's local ENU frame, and \(S\) is a \(3\times 2\) matrix,  
\[S= \begin{bmatrix}I_{2}\\ \mathbf{0}_{1\times 2}\end{bmatrix} . \tag{13}\]  
The transformed measured crater in the lunar planetary frame is calculated by  
\[B_{i}=T_{PB}A_{i}T_{BP}, \tag{14}\]  
where any focal length-related parameters can be related by some scalar \(s_{i}\). The relation between \(A_{i}\) and \(C_{i}\) is then given by  
\[H_{P_{i}}^{T}B_{i}H_{P_{i}}-H_{P_{i}}^{T}B_{i}\mathbf{r}_{P}\mathbf{k}^{T}-\mathbf{k}\mathbf{r }_{P}^{T}B_{i}H_{P_{i}}+\mathbf{r}_{P}^{T}B_{i}\mathbf{r}_{P}\mathbf{k}\mathbf{k}^{T}=s_{i}C_ {i}, \tag{15}\]  
where \(\mathbf{k}^{T}=\left\{0\quad 0\quad 1\right\}\).