# Autonomous and Earth-Independent Orbit Determination for a Lunar Navigation Satellite System
## 3 Proposed Sensors
### Horizon Navigation  
The sextant is an ancient tool for navigation, and in the early days of the space race, it was considered to be one of the primary sources of positioning by astronauts [38]. Modern techniques have vastly improved its performance and applicability [11; 39], being treated in both cislunar and Earth orbits. The application of the techniques developed are treated for the lunar context. Both the Moon and Earth can be modelled as an ellipsoid. This treatment is suitable for distances of more than 100 km above the surface [15; 40], where rugged terrain does not exceed 10 km, with a standard deviation of below 3 km. It is thus not visible on the horizon projection in most wide-field-of-view lens assemblies for systems in orbit. A point on the surface \(\mathbf{p}=\left\{p_{x}\quad p_{y}\quad p_{z}\right\}^{\top}\) of the planetary frame can be expressed by  
\[\frac{p_{x}^{2}}{a^{2}}+\frac{p_{y}^{2}}{b^{2}}+\frac{p_{z}^{2}}{c^{2}}=1, \tag{1}\]  
with radial dimensions \(a\), \(b\), and \(c\). Alternatively, it can be expressed in matrix form:  
\[\left\{p_{x}\quad p_{y}\quad p_{z}\right\} \begin{bmatrix}a^{-2}&0&0\\ 0&b^{-2}&0\\ 0&0&c^{-2}\end{bmatrix} \left\{ \begin{matrix}p_{x}\\ p_{y}\\ p_{z}\end{matrix} \right\}=p_{P}^{\top}\,A_{P}\,\mathbf{p}_{P}=1, \tag{2}\]  
where \(A_{P}\) is the ellipsoid body matrix. The Moon can be approximated as a sphere, where \(a=b=c\), and the Earth as an oblate spheroid, \(a=b\). An illustration of the ellipsoid and the projected shape on a camera sensor is reported in Figure 3. The projected shape may either be a circle, ellipse, parabola, or hyperbola, depending on the conic section created by the ellipsoid. The general quadratic equation describes a conic section in the body frame with measured image coordinates \((u,v)\),  
\[Au^{2}+Buv+Cv^{2}+Du+Fv+G=0, \tag{3}\]  
where \((A,B,C,D,F,G)\) are coefficients. A similar matrix form could be written for this projection, writing a vector \(\mathbf{s}=\left\{u\quad v\quad 1\right\}^{\top}\) that would intersect at a tangent with the ellipsoid surface, as  
\[\mathbf{s}^{\top} \begin{bmatrix}A&B/2&D/2\\ B/2&C&F/2\\ D/2&F/2&G\end{bmatrix} \mathbf{s}=0. \tag{4}\]The vector \(\mathbf{s}\) can be transformed to the planetary frame using the estimated attitude transformation matrix \(T_{PB}\) in Section 3.1. Errors in the estimated horizon point, such as stray light, optical noise, or large rugged terrain, are introduced in the estimated limb pixel \((u,v)\). The estimated performance of a measured horizon coordinate can be assumed to be 0.1 px [11; 36; 40]. It is suggested that upon implementation the optical system hardware developed reflects this estimation resolution. These two expressions can be related to the sensor position \(\mathbf{r}\), measured from the ellipsoid origin, by \(\mathbf{p}=d\mathbf{s}+\mathbf{r}\) where \(d\) is some scalar multiplier. These two functions can be then simplified by an approach given in [39] to  
\[\mathbf{s}^{\mathrm{ T}}M\mathbf{s}=0, \tag{5}\]  
where  
\[M=A\mathbf{r}\mathbf{r}^{\mathrm{ T}}A-(\mathbf{r}^{\mathrm{ T}}A\mathbf{r}-1)A. \tag{6}\]  
Matrix \(M\) describes the shape of the projected ellipsoid on the image sensor, as a function of the ellipsoid matrix \(A\) and the observer position \(\mathbf{r}\). This term is equivalent to the ellipse matrix, as represented in Equation (4), which can be denoted by \(C\), by the following generalised eigen-problem expression,  
\[C\mathbf{s}=\lambda M\mathbf{s}, \tag{7}\]  
where \(\mathbf{s}\) is an eigenvector and \(\lambda\) is an eigenvalue. It should be noted that the frame subscript has been dropped in Equation (6). It is important that all matrices and vectors are in the same frame, i.e., body, inertial, or planetary. This expression can be used for estimation of the position in an implicit least square-based model. Work by [17] treats the celestial horizon in Cholesky space, which reduces the problem from a non-linear to a linear case.