# Autonomous and Earth-Independent Orbit Determination for a Lunar Navigation Satellite System
## 3 Proposed Sensors
### Triangulation  
Triangulation modelling has been the subject of study for millennia, with cases from the Egyptians and Greeks of using the concept of a triangle to measure distance and size. A historical review is provided in [41], treating records of the Greek mathematician [PERSON] estimating pyramid heights and ship distances at sea, and of [PERSON] estimating the distance between the Sun and Moon. Modern understanding is first accounted for in the 1600s. Techniques of triangulation are mutually inclusive to well-known direction or Angle of Arrival (AOA) position estimation, as is common and highly mature in maritime [42] and aviation [43] fields using radio-based measurements. This topic has been considered more recently in the context of indoor and wireless navigation [44]. The measurements are typically confined to a two-dimensional plane. A representation of the celestial sphere through an AOA approach is illustrated in Figure 5. The celestial body position is located at \(\mathbf{p}\) and the observing vector at \(\mathbf{s}\). A familiar relation can then be established, \(\mathbf{p}=d\mathbf{s}+\mathbf{r}\), as in the case of the horizon. The angle of arrival \(\theta\) can be calculated from the dot product relation, \(\mathbf{p}\cdot\mathbf{s}=|\mathbf{p}||\mathbf{s}|\cos\theta\). This relation is used by [41]. To explore the potential performance of triangulation techniques in space, this relationship can be reduced to the two-dimensional plane, like for wireless or indoor positioning. Figure 4: Horizon estimation performance at varying orbital altitudes, with the eccentricity transition and ELFO altitude range highlighted. Using a measured angle \(\theta_{i}\) from the \(i\)th source at \((p_{x,i},p_{y,i})\), the relationship between position \((r_{x},r_{y})\) and the celestial source is a simple tangent trigonometric function [44],  
\[\frac{r_{y}-p_{y,i}}{r_{x}-p_{x,i}}=\tan\theta_{i}. \tag{8}\]  
For a multiple number of sources, this model may then be solved for \(r_{x}\) and \(r_{y}\) as a determinable system. In AOA radio-based techniques, the geometry of the observation sources is typically treated as an indicator of performance. Known as the Dilution of Precision (DOP), it is calculated from the measurement matrix, providing a relationship between the measurement noise \(\sigma_{\theta}\) and the position error \(\sigma_{r}\). For the purposes of this work, the DOP should be expressed in the form of  
\[\sigma_{r}=DOP\sigma_{\theta}. \tag{9}\]  
The DOP may be derived in terms of an angular measurement \(\theta\) by first taking the derivatives of Equation (8) in terms of \(\partial r_{x}\) and \(\partial r_{y}\),  
\[\partial\theta_{i}=\frac{1}{r^{2}}(-(r_{y}-p_{y,i})\partial r_{x}+(r_{x}-p_{ x,i})\partial r_{y}), \tag{10}\]  
where \(r^{2}=(r_{x}-p_{x,i})^{2}+(r_{y}-p_{y,i})^{2}\). This may be written in matrix form as  
\[ \begin{cases}\partial\theta_{1}\\ \vdots\\ \partial\theta_{m}\end{cases} = \begin{cases}\frac{-(r_{y}-p_{y,1})}{r^{2}}& \frac{(r_{x}-p_{x,1})}{r^{2}}\\ \vdots&\vdots\\ \frac{-(r_{y}-p_{y,m})}{r^{2}}&\frac{(r_{x}-p_{y,m})}{r^{2}}\end{cases} \begin{bmatrix}\partial r_{x}\\ \partial r_{y}\end{bmatrix} =H \begin{bmatrix}\partial r_{x}\\ \partial r_{y}\end{bmatrix} , \tag{11}\]  
for \(m\) observed sources. The DOP is then calculated as \(DOP=\sqrt{\text{Tr}((H^{r}H)^{-1})}\). The DOP, and thus calculated position error, can be considered in the solar system plane. In Figure 6, the Sun, Earth, and Mars are treated as an observed distant celestial source at a position \((p_{x,i},p_{y,i})\). In this example, the sources are placed in one quadrant of the solar system plane, with the Sun at the centre. For a spacecraft at some position \((r_{x},r_{y})\) in the plane, the expected position error may be estimated by first calculating the DOP from Equation (11), and then the position error by Equation (9). Figure 5: Distribution of planetary point sources at positions \(\mathbf{p}\) with trigonometric relationships to a spacecraft at \(\mathbf{r}\), measured by \(\mathbf{s}\).