# Autonomous and Earth-Independent Orbit Determination for a Lunar Navigation Satellite System
## 3 Proposed Sensors
### Star Tracker  
The star tracker is a highly accurate and technologically mature attitude determination sensor common to most spacecraft architectures. The star tracker is critical to any autonomous orbit determination architecture, and so will be briefly discussed. Given the high number of books and articles in the literature discussing star trackers [29; 30; 31], the reader is referred to them for further learning. In short, the star tracker captures images of the night sky, and by identifying star patterns using astrometric databases, typically by comparing angles between neighbouring stars, an attitude can be calculated between the sensor's body and the inertial, celestial frame. The process of star identification is complex, with sophisticated algorithms adopted to compare many star patterns, so false identification is avoided. Any small risk of a star being misidentified leads to it being rejected (usually below 0.01%). Popular algorithms include Pyramid [32], which compares star patterns by constructing and measuring a pyramid-like geometry, and Tetra [33], which identifies stars to a catalogue by using a hash code. In this investigation, the star tracker is treated as an accurate attitude determination source with a precision of approximately 10 arc-sec [15]. This will be delivered by an instrument with a 20\({}^{\circ}\) field of view with a pixel resolution of 1024 \(\times\) 1024 px. This is typical of most common star trackers [34]. The attitude of the sensor represents the orientation of the sensor body frame \(B\) with respect to the inertial celestial frame \(I\), expressed in the form of an attitude transformation matrix \(T_{IB}\). This is important in transforming Moon features, such as the horizon or craters, to a local planetary frame of the body, \(P\). Any attitude error will directly impair the navigation performance, as is highlighted in later sections of this work. The transform between the inertial and planetary frame \(T_{IP}\) is known by international scientific working groups and organisations, such as International Earth Rotation and Reference Systems Service (IERS), and so the body to planetary frame transform is calculated by \(T_{PB}=T_{IP}^{r}T_{IB}\). The transformation matrix between the celestial and planetary frame \(T_{IP}\) has been accurately modelled for the Moon, using measurements from recent missions such as the reconnaissance orbiter [27]. In this paper, it is assumed that any error in the model is negligible, as typical inaccuracies are far below the star tracker performance of 10 arc-sec. The calculated attitude will also include a slight angular bias, caused by mechanical misalignment and thermal fluctuations of the satellite platform. This bias will slightly \begin{table}
\begin{tabular}{l l l l} \hline
**Sensor** & **Design Limitations** & **Challenges to Performance** \\ \hline \multirow{4}{*}{Crater Navigation} & \(\bullet\) & Requires pointing to the lunar crater field. & \(\bullet\) & Crater shadow and rim lighting for detecting and differentiating craters. \\  & \(\bullet\) & AI and fitting algorithm are power-consuming. & \(\bullet\) & Variance in the number of observable craters \\  & \(\bullet\) & Requires links to be placed close together due to limited range and potential & \\  & & obstructing Moon. & \\ \multirow{4}{*}{Inter-Satellite Link} & \(\bullet\) & Requires precise pointing to the connected & \(\bullet\) & Careful management and transfer of position, \\  & & satellite if using optical links. & \\ \multirow{4}{*}{Inter-Satellite Link} & \(\bullet\) & Must be used in combination with another & \\  & & sensor providing an absolute source of & \\ \cline{1-1}  & & position reference. & \\ \hline \end{tabular}
\end{table} 
Table 2: _Cont._rotate the attitude transformation matrix by an angle \(\theta_{mis}\), generating an offset between different sensors. The misalignment can be described by multiplying a Direction-Cosine Matrix (DCM) of \(\theta_{mis}\) to \(T_{IB}\), i.e. \(\hat{T}_{IB}=DCM(\theta_{mis})T_{IB}\).