# Exploiting Lunar Navigation Constellation for GNC Enhancement in Landing Missions
## 2 Simulation Environment
### Ground-Truth Dynamics  
To accurately simulate the ground-truth dynamics of the landing user and LNS serviers, the major gravitational and non-gravitational forces acting in the cislunar environment have been modelled. The former include the gravitational attraction of the Moon, the Earth, and the Sun. The remaining solar system bodies have not been included because their gravitational influence (\(10^{-12}\) m s\({}^{-2}\) for Jupiter) is negligible over the short-term. The irregularities of the lunar gravity field are considered by modelling the lunar gravitational potential, \(\mathcal{U}\), in a body-fixed reference frame with a Spherical Harmonic Expansion (SHE) [21], as in Equation (1):  
\[\mathcal{U}=\frac{\mu}{r}+\frac{\mu}{r}\sum_{n=2}^{N}\sum_{m=0}^{n}\left(\frac {R_{0}}{r}\right)^{n}\left[\overline{\mathcal{C}}_{nm}\cos(m\lambda)+\overline {S}_{nm}\sin(m\lambda)\right]\overline{\mathcal{P}}_{nm}(\sin\phi) \tag{1}\]  
where \(\mu\) and \(R_{0}\) are the gravitational parameter and radius of the Moon, \(r\) is the radial distance from the center of mass, \(\lambda\) is the east longitude, \(\phi\) is the latitude, and \(\overline{\mathcal{P}}_{nm}\) are the fully normalised associated Legendre polynomials. The normalised gravity coefficients (\(\overline{\mathcal{C}}_{nm}\) and \(\overline{S}_{nm}\)) are based on the LP165P gravity model [22]. The harmonic expansion is truncated at order and degree 60 and is such that the error introduced by neglecting the higher-order terms at these altitudes is below the sensitivity of traditional on-board accelerometers. For the Sun and Earth contributions, only the point-mass gravitational acceleration is relevant. In particular, Earth's \(J_{2}\) term produces an acceleration in the order of only \(10^{-11}\) m s\({}^{-2}\) for a lunar orbiter in a 10 km LLO. According to the Restricted N-Body Problem formulation [21], the perturbing acceleration induced on the spacecraft, \(\mathcal{P}_{i}\), by the point-mass gravitational sources, \(\mathcal{P}_{j}\), as seen by the central body, \(\mathcal{P}_{k}\), is modelled as per Equation (2):  
\[\bar{\mathbf{r}}_{ki}=\sum_{ \begin{subarray}{c}j=1\\ j\
eq i,k\end{subarray} }^{\mathrm{N}}\mu_{j}\left(\frac{\mathbf{r}_{ij}}{r_{ij} ^{3}}-\frac{\mathbf{r}_{kj}}{r_{kj}^{3}}\right) \tag{2}\]  
where the generic vector \(\mathbf{r}_{ij}\) is the position of the mass \(\mathcal{P}_{i}\) relative to \(\mathcal{P}_{j}\) and is computed as:  
\[\mathbf{r}_{ji}=\mathbf{r}_{kj}-\mathbf{r}_{ki} \tag{3}\]  
In this work, NAIF's SPICE library (see [23]) is exploited to retrieve the exact Earth and Sun positions from the DE440 ([[https://naif.jpl.nasa.gov/pub/naif/generic_kernels/spk/planets/](https://naif.jpl.nasa.gov/pub/naif/generic_kernels/spk/planets/)]([https://naif.jpl.nasa.gov/pub/naif/generic_kernels/spk/planets/](https://naif.jpl.nasa.gov/pub/naif/generic_kernels/spk/planets/)), accessed on 20 July 2023) ephemerides. All the remaining field forces, such as general relativistic effects (\(\sim\)10\({}^{-10}\) m s\({}^{-2}\) for a 10 km LLO), can be safely neglected for short-term propagations. On the other hand, the remaining forces acting on the spacecraft are associated with non-gravitational perturbations and must be accounted for in a high-fidelity lunar dynamical model. These include contributions such as the Solar Radiation Pressure (SRP), the spacecraft thermal emissions, the lunar albedo and thermal infrared emissions, and the navigation antenna thrust for the LNS servicers. In this analysis, only the SRP contribution is taken into account since it is by far the largest non-gravitational acceleration.