## 3 Results and Discussions
### Alfacrux Digital Twin Pilot
#### 3.2.2 Environment Models  
For the proposed DT model, the spacecraft motion is a major concern. The formulation considered in this work uses the SGP4 model for orbit propagation, uses the TLE file as the only orbital data source, and provides a prediction for the satellite position and velocity over time. The SGP4 considers perturbations such as the Earth's ellipsoidal shape effect over the gravitational field, third-body influence (Sun and Moon) and the atmospheric drag.  
As described in [32], the Earth Magnetic Field is modeled as the gradient of the magnetic potential, which is approximated by a spherical harmonic series expansion. One  
Figure 9: CubeSat air-bearing simulator CAD view with the AlfaCrux.  
of the standard models is the International Geomagnetic Reference Field (IGRF), and it consists of the mathematical description along with the coefficient values for the series expansion. The model allows the computation of the geomagnetic field at any point in space given its spherical coordinates. Furthermore, it allows not only computer simulations but also serves as input data for laboratory experiments in Helmholtz cages.  
The Sun position is another important model, once it is the primary power source for most of the spacecrafts. According to [32], a more precise approach to determine the position is through the data set DE430, a set of celestial bodies' ephemerides provided by the Jet Propulsion Laboratory and generated by laser observations and the numerical integration of a dynamic system. However, this model requires access to the data and numerical methods. A simpler and less precise model is based on analytical equations that may be considered in simulations and is also useful for implementing in the onboard software. This model considers the Sun elliptic motion, containing polynomials and trigonometric terms only.  
With respect to the problem of attitude determination, the Earth albedo model may be considered if the spacecraft has Sun sensors onboard, once the solar radiation reflected by the Earth surface can introduce errors in the Sun position vector computing. Albedo is measured as the reflectivity index, which varies according to the cloud coverage, geographic location, season of the year, and type of surface. In [33], the reflectivity index is provided by the NASA Total Ozone Mapping Spectrometer (TOMS) satellite, which records the measurements observations constituting a grid of data points. The total energy reflected to the satellite is formulated as the sum of each data point contribution, that is, illuminated by the Sun and visible by the satellite. The model presented depends on the reflectivity index, the angle of incidence, satellite altitude, and the angle of reflection to the spacecraft.  
For the results obtained so far in the early stage, the Sun position and Geomagnetic field described in the inertial frame are provided by the software System Tool Kit (STK) from Ansys Inc, along with the AlfaCrux orbit propagation with the TLE. The Earth albedo model was implemented based on the Lambertian reflectance model (LRM), which finally guarantees the attitude reconstruction process.