## 4 Mission Orbit and Relative Navigation Filter  
### Real Formation Mission Orbit  
The formation flying orbit analyzed here, quite similar to that in PROBA-3, is a virtual structure SFF mission that will perform in HEO, which we previously referred to as the HEFFD mission. The HEFFD mission is initially designed as an on-orbit development, verification, and validation platform for multiple task forces. The main objective is to test guidance, navigation, and control (GNC) technology in space flight, especially for accurate ranging metrology units and high-performance propulsion systems that enable fine formation maneuvers.  
The HEFFD mission orbit design is a trade-off analysis process that considers the requirements of: (a) high altitude during the formation flying stage since it is a low-gravity-gradient environment; (b) simple Sun-Earth geometry when SFF during Solar observation phase; (c) launch capability; and (d) ground station location.  
Here, we provide the baseline orbit for the HEFFD mission in Table 2 as:  
and the slave spacecraft is supposed to be performing follow-on flight relative to the master spacecraft, with a minimal distance of 10 km around apogee.  
Some preliminary design parameters of master/slave spacecrafts include: mass: 200 kg/400 kg; reference area: 1.81 m\({}^{2}\)/3.34 m\({}^{2}\); dimensional drag coefficient: 1.29/1.5; dimensional solar radiation pressure coefficient: 1.2/3.27 [21; 22].  
Figures 5 and 6 provide the relative range and elevation angle values for three orbit periods. Clearly, the orbit calculation starts from perigee with a relative formation distance of about 120 km, and gradually approaches to a minimum of about 10 km, which is suitable for mission operation. The bolded red lines in Figure 5 demonstrate the relative orbit values of range and elevation angles,  
\begin{table}
\begin{tabular}{c c} \hline
**Items** & **Model** \\ \hline GA—the geopotential effect of the Earth & 20 th order and degree \\ GA—Sun and Moon gravities & high-precision DE405/LE405 planetary ephemerides model \\ GA—solid Earth tides & IERS Conventions 1996 \\ GA—ocean tides & Center for Space Research 3.0 model \\ NGA—the atmospheric drag & NRLMSISE-00 empirical model \\ NGA—the solar radiation pressure & IERS Standards 1992 \\ \hline \end{tabular} Note: GA: gravitational acceleration; NGA: non-gravitational acceleration.  
\end{table}
Table 1: Accelerations of gravitational and non-gravitational models.  
while using dynamic equations of the radar model (11)-(17). The azimuth angle values are not shown here, since the co-orbit formation configuration is used in this simulation. Moreover, the green lines in Figure 5 show the relative range and elevation values by using the propagated perturbation orbit dynamicsmodel in Table 1. The results illustrate that the real perturbed relative formation orbit drifted quickly from the radar model during simulation, and Figure 6 illustrates the drift bias within three orbit periods.  
According to Figures 5 and 6, the relative orbit dramatically drifted. The formation ranging distance drifted gradually near each perigee arc, almost reached \(-40\) km at the end of the third orbit time, and converged quickly during each apogee arc. The formation mission configuration can be carefully designed initially, considering detailed orbit perturbations. However, a sophisticated relative model is not suitable for navigation filter calculation due to the huge on-board computation burden for real space missions. Finally, by careful consideration, apogee arcs were chosen as the mission operation orbit, which is used for high-precision formation GNC technology verification. The mission apogee arcs, occupying about half of the orbit period time, are clearly shown in Figures 5 and 6, using bolded blue star dots. The relative navigation algorithm has to be carefully designed, considering real orbit drift and model uncertainty.