# Modeling and Control of Satellite Formations: A Survey
## 2 Multiple-Input-Multiple-Output Formation Architecture
[PERSON] and [PERSON] [32] demonstrated that the long-term orbital effects of the solar wing's steering input (wing orientation angle) are highly non-linear and show a strong relationship between orbital inclination and ascending node longitude, and if the wing is properly sized, node drift due to solar torque, on average, is compensated by the differential speed of modality, so the formation is preserved without the use of propellant. Numerical results are presented to illustrate the approach. [PERSON] et al. [33] dealt with establishing relationships with the motion of a satellite group in a physical two-dimensional space with visualization problems. A class of movements is presented that satisfies the visualization tasks, and simple solutions to the problem of motion design and formation control for visualization applications are found, allowing to achieve a compromise between image quality and fuel consumption for the proposed class of maneuvers. As part of a study by the European Space Agency (ESA), [PERSON] and [PERSON] [34] discussed the potential advantages, disadvantages, and challenges of moving satellites in a formation in close proximity to implement synthetic aperture radar (SAR) for trajectory interferometry. [PERSON] and [PERSON] [34] noted that the technology of using SAR is well developed. Radar signals reflected from the Earth's surface, water, and ice have provided high-resolution images for areas such as forestry, geology, ocean wave monitoring, and ice mapping (see [35]). In addition, progress in the processing of SAR data has been made through the use of phase differences between SAR images of the same scene, a technique referred to as interferometric SAR (InSAR). Distinguishing SAR images from two laterally-spaced antennas generally yields terrain elevation measurements. In contrast, along-track interferometry (ATI) is not based on a specific interferometric baseline, but on the time interval between two moving SAR devices separated in the direction along the track, and displaying the same terrain, which provides measurements of the speed of objects on the surface. Space ICT was successfully demonstrated using the Shuttle Radar Topography Mission (SRTM) in 2000 [36, 37]. A serious limitation of single-satellite IWT is due to the physical dimensions of the spacecraft. It can be overcome on the basis of the distributed sensor concept, when two SAR antennas are located on different platforms, thus forming variable baseline. This can be conducted with a small satellite that is passive in terms of radar, but active in terms of orbital maneuvers that are conducted relative to the larger active SAR satellite. Alternatively, two similar SAR satellites that are capable of moving in a controlled formation can be used. In IVT interferometry, two antennas are used, separated in the direction of motion along the spacecraft trajectory by a certain amount \(\Delta L\). Considering the common orientation of the antenna, both antennas thus display the same scene with a time lag of \(\Delta t=\Delta L/v_{\text{sc}}\), where \(v_{\text{sc}}\) denotes the spacecraft speed. When one antenna works for both transmit and receive, and the other only for receive, then the two subsequent sets of SAR phase measurements differ by the value  
\[\Delta\varphi=4\pi\frac{v_{\text{r}}}{v_{\text{sc}}}\,\frac{\Delta L}{\lambda}, \tag{15}\]  
where \(v_{\text{r}}\) is the projection of the target speed onto the direction of the line of sight, \(\lambda\) is the radar radiation wavelength. Thus, the measurement accuracy can be increased either by increasing the base \(\Delta L\), or by decreasing the wavelength \(\lambda\). It should be noted that large values of the base cause the effect of phase wrapping and associated ambiguities, if the velocity range of interest is mapped to a phase interval of more than \(2\pi\).