# Modeling and Control of Satellite Formations: A Survey
## 1 Introduction  
Satellite formations find many applications in the various fields, including Earth surveillance, geodesy, astronomy, space exploration, etc. Following [1], the satellite formation flight (SFF) can be defined as the movement of a group of more than one spacecraft whose dynamic states are coupled by a common control law (see also [2]). At the same time, at least one satellite (formation agent) must track the desired state relative to another agent, and the tracking law must minimally depend on the state of this other agent. For example, even if specific relative positions are actively maintained, the Global Positioning System (GPS) satellites form a formation because only the position and speed of an individual satellite need to be used to correct their orbital motion. Let us present some examples. The distributed space missions with application to Earth observation with special emphasis on radar payloads are presented in [3]. [PERSON] et al. [4] described the distributed satellite system Telematics Int. Mission (TIM), which is a formation of nine cooperating small satellites for photogrammetric Earth observation. TIM's application scenario focuses on the characterization of ash clouds from volcano eruptions, providing useful information to plan detour maneuvers for airplanes to avoid damage to engines. The CloudCT mission [5] uses a formation of 10 nano-satellites to detect the 3D properties of clouds. CloudCT integrates interdisciplinary synergies from nano-satellite system engineering, cloud modeling, and tomographic imaging to enable a sensor network approach to innovative Earth observation. As [PERSON] et al. [6] mentioned, nano-satellites offer an affordable way to realize distributed systems for various scientific and commercial applications in the area of Earth observation and global communication services. [PERSON] et al. [6] presented a contact plan design approach that solves the Medium Access Control (MAC) problem by predicting additive multi-node interference and scheduling interference-free links based on these predictions. The UWE-program aimed for in-orbit demonstrations of key technologies to enable formations of cooperating distributed spacecraft at the pico-satellite level is presented in [7]. In this program framework, the CubeSat UWE-3 is used for the experiments of evaluation of real-time attitude determination and control. The spacecraft formation consists of a group of satellites performing a single mission, which is usually difficult to accomplish with a single device [8; 9; 10]. For example, [PERSON] et al. [9] described a joint project between ESA and NASA called the Laser Interferometer Space Antenna (LISA), which will become the first space observatory for gravitational waves. Its main purpose is to detect and observe gravitational waves from massive black holes and binary galaxies in the frequency range \(10^{-4}\)-\(10^{-1}\) Hz. LISA consists of three identical spacecraft located at a distance of \(5\times 10^{6}\) km from each other in the form of an equilateral triangle. LISA tracks the stretching of space-time caused by gravitational waves by measuring the change in distance between two formation agents using laser interferometry. Three agents will provide independent information about the polarization of the gravitational wave. One shoulder is determined by the distance between the test masses at the ends of each branch of the triangle, which are covered by satellites so that they are not affected by non-gravitational forces. The satellites counteract non-gravitational forces using micro-Newtonian thrusters to follow the trial mass path through what is called drag-free control [11]. [PERSON] [12] considered technologies for implementing sensor networks in orbit for joint measurements by small satellites weighing several kilograms. It is shown that through sensor data fusion and post-processing, innovative distributed methods of Earth observation can be obtained. In [12], exemplary nano-satellite missions in the field of formations are also outlined.