# Modeling and Control of Satellite Formations: A Survey
## 4 Leader-Follower Formation Architecture
The Udwadia-Calaba method is used in [130], where under the assumption of unlimited and time-varying throttle starts throughout the entire maneuver, precisely and explicitly without any restrictions on the distance between satellites, the formation retention problem is solved. [PERSON] and [PERSON] [130] discussed a system of two satellites orbiting a central body (around the Earth). One of them is called the \"master\", and the other is the \"deputy\" satellite, which must maintain a given location relative to the head. The equation of motion of two bodies for an inertial frame of reference is used, which is supplemented by the requirement to hold the station as constraints for obtaining an equation of motion that fully takes into account all nonlinearities. From this, the governing force is derived as an explicit function of state and time required to maintain the formation. The results obtained are also applicable to the case when the distance between the satellites is so great that the linearized relative equations of motion are not applicable. The dynamics describedin the inertial frame of reference is then transformed for convenience into a local moving frame of reference through the transformation matrix. As a practical limitation, in [130] a relative configuration that is circular when projected onto the local horizontal plane is chosen. It is commonly referred to as the PCO [129, 131]. Unrestricted motion is relative motion without a control force acting on the satellite system. The master and slave satellites move only under the influence of gravity, and therefore \"unlimited\" and uncontrolled motions have the same meaning. In the case of movement with restrictions, control forces must be applied to the system. Therefore, \"limited\" and \"controlled\" movements also have the same meaning. When a satellite with a point mass orbits the Earth, its acceleration \(\mathbf{a}\) is written in the Earth's inertial coordinate system as follows [132]:  
\[\mathbf{a}=-\frac{GM}{\left(X^{2}+Y^{2}+Z^{2}\right)^{3/2}} \begin{bmatrix}X\\ Y\\ Z\end{bmatrix}  \tag{58}\]  
where \(G\) is the universal gravitational constant, \(M\) is the mass of the central body (the mass of the Earth). The Equation (58) is written in the inertial terrestrial coordinate system originating from the center of the Earth, the \(X\) axis points to the vernal equinox, the \(Z\) axis passes through the North Pole, and the \(Y\) axis completes the right n.v. (see Figure 3). Assuming that the master satellite is in a circular orbit with a fixed inclination without any disturbance, the goal is to maintain a constant distance between it and the follower, for example, when projecting onto the local horizontal plane. In addition, to prevent discrepancy in \(X\) coordinate, a constraint is also applied between the \(X\) and \(Z\) coordinates. Hill's LVLH coordinate system [21, 23] is used to describe these limitations. The origin of this system is located on the leading satellite, \(X\)-axis lies in the radial direction, \(Y\)-axis is in the path direction, and \(Z\)-axis lies along the orbital angular momentum vector, thus completing the right-handed coordinate system see Figure 3. In LVLH coordinate system, the constraint equations can be written as  
\[y^{2}+z^{2}=\rho^{2}, \tag{59}\] \[2x-z=0, \tag{60}\]  
Figure 3: Coordinate systems in [130]. (Earth s.c. (\(X\)–\(Y\)–\(Z\)) and s.c. Hill (\(x\)–\(y\)–\(z\)). The distance between the leading and trailing satellites on the \(yz\)-plane of the local rotating coordinate system \(x\)–\(y\)–\(z\) must be equal to the specified \(\rho\)). where \(\rho\) is the constant distance between satellites. Since the Equations (59) and (60) describe holonomic constraints, they are twice differentiated in time, which gives  
\[y\ddot{y}+z\ddot{z} =-\dot{y}^{2}-\dot{z}^{2}, \tag{61}\] \[2\ddot{x}-\ddot{z} =0. \tag{62}\]  
Further, in [130], transformations of the rotation of variables from the Earth's r.s. to the Hill coordinate system are performed  
\[ \begin{bmatrix}x+r_{0}\\ y\\ z\end{bmatrix} =\mathbf{R}