# Modeling and Control of Satellite Formations: A Survey
## 2 Multiple-Input-Multiple-Output Formation Architecture
Using the quadratic Lyapunov function \(V(\delta e)=\dfrac{1}{2}\delta e^{\mathrm{T}}\delta e\), where \(\delta e=e_{2}-e_{d}\) is the tracking error of the slave satellite (\(e_{2}\)) orbit of the lead satellite (\(e_{d}\)) in terms of mean orbit elements, in [16] the following control law for the slave satellite was obtained  
\[u=-\big{(}[B]^{\mathrm{T}}[B]\big{)}^{-1}[B]^{\mathrm{T}}\Big{(}\big{(}[A(e_{2 })]-[A(e_{d})]\big{)}+[P]\delta e\Big{)}, \tag{13}\]  
where the time-variable matrix of parameters \([P]\) is chosen when synthesizing the algorithm. Ref. [PERSON] et al. [16] recommended one way of doing this. The second control law, proposed in [16], uses the position and velocity errors in Cartesian coordinates \(\mathbf{r}=(x,y,z)\). Based on the mismatch \(\delta\mathbf{r}=\mathbf{r}_{2}-\mathbf{r}_{2d}\) between the current \(\mathbf{r}_{2}\) and the desired \(\mathbf{r}_{2d}\) the position of the slave satellite, based on the Lyapunov method, the control law is obtained  
\[u=-\big{(}\mathbf{f}(\mathbf{r}_{2})-\mathbf{f}(\mathbf{r}_{2d})\big{)}-[K_{ 1}]\delta\mathbf{r}-[K_{2}]\delta\dot{\mathbf{r}}, \tag{14}\]  
where function \(\mathbf{f}(\mathbf{r})\) is the right-hand side of the equation of free motion of a satellite in orbit, \(\dot{\mathbf{r}}=\mathbf{f}(\mathbf{r})\). In [16], comparative results of a numerical study of the proposed control laws are presented, and further in [31], the main method for constructing orbits with different Cartesian coordinates is presented. The numerical study illustrates the accuracy with which the coordinate transformation occurs after linearization. A hybrid law with continuous feedback is proposed, in which the desired orbit geometry is explicitly given in the form of a control error, and the actual orbit is given in local Cartesian coordinates. Numerical simulations illustrate the efficiency and limitations of such feedback control laws. Using a linearized mapping between relative orbital coordinates results in a slight degradation in quality. However, this method gives the advantage of working in the main space of elements in the case when the relative errors in the construction of the orbit are determined. One of the fundamental differences between satellite movement and conventional rendezvous operations, as noted in [32], is the need to increase the duration of the movement. Therefore, if the formation is to be maintained, then long-term disturbances, in particular those caused by the Earth's compression, must be corrected. [PERSON] and [PERSON] [32] explored the use of motionless aids to counteract the differential precession of the orbital plane, which is the primary perturbation affecting the formation caused by the Earth's compression. The approach used is to use the pressure of solar radiation acting on a relatively small surface called the solar wing, which is attached to the satellite. The resulting torque causes the orbit to precess; if the wing is the correct size, this motion will, on average, suppress the motion caused by flattening, thereby maintaining the formation without using propellant. Ref. [PERSON] and [PERSON] [32] considered satellites that are nominally Earth-oriented; most small satellites that are stabilized by the gravitational field gradient or have sensors designed to observe the Earth fall into this category. It is assumed that a relatively small reflective surface is attached to the spacecraft at an angle \(\alpha\) (usually \(45^{\circ}\)) to the velocity vector. The reflector surface illuminated by the Sun will alternate as the satellite moves around its orbit, creating an out-of-plane component of the alternating solar radiation force. Consequently, the torque that is created in orbit by this force will tend to increase rather than decrease during the entire revolution, leading to a total torque along the direction to the Sun. If the wing area is appropriately sized, this torque can be used to zero out the differential nodal regression velocity between the two orbits, thus maintaining the formation without a propulsion system.