# Modeling and Control of Satellite Formations: A Survey
## 4 Leader-Follower Formation Architecture
The agent's kinematics are described by drift-free controlled distributions [115, 116] on manifolds \(M_{i}\) as  
\[\Delta_{i} : M_{i}\times U_{i}\to M_{i},\] \[\Delta_{i} = \sum_{j}X_{j}u_{j}, \tag{47}\]  
where \(U_{i}\) is the control space, the vector fields \(X_{i}\) form the distribution basis. Controlled distributions are general enough to describe an underactuatio. The formation of a group of agents is determined by its graph, which completely describes the kinematics of individual agents and global inter-agency constraints. The graph of the formation \(F=(V,E,C)\), consists of: a finite set \(V\) of vortices, the cardinality of which is number of agents, where each vertex \(v_{i}:M_{i}\times U_{i}\to TM_{i}\) is a distribution \(\Delta_{i}\) describing kinematics of each individual agent according to (47); binary relation \(E\subset V\times V\), representing communication between agents; families of constraints \(C\) indexed by the set \(E\), \(C=\{c_{e}\}_{e\in E}\). Two different types of formation graphs are considered: undirected formations, where \((V,E)\) is an undirected graph, and directed formations, where \((V,E)\) is a directed graph. In [114], conditions are obtained for determining formation feasibility for two types of formations, and also a management abstraction for a group is found, which allows one to describe a formation as a single object. When directional formation is not feasible it is possible to extract a feasible formation by eliminating the degrees of freedom that cannot be worked out by the follower agents. In [117], nonlinear equations of orbital dynamics of relative motion are introduced for the problem of motion in a formation with separation into constant distances. This general equation of orbital dynamics allows for elliptical, non-coplanar maneuvers at large distances between spacecraft, as well as typical near-circular, coplanar maneuvers at short distances. In addition, changes in the equations of position and velocity have been introduced for a path in the plane of a formation with a large, constant separation angle between satellites. A nonlinear control method with the Riccati equation depending on the state is used to solve the problem of controlling the movement of the formation. This new control method for a non-linear system provides a clear design trade-off between control action and state error, similar to the classical linear quadratic control method. Numerical modeling shows the effectiveness of the new control method with the Riccati equation depending on the state with the developed equations of relative motion. [PERSON] [118] presented a general analytical formulation of the problem for optimal trajectories of satellites flying in a formation, based on the circular version of [PERSON]'s problem. Optimization is performed to minimize the displacement energy received from the motor. The optimization problem is the choice of the trajectory along which the satellite should move during the maneuver, while the required time and boundary conditions are fixed. Low thrust engines are suitable for maneuvering in formation. It is assumed that the engine is running throughout the entire maneuver, and the optimal program calculates the magnitude and direction of the thrust as a function of time. The possibility of using your own dynamics to obtain additional fuel economy by changing the boundary conditions is considered. A method for constructing a satellite trajectory capable of creating a formation of a given configuration from identical spacecraft is given in [119]. The method uses a behavior-based approach to implement autonomous and distributed control of relative configuration using limited sensor information. For each satellite, the desired speed is defined as the sum of the various components of the general high-level behavior patterns. Further, the model of behavior is determined by an inverse dynamic calculation called the formation of equilibrium. Introduced several feedbacks for speed control.