Fuzzy logic control system for formation flying

Pedro A.Capó-Lugo, Peter M.Bainum

Howard University

Washington, D.C. 20059, USA

The reconfiguration procedure for the NASA Benchmark Tetrahedron Constellation is complex and may require many mathematical computations to determine an optimal solution for the problem. There are different papers and works, that are showing a numerical approach to solve the deployment and reconfiguration procedure of a tetrahedron constellation, but this mathematical approach is complicated and may take a lot of computational time for the solution of the problem; an optimal reconfiguration of the tetrahedron constellation using the Gauss Pseudospectral Method, but, once more, the mathematical approach may take a long computational time before an actual solution for the system is available; that have tabulated for a simple case the computational time by using the Jacobi Pseudospectral Method, but, in comparison with the numerical procedures of the first two references, the Pseudospectral Methods may take a longer period of time before an actual solution for the reconfiguration procedure is obtained.

The objective of this paper is to present a control system that does not depend on a mathematical approach, but is based on the knowledge extracted from the differential equations. This control scheme can be designed with the use of intelligent systems technology. In recent years, the emergence of intelligent systems revolutionizes the area of control systems. These intelligent systems adapt, learn, and take different decisions based on the particular problem. Among these intelligent systems are the fuzzy logic (FL), neural networks (NN), and genetic algorithms (GA). These systems are widely used in industrial, pattern recognition, and image recognition applications.

The NN is defined as a model of reasoning based on the human brain and learns from experience. GA uses an evolutionary approach to learn through the process, and the evolution consists of selection, mutation and reproduction. The NN and GA use the basis of machine learning which involves an adaptive mechanism that enables computers to learn by example and by analogy. For this reason, NN and GA are mostly used in adaptive control systems, but the disadvantage in both systems is that the data must be known before both systems are used in the actual process, because the NN and GA systems involve training.

FL provides the perfect scenario to be applied as a non-linear control because it is an intelligent system that does not require previous knowledge of the data and depends on the conditions and conclusions imposed on the system. To design a FL system, a data mining process must be implemented. This process is performed to extract data and knowledge from equations, pictures, or data base. Since no complex mathematical approach is needed to develop the fuzzy control system, FL can be used as a non-linear control to reconfigure the NASA Benchmark Tetrahedron Constellation. FL is also used in industrial applications, but, in this paper, it will be used for the first time for a formation flying reconfiguration procedure.

The authors studied the dynamics and control of the NASA Benchmark Tetrahedron Constellation, and before the initial conditions for the tetrahedron constellation were defined for every specific size. This paper will emphasize the reconfiguration procedure and the development of a FL controller between two specific sizes of the NASA Benchmark Tetrahedron Constellation. To reach the next specific size, the dynamics for a single satellite will be given by the Lagrange Planetary equations.

The Lagrange Planetary (LP) equations describe a set of highly non-linear equations. љData must be extracted from these equations to develop the FL control system. This process of extraction of data is called data mining. Through the data mining process, it is found that there are relations between the magnitude of the acceleration thrust and the behavior of the orbital element. In the reconfiguration procedure between the first two specific sizes of the NASA Benchmark Tetrahedron Constellation, an out of plane motion is not needed since the constellation is required to maintain the same inclination angle, but the in-plane orbital elements (semi-major axis and eccentricity) must be changed to a different orbital dimension. The only orbital element that will be varying with respect to an impulse maneuver is the argument of perigee, but it is found through the data mining process that a small impulse can be applied at the perigee point to correct this orbital element to its desired angle.

The fuzzy logic (FL) controller is an intelligent system that can be applied to any mechanical system. For the first time, this paper shows that the FL controller can be used for a space application in which the controller can reconfigure a single satellite and can be applied to the tetrahedron constellation. This system can also be classified as a real system because it takes decisions based on the actual values entered into the fuzzy system. When this controller is applied to the Lagrange Planetary equations, the desired coordinates are satisfied for a small error.

In conclusion, this paper contributes, for the first time, to the reconfiguration process of the NASA Benchmark Tetrahedron Constellation and shows the effectiveness of the fuzzy system to solve the Lagrange Planetary equations. Also, the system can be implemented very easily into a digital computer because it does not required large amounts of mathematical computations.