Satellite constellation design using the q-G global optimization method

The field of optimization is interdisciplinary in nature, and it is an indispensable tool in various fields of science and engineering. Computational intelligence-based techniques, such as neural networks, simulated annealing, stochastic machines, genetic algorithms and others, have been proven to be effective in solving global optimization problems. In this work, the q-Gradient (q-G) method is applied for solving a satellite constellation design problem for two regional navigation satellite systems, one for covering the Brazilian territory and the other for South America. The q-G method is a global optimization algorithm based on the concepts of q-calculus and simulated annealing. The main idea behind the q-G method is the use of the q-gradient vector of the objective function as the search direction. The q-gradient vector is a generalization of the classical gradient vector based on the concept of Jackson´s derivative and its use provides the algorithm an effective mechanism for escaping from local minima. The algorithm has three free parameters, and it is implemented so that the search process gradually shifts from global exploration in the beginning to almost local search in the end. The q-G method was applied on six 10-D multimodal test functions, and its performance was compared with eleven Evolutionary Algorithms (EAs). The q-G method performed well against the EAs arriving in forth position in a direct comparison with them. For the satellite constellation design problem, we are using the basic design assumption of using four satellites in geosynchronous orbits for covering both Brazil and South America. This optimization problem has 12 design variables (12-D) and includes calculations of the Geometric Dilution of Precision (GDOP) as the main metric for the design of the satellite constellations. Comparison of the results with a previous study for the Brazilian constellation indicates that the q-G method is able to solve this problem and that there is considerable room for improvement with the use of an additional satellite for the South America case study.