Legendre Cooperative PSO Strategies for Trajectory Optimization

Particle swarm optimization (PSO) is a population-based stochastic optimization technique in a smooth search space. However, in a category of trajectory optimization problem with arbitrary final time and multiple control variables, the smoothness of variables cannot be satisfied since the linear interpolation is widely used. In the paper, a novel Legendre cooperative PSO (LCPSO) is proposed by introducing Legendre orthogonal polynomials instead of the linear interpolation. An additional control variable is introduced to transcribe the original optimal problem with arbitrary final time to the fixed one. Then, a practical fast one-dimensional interval search algorithm is designed to optimize the additional control variable. Furthermore, to improve the convergence and prevent explosion of the LCPSO, a theorem on how to determine the boundaries of the coefficient of polynomials is given and proven. Finally, in the numeral simulations, compared with the ordinary PSO and other typical intelligent optimization algorithms GA and DE, the proposed LCPSO has traits of lower dimension, faster speed of convergence, and higher accuracy, while providing smoother control variables.