Newton–Kantorovich/Pseudospectral Solution to Perturbed Astrodynamic Two-Point Boundary-Value Problems

Abstract

The Newton–Kantorovich/Chebyshev pseudospectral method is systematically investigated to solve perturbed astrodynamic two-point boundary-value problems. Because many unperturbed astrodynamic problems have been solved and the perturbations are relatively small, the unperturbed solutions can serve as good initial approximations. The differential equations describing the perturbed problem are linearized about the nominal (approximate) solution, and the pseudospectral method is used to solve the linearized differential equations. By successively updating the nominal solution with the Newton–Kantorovich approach, the linearized solving procedure is iterated to achieve high-order precision. Because no numerical integration is needed and only linear algebraic equations are solved, the method is computationally efficient. The perturbed Lambert’s problem, the relative motion of spacecraft, and the optimal control of spacecraft formation flight are solved. High-precision solutions are obtained with only a few iterations. A convergence analysis shows that the method converges for relatively long times of flight. For the spacecraft formation flight applications, the simulation results show that the method is applicable even for large interspacecraft separations of hundreds of kilometers.