Higher Order Algorithm for Solving Lambert’s Problem

• Published in: The Journal of the astronautical sciences Vol. 65; no. 4; pp. 400 - 422
• Main Authors: Alhulayil, Mohammad, Younes, Ahmad Bani, Turner, James D.
• Format: Journal Article
• Language: English
• Published: New York Springer US 15.12.2018; Springer Nature B.V
• Subjects: Aerospace Technology and Astronautics; Algorithms; Convergence; Engineering; Equations of motion; Iterative methods; Mathematical Applications in the Physical Sciences; Mathematical models; Newton methods; Numerical integration; Perturbation methods; Software; Space Exploration and Astronautics; Space Sciences (including Extraterrestrial Physics; Taylor series; Tensors; Thermal expansion; p-iteration; Lambert’s problem; Computational differentiation
• ISSN: 0021-9142; 2195-0571
• DOI: 10.1007/s40295-018-0137-9

This work presents a high-order perturbation expansion method for solving Lambert’s problem. The necessary condition for the problem is defined by a fourth-order Taylor expansion of the terminal error vector. The Taylor expansion partial derivative models are generated by Computational Differentiation (CD) tools. A novel derivative enhanced numerical integration algorithm is presented for computing nonlinear state transition tensors, where only the equation of motion is coded. A high-order successive approximation algorithm is presented for inverting the problems nonlinear necessary condition. Closed-form expressions are obtained for the first, second,third, and fourth order perturbation expansion coefficients. Numerical results are presented that compare the convergence rate and accuracy of first-through fourth-order expansions. The initial p-iteration starting guess is used as the Lambert’s algorithm initial condition. Numerical experiments demonstrate that accelerated convergence is achieved for the second-, third-, and fourth-order expansions, when compared to a classical first-order Newton method.