The Journal of the Astronautical Sciences https://doi.org/10.1007/s40295-018-0137-9 Higher Order Algorithm for Solving Lambert’s Problem Mohammad Alhulayil 1 · Ahmad Bani Younes 2 · James D. Turner 3 © American Astronautical Society 2018 Abstract This work presents a high-order perturbation expansion method for solving Lam- bert’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 deriva- tive 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 pre- sented 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. Keywords Lambert’s problem · Computational differentiation · p-iteration  Mohammad Alhulayil mohammad.alhulayil@hotmail.com Ahmad Bani Younes abaniyounes@sdsu.edu James D. Turner turner@aero.tamu.edu 1 Aerospace Engineering Department, Khalifa University, Abu Dhabi, UAE 2 Department of Aerospace Engineering, San Diego State University, San Diego, CA, USA 3 745H H.R. Bright Building, Manchester, UK