Complete Lambert Solver Including Second-Order Sensitivities

Abstract

The complete parameter space for Lambert’s problem includes all possible geometries, flight times, and revolutions. An earlier work exploited a rectangular form of this parameter space to interpolate the solution for each revolution. Here, new variable transformations are introduced to efficiently allow for any number of revolutions. The iteration variable is fit using the new transformations and customized tree data structures, leading to a single 1-megabyte coefficient file. The interpolated initial guess seeds a root-solver that typically converges in 1–2 iterations. The new vercosine formulation solver is rigorously tested across the complete domain. Compared to Gooding’s state-of-the-art method, the solver is 1.7–2.5 times faster and maintains accuracy even for the so-called ultra-revolution problem. First- and second-order sensitivities of the output terminal velocities with respect to the input flight time and terminal positions are derived via a general method for differentiating a root-solved process. The new sensitivity expressions are exact and 6–17 times faster to compute than finite difference approximations. The new quadratic model of the Lambert problem solution results from an optional postprocessing step and is useful for a variety of estimation and optimization applications.