On the solution to every Lambert problem

Abstract

Lambert’s problem is the two-point boundary value problem for Keplerian dynamics. The parameter and solution space is surveyed for both the zero- and multiple-revolution problems, including a detailed look at the stress cases that typically plague Lambert solvers. The problem domain, independent of formulation, is shown to be rectangular for each revolution case, making the elusive initial guess and the solution itself amenable for interpolation. Biquintic splines are implemented to achieve continuous derivatives and quick evaluation. Resulting functions may be used directly as low-fidelity solutions or used with a single update iteration without safeguards. A concise, improved vercosine formulation of the Lambert problem is presented, including new singularity-free and precision-saving equations. The interpolation scheme is applied for up to 100 revolutions. The domain considered includes all practically conceivable flight times, and every possible geometry except a small region near the only physical singularity of the problem: the equal terminal vector case. The solutions are archived and benchmarked for accuracy, memory footprint, and speed. For typical scenarios, users can expect $$\sim 6$$ or more digits of velocity vector accuracy using an interpolated solution without iteration. Using a single, unguarded iteration leads to solutions with near machine precision accuracy over the full domain, including the most extreme scenarios. Depending on desired resolution, coefficient files vary in size from $$\sim 3$$ to 65 MB for each revolution case. Evaluation runtimes vary from $$\sim 2$$ to 5 times faster than the industry benchmark Gooding algorithm. The coefficient files and driver routines are provided online. While the method is currently demonstrated on the vercosine formulation, the 2D interpolation scheme stands to benefit all Lambert problem formulations.