Linearized Lambert’s Problem Solution

Abstract

Lambert’s problem is the well-known problem to determine the orbit that allows an object to travel between two given position vectors in a given time of flight under Keplerian dynamics. One drawback of solving this problem is that it requires solving a transcendental equation, which involves commonly using an iterative method that can be undesirable for autonomous onboard applications. In this paper, we present a linearization to the solution of Lambert's problem using Lagrange parameters. It is shown that, around a wide variety of nominal transfer trajectories, high-accuracy solutions to the neighboring transfers can be rapidly determined using this linearized solution. Sensitivity studies are presented that show that errors in the terminal velocities are typically well below 1% for cases with 5% perturbations in the terminal positions and time of transfer. Unlike many similar solutions for the relative Lambert’s problem, the linearized results presented here do not suffer in general when the nominal transfer trajectory has high eccentricity. Examples are presented that demonstrate that this linearization can be used effectively for targeting and orbit determination applications. Important for possible onboard applications, it is shown that the cost of computing the partials needed to create a linearized solution is very cheap; in the case tested, computation of the partials takes only 0.1% of the time needed to compute a full Lambert’s solution.