Partial Derivatives of the Lambert Problem

A procedure for deriving analytic partial derivatives of the Lambert problem is presented. Using the universal, cosine based Lambert formulation; first order partial derivatives of the velocities with respect to the positions and times are developed. Taking advantage of inherent symmetries and intermediate variables, the derivatives are expressed in a computationally efficient form. The added cost of computing these partials is found to be ∼10% to ∼60% of the Lambert compute cost. The availability of analytic partial derivatives increases optimization speed, efficiency and allows for trajectory optimization formulations that implicitly enforce continuity constraints via embedded Lambert problems.