Partial Derivatives of the Solution to the Lambert Boundary Value Problem

Two methods for deriving first-order partial derivatives of the outputs with respect to the inputs of the Lambert boundary value problem are presented. The first method assumes the Lambert problem is solved via the universal vercosine formulation. Taking advantage of inherent symmetries and intermediate variables, the derivatives are expressed in a computationally efficient form. The typical added cost of computing these partials is found to be to 35% of the Lambert computed cost. A second set of the same partial derivatives is derived from the fundamental perturbation matrix, also known as the state transition matrix of the Keplerian initial value problem. The equations are formulated in terms of Battin’s partitions of the state transition matrix and its adjoint. This alternative approach works with any Lambert formulation, including one that solves a perturbed Lambert problem, subject to the availability of the associated state transition matrix. The analytic partial derivatives enable fast trajectory optimization formulations that implicitly enforce continuity constraints via embedded Lambert problems.