Partial Derivatives of the Solution to the Lambert Boundary Value Problem

Abstract

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 ∼15 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. Nomenclature a, A = notation for vector (column) and matrix, respectively a b = ∂a∕∂b, where a and b are general variables d = transfer angle parameter (1 for 0 < θ < π, −1 for π < θ < 2π) E = eccentric anomaly F = hyperbolic eccentric anomaly f = Lagrange function (part of f and g functions) g = Lagrange function (part of f and g functions), TU k = universal variable for solving Lambert equation using vercosine formulation k = value of k corresponding to T , TU LU = length unit N rev = number of revolutions q = generic variable name representing a scalar function r 1 ; r 2 = initial and final position vectors, respectively (input to Lambert problem), LU S = geometry parameter appearing in Lambert equation, TU T p = parabolic time of flight, TU T = target time of flight (input to Lambert problem), TU t 1 ; t 2 = initial and final times, respectively, TU TU = time unit v 1 ; v 2 = initial and final velocity vectors, respectively (output to Lambert problem), LU/TU W = auxiliary function appearing in Lambert equation δ; d = variation and differential operators, respectively ϵ = small numerical value (e.g., 0.02) θ = transfer angle μ = gravitational parameter of the primary, LU 3 ∕TU 2 τ = geometry parameter appearing in Lambert equation Φt 2 ; t 1 = state transition matrix from time t 1 to t 2 , with first quadrant units of TU and third quadrant units of TU −1 Subscript LT = expression subjected to the Lambert targeting constraints

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