Uncertain Lambert Problem

Abstract

This analysis addresses the problem of constructing uncertainties in initial and final satellite orbital state vectors that are dynamically consistent in Lambert’s boundary-value problem, given uncertainties in the initial and final position vectors. The structure of the general nonlinear problem is discussed in terms of transformations of the probability density functions of the given positions. For the linear (first-order) variational version of the problem, the covariance matrices of the state variations are presented in explicit terms of the covariance matrix of the given position variations and partitions of the state transition matrix of the nominal orbit. Equivalence is demonstrated between the linear variational results, which involve no iteration, and a weighted batch least-squares differential-correction solution of the orbit determination problem. To illustrate the utility of the working formulas in the linear variational problem, two approaches are taken: 1) an analytic approach to solve directly for the covariance terms, based on linearized dynamics, and 2) a numerical Monte Carlo-based approach that generates a sample covariance. The numerical results for state covariance from the analytic approach agree with results obtained by batch least-squares differential correction, and both agree with results from the Monte Carlo approach to within differences explainable by the limitations of sampling.