Covariance analysis of Lambert's problem via Lagrange's transfer-time formulation

An analytical linear covariance prediction is formulated for Lambert's boundary value problem with navigation errors. The required initial velocity impulse in orbit targeting problems is computed under the initial and final position vector errors, the initial velocity vector error and the transfer time error. These errors are assumed to satisfy Gaussian distributions. The uniqueness of our approach is that first-order partial derivatives of the outputs (boundary velocity vectors) are derived with respect to the Lambert inputs (boundary position vectors and transfer time) using the Lagrange's transfer-time formulation and the Lagrangian F&G solution. This allows establishment of a direct relationship between the partial derivatives for boundary value problems and classical state transition matrix for initial value problems. Then linear covariance matrices of terminal position and velocity vectors are derived. The case of hyperbolic transfer is also studied. Numerical simulations are presented to illustrate and verify the proposed analytical linear covariance technique using Monte Carlo error distributions.