Blair Thompson, U. S. Air Force 319th Combat Training Squadron; Jesse Gossner, U. S. Air Force 319th Combat Training Squadron; Brandon Sais, U. S. Air Force 319th Combat Training Squadron; Elizabeth Cunningham, U. S. Air Force 319th Combat Training Squadron
Keywords: two-line elements, covariance, space situational awareness
Abstract:
We present a method for quickly propagating the covariance associated with a two-line element set (TLE). Based on general perturbation theory, TLEs are reasonably accurate and propagate the orbital state of a resident space object (RSO) relatively quickly over long periods of time, making them useful for certain first-order space situational awareness (SSA) analyses. However, TLEs are generally not published with covariance or other measures of uncertainty, making it difficult to model and determine uncertainty of the analyses results. In cases where the TLE covariance is available or can be estimated by some indirect method, the traditional method of propagating the covariance is numerical integration of the state transition matrix (STM) which involves integration of forty-two equations of motion, six equations for the orbital state (position and velocity) and thirty-six equations for the six-by-six state transition matrix. The numerical integration time significantly limits the usefulness for SSA purposes, especially when longer time spans or multiple space objects are involved. As a result, this type of processing loses much of the advantage of general perturbations techniques, namely generally faster state propagation. Although TLE’s operate on a specific type of mean orbital elements as the input, propagation tools such as Simplified General Perturbations 4 (SGP4) typically generate for output a Cartesian state vector (i.e., position and velocity) in earth-centered inertial (ECI) coordinates. A state estimator (i.e., filter) processing observations to generate a TLE would typically generate the covariance of the mean orbital elements of the TLE, which is generally not as useful as a Cartesian covariance which can be transformed into a three-dimensional position probability ellipsoid for uses such as computing probability of collision. The Cartesian covariance is more useful and preferred over the mean orbital elements covariance.
Because Lambert targeting is the solution of the fundamental boundary value problem of astrodynamics, a Lambert targeting routine can be used to estimate the effects of small position and velocity errors at some reference time on the errors at some future time, which is the essence of the state transition matrix. We use Richard Battin’s Lambert routine because it is universal, robust, and was designed for quick convergence (minimum iterations). Because any practical conjunctional analysis would generally work over multiple days of orbit and covariance predictions, we adapted, in part, Loechler’s extension to Battin’s method to allow for multi-revolution solutions with no increase in computing time.
The method presented in this paper quickly propagates covariance by computing the STM using a Lambert targeting routine in lieu of numerical integration, enabling fast multi-rev (even multi-day) analyses commensurate with TLE accuracy. Furthermore, the fast method enables the estimation of state covariance of space objects from historical TLE data. Covariance of a set of TLEs can be estimated at a future epoch be propagating all previous TLEs to that future time and computing the statistics of the resulting position and velocity vector sets. This future covariance estimate can be propagated back in time to some reference epoch using the new, fast method presented in this paper. The back-propagated covariance can then be transformed into the radial, in-track, cross-track coordinate system, and applied to the TLE at any epoch time as a representative initial state covariance for that object.
Date of Conference: September 17-20, 2019
Track: Astrodynamics