Mark Vincent, Raytheon
Keywords: Covariance propagation, Liouville’s Theorem, Probability of Collision
Abstract:
During an investigation1 of the possibility of using Keplerian or Equinoctial orbital elements to calculate the Probability of Collision (Pc), the need to understand the evolution of the error covariance matrix became apparent. One aspect of this was that there appeared to be a paradox that both the positional and velocity uncertainties increase in time while the total error volume in 6-D phase space remains constant, for conservative system (in accordance with Liouvilles Theorem). This paradox can be explained by starting with a very simple model (Fig. 1) and then applying similar concepts to higher degree systems and force models.
Fig. 1. Preserving Phase Space Volume in Time in the case of a Simple No-acceleration 2-D Model
In the figure the possible initial position and velocity conditions at the starting epoch are represented by the shaded square with +/- unit values at the corners. The position at time t is simply the initial position plus the constant value of the initial velocity times t. All possible position/velocity values at time t are represented by the parallelogram. Thus the possible range or uncertainty of the position has increased (while the uncertainty in velocity has remained the same) however the area of possible phase space values has remained the same.
The evolutions in time of the more complex models (both conservative and non-conservative) were compared to real-life data from non-conservative systems. This led to a new examination of the parabolic and periodic terms in the size of the 6-D volume in the latter and their effect on Pc. The conclusions that are drawn are used to clarify the on-going debate between 3-D and 6-D methods to calculate Pc, including the 6-D process of Coppola2. Finally, this new understanding of the covariance behavior is incorporated into the study of the potential advantages of using Keplerian/ Equinoctial elements. Preliminary results using the error state transition matrices from Vallado3 (p. 811) show the superior behavior for the Keplerian representation in maintaining the 6-D phase space volume compared to the truncation errors inherent in the Cartesian representation. Whether or not Unscented Transforms and non-linearized propagations remove the abovementioned periodic effects is also investigated.
Vincent, M.A., Strengthening the Bridge between Academia and Operations for Orbital Debris Risk Mitigation, AMOS Conference, Maui HI, 2017.
Coppola, V.T., Including Velocity Uncertainty in the Probability of Collision between Space Objects, AAS Conference, Charleston SC, 2012.
Vallado, D.A., Fundamentals of Astrodynamics and Applications, 4th Edition, Microcosm Press, 2013.
Copyright © 2018 Raytheon Company. All rights reserved.
Date of Conference: September 11-14, 2018
Track: Poster