Analytical Uncertainty Propagation for Satellite Relative Motion along Elliptic Orbits

For satellites flying in close proximity, monitoring the uncertainties of neighboring satellites’ states is a crucial task because the uncertainty information is used to compute the collision probability between satellites with the objective of collision avoidance.

In this study, an analytical closed-form solution is developed for uncertainty propagation in the satellite relative motion near general elliptic orbits.

• The Tschauner–Hempel equations are used to describe the linearized relative motion of the deputy satellite where the chief orbit is eccentric.

• Under the assumption of the linearized relative motion and white Gaussian process noise, the uncertainty propagation problem is defined to compute the mean and covariance matrix of the relative states of the deputy satellite.

• The evolution of the mean and covariance matrix is governed by a linear time-varying differential equation, for which the solution requires the integration of the quadratic function of the inverse of the fundamental matrix associated to the Tschauner–Hempel equations.
• The difficulties in evaluating the integration are alleviated by the introduction of an adjoint system to the Tschauner–Hempel equations and the binomial series expansion.

The accuracy of the developed analytical solution is demonstrated in illustrative numerical examples by comparison with a Monte Carlo analysis. The figure below represents the evolution of the probability ellipsoid:

• proposed analytical solution (solid line),
• analytical solution using the Clohessy-Wiltshire (CW) equations (dashed line), and
• Monte Carlo simulation using 500 samples (dots)

=> The proposed analytical solution (solid line) can accurately predict the probability distribution of the states (dots) obtained from Monte Carlo simulation. On the other hand, the solutions using the CWequations fail to capture the distribution. This is because the CWequations assume a circular chief orbit; thus, when applied to the case where the chief orbit is elliptic, their accuracy significantly decreases.

Related Publication

• S. Lee, H. Lyu, and I. Hwang, “Analytical Uncertainty Propagation for Satellite Relative Motion along Elliptic Orbits,” AIAA Journal of Guidance, Control and Dynamics, Vol.39(7), pp. 1593-1601, July 2016,