Sven K. Flegel, Space Environment Research Centre; James Bennett, EOS Space Systems
Keywords: State uncertainty, normality, Henze-Zirkler, entropy
Abstract:
The state uncertainty of space objects is used by different fields including collision probability estimation, manoeuvre detection, track association, multi-target tracking and sensor scheduling. Each of these areas has its own accuracy and realism requirements of the uncertainty. As traditional orbit determination methods produce a state fit to observations based on root-mean-square minimisation, the state uncertainty at and for a time after epoch is Gaussian. Methods building on this information therefore typically rely on this distribution. However, over time normality breaks down, impacting the applicability of these methods. The aim of the current work is to perform an automatic assessment of the timeframe until breakdown of Gaussianity of the state uncertainty of all orbit determination results, which are introduced into the catalogue being set up by the Space Environment Research Centre (SERC Limited). Supplying this information as meta-data will help users chose appropriate methods when working with the state uncertainty supplied through the catalogue.
In the current paper, two fundamentally different approaches are outlined to determine normality of state uncertainty. The first method is the Henze-Zirkler test, which belongs to the family of consistent approaches and operates on a random particle sample. As the test result varies based on the given sample, rigorous assessment requires Monte-Carlo based analyses. Furthermore, non-linearities in the relative motion of particles increase with distance. This creates a dependence of the test results at predicted epochs on the compactness of the initial sample, which translates into sample size. The final test result is based on a hypothesis test for which the confidence level is set to the commonly accepted value of 95 per cent. As the method does not rely on any other assumptions, it is applied here as a baseline for comparison. The second test compares the entropies of the covariances obtained with a linearized and a non-linearized prediction method. The linearized solution is obtained using the error state transition matrix. The non-linearized prediction employs an unscented transform. The states obtained thereby are propagated individually to a common epoch and an updated state covariance is derived from them succinctly. This test does not require Monte-Carlo computation and is therefore more computationally efficient. It does however employ a tuneable decision threshold parameter, which is calibrated here by comparing the two methods based on their results for a select number of test cases. The comparison is performed on the state uncertainty in cartesian space. For the calibration, state and uncertainty predictions rely on two-body motion and assume no process noise. In the comparison, an emphasis is put on the time after epoch at which non-Gaussianity is first detected. The perturbed state prediction is then used for the entropy based test and its results are compared to those of the calibration cases. The paper concludes with a first look at the validity time frame for the Gaussian assumption in state uncertainties from orbit determination results contained in SERCs catalogue.
Date of Conference: September 11-14, 2018
Track: Astrodynamics