Representation of Probability Density Functions from Orbit Determination using the Particle Filter

NASA Goddard Space Flight Center, Greenbelt, MD 20771, 301-286-7526,james.r.carpenter@nasa.govAbstract: Statistical orbit determination enables us to obtain estimates of the state and thestatistical information of its region of uncertainty. In order to obtain an accurate representation ofthe probability density function (PDF) that incorporates higher order statistical information, wepropose the use of nonlinear estimation methods such as the Particle Filter. The Particle Filter (PF)is capable of providing a PDF representation of the state estimates whose accuracy is dependent onthe number of particles or samples used. For this method to be applicable to real case scenarios,we need a way of accurately representing the PDF in a compressed manner with little informationloss. Hence we propose using the Independent Component Analysis (ICA) as a non-Gaussiandimensional reduction method that is capable of maintaining higher order statistical informationobtained using the PF. Methods such as the Principal Component Analysis (PCA) are based onutilizing up to second order statistics, hence will not suffice in maintaining maximum informationcontent. Both the PCA and the ICA are applied to two scenarios that involve a highly eccentricorbit with a lower apriori uncertainty covariance and a less eccentric orbit with a higher a prioriuncertainty covariance, to illustrate the capability of the ICA in relation to the PCA.Keywords: Orbit Determination, Particle Filter, Non-Gaussian, Data Compression, NonlinearEstimation.1. IntroductionThe Statistical Orbit Determination (OD) problem involves the estimation of the states of a spaceobject based on noisy measurements. In recent years, there has been a dramatic increase of spaceobjects (assets and space debris) particularly in Low Earth Orbit (LEO) [?]. This increase of spaceobjects pose a threat to the space assets based on potential collisions as well as the increase inoperational costs during maneuvers required to avoid collisions. For instance, in 2009, CelesTrakpredicted that an Iridium Satellite and the defunct Russian Satellite Cosmos, would have a closeapproach of 584 meters [?, ?], nevertheless they collided with one another. Since 1998, the ISS hasperformed 13 maneuvers to avoid collisions with space debris [?]. The last maneuver was in 2009,performed to avoid the debris from the Iridium and Cosmos satellite collisions. This required 30hours to plan and execute [?] as well as the cost of propellant for the delta-V maneuver implemented.The classical methods of statistical OD, such as the Extended Kalman Filter (EKF) [?] and theUnscented Kalman Filter (UKF) [?], may not always be the best choice of an estimator for allproblems because they only consider the second moments of the state. As long as the observationerror and process noise can be accurately assumed to have a Gaussian distribution, these secondmoments are sufficient to infer all other statistics as well. One example in which the non-Gaussianerrors could arise is when observations are based upon short arcs of tracking data when tracking1