Laura Pirovano, University of Surrey; Daniele Antonio Santeramo, Politecnico di Milano; Alexander Wittig, University of Southampton; Roberto Armellin, Surrey Space Centre; Pierluigi Di Lizia, Politecnico di Milano
Keywords: Data Association, Initial Orbit Determination, Differential Algebra, Space Debris
Abstract:
The problem of determining the state of resident space objects (RSOs) is fundamental to maintain a collision-free environment in space, predict space events and perform activities. Due to the development of new technologies and the ever-growing number of RSOs, the number of observations available is increasing by the day. This calls for more efficient methods able to deal with the amount of data produced. Furthermore, when surveying the sky, the short-arc nature of the observations does not allow for precise orbit determination during a single passage of the object over an observing station: being the detections very close in time, little is known about the curvature of the orbit. Thus, for each observation there is more than one orbit that complies with the observation values. The set of admissible solutions corresponding to a single observational arc is here called the Orbit Set. To reduce the uncertainty on the solution and pinpoint the correct orbit associated with the observation, one needs other independent observations of the same object. The main challenge in this, however, is to determine whether two or more observations pertain to the same object, thus whether they are correlated. This is the problem of data association.
Current approaches for the data association problem suffer from either high computational effort, due to the point-wise based algorithms to keep high accuracy on the dynamical model, or low accuracy, due to the use of simplified dynamics to keep the problem (semi-)analytical and computationally efficient. We propose a novel approach that exploits Differential Algebra (DA) to overcome the issues from literature. DA is a computing technique that uses truncated power series (TPS) instead of numbers to represent variables, thus allowing us to obtain results of complicated functions as high-order Taylor polynomials. For the data association problem, we can thus obtain the definition of the Orbit Set with respect to the observation uncertainty in an analytical form, without the need for point-wise sampling, and propagate the polynomial form independently of the dynamics chosen, thus allowing us to keep accurate dynamics.
The technique consists of three different steps. Supposing an observation is available at a first epoch, in the first step a Differential Algebra Initial Orbit Determination (DAIOD) algorithm is developed to compute the state of the orbiting body. The solution is obtained as a Taylor polynomial that links the uncertainty in the observation to the state of the orbiting body. In this way, by evaluating the polynomial for admissible values of the observations, one obtains a range of possible solutions for the IOD problem, that is the Orbit Set (OS). The accuracy of the Taylor expansion is managed by the Automatic Domain Splitting (ADS) tool which estimates and controls the truncation error of the polynomial.
For the second step, the state computed at the initial epoch is propagated to a second epoch where a new observation is available. The state is then projected onto the second observation domain, to obtain the observations at the second epoch compatible with the propagated state. DA and ADS are also used during the propagation: the former allows the uncertainties to be propagated to the final epoch keeping a Taylor representation, while the latter manages the truncation error. The propagation can be performed with any dynamics desired. Thus, thanks to the use of DA, at the end of this step one has a function that maps the initial observation domain onto the observation domain at the second epoch, allowing for a direct search for overlapping regions, that is possible correlation.
The third step determines the correlation probability of two observation sets, that is the probability that two observations belong to the same object, by exploiting the Subset Simulation (SS). The observation domain is sampled at the first epoch and directly propagated to the second epoch by means of a simple polynomial evaluation, thanks to the analytical function created. The samples created with Markov Chains are ranked using the sum of weighted square residuals. By describing the angular uncertainties in the observation as independent Gaussian variables, the performance function behaves like a Chi-square, allowing us to perform statistical tests to determine the probability.
Unfortunately, not all observations allow for IOD to be performed. For this reason, an alternative way to treat observations is sought: the tracklet data is enclosed in the so called Attributable after performing a linear regression and the Admissible Region approach is used to determine the uncertainty of the state in spherical coordinates rather than in cartesian coordinates following the DAIOD algorithm. However, this method would only be suitable for re-acquisition after a short period, due to the large uncertainty involved.
All algorithms are developed in C++ where the DACE library containing the DA routines is available. Firstly, the observations are simulated to allow for the validation of the algorithms for different observing strategies. This underlines the necessity to treat some tracklets differently due to the impossibility to perform IOD for very short and very imprecise observations. Then, the DA propagator is tested to prove its accuracy in propagating polynomial forms. Lastly, the data association is performed with real observations 24 hours apart to check the success rate of the algorithm. Tests for the Admissible Region approach are also performed on clustered objects re-observed every 2 hours in a 6 hours span.
Date of Conference: September 11-14, 2018
Best Paper & Best Student Paper Award Winner 2018
Track: Astrodynamics