Paul W. Schumacher, Jr. (HSAI-SSA, Air Force Research Laboratory), Matthew P. Wilkins (Schafer Corporation)
Keywords: SSA
Abstract:
The advent of high-sensitivity, high-capacity optical sensors for space surveillance presents us with interesting and challenging tracking problems. Accounting for the origin of every detection made by such systems is generally agreed to belong to the most difficult category of tracking problems. Especially in the early phases of the tracking scenario, when a catalog of targets is being compiled, or when many new objects appear in space because of on-orbit explosion or collision, one faces a combinatorially large number of orbit (data association) hypotheses to evaluate. The number of hypotheses is reduced to a more feasible number if observations close together in time can, with high confidence, be associated by the sensor into extended tracks on single objects. Most current space surveillance techniques are predicated on the sensor systems ability to form such tracks reliably. However, the required operational tempo of space surveillance, the very large number of objects in Earth orbit and the difficulties of detecting dim, fast-moving targets at long ranges means that individual sensor track reports are often inadequate for computing initial orbit hypotheses. In fact, this situation can occur with optical sensors even when the probability of detection is high. For example, the arc of orbit that has been observed may be too short or may have been sampled too sparsely to allow well-conditioned, usable orbit estimates from single tracks. In that case, one has no choice but to solve a data association problem involving an unknown number of targets and many widely spaced observations of uncertain origin. In the present paper, we are motivated by this more difficult aspect of the satellite cataloging problem. However, the results of this analysis may find use in a variety of less stressing tracking applications. The computational complexity of track initiation using only angle measurements is polynomial in time. However, the polynomial degree can be high, always at least cubic and commonly quartic or higher. Therefore, practical implementations require attention to the scalability of the algorithms, when one is dealing with the very large number of observations from large surveillance telescopes. We address two broad categories of algorithms. The first category includes and extends the classical methods of Laplace and Gauss, as well as the more modern method of Gooding, in which one solves explicitly for the apparent range to the target in terms of the given data. In particular, recent ideas offered by Mortari and Karimi allow us to construct a family of range-solution methods that can be scaled to many processors efficiently. We find that the orbit solutions (data association hypotheses) can be ranked by means of a concept we call persistence, in which a simple statistical measure of likelihood is based on the frequency of occurrence of combinations of observations in consistent orbit solutions. Of course, range-solution methods can be expected to perform poorly if the orbit solutions of most interest are not well conditioned. The second category of algorithms addresses this difficulty. Instead of solving for range, these methods attach a set of range hypotheses to each measured line of sight. Then all pair-wise combinations of observations are considered and the family of Lambert problems is solved for each pair. These algorithms also have polynomial complexity, though now the complexity is quadratic in the number of observations and also quadratic in the number of range hypotheses. We offer a novel type of admissible-region analysis, constructing partitions of the orbital element space and deriving rigorous upper and lower bounds on the possible values of the range for each partition. This analysis allows us to parallelize with respect to the element partitions and to reduce the number of range hypotheses that have to be considered in each processor simply by making the partitions smaller. Naturally, there are many ways to combine ideas from both categories of algorithms, so that the subject remains a fertile one. We present numerical results based on simulated data sets and on real data from the PanSTARRS system.
Date of Conference: September 11-14, 2012
Track: Poster