Sequential Probability Ratio Test for Collision Avoidance Maneuver Decisions
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quential Probability Ratio Test for Collision Avoidance Maneuver Decisions J. Russell Carpenter,1 F. Landis Markley,2 and Dara Gold3
Abstract When facing a conjunction between space objects, decision makers must choose whether to maneuver for collision avoidance or not. We apply a well-known decision procedure, the sequential probability ratio test, to this problem. We propose two approaches to the problem solution, one based on a frequentist method, and the other on a Bayesian method. The frequentist method does not require any prior knowledge concerning the conjunction, while the Bayesian method assumes knowledge of prior probability densities. Our results show that the frequentist method is inferior to the Bayesian method.
Introduction We often hear it said that “space is big” by which we think decision makers mean that collisions between space objects are so rare that many collision avoidance maneuver (CAM) decisions are the result of false alarms. Nevertheless, the consequences of a missed detection might be catastrophic. It is our understanding of current CAM decision procedures that they do not explicitly account for the costs and risks of false alarms and missed detections. It seems to us that a utilitarian decision making procedure that balances the costs and risks of false alarms and missed detections would provide some benefit to decision makers. We propose herein such a procedure that is applicable to a common class of CAM decision problems. The remainder of this paper is organized as follows. In the next section, we review how the full CAM decision problem reduces under certain assumptions to the comparison of hypotheses concerning parameters of Rician densities; assumptions on which at present, our proposed method depends. Then, we briefly discuss 1 Navigation and Mission Design Branch, NASA Goddard Space Flight Center, Code 595, Greenbelt, MD 20771. 2 Attitude Control Systems Engineering Branch, NASA Goddard Space Flight Center, Code 591, Greenbelt, MD 20771. 3 Graduate Student, Department of Mathematics and Statistics, Boston University, 11 Cummington Mall, Boston, MA 02215.
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collision avoidance as a decision problem involving equalizing utilities based on cost and risk. Next, we review Wald’s Sequential Probability Ratio Test (SPRT), [1] which forms the basis of our proposed method. While the SPRT involves a likelihood ratio between two simple hypotheses, the CAM decision problem as we define it involves a comparison among two continuous sets of hypotheses. In the subsequent section, which we believe involves the main technical contribution of this paper, we discuss two methods to reduce the compound hypotheses to simple ones. In the penultimate section, we illustrate and compare our methods using a static example based on the Magnetospheric Multi-Scale (MMS) mission, and finally we conclude with a summary and suggestions for follow-on work.
Problem Statement The Simplified CAM Decision Problem Our approach to the CAM decision problem starts with the ideas of Foster an
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