A Numerical Implementation of Spherical Object Collision Probability
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A Numerical Implementation of Spherical Object Coliision Probability Salvatore Alfano! Abstract Collision probability analysis for spherical objects exhibiting linear relative motion is accomplished by combining covariances and physical object dimensions at the point of closest approach. The resulting covariance ellipsoid and hardbody can be projected onto the plane perpendicular to relative velocity when the relative motion is assumed linear. Collision potential is determined from the object footprint on the projected, two-dimensional, covariance ellipse. The resulting double integral can be reduced to a single integral by various methods. This work addresses the numerical computation of this single integral using Simpson's one-third rule to achieve at least two significant figures of accuracy over a wide range of parameters.
Introduction The assumptions involved in the probability formulation are well defined in
references [1- 3] and are summarized here for the reader's convenience. Space object collision probability analysis (COLA) is typically conducted with the objects modeled as spheres. At the point of closest approach, each object's positional uncertainty is combined and their radii summed. By assuming linear relative motion, the resultant is projected onto a plane perpendicular to the relative velocity where the collision probability is calculated. The combined covariance size, shape, and orientation are coupled with physical object sizes to determine collision potential. The projection results in a double integral that can be reduced to a single integral by using the error function or a contour integral. Probabilistic collision dynamics requires computational techniques that are sufficiently efficient, robust, and accurate for decision makers. The probability integral can be numerically approximated in many ways. Although several computational methods have been developed, they remain proprietary and unavailable to the general public. For this effort the contour integral method was implemented and tested, but tended to be more computationally intensive than what follows. This work "Technical Program Manager, Center for Space Standards and Innovation, 7150 Campus Drive, Suite 260, Colorado Springs, CO 80920.
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evaluates the error function integral in Cartesian space using Simpson's one-third rule [4] as originally developed by The Aerospace Corporation. It is currently available in the Satellite Tool Kit Advanced Conjunction Analysis Tool (STK/AdvCAT) feature [5] and is also used to generate the Satellite Orbital Conjunction Reports Assessing Threatening Encounters in Space (SOCRATES) [6]. The number of intermediate steps is adjusted based on combined standard deviations, object size, and miss distance. A slight modification is made at the upper and lower bounds of integration due to the tendency of Simpson's rule to underestimate in those regions. Testing shows the numerical method to be accurate to more than two significant figures for covariance aspect ratios ranging from 1 to 500 in
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