Orbital Data Applications for Space Objects by Lei Chen Xian-Zong Bai Yan-Gang Liang & Ke-Bo Li

Author:Lei Chen, Xian-Zong Bai, Yan-Gang Liang & Ke-Bo Li
Language: eng
Format: epub
Publisher: Springer Singapore, Singapore

The calculation of collision probability can be transformed from 3D integral to 2D integral in the conjunction plane under the assumption of linear relative motion. The conjunction is long-term if the relative velocity is small in the conjunction, both the magnitude and direction of the relative velocity will change during the conjunction. The relative becomes nonlinear. For example, satellites in the formation flight or space rendezvous, or the geosynchronous satellites have low relative velocity (meters per second or less) so that the time they spend in the encounter region is appreciable. The linear relative motion region might be smaller than the encounter region for very low relative velocity. Even if one propagates to the point of closest approach and applies current methods, an incorrect collision probability will be obtained.

The calculation of collision probability is more complex in the nonlinear relative motion than in the linear one. First, differ from the one in linear relative motion, the collision probability is changeable during the nonlinear encounter. Second, the problem can not be transformed to a 2D integration, but requires a numerical integration of the probability density through the volume swept out by the combined hard body of the space vehicles. This integration is complicated by the changing orientation of the hard body and combined position-error ellipsoid throughout the encounter. In addition, integration limits on this volume integral complicate the problem.

Chan investigated when it is justified to assume rectilinear relative motion so as to replace a three-dimensional integral by a two-dimensional integral for computing the collision probability when two spacecraft are in close proximity [1]. Chan also concerned with the modeling of long-term spacecraft encounters for the purpose of computing collision probability [29]. Patera presented a method for calculating the collision probability between two space vehicles when the relative motion is nonlinear, which is developed using contour integration methodology. The method involves transforming the problem to a scaled frame in which the error covariance matrix is symmetric in three dimensions [30, 31]. Slater examined the evolution of the collision probability for satellites in formation flight and determined an effective maneuver algorithm that can minimize the probability of collision while reducing the energy expenditure in the maneuver [32]. Alfano pointed out that the dimension associated with relative velocity must be reintroduced for nonlinear motion. This can be simply done by breaking the collision tube into sufficiently small cylinders such that the sectional motion is nearly linear, computing the linear probability associated with each section, and then summing. An alternate method is that creates a voxel grid in Mahalanobis space, computes the probability of each affected voxel as the combined object passes through the space, and sums [33]. McKinley developed an algorithm that reduces the complex nonlinear relative motion into small linear segments that are easy to evaluate [34]. Xu Xiao-li of the Purple Mountain Observatory of Chinese Academy of Sciences studied the method used to calculate collision probability for nonlinear relative motion based on McKinley’s method [35]. Zhang Ge of Dalian University of Technology

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