Research On Calculational Method Of Spacecraft Collision Probability

With the development of human space technology, the protection of the space environment, the safety of space satellite has become an international issue, the research about the probability of collisions between space objects has also become a hot topic of national space research. This paper to study the collision probability calculation method between the spacecraft and the space objects, make the calculate of collision probability further simplify, improve the calculation speed of the collision probability, in order to facilitate the implementation of the optimal collision avoidance maneuver.This article used each of the spacecraft and space object position vector and velocity vector and error covariance, analysis the calculation of collision probability, derived in detail this article's two collision probability formula. Suppose the spacecraft and space objects are spheres, by definition meet coordinates , relative position error between the spacecraft and the space objects can use a three-dimensional Gaussian probability density function show. In the meet coordinate system, integration of the probability density function along the relative velocity direction, the projection area which the total envelope spherical of the spacecraft and space objects in the encounter plane coordinate system is a circular domain. In the encounter plane coordinate system, the collision probability between the spacecraft and the space object is integration the probability density function of two-dimensional in the projection circle domain. To simplify the integral calculation, rotating the meet coordinate system, and to the rotated coordinate system do coordinate scale transformation, then the collision probability formula is the integral of a two-dimensional probability density function in the elliptic region. Derived the formula for calculating the collision probability with two ways: directly on the ellipse domain make polar transformation, we can be turned the above-mentioned two-dimensional integral into definite integral, this definite integral is easy to programmed calculation in thematlab, and the calculating speed is faster; with area equal circular domain equivalent replacement ellipse regional integration, then do the polar coordinate transformation, use of the infinite series substitute the trigonometric integrals, the collision probability formula can be turned into the form of infinite series, take the first number of infinite series as the approximate collision probability, and analyzed the error in the collision probability formula after the equivalent replacement, Obtain the maximum error and the minimum error and the maximum truncation error, this method are more concise formula for calculating the collision probability, and calculate more faster. Finally, through two fair examples validate the collision probability calculation method.