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Collision detection of two fast moving objects

To model the status of collision of two fast moving objects in the field of space science and computer aided geometric design, one approach is to perform high-precision orbit and forward propagations of two objects, then compare the distance between them at closely spaced instants of time to determine the moment of closest approach, and then compute its minimum distance. The mathematical and physical formulation of two fast moving objects is the corresponding position covariance ellipsoids. In this paper, we assume that the fast moving object's positional uncertainty is along its moving direction and the three principal axes of the covariance ellipsoid are mutually orthogonal. A physics based simulation has been developed in order to detect three different stages of the two objects collision. To detect status of collision, we use a new analytical method to compute eigenvalues and compare the analytical solution with other well-known numerical computation approach. This analytical approach shows that if the characteristic equation has at least two negative real solutions, then the ellipsoids are separated. In addition, if the ellipsoids share a common point, or touch one another, the characteristic equation has a positive double real solution and the discriminant of the quadratic equations is zero. Moreover, two or more complex eigenvalues are obtained when it is a complete penetration of one ellipsoid by another. An advantage of using the analytical approach is not only that it yields direct solutions with numerical approximation and iteration, but also it is computationally efficient and accurate. Furthermore, we introduce the eigenvalues prescreening method to reduce the computing time and to detect the collision of fast moving two objects. The symbolic manipulation tool provides the capability to prescreen results and solves a system of four simultaneous non-linear equations. An advantage of prescreening the eigenvalues is not only that it indicates the status of two objects, but also it is computationally efficient. This prescreening methodology, in conjunction with the analytical approach, provides the detection of collision tool for the fast moving objects.