Addressing Nonlinear Relative Motion For Spacecraft Collision Probability

Linear methods for computing satellite collision probability can be extended to accommodate nonlinear relative motion in the presence of changing position and velocity uncertainties. 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 are projected onto the plane perpendicular to relative velocity by assuming linear relative motion and constant positional uncertainty throughout the brief encounter. Collision potential is determined from the object footprint on the projected, two-dimensional, covariance ellipse. For nonlinear motion, the dimension associated with relative velocity must be reintroduced. 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. The method begins with object position and velocity data at the time of closest approach. Propagation of position, velocity, and covariance is done forward/backward in time until a user limit is reached. An alternate method is presented 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. These general methods are not dependent on a specific propagator or linear probability model.