Effect of the nature of uncertainty on the optimization of collision avoidance maneuvers

Abstract

With the soaring number of space debris, collision avoidance maneuvers have become crucial to ensure the safety of any operating satellite. In real-world scenario, the problem gets a little more complicated due to the presence of uncertainty in the values of the states of the space objects. Also, the uncertainty distribution assigned to a given state variable evolves over time. This paper studies the effects of linear and nonlinear propagation of the uncertainty distributions on the collision avoidance maneuver optimization. These different methods for error propagation lead to differing shapes and sizes of the error distribution, resulting in differing regions to be avoided during the encounter, thus resulting in different optimal Δv requirements. In the method using nonlinear propagation of uncertainty distribution, the uncertainty of the passive object is propagated using Monte Carlo samples while that of the operating spacecraft is evolved using Unscented Transform. The sigma points from the Unscented Transform of the operating spacecraft are required to lie outside the stretch of most of the debris uncertainty while implementing cumulative density function. This also results in defining a new probabilistic measure which is like a modified Mahalanobis distance but for a non-Gaussian distribution. The other approach uses linearly propagating the error covariances and a combined error ellipsoid to ensure the maintenance of a desired Mahalanobis distance of the hard body from the uncertainty distribution to ascertain safety. Both the approaches consider nonlinear state dynamics while incorporating the perturbing forces to find the optimal thrust profile and the corresponding steer angles. The approach implementing nonlinear propagation of uncertainty and using cumulative density to ensure that the sigma points of the satellite avoid the debris’ uncertainty distribution is found to require lower Δv compared to the other method while also representing the error propagation better. However, the method implementing linear propagation of uncertainty and using Mahalanobis distance of the hard body from the combined error distribution to frame the avoidance constraint proves to be computationally much faster.