Convex optimization of collision avoidance maneuvers in the presence of uncertainty

Abstract

Over the past few decades, a considerable increase in the number of space debris has become a serious threat to the safety of the operating satellites. So, collision avoidance maneuvers have become indispensable in providing immediate and cost-effective solutions. The unavoidable presence of uncertainty associated with any physical system adds to the challenge. In this paper, a method is presented to determine the fuel-optimal collision avoidance trajectory using a convex optimization technique while incorporating the evolution of uncertainty associated with the space objects. The safety of the satellite is ensured by the collision avoidance constraints which involve calculating the instantaneous collision probability and using the Mahalanobis distance. An avoidance region is also demarcated around a space object with uncertain states to prevent any collision. Posing the problem as a convex optimization problem helps to eliminate the problem caused by dependency of the optimization variables on the initial guesses used. However, convexification of this problem with evolving uncertainty and collision probability calculation presents a set of challenges. The evolution of the positional error covariance ellipsoid leads to a feasible non-convex space surrounding the avoidance region. Moreover, implementation of a convex optimization formulation here requires convexification of the expression for collision probability, which is originally not purely convex in nature when a Gaussian probability density function is involved. This paper presents an approach to tackle the above-mentioned challenges. The satellite is returned to its nominal orbit in a timely manner, as time spent off the nominal trajectory can often lead to disruption of the space mission. The effects of various parameters on the required Δv are also explored.