Dealing with nonlinearities in orbit determination of resident space objects

Abstract

Space activities are continuously at risk due to the ever-increasing amount of uncontrolled space objects, called space debris, which can collide with operational spacecraft. Therefore, space agencies have started space situational awareness (SSA) programs, which aim to provide timely and accurate data regarding the space environment and hazards to infrastructure in orbit and on the ground. An essential need for SSA is the orbit determination of as many space objects as possible, in order to predict and prevent collisions. In this work, techniques for orbit determination of objects observed on short arcs are developed. This scenario frequently occurs with space debris, which are characterised by long observational gaps due to observability constraints. The classical least squares method, used for orbit determination, is revisited by studying the effect of nonlinearities in the mapping between observations and state. Differential algebra techniques are exploited, which enable the computation of high-order Taylor expansions of the residuals with respect to the state. Approximations of differential correction methods can then be avoided and the confidence region of the solution accurately characterized with a nonlinear approach. This uncertainty region can be reduced when more observations of the same objects are acquired. Thus, a method that both recognizes associated observations and sequentially reduces the solution uncertainty when two or more sets of observations are associated is proposed. The six dimensional ($6D$) association problem is addressed as a series of $2D$ and $4D$ problems to alleviate the computational cost. In case of successful association, part of the initial uncertainty is pruned away. The least squares solution and its uncertainty region can finally be used to initialize a sequential filter. A particle filter is implemented using differential algebra, so that system and measurement equations are substituted by polynomial evaluations and the resulting computational cost of the algorithm is alleviated. A population-based stochastic optimizer, the multiobjective particle swarm optimizer, is exploited to generate particles that best represent the probability distribution over solution orbits.