Three-dimensional relative reachable domain with initial state uncertainty in Gaussian distribution

Abstract

A probability threshold is a confidence level defining a bound outside which the occurrence of a random variable is considered as a rare event. Upon providing such a probability threshold on the initial state uncertainty, relative reachable domain is defined as the minimum positional volume enclosing all the possible relative trajectories resulting from the initial state uncertainty. In this study, the conventional coplanar relative reachable domain with initial state uncertainty in isotropic Gaussian distribution is extended to the three-dimensional case with uncertainty in arbitrary Gaussian distribution. Positional errors in Gaussian distribution are thought to be confined within an error ellipsoid depending on the given probability threshold. Such an error ellipsoid will evolve with time and consequently sweep out a volume, which is the relative reachable domain to be determined. This paper proposed an algorithm of solving the envelope surface of the relative reachable domain for close range relative motion based on the linearized dynamical model and the mathematical definition of an envelope. Moreover, the algorithm is modified to improve the accuracy for long range relative motion. Comparisons between the solved relative reachable domain and the result of 10,000 Monte Carlo runs, which can be regarded as the true result, for different scenarios on circular reference orbits demonstrated the feasibility of the proposed method.