Abstract

The reachable and feasible sets of spacecraft are important tools for many areas of astrodynamics such as mission design, space situational awareness, and assessment of potential threats. Whilst these may be obtained by solving many optimal control problems in the indirect formulation, this process is slow and can suffer from convergence issues with a lack of a good initial guess. This work presents an analytical method of rapidly estimating the reachable and feasible sets utilizing the D matrix, an error state transition matrix under linearity assumptions for Keplerian motion which employs time as a state variable and true anomaly as the reference variable. Analytical expressions for the control laws are derived to produce the maximum/minimum change in orbital radius, time-offlight, and out-of-plane height. Moreover, an analytical expression for the control efficiency is derived to allow thrust activation only when it is efficient to do so, allowing a target fuel consumption to be attained. By mixing the contributions of these controls, the complete reachable or feasible sets can then be rapidly swept over with minimal effort, which is demonstrated in a series of numerical simulations. The results show that the optimal control problem solution presents only minor improvements, highlighting the capacity of the D matrix to provide marginally sub-optimal results within a fraction of the computational time.