Reachable domain for spacecraft with ellipsoidal Delta-V distribution

Conventional reachable domain (RD) problem with an admissible velocity increment, $\mathrm{\Delta }𝐯$, in an isotropic distribution, was extended to the general case with $\mathrm{\Delta }𝐯$ in an anisotropic ellipsoidal distribution. Such an extension enables RD to describe the effect of initial velocity uncertainty because a Gaussian form of velocity uncertainty can be regarded as possible velocity deviations that are confined within an error ellipsoid. To specify RD in space, the boundary surface of RD, also known as the envelope, should be determined. In this study, the envelope is divided into two parts: inner and outer envelopes. Thus, the problem of solving the RD envelope is formulated into an optimization problem. The inner and outer reachable boundaries that are closest to and farthest away from the center of the Earth, respectively, were found in each direction. An optimal control policy is then formulated by using the necessary condition for an optimum; that is, the first-order derivative of the performance function with respect to the control variable becomes zero. Mathematical properties regarding the optimal control policy is discussed. Finally, an algorithm to solve the RD envelope is proposed. In general, the proposed algorithm does not require any iteration, and therefore benefits from quick computation. Numerical examples, including two coplanar cases and two 3D cases, are provided, which demonstrate that the proposed algorithm works efficiently.