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Conventional reachable domain (RD) problem with an admissible velocity increment, Δv, in an isotropic distribution, was extended to the general case with Δv 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. KEYWORDS reachable domain maneuverability orbit uncertaintyResearch Article

R-bar refers to the local vertical axis pointing radially upward in a satellite-fixed reference frame. Approaching a satellite along the R-bar, especially for rendezvous and docking to geostationary satellites, is advantageous in terms of safety considerations and flight time compared to other options. In this paper, a specialized study on autonomous R-bar proximity operations with respect to a geostationary target from a separation of several kilometers to a few hundreds of meters, commonly referred to as the closing phase, is carried out and a comprehensive solution for both attitude and orbit control in this scenario is proposed. An integrative design of the guidance, navigation, and control for R-bar proximity operations is presented. Impulsive R-bar hopping maneuvers are developed for the trajectory guidance. This method is shown to be passively safe and time efficient. The onboard sensors provide measurements of the line-of-sight, range to the target, attitude and angular velocity in the inertial frame. Due to the sensitivity of the sensor’s pointing in the far-range phase, a sliding mode attitude control law is introduced to align the optical axis with the line-of-sight to the target. Sensor measurements are fused and processed by an extended Kalman filter. Simulation results indicate that the proposed integrative guidance, navigation, and control algorithms are robust to uncertainties and noise, and can be used as a comprehensive solution for R-bar rendezvous and docking mission design during the closing phase.