Continuous-Time Modeling and Control Using Nonsingular Linearized Relative-Orbit Elements

Applications, such as autonomous rendezvous and docking for CubeSats, on-orbit assembly of space stations, and orbital-debris harvesting or removal technologies, require relative-motion guidance-and-control-approach application for close-proximity operations with frequent trajectory controls. This study expands upon current relative-orbit approaches in delivering a nonsingular osculating linearized-relative-orbit-element state extracted from the Clohessy–Wiltshire equations’ integration constants capable of including perturbation accelerations and control. The Lagrangian brackets, used in Lagrange’s planetary equations and other osculating forms, are applied to the acquired state vector to obtain the elegantly simple kinematics. The proposed relative-motion state vector is promising for a range of proximity operations, as it provides the capability to include perturbation accelerations and control without altering the formulation and no loss of geometric insight. The linearized-relative-orbit-element variational equations are validated with a differential drag example. The Lyapunov control theory is applied to develop a linearized-relative-orbit-element feedback-control law demonstrating transitioning between relative orbits. The linearized-relative-orbit-element control formulation is implemented for two relative-orbit reconfigurations illustrating the geometric insight inherent in the developed approach.