Trajectory Guidance Using Periodic Relative Orbital Motion

Abstract

Relative motion of one satellite about another in circular orbit, where the two objects have the same semimajor axis, is periodic in the linearized approximation. A set of orbital elements, the geometric relative orbital elements, which are an exact geometric analogy to the classical orbital elements, can be defined. The relative orbit is manifestly seen to be an ellipse or circle in apocentral coordinates, analogous to perifocal coordinates in inertial motion and different from the local-vertical local-horizontal Cartesian coordinates customarily used for analysis of relative motion problems. These provide a way to do relative motion trajectory design and guidance that stands in contrast to the use of Cartesian coordinates. Algorithms are presented for computing the delta-v and timing for impulsive maneuvers to change a single element: two that change the plane and one that changes the size of the relative orbit. The change in size is a two-maneuver transfer and uses the solution of the three-point periodic boundary value problem, also solved and presented here. This solution permits the direct computation, with no iteration or searching, of the periodic relative orbit that connects two points. Optimization to minimize fuel use can be done with a one-dimensional search.

Comprehensive Survey and Assessment of Spacecraft Relative Motion Dynamics Models

“…In their research studies, the authors demonstrate the improved geometric insight and guidance control simplicity afforded by working with these new HCW invariant parameters but do not develop a framework for including other perturbations. Additional examples of parameterizing the relative motion with invariant trajectory parameters are provided by Ichikawa and Ichimura [16] in the context of relative orbit transfer using the HCW equations as well as by Healy and Henshaw [17], who define a set of geometric relative orbital parameters that are analogous to the classical orbital elements. The latter provide a detailed methodology for computing the geometric parameters based on constraints imposed by the unperturbed relative motion, and just as [14], provide a compelling argument for the simplicity and geometric insight provided by the new relative motion description through straightforward impulsive maneuvers to reconfigure the orbital parameters.…”

Study the Influence of Atmospheric Drag and J2 Effect in a Close-proximity Operation at LEO

“…A very common method is follow to identify the position of a satellites by using the six quantities of orbital elements. [1,21]. The following equation is described the state of X in the ECI reference frame.…”