Describing relative motion with differences of orbital elements instead of Cartesian coordinates. Subscript : primary (reference), : secondary.
Relative Eccentricity and Inclination Vectors
- Relative orbital elements include:
- the relative eccentricity vector
- the relative inclination vector
- = angle between the orbital planes; the relative inclination vector is
- : argument of latitude of the relative ascending node (the point where the second spacecraft crosses the orbital plane of the first going in the direction)
Orbital Element Representations
- Three families:
- Relative orbital elements
- Orbital element differences
- Geometric ROE (constants of motion of HCW)
Orbital Element Differences
- Classical elements: take the difference between primary and secondary:
- Easy to understand, but have singularities (the same ones as the classical elements)
- Alternatively: equinoctial element differences
Relative Orbital Elements
- Relative semimajor axis:
- Relative mean longitude, using the mean argument of latitude :
- Relative eccentricity vector:
- Relative inclination vector:
- These are commonly used for control of formation-flying or cluster-flight spacecraft
Mapping to Cartesian
- : radial, along-track, cross-track; : argument of latitude of the primary
- : a semimajor-axis difference makes the secondary drift along-track
Geometric Relative Orbital Elements
- Geometric elements/parameters that define constants of the motion (from the HCW state ):
- Relative ellipse size:
- Center of the projected ellipse:
- Relative in-plane phase:
- Cross-track amplitude and phase: