Relative Motion

Motion of one spacecraft as seen from another: rendezvous, formation flying, comparing orbits.

Comparison of Orbits

• It is often useful to compare orbits by looking at the magnitude of the position differences
• Relative motion makes this systematic: describe one orbit relative to the other

Setup

• Plan:
  1. General derivation and circular reference orbits (Curtis chapter 7): Hill-Clohessy-Wiltshire Equations
  2. Curvilinear coordinates, elliptical reference orbits: Tschauner-Hempel Equations
• Reference orbit: the primary satellite; the other is the secondary
• Frame: the RSW frame of the primary
• Relative position:
• Alternative description: Relative Orbital Elements

Relative State from Inertial State Vectors

• Finding the relative coordinates from the inertial state vectors of each satellite
• Find the RSW coordinate axes of the reference satellite:
• Write these as a matrix, with each of these vectors being rows:
• Angular velocity of the reference frame, computed from the angular momentum:

Relative Position and Velocity

• The term removes the apparent velocity due to the rotating frame

• Examples in the book: 7.1, plot in 7.2

Advanced Techniques

• References:
  • Sullivan, Grimberg, D’Amico, “Comprehensive survey and assessment of spacecraft relative motion dynamics models”, JGCD 2017, doi: 10.2514/1.G002309
  • Sinclair, Sherrill, Lovell, “Geometric interpretation of the Tschauner-Hempel solutions for satellite relative motion”, Advances in Space Research 55 (2015) 2268-2279, doi: 10.1016/j.asr.2015.01.032
• The trajectory is curved: use spherical coordinates instead of rectilinear
• Dynamics:
  • Quadratic Volterra: second-order Taylor series expansion, rectilinear
  • Double Transformation: nonlinear coordinate mapping of linear solutions, cylindrical or spherical coordinates

Elliptical Reference Orbits

1. Tschauner-Hempel (TH) equations: do a variable transformation; the independent variable is the true anomaly of the reference orbit instead of time
2. Yamanaka-Ankersen (YA) STM: singularity-free, structurally simple STM, widely used
3. Time-explicit linear models (Broucke, Melton): time-explicit solutions that avoid the need to compute the true anomaly directly
4. Lyapunov-Floquet (LF) transformation: a method to relate the linear time-varying system of an elliptic orbit to the underlying LTI system (like HCW) through periodic coordinate transformations

• TH and YA: Tschauner-Hempel Equations

Time-Explicit Linear Models

• Limited to
• Broucke: relative state expressed as variations of the chief’s orbital parameters
• Melton: expansion in powers of eccentricity

Lyapunov-Floquet Transformations

• Provides a periodic coordinate transformation that relates the time-varying elliptic system to a time-invariant circular system (HCW)

Comparison

Method	Independent variable	Coordinate basis	Primary advantage	Key limitations / complexity
Tschauner-Hempel (TH)	True anomaly ()	Scaled rectilinear	Fundamental linear framework for arbitrary eccentricity	Original solutions contain singularities at perigee/apogee
Yamanaka-Ankersen (YA) STM	True anomaly ()	Rectilinear	State-of-the-art singularity-free analytical solution	High computational cost due to transcendental functions and matrix inverses
Time-Explicit (Melton/Broucke)	Time ()	Cylindrical or rectilinear	Avoids the overhead of solving Kepler’s equation to find	Accuracy often depends on series expansions in (e.g., Melton retains )
Lyapunov-Floquet (LF)	Time () or true anomaly ()	Virtual circular to elliptic mapping	Relates complex elliptic dynamics to simple HCW intuition	Choice of transformation (e.g., periapse vs apoapse matching) affects local accuracy