Tool 10 · Rendezvous & Relative Motion

HCW Relative Motion Visualizer

Visualize chaser motion around a target spacecraft in the LVLH frame using the Hill–Clohessy–Wiltshire equations. Explore natural relative ellipses, along-track drift, radial motion, cross-track oscillation, and why timing matters in rendezvous design.

What this tool computes

The HCW equations describe the linearized relative motion of a chaser spacecraft near a target in a circular reference orbit. This is one of the most important first models for rendezvous, proximity operations, formation flying, and inspection missions.

  • Relative position in the LVLH frame
  • Radial, along-track, and cross-track motion
  • Natural elliptical relative trajectories
  • Along-track drift caused by initial radial offset or velocity mismatch
  • Range and closest approach over the simulation window

LVLH frame convention

This page uses a standard local orbital frame around the target spacecraft:

  • x: radial direction, outward from Earth
  • y: along-track direction, approximately direction of orbital motion
  • z: cross-track direction, normal to the orbital plane
\[ \mathbf\rho = [x,\ y,\ z]^T \]

HCW equations

For a target in a circular orbit with mean motion \(n\), the linearized relative dynamics are:

\[ \ddot{x} - 2n\dot{y} - 3n^2x = 0 \] \[ \ddot{y} + 2n\dot{x} = 0 \] \[ \ddot{z} + n^2z = 0 \]

These equations show why relative motion is not simply straight-line motion. The rotating frame creates coupling between radial and along-track motion.

Engineering insight

  • A pure radial offset can create along-track drift.
  • Cross-track motion behaves like a simple harmonic oscillation.
  • Some initial conditions form closed relative ellipses.
  • Small velocity errors can grow into large along-track separation.

Important limitation

HCW is a linear model. It is most useful for small relative distances near a circular target orbit. It does not include J2, drag, eccentric target motion, finite burns, or navigation uncertainty.

Interactive HCW visualizer

Initial relative position (m)

Initial relative velocity (m/s)

Presets are designed to show common rendezvous behaviours.

Results

Target orbital period

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Mean motion n

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Initial range

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Closest range

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Final relative position

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Drift indicator

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Interpretation

Run the simulation to visualize the relative trajectory in LVLH coordinates.

3D relative trajectory in LVLH frame
In-plane relative motion: radial x vs along-track y
Relative position history

Assumptions and limitations

This tool assumes:

  • Circular target orbit
  • Small relative distance compared with orbital radius
  • Linear HCW dynamics
  • No burns during propagation
  • No perturbations, drag, navigation errors, or collision constraints
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