Spacecraft Docking - 2D Hill-Frame Rendezvous Map and Delta-V

Controls

Live readouts

...

Numerical check

running...

Performance graph

How to use

A chaser spacecraft flies near a target (the origin) on almost the same orbit. The map is the target's rotating frame: horizontal = along the orbit (flight direction to the right), vertical = radial (up is away from Earth). Both axes use the same scale, so the picture is true to shape.

With the plan switched on, burn 1 sends the chaser on a free-flight arc that passes through the target after the chosen transfer time, and burn 2 cancels the closing speed there. The dashed amber line is the free drift the chaser would follow from its current state; the red dashed circle is a 200 m keep-out sphere. Use the manual burn buttons to nudge it and watch the orbit mechanics: a forward burn raises the orbit, its period grows, and the chaser drifts backwards relative to the target.

Try: V-bar from 5 km with 0.75 orbits, then raise to 1.25 orbits and compare delta-v; turn the plan off and apply +0.1 m/s radial and along-track burns separately; switch to the R-bar preset and compare how the approach looks.

The science

Artwork: the station, chaser and Earth backdrop are Blender-rendered images. They are icons, not to scale (the real chaser is metres across on a map that spans kilometres), the chaser is drawn nose-toward-target as an illustration (attitude is not simulated), and the Earth limb is only a direction cue for radial-down.

For a target on a circular orbit with mean motion n = √(μ / a³), the chaser's relative position (x radial, y along-track) obeys the Clohessy-Wiltshire (Hill) equations:

ẍ = 3n²x + 2n ẏ + aₓ, ÿ = −2n ẋ + a_y

with the closed-form free solution x(t) = (4 − 3cos nt) x₀ + (sin nt / n) ẋ₀ + (2/n)(1 − cos nt) ẏ₀ y(t) = 6(sin nt − nt) x₀ + y₀ − (2/n)(1 − cos nt) ẋ₀ + ((4 sin nt − 3nt)/n) ẏ₀.

Two-impulse rendezvous: write the free solution as r(τ) = Φᵣᵣ r₀ + Φᵣᵥ v₀ and demand r(τ) = 0, so v₀ = −Φᵣᵥ⁻¹ Φᵣᵣ r₀; burn 1 is v₀ minus the current velocity and burn 2 is minus the arrival velocity. For τ equal to a whole number of orbits Φᵣᵥ is singular (no solution), which is why the slider avoids it. The combination ẏ + 2n x is constant in free flight, and a chaser with ẏ = −2n x stays in a closed 2:1 ellipse around the target.

Honest simplifications: linearised relative motion (valid while the range is much smaller than the orbit radius, a few tens of km), circular target orbit, point-mass gravity (no J2 or drag), impulsive burns, in-plane motion only, no attitude or thruster geometry, no sensors or safety logic. Real rendezvous adds those; this is an educational model.

Keyboard, touch and accessibility

Space pauses or resumes the simulation.
R resets it to its initial state.
C re-runs the numerical check.
All sliders are native range inputs and work with arrow keys, Home and End; on touch devices drag on the canvas where the sim supports pointer input.
Rendering is capped at a device pixel ratio of 2 so phones stay smooth; the frame-time graph above shows the cost of the current settings (green line is 16.7 ms, i.e. 60 frames per second).

What is checked and why you can trust it

Every time the page loads, and whenever you press Re-run check, the simulation evaluates its own mathematics against an independent reference, such as a closed-form formula, a finite-difference derivative or an exhaustive count. Each line above reports PASS or FAIL together with the measured numbers, so you can see by how much it agrees. Results are also published on window.__msChecks for automated testing. The check never uses the same code path as the animation it validates, otherwise it would only confirm itself.

How to read a failure: a FAIL line means the measured discrepancy exceeded the stated tolerance for the current slider values. Try Reset, then move the sliders back toward their defaults; if a check still fails at the defaults it is a bug worth reporting. Tolerances are chosen from the numerical method itself, never tuned to make a test pass: round-off level (1e-9) for exact algebra, about 1e-3 for finite differences and Monte Carlo estimates with a few hundred thousand samples, and a documented margin for stochastic learning runs.

Reproducibility: the particle and pen simulations use the browser random generator for initial positions, so exact pixels differ between visits, but the checks depend only on mathematics and not on those positions. Fixed wave tables (curl noise) and fixed grammars (L-system) are deterministic.

This is an educational model. It shows the mechanism clearly, but it is deliberately simplified; the science panel lists what was left out.