M075 · Relative orbits and docking

Orbital Rendezvous in the Hill Frame

Does reaching the target also mean matching its velocity? No: getting there and matching speed are separate jobs. The first burn only aims the chaser at the target's position; it arrives with relative velocity left over, and a second burn at arrival must cancel it. Some flight times, such as exactly one orbit of the target, leave no usable aim. The simple linear model also drifts from true orbital motion, and arriving too fast still fails.

Drag the chaser in the target’s rotating frame and solve for a two-burn rendezvous. Compare a coast with no burns, the Hill plan and a full two-body propagation, then try burn trims, a docking corridor, approach-speed limits and singular flight times.

Elapsed $\text{Pending}$

Hill separation $\text{Pending}$

Endpoint solve $\text{Pending}$

Total planned impulse $\text{Pending}$

Endpoint condition number $\text{Pending}$

Hill position miss $\text{Pending}$

Full two-body miss $\text{Pending}$

Full residual speed $\text{Pending}$

Acquired model difference $\text{Pending}$

Full separation / radius $\text{Pending}$

Acquired corridor check $\text{Pending}$

Hill incoming speed $\text{Pending}$

Relative Hill invariant defect $\text{Pending}$

Planned flight time $\text{Pending}$

Plan the burns and compare the results

Does reaching the target also mean matching its velocity?

Target orbit radius

r_c

1.10\,R_\oplus

Initial radial position

x_0

60.00\,m

Initial along-track position

y_0

-100.00\,m

Initial radial velocity

\dot x_0

0.00\,m\,s^{-1}

Initial along-track velocity

\dot y_0

0.00\,m\,s^{-1}

Flight time / target period

t_f/P_c

0.10\,

Radial departure trim

\delta v_x

0.00\,m\,s^{-1}

Along-track departure trim

\delta v_y

0.00\,m\,s^{-1}

Keep-out radius

r_{\mathrm{KO}}

20.00\,m

Approach half-angle

\alpha

30.00\,{}^\circ

Incoming speed limit

v_{\mathrm{lim}}

0.50\,m\,s^{-1}

Acquired time

t/t_{\mathrm{obs}}

0.00\,\%

Drag the orange release ring to set both relative coordinates; the sliders work from the keyboard. The planner rejects an endpoint map whose dimensionless condition number is above one thousand. It then shows a coast with no burns and produces no plan. Orange samples in the docking plot break the approach corridor or the incoming-speed limit. The check uses nine hundred intervals, so it flags problems but does not guarantee collision avoidance between them. The final braking burn cannot make up for an approach that is too fast.

The planar Clohessy–Wiltshire model assumes the target is on a circular orbit and the separation is small compared with the orbit radius. Radial points right and along-track points up. Relative positions are in metres and time is in seconds. The purple propagation keeps full inverse-square gravity; it treats the central body as a point mass and ignores collisions with the planet. Trims are added to the solved departure velocity, and the final Hill braking burn is also applied to the nonlinear arrival.

Physics tutorial

A rotating frame makes orbital rendezvous local

BackgroundA nearby chaser and a target on a circular orbit are both falling under gravity. In the target’s rotating frame, the difference in radial gravity and the Coriolis terms couple the two relative coordinates. A forward burn can first carry the chaser forward while changing its orbit so that, over the long run, its along-track drift reverses.

Why it mattersMIT’s astrodynamics notes derive the linear Clohessy–Wiltshire equations. This Lab uses one coordinate pointing radially outward and one pointing forward along the track. It solves the planar endpoint map and, as an independent check, propagates the same starting state in full inverse-square gravity.

Start with the essentials

Focus question: Can a two-burn plan reach the target and stay inside the approach corridor?
One-sentence intuition: Solving for the endpoint puts the chaser at the target and a separate burn on arrival matches its velocity, but the plan can still fail if the endpoint map is singular, the nonlinear error is large or the chaser comes in too fast.

Core mathematical model

Coupled equations of relative motion

n=\sqrt{\frac{\mu}{r_c^3}},\qquad \ddot x-2n\dot y-3n^2x=0,\quad\ddot y+2n\dot x=0

The target’s orbit must be circular. The Coriolis coupling is why the intuition of chasing in a straight line fails. Motion out of the orbital plane is left out.

Solving for the departure velocity

\mathbf r_f=A(t_f)\mathbf r_0+B(t_f)\mathbf v_0^+,\quad\mathbf v_0^+=-B(t_f)^{-1}A(t_f)\mathbf r_0

The departure velocity is chosen so that the chaser ends up at the target position. The chaser’s initial velocity changes the cost of the departure burn, not the solved velocity itself.

Velocity-to-position map

nB(t)=\begin{pmatrix}\sin nt&2(1-\cos nt)\\-2(1-\cos nt)&4\sin nt-3nt\end{pmatrix}

After exactly one target period this matrix loses rank. The code scales it by the mean motion and rejects plans whose condition number is above one thousand.

The two burns and their total cost

\Delta\mathbf v_1=\mathbf v_0^+-\mathbf v_0^-,\quad\Delta\mathbf v_2=-\mathbf v_f^-,\qquad\Delta v=\lVert\Delta\mathbf v_1\rVert+\lVert\Delta\mathbf v_2\rVert

The second burn cancels the actual Hill arrival velocity, trim included. Applied to the nonlinear arrival, the same vector leaves a residual speed that you can measure.

A quantity conserved while coasting

\mathcal C=\dot x^2+\dot y^2-3n^2x^2,\qquad \frac{d\mathcal C}{dt}=0

This quadratic integral belongs to the Hill model with no burns. It is not the spacecraft’s energy in the inertial frame.

Converting to the inertial frame for the two-body check

\mathbf R=Q(nt)\left[\begin{pmatrix}r_c\\0\end{pmatrix}+\mathbf r\right],\quad\mathbf V=Q(nt)\left[\dot{\mathbf r}+nJ\left(\begin{pmatrix}r_c\\0\end{pmatrix}+\mathbf r\right)\right],\quad J=\begin{pmatrix}0&-1\\1&0\end{pmatrix}

Rotating the velocity must include the term from the frame’s own rotation; leaving it out gives the wrong inertial starting state. Units are converted before propagation.

Common difficulties

Meeting is not docking

Typical misconceptionIf the position error is zero, the relative speed is zero too.

Better mental modelThe chaser usually still has some velocity on arrival until the braking burn. Check the incoming speed separately.

The linear model is not exact

Typical misconceptionAn exact Hill endpoint means an exact two-body rendezvous.

Better mental modelThe purple trajectory starts from the same physical state but keeps nonlinear gravity. Its position and residual-velocity errors grow with separation.

Some flight times have no unique burn

Typical misconceptionEvery flight time gives a unique, finite burn.

Better mental modelAt certain times the endpoint map loses rank. When a plan is rejected, the coast with no burn is shown instead.

Passing sampled checks does not prove safety

Typical misconceptionA green sample check guarantees collision avoidance at every instant.

Better mental modelThe cone includes a one-metre allowance at the port, and the check covers nine hundred intervals. It cannot catch violations between samples or operational physics that the model leaves out.

Run the experiment

01
Run the nominal approach
Acquire the nominal approach and compare the Hill miss, the full-gravity miss and the residual speed.
What to observe: The Hill plan reaches the target exactly, while the nonlinear residual stays small but not zero.
02
Compare the coasts
Choose the forward-impulse preset and look at the gray no-burn trajectory.
What to observe: A forward kick sets off coupled radial motion and changes the along-track drift. The departure burn has to pay to bring the chaser back to the planned start.
03
Break the plan
Select a flight time of one whole period, or add a small along-track trim.
What to observe: The singular plan is rejected. A trim leaves a finite position miss that remains even after braking.
04
Check the approach
Tighten the speed limit, then increase the release separation.
What to observe: A zero miss at the endpoint does not stop the approach from being too fast, and it does not stop the linearisation error from growing.