A burn that moves the wrong way

The first rung of this ladder settled the phasing problem: to catch a target ahead in the same orbit, drop into a lower one, go faster, come back up. That is a manoeuvre measured in degrees of orbital angle and hours of waiting.

The last kilometre is a different problem. At a hundred metres from a station the two vehicles are on almost the same orbit, the relative speeds are centimetres per second, and reasoning about it in terms of two separate conics is hopeless — the difference between two nearly identical ellipses is where all the information is, and it is lost in the rounding.

What is needed is a set of equations for the difference, and they turn out to be simple, linear, and thoroughly counterintuitive.

The frame, and the equations in it #

Put the origin on the target, which is in a circular orbit of radius $a_0$ and mean motion $n$. Let $x$ point radially outwards, $y$ along the direction of travel, and $z$ out of the orbital plane. This is the local-vertical local-horizontal frame, and it rotates once per orbit.

A chaser at a small displacement from the origin feels the difference between the gravity at its position and the centripetal acceleration the frame supplies, plus the Coriolis terms the rotation introduces. Linearising in the displacement gives

$$\ddot x - 3n^2x - 2n\dot y = 0,\qquad \ddot y + 2n\dot x = 0,\qquad \ddot z + n^2 z = 0.$$

Clohessy and Wiltshire published these in 1960 for exactly the problem of terminal rendezvous; Hill had written them in 1878 for the lunar theory, which is why they carry both names.

Three things about them are worth noticing before any solution is written down.

The out-of-plane motion decouples completely. $z$ obeys a harmonic oscillator at the orbital frequency, independent of everything else, and a cross-track offset simply oscillates through zero twice per orbit. That is why out-of-plane errors are cheap to remove and why they are removed at a node crossing.

The in-plane equations are coupled by Coriolis terms, not by gravity. The $-3n^2x$ term is the tidal difference in gravity, and the $2n\dot y$ and $-2n\dot x$ terms are the rotation of the frame. Most of what is strange below comes from those.

And there is no $y$ in the second equation. The along-track direction has no restoring force at all: a displacement along the track is a displacement to a point on the same orbit, and nothing pushes it back.

The solution, and the ellipse in it #

Integrating the in-plane equations gives, for initial state $(x_0, y_0, \dot x_0, \dot y_0)$,

$$x(t) = (4-3\cos nt)\,x_0 + \frac{\sin nt}{n}\,\dot x_0 + \frac{2}{n}(1-\cos nt)\,\dot y_0,$$

$$y(t) = 6(\sin nt - nt)\,x_0 + y_0 - \frac{2}{n}(1-\cos nt)\,\dot x_0 + \frac{1}{n}(4\sin nt - 3nt)\,\dot y_0.$$

Everything about close-approach operations is in those two lines.

Set every initial condition to zero except $\dot y_0 = \Delta v$ — a burn along the direction of travel — and evaluate after one full revolution, $nt = 2\pi$:

$$x = 0,\qquad y = -\frac{6\pi\,\Delta v}{n}.$$

The radial offset has returned to zero and the along-track offset is negative: the chaser is behind. A burn in the direction of travel moved the vehicle backwards.

There is no paradox and the explanation is the vis-viva relation in disguise: the burn raised the chaser’s orbital energy, so its semi-major axis is larger, so its period is longer, so after one lap of the target it has not quite finished its own. The instantaneous effect is forwards and the effect after a revolution is backwards, and the crossover is at three-quarters of a revolution.

The manoeuvre flown wrong, once, in public #

The result has a demonstration that predates any use of the equations in flight, and it cost most of a spacecraft’s manoeuvring propellant.

On Gemini 4, in June 1965, the crew were to close on the spent second stage of their own launch vehicle, drifting a few hundred metres away and plainly visible through the window. The natural action — and the one taken — was to thrust towards it.

Thrusting towards an object ahead in the same orbit is a prograde burn. It raised the spacecraft’s orbit, lengthened its period, and the range began to open rather than close. Further thrust made it worse at an increasing rate, and every correction was made against a target visibly moving the wrong way. The attempt was abandoned after consuming a large fraction of the propellant, and the rest of the mission was flown without it.

Nothing was broken and nobody had been misinformed. The equations above had been published five years earlier, and the phasing physics underneath them had been understood since the seventeenth century. What failed was the translation from a written result into a control input, under time pressure, with the target in view and an intuition trained entirely on aircraft.

The response was procedural rather than theoretical: rendezvous was thereafter flown from a computed solution rather than by eye, and Gemini 6A closed on Gemini 7 six months later to within thirty centimetres. What changed was not what was known but who was permitted to choose the direction of a burn. The episode is still taught because the correct action here is not merely unobvious — it is the reverse of the obvious, and looking harder at the target makes it worse.

The 2:1 ellipse #

Now ask which initial conditions produce motion that does not drift. Inspecting the $y$ solution, the terms proportional to $t$ are $-6nx_0 t$ and $-3\dot y_0 t$; the drift vanishes when

$$\dot y_0 = -2n\,x_0.$$

Impose that, and the motion is a closed ellipse traversed once per orbit — with semi-axis $x_0$ radially and $2x_0$ along the track.

Exactly twice, for every offset, every vehicle and every orbit. The ratio is a property of the equations rather than of anything physical about the spacecraft, and it comes from the factor of 2 in the Coriolis terms.

The closed ellipse is not a curiosity; it is the standard configuration for a vehicle waiting near a station. A chaser parked on one is in a passively safe state: without any further action it will keep returning to the same relative positions, never approaching closer than the ellipse’s minimum and never drifting away. Adding a cross-track oscillation of the right phase turns it into a safety ellipse, a three-dimensional loop whose projection on the target’s orbital plane never passes through the origin — so a total loss of control does not produce a collision.

There is a compact way to say which initial conditions drift and which do not, and it turns out not to be a statement about the rotating frame at all.

Combine the radial offset and the along-track rate into the difference in semi-major axis they imply, and the secular along-track drift is $-\tfrac{3}{2}n\,\Delta a$. That is the derivative of the harmonic law and nothing else. Two orbits with the same semi-major axis have the same period and cannot separate secularly, whatever else differs between them; two with different semi-major axes separate at a rate fixed by the period difference alone.

So the closed-ellipse condition $\dot y_0 = -2nx_0$ is the linearised statement that $\Delta a = 0$. Every non-drifting relative orbit is a pair of orbits of equal energy, and the ellipse is the shape that the difference between two equal-energy orbits takes when they differ in eccentricity or in phase.

That reading is worth carrying because it outlives the linearisation. The Clohessy–Wiltshire equations fail at large separations, at eccentric targets and under oblateness; the statement that secular along-track drift requires a difference in orbital energy does not. It remains the first thing to check when a formation is opening, and the size of the mismatch follows from the drift rate in one line — a drift of one metre per second at four hundred kilometres altitude is a semi-major-axis difference of about six hundred metres, which is a fifth of a metre per second of the wrong burn, delivered once.

Two ways in #

The approach itself is chosen from two families, and the difference between them is an argument about what happens if the thrusters stop.

The V-bar approach comes in along the velocity vector, from ahead or behind. It is intuitive to fly and it has an unpleasant property: a vehicle on the V-bar with no relative velocity does not stay there, because being at a different along-track position is being at a different place on the same orbit, and any residual radial velocity opens an ellipse. Worse, closing along the V-bar requires a retrograde burn to move forwards, which is the counterintuitive result above and which has to be flown correctly every time.

The R-bar approach comes in along the radius vector, from below. Its virtue is the third trajectory in the figure at the top of this essay: a purely radial displacement with no along-track velocity produces a closed teardrop that returns to its start. A vehicle on the R-bar that stops thrusting drifts away from the target rather than towards it, because the same gravity gradient that destabilises the radial direction pushes an inner vehicle further in and ahead. Failure carries the vehicle to safety rather than into the station, and that is why the Space Shuttle’s later approaches to the International Space Station and every commercial cargo vehicle since have come in from below.

The arithmetic of a real approach #

It is worth working one case through, because the numbers are small in a way that is easy to disbelieve.

A cargo vehicle holds at a waypoint 200 metres behind the station on the V-bar and is to close to 30 metres. The along-track motion needed is 170 metres. Flying it as a two-impulse transfer over a quarter of an orbit, the required initial velocity comes out of inverting the Clohessy–Wiltshire solution for the desired end state, and it is about six centimetres a second. The braking impulse at the end is of the same size. The total is under fifteen centimetres a second for the whole approach.

That is less than a walking pace divided by twenty, delivered by thrusters producing hundreds of newtons on a vehicle of ten tonnes, in pulses of a few tens of milliseconds. The control problem is not the size of the manoeuvre but the size of the smallest manoeuvre the hardware can make, and a thruster whose minimum impulse bit corresponds to two centimetres a second cannot fly this approach smoothly at all.

The one manoeuvre the equations cannot compute #

Guidance in this frame is usually posed as a two-impulse transfer: given a relative state now and a desired one at a time $T$ from now, find the two burns. The answer is a matrix inversion — the position block of the state transition matrix, inverted to give the required initial velocity — and within the linearisation it is exact.

It is also singular, at $nT = 2\pi$ and at every multiple of it. After exactly one revolution the radial coordinate has returned to a value that no longer depends on the radial velocity that was given to it, so the position block maps a smaller set than it started from and the transfer becomes unsolvable for a general target state.

In practice the required velocity change, plotted against transfer time, is a sequence of shallow wells separated by poles: a transfer scheduled near a whole number of revolutions demands an enormous impulse, and one at exactly a whole number demands an infinite one.

That is not a physical obstruction. The vehicle can reach the state, with three impulses instead of two, or with two at a different timing. It is nevertheless a real constraint on how an approach is scheduled, and it is why waypoint-to-waypoint transfers are flown over quarter and half revolutions rather than whole ones — a plan calling for a hold of one orbit followed by a move has specified the two operations that cannot be combined.

A singularity in a solution method is not always a singularity in the problem, and telling the two apart is most of what makes a guidance algorithm trustworthy. The physics is perfectly well behaved at $nT = 2\pi$; what fails is the demand that exactly two impulses do the work.

The phasing arithmetic runs over a wide range of lead angles, and the two ends of it look nothing alike.

Formation flying, and the cost of a shape #

The closed-ellipse solution is what makes a formation possible without continuous thrust.

Two spacecraft on the same closed relative ellipse with different phases maintain a separation that varies through the orbit and repeats exactly. A constellation of several, phased evenly, traces a rotating configuration that costs nothing to maintain in the ideal problem. GRACE’s two spacecraft flew in a simple along-track formation 220 kilometres apart; the proposed interferometric missions want three or more on a closed ellipse whose plane is fixed on the sky.

The cost comes from everything the linearisation left out. The Earth’s oblateness makes the two vehicles’ nodes regress at slightly different rates if their orbits are not identical, which opens the formation on a timescale of days. Differential drag does the same, at a rate proportional to the difference in ballistic coefficient. A formation’s fuel budget is not for holding the shape but for cancelling the differential perturbations, and it is the same accounting that a geostationary satellite’s station-keeping budget is written in.

What the equations cost #

The linearisation is the source of the simplicity and the source of every limitation, and its terms are worth listing.

The separation must be small compared with the orbital radius. At 400 kilometres altitude the orbital radius is 6,778 kilometres, so a separation of ten kilometres is a part in seven hundred and the neglected quadratic terms are of that order — fine. At a hundred kilometres it is a part in seventy, and the errors become visible over a few orbits.

The target orbit must be circular. For an eccentric target the coefficients are no longer constant and the Tschauner–Hempel equations replace them; they have a closed-form solution but not a pretty one, and the qualitative structure survives with the 2:1 ratio becoming a function of true anomaly.

Only two-body gravity is included. Oblateness, drag and solar radiation pressure all enter as differential accelerations, and over more than a few orbits they dominate.

And the impulses are instantaneous. A real thruster fires for tens of seconds, during which the vehicle moves, and the difference between the impulsive result and the finite-burn one is second-order in the burn duration over the orbital period — negligible for a proximity manoeuvre and not for anything larger.

Where the model stops being a model #

There is a point at which none of this applies, and it is worth naming because it is where the actual difficulty of docking lives.

Inside about ten metres, the relevant dynamics are no longer orbital. The relative acceleration from the gravity gradient at ten metres and 400 kilometres altitude is $3n^2x \approx 4\times10^{-5}$ m/s², which over a sixty-second final approach produces a displacement of seven centimetres. The thruster minimum impulse bit, the plume impinging on the target, the attitude control deadband and the sensor noise all produce larger effects.

The equations that govern a rendezvous do not govern a docking. The transition is somewhere around a hundred metres, and it is where the problem stops being celestial mechanics and becomes control engineering. And the relative motion at a much smaller impulse, which is the regime the final approach is flown in.

Where this ladder goes next #

This rung establishes the relative-motion frame, the closed ellipse in it, and why an approach is flown from below.

Above it lies the guidance problem: choosing a sequence of impulses that takes a vehicle from one relative state to another at minimum cost, subject to keeping it outside an approach corridor and passively safe throughout. That is a linear-programming problem in the Clohessy–Wiltshire solution, and it is one of the few places in spaceflight where the optimal answer is computable in closed form.

Beside it lies the eccentric-target case, which is what a rendezvous with an asteroid or a comet actually is, and where the relative motion has a shape that changes through the orbit.

And below it, as the thing this rung is really about: an intuition trained on translation is wrong in a rotating frame, and the frame is rotating whenever the reference is in orbit. Every one of the surprises above is a Coriolis term, and every one of them has cost somebody a manoeuvre.