A grapple point that moves faster than the orbit

Rendezvous is taught as orbital mechanics, and up to the last few metres that is what it is. A chaser in nearly the same orbit as its target moves, in the target’s frame, under the pull of a gravity field that varies slightly across the gap between them — a relative orbit that breathes once per revolution, drifts along the orbit if the two periods differ, and responds to a push along the direction of motion by moving the opposite way. The equations are linear, they are solved in closed form, and every manoeuvre of an approach — the phasing burns, the hops, the final closing — is designed with them. Holding station a few metres from a target costs millimetres per second per minute, because the forces involved are the differences of gravity over a few metres.

All of that assumes the target holds still, in the sense that matters: that the point on it where the chaser must make contact stays in the same place relative to the target’s centre. A working satellite, whose attitude is controlled, does. A dead one does not. Satellites and rocket stages left in orbit tumble, set spinning by whatever happened at the end of their lives — a residual spin at separation, a stuck thruster, outgassing, the torque from sunlight on an asymmetric shape — and over years that tumble evolves under the Earth’s magnetic field and the Sun’s pressure. Removing them from orbit, which the growth of the debris population makes increasingly urgent, means capturing objects that are turning at anywhere from a fraction of a degree to several degrees per second. For those, the last few metres are a different problem.

Following a point that turns #

The simplest tumble is a flat spin, a steady rotation about the axis of the body’s largest moment of inertia, which is where a body that dissipates energy internally ends up. A grapple point a distance $r$ from the spin axis moves on a circle at speed $\omega r$ and accelerates towards the axis at $\omega^2 r$. A chaser that wants to stay in contact position with that point — to follow it round so that the relative geometry is fixed while the capture mechanism closes — must supply that acceleration itself, continuously, with its thrusters.

At a degree per second, a grapple point three metres from the axis costs five centimetres per second every minute to follow — twenty-four times what the orbital dynamics would cost to hold the same chaser ten metres away. At five degrees per second, the cost is 1.4 metres per second per minute, six hundred times the orbital term. Because it rises as the square of the rate, every doubling of the tumble quadruples the bill. Above a few tenths of a degree per second, the orbit — the entire subject of the approach until this point — is the small term, a slowly varying bias on a much larger rotational one.

That inversion changes what the problem is. The Clohessy–Wiltshire equations describe the relative motion superbly, and they are still correct; but the controller that closes the last metres is now dominated by the target’s rotation, and the quantities it most needs to know are the target’s spin rate, its axis, and where on the target the grapple point is — not the relative orbit.

How fast is too fast #

A chaser does not need to follow the grapple point for long — only for as long as it takes to close a mechanism — and it has a budget of propellant set aside for that phase. The two together set the fastest tumble it can handle.

The followable rate goes as the square root of the budget divided by the product of the radius and the time. The square roots make the limit forgiving in one direction and unforgiving in another. Five times the propellant buys only 2.2 times the rate; five times the time costs only a factor of 2.2 in the rate. For a five-minute capture of a point three metres out, a budget of two metres per second handles a tumble of 2.7 degrees per second, and ten metres per second handles six. Five minutes is a short time to approach the last metre, confirm alignment, and close a mechanism on an object never designed to be grasped, and a real capture sequence that includes inspection and alignment would take longer — so the practical ceiling for following, as opposed to grabbing on the fly, is a few degrees per second.

That puts the measured tumbles of large derelicts at or just beyond the edge. The large Earth-observation satellite that failed in orbit in 2012, and which is one of the best-studied candidates for removal, was measured by laser ranging a year later rotating at about 2.7 degrees per second, a rate that was increasing at the time. Old rocket bodies tumble at rates from a small fraction of a degree to tens of degrees per second, and many have settled into flat spins. For the fast ones the options are to grab them in a single brief contact without synchronising, to use a capture method that does not need a particular grapple point — a net, a harpoon, a tether — or to slow them down first.

A window per revolution #

A chaser that does not follow the rotation waits for the grapple point to come round.

The window is simply the angle of tolerance divided by the rate. At 2.7 degrees per second, a capture mechanism that tolerates ±5 degrees of misalignment has 3.7 seconds each revolution — once every 133 seconds — in which to close. That is ample for a mechanism that is already in position and needs only to be triggered, and it is the basis of capture schemes in which the chaser holds station at a fixed point in space near the target and a manipulator arm reaches in at the right moment. It is much less comfortable for a scheme in which the chaser itself must move into contact, because the approach of the last half-metre takes longer than the window, and the chaser must either commit before it can see that the window will be good or wait out many revolutions for a favourable one.

The window argument and the cost argument meet at a single number, the rate. Below about a degree per second, following is cheap and windows are long, and almost any method works. Above five, following is expensive and windows are a second or two, and only methods that need neither survive. Between them lies the regime of most of the objects that matter.

A tumble that never repeats #

A flat spin is the easy case. A body rotating about no principal axis — which is what a fresh tumble usually is, before internal damping has had years to settle it — does not spin; it tumbles, and the difference is visible in the motion of any point on it.

The body’s angular momentum is fixed in space, and its rotation axis wanders around it: the body nods as it turns, in a motion governed by Euler’s equations for a rigid body with three different moments of inertia. The grapple point therefore moves on a path that is nearly circular in projection but shifts from one revolution to the next, and moves in and out of the plane by metres. The motion is quasi-periodic — it combines the rotation about the angular momentum with the nodding at a different, incommensurate frequency — and it never exactly repeats. A wingnut spun about its intermediate axis turns over on its own for the same reason: three unequal moments of inertia make the free motion of a rigid body far richer than rotation.

For a chaser following the point, the consequence is that the acceleration it must supply is not constant. It swings by a factor of six over each nodding period, from a fifth of a centimetre per second squared to more than a centimetre, in a pattern that depends on the target’s moments of inertia and on its full rotational state. The thrusters must be sized for the peak and commanded to follow the pattern, and the pattern can be predicted only by a model of the target’s rotation — which means knowing, or estimating, its moments of inertia.

Why the dead satellites tumble the way they do #

A tumble is not random; its history can be read from its state, and the reading matters for capture. A satellite that dies with its attitude control running out usually starts with a small residual rotation, and then several slow torques take over. Sunlight pressing unevenly on an asymmetric body — a large solar array on one side — exerts a steady torque that can spin it up over months, in the rotational counterpart of the push of sunlight and re-radiated heat that drifts small asteroids’ orbits. The Earth’s magnetic field induces eddy currents that slow any conducting body. The gravity gradient pulls the long axis of an elongated body towards the vertical. Which wins depends on the body’s shape, size and orbit, and different derelicts are measured spinning up, slowing down or settling into steady states.

Internal dissipation shapes the tumble whatever the external torques do. Fuel sloshing in tanks, flexing appendages and loose components all turn rotational energy into heat while leaving the angular momentum untouched, and a rigid body with fixed angular momentum has its lowest energy when it spins about its axis of largest moment of inertia. Given enough time, a derelict therefore tends towards a flat spin about that axis — the simplest case to capture — and the tumbling, nodding motion of the figures is characteristic of objects that are young as derelicts, or that have been disturbed. An old rocket body in a flat spin at a high rate and a newly dead satellite tumbling irregularly at a low one present almost opposite difficulties, and a removal campaign has to be prepared for both.

Measuring the spin from the ground #

Much of what is known about derelicts’ rotation comes from the ground, and the methods are indirect in instructive ways. A tumbling object’s brightness varies as it turns, because it presents different areas and angles to the Sun and the observer, and a light curve taken over a pass gives the apparent period — though not uniquely, because a body with two similar faces flashes twice per rotation. Satellites that carry laser retroreflectors can be ranged with lasers from the ground; as the body turns, the retroreflector’s distance varies by centimetres, and the pattern gives the rotation period and, over months of passes, the direction of the spin axis. The 2.7-degree-per-second figure came from exactly such ranging. Radar imaging from large ground antennas resolves the shape of the largest objects and shows their rotation directly.

Each of these gives the rotation of a body that is kilometres to thousands of kilometres away, averaged over a pass, and each misses some of what a capture needs: the nodding, the exact position of a grapple point, the inertia ratios. They are enough to choose targets and plan missions, and not enough to close a mechanism; the last estimate has to be made by the chaser, from a few metres, in the minutes before contact.

A problem of estimation #

This is where the tumbling target meets the camera that cannot see how far. An approach to a cooperative target is navigated with the target’s help: it carries reflectors, lights or radio beacons, and its attitude is controlled and known. A derelict provides none of that. Its rotation must be estimated from images — the apparent motion of recognisable features on its surface — and the estimate must include the rotation rate, the orientation of the angular momentum, the phase of the nodding and, for a body tumbling about no principal axis, the ratios of its moments of inertia, because those set the future of the motion. The images give directions, not distances, and a feature seen to move could be close to the axis and moving slowly or far from it and moving fast.

The estimation is possible because the rigid-body motion is so constrained. Euler’s equations with fixed moments of inertia allow only a small family of motions, and a sequence of images over several nodding periods pins down which one is happening. The filters that do this — tracking features, fitting a rotational state and the inertia ratios together — have been demonstrated on image sequences of real tumbling objects taken from the ground and from inspection spacecraft. But their errors are what the capture’s success depends on, and an estimate of the grapple point’s position a minute ahead is only as good as the estimate of the inertia ratios, which the first few minutes of observation constrain weakly.

Slowing the target first #

If following is expensive and grabbing on the fly is risky, the tumble can be removed before capture. Several ways have been studied. A conducting body tumbling through the Earth’s magnetic field is slowed by the eddy currents induced in it, naturally, over years; a chaser carrying a strong magnet could induce the same currents faster, without contact. A chaser could push gently on the target with a stream of ions from an electric thruster, applying a torque at a distance. Or the chaser could make brief, repeated contacts — pads touching the tumbling body at the right moments — each removing a little of its angular momentum.

Every one of these has to put the target’s angular momentum somewhere. A tumbling three-tonne body with a moment of inertia of ten thousand kilogram square metres, turning at three degrees per second, carries about five hundred newton-metre-seconds of angular momentum, and the spin a spacecraft removes has to be put somewhere — into the chaser’s own reaction wheels, which saturate far below that, or out into space as torque from thrusters, at a propellant cost that depends on the lever arm. Detumbling is therefore a propellant and time budget of its own, and whether it is cheaper than following depends on the rate and on how long the capture must last.

What the arithmetic leaves out #

The cost figures treat the chaser as holding a fixed position relative to a rotating point, which is the most demanding case. A chaser can instead fly a path that intercepts the point at the right moment without matching it continuously, spending far less; the trade is that the relative velocity at contact must be controlled, and a contact at the wrong velocity transfers momentum to the target and can set it tumbling faster. The tumble is treated as rigid-body motion, but many derelicts carry flexible appendages — solar arrays, antennas — whose flexing couples to the rotation and changes it, and fluid left in their tanks sloshes and dissipates energy on timescales of hours to days. And the orbital term is taken for a target in a low circular orbit; in an eccentric orbit, or at the close ranges where differential drag matters, it varies around the orbit and with the solar cycle, but it remains the small term at any rate of tumble worth worrying about.

Where the orbit went #

The progression of this whole subject has been from the global to the local. Launch windows and phasing are about the orbit as a whole; catching up with a target is about the relative drift over many revolutions; the close approach is about the relative orbit over one. At the end, a few metres from a tumbling object, the orbit has become a small, slowly varying bias, and the problem is the target’s rotation — a rigid body’s free motion, estimated from pictures and matched by thrusters. Whether the objects that most need removing can be removed depends much less on the orbital mechanics, which is solved, than on the rotation, which must be measured each time.

Still open: whether the fastest derelicts can be grasped at all #

The objects whose removal would most reduce the risk of further collisions are the largest, and among them are some of the fastest tumblers. The satellite measured at 2.7 degrees per second was, by later measurements, spinning faster still, and some rocket bodies spin at tens of degrees per second in flat spins that no chaser could follow. For those, the choice is between capture methods that need no synchronisation — nets, harpoons, tethers, each of which has been tested in orbit only on small cooperative targets — and detumbling at a distance, which has been studied in simulation and the laboratory but never demonstrated on a real derelict. The first demonstrations of removing uncooperative objects, now being prepared, will test both; which of them works on a large, fast-tumbling body with flexible appendages, and at what cost, is what they will find out.