A camera that cannot see how far

Everything so far has assumed that a chaser knows where its target is. The relative frame and what happens to it when the reference orbit is eccentric are both statements about how a relative state evolves, and they take the state as given.

It is not given. It is estimated, from whatever a spacecraft’s own sensors deliver, and the cheapest, lightest and most reliable of those sensors delivers two angles and no distance.

That turns out not to be a limitation of precision.

Why the blindness is exact #

The relative equations of motion in the close-approach frame are

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

and every term is linear in the state with no constant anywhere. A system of that form is homogeneous: if $(x(t), y(t), z(t))$ is a solution, so is $(\lambda x, \lambda y, \lambda z)$ for any constant $\lambda$.

A camera measures the direction to the target, which is the unit vector along the relative position. Scaling the whole position vector leaves that unit vector untouched. So the entire one-parameter family of trajectories obtained by scaling a given one produces the identical measurement at every instant, for all time.

The consequence is stronger than “the range is poorly constrained”. The likelihood is flat along that direction in the parameter space. A filter fed bearings alone will report whatever range its initial guess contained, the covariance along that direction will not shrink, and adding a hundred times as many observations changes nothing. An unobservable direction is not a noisy one.

This is the same structure as a closure phase’s immunity to a per-antenna error, read the other way round: a measurement insensitive to a quantity is a measurement that cannot determine it, and whether that is a strength or a defect depends entirely on which quantity it is.

What breaks it #

The degeneracy is a property of a homogeneous system, so anything that puts a length into the problem destroys it.

A manoeuvre. An impulse is specified in metres per second, not in units of the trajectory’s own size. Applying the same impulse to a trajectory and to its scaled copy produces two trajectories that are no longer related by any scaling, and their bearing histories separate immediately. This is the mechanism a real vehicle uses, and it is why an angles-only approach is flown as a sequence of deliberate offsets rather than as a straight line.

A known size. If the target’s physical extent is known and it is resolved in the camera, its angular size gives the range directly. That works inside a few hundred metres for a large target and not at all beyond it, which is why angles-only navigation is a far-field technique and something else takes over at close range.

A second observer. Two cameras a known distance apart triangulate. This is the reason formation-flying missions with several vehicles have a navigation problem that a single chaser does not.

And the non-linear terms. The exact relative dynamics are not homogeneous — the gravity difference between two points goes as the inverse square of each one’s distance from the centre, and that is not linear in the separation. So the degeneracy is broken by the terms the linearisation threw away, which means the range is technically observable from bearings alone. The effect is of order the separation divided by the orbital radius, which at ten kilometres from a target in low Earth orbit is one part in seven hundred, and the range becomes observable on a timescale of many orbits — the same weak signal an eccentric reference orbit supplies, and for the same reason. It is a real effect and it is not a method.

The manoeuvre is chosen for what it reveals #

Once the range is known to be purchasable, the question becomes what to buy.

The standard treatment writes the observability as a matrix and asks which manoeuvre maximises its smallest eigenvalue — which is to say, which impulse most effectively separates the trajectories that the measurements currently cannot tell apart. Three results come out of that analysis and all three are unobvious.

A radial impulse is worth more than an along-track one of the same size. An along-track impulse produces a large along-track displacement and a small change in the direction, and it is the direction that is measured. A radial impulse produces a closed teardrop that swings the line of sight through a large angle.

The best time to manoeuvre is not the start. An impulse made immediately gives the longest baseline over which to watch its effect, which argues for early; but the range estimate is worst at the start, so the manoeuvre’s own predicted effect is least well known then, which argues for later. The optimum is in between and depends on the initial uncertainty.

And a manoeuvre that is good for navigation is usually bad for the approach. The impulse that best reveals the range is one that moves the vehicle across the line of sight, and the approach wants motion along it. So an angles-only approach spends propellant going sideways, and the total cost is measurably higher than the same approach flown with a ranging sensor — the premium being the price of not carrying the sensor.

The arithmetic of one approach #

A worked case fixes how much this costs, and the numbers are small in the way everything in proximity operations is small.

Take a chaser a hundred kilometres behind a target in a four-hundred-kilometre orbit, with a camera good to fifty microradians and no idea of the range beyond a factor of two. Fifty microradians at a hundred kilometres is five metres of cross-track position — an excellent measurement of a direction and no measurement at all of a distance.

Fire a radial impulse of a tenth of a metre a second. Over the following quarter of an orbit the resulting displacement is of order $\Delta v/n$, which at that altitude is about ninety metres, and it is perpendicular to the line of sight. At a range of a hundred kilometres that is nine hundred microradians — eighteen times the measurement error — and at a range of two hundred kilometres it is half as many. The two hypotheses have been separated by nine sigma, from one impulse of a tenth of a metre a second.

That is the whole mechanism, and its efficiency is the reason the technique is used rather than merely studied. The information content of a manoeuvre scales as $\Delta v$ divided by the range and divided by the angular error, so the leverage is enormous at long range where the geometry is otherwise hopeless, and falls away at close range where it is no longer needed.

The awkward regime is the one in between. At a few kilometres the target is beginning to be resolved but not well, the angular displacement from an affordable impulse is comparable with the centroid systematic, and the approach is being flown fast enough that there is little time to watch. That band — roughly one to ten kilometres — is where the published analyses spend most of their effort, and it is where a mission that can afford a ranging sensor stops pretending it cannot.

Why anybody accepts this #

A ranging sensor is not exotic. Radar, lidar and differential satellite navigation all give range directly, and all three are flown routinely. The reason angles-only navigation is studied at all is that each of them has a condition it fails under.

Radar and lidar need power and mass, and both scale badly for a small vehicle. On a spacecraft of a few hundred kilograms the ranging sensor can be a substantial fraction of the payload.

Differential navigation needs the target to cooperate. It works by both vehicles receiving the same navigation satellites and differencing, which requires the target to have a receiver and a radio link. A defunct satellite has neither.

And every active sensor needs the target to be a good reflector. Lidar returns from an unprepared surface — one without retroreflectors, at an unknown attitude, possibly specular — are unreliable in a way that a retroreflector-equipped docking target is not.

A camera has none of those requirements. It is passive, it is light, it works on any target bright enough to see, and it works at ranges of hundreds of kilometres where nothing else does. So the far phase of an approach to an uncooperative object is flown on a camera essentially by default, and the price is the manoeuvres this essay is about.

What was actually measured #

A bearing, from a camera, is two numbers: the position of a point of light on a detector, converted into two angles through a calibration of the optics and combined with the spacecraft’s own attitude.

Three error sources dominate, and only one of them is about the camera.

The attitude. A bearing in the spacecraft’s body frame is useless; what the filter needs is a bearing in the orbital frame, which requires knowing where the spacecraft is pointing. A star tracker delivers that to a few arcseconds by fitting a measured pattern of stars against a catalogue, and its error propagates directly into the bearing.

The centroid. At long range the target is a point source and its position on the detector is estimated to a fraction of a pixel — better than the pixel size by a factor of five or ten, in the same way a photometric centroid is measured to a fraction of a pixel. At shorter range the target is resolved, and the centroid of a resolved, irregularly illuminated, tumbling body is not its centre of mass. The transition between the two regimes is a systematic that changes sign.

And the lighting. A camera sees a target only when the Sun does. An approach in eclipse produces no measurements at all, and an approach with the Sun behind the target produces a thin crescent whose centroid is displaced toward the lit limb by a substantial fraction of the target’s size.

What is verified, when such a system is flown, is the filter’s own consistency: whether the residuals are the size the covariance says they should be. A filter with an unobservable direction and an overconfident prior produces beautiful residuals and a wrong answer, which is the specific failure this essay is about.

Where the model stops #

The homogeneity is a property of the linearisation. Stated exactly, the range is weakly observable without a manoeuvre, through the non-linear terms and through the eccentricity of the target’s own orbit, which breaks the scaling in the same way that it breaks the two-to-one ellipse. Both effects are small at close range and neither is a design basis.

One camera, one target. If the target is resolved and several features on it are tracked, the relative attitude becomes observable as well, and the problem changes into something better conditioned. That is how the close phase is actually flown, and it is a different problem rather than a refinement of this one.

The estimator is not in the figures. What is drawn is the ambiguity in the dynamics. A real filter carries a covariance, a process noise and a prior, and how badly an unobservable direction behaves depends on all three — a filter with a tight prior on the range reports that prior confidently, and one with a loose prior reports a covariance that never shrinks. Neither is wrong about the data.

And nothing here is about the target’s motion. An uncooperative object is usually tumbling, and a bearing to its centroid is a bearing to something that is moving with respect to its centre of mass at a rate set by its own rotation. Separating that from the orbital relative motion is the hardest part of the real problem and none of it is in this essay.

The same blindness in an orbit determination #

The structure is not peculiar to a relative frame, and the version of it that predates spaceflight by two centuries is worth putting beside this one.

Determining a heliocentric orbit from astrometry alone is the same problem at a larger scale: a telescope measures two angles per observation and no distance, and the orbit has six parameters. The classical methods — Laplace’s and Gauss’s — recover all six, and the reason they can is that the dynamics there are not homogeneous. The Sun’s gravity introduces a length, through the relation between an orbit’s size and its period, so two candidate orbits differing by a scale factor move differently against the sky and the angles separate them.

That is a useful contrast rather than a digression. Where the dynamics contain a length, angles alone are enough; where they do not, angles alone can never be enough, and the difference between the two cases is exactly whether the equations of motion are homogeneous in the position.

The relative problem sits on the wrong side of that line because it has been linearised about a reference orbit, and linearising removed the length. The exact relative problem has the Earth’s gravitational parameter in it and is therefore weakly observable — which is the sense in which the blindness of this essay is a property of the model rather than of the universe, and also the sense in which that distinction does not help anybody on a two-day approach.

Aiming at a plane instead of at a planet is the same accounting on an interplanetary trajectory, where the quantity determined well and the quantity determined badly are set by the geometry of the approach rather than by the quality of the tracking. In each case the question to ask of a navigation problem is not how accurate the instrument is but which combinations of the state it is sensitive to at all.

The generalisation #

The structure here appears wherever a measurement is a ratio and a dynamics is linear, and it is worth stating in a form that does not mention spacecraft.

A homogeneous system has a scaling symmetry. A measurement that is a direction, a ratio, a colour or a shape is invariant under that symmetry. Put the two together and the scale is unobservable — exactly, not approximately, and no accumulation of data helps. The symmetry of the dynamics and the invariance of the instrument have to be compared before an experiment is designed, because if they match, one parameter is simply absent from the problem.

The cure is always the same in form: introduce something with the missing dimension in it. Here it is an impulse with a length in it. In astronomy it is more often a physical size or a luminosity — the reason a brightness is a distance only if something is known is precisely this argument, with the inverse-square law playing the part of the homogeneous dynamics and the standard candle playing the part of the manoeuvre.

There is a second reading, about cost. The information that resolves the degeneracy is not free and its price is in the currency the mission is most short of. A rendezvous spends propellant to learn where its target is, and the propellant spent learning is propellant not spent approaching. That trade — paying for information out of the same budget as the action — is unusual in observational work, where an observation is generally free of the thing being observed, and it is the reason angles-only guidance is posed as an optimisation rather than as an estimation.

Still open: the target that will not hold still #

What comes next is the case the last two sections keep running into. A target that tumbles presents a docking interface whose position in the relative frame is the sum of two motions on completely different timescales, and capturing it requires matching the faster one — which is a control problem with the orbital mechanics as a boundary condition rather than as the subject.

Beside it lies the filter itself: what a covariance means when one of its directions is unobservable, how a consistency test detects that, and why an estimator that is confidently wrong is a more dangerous failure than one that is honestly uncertain. That is not a question about orbits at all, and it is where the practical difficulty of an autonomous approach actually lives.