How many poses are enough

The question sounds like it should have a rule of thumb attached to it and it has two different answers, which is why the rules of thumb disagree.

How many poses are needed for the parameters to be determined at all? As many as there are parameters, and not one more. Three readings of a four-bar’s input and output angles fix its three invariants exactly, by a linear solve, with no residual and nothing left over.

How many are needed for the answer to be worth having? Enough that the instrument’s error does not arrive in the answer multiplied by something large — and that is a question about conditioning, which is a different quantity from rank, improves for a different reason, and flattens out sooner than anybody expects.

Rank first, and it arrives immediately #

A four-bar read by protractor has three identifiable parameters. Its identification Jacobian has four columns, one combination of which is exactly zero, so the rank cannot exceed three however many rows are stacked.

It reaches three at three rows. Three poses of the site’s four-bar give a matrix whose singular values are 0.774, 0.497, 0.117 and 1.5 × 10⁻¹⁵⁷ — full rank on the identifiable subspace, at the smallest number of readings that could possibly do it.

That is not a surprise and it is worth saying anyway, because it disposes of a common way of thinking about calibration. More data does not increase rank. A deficiency at three poses is a deficiency at three hundred, and a rank that is full at three stays full. Rank is a property of the columns, and adding rows never changes a dependency among columns.

So if the question is can these parameters be recovered, the answer is available from a handful of poses and does not improve. Everything else is about how well.

Then conditioning, and it improves slowly #

The number to watch after that is the smallest singular value on the identifiable subspace — the observability of whichever combination of parameters the readings determine worst.

At three poses it is 0.1183. At six, 0.2077. At ten, 0.2680. At twenty, 0.3790.

Two things about that sequence. It rises steadily rather than jumping, and it rises slowly: doubling the poses from ten to twenty buys a factor of 1.41, which is exactly the square root of two and is the signature of averaging rather than of new information. The interesting part of the curve is over by about ten poses; after that the calibration is not learning about the machine, it is beating down noise.

The condition number tells the same story from the other side. Over the recovered directions it is 6.5 at three poses, 5.5 at four, and 5.2 from five onwards — flat to the printed digits for every pose count after that. The shape of the problem is fixed by five poses. What later poses change is the overall scale of the matrix, not the ratio between its best and worst directions.

Why it flattens #

The reason is worth having because it says when the flattening will happen on a different machine.

Each pose contributes one row, and a row is a direction in a four-dimensional space. Three parameters means three directions have to be spanned, and after a handful of well-spread poses they are spanned — a fourth row lands in a space the first three already cover, so it adds no direction, only weight.

The information in a pose is a direction in a space of fixed dimension. That dimension is the number of parameters, not the number of readings, so the number of poses at which the curve flattens is set by how many parameters there are and not by how big the machine is or how long the measurement takes.

For a three-parameter problem the flattening is at five or six. For a six-parameter one it is later, and for a serial arm’s thirty later still — but always at some small multiple of the parameter count, and never at the hundreds that a routine calibration procedure asks for.

Where beats how many #

The upper curve in the first figure is what happens when each next pose is chosen rather than taken in turn.

At eight chosen poses the observability is 0.2957. At twelve evenly-spaced poses it is 0.2935. Eight chosen readings are worth twelve taken in order — a third fewer measurements for the same answer, from a machine and an instrument that have not changed.

The gap is largest in the middle of the range and closes at both ends. At three poses choosing is worth a great deal, because three badly-placed poses can be nearly dependent and three well-placed ones cannot. At twenty poses both sets have covered the turn and both read 0.3790, identically.

So the advice that falls out is not “measure more” and it is not “measure less”. It is that where the poses are matters more than how many there are, over exactly the range where a calibration is actually planned, and that the choosing is cheap: it is a few hundred decompositions of a matrix with four columns, which takes less time than moving the machine once.

What the extra poses do buy #

Averaging, and averaging is worth having — it is just worth knowing what kind of gain it is.

With independent readings each carrying an error of size ε, the error in the recovered parameters falls as ε/√n. Going from ten poses to forty halves it. That is a real improvement and it is available without thinking about pose placement at all.

It is also the only thing extra poses buy once the directions are spanned, and it obeys a law with no surprises in it. Sixteen times the measurements for four times the accuracy; a hundred times for ten. Anybody weighing an afternoon of measurement against a better instrument should be comparing √n against the instrument’s own factor, and the instrument’s error arrives multiplied by a constant that pose placement can change and pose count cannot.

Put the two together and the shape of a good measurement plan is clear. Choose a small number of poses well, then repeat them, rather than taking many poses badly. Repetition buys the √n and costs nothing in placement; more distinct poses buy the same √n and can be worse placed than the ones already taken.

Three poses, and what they cannot settle #

The rank arrives at three and the residual does not, and that difference decides whether a three-pose identification is a good idea.

Three poses and three parameters is a square system. It has one solution, the residual is zero by construction, and nothing about the answer says whether the readings were any good. An instrument with a fault, a machine that moved between readings, a transcription error in one number: all of them produce a perfectly consistent set of three K’s and no signal at all.

A fourth pose changes that completely. Four rows against three columns is over-determined, the residual is no longer zero, and its size is a statement about the data. That is the cheapest quality check in the whole field and it costs one measurement.

This is the practical reason the minimum is never the right answer even when it is sufficient. Three poses determine the parameters and cannot be checked; five determine them and can. Reading a residual is a subject of its own, and the precondition for having one at all is measuring more than the minimum.

A number that is not the answer #

There is a temptation to convert all of this into one figure — measure four times as many poses as parameters or something like it — and the temptation should be resisted for a reason the curves make visible.

The flattening point depends on the spread of the poses, not only on how many there are. Six poses crowded into fifty degrees of crank travel give an observability of a fraction of what six spread over the turn give, because their rows are nearly parallel. A rule in poses-per-parameter is silent about the one variable that matters most.

What can be quoted as a rule is the shape: rank at p poses, most of the conditioning by about 2p, and √n thereafter, where p is the parameter count. Any specific number of poses that follows from that depends on the mechanism and on how much of its travel is reachable, and both are computable in advance from the drawing.

The machine’s own travel comes into it #

A four-bar whose crank turns all the way round can be measured anywhere on the circle, and every pose count above is quoted for one of those. A machine that rocks cannot.

A double rocker at 4, 3.2, 1.4, 3.0 reaches seven of the twenty-four positions in the sweep this field uses, so a twenty-four-pose plan on that machine returns seven rows. The rank is still three, because seven rows are more than three columns.

What is surprising is the conditioning. Those seven poses give a condition number of 2.77 and a σ₃ of 0.649, against the crank rocker’s twenty-four at 5.21 and 0.415 — the machine that reaches less is the better-conditioned measurement, by a factor of one and a half in the quantity that matters.

The reason is that conditioning is about the rows’ directions and lengths rather than about their angular spread, and a double rocker’s output angle is more sensitive to its lengths over the arc it does reach. An arc is not automatically a bad pose set, which is worth having because the instinct says otherwise.

What is automatically bad is crowding, and the distinction is worth measuring rather than assuming. Six poses spread round the turn give κ 5.22 and σ₃ 0.207; six poses crowded into 0.9 radians of the same machine give κ 47.2 and σ₃ 0.033, six times worse in the ratio and six times worse in the worst direction.

So the quantity to avoid is not a short travel but a narrow one relative to how fast the mechanism’s derivatives change. A double rocker’s arc is short in angle and wide in what the rows do; a crowded set on a crank rocker is the opposite.

The count that is actually decided #

Bringing the two answers together, here is what a plan looks like on the site’s four-bar read by protractor.

Three poses determine the shape and cannot be checked. Five give a residual and reach the flat part of the condition number. Eight, chosen, give the observability that twelve evenly spaced ones give. Twenty give 0.3790, and forty give the same 0.3790 with the noise halved.

So the answer to the essay’s title is about eight, plus as many repeats as patience allows, and the eight matter far more than the repeats. That is a smaller number than most calibration procedures specify, and the reason procedures specify more is that they are written for models with far more parameters — which is a good reason, and one that scales with the parameter count rather than with a habit.

Two poses, and the shape of failing #

It is worth looking at what happens below the minimum, because the failure is not the one a reader expects.

Two poses of a four-bar give a matrix with two rows and four columns. Its rank is two, so two combinations of the parameters are determined and two are not — one of them the scale direction, which was never going to be determined, and one of them a combination that would have been determined by a third reading.

A fit handed that data does not fail. It converges, the residual goes to zero because two equations in four unknowns always can, and four lengths come back. They fit the two readings exactly and they are not the machine’s, and nothing in the output distinguishes this case from a successful calibration except a rank the fit never computed.

That is the argument for computing the rank rather than trusting the fit, and it is why this field’s routines refuse an identification with fewer readings than parameters rather than answering it. A least-squares problem with more unknowns than equations is not ill-posed in a way that shows up as an error; it is under-determined in a way that shows up as a confident answer.

Repeats are not poses #

The distinction between measuring more positions and measuring the same positions more often is easy to lose and it has a clean statement.

Measuring a new pose adds a row that may point in a new direction. Measuring the same pose again adds a row that points in exactly the direction of one already there. The first can raise the rank and improve the conditioning; the second can do neither, and its whole contribution is to average that row’s noise.

So the two are different operations on the same matrix, and the curve in the first figure measures only the first of them. A twenty-pose plan measured once and a five-pose plan measured four times take the same time and give different answers: the first has better conditioning, the second has the same conditioning with half the noise on each row. Which is better depends on whether the mechanism’s worst direction is at 0.19 or at 0.38, and that is exactly what the curve says.

The arrangement nobody should choose is the one that happens by accident: many poses, crowded, measured once. It buys neither the directions nor the averaging, and it takes as long as either.

When the parameter count goes up #

The last figure is the check that this essay’s rule is about the parameter count rather than about four-bars.

Read the same machine with a coordinate machine and there are six parameters instead of three. The curve has the same shape — steep, then flat — and the knee moves out, because six directions take longer to span than three. The rank arrives at three poses rather than at three readings, since each pose now contributes two rows; the conditioning keeps improving usefully to around a dozen poses rather than around six.

Nothing else changes. The plateau is still a plateau, the gain past it is still √n, and choosing still beats not choosing by about the same fraction.

That is the sense in which the rule generalises. The knee is at a small multiple of the parameter count, wherever the parameter count comes from. A model with thirty parameters wants scores of poses and not thousands, and a procedure that asks for thousands is either averaging deliberately or has not been thought about.

The plan is drawn on a machine that does not exist #

One last complication, and it is the reason this essay’s numbers should be read as a shape rather than as a specification.

Every curve here is computed from the nominal four-bar. The machine being measured is not that machine — if it were, there would be nothing to calibrate. So the observability a plan promises is the observability the nominal machine would have given, and the real one gives something slightly different.

How different is easy to bound and reassuringly small. A machine out of true by a per cent has an identification Jacobian whose entries are out by a per cent, so its singular values are out by a per cent, and a plan that promised 0.2957 delivers something within about a per cent of it. The plan is not fragile.

What is not small is the effect on reachability, and that is a different quantity with a step in it rather than a slope. A crank rocker whose rocker is one per cent shorter than nominal has limit positions a degree or two away from where the plan put them, and a planned pose that sat inside the travel by half a degree is now outside it. The observability moves by a per cent and the pose disappears entirely.

That asymmetry — smooth in the conditioning, discontinuous in the reachability — is why a plan should keep its poses away from the limits even though the limits are not where the objective would avoid anyway. It is also an essay of its own.

What this does not settle #

Two things, both of which the following essays take up.

Which poses. The upper curve above comes from a selection rule, and the rule maximises one particular number. Four other numbers are in use and they disagree about which set of eight is best, which means “chosen” is not a single thing.

And whether the poses exist. A pose plan is drawn up against the nominal machine, and the machine that gets measured is not the nominal one. A crank rocker out of true by enough to matter reaches a slightly different set of angles, so a plan can specify a pose the machine declines to visit — which is not an error, and is a piece of information the plan did not expect to get.