Where a calibration should measure

A calibration has to be told where to measure and the usual instruction is spread them out. That instruction is nearly right, which is the most difficult kind of instruction to improve on.

Here is what happens when the choice is made by arithmetic instead.

The rule, stated #

Take a pool of candidate poses — thirty-six evenly spaced round the crank, say. Start with none. Repeatedly add whichever candidate makes the smallest singular value of the resulting identification Jacobian as large as it can be, restricted to the identifiable subspace.

The restriction matters and it is not a technicality. On an angle-only problem the smallest singular value of the whole matrix is exactly zero for every pose set whatever, because one direction is invisible no matter what is measured. Maximising it would be maximising nought. So the objective is the smallest singular value over the directions that are recoverable, which for a four-bar read by protractor is the third of four.

That is the whole rule. It is greedy, it is not optimal, and it is close: against four hundred random subsets of the same size it comes within three per cent of the best any of them found, which is measured rather than assumed and is not a proof of anything.

What it does #

The first three poses land at 17.2°, 287.2° and 137.2°. Their gaps round the circle are 120°, 150° and 90° — about as spread as three points on a circle get, and not evenly so.

They have to be. Three parameters need three independent rows, and rows of the identification Jacobian at nearby poses are nearly the same row — the derivative of the output angle with respect to a length does not change much over five degrees of crank. Three crowded poses give a matrix that is technically rank three and practically rank one and a half, and the objective sees that immediately.

Then 327.2°, 127.2°, 317.2°, 117.2° and 147.2°. Each of those goes into a gap the earlier ones left, and the later ones cluster into the two regions the first three bounded most widely rather than continuing to bisect evenly.

Nothing in the rule mentions spreading. It maximises a number computed from a decomposition. Spreading is what maximising that number turns out to mean, on this mechanism, and arriving at a well-known experimental-design pattern from a mechanism’s own conditioning is a better argument for the pattern than the pattern is for itself.

What each one is worth #

The trail of singular values as the set grows, from a pool of thirty-six: 0.1515 at three, then 0.1843, 0.2259, 0.2516, 0.2809, and 0.3017 at eight.

The steps are 0.033, 0.042, 0.026, 0.029, 0.021. Remarkably even for a greedy rule — each of the first eight poses is worth roughly what the last was, which is what a greedy rule buys by always taking the best remaining candidate, and is not what the evenly spaced curve does.

The comparison the figure above makes uses a smaller pool and gives 0.2957 at eight chosen against 0.2935 at twelve evenly spaced. Both numbers are for the same mechanism and the difference between them is the pool — which is worth noticing, because it says a selection’s value depends on what it was allowed to choose from as well as on how it chose.

Past about fifteen poses the increments fall under 0.005 whatever the pool. By then there is no gap left worth bisecting and the remaining candidates all sit near something already taken, so choosing and not choosing converge: at twenty out of thirty-six they agree to four decimal places, because twenty poses spread over a turn is twenty poses spread over a turn however they were picked.

Where it is worth the trouble #

The honest summary of the numbers is that pose selection is worth about a third of the measurements, over the range where a calibration is planned, and nothing at all outside it.

Below four poses there is not much to choose: any three well-separated poses are about as good as any other three. Above about fifteen the pool is saturated. In between — five to twelve poses, which is exactly the range the previous essay argues is the right range — the chosen set is worth around four extra evenly-spaced poses, consistently.

There is a second case where it is worth much more and it does not show in these curves. When the machine’s travel is an arc rather than a turn, the candidate poses are crowded to begin with, their rows are all nearly parallel, and the difference between the best five and a careless five is large. A crank rocker measured over its rocker’s swing, or a double rocker which never has more, is that case.

The objective is a choice #

The rule above maximises the smallest singular value. That is one of at least five numbers in use for exactly this purpose, and they do not agree.

The geometric mean of the singular values rewards a set that is broadly informative and tolerates one weak direction. The reciprocal condition number rewards evenness and is indifferent to overall size, so it will happily prefer a set that recovers everything equally badly. The smallest singular value alone — the one used here — rewards the worst-recovered parameter and is indifferent to how good the others are.

Which is right depends on what the answer is for. If one parameter matters more than the others, none of the five is right and the objective should be that parameter’s own variance. If the report carries all of them, the smallest is the defensible default, because it is the one that bounds the worst thing that can happen.

The disagreement between the five is measurable and it is not small: at eight poses they differ by a factor of 1.5 about how much of the job has been done, and two of the four say it is already finished.

Greedy, and what that costs #

A greedy selection is not the best selection and it is worth being exact about the gap.

The exhaustive problem is a choice of k poses from a pool of n, which at eight from thirty-six is thirty million subsets and a decomposition each. Nobody does that. The greedy answer takes k passes over the pool, which at eight from thirty-six is about two hundred and eighty decompositions of a matrix with four columns — under a second.

Against four hundred random subsets of size five from a pool of twenty-four, the greedy set scores 0.2329 and the best random one scores 0.2396. The greedy set is 2.8% worse than the best of four hundred random tries.

That is the honest statement and it is not a bound. It says the greedy answer is in the right region and it does not say no better set exists; four hundred random subsets out of forty thousand is a sample. A selection rule this cheap being within a few per cent of anything is the useful fact, because the alternative it is competing with is not the optimum, it is whatever a person would have chosen.

What the pattern is avoiding #

Looking at which poses the rule never takes early is as informative as looking at which it takes.

It does not take poses near the dead centres. At a limit position the output link stops and reverses, so its angle is stationary in the crank angle and nearly stationary in the lengths too — the row there is short, and a short row contributes little whatever direction it points in.

It does not take pairs of poses symmetric about a dead centre, either, until it has run out of other options. Those two rows are nearly the same row with the sign of one term flipped, so the pair carries not much more than one of them.

Both of those are statements about the mechanism, arrived at by a routine that knows nothing about mechanisms. That is the recurring pleasure of this field: the arithmetic keeps rediscovering things the subject already knew, and the rediscovery is a check on both.

A pool is a decision too #

The rule chooses from a pool and the pool is supplied. That is a second decision hiding inside the first, and it has a size to it.

A pool of thirty-six evenly spaced candidates on a full turn is ten degrees apart, which is finer than the difference the objective can see between neighbouring poses — so refining the pool further changes nothing, and the chosen set from a pool of seventy-two is the same set to within one pool spacing. Coarsening it does bite: a pool of twelve forces the first three poses onto a thirty-degree grid, and the set that results is measurably worse than one from a pool of twenty-four.

There is a practical reading. If the machine can be positioned to any angle, a pool of two or three dozen is enough and more is waste. If the machine indexes — a Geneva drive or a ratchet, where the positions are what they are — then the pool is the machine’s own stops, and the selection is genuinely constrained rather than merely discretised.

The second case is the one where the arithmetic earns most, because a person choosing by eye from a list of stops has no way to tell that two of them contribute nearly the same row.

Where the poses go on a different machine #

The pattern above — spread, then bisect — is not universal, and it is worth seeing one case where it is not.

The site’s crank rocker turns all the way round and its output angle responds to every length at every position, so no part of the turn is uninformative and the objective’s only lever is spread. A drag link, whose crank and coupler both rotate fully, behaves the same way.

A machine whose response is strongly uneven does not. A four-bar close to a change point has a small region of crank angles in which the output moves very fast and the derivatives are large, and rows from that region are long. The objective then wants some poses from the fast region — because long rows carry weight — and some from elsewhere, because rows from one region all point the same way. The pattern is two clusters rather than an even spread, and a person told to spread the poses out would have missed it.

That is the general form of what the selection is for. When intuition and the objective agree, the objective is a check. When they disagree, the objective is right about the matrix and the question is whether the matrix is the right thing to be right about — which is a question about the objective rather than about the rule.

Choosing against a parameter rather than against all of them #

The objective used here treats the parameters as equally interesting, and often they are not.

A designer who cares about the crank’s length and not about the frame’s has a different question: not how observable is the worst combination but how well is this one parameter recovered. That is a different number off the same decomposition — the length of the corresponding row of the pseudo-inverse — and maximising it selects a different pose set.

The two rules agree less than one would hope. A set chosen for the worst combination spreads; a set chosen for one parameter concentrates wherever that parameter’s column is longest, which for the crank on this machine is near the positions where the crank is perpendicular to the coupler. Four poses chosen the second way recover the crank better than four chosen the first way and recover the frame length considerably worse.

That is not a defect of either rule. It is the fact that a pose set is a design and a design needs a specification, and calibrate the machine is not one. The specification that makes the smallest-singular-value rule correct is: the report will carry every parameter and no one of them is more important. Most calibrations are written as though that were true and few of them mean it.

Two rules that are not on this list #

Worth naming, since both are common and neither is what the arithmetic above does.

Poses chosen for coverage of the workspace. A plan that spreads the tool point evenly through the reachable region is spreading in the wrong space: what has to be spanned is the space of parameter directions, and two configurations with the tool in very different places can contribute nearly the same row. On a serial arm this is the standard mistake, because the workspace is what a person can see and the row space is not.

Poses chosen at the extremes. The instinct that the ends of travel are informative is right about many measurements and wrong about this one. A limit position is where the output is stationary, so its row is short. The selection here takes poses near the limits late and reluctantly, and it is right to.

Both mistakes have the same shape: they optimise something visible in the machine rather than something visible in the matrix. The matrix is the thing the answer comes out of.

The rule is cheap enough to be a habit #

A last practical point, because a technique that costs anything gets skipped.

Selecting eight poses from thirty-six candidates on a four-bar is two hundred and eighty decompositions of a matrix with at most eight rows and four columns. Each decomposition is a handful of Jacobi rotations. The whole selection is well under a second, and the identification Jacobian it needs is built from the nominal dimensions the drawing already carries.

So there is no case in which a plan should be drawn up by eye. The comparison is not between an expensive optimisation and a quick judgement; it is between a second of arithmetic and a guess, and the arithmetic wins by a third of the measurements over exactly the range where measurements are being planned.

What makes this worth writing down rather than assuming is that the cost was not always this low. An exhaustive search over eight from thirty-six is thirty million subsets and is genuinely expensive; a greedy pass is two hundred and eighty. The reason pose selection is a habit rather than a project is the greedy rule, and the reason the greedy rule is acceptable is the measurement against four hundred random subsets rather than a theorem.

What a plan is #

Putting it together, a measurement plan for a four-bar read by protractor is eight numbers and it can be written down before the machine is touched.

Take the machine’s travel and place a pool of candidates across it, three dozen or so, or the machine’s own stops if it indexes. Compute the identification Jacobian at each candidate pose from the nominal dimensions — which is all that is available, and is close enough, because the machine is nearly what its drawing says. Run the selection. Print the eight crank angles and the observability the set achieves. Take the readings in a scrambled order, repeat the first pose last as a drift check, repeat the whole set for the √n, and fit.

One more thing belongs in the plan and it is easy to leave out: the order. Poses taken in sequence round the turn make a machine that drifts during the measurement look like a machine with a geometric defect, because both produce a residual correlated with the crank angle. Taking them in a scrambled order converts a drift into scatter, which is a much less confusing thing to find, and it costs nothing at the bench.

The one part of the plan that can go wrong is the first step: it is computed on the nominal machine and executed on the real one. Some of the planned poses may not exist, which is the subject of a later essay and is not a disaster — a pose that cannot be reached is dropped and counted, and eight poses that become seven are still seven well-chosen ones.