Two instruments disagree about the worst

The amplification is one number: the factor by which the instrument’s error reaches the answer. On this site’s four-bar with thirty poses read by protractor it is 1.90, bounded by 2.16.

It is the worst direction’s, and a report usually wants the parameters’.

What a single number was standing in for #

Before the arithmetic, the practice this essay is refining, because the single amplification is not a bad number and is doing a job.

It bounds everything. Whatever combination of the parameters a reader cares about, its error is at most the reading error times the amplification, so a report quoting one number has made a claim that cannot be violated. That is worth a great deal and is why the practice exists.

It is also, on any problem with a spread in its spectrum, wrong by the spread. A coordinate machine on this four-bar recovers its crank thirty times better than its rocker; quoting the worst for all six is conservative by up to thirty.

Conservatism has a cost here that it does not have elsewhere. A calibration’s numbers feed a decision — whether the machine is good enough, whether to tighten a tolerance, whether to remeasure — and an error bar thirty times too large can turn a machine that is fine into one that needs work.

So the single number is a valid bound and a poor estimate, and the two are wanted for different things.

From a spectrum to a parameter #

The error in one parameter is not one singular value. It is a combination of all of them, weighted by how much that parameter appears in each right singular vector.

Concretely: the covariance of the recovered parameters goes as (MᵀM)⁻¹, whose diagonal entries are Σ vᵢⱼ²/σⱼ² — for parameter i, a sum over the directions j, each contributing the square of that parameter’s component in direction j divided by the square of that direction’s singular value.

A parameter that lies mostly in well-conditioned directions comes back well however bad the worst direction is. One that lies mostly in the weak direction comes back badly.

So the per-parameter errors are a list of the same length as the parameter list, and the single amplification is the worst entry of a related list rather than a summary of this one.

The protractor’s four #

For a four-bar read by protractor, the four lengths’ errors relative to the reading error — computed from the pseudo-inverse over the three recoverable directions — are

g    1.413
a    0.989
b    0.579
c    2.150

The worst is the rocker at 2.150 and the best is the coupler at 0.579, a spread of 3.7. A single amplification quoted for the set would be the worst of them, and it would over-state the coupler’s error by nearly four.

In the invariants’ own coordinates the same calibration gives 1.219, 0.356 and 0.673 for K₁, K₂ and K₃ — a different list of a different length, because they are different quantities, and a report has to say which it is quoting.

The coordinate machine’s six #

Read the same machine with a coordinate machine and the spread is much larger.

The six singular values run 16.24, 15.99, 4.588, 1.017, 0.397 and 0.100 — two and a half orders — and the per-parameter errors run

g 2.440   a 0.317   b 2.592   c 9.471   u 1.416   v 1.624

A spread of thirty between the crank at 0.317 and the rocker at 9.471. Quoting the worst for all six over-states the crank’s error by a factor of thirty.

The instrument that recovers the most parameters recovers them most unequally. That is not a coincidence: two of its columns are far longer than the rest, they dominate the spectrum, and everything else has to be distinguished against them.

Why the spread is what it is #

The coordinate machine’s two-and-a-half-order spread is not bad luck and understanding where it comes from says how to reduce it.

The two largest singular values are nearly equal and an order above the rest. That pattern — two large, nearly equal — is what a pair of columns looks like when the observable responds to them strongly and almost identically at every configuration. Here the pair is the tracing point’s two coordinates: moving the tracer along the coupler moves the observed position by the coupler’s length times the change, and moving it across does the same in the other direction.

Both effects are large and both are nearly independent of the crank angle. So those two columns are long and nearly orthogonal to everything else, and the four length columns have to be distinguished from each other in what is left.

A parameter that is very easy to see makes the others harder to see, because a condition number is a ratio. Removing the two easy ones — fixing the tracing point at a known position, if it is a machined feature with its own drawing — takes the model from six parameters at 162 to four at 9.8.

Sixteen times better conditioned, by removing two parameters that were perfectly identifiable. That is not a trick and it is not free: fixing a parameter is a claim that its nominal value is right, and if it is not the error is absorbed into the four that remain.

They disagree about which is worst #

Here is the result the title is about.

Ask a protractor which of a four-bar’s four lengths is least well recovered and the answer is the rocker, at 2.150 against the coupler’s 0.579. Ask a coordinate machine and the answer is also the rocker, at 9.471 — but ask both instruments together and the answer is the ground length, at 1.597 against the rocker’s 1.147.

So the third row of the table reverses the ranking of the first two. Adding a protractor to a coordinate machine improves the rocker by a factor of eight and the ground length by only a third, which moves the ground length to the bottom of the list.

A plan built on the second instrument’s answer — measure more poses where the rocker’s column is strongest — is not the plan the third arrangement wants.

Which parameter is worst is a property of the pairing rather than of the machine, which is the same sentence the rank essay makes about identifiability and the units essay makes about the null space. It is now three quantities in a row that belong to the pairing, and it is worth taking as the field’s general shape rather than as three coincidences.

The condition number is the worst pair #

One more reading of the summary, because it makes the disagreement easier to picture.

A condition number is the ratio of the largest singular value to the smallest, so it is a statement about two directions and nothing about the ones in between. Two spectra with the same first and last values have the same condition number and can be entirely different in the middle.

The protractor’s three recoverable values are 1.975, 1.183, 0.379 — a ratio of 5.2, with the middle value comfortably placed. A hypothetical instrument with 1.975, 0.400, 0.379 has the same condition number and two weak directions rather than one.

Those two would be reported identically by a condition number and would behave quite differently: the second recovers two combinations badly, so twice as much of the answer is uncertain.

A condition number is a summary of the extremes, which is a strange statistic to summarise a list with when the list is four numbers long. It survives because in numerical analysis, where the term comes from, the matrix is usually far too large to print.

What a plan should be built on #

If a report carries several parameters and one of them matters most, the objective for a pose selection is not any of the five observability indices.

It is that parameter’s own variance: the corresponding diagonal entry of (MᵀM)⁻¹. Maximising its reciprocal selects poses where that parameter’s column is most nearly orthogonal to the others, which is a different set from the one that maximises the smallest singular value.

On this machine the two selections agree on the first three poses — three directions have to be spanned and both rules find them — and part company at the fourth. The single-parameter rule then concentrates where that parameter’s column is longest; the σ_min rule keeps bisecting gaps.

A plan chosen for one parameter recovers that parameter better than a σ_min plan does and recovers the others worse, which is what optimising for one thing means.

A pose set is a design and a design needs a specification, and calibrate the machine is not one.

Both instruments beats either #

The third row of the table is the interesting one and it is the one nobody would predict.

A protractor recovers three parameters and cannot see the fourth. A coordinate machine recovers six. Adding the protractor’s readings to the coordinate machine’s therefore adds nothing in terms of what is recoverable — the same six, no more.

It adds a great deal in terms of how well. The condition number goes from 162 to 26, six times better; the per-parameter list tightens from a spread of thirty to a spread of under seven; and the worst entry falls from 9.471 to 1.597, a factor of six.

The reason is that observability is a property of a direction rather than of a parameter. The coordinate machine’s weakest direction is some combination of the six that its rows barely distinguish, and the protractor’s rows — which carry no size information at all — happen to be highly informative in exactly that direction.

Two instruments in different units are not redundant. A direction one of them cannot see at all may be a direction the other sees best, and the union is better than either.

That is worth having as a design rule for a measurement rig. The instinct is to buy the most capable instrument and use it alone; the arithmetic says to add a cheap one whose readings are of a different kind.

The list is short #

The recommendation that follows from all of this is the same one the indices essay reaches, and it is worth arriving at twice from different directions.

A four-parameter problem has four numbers in its per-parameter error list. A six-parameter one has six. Those fit on a line, they are exactly what the decomposition already produced, and every question this essay raises is answered by looking at them.

                        g      a      b      c      u      v
protractor, 20 poses    1.413  0.989  0.579  2.150
coordinate machine      2.440  0.317  2.592  9.471  1.416  1.624
both instruments        1.597  0.342  1.341  1.147  0.238  0.255

A reader given those three rows can see which instrument recovers what, which parameters are the weak ones, and which plan dominates which — without being told any single amplification.

Compression is what created the disagreement, and the object being compressed is small enough not to need it.

An amplification is not a standard deviation #

A precision that keeps the numbers usable, because the table looks like a set of error bars and is not quite.

The entries are ratios: how much the answer’s error is, relative to the instrument’s. Multiplying by the instrument’s actual reading error gives an error on the parameter, and that step needs the instrument’s error to be known — which it usually is not, directly.

What is available is the calibration’s own residual, which estimates the data’s inconsistency and is generally an excellent estimate of the reading error. So the chain is: residual, times amplification, gives the parameter’s error.

Two cautions about that chain. It assumes the readings’ errors are independent and of similar size, which a systematic error violates — and a systematic error also displaces the answer by the amplification without inflating the residual, so the chain under-states the total error whenever one is present.

The residual measures the random part and the amplification converts it. The systematic part is invisible to both and is what a residual’s distribution is for.

Selecting for one parameter, in practice #

The recommendation to select poses for one parameter’s variance deserves a note on what it costs, since it is the least standard thing in this essay.

The objective is one diagonal entry of the inverse of MᵀM, which means an inverse of a small matrix per candidate rather than a decomposition. For a four-parameter problem that is a 4 × 4 inverse, which is faster than the decomposition the σ_min rule needs.

So it is cheaper, not more expensive, and the reason it is not standard is that it requires somebody to have decided which parameter matters. That decision is the same one an observability index quietly makes by choosing to summarise the list one way rather than another, and making it explicitly is strictly better than making it by choosing a formula from a paper.

The hard part of a measurement plan is the specification, not the optimisation, and both the σ_min rule and this one are cheap once there is one.

What the list does not settle #

Two things, and they are the same two that limit every derivative-based instrument in this field.

It assumes the model is right. A per-parameter error is computed from the columns that are in the model, and a model missing a parameter produces an excellent list and returns numbers that are not the machine’s.

And it is a local statement. Two well-separated parameter vectors both fitting the data have ordinary error lists at each, and nothing in the list detects the other.

So the list answers the narrow question well — given the model and given uniqueness, how much does each parameter’s error come to — and it is one of the three or four questions a calibration has to answer.

The same disagreement between pose sets #

Instruments are the sharpest case and the phenomenon is more general: any two things that change the identification Jacobian can disagree about which parameter is worst.

Two pose sets on the same machine with the same instrument can. A plan optimised for the smallest singular value is optimising one direction of the row space, and which parameters that direction mostly consists of depends on the mechanism — so improving the worst combination can move the ranking of the individual parameters without being asked to.

That is harmless when the report carries all of them and it is not harmless when a downstream calculation leans on one.

A single figure of merit hides a rearrangement, and the rearrangement is what the list shows. Two plans with the same headline number and different lists are different plans, and the headline is precisely the statistic that cannot tell them apart.

That completes the pattern this essay is about. Change the instrument and the worst parameter changes; change the poses and the ordering changes; change the summary and the ranking of two plans changes. Three ways for one number to conceal a difference, all of them fixed by printing four.

Where the numbers came from #

For anybody reproducing this: the per-parameter figures are diagonal entries of the inverse of MᵀM, square-rooted, on the identification Jacobian at the answer, with each block of rows scaled to unit root-mean-square where an instrument reads more than one kind of quantity.

That last step is not cosmetic. Rows in radians and rows in millimetres cannot be compared without it, and the per-parameter errors afterwards live in the scaled frame — so they are ratios against the instrument’s own error rather than absolute figures.

Quoting them as ratios is the right choice anyway. An amplification is the useful form, because it lets a reader with a better instrument compute their own answer by multiplying, and because it is a property of the mechanism and the poses rather than of the equipment.

That is also why the list can be published with a machine rather than with a measurement. A four-bar of these proportions, measured at these poses by an instrument of this kind, amplifies by these amounts — a statement about a design, computable from a drawing, and true for every machine built to it. Nothing in the list depends on a machine having been built, which makes it a specification rather than a result and puts it where a designer can use it.