What this field cannot measure

The practice field’s boundary essay exists because a limit stated in eight places looks like one limit and was four. This field has three, they are of different kinds, and only one of them is what a reader would guess.

One: a direction the readings cannot span #

The first is exact, computable and completely understood.

An instrument whose readings are dimensionless in the lengths cannot see a mechanism’s size. The parameter vector is a null vector of its own identification Jacobian, measured at 7.6 × 10⁻¹⁵, and the fourth singular value is 4.5 × 10⁻¹⁵ against a largest of 2.16.

This limit is the best-behaved thing in the field. It is decided by a rank, the rank decision has a gap of fourteen orders, the missing direction is named in closed form, and the repair — one reading with a length in it — is cheap and complete.

A limit that can be computed, named and priced is not much of a limit, and this one is properly a feature of the measurement rather than a boundary of the field. It is here to be contrasted with the two that follow.

The three are not the same kind of thing #

Before taking them in turn, the distinction that makes the essay worth writing.

The first is a limit of a particular measurement: change the instrument and it disappears entirely. It is not a boundary of the field at all, and it is here because it is the one a reader expects and because contrasting it with the others is the quickest way to see what a real boundary looks like.

The second is a limit of an apparatus: the derivative-based instruments this field is built on cannot see it, and other instruments can, at a cost. It is a boundary of the method rather than of the subject.

The third is a limit of evidence: no instrument, no method and no amount of data reaches it. It is the same kind of statement as the site’s exclusion of friction — a place where the standard of evidence runs out rather than where the ambition does.

Filing all three under what a calibration cannot determine loses that, and losing it is how a boundary comes to look like a complaint about difficulty.

Two: an alternative no derivative detects #

The second is exact and is not computable by the field’s main instrument.

Three cognate linkages trace one coupler curve. Their cranks differ by sixty per cent, each fits a path measurement exactly, and the identification Jacobian at each of the three is of full rank with an unremarkable condition number.

A derivative describes a neighbourhood. An alternative that is not in the neighbourhood leaves no trace on any derivative, so the rank, the null space, the condition number and all five observability indices pass at each of the three.

The available instruments are a multi-start search, which samples and does not prove, and an algebraic construction where one exists — Roberts’s here, homotopy continuation in general, which is complete and expensive.

So the second limit is a limit of the derivative-based apparatus rather than of measurement. It is resolvable, by information of a different kind: knowing where the ground pivots are settles the cognates, and one photograph settles a branch.

Three: a model nobody thought of #

The third is the real boundary and nothing in the field touches it.

Every instrument here works on the columns that are in the model. A property the machine has and the model does not has no column, so no rank sees it, no condition number sees it, and no search over parameter space finds it — the search is over a space that does not contain it.

What is left is the residual, and the residual’s power depends on how nearly the missing property resembles something in the model. A tracing point 0.198 units off produces a residual of 4.2 × 10⁻³, which is visible against a good instrument. A missing property whose effect is nearly parallel to a parameter already present produces almost no residual at all, and the parameter it is absorbed into is wrong by an amount nothing reveals. There is no bound on how wrong: the absorption is exact in the limit of parallel columns, so an arbitrarily large error in the machine can produce an arbitrarily small residual.

That case is not detectable from data by any method. It is a limit of the same kind as the site’s exclusion of friction: not a difficulty to be worked around, but a statement about where the evidence runs out.

Three limits, three instruments #

Setting them side by side makes the shape of the field’s evidence visible, and the shape is that the three are not comparable in strength.

limit                         instrument            what it gives
a direction not spanned       a rank                a proof
an alternative elsewhere      a search              a sample
a model not thought of        nothing               nothing

The first column is what can go wrong; the second is what looks for it; the third is what the looking is worth.

A rank proves. Its decision is made across a gap of fourteen orders, the missing direction is named in closed form, and no amount of further measurement changes it.

A search samples. Twenty fits from scattered starts finding three answers is evidence that there are at least three and no evidence that there are exactly three, unless an algebraic construction supplies the count.

And the third has no instrument at all. That asymmetry is the most important thing in this essay, because a reader who has seen the field’s exact results — a null space at 10⁻¹⁵, a rank across fourteen orders, a bound checked against a measurement — will reasonably expect its limits to be equally sharp, and one of them is not a limit that can be measured up to.

What is established, precisely #

Collecting the qualifications into one sentence, because the sentence is what a report is entitled to say.

Among the models supplied, near the answer found, and given that the readings are what they claim, these combinations of parameters are determined to these accuracies.

Each clause is doing work. Among the models supplied excludes the third limit. Near the answer found excludes the second. Given that the readings are what they claim excludes a systematic instrument error, which is indistinguishable from a missing parameter without moving the instrument.

What remains is a strong claim and a narrower one than the machine’s dimensions are these.

Why the third limit is not a failure of rigour #

It would be easy to read the third limit as an admission that the field’s apparatus is incomplete, and it is worth resisting that reading because the same limit applies to every measurement of anything.

A measurement compares a model’s predictions against readings. If a model cannot express what the machine does, the comparison detects it only through the leftover, and if the leftover is small the comparison detects nothing. That is not a property of identification Jacobians or of mechanisms; it is what fitting a model is.

What this field can add is a statement of how large the leftover would be for each candidate property. A tracing point offset of 0.198 gives 4.2 × 10⁻³ of residual. A joint axis half a degree out gives some other number. Those are computable in advance, from the drawing, for any property somebody names.

So the honest position is: for every property considered, the field can say how visible it would be; for properties not considered, it can say nothing. That converts an open-ended worry into a checklist with numbers on it, which is as far as the apparatus goes and further than it might appear.

What the field does not model #

Beyond the three limits, a list of things a calibration here does not attempt, so the exclusions are as clear as the practice field’s.

Anything needing a material property. No friction, no stiffness, no wear. That is the site’s standing boundary and this field respects it: every input here is a reading, a hole position or a stated process fraction, and every one is checkable by anybody with the parts.

The instrument itself. An encoder’s eccentricity, a coordinate machine’s own error map, an interpolation nonlinearity — all real, all properties of a device rather than of a geometry, and all outside. That is why a self-calibrating machine is self-consistent rather than correct.

Statistics beyond a variance. Errors are combined in quadrature and worst cases are summed. Whether a shop’s positioning errors are Gaussian and what happens in their tails is a question this site hands elsewhere.

And time. A machine that changed during the measurement is not one machine, the field’s models assume it is, and the only defence offered is a habit — scramble the pose order and repeat the first pose last.

What a residual can and cannot rule out #

The residual is the only instrument that looks outside the model, and it is worth being precise about what a small one licenses.

A residual at the instrument’s own repeatability says: no unmodelled effect larger than the instrument’s precision, in any direction the instrument can see, at the poses measured. That is a real and useful exclusion.

It says nothing about an effect the instrument cannot see — a property that changes something not being read. It says nothing about the poses not measured, which is why measuring a few poses outside the fitted range is the cheapest available extension of the exclusion. And it says nothing about an effect nearly parallel to a parameter in the model, because that effect has been absorbed rather than left over.

So the exclusion is bounded in three ways, and each bound is closable a little: a second instrument widens the first, extra poses widen the second, and only thinking widens the third.

A small residual is evidence of absence, over a stated range, for effects of a stated kind. Reporting it without the range and the kind is what turns a bounded exclusion into an unbounded claim.

Where the boundary moved #

The tolerance field’s boundary essay ends by naming the first thing the work after it should do: take the feature positions as the variables and derive the lengths from them, since two holes bored in one setup share an error and no analysis on the lengths can see it.

That is done, and it turned out to be the same subject as identification rather than a neighbour of it. Both ask where a number on a drawing comes from and what could confirm it. A length is derived from holes and a parameter is derived from readings, and neither is the thing that exists.

So the line the practice field drew has moved once more, in the same direction and for the same reason: the quantity turned out to need no material property, only a willingness to compute one layer further out.

What has not moved is the test. Can every input be read off a drawing or an instrument? A hole position can. A setup’s shared fraction can, by measuring twenty parts. A pose can. A friction coefficient cannot, and it is still outside.

What a reader should hold the field to #

The inventory a sceptical reader should check every claim in this field against, in the shape the practice field’s boundary essay uses.

Computed and asserted here. The rank of an identification, exactly, with its gap. The null directions, named and measured. The condition number and the per-parameter amplifications. The observability of a pose set, before measurement. The band a feature model gives, by two routes. The Denavit–Hartenberg parameters of a pair of axes, by construction and in closed form.

Named and not computed. Whether a candidate property the model omits is present. How many exact alternatives a non-linear identification has, except where an algebraic construction supplies the count. The derivative of a limit position with respect to a length. A clearance’s own provenance.

Computed elsewhere and taken as input. A shared setup fraction, which is a process measurement, and stated wherever used. Nothing else: every other number here is computed from a stated rule by the machinery that draws the figures.

That third row is what makes the boundary drawable at all, and it is the same row the practice field reports. A field that imported a figure would have to import its uncertainty with it.

If a claim here asserts something in the second list, it is wrong and should be reported as such.

What would move it again #

Two candidates, both named and neither taken.

A clearance’s own provenance. A clearance is a hole radius minus a pin radius, both of which are made, and whether the two were made together decides whether their errors cancel. That is exactly this field’s feature argument one component down, it needs no new machinery, and it is not done.

And a limit position as a reading. A pose the machine refuses locates a limit, a limit is a function of the parameters, and that function is a row of a kind no pose inside the travel provides. It is answerable — the derivative of a limit angle with respect to a length is elementary — and it is not answered, because the limit is where the implicit-function theorem the whole field rests on does not apply.

Both are geometry, both pass the test, and both are recorded here rather than claimed.

What the field is for, given all that #

A closing statement of the positive claim, since an essay of limits can leave the impression there is nothing left.

For a mechanism whose model is right, measured by an instrument whose errors are random and of known size, at poses that span the parameter directions, this field determines exactly which combinations of the parameters the readings fix, how accurately each of them comes back, and which of them do not come back at all — and it determines all of that before the machine is touched, from the drawing.

That is a large amount and it is more than a calibration usually reports. The rank is exact. The amplification is bounded and measured. The pose plan is optimised. The null directions are named in closed form.

What the limits say is that those results are conditional, and that the conditions are of three different strengths: one provable, one samplable, one not checkable at all. A field that states which of its claims are which is more useful than one that states only the claims, which is the argument the boundary essay this one answers makes for the whole site.

The useful way to read a calibration #

Not as a set of dimensions, and the alternative is better rather than weaker.

A calibration returns a shape, an accuracy for each part of it, and a list of what was assumed. The shape is what the readings determined; the accuracy is the residual times the amplification, per parameter; the assumptions are the fixed parameters, the branch, and the model’s own list.

Read that way it is a strong and checkable result. Read as four dimensions it is three measurements and something else, with nothing on the page to say which is which — and the whole of this field is the difference between those two readings of one output.

The same sentence would serve for the site’s other twenty-five fields with a word changed. A mobility read as a count is a fact; read as a guarantee that a mechanism moves it is sometimes wrong. A tolerance band read as a band on this machine is exact; read as a property of the design it is a size in disguise. In every case the arithmetic is right and the reading is where the error enters, which is why a boundary essay is a piece of arithmetic about how to read arithmetic.