A ruler and a protractor

The result the last essay left standing is that a four-bar read by a protractor has one parameter no measurement can reach. The obvious next question is what to do about it, and the answer is short enough to give straight away: read something with a length in it.

The interesting part is why that works, because the reason is not about mechanisms at all.

Dimensions decide the rank #

Take one entry of the identification Jacobian and multiply it by the parameter it differentiates against. That product is what Euler’s relation sums over, and whether the sum can vanish is decided by what kind of number it is.

Angle readings. ψ is dimensionless. ∂ψ/∂ℓ therefore has dimensions of one over a length, and ℓ · ∂ψ/∂ℓ is a pure number. Scaling the machine multiplies each ℓ by k and divides each derivative by k, and the product is unchanged — so the four products are the same at every size, and a sum of them that vanishes at one size vanishes at every size. That is the null space.

Position readings. Pₓ has a length in it. ∂Pₓ/∂ℓ is dimensionless, and ℓ · ∂Pₓ/∂ℓ is a length. Scaling multiplies each of those by k. The sum scales rather than cancelling, and there is nothing to make it zero.

That is the whole mechanism. The null space is a fact about the units of the readings, and the machine is a spectator. It is also why the result is robust in a way a numerical fact would not be: no choice of linkage, no choice of pose, and no improvement in the instrument changes which side of the argument a reading falls on.

What the numbers do #

On twenty poses of the site’s four-bar, with the tracing point at (0.45, 0.62) in coupler coordinates:

A protractor on the output link gives singular values 1.975, 1.183, 0.379 and 3.9 × 10⁻¹⁵. Rank three of four, condition number 5.2 over the recovered directions.

A coordinate machine on the tracing point, asked for the same four lengths, gives 4.602, 4.374, 3.368 and 0.469. Rank four of four. The fourth value is not marginal and not rescued by luck: it is one part in ten of the largest, against one part in 10¹⁴ for the protractor.

That factor of 10¹³ between the two fourth singular values is the number to carry. It is not the difference between a hard problem and an easy one. It is the difference between a problem and no problem.

Six parameters, not four #

A coordinate machine that watches the tracing point raises a question a protractor never does: how does the model know where the tracing point is?

It does not, and pretending it does is a mistake with its own essay. The point sits at some (u, v) in the coupler’s own frame, those two numbers are as much a property of the built machine as the four lengths are, and a fit that fixes them at their nominal values while the machine has them somewhere else absorbs the difference into the lengths.

So the honest model for a coordinate machine has six parameters. Its identification Jacobian has two rows per pose and six columns, forty rows against six on this sweep, and its singular values run 16.24, 15.99, 4.588, 1.017, 0.397 and 0.100. Rank six, no null space, condition number 162.

That condition number is the cost of honesty and it is a real cost: the worst-recovered combination of parameters is a hundred and sixty times less observable than the best, so whatever error the instrument has arrives in that combination multiplied by a hundred and sixty rather than by one.

Both at once #

Now put both instruments on the machine. Three rows per pose, sixty rows against six columns, and the singular values run 11.07, 10.79, 8.837, 6.282, 1.835 and 0.425. Rank six. Condition number 26.

The same six parameters, six times better conditioned, from adding readings that on their own could not recover four of them.

That is worth stating plainly because the intuition runs the other way. A protractor cannot see the size; a coordinate machine can see everything; so surely the protractor adds nothing. It adds a great deal, and the reason is that observability is not a property of a parameter but of a direction. 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. The information they carry lies in different directions of the same space, and a direction one of them cannot see at all may be a direction the other sees best.

The trap in comparing them #

There is a way to get all of this wrong and it is the first thing anybody does.

A decomposition of a matrix with rows in radians and rows in millimetres compares them, whether or not that means anything. Change the length unit from millimetres to metres and every position row is divided by a thousand while every angle row is not: the singular values move, the condition number moves, and nothing about the machine or the instruments has changed.

So a block of rows in one unit has to be scaled against a block in another before any condition number is quoted, and the scaling is a choice. The one used here is to bring each block to unit root-mean-square, which makes the comparison a statement about the mechanism rather than about a choice of unit — and it is reported rather than done quietly, because the singular values afterwards live in the scaled frame and a reader comparing one against a sensitivity in radians would be comparing two different matrices.

The rank is safe from all of this: scaling rows by non-zero factors cannot change which combinations of columns are zero. A rank is a fact about the pairing; a condition number is a fact about the pairing and the units. That is why this field states ranks flatly and always says what a condition number was computed in.

Which instrument to reach for #

The practical reading is not “always use a coordinate machine”, and the numbers say why.

A coordinate machine recovers six parameters at 162. A protractor recovers three at 5.2. If what is wanted is the machine’s shape — its Grashof class, its transmission angle through the turn, its velocity ratio, everything that is a ratio — the protractor recovers it better conditioned by a factor of thirty, from an instrument that costs nothing and needs no fixture.

Reaching for the more capable instrument because it is more capable is a real mistake here. It recovers more and it recovers each thing worse, and if the extra parameters are not wanted then the extra rows have only made the answer noisier.

The right question is which parameters the report has to carry. Three invariants and a nominal size is a complete and well-conditioned answer to most questions anybody asks about a linkage. Six parameters at 162 is the answer to a question about where the tracing point actually goes, which is a different question and sometimes the one that matters.

An instrument this field does not model #

Worth naming, because the argument above makes it sound as though any reading with a length in it will do.

A reading has to be a reading of the mechanism’s own geometry. A dial gauge against the coupler measures a distance from the gauge’s mount to the coupler, so the mount’s position is now a parameter too — two more numbers in the column list, and whether they are identifiable is a fresh question with its own rank. An instrument brought to a machine brings parameters with it, and a calibration that forgets to count them has a model missing a parameter in the most literal sense.

The coordinate machine above escapes this only because its frame is taken as the mechanism’s frame — the two ground pivots define the coordinates the readings are in. Change that and the six parameters become eight, and the two new ones are a rigid transform whose identifiability is decidable by exactly the machinery in this essay and is not automatic.

That is a general caution and it is the reason this field states three things in every result: the mechanism, what is read, and where from. Drop any one and the rank is not defined.

The same statement for every mechanism on the site #

The argument above never used a four-bar. It used two facts: that the observable is a function of the parameters, and that the parameters include some lengths. So it applies wherever both hold, which is everywhere on this site.

A Watt six-bar read by a protractor on its output link has rank seven of nine, and one of the two missing directions is the whole-machine scaling this argument predicts. A serial arm measured by joint encoders alone would have the same deficiency, which is why nobody measures one that way — a robot’s calibration is done with a tracker precisely because what is wanted is a position. A cam measured by the angular position of its follower recovers the follower’s motion law and not the cam’s size.

The pattern is worth stating as an instruction rather than as an observation. Before designing a measurement, ask what its readings are made of. If every reading is dimensionless in the parameters, one direction of the parameter space is already gone and no amount of care recovers it. That is a five-minute check on a piece of paper and it is available before any instrument is bought.

The check has a positive form too. If a parameter set contains quantities of more than one kind — lengths and angles, say, as a spatial loop’s does — then a scaling acts on only some of them, and the null direction has zeros in it. The six-bar’s does, against the parameter that is a fraction rather than a length. Reading a zero off a measured null vector is dimensional analysis performed by a decomposition, and it is a pleasant way to confirm that a parameter list has been classified correctly.

Why the coordinate machine is badly conditioned #

The condition number of 162 deserves an explanation rather than a shrug, because it is the largest of the three and the instrument is the most capable.

Look at the six singular values: 16.24, 15.99, 4.588, 1.017, 0.397, 0.100. The first two are an order above the rest and they are nearly equal. That pattern — two large, nearly equal values — is what a pair of columns looks like when the observable responds to them strongly and almost identically, and here the pair is the two coordinates of the tracing point.

Moving the tracing point along the coupler moves the observed position by roughly the coupler’s length times the change; moving it across moves it by the same amount in the other direction. Both are large effects and both are nearly independent of the crank angle, so those two columns are big and nearly orthogonal to everything else. The four length columns are what is left, and they have to be distinguished from each other against a background of two much louder parameters.

A parameter that is very easy to see makes the others harder to see, because the condition number is a ratio. That is not intuitive and it has a design consequence: if the tracing point’s position happens to be known independently — because it is a machined feature with its own drawing, say — then fixing it rather than fitting it 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.

Fixing a parameter is a claim #

That last observation is a lever and it has to be used carefully, because fixing a parameter is not free.

A parameter held at its nominal value is a claim that the machine’s value is the nominal one. If it is not, the difference does not disappear — the fit absorbs it into the parameters that are still free, exactly as an unmodelled parameter is absorbed, and with the same symptoms: a residual that will not fall to the noise, and lengths that move to compensate.

So the trade is between conditioning and honesty, and the two numbers that decide it are both computable in advance. Fixing a parameter improves the condition number by a factor that comes out of the decomposition. Getting it wrong by δ moves the other parameters by δ times a coefficient that comes out of the same matrix. The question is whether the drawing’s value is good to better than that, and it is a question with a number rather than a matter of taste.

The version of this trade that is always wrong is the silent one: fixing a parameter because the fit converges better with it fixed. The fit converges better because the objective has fewer directions to be flat in, which is true whether or not the fixed value is right.

An arithmetic worth doing before buying anything #

Put the three rows of numbers side by side and a purchasing decision falls out of them.

protractor          rank 3 of 4     κ 5.2      no fixture, no datum
coordinate machine  rank 6 of 6     κ 162      a frame, a datum, a fixture
both                rank 6 of 6     κ 26       all of the above, plus an encoder

The middle row is the expensive one and it is the one most likely to be chosen, because it is the instrument that measures everything. The bottom row costs one encoder more than the middle row and is six times better. The top row costs almost nothing and answers most of the questions.

None of that is a claim about instruments in general. It is the output of one decomposition of one mechanism’s identification Jacobian at one pose set, and the same three rows on a different linkage would have different numbers in them. The point is that the three rows are computable before the machine is touched, from geometry the drawing already contains, and that the ranking they produce is not the ranking intuition produces.

That is the most useful thing this field offers anybody who is not reading it for the algebra. A measurement plan is a design, it can be evaluated the way a mechanism is evaluated, and the evaluation is a rank and two singular values.

One reading is enough for the size #

A last practical note, and it makes the repair smaller than the essay so far suggests.

Nothing about closing the null space requires a coordinate machine, a fixture or a datum. The direction that is missing is one-dimensional, so one number with a length in it closes it. A rule across the frame bar does it. A caliper on the crank does it. A single photograph with a scale bar in it does it.

What the coordinate machine buys is not the size but the tracing point’s actual path, which is a different thing to want. If the report is three invariants plus a size, the cheapest honest measurement on this list is a protractor and a rule — rank four, and the conditioning of the three invariants is the protractor’s own 5.2 rather than the coordinate machine’s 162, because the size arrives from a separate reading rather than having to be untangled from the others.

That arrangement does not appear in the three-row table above because it is not one instrument watching a motion; it is a motion measurement and a static one, combined. It is also, on most benches, what somebody would actually do.

What this does not fix #

The null space closes and two other things do not, and it is worth separating them here since they arrive in later essays.

Conditioning is not rank. A parameter with a singular value of 0.100 against a largest of 16.24 is recoverable in the sense that matters to a rank and barely recoverable in the sense that matters to a report. Nothing in this essay improves it; choosing where to measure does.

And a discrete ambiguity is invisible to both instruments. A rank is a local object, computed from derivatives at one point of parameter space. Two separated parameter vectors that both fit the data exactly are not a direction, no derivative detects them, and adding a second instrument does not help unless it happens to read something that differs between them. Three linkages tracing one coupler curve is that case, and a coordinate machine watching the traced path alone is exactly the instrument that cannot tell them apart.