Three machines, one curve

Every ambiguity so far in this field has been a direction: a line in parameter space along which the readings do not change, found by the rank of a matrix. This one is not a direction. It is two other machines, a long way off, each fitting the data exactly.

The theorem, from the other side #

Roberts’s theorem has been on this site since the synthesis field: given a four-bar and a tracing point on its coupler, there is a construction — on the original’s own lengths — producing two more four-bars whose tracing points describe the same curve.

The synthesis field uses it as a design tool. A linkage whose curve is right but whose pivots are in the way has two alternatives with the same curve and different pivots, which is a genuinely useful thing to be handed.

Read as an identification result it says something else. If the measurement is of the curve, three parameter vectors fit it exactly, and the fit has no basis for preferring one.

How far apart they are #

The three are not near each other. For the linkage drawn above — g = 4, a = 1.6, b = 3.2, c = 2.8, tracing point at (0.42, 0.55) — the three sets of lengths are

original     g 4.000    a 1.600    b 3.200    c 2.800
cognate 1    g 2.768    a 2.214    b 1.107    c 1.938
cognate 2    g 3.197    a 2.558    b 2.238    c 1.279

The cranks run from 1.600 to 2.558, a spread of sixty per cent. The couplers run from 1.107 to 3.200, a factor of nearly three. The ground pivots are in three different places.

This is not a flat valley with a broad minimum. It is three separated points, and the objective rises steeply between them: perturb the original towards cognate 1 by ten per cent of the distance and the fit is far worse than at either end.

They trace the same curve #

The claim needs a measurement and the measurement needs care, because comparing two curves means comparing a set of points against a polyline and a polyline cuts the corners of the curve it samples.

So the comparison is quoted against its own resolution. At 240 samples the largest departure of any cognate’s traced point from the original’s polyline is 2.06 × 10⁻⁴, and the chord sag — how far a point known to be exactly on the curve falls from the polyline — is 2.23 × 10⁻⁴. The ratio is 0.93.

At 480 samples the departure is 5.65 × 10⁻⁵ and the sag is 5.65 × 10⁻⁵: a ratio of 1.00. At 960 the departure is 1.40 × 10⁻⁵ against a sag of 1.41 × 10⁻⁵.

Both fall as the square of the sampling, together, and the ratio stays at one. The curves agree to the resolution of the comparison, and the resolution is what improves. Asking for agreement below the sag would be asking a polyline to be a curve, which is the trap the synthesis field’s own version of this check fell into the first time it was written.

What the rank says #

Nothing, and that is the point.

Build the identification Jacobian at the original — six parameters, four lengths and the two coupler-point coordinates, with the tracing point’s position as the observable. Its rank is six. Its singular values are ordinary. Its condition number is unremarkable.

Do the same at cognate 1. Rank six, ordinary singular values, unremarkable condition number.

A derivative is a local object. It says how the observable responds to small changes in the parameters here, and the existence of another exact solution a long way off leaves no trace on it whatever. Every diagnostic in this field that is built from the identification Jacobian — the rank, the null space, the condition number, all five observability indices — passes at each of the three.

That is worth stating as a limitation of the whole apparatus rather than as a curiosity about coupler curves. The matrix answers is this answer locally determined, and it cannot answer is this answer unique.

What the instrument sees #

The ambiguity is not a property of coupler curves; it is a property of a particular measurement of them, and being exact about which measurement is what makes the result usable.

Measure only the traced path, as a set of points in the plane, with no ground pivots and no timing. Three answers. Nothing separates them.

Measure the path and the ground pivots. One answer. The three cognates have their pivots in three different places — O₂ at the origin for all three, but O₄ at (4, 0), at O₃, and at a third point — so knowing where the frame’s holes are settles it immediately.

Measure the path and the crank angle at each point. One answer, for a similar reason: the three cognates traverse the curve with their cranks at different phases, and knowing which point of the curve corresponds to which input angle picks out one of them.

So the resolution is the same as the branch ambiguity’s: information of a different kind rather than more of the same. A better coordinate machine, more points, tighter tolerances — none of it helps. A tape measure across the frame does.

Three exact minima, measured as such #

The claim that the objective has three separate global minima deserves the treatment every other claim here gets, which is a run rather than an argument.

Generate a coupler curve from the original linkage. Hand the points to a fit with six free parameters — four lengths and two coupler coordinates — and start it near each of the three cognates in turn. Each fit converges, each residual reaches the solver’s floor, and each returns the cognate it started near, to fourteen figures.

Start it midway between two of them and it goes to one or the other, depending on which side of a boundary the start fell. Between the three there is a surface separating basins, and on that surface the objective is far from zero.

So this is not a broad flat region misread as three points. It is three points, with an ordinary hill between them, and the fit behaves exactly as a fit on a multi-modal objective behaves — which is to say it finds a minimum, reports success, and says nothing about the other two.

Why this is the realistic case #

It would be easy to dismiss path-only measurement as artificial. It is the natural thing to do in two circumstances that come up.

Reverse-engineering. A machine exists, its motion can be watched, and it cannot be taken apart — a mechanism inside a housing, a competitor’s product, an artefact. What is available is where the visible point goes.

Design from a demanded curve. A curve is specified and the question is which linkage draws it, which is path synthesis rather than identification and has exactly the same three-fold answer. That the same theorem governs both is unsurprising once said and is worth saying: synthesis from a path and identification from a path are the same problem with the data coming from different places.

In both cases the three answers are a gift rather than a problem, because they are three buildable machines and the designer can choose whichever has its pivots somewhere convenient. The problem arises only when somebody reports one of the three as the answer.

The pivots are the missing information #

Of the three resolutions listed above, the ground pivots are the one worth dwelling on, because they say what kind of thing a path measurement leaves out.

A coupler curve is a set of points in the plane. It carries no information about where the machine is beyond the fact that the tracing point visited those places. The three cognates put their frames in three different places, and the curve is indifferent to all three.

That is the same shape of gap as the scale null space, one level up: there the missing information was a size and the readings were dimensionless; here the missing information is a placement and the readings are about one point. In both cases what is absent from the measurement is a whole aspect of the machine that the observable happens not to encode.

There is a neat way to say it. The curve determines the mechanism up to the group of transformations that preserve the curve and permute the cognates, which for a generic curve is a three-element set rather than a continuous group. A finite group gives a finite ambiguity; a continuous one gives a null space. The instruments differ accordingly — a rank finds the second and only a search finds the first.

Finding them #

Since no derivative detects the alternatives, the instrument has to be something else, and there are two.

Run the fit from many starts. Scatter initial parameter vectors over a plausible region and see how many distinct answers come back. On this problem three come back, each with a residual at the solver’s floor. That is crude, it is not exhaustive, and it is the only general method available.

Or use the algebra. For the coupler curve there is a construction — Roberts’s — that produces the other two exactly, with no search. Where such a construction exists it is far better than any search, because it is complete: three, and not three-that-were-found.

The site has both. The homotopy field counts the solutions of a closure system exactly, by following every path, and is the general version of the second method. It is expensive and it answers the question a multi-start search only samples.

The reuse of an old result #

It is worth pausing on how this essay got its content, because the pattern has repeated three times in this field and says something about why the field belongs here.

Nothing new was computed. Roberts’s construction was implemented in the synthesis field long before this one opened, along with the machinery to trace each cognate and compare the traces against a polyline — including the chord-sag correction, which that field discovered the hard way when a correct pair of cognates failed a check demanding more precision than the comparison had.

All that changed is the question. The synthesis field asked are these three curves the same, because a designer wants to know the alternatives are genuine. This field asks can a measurement of the curve tell these three apart, which is the same computation and a different sentence.

A result about design read as a result about measurement is what happened with Freudenstein’s coefficients and with the assembly count as well. Three times is a pattern, and the explanation is the one that essay gives: minimality and identifiability are one property read from opposite ends, so a compact classical result about what determines a machine is automatically a result about what a measurement of that machine determines.

A count, and where it comes from #

Three is not a coincidence and it is not specific to this linkage.

Roberts’s construction is built from a similarity of the coupler triangle, and the three linkages correspond to the three vertices of that triangle. Every four-bar coupler curve therefore has exactly three, and the count does not depend on the proportions.

The degenerate cases are worth naming. If the tracing point is on the line through A and B, the coupler triangle is flat and the three cognates collapse — two of them coincide with the original. That is the one arrangement where a path-only measurement is unambiguous, and it is the arrangement in which the coupler curve is least interesting.

The ambiguity and the usefulness of the curve arrive together, which is a pattern worth noticing. A tracing point off the coupler line gives a curve worth drawing and three machines that draw it; one on the line gives a curve nobody wants and one machine.

Three answers is not three times the uncertainty #

A distinction that decides how the result should be reported.

A continuous ambiguity — the scale of a four-bar — means the answer is a family and any statement about a member of it is arbitrary. There is nothing to report but the invariants.

A discrete ambiguity means the answer is one of three specific machines, each of which is completely determined. That is far better than it sounds: three exact alternatives is a much stronger result than a one-parameter family, and each of the three can be checked against any additional evidence at all.

So the report is a list of three, not a hedge. The traced path is consistent with exactly these three linkages; the first has its second ground pivot at (4, 0), the second at (1.16, 2.51), the third at (2.84, 1.47). Anybody with access to the machine settles it in a second, and anybody without it knows precisely what the alternatives are.

A finite ambiguity is a result and an infinite one is a gap, and running them together under the word unidentifiable loses the difference.

There is a third case between them and it is the worst of the three: a direction that is nearly flat rather than exactly flat, along which the answer is neither determined nor a clean family. It moves with the noise, it moves differently on each repeat, and its size cannot be stated. The seventh singular value of a six-bar is one of those, at one part in thirteen thousand. Given the choice, an exact ambiguity is much the better thing to have.

What it costs to ignore #

A last practical note, since the ambiguity is easy to shrug at when the machine is in the room.

The cost lands on whoever receives the numbers. A path measured, a linkage reported, a drawing produced from it, a part made — and one time in three the linkage is not the one that was measured. Its curve is right, so anything that only cares about the curve is fine; its pivots are elsewhere, so anything that cares about mounting, envelope or interference is wrong by a large amount rather than a small one.

That is a different failure from the ones elsewhere in this field. A wrong size is out by a per cent; an absorbed parameter is out by eight; a cognate reported as the original has a crank sixty per cent out and its second pivot in a different place entirely.

A discrete ambiguity resolved wrongly is not a small error, and it is the one case in this field where the failure is loud in the world and silent in the data.

What this adds to the field #

Two things, both of which change how a report should read.

A rank of six does not mean the answer is unique. Everything in this field built on the identification Jacobian is a statement about a neighbourhood, and it is worth saying so wherever such a statement is made. This site’s own routines report the rank and the condition number; neither is evidence about global uniqueness and neither should be read as such.

And a multi-start check is cheap. It is the only routinely available detector of an alternative that is not nearby, and it belongs in a procedure beside the residual plot.

That is the practical close, and there is a larger one behind it. This field’s instruments divide into two kinds: those built from derivatives, which describe a neighbourhood exhaustively and say nothing about anywhere else, and those built from search, which sample everywhere badly. The first kind is where all the exact results are — a null space measured to 10⁻¹⁵, a rank decided across fourteen orders — and the second kind is the only one that can see this essay’s subject. Knowing which questions belong to which is most of knowing what a calibration has established.

And a second detail worth carrying. Twenty fits from scattered starts costs seconds on a four-bar and is the only routinely available detector of a discrete alternative. It belongs in a calibration procedure for the same reason a residual plot does: it is a test the answer could fail, and one that no other test in the procedure performs.