Three linkages, one curve

Here is a result that has no business being true.

Take a four-bar with a point marked on its coupler. That point traces a curve — a sextic, generally with two branches, and its shape is a sensitive function of the four lengths and of where the point sits.

There are exactly two other four-bars, with different ground pivots and different link lengths, that trace the same curve. Not a similar curve, not a curve that agrees near a few points: the same curve, every point of it.

Samuel Roberts published this in 1875. Chebyshev found it independently, and the standard name gives them both.

It is not an approximate statement, a statement about a family of similar curves, or a statement that holds near some configuration. Three specific mechanisms; one curve; every point.

The construction #

The theorem is usually presented as a compass-and-straightedge construction with parallelograms, which is how Roberts gave it and is genuinely hard to follow. In complex numbers it is one multiplication.

Write the coupler point in the coupler’s own frame:

$$\lambda = \frac{P - A}{B - A}$$

One complex number. Its modulus says how far along the coupler the point sits relative to the coupler’s length; its argument says how far off the line of the coupler it is. Everything about where the tracing point is attached is in that one number.

Then the third fixed pivot is

$$O_3 = O_2 + \lambda\,(O_4 - O_2)$$

which says that the triangle O₂O₄O₃ formed by the three ground pivots is similar to the coupler triangle ABP. That similarity is the whole of Roberts’s construction, restated. The famous “Cayley diagram” with its parallelograms is a way of drawing that statement with instruments.

The two other linkages then fall out:

Cognate 1 grounds on O₂O₃. Its four lengths are the original’s, each multiplied by |λ|, with the roles permuted: what was the coupler becomes a crank, and what was a crank becomes the coupler.

Cognate 2 grounds on O₄O₃, scaled by |1 − λ|, with a different permutation again.

The permutation is the interesting part, and it is why the theorem is useful rather than merely decorative. A four-bar’s behaviour depends enormously on which of its links is which — Grashof’s classification is exactly a statement about that — so three linkages with permuted roles can be completely different mechanisms. The original in the figure is a crank-rocker whose input goes round; its first cognate is a double-rocker whose input link swings through about fifty degrees. Same curve, different machines.

The collinear coupler #

Both cognates have a feature that looks like a mistake in the figure and is not: their tracing point lies on the line of the coupler rather than off it.

This is a consequence rather than an assumption. In a cognate, the two links reaching the tracing point from the two cranks both turn with the same one of the original mechanism’s angles, so they are parallel, and a point reached by two parallel offsets from two pins is on the line through both pins.

It is worth stating because a coupler point on the coupler’s own line is unusual — the original four-bar’s point is deliberately off the line, since a point on the line traces a much less interesting curve. The cognates are the exception, and the reason they still draw an interesting curve is that their coupler is short and their cranks are long, which is exactly the permutation the construction performs.

Where the third pivot comes from #

The formula O₃ = O₂ + λ(O₄ − O₂) deserves unpacking, because it is doing something that looks like sleight of hand and is not.

Write the mechanism as a chain of complex numbers. Let z₁ be the crank vector from O₂ to A, z₂ the coupler vector from A to B, and z₃ the rocker vector from O₄ to B. The loop closes: z₁ + z₂ − z₃ = g, where g is the ground vector from O₂ to O₄.

The tracing point, measured from O₂, is P − O₂ = z₁ + λz₂. That is the original mechanism’s route to P: along the crank, then a fraction λ of the way along the coupler.

Now measure P from O₄ instead. Substituting the loop equation gives P − O₄ = z₃ − (1 − λ)z₂ — a route along the rocker and then back along part of the coupler, which is the same mechanism described from the other ground pivot.

And now measure it from the point O₂ + λg. Substituting again gives

$$P - O_3 = (1 - \lambda)\,z_1 + \lambda\,z_3$$

which is a route consisting of a fixed multiple of the crank vector followed by a fixed multiple of the rocker vector. Both of those are constant-length links turning with angles the original mechanism already has. So there is a third dyad reaching the tracing point, from a pivot that was not on the mechanism, using bars that are scaled copies of the two cranks.

Three routes to P, from three pivots. Pair them off two at a time and there are three four-bars, and the third pivot is where the algebra put it rather than where a construction happened to lead.

That derivation also says why the scale factors are |λ| and |1 − λ|: they are the coefficients that appear when the loop equation is substituted, and they are the reason a coupler point close to one of the pins gives one very small cognate and one nearly the size of the original.

Checked, and against what #

The three curves are compared by tracing each one independently — its own ground pivots, its own crank, its own solver runs — and measuring how far each point of a cognate’s trace falls from the original’s curve.

That comparison needs a floor, and finding the floor is most of the work.

The reference curve is a polyline through solved positions, and a polyline cuts the corners of the curve it samples. So a cognate point that is exactly on the true curve still measures a small distance from the polyline — the chord sag. Asking for agreement below that is asking for the impossible, and the first version of this check did exactly that: it demanded 10⁻⁶ from a comparison whose resolution was 1.4 × 10⁻⁵, and reported that two correct cognates disagreed.

With the floor measured rather than assumed, the two cognates come in at 0.94 and 0.86 times it — which is to say, indistinguishable from exact at the resolution available. A cognate with one bar 5% wrong measures about eleven thousand times the floor, so the comparison is capable of noticing a difference when there is one.

The general lesson is one this site keeps relearning from different directions: a tolerance quoted without the resolution it is measured against is not a claim about the thing being measured. The same discipline appears in the centrode rolling check, where the arc-length agreement is quoted against how it changes when the sampling is doubled, and in the curvature check on a cam profile, where a three-point circle fit gets worse with finer sampling and a check written to expect convergence failed on a correct answer.

Tracing a cognate is not tracing the original #

There is a practical difficulty in the comparison that turned out to be the whole reason the first attempt found nothing.

The original four-bar in these figures is a crank-rocker: its input goes round, so sweeping the crank from 0° to 360° visits every configuration. Its cognates need not be, and the first one is a double-rocker whose input link swings through about fifty degrees. Driving it from 0° to 360° found two positions out of three hundred and sixty-one, and the cognate appeared to draw almost nothing.

The fix is to remember what a coupler curve is. It is a single closed curve, and a mechanism whose input rocks covers it by going out along one assembly branch and coming back along the other. So every crank angle is tried on both branches, and the positions that assemble are kept. What comes back is the whole curve.

That is not a special measure for cognates. It is the correct way to trace any coupler curve whose mechanism does not have a full rotator, and the reason it does not come up in the coupler-curve essay is that the linkages there were all chosen to have one.

What the theorem does not say #

Three clarifications, because the result is easy to over-read.

It is about the curve, not the timing. The three linkages trace the same set of points. They do not trace them at the same rate: drive each cognate’s crank at constant speed and the tracing point moves round the curve on a different schedule. For a mechanism that has to be somewhere at a particular time — a film pulldown, an indexing mechanism — the cognates are not interchangeable, and choosing among them is a real decision rather than a free one. This is the same distinction the universal joint essay turns on: the same positions in a different order in time is a different machine.

It is about coupler points, not coupler bodies. A cognate reproduces the path of the point. The orientation of the coupler at each point along that path is generally different, so a mechanism synthesised for rigid-body guidance cannot have its cognates substituted — they will put the body in the right place at the wrong angle. Roberts’s theorem belongs to path generation and does not extend to motion generation.

Degenerate cases exist. If the coupler point lies on the line AB, then λ is real, O₃ is on the line O₂O₄, and the three linkages collapse in ways that depend on where exactly. If λ is 0 or 1 the point is one of the pins, its “curve” is a circle, and the cognate construction returns nothing useful. These are the cases the figure avoids by using a coupler point genuinely off the line, and they are worth knowing about because a numerical search over coupler points will walk into them.

What it is for #

Three uses, in increasing order of how surprising they are.

Packaging. A designer who has found the linkage that draws the required curve, and discovers it will not fit in the space available, has two more to try. The ground pivots are in different places and the proportions are different, so one of the three may fit where the others do not. This is the standard textbook justification and it is real.

Reading the atlases. Hrones and Nelson’s atlas contains some seven thousand coupler curves. Every one of them is drawn by three linkages, so the book is implicitly three times larger than its page count suggests, and a designer who found the right curve with the wrong proportions had two more sets of dimensions available without further searching.

Straight-line mechanisms. This is the elegant one. Watt’s linkage and Chebyshev’s both trace approximately straight coupler curves, and they are cognates of each other — or more precisely, the cognates of one are related to the other, which is why two mechanisms invented eighty years apart for the same purpose turn out to be the same mechanism seen differently. The same relation explains the Roberts straight-line linkage and several others in the family: they are not independent inventions so much as different cognates of one curve.

Six-bars for free #

There is a use of the theorem that is less well known than the packaging argument and is more interesting, because it produces a mechanism nobody designed directly.

Take the original four-bar and one of its cognates. Both draw the same curve, and both have their tracing point at the same place at the same instant — they are, in a real sense, one mechanism drawn twice. So the two can be combined: keep both, join their tracing points, and delete a redundant link.

What comes out is a six-bar with a useful property. The tracing point is now carried by two independent dyad chains rather than one, and the whole assembly is driven from a single input. The classical use is to produce parallel motion: a suitably chosen combination gives a link that translates without rotating over the whole cycle, which is what a drawing-board parallel rule and a good many lamp arms do.

The construction is worth knowing because it inverts the usual relationship between mechanism complexity and design effort. A six-bar with a prescribed property normally requires six-bar synthesis, which is substantially harder than four-bar synthesis. This one falls out of a four-bar and a theorem.

What it says about synthesis #

The theorem has a consequence for the field around it that is worth stating plainly.

Path-generation solutions come in threes. Any method that synthesises a four-bar to trace a curve is, without knowing it, finding one member of a triple. A designer who searches the space of four-bars for the best fit to a desired curve is searching a space in which every point has two twins, so the search is three times as redundant as it looks, and an optimiser that has found a local minimum has found three.

That cuts both ways. It means the effective size of the design space is a third of what it appears, which is bad for anyone hoping the space is rich. It also means that any solution found comes with two alternatives for free, which is good for anyone whose problem is packaging rather than existence.

And it means that the coupler curve, not the linkage, is the natural object. Roberts’s theorem says that the map from four-bars to curves is three-to-one, so a curve is the invariant and a linkage is a way of realising it. That is the same shift in viewpoint that the centrodes make for a coupler’s motion — the bars are one way of producing the motion and not the motion itself — and it is probably the most useful idea in this field for someone coming from analysis rather than design.

Why the theorem is not obvious #

It is worth asking why a result this clean took until 1875, when four-bar coupler curves had been studied for a century, and the answer is instructive about how the subject was done.

A coupler curve is a sextic — degree six — and its equation in the coupler point’s coordinates is long and unilluminating. Two four-bars draw the same curve if their two sextics are the same polynomial up to a constant, and checking that by comparing coefficients is a page of algebra that reveals nothing about why it might be true.

Roberts found it by construction rather than by algebra: parallelograms, similar triangles, and a diagram. Chebyshev came at it from the straight-line problem, where he was already comparing linkages that did the same job. Neither route is one a coefficient comparison suggests.

The complex-number form used above is later and is what makes it look easy. Writing the coupler point as λ = (P − A)/(B − A) puts the whole of “where the tracing point is attached” into one number, and the construction is then multiplication by that number. That is a change of notation and not of content — Roberts’s parallelograms are drawing the same multiplication with instruments — but it is the difference between a result that has to be verified and one that can be seen.

The general lesson is one this site keeps meeting: the representation decides what is obvious. A mechanism’s mobility is a rank and is invisible in a formula that counts joints; a coupler point is one complex number and is four real ones in any other notation.

Roberts’s theorem is a piece of classical geometry and its most useful modern reading is as a design freedom rather than a curiosity. Any coupler curve a designer has settled on comes with two other linkages that draw it exactly — different proportions, different ground pivots, different bars playing different roles — and all three are available for the price of one complex multiplication. So when a synthesised linkage is unbuildable for a reason that has nothing to do with its curve — a pivot in the wrong place, a link that fouls, a frame that cannot be reached — the first thing to try is not a re-synthesis but a cognate. Same curve, exactly, and a completely different physical arrangement. That is an unusually cheap escape from a common failure, it needs no optimisation and no compromise on the curve, and it is available at every stage of a design. A method that returns one linkage per curve has been quietly discarding two thirds of its answers.