Exactly right, and unbuildable

Burmester’s three-position construction is exact. Choose any two points of the moving body, take the circumcentre of each one’s three images, and the resulting four-bar puts the body in all three prescribed positions to the last decimal place.

That is a theorem, it is not approximately true, and the forward solver confirms it every time to about 10⁻¹⁵.

It is also, for most choices of the two points, useless.

Reaching and getting to #

The gap is between two verbs.

Reaching a position means the mechanism has a configuration in which the body is there. That is what the construction guarantees and what the verification confirms: drive the crank to the right angle, close the loop, and the pins are where the pose says.

Getting to a position means the mechanism can move there from where it is, continuously, without anything being disassembled. That is a statement about a path through the mechanism’s configuration space, and the construction never looks at one.

The distinction is not academic. A four-bar has two assembly branches — for any crank angle the linkage can reach, the coupler and rocker can be reflected about the diagonal joining the crank pin to the far ground pivot, and both configurations satisfy every constraint. A physical four-bar cannot pass between them: doing so would require the coupler and rocker to become collinear and pass through, which is a configuration the lengths may not allow, or a pin to come out.

So a synthesis whose three poses land on two different branches is exactly correct and physically useless. The machine would have to be dismantled halfway through its cycle.

What a branch is, exactly #

The word “branch” gets used loosely and the geometry is worth pinning down, because the defect is easier to reason about once it is clear.

Fix the crank angle of a four-bar. The crank pin A is then determined — it is a point on a circle. The rocker pin B must be at a fixed distance from A and at a fixed distance from the ground pivot O₄, so it lies at the intersection of two circles.

Two circles meet at two points, one point, or none. Two points is the ordinary case, and those two points are the two branches: B above the line AO₄, or B below it. One point is the boundary — the two circles are tangent, the three links A, B and O₄ are collinear, and the mechanism is at a toggle. None means the configuration does not exist and the linkage cannot be assembled at that crank angle at all.

So a branch is a choice of which intersection to take, and the choice is fixed at assembly. Moving between branches requires passing through the tangency, which is the toggle — and at a toggle the mechanism’s Jacobian loses rank, the input has no leverage on the output whatever, and a crank-driven linkage cannot pass through it under control even when the geometry permits.

For a crank-rocker, Grashof’s condition guarantees the crank rotates fully, which is precisely the condition that the two circles always meet in two points, which is precisely the condition that there are no toggles. Its two branches are therefore entirely separate for the whole revolution: a crank-rocker on one branch stays on it forever. That is the cleanest case of a branch defect, and it is the commonest one in this survey.

Three defects with three names #

The literature distinguishes three failures and they are worth keeping apart.

A branch defect is the one above: the prescribed positions require configurations on different assembly branches. The mechanism can be assembled at each of them and cannot move between them.

A circuit defect is subtler and often lumped in with the first. A mechanism’s configuration space may have several connected components — circuits — and a linkage whose input rocks rather than rotates has a travel bounded by toggle positions at each end. Two poses can be on the same branch in the reflection sense and still be on different circuits, separated by a toggle the mechanism cannot pass.

An order defect is the one that catches people who have been careful about the first two. The mechanism reaches all three poses, on one circuit, and meets them in the wrong sequence as the input turns: 1, then 3, then 2. For a machine that has to do three things in order, that is as fatal as not reaching them at all, and it is completely invisible in any static check.

The survey measures all three by the same method. Drive the mechanism from the first pose in both directions, carrying each solved configuration forward as the next guess exactly as a built mechanism does, and record where along that travel each other pose appears. A pose that never appears is on another branch or beyond a toggle. A pose that appears out of turn is an order defect.

For the three poses in the figure: 1,176 exact solutions, 810 with a branch or circuit defect, 190 with an order defect, and 176 — fifteen per cent — that a machine could use.

Every one of the 1,000 disqualified solutions passes the exactness check. The construction did its job on all of them, and nothing about the arithmetic distinguishes the 176 from the rest.

Why the number is not universal #

Fifteen per cent is a number about these three poses and not a general law, and the survey was run on several pose sets before one was chosen for the figures.

Some sets give no order defects at all: every solution that stays on one circuit happens to meet the poses in sequence, and an essay written on one of those would be describing a failure its own figures never showed. Others give almost nothing usable — one set tried during development produced 1,173 defective solutions out of 1,176 and three usable ones, all of them with pivots outside the machine.

The poses used here were chosen because they exhibit all three outcomes in useful proportions. That is a deliberate choice and it is stated because the alternative — presenting whichever set happened to be tried first as though the percentages were a property of synthesis — would be exactly the kind of thing this site’s figures exist to avoid.

What is general is that the fraction is well below one, for every pose set tried, and that nothing in the construction correlates with it.

Measuring it properly #

Getting the survey to say anything true took three corrections, and all of them are the kind that produce a confident wrong answer rather than an error message. The third was found a phase later and changed the headline number, so it goes first.

This essay said 111 for two phases, and the answer is 176. The count is produced by sweeping each synthesised linkage and recording which poses it reaches, so it depends entirely on the sweep completing — and the sweep was not completing. The analytic Jacobian’s rows for an attached coupler point had the wrong sign on every term involving the offset from the coupler line, which is a defect that never draws a wrong picture and instead makes Newton crawl until the solver’s stall rule declares the position unreachable. An unreachable pose is exactly what a branch defect looks like. So 65 linkages that reach all three poses perfectly well were counted as unable to, and another 59 were filed under the wrong defect: 934 branch defects and 131 order defects were really 810 and 190.

Fifteen per cent, not nine. The shape of the argument is unchanged and one of its numbers was wrong by two thirds, which is the ordinary way a measurement fails when the instrument is the thing at fault.

A pose is met only at its own crank angle, not at the nearest sampled one. The first version compared the rocker pin’s position at whichever swept angle came closest to the pose’s. At 720 steps that is up to a quarter of a degree away, which on a linkage with links of order one is about a centimetre of pin travel. Against a tolerance of 2 × 10⁻³ nothing ever matched, so every one of 1,176 exactly correct syntheses was reported as having a branch defect. Driving to the pose’s exact angle from the carried configuration removes the sampling from the measurement entirely, and the tolerance can then be the solver’s rather than the sweep’s.

Both directions, because a rocking input is not a defect. A synthesised four-bar often has an input that swings rather than rotates — nothing in the construction attends to Grashof’s condition — and sweeping only forwards would call every pose behind the starting one unreachable. What matters is whether the poses lie on the same circuit, and running the sweep out in both directions is what measures that.

There is a third decision that is not a correction but is worth stating: reversal is not an order defect. A mechanism that meets the poses 3, 2, 1 is meeting them in order with the input running the other way, which is a wiring decision rather than a design failure. So the order test accepts a monotone sequence in either direction.

The defects are not independent of each other #

A designer trying to avoid all three at once will notice that they pull in different directions, and the interaction is worth a paragraph.

Avoiding branch defects favours mechanisms whose whole travel is on one branch with plenty of room, which tends to mean a crank-rocker with a comfortable transmission angle — a well-proportioned, unadventurous linkage.

Avoiding circuit defects favours mechanisms with a rotating input, because a rotating input has one circuit covering the whole revolution while a rocking one has its travel bounded at both ends by toggles that the poses might straddle.

Avoiding order defects is a constraint on where the prescribed crank angles fall around the circle, and it has nothing to do with the linkage’s proportions at all — it is decided by the arrangement of the poses relative to the fixed pivot the construction happened to produce.

So the first two favour a particular kind of mechanism and the third is close to a coin toss. That is consistent with the survey’s numbers: the branch and circuit failures account for the great majority, and the order failures are a smaller, largely uncorrelated tax on what survives.

It also explains why generating many solutions works better than reasoning about one. The properties that matter are not smooth functions of the choice of coupler points — a small move of one pin can flip a pose from one branch to the other — so a local search from a defective solution is not reliable, and a grid is.

Why the construction cannot see any of it #

It is worth being clear about why this is not simply an oversight in Burmester’s method.

The construction operates on positions. It takes three images of a point, fits a circle, and returns a centre. Everything it knows is contained in nine numbers describing where the body has to be, and nothing in those nine numbers says anything about paths — because the poses are a set, not a sequence, and the specification did not include one.

Adding the sequence would not help either, because the branch question is not about the poses at all. It is about the mechanism: which configurations it can move between is decided by the four lengths, and the four lengths are the construction’s output rather than its input. There is no way to check a property of the answer before the answer exists.

That is a general shape and it is worth recognising: a construction that solves a problem exactly may be solving a different problem from the one asked. The one asked was “find a four-bar that takes the body through these poses”. The one solved was “find a four-bar that has a configuration at each pose”. Those differ by the word “takes”, and the word “takes” is about motion.

What it looks like in a machine #

The abstract statement — “the poses are on different branches” — describes something with a very concrete failure mode, and it is worth naming because it is how the defect gets discovered in practice.

The mechanism is built. It is assembled with the coupler on one side of the diagonal, because that is how the parts fitted together on the bench. It runs, it reaches the first two prescribed positions perfectly, and it never reaches the third — the crank turns and the output goes somewhere else.

The natural diagnosis is that something is made wrong, so the parts get measured. They are correct. The linkage is then reassembled the other way round, and now it reaches the third position and misses the second.

What has gone wrong is not manufacturing and not the arithmetic. It is that the design was checked by driving to each position and confirming the geometry, which is exactly the verification the construction essay describes — and that check passes on a branch-defective linkage, because each position individually is fine.

This is a good argument for the discipline the site applies to figures being applied to design review as well. A check that examines configurations one at a time cannot see a property of the path between them, and the only fix is to compute the path.

What a designer does about it #

The practical answer is not to be cleverer about the construction. It is to generate many solutions and test them.

That is what the survey is: a grid over the coupler plane, every pair synthesised, every one swept. It is cheap — the construction is a circumcentre and the sweep is a few hundred solves — and it turns a two-parameter family of exact answers into a much smaller set of usable ones, which is what the designer wanted in the first place.

The pattern generalises past synthesis. The exact linkage that cannot be assembled through its travel, the mobility formula that declares a working mechanism immobile, the spatial four-bar whose count says −2 — in every case a closed-form result is right about what it computes and silent about something else, and in every case the resolution is to compute the something else rather than to improve the formula.

This site’s version of that is stated once and applied everywhere: a claim gets a test it could fail, and the test is taken by a route the claim does not control. For synthesis, the claim is a theorem and the route is the forward solver. The theorem passes every time. The solver disqualifies nine solutions in ten, and nothing in the theorem is wrong.

The same failure in the other fields #

Once the shape is named it turns up repeatedly on this site, and collecting the instances is worth doing because the common feature is not obvious from any one of them.

The transmission angle is a function of configuration, and a mechanism whose average transmission angle is comfortable may have a worst case that jams. The average is correct and answers a question nobody asked.

Grübler’s formula counts links and joints, and is right about the count while being silent about whether the constraints are independent. Its answer is not approximate; it is a correct answer to a different question.

The Grashof condition predicts which link rotates fully and says nothing about whether the mechanism is any good when it does.

And a synthesis construction puts the body at three prescribed poses without addressing whether it can be carried between them.

In every case a closed-form result is exactly right about the quantity it computes, and the thing that matters is somewhere else. The failure is never that the mathematics is wrong; it is that the mathematics answers the question it was given, and the question was a proxy.

That is why this site pairs every claim with a measurement taken by a different route rather than with a more careful derivation. A better derivation of the wrong quantity is still the wrong quantity, and only running the mechanism finds out.