Concept

Branch defect — where it appears

A synthesised mechanism's failure to reach its prescribed positions without being taken apart, because they lie on different assemblies. It is invisible to the equations that placed the positions and is one of the reasons most exact syntheses are unusable.

Named by 6 essays across 4 fields — each of them below, with the objects they name alongside it.

Of 1176 exactly correct syntheses, 176 could be built. Every pair of points on a 7×7 grid over the moving body, each pair synthesised into a four-bar and each four-bar verified by the forward solver as reaching all three prescribed poses. 1176 of them do, exactly. Then each is swept from the first pose in both directions, carrying the branch the way a built mechanism must: 810 cannot reach all three without being taken apart and reassembled, and 190 reach them in the wrong order. 176 — 15% — are mechanisms rather than theorems. Nothing in the construction distinguishes them.

Exactly right, and unbuildable

A linkage synthesised through three prescribed positions reaches all three. That is a theorem and it holds exactly. Whether it reaches them in one piece, without being taken apart, and in the order asked for, are separate questions the construction says nothing about — and of 1,176 exactly correct solutions, 176 could be built.

wrong · Misconception

Five positions, and what is left

Three prescribed poses leave a whole plane of choices. Four leave a curve. Five leave four points, and finding them is the first thing in this site's synthesis field that a compass cannot do — it needs two cubics intersected, which is algebra rather than construction. Four points give six four-bars, and two of them can be built.

synthesis · Synthesis

One linkage, two curves. The same bars, the same lengths, the same driving angle — assembled two ways. One trace is where the polynomial vanishes and the other is not: the worst value of xy − 0.5 along the second is 8.7e-1, against 4.7e-14 along the first. Every position on both was solved to 9.5e-14. Nothing about the second linkage is defective; one of its parallelograms is a crossed one, so a direction is being carried wrongly, and the machine is faithfully computing a different function.

The proof drew more than the curve

Sixteen ways to assemble one linkage. Eight of them close. Four put the tracing point on the curve and four put it somewhere else — at a closure residual of 9.6 × 10⁻¹⁵, which is the same floor the right ones reach. No tolerance on the closure could ever have told them apart.

computing · Compute

The error between the precision points. chebyshev: worst 0.2232°, RMS 0.1475°; least-squares: worst 0.3900°, RMS 0.1203°. The error is zero at each precision point by construction and nowhere else. chebyshev has the smallest maximum here, and which curve is best depends entirely on which measure is asked for.

Where an optimiser starts

An approximate synthesis is a local search on an objective that is non-convex, disconnected and not everywhere defined, so the answer depends on where the search began. Nothing in the optimisation supplies that. What supplies it is the exact constructions the field spent four rungs on, and an atlas of coupler curves — which is why a method superseded by computers is still the thing that feeds them.

synthesis · Synthesis $What a pin's size costs a synthesis. Every one of the 1,176 exactly correct three-position syntheses the synthesis field's survey produces, asked a question its three verdicts cannot ask: are any two of its pins closer together than the material round them? The poses span about two units, so the horizontal axis is a pin as a fraction of the mechanism. At a boss radius of 0.05 nothing is lost; at 0.4 — a pin nearly a fifth of the pose span — 27% of the exact solutions cannot be built, and 15% of the ones that had already passed the branch, circuit and order tests go with them. The defect is real, it is rarer than the kinematic ones, and it is not correlated with them.$

A defect that is not kinematic

The synthesis field's survey ends in three verdicts and all three are about which solutions a sweep visits. Here is a fourth, found by asking how much room two pins need — and it is the first defect on this site that a simulation cannot find, because in the equations a pin is a point.

synthesis · Synthesis

The configuration space of a crank rocker. Every point of the square is a pair of angles — the crank's and the rocker's — and the curve is where the coupler is exactly the right length to join them. That curve is the mechanism: one equation in two angles leaves one freedom, which is the mobility. It has 2 components, and that is the two assembly branches. A built linkage cannot cross between them, because there is no path in the set to cross by — and each goes all the way round, which is what makes the crank a crank. The square's left and right edges are the same line, and so are its top and bottom.

Branches were components all along

Seven words have been used for one thing. An assembly branch, a circuit, an assembly mode, a working mode, a posture and a branch defect are all statements about the connected components of a mechanism's configuration space — and once that is said, a four-bar's two circles, a platform's six modes and a synthesis defect stop being three subjects.

serial · Algebra

Named alongside it

The objects these essays reach for when they reach for this one.

Assembly branch Kinematic synthesis Grashof's condition Antiparallelogram Approximate synthesis Atlas Bezout Boss Burmester point Burmester theory Centre-point curve Circle-point curve

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