One chain, four mechanisms

A four-bar linkage is four bars and four pins. One of the bars is bolted to the bench and the others move.

Which one? The question sounds like it has an obvious answer — the ground link, obviously — and the answer is that the chain does not contain the information. Four bars pinned in a loop is a kinematic chain, and it becomes a mechanism only when a link is chosen to be the frame. Choose a different one and the result is a different mechanism made of the same parts.

What changes and what does not #

The relative motion between any two links is untouched. If link 2 rotates 30° relative to link 3 when the mechanism is in a particular configuration, that stays true whichever link is grounded, because relative motion is a statement about the two links and not about the bench.

What changes is everything absolute. Which link traces which path, which one rotates fully, what the output looks like, whether there is a useful input at all — all of that is a statement about motion relative to the frame, and the frame is exactly what has been changed.

That is the whole of kinematic inversion, and it accounts for a surprising fraction of the mechanisms in the classical catalogue being the same mechanism.

The four inversions of a four-bar #

Take a crank-rocker: ground 4, crank 1, coupler 3.5, rocker 3. Ground each of the four in turn.

Ground the long link. The original: crank rotates fully, rocker swings. This is the crank-rocker, and it is what the site’s other essays draw.

Ground the crank. Grashof’s classification says that grounding the link adjacent to the shortest gives a crank-rocker and grounding the shortest itself gives a double-crank, so this one is a double crank — both links adjacent to the frame rotate fully. A drag-link mechanism, and the useful thing about it is that it converts a constant input rotation into a non-constant output rotation, both going all the way round.

Ground the coupler. Another crank-rocker, by the same rule applied from the other side.

Ground the rocker. A double rocker: neither link adjacent to the frame can turn fully, so this one has no rotating input at all and must be driven from the coupler.

Four mechanisms, three Grashof types, one chain. The classification is not a property of the four lengths alone — it is a property of the four lengths and which one is the frame — and Grashof’s rule is precisely the statement of how the type depends on that choice.

Why the mobility does not change #

Before the invariant, a small check that the idea is coherent at all: an inversion has the same mobility as the mechanism it came from, and the reason is visible in the formula.

Grübler counts 3(n − 1) − 2j₁, where n includes the frame. Inversion does not change how many links there are, nor how many joints, nor what kind they are. It changes only which link the (n − 1) subtracts, and the formula does not care which. So all four inversions have mobility 1.

The Jacobian route agrees for a slightly better reason. Grounding a link amounts to fixing three of the system’s coordinates — two of position and one of orientation — and the constraint equations relating the rest are unchanged. Removing three coordinates and leaving the constraints alone reduces both the unknown count and nothing else, so the rank is unchanged and the difference is unchanged. What is being fixed is a frame, not a constraint.

That distinction is worth having. A mechanism can be over-constrained or under-constrained by adding or removing joints; grounding a different link does neither, and an inversion is therefore always a mechanism if the original was.

The invariant #

If the four inversions are genuinely the same chain, something should be the same about them, and it should be measurable.

The thing that cannot change is the shape of the closed loop. Four bars pinned in a loop form a quadrilateral, and as the mechanism moves that quadrilateral flexes through a one-parameter family of shapes. Grounding a different link does not change which shapes are available; it changes only which vertex is nailed down and which edge is horizontal.

A quadrilateral’s shape, given its four sides, is fixed by one more number, and the natural choice is a diagonal — the distance between two opposite pins. There are two of them. The pair (d₁, d₂) determines the shape completely, and it makes no reference to where anything is on the page.

So the check is: sweep each inversion independently, with its own solver runs and its own crank, and record the pair of diagonals at every position. If the four are the same chain, they visit the same set of pairs.

They do, and the agreement is at half the sampling resolution — which is the answer to expect when a point lands between two samples, and which is quoted against the measured spacing of the reference cloud rather than against an absolute tolerance.

Two details in that measurement are worth recording.

The diagonals must be compared as an unordered pair. An inversion renames the joints, so which diagonal a given inversion calls “the one from O₂ to B” depends on where it starts counting. Sorting the pair factors out exactly the arbitrary thing being tested.

Half the inversions cannot turn their crank. The double-rocker’s input swings through a limited range, so sweeping it from 0° to 360° and keeping what converges found nothing at all — and an inversion that visits no shapes trivially agrees with every other one. Trying each crank angle on both assembly branches finds the whole locus regardless of which links happen to rotate. This is the same correction tracing a cognate needs, for the same reason, and it is a general point about sweeping a mechanism whose input is not a full rotator. A double rocker’s travel is bounded by toggle positions, and beyond them the configuration does not exist rather than the solve being poor — telling those two apart is what the whole apparatus is for.

There was a third detail, and it is the kind of thing that makes a check silently meaningless. The comparison quotes the disagreement against the reference cloud’s nearest-neighbour spacing, and the two assembly branches at one crank angle are reflections of each other in the ground line — so both give exactly the same pair of diagonals, and every point in the cloud has a duplicate at distance zero. The spacing came out at 2 × 10⁻¹⁵, and the agreement looked a million million times worse than the sampling. Coincident samples are not neighbours.

Reading the four panels #

The figure’s four panels repay a moment’s attention because two of them look wrong at first.

The crank-fixed panel shows a mechanism whose links sweep round each other in a way that reads as a tangle. That is the double crank, and both of its frame-adjacent links really do go all the way round: what looks like a tangle is two links passing each other, which they must, and the panel’s ghost outlines show it happening.

The rocker-fixed panel shows fewer positions than the others — 640 of 2,880 attempts — and that is not a failure of the sweep. It is a double rocker; its input link genuinely cannot turn past a limit, and the positions that are missing are configurations the mechanism does not have. The count is printed under each panel for exactly this reason: a figure that showed only what assembled, without saying how much did not, would let a reader assume the ranges were comparable.

The panels are also drawn to their own scale rather than to a common one, because grounding a different link puts the mechanism in a different place and at a different size on the page. That is a drawing decision, it makes the four shapes comparable and the four sizes not, and it is worth knowing before comparing panel to panel.

Why this matters more for the slider-crank #

The four-bar’s inversions are instructive. The slider-crank’s are famous, and they are the reason the idea has a name.

A slider-crank is a four-bar with one pin replaced by a slide. Its four inversions are:

Ground the frame. The engine mechanism: crank, connecting rod, piston in a cylinder. This is what an internal combustion engine is.

Ground the crank. The Whitworth quick-return. The slider now runs in a link that rotates, and the geometry produces a large time ratio — the number a plain crank-rocker struggles to get past 1.2 — which is why shapers use this and not the direct form.

Ground the connecting rod. The oscillating-cylinder engine, in which the cylinder itself rocks on a trunnion while the piston reciprocates inside it. Popular in small marine and toy steam engines because it needs no valve gear: the cylinder’s rocking uncovers the ports.

Ground the slider. The hand pump: the cylinder is fixed, and what was the frame now swings.

Four machines with different names, different uses and different centuries, and they are one chain. That is the observation the concept exists for, and it is genuinely useful rather than merely tidy: a designer who needs a large time ratio does not invent the Whitworth mechanism, they invert a slider-crank and read off what happens.

The chain, and what it is #

Kinematic inversion is the reason the word “chain” earns its place in the vocabulary alongside “mechanism”, and it is worth being precise about the difference because most treatments blur it.

A kinematic chain is a set of links and the joints between them. It says what is connected to what, and how, and with what dimensions. It does not say what is stationary and it does not say what is driven.

A mechanism is a chain with a link designated the frame.

A machine is a mechanism with a link designated the input, and with forces in it.

Each step adds information that the previous level genuinely does not contain, and the arithmetic follows the same pattern. Grashof’s condition is a statement about a chain — the sums of the lengths — but its conclusion, that a particular link rotates fully, is a statement about a mechanism, because “rotates fully” means relative to the frame. So Grashof’s rule has to be quoted with the frame named, which is exactly how the classical statement does it: the shortest link rotates fully if it is adjacent to the frame or is the frame.

This is also why the site’s data model declares the topology of each mechanism explicitly rather than inferring it from the constraint list. A coupler carrying three pins can be written as two distance constraints or as one plus an attachment, and those are the same physical chain described two ways; the constraints do not determine the chain, so the chain has to be stated.

Inversion as a design move #

The practical use is worth stating directly, because it is easy to read all this as taxonomy.

A mechanism that will not fit can be inverted. The four inversions put the frame in four different places and therefore need bearings in four different places. If the packaging says a bearing cannot go at O₄, an inversion may put the required motion somewhere it can.

A mechanism whose input cannot rotate can sometimes be fixed by inversion. A double-rocker has no rotating input, which is awkward if the power comes from a motor. Grounding a different link of the same chain may give a crank — and the relative motion, which is what the machine was designed for, is unchanged.

A relation that is hard to see in one inversion may be obvious in another. This is how function generation is solved with a motion-generation construction: invert so that the coupler is the frame, and a specification about two angles becomes a specification about positions.

The last one generalises. Inversion is a change of reference frame, and changing reference frame is the standard move for making a hard kinematics problem easy — the same move that turns the analysis of a planetary gear train into an ordinary train seen from the carrier.

Inversion in space #

The idea is not planar, and one spatial case is worth naming because it shows the concept doing work where it is not obvious.

The universal joint is a spherical four-bar — four revolute axes through a common point. Its normal inversion grounds the link that carries the two shaft bearings, and the input and output are the two shafts.

Ground one of the cross’s links instead and the same chain becomes a mechanism in which the two shafts both move relative to the cross. That is not a machine anyone builds, and its use is analytical: the velocity relation between the two shafts is easier to derive in the inverted frame, where the cross is stationary and the two shafts are simply rotating about axes fixed in it.

The general point is that inversion costs nothing and changes what is easy. A relation that requires a page of trigonometry in one frame may be a single observation in another, and the frames are all equally valid because the relative motion — which is what a kinematic relation is about — is the same in all of them.

What inversion is not #

Two things it is easy to conflate with it.

It is not the same as running the mechanism backwards. Driving a four-bar from the rocker instead of the crank changes which link is the input and leaves the frame alone; the mechanism is the same mechanism, and the only thing that changes is which end of it the power comes in. Inversion changes the frame.

It is not a rigid-body transformation of the whole thing. Picking a mechanism up and putting it down rotated changes nothing at all — every absolute motion is transformed the same way and the mechanism is the same mechanism. Inversion changes which link is stationary, which is a change to the mechanism’s structure and not to its placement.

The test that distinguishes all three is the one the figure runs: sweep each and compare the loci of loop shapes. A rotated mechanism gives the same locus for the trivial reason that it is the same mechanism. A mechanism driven from the other end gives the same locus for the same reason. Four inversions give the same locus for a reason that is not trivial at all — they are four different mechanisms, with different Grashof types and different behaviours, and what they share is the chain underneath.

What the panels do not show #

The figure shows four mechanisms and one invariant, and there is a third thing that changes under inversion which no static panel can display: the timing.

Each inversion’s output is a different function of its input’s angle. The crank-rocker’s rocker swings out and back with one profile; the double crank’s output goes all the way round at a varying rate; the double rocker’s output does something else again. So two inversions that visit the same set of loop shapes visit them on different schedules, and for a machine that has to be somewhere at a particular moment they are not interchangeable.

That is why the quick-return mechanisms are inversions rather than reproportioned direct forms. What a designer wants from them is precisely the altered timing — a large ratio between the working stroke and the return — and the reason inversion supplies it is that the relationship between the input’s constant rotation and the output’s travel is exactly what grounding a different link changes.

The invariant and the variable are therefore complementary: the diagonals say the four mechanisms are one chain, and the timing says they are four machines. Neither statement is more true than the other, and a treatment that gave only the first would make inversion sound like a classification exercise rather than a design move.