Eight kinds of four-bar

Grashof’s condition is the oldest classification in this subject, and its shape is worth looking at before anything is added to it. It compares the shortest bar plus the longest against the other two. When the comparison comes out one way it gives the linkage one of three names, according to where the shortest bar sits: crank-rocker, double crank or double rocker. When it comes out the other way it gives a single name, triple rocker, and stops.

That is a classification with a remainder. Three names for the linkages that pass, one for the linkages that fail, and nothing in the rule to say whether the ones that fail are one kind of thing or several.

The name that covers nearly half of length space #

How much the remainder holds depends on how lengths are chosen, but it is not a corner case under any reasonable choice. Draw four lengths uniformly between 0.2 and 5, throw away the sets that cannot be assembled at all, and 1,771 of the first 4,000 fail Grashof’s condition. Forty-four per cent of what a random draw produces shares one name.

There is a second, quieter defect on the passing side. The rule names a Grashof linkage by where its shortest bar is, and a crank-rocker is any linkage whose shortest bar is adjacent to the frame. So a four-bar whose shortest bar is the input and one whose shortest bar is the output are both crank-rockers. As chains they are the same object, since which bar is bolted to the bench is a decision rather than a property. As machines they are opposites: in one the motor turns a full circle and the far link swings, in the other the link the motor is on can only swing. A classification meant for a designer, who has already decided which shaft carries the motor, ought to tell those apart, and this one gives them the same name.

Both defects have one cause. Grashof’s condition is a single inequality, so it can only say yes or no, and the names are attached afterwards from a separate observation about the shortest bar. A finer classification needs more than one comparison, and the right ones turn out to be sitting in the geometry already.

Three sums instead of one inequality #

Write the ground as g, the input as a, the coupler as b and the output as c, and form three sums:

$$T_1 = g + c - a - b, \qquad T_2 = g + b - a - c, \qquad T_3 = b + c - a - g$$

Each has the same physical meaning. A four-bar can be laid completely flat, all four bars along the ground line, in exactly three arrangements that close. The input can point at the output’s pivot while the output points away from the input’s; both can point at each other; or the input can point away while the output points in. Ask how long the coupler must be to span the gap in each arrangement, and the three answers are T₁ = 0, T₂ = 0 and T₃ = 0 in turn. The fourth arrangement, both links pointing outward, needs a coupler as long as the other three bars together, and that is the boundary of assembling the chain at all rather than a wall inside the region where it assembles.

A flat four-bar is a change point: the one configuration where the chain can fold either way and the motion is not decided by the geometry. So the three sums are signed distances from the three kinds of change point. A linkage with every sum well away from zero is a long way from ever lying flat, and a linkage with one sum at zero lies flat somewhere in its cycle.

Grashof’s condition is recovered from these signs rather than added to them. The product T₁T₂T₃ is positive exactly when the shortest bar plus the longest is less than the other two, and it is an identity rather than a coincidence: of the three sums, one sets the shortest and longest bars against the other pair, and the signs of the remaining two are forced by where the input sits among the lengths, so the product can only report that one comparison. Checked on every linkage in the census below, the two statements disagree nowhere.

But a product discards what the individual signs carry. Three signs make eight patterns, and Grashof’s condition reads them as two.

Sorted by sign, then measured #

A prediction from signs needs a measurement that does not use them. The one used here asks the chain itself. Hold the input at an angle; its pin is then a definite distance from the output’s pivot, and the coupler and output can close that distance exactly when it lies between the difference of their lengths and their sum. Ask the same question at 360 angles of the input, and again at 360 angles of the output with the input free, and the result is what each grounded link can do: turn all the way, or swing through some range, and if it swings, which positions along the ground line that range contains.

Nothing in that measurement mentions T₁, T₂ or T₃. It knows about triangles and nothing else.

Every one of the 4,000 moves as its region says. The crank-rockers, 579 of them, have an input that turns and an output that swings between two positions without reaching either end of the ground line. The 540 double cranks turn at both ends. The 530 linkages in the region the census calls rocker-cranks are crank-rockers read backwards, with the output turning and the input swinging — and the usual classifier, asked about the same 530, calls every one of them a crank-rocker. The 580 double rockers swing at both ends without reaching the ground line at either. The four regions that fail Grashof’s condition hold 459, 428, 456 and 428 linkages, and each is a different kind of triple rocker.

Four names for eight regions, then, and the two places where the names run out are exactly the two defects above.

What the four triple rockers are #

The panels above draw one linkage from each region, solved at a crank angle in the middle of its range, with the reachable part of each grounded pin’s circle drawn thick. The four panels on the second row are the triple rockers, and they differ in something a drawing of a single position cannot show: which way along the ground line each rocker is able to swing.

Every rocker has two special positions. It can point along the ground line at the other pivot, which here is called the inner position, or along the ground line away from it, the outer one. A rocker’s range of motion is a single arc of its circle, and that arc can contain the inner position, the outer one, or neither. It cannot contain both, because a link whose range contained both ends of the ground line would have swept through everything between them on one side and could turn fully.

In the 0–π rocker, the input’s range contains its inner position and so does the output’s: the input can swing across the line towards the output pivot and the output can swing across it towards the input pivot, and the two sweep 76% and 72% of a circle. In the π–π rocker the input swings through its outer position and the output through its inner one, over 84% and 51%. In the π–0 rocker both swing through their outer positions, 84% and 85%. In the 0–0 rocker the input swings through its inner position and the output through its outer one, 67% and 91%. The names are McCarthy’s, and they read the same facts as angles measured from the ground line: 0 for a rocker that can lie along the line in the direction from input pivot to output pivot, π for one that can lie along it the other way.

Why a designer should care is that the position a rocker swings through is where its useful stroke is centred. A windscreen wiper’s arm sweeps symmetrically about a line it passes through; a clamp’s lever wants its stroke on one side. The four triple rockers put their swings in four different places relative to the frame, and a design that needs the output to sweep through the direction of the other pivot is restricted to two of the four regions before any length has been chosen.

The Grashof rows have a different signature. The double rocker’s two links swing 23% and 21% of a circle and reach neither the inner nor the outer position. Its coupler is the link that turns fully, so its rockers swing on one side of the ground line, and the other side belongs to the linkage’s other assembly, which the chain cannot reach without being taken apart.

Two routes to the same ranges #

The census is cheap because the closure test is a comparison of three numbers. It is only worth trusting if it says what the solver says, and the solver is a different kind of computation entirely: it drives a link round, runs Newton’s method on the loop-closure equations at every angle, and records which positions it can find.

There is a hazard in the solver route that the closure test does not share. A sweep that fails at an unreachable angle leaves the joints wherever the failed iteration put them, and the next angle, which may be perfectly reachable, then starts from nonsense and fails too. A careless solver sweep therefore reports ranges that are too short, on exactly the linkages whose ranges have gaps, which are the interesting ones. The sweep here retries every failure from both mirror-image guesses before giving an angle up.

With that done, the two routes refuse the same angles on all eight linkages: no disagreement anywhere in 5,760 decisions. That is what licenses running the census on the cheap route alone. The solver established that the triangle test measures what the mechanism does; the census then asks it 2,880,000 times.

The walls, drawn #

Each sum is linear in the four lengths. Hold two lengths fixed and vary the other two, and every boundary in the plane that is left is a straight line: the three walls where a sum vanishes, and the four lines beyond which the chain cannot be assembled. So a region’s share of such a slice is a convex polygon, and it can be computed exactly by clipping a rectangle against seven half-planes instead of being coloured in by sampling.

The slice shows something the census could not. Only six regions appear on each side, and not the same six. The reason is a line of arithmetic: T₂ − T₃ = 2(g − c), so once the ground and the output are held, which of those two sums is larger is decided before the input or coupler is chosen, and the two sign patterns that would need the other order cannot occur anywhere on the slice. Holding g = 4 and c = 3 removes the double crank and the π–π rocker. Holding g = 3 and c = 4 removes the rocker-crank and the 0–π rocker. Between them the two slices show all eight.

The same arithmetic gives T₁ − T₃ = 2(g − b) and T₁ − T₂ = 2(c − b). The three sums are not independent of the order of the lengths, and that is where Grashof’s attention to the shortest and longest bars comes from: the sign pattern is largely a statement about which bars are longer than which.

There is also a statement about size. Multiplying every length by the same factor multiplies every sum by that factor and changes no sign, so the region is a property of a linkage’s shape. That is the classification-sized version of the observation that Grashof’s condition survives a scaling: eight kinds of shape, and no kind of size.

Lengthening one bar #

A slice shows where the walls are. What it does not show is what crossing one does to the machine, and the direct way to see that is to take one linkage and change one length slowly.

Starting from the standard crank-rocker, with g = 4, b = 3.5 and c = 3, and lengthening the input, the linkage stays a crank-rocker until a = 2.5, where T₃ passes through zero and it becomes a 0–π rocker. At a = 3.5, T₁ passes through zero and it becomes a rocker-crank. At a = 4.5, T₂ does and it becomes a 0–0 rocker. Three walls, four kinds of machine, and the order is the order of the three lengths at which the chain can be laid flat.

Inside a band the share of the turn that assembles varies continuously, falling from a full circle as the input grows past the point where the far side of the chain can follow it. At a wall it jumps in kind: two circuits become one, a full turn becomes a swing, a swing that contained the inner position becomes one that contains the outer. Nothing between the walls changes what the linkage is, however much it changes how much of each thing it does.

Why the kind can only change at a wall #

That is not a numerical accident of one walk, and the reason connects this classification to a claim the serial field made about branches: that an assembly branch is a connected component of the configuration space.

A four-bar’s configuration space is a curve. Plot the output’s angle against the input’s, both assemblies together, and a Grashof linkage draws two closed loops that never meet, while a triple rocker draws one. How a link can move is read off that curve — whether a loop runs all the way across the input axis, whether it crosses the line where the output points at the input pivot. So the kind of motion can change only when the curve changes its topology, and a smooth curve changes topology only by passing through a point where it crosses itself.

A self-crossing of the configuration curve is a point where the loop-closure equations lose rank, which is the event measured as a drop in mobility since the first essays on constraint. For a four-bar the only configurations where that happens are the ones with all four bars in a line. And a flat arrangement exists precisely when one of the three sums is zero.

So the chain of reasoning closes. The walls are the flat linkages; the flat linkages are the only ones whose configuration curve has a crossing; a crossing is the only place circuits can join or part; and the circuits decide the motion. At a = 2.3 the curve is two loops, at a = 2.5 the loops touch at the one position where the input points outward and all four bars lie on the ground line, and at a = 2.7 they have merged into one. Between walls there is nowhere for the topology to change, which is why the census found no exception and could not have found one.

What the eight regions are for #

Read as a design tool, the classification reorders the first decisions. Grashof’s condition asks for four lengths and then reports a name. The signs can be chosen first: a designer who wants an output that swings through the direction of the input pivot and an input that swings through its outer position has asked for T₁ > 0, T₂ < 0 and T₃ > 0, three linear inequalities, and every set of lengths inside them will do it.

The inequalities also say how much margin a design has, and in a more useful form than Grashof’s single margin. The smallest of the three distances to a wall is how far the linkage is from lying flat somewhere, and a tolerance smaller than that cannot change its kind. That is the per-region version of the margin that decides whether a classification survives manufacture.

Re-grounding a chain permutes the roles of the four bars, so it permutes the three sums among themselves and flips some of their signs. A crank-rocker grounded on its coupler is still a crank-rocker and grounded on its shortest bar is a double crank, as the inversion essay found by sweeping; in these terms the chain carries one unordered set of three absolute values, and the choice of ground decides which signs they wear.

What the regions do not say anything about is the quality of the motion. Two linkages in the same region can differ by a factor of three in their worst transmission angle, and the region is the first decision rather than the design.

What the census cannot see #

Three limits, of different strengths.

It does not sample the walls. Linkages within two per cent of the longest bar of a wall were redrawn, because a sampled closure test cannot resolve a range whose gap is narrower than its step. The change-point linkages themselves, where a sum is exactly zero, are therefore not in the census at all. What happens on a wall is shown by the configuration curves and by the argument about crossings, not by counting.

Its proportions are the proportions of a draw. That the eight regions hold between 428 and 580 linkages each is a fact about lengths drawn uniformly between 0.2 and 5, not about linkages anybody builds. A different draw would fill the regions differently and would not move a wall.

It classifies, and nothing else. The measurement asks which positions each grounded link reaches, and the prediction is about exactly that. It says nothing about the coupler curve, the speed of the output or the forces, all of which vary continuously inside a region and are where the rest of a design happens.

What comes next: the slider-crank’s regions #

The next case is the slider-crank, and it is the limit of this one rather than a separate case. Let the output bar and the ground grow together, keeping their difference fixed at e. The output pin then moves on a circle so large that near the input it is a straight line, square to the ground line and a distance e from the crank’s pivot: an offset slider-crank.

In that limit T₁ grows without bound and stays positive, while T₂ tends to b − a + e and T₃ to b − a − e. Two signs are left, so there are four regions rather than eight, and the textbook condition for an offset slider-crank’s crank to turn fully — a connecting rod longer than the crank plus the offset — is the region where both remaining sums are positive. What that case has to establish, by the same census and the same two routes, is whether the slider-crank’s four kinds are exactly four of these eight carried to the limit, and which four disappear on the way.