The problem the other way round

Here is the shape of every essay on this site so far. Four lengths are given. The solver drives the crank, closes the loop, and reports where everything went. Some quantity is measured — a transmission angle, a coupler curve, a velocity ratio — and a claim about it is checked.

That is the analysis problem: given the mechanism, find the motion. It is the right problem for a reader, because a mechanism is what a picture shows, and it is the problem the whole apparatus of this site is built to solve.

It is not the problem a designer has.

A designer knows where something has to go. A bucket must tip through these three attitudes. A hatch must lie flat here and stand upright there. A film must be pulled down by exactly one frame and held still for the rest of the cycle. The motion is the specification, and the lengths are what has to be found.

That is synthesis, and it is genuinely a different problem. Not harder in a vague sense — different in structure, with its own theorems, its own failure modes, and a two-hundred-year history of people solving it by means that look nothing like solving equations.

Why it is not just analysis backwards #

The obvious approach is to write the analysis equations and solve them for the lengths instead of for the angles. That works for the smallest problems and stops working quickly, for three reasons that are worth separating.

The equations are not square. A four-bar has four lengths, two ground-pivot positions and a coupler-point position — nine numbers, less the three that only place the whole mechanism on the page, so six that matter. A prescribed pose is three numbers. Three poses is nine conditions on six unknowns, which is over-determined; two poses is six conditions on six unknowns, which sounds right and turns out to leave a two-parameter family. The counting is not the obstacle by itself, but it means the problem’s character changes with the number of positions rather than being one problem with more or fewer data.

They are not linear, and their non-linearity is the mechanism’s. The loop-closure equations involve distances, so a synthesis equation involves a square root of a sum of squares of unknowns. Newton on that lands somewhere, and where it lands depends on where it started, and there is no reason the answer it finds should be the one a designer wants.

Most solutions are useless, and nothing in the equations says which. This is the one that matters most and it gets its own essay. A synthesis can satisfy every prescribed position exactly and produce a linkage that cannot pass between them without being dismantled. The equations are silent about this because it is not a statement about positions; it is a statement about paths.

What a designer actually asks for #

Synthesis problems come in three kinds, and they are different enough that a method for one is often useless for another.

Function generation. The output angle must be a specified function of the input angle: turn the input by θ and the output should be at f(θ). This is what a mechanical computing linkage does, and it is what a linkage that has to approximate a square root or a logarithm was doing before electronics. The specification is a curve on a graph, and the four-bar is being asked to approximate it.

Path generation. A point on the mechanism must trace a specified curve. This is the classical coupler-curve problem, and it is where the atlases came from.

Motion generation, also called rigid-body guidance. A whole body — not a point — must pass through specified positions and orientations. This is what the figure above shows and what most of the mechanisms people actually design are doing: the bucket has to arrive at the right place and the right angle, and a linkage that gets the point right and the angle wrong has not solved the problem.

The three are not variants of one problem. Function generation constrains a relation between two angles and says nothing about where anything is. Path generation constrains a locus and says nothing about orientation. Motion generation constrains both. The methods differ accordingly, and this field’s essays are about the third, because it is the one with the cleanest theory and because Burmester’s construction for it is exact.

Precision positions #

The unifying idea across all three is the precision position: a finite number of places where the mechanism is required to be exactly right, with no requirement at all in between.

That is a strange specification to meet for the first time and it is worth defending, because the natural request is “follow this curve”, not “pass through these four points on it”.

The defence is that “follow this curve” is generally impossible. A four-bar’s coupler point traces a curve of degree six, and a designer’s desired curve is very unlikely to be one of those. So the choice is between an exact solution to a reduced problem and an approximate solution to the real one, and classical synthesis takes the first: satisfy a handful of positions exactly, accept whatever happens between them, and check afterwards.

The number of positions that can be satisfied exactly is not a matter of effort. It is fixed by counting, and the fourth position is where the counting starts to bite: three positions leave a designer the whole coupler plane to choose from, four leave a curve, five leave finitely many solutions and six generally leave none.

Two positions, and why nobody stops there #

The smallest interesting case is two prescribed positions, and it is worth doing because it explains the shape of everything above.

A rigid body moved from one position to another can always be got there by a single rotation about a single point — Chasles’s theorem in the plane, and the point is the pole of the displacement. Find the perpendicular bisector of the segment joining any point’s two images; every such bisector passes through the pole, so two of them locate it.

So for two positions the answer is: put a fixed pivot at the pole and a moving pin anywhere, and a single link carries the body from one position to the other exactly. There are infinitely many solutions and they are all trivial, because a single revolute joint between the body and the frame is not a mechanism with an input and an output — it is a hinge.

Two positions are therefore a solved problem with an uninteresting answer, and the reason to state it is that the pole survives into the harder cases. Three positions have three poles, one for each pair, and those three poles form a triangle whose properties carry most of Burmester’s classical results. The construction this field uses does not need the pole triangle, because a circumcentre is easier to compute than to construct with compasses, but the two are the same geometry seen from different centuries.

When the positions cannot be met exactly #

Everything above is exact synthesis: a finite number of positions, satisfied to the last decimal place. There is a whole other branch, and a designer meets it as soon as the specification has more than five positions in it.

Approximate synthesis gives up on exactness and minimises an error instead. Specify twenty positions along a desired path, define a cost — sum of squared distances between where the coupler point goes and where it should — and search the space of link lengths for a minimum. This is an optimisation problem, and it behaves like one: it has local minima, its answer depends on the starting guess, and it gives no guarantee that a better solution does not exist elsewhere.

What it does give is a mechanism for specifications that exact methods cannot touch. It also gives control over where the error goes, which exact synthesis does not: a designer can weight the positions that matter and let the others drift, and that is often exactly what is wanted.

The two approaches are not rivals so much as different regimes, and the sensible practice uses both — exact synthesis to find a family of candidates that satisfy the critical positions, then optimisation from those starting points to trim the behaviour in between. The exact construction’s real value in that workflow is that it supplies good starting guesses in quantity, which is precisely what an optimiser on a non-convex problem needs and cannot generate for itself.

The curve that four-bar draws is not unique to it either: two other four-bars draw the same curve exactly, and knowing that is part of knowing what a synthesis has actually delivered.

The atlases #

Before any of this could be computed, the practical method was to look it up.

Hrones and Nelson’s Analysis of the Four-Bar Linkage (1951) is the famous one: some seven thousand coupler curves, drawn by machine, printed on card, bound in a large book. A designer with a curve to produce would leaf through until something resembled it, read off the proportions, and scale.

There were others, and the genre is worth knowing about. Kinematic synthesis produced whole libraries of this kind: atlases of coupler curves, tables of Burmester solutions, nomograms for function generation, and — in the Soviet tradition — catalogues organised by the task rather than by the proportions, so that a designer looked up “approximate straight line over 40% of the stroke” and got the linkages that do it.

It is easy to be condescending about that and it would be wrong. The atlas is a solution to the path-generation problem that is complete in a way no equation-solving method is: it covers the space of four-bar proportions systematically, it shows the designer what is achievable rather than only whether one guess works, and it makes the relationship between proportions and curve shape visible. A designer who has spent an afternoon with it has an intuition that a solver does not provide.

What it cannot do is satisfy a motion specification, because a curve on a page shows where the coupler point goes and not what the coupler is doing. And it cannot answer the question a designer actually has, which is not “what curve does this linkage draw” but “which linkage draws this curve” — the atlas answers the first question seven thousand times and leaves the reader to do the search.

The atlas is the reason Roberts’s cognate theorem is practically useful rather than merely surprising. Every curve in the book is drawn by three different linkages, so the book is implicitly three times larger than it looks, and a designer who found the right curve with the wrong proportions had two more sets to try.

Which mechanism, before which dimensions #

There is a decision that comes before any of this and that the classical theory mostly assumes away: type synthesis — choosing what kind of mechanism to use at all.

A four-bar is one answer. A slider-crank is another, and it is the same chain with a joint changed. A six-bar has more freedom and can satisfy more positions. A cam can produce any motion whatever, at the cost of accelerations nobody asked for and a part that must be machined rather than assembled from bars. A gear train gives an exact ratio and no path at all.

Everything in this field is dimensional synthesis: the type has already been chosen, and what remains is to find the numbers. That is the part with theorems in it, and it is the part a computer is good at.

Type synthesis has far less theory and is where the design actually happens. The rule of thumb worth stating is that the number of prescribed positions drives it: up to five, a four-bar will generally do; beyond that, either a six-bar or an admission that the specification wants a cam. And the trade running underneath is the one this whole site keeps meeting — a linkage gives the motion its geometry allows, and a cam gives the motion that was asked for, at a price paid in the derivatives.

What this field does #

Four essays, and they follow the problem rather than the history.

This one has said what the problem is. The next gives Burmester’s construction for three prescribed positions, which is exact, takes six lines of arithmetic, and has no iteration in it at all — the fixed pivot of any moving point is the circumcentre of its three images, and that is the whole method.

The third adds a position and watches the freedom collapse: with four poses only points on a particular cubic will do, and the cubic is drawn here as a contour of a measured quantity rather than plotted from its equation.

The fourth is Roberts’s theorem, which says that whatever linkage a synthesis ends with, there are two others that draw the same coupler curve — so a solution that will not fit in the space available has two alternatives that will.

Running underneath all four is the discipline the rest of the site uses, applied to a problem where it is unusually necessary. A synthesis is a construction, and a construction cannot check itself. So every linkage these essays produce is handed straight back to the forward solver: driven to each prescribed position, the pins are compared with where the specification says they should be, and the linkage is then swept from the first position in both directions to see whether it can actually get to the others.

The construction always passes the first test, because it is a theorem. A substantial fraction fail the second, and that is the field’s real subject.

Why the forward solver is not optional here #

It would be reasonable to ask why a construction that is provably exact needs checking with a numerical solver at all, and the answer has two parts.

The first is the ordinary one this site applies everywhere: a proof is about the mathematics and the code is about the arithmetic, and the gap between them is where mistakes live. The check caught one immediately, and it was in the check rather than the construction. The first version built one mechanism and drove it to each pose in turn, carrying the previous solution forward as the next guess. That is exactly right for a sweep, because a built mechanism cannot jump between assembly branches and following it is the physically honest thing to do. Three prescribed poses are not a sweep. They are three separate questions, and dragging the guess across the plane from the answer to the first left the crank pin far from where the crank constraint was about to put it. The solve stalled at a residual of 0.70 on a linkage whose construction was exact to fourteen decimal places, and the report said the synthesis had failed.

The second part is more interesting and is the reason this field exists on a site about analysis. The construction answers a different question from the one the designer asked. It answers “is there a four-bar whose coupler occupies these three poses”, and the answer is yes with a two-parameter family to choose from. The designer asked “is there a four-bar that takes the coupler through these three poses”, and that question has the word “takes” in it — it is about motion, and motion is what a solver computes and a construction does not.

Separating those two questions is most of what the rest of this field does. The construction is the cheap part; the sweep that follows is where the answer is.

The reversal has a consequence for what a site like this can honestly offer, and it is worth naming. Analysis has one answer and synthesis has a family, so an essay about analysis ends with a number and an essay about synthesis ends with a set of candidates and a filter. That is why the synthesis field’s rungs are about counts, defects and searches rather than about mechanisms — the object being computed is not a linkage but the space the linkages live in. It also explains the atlases, which look like a historical curiosity and were a rational response to exactly this. If the answer is a family and the family cannot be searched by hand, then somebody enumerating it once and printing it is doing the only thing available. What replaced the atlas is not a better method but a cheaper search, and the family is the same family it always was.