Bennett's condition is a ratio

Every mechanism in the planar fields is described entirely by lengths. Scale them all and the result is a similar machine, and every angle it reaches is the angle the original reached.

A spatial loop is not like that, and the difference is the subject of this essay.

Two kinds of parameter #

A spatial four-bar is described by, per link, a length — the common normal’s length between two joint axes — and a twist, the angle between them about that normal.

Four lengths and four twists, and the two kinds behave completely differently under a scaling. The lengths are multiplied. The twists are not, because a twist is an angle and an angle has no size.

So the scaling group acts on a four-dimensional subspace of an eight-dimensional parameter space. That is a smaller group action than a planar mechanism’s, where it acts on everything, and the consequences are different in a way worth working out rather than assuming.

What a scaling of half a parameter space means #

Before the condition, it is worth being careful about what is even being claimed, because scale the machine is ambiguous the moment there are two kinds of parameter.

Multiplying every length and holding every twist produces a similar machine in the ordinary geometric sense: the whole assembly is the original enlarged about a point, every axis points the same way, every angle between any two features is unchanged. That is the operation this essay means.

Multiplying every parameter, including the twists, produces something else entirely — a machine whose axes point in different directions, which is not similar to the original and generally does not move at all. That operation has no geometric meaning and it is what an unclassified probe would do.

So the scaling group acting on a spatial loop is not multiply the parameter vector. It is a group acting on a subspace, chosen by which coordinates carry a dimension, and the choice is dimensional analysis performed by a person before any arithmetic runs.

On a planar mechanism the two operations coincide, which is why twenty-five fields of this site have never had to make the distinction and why it is easy to carry the reflex into a field where it is wrong.

What Bennett’s condition says #

Bennett’s linkage is the four-revolute spatial loop that moves when Kutzbach’s count says it cannot, and it does so only for particular combinations of its eight parameters. The condition is

a / sin α  =  b / sin β

with opposite links equal in both length and twist.

The left side has a length divided by a dimensionless number, so it is a length. The right side likewise. The condition equates two lengths.

Scale the machine — multiply a and b, leave α and β — and both sides are multiplied by the same factor, so the equality survives. Bennett’s condition is preserved by a scaling, and a scaled Bennett linkage is a Bennett linkage.

That is not automatic and it is worth seeing why it works. The condition happens to be homogeneous of degree one in the lengths, so a uniform scaling multiplies both sides equally. A condition that mixed degrees — say a + b = sin α — would not survive, and would be a condition that could only hold at one size.

The quantity that is neither #

The ratio a/sin α is a length, and it is the loop’s own characteristic size. Bennett’s condition says the two independent link pairs agree on it.

That is worth dwelling on because it is a quantity a reader would not have thought to look for. A planar four-bar has no characteristic size — its four lengths are four independent numbers and no combination of them is singled out. A Bennett loop has one, produced by the condition, and it is what the two halves of the parameter space agree on.

Its value on a loop with a = 1 and a twist of 30° is 1 / sin 30° = 2.000 units, and every other link pair of that loop must return the same 2.000. A loop whose second pair returns 2.05 is not a Bennett linkage and does not move, whatever its individual numbers look like.

That gives the condition a shape a designer can hold: compute one number per link pair and require them to agree. It is a single scalar test rather than a relation to be checked, and the amount by which two pairs disagree is a direct measure of how far the machine is from being a Bennett loop — a length, and one whose dimensionless form is given below.

That gives a clean split. The twists are the loop’s shape and the ratio is its size, and Bennett’s condition is a statement that the shape determines the lengths up to one common factor.

Concretely: choose α and β freely, choose a freely, and b is then forced to a sin β / sin α. Three free numbers, of which two are dimensionless and one is a length. A Bennett linkage is therefore a two-parameter family of shapes times a one-parameter family of sizes.

That is a much tighter description than a planar four-bar’s, which has four lengths and three shape parameters. Overconstraint shows up as parameters removed, and here it removes one from a shape space that started with three.

Angles that are parameters and angles that are configurations #

A distinction that has to be made carefully here and does not arise in a planar field, because a spatial loop has angles of two entirely different kinds.

A twist is a parameter. It describes the machine: how one joint axis sits relative to the next, fixed once the parts are made, and it appears in the drawing.

A joint angle is a configuration. It describes where the machine is at an instant, it changes as the mechanism moves, and it is what a sensor reads.

Both are angles and both are dimensionless, and confusing them makes nonsense of everything above. The scaling acts on parameters; it does not act on configurations at all, because a configuration is not part of the machine.

In a planar mechanism the distinction is easy because every parameter is a length and every configuration is an angle. In a spatial one it is a real hazard: a table of eight parameters with four angles in it, beside a set of readings that are also angles, and only the modelling says which is which.

A parameter list has to be classified before it can be scaled, and the classification here is by role rather than by units. That is the practical difficulty of extending the whole scaling apparatus from the planar fields to the spatial one.

Seven of eight, and why so many #

Apply the field’s own question and the answer differs from the planar case in an instructive way.

Measure a spatial loop’s joint angles as it moves. Those readings are dimensionless, so by the standing argument they cannot recover a size — the four lengths come back only up to a common factor, exactly as a planar four-bar’s do.

But the twists are already dimensionless, so they are not in the invisible direction at all. Scaling the machine leaves them untouched, so the readings’ inability to see the scaling costs nothing in the twists.

So an angle-only measurement of a spatial loop recovers all four twists exactly and the four lengths up to one factor. Seven of eight parameters, rather than three of four.

That is a better ratio than the planar case and it is the same statement: the invisible direction is one-dimensional, and a parameter space with more dimensionless parameters in it loses proportionally less to it.

The measurement, and what it took #

The exponents here are less straightforward to obtain than a planar mechanism’s and the reason is the two kinds of parameter.

The probe scales a stated list of parameters and holds the rest. For a planar four-bar that list is every parameter it has. For a spatial loop it is the four lengths, and the four twists must be excluded explicitly — a probe handed the whole parameter vector would be scaling angles, which is not a similarity of anything and would produce a different mechanism rather than a bigger one.

That exclusion is a piece of knowledge the probe cannot supply. It comes from knowing which parameters carry a dimension, which is dimensional analysis and is the one input this method needs from a person.

Getting it wrong is not subtle: scale a twist by 1.6 and the loop stops satisfying Bennett’s condition, stops moving, and the probe returns nothing rather than a wrong exponent. The failure is loud, which is fortunate, and it would not be on a mechanism whose conditions are less exacting.

The six-bar’s null space has the same structure and the site found it the other way round: a measured null vector with a zero in it, at the one parameter that was a fraction rather than a length. There the classification was read off the arithmetic; here it has to be supplied to it.

Where the two halves interact #

The interesting behaviour is in the conditions rather than in the parameters, and Bennett’s is the site’s sharpest example.

A condition that involves only twists — these two axes are parallel — is scale-free and says nothing about the lengths. A condition that involves only lengths — these two links are equal — is scale-covariant and says nothing about the twists.

Bennett’s involves both, and what it demands is that a ratio of lengths match a ratio of sines. That is a relation between the shape and the proportions, and it is exactly the kind of relation a machine can be built to and can fail to hold.

Which is why the linkage is fragile. The site has measured that: a Bennett loop with a length out by a small amount stops moving, because the overconstraint that lets it move at all depends on two conditions agreeing exactly.

The scaling result says what the fragility is not. It is not a fragility to size — scale the whole loop by any factor and it still moves — and it is a fragility to the relation between the halves.

The other spatial loops #

Running the same question over the field’s other objects.

Sarrus is a pure translator, and its condition is about the directions of its joint axes: two sets of three parallel axes, in perpendicular planes. That is entirely a statement about twists, so it is scale-free outright, and a Sarrus linkage at any size is a Sarrus linkage.

The universal joint has one length that does not matter — the joint’s overall size — and one angle that decides everything. Its velocity ratio varies through each turn as a function of the shaft angle alone, so the whole of what the field says about it is scale-free.

Bricard’s loops need symmetry conditions on both lengths and twists, so they are mixed in the same way Bennett’s is.

The pattern: the loops whose conditions are on twists alone are unconditionally scale-free, and the ones whose conditions mix are the ones with a characteristic size. A designer scaling one of the second kind has to scale the lengths and hold the twists, which is obvious once said and is not what “scale the machine” ordinarily means.

What the overconstraint costs in parameters #

The counting is worth doing because it puts a number on what overconstraint means.

A general four-revolute spatial loop has four lengths and four twists — eight parameters — and Kutzbach’s count says it cannot move: six freedoms per body, five removed by each revolute, four bodies and four joints gives a mobility of −2.

Bennett’s loop moves, and it does so by satisfying conditions that remove parameters. Opposite links equal in length and in twist is four conditions; the ratio condition is one more. Eight parameters less five is three, of which two are dimensionless and one is a length.

So the family of Bennett linkages is three-dimensional inside an eight-dimensional space, and a machine drawn at random from that space does not move. That is the sharpest available statement of what makes the linkage remarkable, and it is a count rather than an adjective.

It also says why the linkage is fragile in manufacture. A machine has to sit on a three-dimensional surface in eight dimensions, and any error takes it off. The overconstraint that lets it move is the same thing that makes an error fatal, which the site has measured directly and which the parameter count explains.

The planar loop as a spatial one #

The site checks that a planar four-bar counted as a spatial loop is overconstrained, and the scaling result adds a footnote to that.

A planar four-bar viewed spatially has all four twists at zero. That is a condition on twists alone, so it is scale-free — every scaled copy is still planar, which is a fact so obvious that it is worth noticing it is a special case of something.

And the four lengths are then free: no condition relates them, because the condition Bennett’s linkage satisfies is degenerate when every sine is zero. So the planar case sits at a boundary of the spatial family where the mixed condition disappears and the loop becomes a pure length problem again.

The planar field is the spatial field with the dimensionless half of the parameter space set to zero, and everything in the planar field that is a shape is a ratio of lengths because the angles were used up.

A dimensionless form of the condition #

Since the condition equates two lengths, dividing through gives it a scale-free form, and the form is more useful than the original.

(a / b) = (sin α / sin β)

Both sides dimensionless, both sides shapes, and the condition now reads as a statement that one ratio equals another. A designer choosing twists has thereby chosen a length ratio, and the machine’s size is left over.

That form also says how to measure compliance. A built loop’s departure from Bennett’s condition can be quoted as the difference between those two ratios, which is dimensionless and comparable between machines of any size. Quoted in the original form the departure is a length and is not comparable — a loop of twice the size with the same proportional error has twice the departure.

A condition written as an equality of lengths has a scale-free form and the scale-free form is the one to report, which is the same recommendation the Grashof margin gets and the curvature field needs. Three fields, one recommendation, and the site currently follows it in none of them.

Where “scale the machine” stops being an instruction #

Every other mechanism in this survey answers the instruction multiply every parameter by k without hesitating, because every parameter is a length. A spatial loop is the first one on this site where the instruction is not well posed, and that is worth more than any of the numbers it produces.

A Bennett linkage has four link lengths and four twist angles. A scaling multiplies the first four and does nothing to the second four, so it acts on half the parameter space and fixes the other half pointwise. There is no sense in which the twists get bigger. The parameter space is a product of a cone of lengths and a torus of angles, and the scaling is a dilation of one factor and the identity on the other.

That is a fact about spatial mechanisms generally rather than about Bennett’s in particular, and it means the reflex twenty-five other fields have trained does not transfer intact. Scale the machine has to be read as scale the lengths here, and somebody who reads it as scale the parameters has scaled a twist angle, which is a different mechanism rather than a larger one.

Bennett’s condition survives the operation — a/sin α = b/sin β is homogeneous of degree one in the lengths and untouched in the angles — so the linkage is a family of shapes times a family of sizes rather than a set of isolated machines. That is a property of the condition and not a necessity: a condition mixing a length with an angle additively would not survive, and nothing forbids one.

The measurement consequence follows directly and it is the cheerful one. An angle-only measurement of a spatial loop recovers seven of its eight parameters, against three of four on a planar four-bar, because half of what is being measured had no size in it to begin with. The invisible direction is one-dimensional whatever the mechanism, so a machine with more parameters loses proportionally less, and a machine whose parameters are mostly angles loses almost nothing.

That inverts a common impression worth naming. Spatial mechanisms are described as harder to calibrate than planar ones, and in conditioning, in pose selection and in the sheer number of readings they are. In what can be recovered at all they are easier, and the two statements are about different questions. A reader who has absorbed the first will guess wrong about the second.

The lesson is to count the dimensionless parameters before estimating what a protractor can do, and it is a count anybody can do from the parameter list without any linear algebra at all. Every dimensionless parameter is one a protractor recovers outright. The lengths are where the loss lives, and the loss is one direction however many of them there are.