A band with a direction in it

The tolerance field’s sensitivities at a crank angle of one radian are

∂ψ/∂g = +0.2931    ∂ψ/∂a = −0.3416    ∂ψ/∂b = −0.3651    ∂ψ/∂c = +0.1491

Multiply each by its own length and add: 0.2931×4 − 0.3416×1 − 0.3651×3.5 + 0.1491×3 = 3.3 × 10⁻¹⁶.

Zero. And it is zero at every crank angle, exactly.

What that means for a tolerance #

The four sensitivities are the gradient of the output angle in the space of the four lengths, and a gradient orthogonal to a direction means the output does not change along it.

The orthogonality is not approximate and it is not a property of this configuration. It holds at every crank angle, on every four-bar, and the reason is one line: the output angle is a function of the ratios of the lengths, a function homogeneous of degree zero is annihilated by its own argument, and the gradient dotted into the parameter vector is exactly that annihilation.

So the tolerance field has been computing, at every configuration since the site first wrote about tolerance, a gradient with a known exact null direction in it, and has never used the fact.

The direction is the vector of the lengths themselves. A machine made uniformly too big or too small has no output error at all.

Measured: add a quarter of a per cent to all four lengths and the output angle at one radian moves by 0.000°, to the solver’s floor. Add the same quarter of a per cent to the ground length alone and it moves by 0.168°; to the crank alone, −0.049°; to the coupler alone, −0.183°; to the rocker alone, +0.064°.

So of all the ways a four-bar can be mis-made, there is one whole direction in which it cannot go wrong, and it is a direction a shop could plausibly wander along without meaning to.

The direction is the vector of the lengths themselves. A machine made uniformly too big or too small has no output error at all.

A quarter of the box is harmless #

That is not a curiosity and it can be quantified.

Take a tolerance box of ±δ on each of the four lengths, with the errors independent and equally likely anywhere in the box. Decompose each error vector into its component along the scale direction and the rest.

The mean squared component along the scale direction, as a fraction of the total mean square, is

Σ pᵢ² δ² / (|p|² · Σ δ²)  =  1/4

exactly, when the four tolerances are equal. A quarter of the mean squared manufacturing error does nothing to the output.

That is a fact about the dimension rather than about the machine: one direction of four. With unequal tolerances the fraction moves, since the box is then not isotropic and the scale direction is not equally represented in it.

It is also not a saving to be exploited. A tolerance specifies a box and a box contains the harmless direction and the harmful ones together; nothing about tightening or loosening a tolerance changes which quarter is which.

Where it can be exploited #

There is one arrangement where the direction is exploitable, and it is the practically important half of this essay.

If the four lengths’ errors are correlated so as to lie along the scale direction, they are harmless. That happens when a single common cause scales all four together, and there are two such causes worth naming.

The first is a measuring instrument with a scale error. A shop whose gauge reads one part in a thousand long makes every length one part in a thousand long, and the machine that results is kinematically perfect. That is an error a quality system exists to catch and a mechanism does not care about, which is an uncomfortable pairing and a true one.

The second is thermal, and it is the one with numbers.

Uniform thermal expansion.

There is one arrangement where the direction is exploitable, and it is the practically important half of this essay.

If the four lengths’ errors are correlated so as to lie along the scale direction, they are harmless. That happens when a single common cause scales all four together, and there is a common cause that does exactly that.

Uniform thermal expansion.

A four-bar made entirely of one material, at a uniform temperature, has all four of its lengths multiplied by 1 + αΔT. That is a pure scaling, so the output angle does not move.

Measured on an all-aluminium machine, α = 23 × 10⁻⁶ per kelvin:

ΔT = 10 K     scale 1.000230     Δψ = 0.000°
ΔT = 50 K     scale 1.001150     Δψ = 1.3 × 10⁻¹⁴°
ΔT = 100 K    scale 1.002300     Δψ = 0.000°

A hundred degrees, a scale change of a quarter of a per cent, and no kinematic error whatever.

Both of those read the box from outside, as a band. The box is easier to understand one corner at a time, because a corner is a definite machine with a definite sign on each of its four lengths — and two of the sixteen lie along the scale direction, which means two of the corners a tolerance budget is paying for cost the output nothing at all.

Two corners of the box are free #

The corner enumeration makes the result concrete in a way the algebra does not.

A ±δ box on four lengths has sixteen corners, each a combination of signs. Two of them — all plus and all minus — are nearly along the scale direction, and they would be exactly along it if the four lengths were equal.

They are not equal here, so no corner lies exactly on the null direction and the two all-same-sign corners are the ones nearest it. Their output errors are correspondingly the smallest of the sixteen: the departure that survives is only the part of the perturbation that is not a common scaling, which for a ±0.01 box on lengths of 4, 1, 3.5 and 3 is what is left after removing the best-fitting uniform change.

That gives a design lever that a box tolerance cannot express and a proportional one can. Specify each length to ±0.25% rather than to ±0.01 and the all-plus corner becomes exactly the scale direction — output error zero — while the mixed-sign corners are as bad as before.

A proportional tolerance puts two of the sixteen corners exactly on the nominal output, and an absolute one does not. That is a reason to write a tolerance proportionally that has nothing to do with the reasons the survey gives and points the same way.

And where it stops #

The result is exact and it depends on the expansion being uniform in both senses: one material, one temperature.

Take a machine with a steel frame and aluminium links — α = 11.7 × 10⁻⁶ against 23 × 10⁻⁶, a perfectly ordinary combination — and heat it uniformly:

ΔT = 10 K     Δψ = −0.0076°
ΔT = 50 K     Δψ = −0.0379°
ΔT = 100 K    Δψ = −0.0758°

Linear in the temperature, as a first-order effect should be, and not zero.

Put that beside the manufacturing band. A ±0.01 tolerance on all four lengths gives a worst-case output band of 0.658° at the same configuration. So a hundred-degree swing on a mixed-material machine costs about a ninth of the whole manufacturing budget, and on a single-material machine it costs nothing.

The difference between the two is entirely whether the errors lie along the null direction.

The thermal result is not the usual one #

It is worth separating this from the thermal effect everybody already knows about, because they are different and the familiar one is not what is being claimed.

The familiar thermal problem in a mechanism is binding: parts of different materials expanding by different amounts, closing up a clearance, and jamming a joint. That is real, it is a bodies-field question about material and clearance, and nothing here bears on it.

What is claimed here is about the kinematics: where the output link points. A machine that expands uniformly points its output link in exactly the same direction at every crank angle, however hot it is, because its four lengths are still in the same ratios and an output angle is a function of ratios.

Those two are independent. A uniformly-expanding machine can still bind, if a clearance closes; a mixed-material machine can be free of binding and still have a kinematic error.

Thermal binding is about absolute expansion and thermal kinematic error is about differential expansion, and the second is zero for a single-material machine at any temperature whatever. That is a stronger statement than the usual advice to match coefficients and it says why matching them is worth doing.

What a designer should take #

Three things follow and none of them is obvious from a conventional tolerance study.

Make the four links of one material if the machine will see temperature. Not for the usual reason — differential expansion causing binding — but because a uniform expansion is kinematically free and a differential one is not. A tolerance study on the lengths cannot see the distinction, because it treats the four errors as independent and the whole point is that they are not.

A machine’s own thermal error is a shape error, not a size error. What survives is the departure from a common scaling, so the quantity to control is the spread of the expansion coefficients rather than their magnitude. A machine of one exotic material with a huge coefficient is thermally perfect; one of two ordinary materials is not.

And a tolerance budget spent on holding all four lengths in proportion buys nothing. A shop that could hold the four lengths to a common scale error of ±0.5% and their ratios to ±0.01% would produce a machine indistinguishable from perfect, and no drawing asks for that because no drawing has a place to say it.

What a mixed machine’s error actually is #

The steel-and-aluminium number deserves unpacking, because the way to reduce it is not obvious from the number alone.

The error is not caused by the frame expanding. It is caused by the frame expanding differently from the links. Subtract a common scaling — the one that best fits the four expansions — and what remains is the departure, and the departure is what the sensitivities act on.

For a steel frame and aluminium links, the common part is whatever a weighted average of the two coefficients gives, and the departure is the frame short by (23 − 11.7) × 10⁻⁶ per kelvin relative to it, with the three links long by a smaller amount each.

So a machine of two materials has a thermal error proportional to the difference of their coefficients, and choosing two materials whose coefficients are close is worth exactly as much as the closeness. Steel and cast iron differ by about 10%; steel and aluminium by 100%; steel and a polymer by an order.

The design rule is match the coefficients, and the reason is that the common part is free. That is a stronger statement than thermal expansion causes errors, and it explains why a machine made entirely of a high-expansion material can be better than one made mostly of a low-expansion one.

Why the conventional study cannot see it #

Worth being precise about the gap, because the conventional study is not wrong.

The tolerance field’s worst case is Σ|∂ψ/∂ℓᵢ|·δᵢ, which takes each length to its extreme independently and adds the magnitudes. That is correct for independent errors and it is an upper bound for any correlation whatever.

What it cannot express is that a particular correlated combination gives zero. Taking absolute values throws away the signs, and the cancellation that makes the scale direction harmless is entirely in the signs — two of the four sensitivities are positive and two are negative, and the weighted sum cancels.

The root-sum-square version has the same problem for the same reason: squaring throws the signs away too.

A correlation structure needs a covariance, not a box, and the field’s own feature-based extension is exactly the machinery for that — a shared error that cancels out of a distance is the same idea as a common scaling that cancels out of an output.

Overconstrained loops are the exception #

One class of mechanism where the harmless direction is not harmless, and it is the class the spatial field is about.

An overconstrained loop moves because two constraints agree exactly. Scale the whole loop and they still agree, so a uniform scaling is harmless there too — the argument is unchanged.

What is different is everything else. A general perturbation of an overconstrained loop stops it moving entirely rather than moving its output a little, so its “band” is not a band at all: the machine either assembles or does not. The site measures that: a Bennett loop with a length out by a thousandth stops.

So for those mechanisms the four-dimensional picture is wrong. The set of parameter vectors that produce a working machine is a lower-dimensional surface rather than an open region, and the harmless directions are the ones along that surface — of which the scaling is one and there are others.

The scale direction is harmless for every mechanism; for an overconstrained one it is one of a larger set of harmless directions and everything off them is fatal rather than inaccurate. That is a much sharper tolerance situation and it is the reason the spatial field’s machines need a different treatment.

Reading it as an identification result #

The whole of this is the scale null space read in the other direction, and putting the two side by side is the neatest statement of what a null direction is.

Read forwards, from parameters to output: the output does not change along that direction, so a manufacturing error along it is harmless.

Read backwards, from output to parameters: the readings do not constrain that direction, so a measurement cannot recover it.

Same matrix, same null vector, two conclusions that sound unrelated and are one fact. A direction the output cannot see is a direction that cannot see the output.

That equivalence is worth keeping, because it says where else to look. Every null direction anywhere on this site is simultaneously an unrecoverable parameter and a harmless error, and the six-bar’s second one — scaling the second loop alone about O₄ — is therefore also a free direction in that machine’s tolerance box that nobody has ever exploited.

A catalogue of null directions is a catalogue of free errors #

The finding here is small and its generalisation is not, so the generalisation is worth stating carefully.

One direction of a four-bar’s tolerance box does nothing to the output. Exactly nothing, at every configuration, because the output angle depends only on the ratios of the lengths and that direction changes no ratio. With equal tolerances on all four links it accounts for precisely a quarter of the mean squared parameter error, and a tolerance budget that treats the four lengths as independent is spending a quarter of itself on an error that cannot appear.

The physical instance is the one that convinces. A single-material machine that gets hot expands along exactly that direction. A steel four-bar at a hundred kelvin above nominal is 0.23% larger in every link and its output angle has moved by 1.3 × 10⁻¹⁴ degrees, which is the solver’s noise floor and not a number. Make the frame steel and the links aluminium and the same hundred kelvin gives 0.076° — a ninth of what the manufacturing tolerances produce, and now a real error, because two materials do not move along the null direction.

That is a design rule falling out of a rank computation: for thermal purposes, one material is qualitatively different from two, and the difference is not a matter of degree.

Neither a worst-case nor a root-sum-square study can see any of it. Worst-case takes absolute values and root-sum-square squares them, and both discard the signs the cancellation lives in. The machinery that can see it is the covariance the feature-based work already introduced, which keeps the correlations and is the same apparatus for both jobs.

And the generalisation is the part to carry. Every null direction of an identification Jacobian is simultaneously two things: a combination of parameters no measurement can recover, and a combination of manufacturing errors that costs nothing. Those are the same statement read from opposite ends. So the site’s whole catalogue of unrecoverable directions is also a catalogue of free errors, and it has been sitting there being read one way only.

The Watt six-bar’s second null direction is the case worth checking, because it is not a uniform scaling of anything. It is a scaling of the output loop alone about the third ground pivot — five parameters multiplied and four left alone — which means a six-bar whose second loop is uniformly oversized has no output error whatever. That is a free direction in a nine-dimensional tolerance box, it is not a symmetry anybody would have guessed, and nobody has ever exploited it.

Whether it is usable is a separate question and the honest answer is that it depends on the workshop. A free direction is only worth money if parts can be made cheaper along it, and uniformly oversize this sub-assembly is a strange instruction to give a machinist. But it is free, it is exact, and knowing which errors cost nothing is the first half of knowing where to spend.