A joint is a surface that slides on itself

Every field on this site declares its joints. A four-bar arrives with four pins; a Gough platform arrives with six spherical joints and six sliders; Bennett’s linkage arrives with four revolutes at stated twists. The joints are the input, and everything the site asks is downstream of them — what the count says, what the rank measures, where the mechanism goes.

This field asks what a joint is.

The answer the constraint field has been using is a number. A revolute takes two freedoms away in the plane and five in space; so does a prismatic; so does a helical. Grübler and Kutzbach add those numbers up, and the sum is a mobility. It works, and it is the right instrument for the question it answers.

It is also blind in a way worth stating plainly. A count cannot tell a pin from a slide. Both are one freedom, both take five away in space, and the two mechanisms that come of swapping one for the other have nothing in common. The number is a shadow of something larger, and the something larger is the subject of this field.

What the joint permits #

Start with the thing itself: two bodies, touching, one able to move relative to the other. What it may do is a set of displacements — every rigid motion that keeps the two bodies in the contact they are in.

That set has a property the number does not record. Perform one permitted displacement and then another, and the result is a permitted displacement: the bodies were in contact after the first and in contact after the second, so they are in contact after both. Undo a permitted displacement and that is permitted too. Doing nothing is permitted. A joint’s freedom is a group, and the count that the constraint field has been adding up is that group’s dimension.

The word for the ordinary kind of joint is a lower pair: two bodies touching over a surface rather than along a line or at a point. That distinction is usually presented as being about wear and contact stress, and it has a purely geometric consequence that is much more interesting.

If two bodies touch over a surface, then a displacement keeps them in contact exactly when it carries that surface onto itself. Anything else opens a gap or drives one body into the other. So the set of displacements a lower pair permits is the symmetry group of its own surface, and the question which joints exist becomes the question which surfaces slide on themselves.

That second question has an answer, it is short, and it can be computed.

One row per point #

A twist $\xi = [\boldsymbol\omega ; \mathbf{v}]$ — the six-vector this site has used since the spatial field — moves the point $\mathbf{p}$ at velocity $\boldsymbol\omega \times \mathbf{p} + \mathbf{v}$. It keeps a surface if that velocity is tangent to the surface at every point of it, which is one linear condition per point:

$$\nabla f(\mathbf p) \cdot (\boldsymbol\omega \times \mathbf p + \mathbf v) = 0 \quad\Longleftrightarrow\quad \big[\,\mathbf p \times \nabla f(\mathbf p)\;;\;\nabla f(\mathbf p)\,\big] \cdot \xi = 0 .$$

Sample the surface, write one row per sample, and the twists that carry the surface onto itself are the null space of the matrix. Its dimension is six minus a rank.

There is nothing fitted here and nothing recognised. No routine looks at a surface and decides it is a cylinder; the surface’s own gradient writes the rows down and a rank decision does the rest. That matters because the result is going to be a classification, and a classification produced by a program that already knows the answer is a lookup table with extra steps.

The check on the whole apparatus is the second line in that figure. A twist the surface does not permit is chosen mechanically — the candidate furthest outside the computed algebra — and applied at the same magnitudes. The surface moves by the size of the motion, as it must. The first line stays at $10^{-16}$.

What comes out #

Run it, and the classical table appears without having been consulted.

A plane gives three. One rotation, about its own normal, and two translations in it. Two flat faces resting on each other slide and turn and cannot do anything else.

A sphere gives three. Three rotations, about its centre, and no translation at all. Both answers are three and they are different threes, which is the whole of this field’s first argument in one comparison: planar motion and spherical motion share a dimension and nothing else.

A circular cylinder gives two — a rotation about its axis and a translation along it, which is the cylindrical pair, a shaft in a plain bore with nothing stopping it sliding out.

A prism gives one translation. Any cylinder whose cross-section is not a circle: an ellipse here, a square in a machine tool’s ways. The moment the section stops being round the rotation goes and the slide is left.

A thread gives one screw, and reports its own lead. This is the strongest single check the machinery gets, because it produces a quantity rather than a dimension. The helicoid is generated with a lead of $0.4$; nothing tells the computation that; and the pitch of the twist that comes back is $0.400000000000$, out by $5 \times 10^{-16}$. A dimension can come out right for the wrong reason and a twelfth decimal place cannot.

The prism and the thread are worth putting side by side, because the difference between them is the whole of what a pitch is. A prism’s cross-section repeats along the axis and does not turn as it goes; a thread’s repeats along the axis and turns, at a fixed rate. So the prism’s surface is carried onto itself by a translation and the thread’s by a translation and a rotation together, in a fixed ratio, and neither one alone. A helical pair is not a joint with two freedoms that happen to be linked; it is a joint with one, and the one is a screw.

And three surfaces that look nothing alike give the same group. A cone, a torus and an ellipsoid of revolution each leave exactly one rotation, about their axis. So does a shaft with two collars on it — a cylinder that would have slid, with the slide taken away by two annular faces. Four different objects, one group: the revolute pair, which is the joint every four-bar on this site is made of.

Why the sampling cannot be a grid #

One detail of the computation is worth recording, because getting it wrong produced a wrong answer that looked entirely reasonable.

The rows come from sampled points, and the first version sampled on a grid in the surface’s own parameters. On a surface of revolution that puts every sample on a handful of meridians, and a matrix built from a handful of meridians is rank-deficient for a reason that has nothing to do with the surface: the samples are not general enough to write down every condition the surface imposes. A cone came back with two freedoms, which is the answer for a circular cylinder, and the extra freedom was a translation the cone does not have.

Nothing about the output said so. The dimension was an integer, the null space had a basis, and the basis classified cleanly as a cylindrical pair. What caught it was the rank gap — the ratio of the smallest singular value kept to the largest discarded — which on a genuine answer here runs to $10^{15}$ and on that one ran to about $10^{2}$. A rank decision with a gap of a hundred is a decision somebody’s tolerance is making; a rank decision with a gap of $10^{15}$ is the surface making it.

The samples are a golden-ratio additive sequence in the two parameters now, which spreads them over the whole surface with no structure for the matrix to inherit, and every gap in the table above is above $6 \times 10^{15}$. The general lesson is one this site keeps relearning in different fields: a numerical rank is a measurement, and a measurement without its own error bar is an assertion.

Almost no surface is a joint #

Two of the eleven surfaces give nought, and it is worth dwelling on which two.

Squash a sphere along one axis and two of its three rotations go, leaving the revolute. Squash it along a second axis, by a different amount, and the last one goes too. The surface with no symmetry is not exotic and not badly behaved: it is what a surface is like. A wobbly blob gives nought for the same reason, and so would almost anything anybody machined without meaning to.

That is the shape of the whole result. A joint is a coincidence. The surfaces that slide on themselves are a vanishingly thin set inside the surfaces there are, and the six groups they produce are the six joints every textbook lists — not because five generations of engineers agreed on a convention, but because there is nothing else for a surface contact to permit.

The machinery has to be able to lose a symmetry for that claim to mean anything, so it is checked in the failing direction as well: a sphere with a single small bump on it goes from three to nought. Not to two, and not to one.

Six, from eleven #

Six distinct groups, from eleven surfaces, and adding more surfaces does not add more groups. That is the claim the next rung is about, and it is the reason this field exists at all: the joints are not a list, they are a classification, and the classification is short.

The six have names, and the names are the ones on every kinematics table:

• the planar pair, from a plane, three freedoms;
• the spherical pair, from a sphere, three;
• the cylindrical pair, from a circular cylinder, two;
• the revolute pair, from any surface of revolution, one;
• the prismatic pair, from any prism, one;
• the helical pair, from a thread, one, at the thread’s own pitch.

What a point sees #

A group is an abstraction, and a reader is owed a picture. The picture is the orbit: fix one point of the moving body and draw everywhere the group can send it.

The orbits are the surfaces. That is not a coincidence and it is this argument read backwards: a lower pair’s surface is an orbit of its own group, which is precisely why the surface can slide on itself. Everything the group does to the surface it also does to any point of it, and a point’s worth of the group is all a reader needs to see what the joint is.

It also settles the pin-and-slide comparison the essay opened with. A revolute and a prismatic both have dimension one. One sends a point round a circle and the other along a line, and no amount of counting recovers that.

The one this does not cover #

Two bodies can touch without touching over a surface. A cam against a follower, a tooth against a tooth, a wheel on a rail: contact along a line or at a point, which is a higher pair, and the constraint field has counted those too — a planar higher pair takes one freedom away instead of two.

The count survives. The group does not.

Both freedoms are real, both keep the contact exactly, and their composite does not. So a higher pair’s permitted set is not closed under composition, is not a group, and the freedom count is the dimension of nothing at all. That is the last rung of this field and it is where the field’s own boundary sits.

The freedom is also the way the surface is made #

There is a consequence of the surfaces-that-slide-on-themselves characterisation that reaches outside kinematics entirely, and it explains a coincidence between this list of six and a workshop’s list of operations.

A surface that carries onto itself under a continuous family of displacements can be generated by that family. Take a curve, sweep it under the motion the surface permits, and the surface is what the sweep produces — necessarily, because the motion leaves the surface where it is, so any point of it traces a curve that stays on it.

Run that over the six. A plane is generated by a straight sweep, which is a planer or a milling pass. A cylinder is generated by rotating work against a fixed tool, which is turning. A sphere is generated by rotating about two axes, or by lapping a pair against each other. A thread is generated by a rotation coordinated with a translation, which is exactly what a lathe’s leadscrew does and is why the machine that makes threads is the machine whose own motion is a screw. A prism is generated by drawing or extruding along its axis.

So the six lower pairs are also, and not by coincidence, the six surfaces a workshop can make by moving a tool in a straight line, in a circle, or in a fixed combination of the two. A joint’s own freedom is the motion of the machine that cuts it, and the reason those are the accurate surfaces is that generating a surface by a motion makes it as accurate as the motion rather than as accurate as a tool path.

That inverts the usual account of why these six are the joints in every mechanism. The standard reason given is convenience or tradition; the real reason is that a surface which slides on itself is a surface that can be generated, a generated surface can be made accurately, and a joint that is inaccurate is not much of a joint. The classification and the manufacturability are the same fact.

It also predicts the exceptions correctly. Surfaces that are not on the list — the ellipsoid, the general torus, an arbitrary swept shape — are exactly the ones that need a tool path rather than a generating motion, and are exactly the ones that are not used as joints. When one is used as a bearing surface it is used as a higher pair, touching at a point or along a line, which is the case this essay’s last section sets aside.

Which is the strongest reason to hold the joint as a group rather than as a number. A number says how many freedoms a joint has. The group says which motions they are, and therefore what surface can carry them, and therefore how the part gets made — three questions the count cannot distinguish and one answer covering all of them.

What the rest of the ladder does #

Having a group where the site had a number is worth something only if the group is used, so the shape of what follows is worth stating.

The next two rungs finish the classification: the census that gives six and no others, and the difference between two joints of the same dimension. Then the twelve groups of rigid displacements, of which these six are the ones a single pair gives, and the measurement that says how rare being a group is at all — not one subspace of the twists in eighty thousand, above dimension one, is closed.

Then composition. A chain’s displacement set is the product of its joints’ groups, and a product of groups is usually not a group; a parallel machine’s platform gets the intersection, and an intersection always is. That is where Sarrus’s linkage stops being a surprise: it is two planar groups meeting in a line, and its platform translates for the same reason two planes meet in one.

And last, the measurement this field was built to make. Take the displacements a mechanism actually reaches, take their logarithms, and close them under the bracket. A planar four-bar gives three. Sarrus gives one. Bennett’s linkage gives six — its motion is inside no proper group at all, which is what paradoxical has meant in this collection for a long time and has never before been an integer.

None of that is available from a number. All of it is available from the object the number was the dimension of, and the object was sitting in the shape of the surfaces all along.