Machine Kinematics
solved, then drawn
Contacts that only push

Six hold nothing

Every exact-constraint coupling on this site — Kelvin, Maxwell, three-two-one, and a Kelvin clamp with a seventh pad added — has rank six and holds the part not at all. The escape a Maxwell coupling leaves is a pure vertical translation with nothing else in it, which is not a defect: it is what a coupling is, and gravity is the seventh contact nobody draws.

Assumes Six points and no more and Four in the plane and seven in space.

14 min read 8 figures Two routes to the same numberThe constant that is not constantSomebody had a problem

This site has a field about exact constraint. Six contacts, six freedoms removed, a part that returns to the same place every time it is put down, and a set of arrangements — Kelvin, Maxwell, three-two-one — whose ranks are computed and checked. Every number in it is right.

Run the same arrangements through the routine of the field that treats a contact as an inequality and not one of them holds anything.

The audit

The couplings, with the same contacts and the same normals as the seating essays use, read as unilateral:

arrangement contacts rank holds escape
Kelvin 6 6 no a lift, with a small twist in it
Maxwell 6 6 no a pure vertical translation
three-two-one 6 6 no a lift and a rotation
parallel grooves 6 5 no a lift
three legs 3 3 no a pure vertical translation
four legs 4 3 no a pure vertical translation
Kelvin plus a pad 7 6 no identical to Kelvin’s

Every row fails, and the last row fails in the most interesting way: adding a seventh contact — which is the count this field says is the minimum in space — changes the escape by nothing at all, because the added pad’s normal points the same way as three others already there.

maxwell: 6 contacts, 6 of the six freedoms taken. A plan view of the arrangement, with each contact drawn where it acts and an arrow along the direction of the force it can carry. A ball's contact force passes through the ball's centre whatever surface it rests on, so the whole constraint system is these points and these directions and nothing else. Written as wrenches — a force along a line, which is a screw of zero pitch — their rank is 6. Six independent wrenches leave nothing free: the part has one place to be, and putting it down twice puts it in the same place.
Fig. 1 Three balls in three radial grooves. Six contacts, rank six, and a pure vertical escape.

Why the rank was never the answer

The rank is six on five of those seven rows, and that number is what the exact-constraint argument is built from: six independent constraint wrenches, six freedoms removed, nothing left.

The step that is doing the work is nothing left, and it is a step from a rank to a mobility, which is legitimate when the constraints are equations. A part on six pins has one configuration because six equations in six unknowns have one solution. A part on six pads has a set of configurations, because six inequalities in six unknowns bound a cone, and the cone has an interior.

The counting rung states why: dd vectors can span dd dimensions and can never positively span them. So no arrangement of six contacts on a body in space is a hold, and the couplings are not near misses — they are on the wrong side of an arithmetic fact.

And it is not a defect

Which is the point of this rung rather than an accusation against a coupling.

A kinematic coupling is designed to be put down onto something and taken off again. The escape it leaves is the direction it is lifted in, and if it did not have one it could not be used. A Maxwell coupling’s escape is a pure vertical translation with nothing else in it: not a twist, not a rotation, a clean lift — which is exactly the specification, and it is a rather beautiful thing for the machinery to return unprompted.

kelvin: 6 contacts, 6 of the six freedoms taken. A plan view of the arrangement, with each contact drawn where it acts and an arrow along the direction of the force it can carry. A ball's contact force passes through the ball's centre whatever surface it rests on, so the whole constraint system is these points and these directions and nothing else. Written as wrenches — a force along a line, which is a screw of zero pitch — their rank is 6. Six independent wrenches leave nothing free: the part has one place to be, and putting it down twice puts it in the same place.
Fig. 2 A Kelvin clamp: one ball in a socket, one in a groove, one on a flat. Its escape is a lift with a small twist in it, which is the asymmetry of the arrangement showing.

The difference between Kelvin’s escape and Maxwell’s is worth reading. Maxwell’s is symmetric — three balls, three radial grooves, and the escape is a pure translation up the axis. Kelvin’s is not: a socket, a groove and a flat are three different things, and the escape comes out as a lift with a rotation component in it. So the escape direction is a fingerprint of the arrangement’s own symmetry, computed without being asked for.

What holds either of them down is gravity, and gravity is a force, which is outside this field. So the honest description of a kinematic coupling is: six contacts that locate exactly, plus one force that this site does not compute, and the seventh contact in the count is the one nobody draws.

The seventh pad, twice

The last row of the audit is worth its own paragraph because this site has an essay about it and the two readings are complementary.

The seventh contact adds a steady pad under a Kelvin clamp and computes what it costs: no rank, so no freedom removed, and a gap of 3.5 µm with contact errors of ten microns. The added pad is redundant and becomes the one that cannot touch.

Read as a hold, the same pad changes the escape by nothing. Not by a little: the escape twist with the seventh pad is the same six numbers as the escape twist without it, to every figure, because the pad’s normal points up like three others already in the set and its row is inside the hull of theirs.

The contacts that fight each other — kelvin-and-a-pad. With more contacts than freedoms, some weighted sum of the contact forces is zero: those contacts push against each other and not against the part. The weights are the left null space of the wrench matrix and they are drawn here. For four legs on a floor they come out as −−−−−−+ — the alternating sum — which is why a table rocks about a diagonal and never sideways, and why the gap under the fourth leg is δ₁ − δ₂ + δ₃ − δ₄ exactly.
Fig. 3 The added pad, from the earlier essay’s side. It adds no rank there and it adds no restraint here, and the two facts have the same cause.

So the two analyses agree about the pad and disagree about what they are measuring, which is the pattern of this whole rung. The bilateral reading says the pad adds no constraint; the unilateral reading says it adds no restraint; and both are consequences of its row lying inside the hull of the others, which is a single geometric fact stated in two vocabularies.

How the escape is found in six dimensions

The planar rungs of this field enumerate a cone’s extreme rays exactly, and in space that is not what happens, so it is worth saying what does.

An extreme ray of a cone in six dimensions lies on five facets, so the candidates are (N5)\binom{N}{5} rather than (N2)\binom{N}{2} — affordable for seven contacts and not what a reader needs. What a reader needs is one escape, named. So the routine asks the program for the twist that separates at every contact by as much as it can inside a unit box:

maximise ssubject toai⋅t≥s,    −1≤tj≤1.\text{maximise } s \quad\text{subject to}\quad a_i \cdot t \ge s, \;\; -1 \le t_j \le 1 .

A positive optimum is a proof that the part escapes, with the escaping twist attached. A zero optimum proves nothing: it says no twist leaves every contact at once, and the part may still slide along some of them.

That asymmetry matters here because every row of the audit returns a positive optimum — between 0.500 and 0.707 — so every one of them is proved rather than merely unrefuted. The couplings do not slide out; they lift off, cleanly, all at once.

The same asymmetry appears in the planar rungs from the other end: an ellipse in its own bounding box has an escape the enumeration finds and the program does not, because its spin maintains every contact rather than leaving any. Two instruments, two blind spots, and the reason both are kept.

What a locating scheme is worth

The distinction has a use beyond tidiness and it is worth extracting, because it changes what a fixture drawing means.

A drawing showing six locators is a drawing of a locating scheme, and the thing it specifies is where the part will be given that something presses it into the locators. That something is a separate design problem with its own drawing, and a fixture designer who has drawn six pads and stopped has drawn half a fixture.

The two halves can be done in either order and they interact, which is what the rest of this field is about: adding the clamp changes the arrangement, so it changes the pose set a clearance leaves and adds a condition on the tolerances that the six-pad drawing did not have.

What a coupling would have to be

It is a fair question what a kinematic coupling that was a hold would look like, since the field says seven contacts can do it and a coupling has six.

The answer is that it would not be a coupling. Adding a seventh contact with a downward normal — a pad above the part, or a hook — gives an arrangement that can hold, and it also gives an arrangement that cannot be assembled by putting the part down: the seventh contact has to be brought to the part after the part is in place, which makes it a clamp rather than a locator.

That is the same trade the removal-cone rung states for joints: a thing designed to be assembled needs a direction in, and a hold is precisely a thing with none. A coupling is a locating scheme that has been designed to be assembled, so it must have an escape, and the escape is the thing it is used through.

Three assemblies, and the three answers a removal cone gives. A cone in the plane can be an arc, a single ray, or nothing, and each of the three is a different kind of joint. An arc is a rest: the part is put there and lifted off, and nothing about the faces decides where it goes. A single direction is a slot: the part has one way in and one way out and is located in every other respect. Nothing at all is a joint in the sense a woodworker means it — the pieces cannot be separated in this plane and the assembly happens somewhere the drawing does not show. The whole classification is the width of one cone, computed from the faces that touch.
Fig. 4 The classification a coupling sits in: an arc of directions out, which makes it a rest rather than a joint or a slot.

Which suggests the honest way to specify one. A coupling wants an escape cone that is a single clean direction — so that the part goes on and comes off along one line and is restrained in everything else — and Maxwell’s is exactly that. Kelvin’s is not quite: its escape carries a rotation component, so lifting a Kelvin clamp straight up is not the motion its own geometry prefers. That is a small thing and it is a real one, and it comes out of the field’s routine rather than out of anybody’s judgement.

Three legs, and four

The bottom rows of the audit are the smallest cases and they say something the larger ones hide.

Three legs on a floor is three contacts with parallel normals: rank three, and the escape is a pure vertical translation. That is a stool, and nobody expects a stool to be held.

Four legs is four contacts with the same three-dimensional row space — rank still three, since a fourth parallel normal adds nothing but a moment already spanned — and the same escape. The fourth leg buys no restraint whatever, which is the unilateral reading of why a four-legged table rocks.

three-legs: 3 contacts, 3 of the six freedoms taken. A plan view of the arrangement, with each contact drawn where it acts and an arrow along the direction of the force it can carry. A ball's contact force passes through the ball's centre whatever surface it rests on, so the whole constraint system is these points and these directions and nothing else. Written as wrenches — a force along a line, which is a screw of zero pitch — their rank is 3. 3 is not six, so 3 freedoms are left, and the figure below names them: a rotation along (0.00, 0.00, 1.00). Six contacts is not the condition; six independent contacts is.
Fig. 5 Three legs, three parallel normals, rank three. The escape is straight up and no number of further legs removes it.

Both of those are obvious and neither is a waste of a row, because between them they show that the failure is about directions rather than counts. Six contacts fail; three fail; a hundred parallel ones would fail. What matters is whether the normals surround, and every locating scheme on this site is designed so that they do not — a locating scheme’s normals face a workpiece from the directions a machinist can reach it from, which is never all of them.

The one that is short of rank

One row of the audit has rank five rather than six, and it is worth naming because it is the only place the two analyses agree about a failure.

The parallel-grooves arrangement is three balls in three grooves whose axes are all parallel instead of radial. Six contacts, and the seating essay’s own verdict is that it is not a coupling: the part can still move. The rank is five, so it leaves a freedom in the bilateral sense too — a rotation about a particular screw axis, which the earlier essay names.

parallel-grooves: 6 contacts, 5 of the six freedoms taken. A plan view of the arrangement, with each contact drawn where it acts and an arrow along the direction of the force it can carry. A ball's contact force passes through the ball's centre whatever surface it rests on, so the whole constraint system is these points and these directions and nothing else. Written as wrenches — a force along a line, which is a screw of zero pitch — their rank is 5. 5 is not six, so 1 freedom is left, and the figure below names it: a translation along (1.00, 0.00, 0.00). Six contacts is not the condition; six independent contacts is.
Fig. 6 Six contacts that do not locate. The only row of the audit where the bilateral analysis fails as well as the unilateral one.

That row is the control, and a field’s audit needs one. Every other row is a case where the bilateral analysis passes and the unilateral one fails, and a reader is entitled to wonder whether the unilateral routine simply fails on everything. It does not: it fails on the arrangements that are not holds and it would report a hold if it met one, which the seven-contact box is.

The units, and the thing they nearly hid

The audit’s rows are built from contacts specified in millimetres — the coupling radius is 40 in the seating library’s own units — and that is exactly the situation the field’s units convention was written for.

A row mixes a moment with a direction. Feed it millimetres and the moment entries are forty times the direction entries; feed it metres and they are forty times smaller. Neither is a property of the coupling, and a hull taken from a stack of such rows would be a statement about the drawing’s units. Every row here is built with the moment divided by the coupling’s own radius, and the check is to build the same arrangement at a thousandfold difference and require the same margin.

That is the same discipline the seating essays adopted for the same matrix read as wrenches, and the two conventions have to agree or the audit above would be comparing objects of different shapes. They do: the characteristic length is the coupling’s radius in both, chosen once and used by both readings.

How much a contact error is magnified, 60 mm out. Each coupling's worst-case magnification: move the six contacts by a unit vector of errors — a ball a micron oversize, a groove a micron deep — and measure how far a point 60 mm from the centre moves. It is a worst case over directions of error, which is the largest singular value of the map from the six errors to the three components of the displacement, so it cannot be improved by choosing a flattering error pattern. Kelvin's clamp magnifies by 2.55 at the rim against Maxwell's 1.35, because its socket is a fixed point and everything else turns about it. Both are exactly constrained; only one of them is symmetric.
Fig. 7 The other quantity the same matrix produces: how far a part moves when a contact moves, which is a bilateral question and has the same units problem.

What the audit is evidence for

It is worth being explicit about what a table of seven failures establishes, because a routine that failed everything would produce the same table.

The routine passes things. A box on seven contacts comes back held with a margin of 0.116, from the same code with the same tolerances, and so do two of the seven planar arrangements in the field’s own ledger. The parallel-grooves row is the other kind of control: it is the one arrangement the bilateral analysis also fails, and it fails both ways for the same visible reason.

So the audit is not the routine saying no to everything. It is a class of arrangements — every one designed to be assembled by putting a part down — that fails a test they were never built to pass, and the interesting thing is that nothing in the design tradition around them says so.

That is worth one more sentence because it is the useful half. The exact-constraint literature is careful, quantitative and correct, and it has a vocabulary for over-constraint, for sensitivity, for repeatability and for the seventh pad. It has no word for the thing this rung measures, because the preload is assumed everywhere and an assumption that is never stated is one that cannot be checked.

The preload has a direction, and the direction is geometric

Gravity is named above as the seventh contact and set outside the field because it is a force. Its direction is not a force, and keeping the direction turns the audit’s seven failures into a usable design condition rather than a boundary marker.

A coupling escapes along a particular twist — Maxwell’s along a pure vertical translation, Kelvin’s along something with a rotation in it, each of them computed and reported by the routine. Whatever holds the part down has to oppose that twist, which is a statement about direction alone: the preload must do negative work on the escape twist. Magnitude decides whether it wins against whatever is trying to lift the part; direction decides whether it is even pointed at the problem.

That is checkable with the machinery this field already has, and it is a stronger check than it sounds. Compute the escape twist, take the preload’s own screw — gravity is a force through the part’s centre of mass, a spring is a force along its axis, a magnet is a force along its normal — and form the reciprocal product. Negative means the preload opposes the escape. Zero means it does not touch it, and a coupling preloaded orthogonally to its own escape is not held at all however hard it is pressed.

The consequence a maker meets is the orientation one. A Maxwell coupling escaping along a pure vertical translation is held perfectly by gravity on a bench and not at all on its side, where gravity has no component along the escape and the part is free to lift off along a direction nothing opposes. That is not a subtlety about preload magnitude; it is a coupling that has stopped being a hold because it was rotated.

So the honest specification of a kinematic coupling has three parts rather than two: the contacts, the preload, and the direction the preload must come from. The first is what every drawing shows, the second is usually mentioned, and the third is almost never stated — and it is the one that decides whether the assembly works in the orientation it is actually used in.

That also gives the field’s boundary its proper shape. This field cannot say how hard to press, and it can say which way, exactly, for every arrangement in the audit — because the escape twist is geometry and so is the line a force acts along. What is outside is one scalar, and what is inside is everything that decides whether the scalar has anything to act on.

What this rung actually adds

It is worth being clear about what changes and what does not, because the answer is nothing about the couplings and a good deal about what the numbers mean.

The couplings are correct, their ranks are correct, their sensitivities are correct, and the essays about them do not need amending. What changes is one sentence that has been doing more work than it can bear: six independent constraints leave no freedom. That is true of equations and false of inequalities, and every arrangement in the exact-constraint field is made of inequalities that are treated as equations because something presses them together.

Seven arrangements, one routine, and the two that hold. Every row is the same three steps: write down one row per contact — the moment of its normal about the origin, then the normal itself — take the convex hull of those rows, and ask whether the origin is inside it. The parts differ, the numbers of contacts differ, and the routine does not. Two of the seven hold. The other five leave the part something, and the interesting column is what: four rays of rotation for the pinwheel, a translation straight out of the vee, and for the last two a whole line rather than any number of rays, which is what a rank below three means and is the case a reader has to be warned about. Note that the four contacts of the second row are the four of the first row, on the same four edges of the same square, at the same distance along each. positioned by solving, not by drawing.
Fig. 8 The field’s own ledger, none of whose seven rows is a coupling. The two vocabularies were kept apart for eight rungs and this is where they meet.

Naming the assumption is the whole of the rung. A locating scheme assumes a preload; a hold does not; and the arithmetic changes from a rank to a positive span the moment the assumption is dropped. Which of the two a designer wants is a decision, and the failure this rung exists to prevent is making it by accident — drawing six pads, counting to six, and believing a number that is about the other question.

Where this goes next

kelvin: 6 contacts, 6 of the six freedoms taken. A plan view of the arrangement, with each contact drawn where it acts and an arrow along the direction of the force it can carry. A ball's contact force passes through the ball's centre whatever surface it rests on, so the whole constraint system is these points and these directions and nothing else. Written as wrenches — a force along a line, which is a screw of zero pitch — their rank is 6. Six independent wrenches leave nothing free: the part has one place to be, and putting it down twice puts it in the same place. Six points and no more A ball resting on a surface is a joint: it takes one freedom away, and the force it can carry is a line through the ball's centre. Six of them, arranged well, take all six freedoms and leave a part with one place to be. Six arranged badly take five, and the sixth freedom is a screw with an axis this site can name. The contacts that fight each other — four-legs. With more contacts than freedoms, some weighted sum of the contact forces is zero: those contacts push against each other and not against the part. The weights are the left null space of the wrench matrix and they are drawn here. For four legs on a floor they come out as +−+− — the alternating sum — which is why a table rocks about a diagonal and never sideways, and why the gap under the fourth leg is δ₁ − δ₂ + δ₃ − δ₄ exactly. The seventh contact Add a contact to a part that is already exactly constrained and it adds no rank, so it constrains nothing — and it is the only contact in the set that can fail to touch. For four legs on a floor the combination that constrains nothing is the alternating sum of the four, which is why a table rocks about a diagonal and never sideways. Of 1176 exactly correct syntheses, 176 could be built. Every pair of points on a 7×7 grid over the moving body, each pair synthesised into a four-bar and each four-bar verified by the forward solver as reaching all three prescribed poses. 1176 of them do, exactly. Then each is swept from the first pose in both directions, carrying the branch the way a built mechanism must: 810 cannot reach all three without being taken apart and reassembled, and 190 reach them in the wrong order. 176 — 15% — are mechanisms rather than theorems. Nothing in the construction distinguishes them. Exactly right, and unbuildable A linkage synthesised through three prescribed positions reaches all three. That is a theorem and it holds exactly. Whether it reaches them in one piece, without being taken apart, and in the order asked for, are separate questions the construction says nothing about — and of 1,176 exactly correct solutions, 176 could be built.

About the same objects

Not linked from either essay — found by the objects both name.

What links here

Essays that link to this one from their own argument.

The objects this essay names

Each one links to every other essay that touches it.

Degrees of freedomEscape coneExact-constraintForm closureLocating schemeMobilityOverconstraintRankUnilateral constraintWrench