Machine Kinematics
solved, then drawn
What a joint is

The count cannot tell a pin from a slide

A revolute, a prismatic and a helical pair all take five freedoms away in space and leave one. Grübler adds the same number for each, the constraint rank measures the same number for each, and the three joints have nothing whatever in common — one sends a point round a circle, one along a line, and one along a helix at a rate the joint decides.

Assumes A joint is a surface that slides on itself and Counting and measuring mobility.

15 min read 7 figures Two routes to the same numberThe singularities are the point

The constraint field’s first essay asks what decides whether a mechanism moves, and answers it with an arithmetic: count the links, count what each joint takes away, subtract. The second checks that arithmetic against the rank of the constraint Jacobian, which is a completely independent route to the same number, and the two have been run side by side on every mechanism this site has drawn since.

Both of them read a joint as an integer. In space a revolute takes five freedoms away; so does a prismatic; so does a helical. The table of what each pair costs has one number per row and the three rows are identical.

This rung is about what that costs.

Three joints, one number

Take the three one-freedom pairs and put them side by side as the surfaces they are.

A shaft with collars, carried by its own groupA shaft with collars, drawn faint where it started and solid where a displacement of its own symmetry group has carried it. **The two drawings are the same set of points.** The same shaft with two flanges on it. The flanges are annuli, they forbid the slide, and what is left is one rotation — which is the joint every four-bar on this site is made of. The permitted twists are computed from the surface's own normals — one linear condition per sample point, saying that the velocity the twist gives that point is tangent — and the answer here is 1 freedom — a rotation. Every point of the displaced surface satisfies the original surface's own equation to 2.2e-16, which is what "the surface slides on itself" means as a number. A lower pair is two bodies touching over a surface, so this group is exactly what the joint permits, and its dimension is the freedom count the constraint field has been adding up since the foundation.a shaft with collars · 1 of 6 freedomsa rotation · off by 2.2e-16
Fig. 1 A revolute pair: a shaft with collars, carried by the one rotation its surface permits. The collars are what forbid the slide, and without them this is a different joint.

Three surfaces, three groups, one number. Every count on this site reports one for each of them, and every rank measurement agrees, because the rank is measuring the same dimension by a different route. The two-route check that has caught real errors elsewhere — a velocity solve with its right-hand side double-negated, a mobility formula declaring a working mechanism immobile — is a check on the dimension. It cannot see the type, because both of its routes compute the dimension.

So a third instrument is needed, and the previous two rungs built it. The joint’s permitted set is a group; the group’s dimension is the count; and what the count throws away is which group of that dimension it is.

What a point sees

The quickest way to see that three groups of the same dimension are three different things is to watch one point.

One point, six groups, six shapes. A group has no picture, so here is the next best thing: fix one point of the moving body — the marked one — and draw everywhere the group can send it. A prismatic pair sends it along a line, a revolute round a circle, a helical along a helix, a cylindrical over a cylinder, a spherical over a sphere, a planar over a plane. Those six shapes are the six surfaces the previous figures drew, which is not a coincidence and is the field's first argument read backwards: a lower pair's surface is an orbit of its own group, which is exactly why the surface can slide on itself.
Fig. 2 One point of the moving body, under each of the six lower pairs. The three one-freedom pairs are the first three, and they are a line, a circle and a helix.

A prismatic pair sends the point along a straight line. A revolute sends it round a circle. A helical sends it along a helix, and the helix’s pitch is a property of the joint rather than of the point.

Those are not three descriptions of one thing. They are three different curves, and a mechanism built with one of them is not related to the mechanism built with another.

The orbit is worth taking seriously as the picture of a joint, because it is the only one that is honest about what a group is. A group is a set of displacements, and a set of displacements has no shape — it lives in a six-dimensional space nobody can draw. What it does to a point lives in the room. So the six pictures above are not illustrations of the six pairs; they are the six pairs, reported by the only witness available.

They also close a loop with the previous rung. The orbit of a point under a lower pair’s group is that pair’s surface: a circle for a revolute lies on the shaft, a line for a prismatic lies along the way, a helix for a screw lies on the thread. That is why the surface can slide on itself — the group moves every point of the surface along a curve that is still in the surface — and it is why the classification of joints and the classification of self-sliding surfaces are the same classification asked in two directions.

Where H can send one point. The orbit of a single point of the moving body under a screw, which is a helix. The helical pair; the surface is a thread, and its pitch is the surface's own lead. The orbit is the only honest picture of a group: the group itself is a set of displacements and has no shape, and what a reader can see is what it does to something.
Fig. 3 The helical pair on its own. Its orbit is the only one of the six that carries a length and an angle in a fixed ratio, and the ratio is the joint’s pitch — one number that no count records and that changes every displacement the joint produces.

The same mechanism, three times

The abstraction becomes concrete the moment a joint is swapped inside a working mechanism.

Take a four-bar: four links, four revolutes, mobility one. Replace one pin with a slide and it becomes a slider-crank — four links, three revolutes and a prismatic, mobility one. Every count is unchanged. The constraint rank is unchanged. And the two mechanisms do not resemble each other: one is a closed loop of four bars whose coupler traces a sextic, the other is a piston mechanism whose output is a reciprocation with a stroke and a quick return.

It is tempting to say that a slider-crank is a four-bar with one link made infinitely long, and as a limit of shapes that is true. As a statement about joints it is false and the falsity is the point: a prismatic pair is not a revolute pair with a large radius, it is a different group, and its orbit is a line rather than a very large circle. The limit is a limit of mechanisms, taken in the space of link lengths, and it passes through no configuration in which the joint changes type. What the joint is does not vary continuously with anything.

Replace one pin with a screw instead and something stranger happens. The mechanism still has four links and four one-freedom pairs and mobility one, and its configuration does not close: turning the crank advances the nut along the thread, and after a full turn the mechanism is not where it started. A loop of four one-freedom joints whose mobility count is one and whose motion is not periodic. Nothing in the count anticipates that; the pitch does all of it, and the pitch is not in the count.

Four instruments, and only the last one names the group. Every instrument this site has for an overconstrained loop, on the same six mechanisms. Kutzbach's count gives −2 for a planar four-bar and −2 for Bennett's. The rank of the constraint Jacobian gives three and three. Both are right and neither separates them. The last two columns are this field's: the span is how many dimensions the logarithms of the displacements the moving link actually reaches occupy, and closes at is the dimension after those are closed under the bracket. A planar four-bar closes at three and the three are planar motion; Sarrus closes at one, a translation, which is the exact straight line the spatial field measured by solving the mechanism sixty times. Bennett closes at six: its displacements occupy four dimensions and no group smaller than all of them contains those four. That is what "paradoxical" has meant on this site for six phases, stated as an integer.
Fig. 4 Six loops, four instruments. The first two columns are the count and the rank, and their job is to agree with each other about a dimension. The last two are this field’s, and they are what a dimension leaves out.

What the count is right about

None of this makes the count wrong, and it is worth being clear about what it does.

Mobility is a statement about how many parameters a mechanism’s configuration needs, and the joints enter it only through their dimensions. That is the correct level of abstraction for the question, and it is why the formula is so short. A mechanism with mobility one has a one-parameter family of configurations whether its joints are pins or slides or screws, and every argument this site makes about whether a crank turns fully or where a solve loses a direction is downstream of that count being right.

The count is also the only instrument that is cheap. It reads two integers off a drawing. The group needs a surface, or a sweep, or a bracket closure, and none of those is available at the back of an envelope.

What is not true is that the count is sufficient. Two mechanisms agreeing on it agree about one number and may be unrelated, and the site has been quietly relying on the reader to supply the difference from the picture. This field supplies it from a computation instead.

Where the difference shows up

Three places, in increasing order of how much it matters.

In what the mechanism does. The obvious one. A pin gives a rotation and a slide gives a translation, and a designer choosing between them is choosing between two shapes of output. Nobody is confused by this in a single joint.

In what a chain of them produces. Much less obvious. Three parallel revolutes give planar motion. Three revolutes whose axes meet at a point give spherical motion. Three revolutes at random give a three-parameter set that is inside no proper group at all. Same joints, same count, same rank — and one of them is a wrist and one is a mechanism whose reachable poses cannot be described by anything smaller than the whole of the rigid displacements. That comparison is a rung of its own, and the count is silent on all of it.

How many dimensions each chain's displacements occupy. Every chain in the field, with the dimension its reached displacements' logarithms occupy. A chain of n joints always has n freedoms; what varies is whether those freedoms compose. Where the bar equals the joint count the motion is inside a group and the group is named; where it reaches six there is no proper group containing the motion, and the two chains that do are the ones whose axes were chosen at random. Nothing about the joints themselves differs — three pins are three pins, and the two rows differ only in where the axes point.
Fig. 5 Every chain in this field, with the dimension its reached displacements occupy. Where the bar matches the joint count, the motion lies in a group and the group is named; where it reaches six there is nothing to be inside. The joint counts do not distinguish the two cases.

In whether an overconstrained mechanism is a subgroup or a paradox. The deepest one, and the reason this field exists. The spatial field established years of this site’s work ago that Kutzbach declares several working mechanisms immobile and that the redundancy splits into two kinds which every count reports identically. That split is a statement about groups, and until this field there was no way to say which group.

What the site has been reading off the picture

There is a quieter version of this loss that runs through every essay on the site, and naming it is most of what this rung is for.

When a figure here shows a four-bar, the reader knows the pins are pins because they are drawn as pins. When it shows a slider-crank, the slide is drawn as a block in a guide. The type of every joint on this site has been carried by the drawing and by the name the mechanism is asked for by — a four-bar, a slider-crank, a Sarrus linkage — and never by a quantity the site computes. This collection’s standing rule is that a mechanism’s connections are declared rather than inferred, and the same is true one level down: the joint types are declared too, and nothing has ever measured them.

That is not a defect in the earlier work; a mechanism’s joints are an input and inputs are declared. It does become a defect the moment a claim depends on the types rather than on the counts, and several do. The claim that a planar four-bar is overconstrained because its axes are parallel is a claim about a group. The claim that a Gough platform’s platform can reach a general pose is a claim about the product of its legs’ groups. The claim that Sarrus’s linkage translates exactly is a claim about an intersection. All three were made on this site before there was any instrument that could see a group, and all three were argued from a rank, a sweep or a solve — which is to say, from evidence about a dimension plus a picture.

This field replaces the picture with a computation. That is the entire upgrade, and it is worth stating in exactly those terms, because the earlier arguments were not wrong: they were under-instrumented, and the reader was doing part of the work.

The rank sees more than the count and still not this

A fair objection at this point is that the count is a straw man. The site does not rely on the count alone — every mechanism is also measured, by taking the rank of the constraint Jacobian, and a rank reads actual positions rather than two integers. Surely the measurement sees what the formula cannot.

It sees more, and not this. The rank of a constraint Jacobian is the number of independent conditions the joints impose at the configuration it is evaluated at. It knows where the axes are, which the count does not, and that is exactly why it catches the parallelogram with a third parallel bar where the count does not. What it returns is still an integer, and the integer is still a dimension.

The screw system behind the rank goes further again. The subspace itself is a geometric object with an axis and a pitch, and measuring how far it turns through a mechanism’s motion is what separates the two overconstraints. That is a real instrument and it is this field’s nearest neighbour. But it is a derivative: it reads the velocities available at an instant, and it is evaluated configuration by configuration. A group is a statement about displacements that can be composed, and no amount of information about velocities at one instant is a statement about that. How far apart the two are is measurable, and it is a rung of this field.

A pitch is not a count either

The helical pair deserves a paragraph of its own, because it is where the loss is largest.

A revolute and a prismatic are at least qualitatively different in a way a reader can see. A helical pair looks like a revolute with an extra property, and the extra property is a real number. Two threads of different lead are two different joints — not two settings of one joint — and every displacement either of them permits is different from every displacement the other permits.

Carried by its own group, and by one that is not. The definition of a joint, measured. The lower line is a screw thread carried along a twist its own normals say it permits; the upper line is the same surface carried along a twist from the complement of that algebra, chosen mechanically rather than by hand. The vertical axis is how far the worst displaced point ends up from the original surface, and the two answers are sixteen orders of magnitude apart — the lower one is at the floor of double precision at every stop and the upper one is simply the size of the motion. There is no threshold in this figure and none is needed.
Fig. 6 A thread carried along its own screw and along a twist it does not permit. What the lower line certifies is not merely that a screw motion is allowed: it is that this particular ratio of advance to turn is, and no other.

A count records none of that. It records a one. And the mechanisms that live on the difference are not obscure: a lead screw, a differential screw, the hundred-to-one reduction that comes from two leads differing by a hair — all of them are pitch arithmetic, and all of them are invisible to the arithmetic the site counts mobility with.

The classification handles it cleanly. The helical group is one type with a real parameter; the classifier reads the pitch off the null space and reports it; and the revolute is the pitch-nought member of the family while the prismatic is the limit as the pitch runs to infinity. The three one-freedom pairs are one family with a parameter, which is a much better description than three unrelated table rows, and it is a description a count cannot express because a count has nowhere to put a real number.

Where R can send one point. The orbit of a single point of the moving body under a rotation, which is a circle. The revolute pair; the surface is any surface of revolution. The orbit is the only honest picture of a group: the group itself is a set of displacements and has no shape, and what a reader can see is what it does to something.
Fig. 7 The pitch-nought end of that family: a circle. The prismatic pair is the other end and its orbit is a straight line, which is the same helix with the pitch run to infinity.

The one place the count and the group must agree

A last check, and it is the sort this site prefers to any argument.

The group’s dimension and the count are supposed to be the same number, computed from disjoint inputs: the count reads how many joints there are and what kind, the group is a null space of a matrix built from a surface’s normals. If they ever disagreed, one of them would be wrong.

They do not. Every row of the census reports a freedom count that is the number the constraint field’s table has carried since the foundation — three for a planar pair, three for a spherical, two for a cylindrical, one each for the revolute, prismatic and helical. Eleven surfaces, eleven agreements, with the rank decisions separated by fifteen orders of magnitude.

That is the site’s standing arrangement — two routes that share no inputs, required to agree — applied to a quantity nobody had thought to check, and it earns the field the right to say something the count cannot. The dimension is common ground. Everything above the dimension is new information, and the next rung is about how much of it there is.

One of the twelve carries a real number

There is a feature of the helical pair that separates it from every other entry in the classification, and it is worth stating here rather than in the census, because it is the sharpest form of what a count cannot see.

The twelve types are twelve up to where they point — a revolute about one axis and a revolute about another are the same type, related by a rigid displacement. That bookkeeping works for eleven of them. It does not work for the screw, because moving a helical pair about in space does not change its pitch: a screw of 2 mm per turn and a screw of 5 mm per turn are not related by any rigid displacement whatever, and they are different subgroups.

So the classification is not twelve types. It is eleven types and a one-parameter family, and the parameter is a real number a designer chooses. Every other joint’s identity is exhausted by naming it and saying where it sits; a helical pair needs a third piece of information that is neither a name nor a placement.

That is why the pitch is on the drawing and the others’ equivalents are not. A revolute’s specification is a size and a position; a prismatic’s is a direction and a length of travel; a helical pair’s is a size, a position and a number that decides what the joint does. It is the only lower pair whose behaviour is a design variable rather than a consequence of its geometry being that geometry.

The limiting cases sit at the two ends of the family and are the other two one-freedom pairs. Pitch zero is a revolute — turn without advance. Pitch infinite is a prismatic — advance without turn. So the three one-freedom joints are not three unrelated things after all: they are the two ends and the interior of a single family, and the count that reports all three as one is collapsing a line to a point.

Which is the most concrete answer available to the question this rung poses. What the count loses is not merely which of twelve; on the one-freedom row it loses a continuum. Three joints, one number, and between two of them a real parameter that decides how far a nut travels per turn — the quantity every threaded fastener in the world is specified by, and the one thing a mobility count cannot record.

What to take to the next rung

The count is a dimension. The dimension belongs to a group. Groups of the same dimension are not interchangeable, and the difference between them decides what a chain of them produces, what a loop of them can do, and whether a mechanism that ought not to move does.

So the next question is how many groups there are to choose from. The answer is twelve, six of which are joints — and the other six are motions a designer may want and cannot buy.

Where this goes next

Every surface tried, and the group it permits. The census the six lower pairs come out of. Each row is a surface, sampled at 240 points; the freedoms column is six minus the rank of a matrix with one row per point, saying that the velocity a twist gives that point is tangent to the surface. Nothing is fitted and no shape is recognised — the surface's own normals write the matrix down. Three different surfaces of revolution give the same group, which is the content of the classification; two surfaces give nothing, which is what almost every surface gives. Eleven surfaces, six groups. The last column is the ratio of the smallest singular value kept to the largest discarded, so a row reading 10¹⁵ is not near being reclassified by anybody's tolerance. Six, and no others Eleven surfaces were handed to the same computation and six groups came out. A cone, a torus and an ellipsoid of revolution give the same joint; a scalene ellipsoid gives none; and the list does not grow when more surfaces are added, because a surface's symmetry group has to leave a two-dimensional set alone and only six groups can. The twelve kinds of freedom, and which are joints. Every connected group of rigid displacements, up to where its axis points and where its origin sits. There are twelve, the height on the page is the dimension, and a line means the lower one is contained in the upper — computed by asking whether each generator of the smaller lies in the span of the larger, with all twelve built about a common axis. Filled discs are joints: six of the twelve are the symmetry group of a surface and can be a single pair, and six are not and have to be built out of a chain. There is nothing at dimension five, which is not obvious and is checked rather than assumed: twenty thousand random five-dimensional subspaces of the twists were closed under the bracket, and every one generated the whole of the six. Twelve kinds of freedom Every set of displacements that is closed under composition is one of twelve, up to where its axis points. Six of them are joints somebody sells. Four are motions a designer may perfectly well want and cannot buy at any price. And there is nothing at all of dimension five — checked here on twenty thousand random subspaces, every one of which generated the whole of the six. What each kind of joint takes away. Grübler's formula is M = 3(n − 1) − 2j₁ − j₂, and the 2 and the 1 in it are not conventions. A lower pair — a pin or a slide — holds two bodies together over a surface and leaves one relative freedom, so it costs 2. A higher pair — a cam against a follower, a wheel on a rail — touches at a point, the contact travels along both surfaces, and it costs 1. Five chains, each built and each measured from the rank of its constraint Jacobian, which has never heard of the formula. The last row is the one worth having: count that cam contact as a pin, as is very easily done, and the formula returns 0 where the mechanism has 1. The Jacobian does not move. What each joint takes away Grübler's formula has a 2 in it for pins and a 1 for cam contacts, and those numbers are not conventions to be memorised. They are the number of constraints each kind of joint imposes, they are measurable as the rank of a matrix, and miscounting one of them is the commonest way the formula is got wrong.

What this makes readable

Essays that name this one as a prerequisite.

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.

ConstraintDegrees of freedomDisplacement subgroupKinematic pairLower pairMobilityOrbitPitchRankScrew