Machine Kinematics
solved, then drawn
What can move

The freedom that is a set

Grübler's rule has been on this site since its first essay, and it adds up numbers. Each of those numbers is the dimension of a group of displacements, and the group has eleven siblings the number cannot distinguish. The count is not wrong; it is a projection, and this is what the projection discards.

Assumes What each joint takes away and A joint is a surface that slides on itself.

17 min read 9 figures Two routes to the same numberThe singularities are the point

The site’s first essay asks what decides whether a mechanism moves and answers with an arithmetic. Count the links, count what each joint takes away, subtract:

M=3(n−1)−2j1−j2in the plane,M=6(n−1)−5j1−⋯in space.M = 3(n-1) - 2j_1 - j_2 \qquad\text{in the plane,}\qquad M = 6(n-1) - 5j_1 - \cdots \qquad\text{in space.}

Every term in that expression is a dimension, and until the essays on joints there was nothing on this site that said what any of them was the dimension of.

Now there is. A lower pair’s permitted displacements are the symmetry group of its own surface, and there are exactly six such groups sitting inside a classification of twelve. The number a joint contributes to Grübler’s rule is that group’s dimension.

This essay is about what follows from reading the rule that way.

What it takes to build each of the twelve. The same twelve, read as a bill of materials. Six of them are one joint, because a lower pair permits the whole symmetry group of its surface and those six groups are exactly the symmetry groups surfaces have. The other five with a dimension take a chain: two slides for planar translation, three for Cartesian motion, a thread and two slides for the screw-in-a-plane group, and three parallel pins with a slide along them for Schoenflies motion — which is a SCARA arm, and is why a pick-and-place machine has four joints and not one. The group each chain produces is measured from four hundred sampled poses rather than declared, and every row agrees.
Fig. 1 The twelve, with each one’s dimension and what it costs in joints. The dimension column is the whole of what the counting rule reads.

What the rule keeps

The good news first, because the rule is not being demoted.

A mechanism’s configuration set is a manifold, and its dimension is what mobility means. The joints enter that dimension only through their own dimensions: a mechanism with nn links and joints of dimensions f1,…,fjf_1, \ldots, f_j has, generically, ∑fi−6(loops)\sum f_i - 6(\text{loops}) freedoms in space, and that is the counting rule written another way.

So the projection the rule performs — from a group to its dimension — is exactly the right projection for the question. Nothing about which group a joint gives changes the count, and a rule that read the groups would be a rule with more input than the answer needs.

One more thing survives, and it is the one the site relies on most. The count is compositional: a mechanism’s mobility is built from its parts’ numbers by addition and subtraction, so a change to one joint changes the answer by a known amount. Nothing about the group view is compositional in that way — a chain’s group is not built from its joints’ groups by any arithmetic, and whether a product of groups is a group is a question that has to be asked afresh each time.

It is also the only instrument that is cheap. The count reads two integers off a drawing. The rank of the constraint Jacobian, which is the site’s second route, reads a matrix of positions and needs a configuration. The group needs a surface, or a sweep of solved configurations, or a bracket closure.

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. 2 Four instruments on six mechanisms. The first column is a count read from two integers and the last is a bracket closure over thirty-one solved configurations; they cost about four orders of magnitude apart in work.
How many freedoms each surface leaves. The same census as a picture, painted by which group each answer is. A plane and a sphere both leave three and they are different threes — planar motion and spherical motion have nothing in common but their dimension, and a count that reports both as three is throwing away the part that decides what mechanism you can build. That is the field's first argument, and the colours are the argument: two bars of equal length in different colours.
Fig. 3 Every surface’s freedom count, painted by which group it is. The heights are what the counting rule adds up; the colours are what it discards.

Three ways it goes wrong, and only one is about geometry

The site has known since its foundation that the rule can fail, and the failure it knew about was the parallelogram with a third parallel bar: three parallel bars, five links, six pins, zero degrees of freedom by the formula and one by the Jacobian. The lesson taken was that special geometry breaks the count.

There are three failure modes, they are structurally different, and special geometry is the mildest.

The geometry is special. The rule assumes the constraints are independent, and at special dimensions they are not. The rank catches it, which is why the site has run the two routes side by side since the foundation.

The graph contains a rigid subchain. The topology field found this one: at ten links, 1,878 graphs pass every arithmetic test and 230 are mechanisms, and the other 1,648 carry a subchain that is already a structure. This is not about geometry at all — it is a fact about which links the pins run between, at generic dimensions, and neither the count nor the rank can see it. The rank returns one and is right about an assembly that does not have ten links.

The joints are not what the number says. The newest of the three. Three joints of dimension one are three different groups, and the count reports the same number for all three. There is no configuration at which this shows up as an error, because it is not an error: the count is right and it is silent.

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. 4 The silence, drawn. Three of these six have a freedom count of one, and the mechanisms you build from them are not related.

What the third failure costs

It costs nothing in the mobility and a great deal in everything downstream, and the cleanest demonstration is a swap.

Take a four-bar. Four links, four revolutes, mobility one. Replace one pin with a slide: it becomes a slider-crank, still four links, still four one-freedom pairs, still mobility one, and a different machine. Replace one pin with a screw instead: still four links, still four one-freedom pairs, still mobility one — and turning the crank through a full revolution does not return the mechanism to where it started, because the nut has advanced along the thread.

A loop of four one-freedom joints whose mobility is one and whose motion is not periodic. Nothing in the counting rule anticipates that, and nothing in it could: the pitch is a real number and the rule has nowhere to put one.

The same swap in a spatial loop is more consequential still. Four revolutes with parallel axes give a planar four-bar built as a spatial loop — Kutzbach −2-2, and it moves. Four revolutes with axes at random give a loop that does not close at all. Four revolutes satisfying Bennett’s condition give a mechanism that moves and is inside no group. The count is −2-2 for all three.

The two routes were both measuring one coordinate

The site’s standing discipline is two routes to every number, computed from disjoint inputs and required to agree. Grübler reads two integers; the rank reads a matrix of positions; where they disagree the formula is the one that is wrong.

That discipline is worth re-examining in the light of the third failure mode, because the obvious worry is that two agreeing instruments gave false confidence.

They did not, and the reason is that they were never claiming more than they measured. Both routes compute the mobility, which is a dimension, and their agreement is evidence about that dimension and nothing else. Nobody on this site ever wrote that a count and a rank agreeing meant two mechanisms were the same.

What did happen — and it is the honest version of the worry — is that the type of every joint was carried by the drawing and by the name the mechanism was asked for by, never by a quantity. This collection’s standing rule is that a mechanism’s connections are declared rather than inferred, and the joint types were declared too. That was correct as far as it went; what this field adds is a measurement where there was a declaration, which is the same upgrade the essays on the graph made for the connections.

The count as a shadow

There is a clean way to say what the count is, and it is worth stating once because it makes the relationship between this field and the constraint field precise.

A joint’s permitted set is a group GiG_i. The count reads dim⁡Gi\dim G_i. A mechanism’s mobility is the dimension of its configuration manifold, which the count estimates from the dim⁡Gi\dim G_i and the loop structure.

Every quantity in the counting rule is a dimension, and the rule is arithmetic on dimensions. That is a complete description of it, it explains all three failure modes at once, and it says what a correction would have to look like.

The first failure mode is arithmetic on dimensions being wrong about a rank, and the fix is to measure the rank. The second is arithmetic on dimensions being blind to a subgraph, and the fix is a mobility counted over every subset of the links. The third is arithmetic on dimensions being blind to which group, and the fix is the closure test.

Two pins, and what their bracket costs. The bracket, on the smallest case there is. Two revolute joints span a two-dimensional set of twists whichever way they are arranged, and no count on this site can tell the two arrangements apart. Their brackets can. Two parallel pins bracket to a translation, which was not in the span, and the span closes at three — planar motion, which is the group the pair of them lives in. Two skew pins bracket to something that closes at six: nothing smaller than the whole of the rigid displacements contains them. The defect column is how far the bracket lies outside the original span as a fraction of its own length, and in both cases it is of order one — which is the ordinary case, and is why a mechanism confined to a subgroup is the exception.
Fig. 5 The third fix at its smallest. Two revolute joints have the same dimensions whichever way their axes point, and the closure of what they generate is three or six.

Three fixes, three different instruments, and the site now has all three. None of them replaces the rule.

A rule that reads six letters instead of one

One consequence of the re-reading reaches into the site’s most recent field and is worth naming.

The topology census enumerates kinematic chains as graphs: a vertex per link, an edge per pin. It counts 1, 2, 16 and 230 chains at four, six, eight and ten links, and every edge in it is a revolute.

With six kinds of joint, a chain is a graph whose edges are labelled, and two labellings of one graph are two different mechanisms — not two drawings of one, as two relabellings of the vertices are. That is a strictly larger census, and it is larger by a factor that grows fast: sixteen eight-link chains with ten pins each become sixteen graphs times however many labellings survive the isomorphism test, which for six labels and ten edges is a much bigger search than the one that produced 230.

The topology field scoped itself to all-revolute chains and said so explicitly. This is the reason that scoping mattered: it is not a simplification of a slightly harder problem, it is a different problem, and the difference is exactly the classification in this essay.

Where the group changes the count’s answer

There is one case where the group view does not merely add information but changes what the count should have said, and it is worth isolating.

Kutzbach’s rule in space assumes each loop imposes six independent conditions. If every joint of a loop lies inside a proper subgroup of dimension dd, the loop imposes at most dd independent conditions rather than six, and the correct arithmetic uses dd.

For a planar four-bar built as a spatial loop, d=3d = 3: 3(4−1)−2×4=13(4-1) - 2 \times 4 = 1, which is right. For a spherical four-bar, d=3d = 3 again, and again the answer is one. For Sarrus’s linkage the two arms are each in a three-dimensional group, and the platform’s is the intersection.

That correction is exactly the number Kutzbach is missing, which the spatial field derived from the rank of the loop’s screw system. The group view derives the same correction from the geometry of the axes, before any mechanism is assembled — and it says which dd to use rather than measuring it afterwards.

A planar four-bar, and the group it moves inFour pins with parallel axes. Kutzbach counts −2 and it moves. The dashed stubs are the joint axes, the solid marker is one point of link 1 and the faint curve is everywhere that point goes. Every frame is a **solve**: the joint angles are the unknowns, the closure of the loop is the equation, and a frame is drawn only where the residual comes below 10⁻⁹. What this field adds to the picture is one number. Take the displacements this link reaches, take their logarithms, and close them under the bracket: the answer is **3**, so the motion lies inside planar motion and composing two of its displacements gives a third one it also reaches, to 4.4e-16. positioned by solving, not by drawing.4 joints · a planar four-barinside G
Fig. 6 A planar four-bar as a spatial loop. Kutzbach’s rule with six conditions per loop gives −2; with three, which is the dimension of the group all four joints lie in, it gives one.

Where the count is better than the group

It is worth putting one case the other way round, because this essay could otherwise read as a demotion and the count has a genuine advantage.

The count applies to mechanisms whose motion is not a group, which is almost all of them. A six-axis industrial robot’s displacement set is six-dimensional and inside nothing; a general spatial four-bar does not close; a Gough platform’s platform reaches a six-dimensional set with no type at all. For every one of those the count is informative and the group view has nothing to say beyond six.

And the count applies before anything is built. A group measured by closure needs solved configurations, which needs a mechanism that assembles, which needs dimensions. The count needs a sketch. Type synthesis — choosing the chain before the lengths — happens at the sketch stage, and the counting rule is what filters the candidates there.

So the division is clean. The count is a cheap universal instrument that answers one question; the group is an expensive instrument that answers a different question about a minority of mechanisms. The minority happens to include most of the mechanisms anybody has bothered to name, which is exactly why the field was worth building.

What still has no group

The rule’s j2j_2 term counts higher pairs, and this is where the re-reading stops.

A higher pair’s permitted set is a manifold and not a group: a disc on an edge permits a slide and a turn, and the composite of the two lifts the contact off the edge by up to 1.2 radii. So its freedom count is the dimension of a set with no closure, and none of the group vocabulary applies to it.

The count is unaffected. A roller is not a slider and the site has counted the difference correctly since the applied field was written; what the group view adds there is not a correction but a boundary, and it is a real one — every argument in the pairs field is about mechanisms built from lower pairs only.

Two permitted motions, and a composite that lifts offA disc resting on a straight edge — a roller follower on a flat-faced cam, and the simplest **higher pair** there is. Two bodies touching at a point rather than over a surface, two freedoms: slide along the edge, and turn, because a disc is its own symmetry group about its centre. Both are permitted and both keep the contact exactly. **Their composite does not.** The faint discs are the two permitted displacements taken separately; the solid one is one followed by the other, and its centre sits 1.049 radii off the dashed line where a tangent disc's centre has to be. The excursion is exactly |t₂ sin φ₁| — the second displacement's slide times the sine of the first one's turn — derived from the two displacements rather than from the composition and agreeing to 10⁻¹⁶. A lower pair's freedoms compose and a higher pair's do not, which is why a joint's freedom count is the dimension of a group in one case and the dimension of nothing in the other.where a tangent disc's centre must be1.049 rtwo freedoms, both permittedcomposite off the edge by 1.049 radii
Fig. 7 The boundary. Two permitted motions of a higher pair and a composite that is not permitted; the count that says two is right and there is nothing for it to be the dimension of.

The rule’s own terms, reinterpreted

Finally, the terms of the formula themselves are worth walking through once in the new vocabulary, because each of them turns out to name something.

3(n−1)3(n-1) or 6(n−1)6(n-1) is the dimension of the group of rigid displacements — three in the plane, six in space — times the number of moving links. It is the dimension of the configuration space before any joint is imposed, and it is the dimension of the whole displacement group, which the classification lists as its last entry.

−2j1-2j_1 or −5j1-5j_1 is what each lower pair removes, and the number removed is six minus the joint’s group’s dimension in space, three minus it in the plane. So the term is not really a subtraction of a constant per joint: it is a subtraction of the codimension of each joint’s group. Writing it as a constant works only because the site’s planar mechanisms all use one-freedom joints; a planar pair in a spatial mechanism removes three rather than five, and the formula’s general form has always allowed for that.

−j2-j_2 is a higher pair, removing one, and it is the only term with no group behind it.

Reading the formula that way makes its scope obvious. It is a sum of codimensions, it assumes the constraints those codimensions represent are independent, and every one of its failure modes is a case where that assumption fails for a reason the sum cannot see.

Relabelling never changes the count

The labelled census the essay names as unbuilt is worth pricing, because the price says something about why nobody has built it and what it would and would not add.

Six joint types on a chain with ten pins is up to 6106^{10} labellings — sixty million — before any reduction by the chain’s own symmetries, and that is per chain, on a census with sixteen of them at eight links and two hundred and thirty at ten. So the labelled census is not the same work again; it is several orders larger, and it is the reason the topology field scoped itself to all-revolute chains and said so.

The interesting question is what that scoping loses, and the answer is unusually clean. Swapping one one-freedom pair for another never changes the count. A revolute, a prismatic and a helical pair all remove two freedoms in the plane and five in space, so any relabelling among the three leaves Grübler’s arithmetic untouched, on any chain, at any size.

So the census loses no mobility information at all by fixing every joint as a pin. Every chain in it stands for its whole family of one-freedom relabellings, and each member of the family has the same count, the same rank at generic dimensions, and the same graph.

And it loses all the behavioural information, which is this essay’s whole point. A four-bar and a slider-crank are the same row of the census with one label changed; so is a four-bar with a screw in it, which does not even return to its own configuration. Three mechanisms, one row, and nothing the census computes separates them.

That is a sharper account of the scoping than the field restricted itself for tractability. The restriction is exactly the projection this essay is about, applied one level up: a census over dimensions rather than over groups, complete about what it measures, and silent about the six letters that would have to be added to say what any of its entries actually does.

Which also says what a labelled census would be for. Not for counting mechanisms, since the count would not move; for enumerating the ones whose behaviour differs, which means enumerating labellings up to the chain’s symmetry and then asking a behavioural question of each. That is a different enumeration with a different purpose, and the reason to want it is everything in this essay rather than anything in the count.

What to carry back to the count

Three sentences, and they are the whole of the re-reading.

The number a joint contributes is the dimension of a group. There are twelve groups; six are joints; the number is the same for the three of dimension one.

Arithmetic on dimensions cannot see a rank, a subgraph or a group. The site has an instrument for each, and the counting rule remains the right first thing to do.

And when a count and a rank agree, they have agreed about a dimension. That is the site’s oldest discipline, it has caught real errors, and it is a check on one coordinate of an object with two.

The practical upshot for anybody reading a mobility number, here or anywhere: the number says how many parameters the motion has and nothing about what kind of motion it is. Those are two questions, they have two instruments, and a great deal of kinematics is written as though the first answered the second.

Span, and what it closes to. Two bars per loop: how many dimensions the reached displacements occupy, and how many they occupy after the brackets are added. For the four trivial loops the two bars are equal — the motion is already inside a group and bracketing adds nothing. For Bennett's linkage and the Bricard six-bar the first bar is four and the second is six, and the gap between the two bars is the whole of what paradoxical means: a one-degree-of-freedom motion that occupies four dimensions of displacement and generates all six.
Fig. 8 The second coordinate on the loops rather than the chains: two bars each, the dimensions occupied and the dimensions generated, with equal bars meaning the mechanism’s motion is a group.
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. 9 The second coordinate, for the chains in this field. A chain of n joints has n freedoms whatever the bar says; what the bar says is which group, if any, the freedoms compose into.

Where this goes next

Mobility, counted and measured. Grübler's criterion counts links and joints and knows nothing about the dimensions; the rank of the constraint Jacobian measures the dimensions and knows nothing about the topology. They agree for four of these five. The parallelogram with a redundant third bar is the exception: the formula declares it a structure with zero degrees of freedom, and it moves. The formula is the one that is wrong, because it cannot see that the third bar's constraint equations are already implied by the other two. Counting and measuring mobility Grübler's criterion counts links and joints and never asks how long anything is. The rank of the constraint Jacobian measures the lengths and never asks what a joint is. Two calculations with no inputs in common, producing one number — which is the only arrangement under which agreement is evidence. 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. 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. 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.

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 subgroupGrübler's criterionHigher pairKutzbach's criterionLower pairMobilityRank