Machine Kinematics
solved, then drawn
Concept

Rank — where it appears

The number of genuinely independent rows in a constraint matrix, which is how many freedoms are removed rather than how many equations were written. It is the measured half of every mobility question on this site, and the gap between it and the counted number is what overconstraint is.

Named by 58 essays across 14 fields — each of them below, with the objects they name alongside it.

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.

A constraint that only pushes

Every constraint on this site so far has been an equation: a pin holds two points together, a bar holds two apart, a mesh holds a ratio. A part resting against another part says only *do not come closer* — so what it may do is a cone rather than a subspace, and whether it can move at all stops being a rank.

holding · Restraint Six assemblies, one routine, three disagreements and one accident. Every row is the same three steps: write down the constraint Jacobian, take its rank, and subtract it from the number of unknowns. The representations differ — bars between points, bodies joined by pins, panels joined by creases, one cell of a pattern that repeats for ever — and the routine does not. The counted column is the arithmetic on the numbers of bodies and joints; the measured column is the nullity of the matrix. They agree on the lazy tong and on the kagome cell and disagree on the other four, most sharply on the deployable ring, which the count declares immobile and which is sold as a mechanism that opens. The right-hand column is the reason: constraints that repeat what another constraint has already said, which the count has no way of seeing and the rank cannot help seeing. The fourth row is worth reading twice: the count says nothing can move and nothing can, so the two agree — and they agree for the wrong reason, because that pattern's flat state shows four freedoms and not one of them is a motion.

Many loops, one freedom

A scissor lift, a folded sheet and a deployable ring are one small unit repeated thirty times, and three things change at once: the count of bodies becomes a parameter, mobility becomes the rank of a matrix, and a unit that moves can be rigid the moment it is joined to another of itself.

networks · Network Kutzbach's count against the measured mobility. Five closed loops of revolute joints. The count is 6(L − 1) − 5j, a statement about how many links and joints there are; the measurement is the number of joints minus the rank of the loop's screw system, which knows only where the axes point. They disagree for four of the five, and the one they agree on is the generic seven-joint loop — so the formula is not broken, it is blind to the special geometry that makes the other four work. The universal joint is counted at -2 degrees of freedom and is in every car built.

Six freedoms, not three

Every mechanism on this site so far has been flat, and flatness is not a simplification made for teaching — it is a special case that hides the most interesting thing constraint counting does, which is get the answer wrong about mechanisms that are in daily use.

spatial · Spatial One, two, sixteen, two hundred and thirty. Every planar chain of mobility one, up to ten links, counted by enumeration rather than quoted. The pins column is forced: a chain of 10 links has one degree of freedom only if it has exactly (3n − 4)/2 pins, which is why no odd link count appears. Pass the count is how many graphs satisfy Grübler's rule, are connected, are simple and give every link at least two pins. Are chains is how many of those survive the fourth condition, that no proper subchain is already a structure — and the gap between the two columns is the whole of this field's first argument: at ten links 1,878 graphs pass a rule that 230 of them deserve. Mechanisms is larger again, because a chain is not a mechanism until a link is held still, and how many different mechanisms that gives is a question about the chain's own symmetry.

The mechanism is the graph

Twenty-one fields of this site have been handed a mechanism and asked what it does. Take the mechanism away and keep only which link is pinned to which, and there is still a finite list of answers: one chain of four links, two of six, sixteen of eight, two hundred and thirty of ten — and 1,878 graphs at ten links that pass every count and are not among them.

topology · Topology Three bars and four bars. On the left, two bars to a common point: three links, three joints, and Grübler gives 3(3−1) − 2(3) = 0. The Jacobian agrees — two free coordinates, rank 2, nothing left over — and the shape cannot change without a bar changing length. On the right, one more bar and one more joint gives mobility 1, and the whole of this site follows from that difference. The triangle is why bridges are triangulated and the quadrilateral is why machines are not.

What decides whether it moves

Before a mechanism does anything it has to be able to. Two bars pinned to a point cannot move; three can. The count that separates them is one subtraction, it is the first thing anybody computes about a machine, and it can be wrong in a way that no amount of care with the arithmetic will catch.

constraint · Mobility A simple planetary, as a graph. Members are vertices and meshes are edges. Each edge carries the two tooth counts and the body the two axes are stationary in — its carrier — and that third label is the whole of what makes an epicyclic different from an ordinary train. Write the mesh relation relative to the carrier and one formula covers both: an ordinary train is the case where every carrier is the frame. This train has 2 meshes across 5 members and 2 freedoms.

A ratio is a null space

Write a gear train as a graph — bodies for vertices, meshes for edges, and on every edge the body the two axes are stationary in — and one formula covers a countershaft gearbox, a planetary, a harmonic drive and a car's differential. The ratio is the null space of a matrix whose entries are tooth counts, so it comes out as a fraction and not as a number that is nearly one.

transmission · Transmission A 3-stage stack at 35°. Every bar is the same length and every stage is at the same angle, which is a consequence of the solve rather than an assumption in the drawing: the pins at the crossings and the roller on the ground leave one freedom, and it has been used. The platform is at 1547 mm and the ram — the heavy line from the ground pin — is 428 mm long. Moving the ram one millimetre from here raises the platform 5.0 mm; at the bottom of the travel the same millimetre is worth 17.6 mm.

A roller is not a slider

A scissor lift has one degree of freedom, at every height and for any number of stages. Grübler's criterion agrees — if the rollers under it are counted as pins in slots. Count them as slider blocks, which is how every textbook draws a slider, and the same formula declares a machine holding a car in the air to be a structure with minus one.

applied · Mobility 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.

constraint · Mobility Losing a freedom and gaining one. Left, the nearest inverse-kinematic singularity: a leg is straight to 1.1e-5 and cannot reach further, so the platform has lost a freedom. That is the workspace boundary and it is a serial arm's singularity. Right, the nearest direct-kinematic singularity: the three leg lines pass through one point to 0.0001, so the three forces the legs transmit are no longer independent and the platform can turn about that point with every actuator locked. It has gained a freedom, and it is 1.97 from the other configuration and nowhere near the edge of the reach.

Locked, and still moving

A serial arm goes singular at the edge of its reach, where it loses a freedom, and the failure is visible as an arm gone straight. A parallel mechanism has a second kind with no serial counterpart — it gains a freedom, in the middle of the workspace, at poses nothing about the legs' reach marks out.

parallel · Parallel 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.

pairs · Pair The same rise, three ways. A 20-unit rise over 120°, then a dwell. Above, the displacement — all three do the same job and are hard to tell apart. Below, the acceleration, differentiated from those same curves. Constant acceleration peaks lowest at 5.556e-3 per degree² and arrives at the dwell with a step; cycloidal peaks highest at 8.727e-3 and arrives at zero. The smoothest law has the largest peak, which is the trade the whole of cam design turns on and which the displacement plot gives no hint of.

The law that costs least is not the smoothest

Constant acceleration gives the lowest peak acceleration of any motion law and an impulsive jerk. Cycloidal motion has finite jerk everywhere and a peak acceleration 57% higher. The trade is real, it is measurable, and the displacement curves that everyone plots give no hint of it.

cams · Cam Two routes to the same derivative. How much the output angle moves when the coupler length moves, through one turn, computed twice. One route rebuilds the mechanism at b ± 10⁻⁶ and solves both from scratch; the other differentiates the constraint equations and solves one linear system against the analytic Jacobian. They lie on top of each other — the strip beneath plots the difference on a four-decade log scale, and its largest value anywhere in the turn is 2.9e-10 against a sensitivity of order 0.41. That is the only independent check there is of the Jacobian itself, whose coupler rows carried four wrong signs from the foundation phase to 2026-08-12 without ever drawing anything wrong.

Two routes to a sensitivity

How far the output moves when a link length moves can be found by rebuilding the mechanism and solving it again, or by differentiating the constraint equations and solving one linear system. The two agree to two parts in a hundred million across a whole turn — and the second route is an independent test of the constraint Jacobian itself, whose coupler rows had carried wrong signs unnoticed.

practice · Tolerance Two things that are not the same configuration. The mechanical advantage of a four-bar, from the velocity solution: for a lossless mechanism it is the reciprocal of the output-to-input speed ratio, so wherever the output momentarily stops the force ratio diverges. It reaches 1673 at 222° — a toggle, where crank and coupler line up, and the reason a knee-joint clamp holds with almost no effort. The transmission angle is at its worst somewhere else entirely: 46.5° at 0°, 222° away, where coupler and rocker line up instead. Both get called "the mechanism jamming" and they are opposite situations — one is enormous force output, the other is force going into the bearings.

Two things called jamming

A four-bar's mechanical advantage peaks at 1,673 in one configuration and its transmission angle collapses in another, 222° away. Both get described as the mechanism jamming. One is enormous force output and the other is force disappearing into the bearings, and they are opposite situations.

wrong · Misconception 6 contacts, and the box lifts straight off. 3-2-1, six contacts. Each pad is a contact and each arrow the direction the box is free to move there — the inward normal, and the whole of what the contact contributes. The rank is 6, which is full: these 6 contacts are 6 independent constraints and a bilateral version of them would fix the box completely. The margin is nought, so the box is free to leave, and no amount of tightening the tolerances on where the pads are would change that. Rank and hold are different questions and this is the pair of pictures that separates them. positioned by solving, not by drawing.

Four in the plane and seven in space

Six independent constraints fix a body in space and six contacts fix nothing, because d vectors can span d dimensions and can never positively span them. The minimum is one more than the dimension — and it is a floor rather than an answer: six contacts on a box held it in none of four thousand random arrangements and seven held it in twenty-one.

holding · Restraint 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.

applied · Mobility 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.

pairs · Pair Three parallel bars, and a formula that says this cannot move. Five links and six pin joints, so Grübler's criterion gives 3(5−1) − 2(6) = 0 and calls it a structure. The Jacobian has rank 5 against 6 free coordinates, so it measures one degree of freedom — and the sweep assembles 59 of 60 positions, which settles the matter. The third bar removes no freedom because its constraint is already implied by the other two, and a formula that counts joints cannot notice that they happen to be parallel. Mechanisms of exactly this kind carry drafting machines, anglepoise lamps and locomotive coupling rods, where the redundant bar is there for load sharing and for keeping the linkage out of its change point.

The mechanism Grübler says cannot move

Three parallel bars between two frames. Five links, six pins, and the criterion every engineering course teaches gives zero degrees of freedom — a structure. It is a mechanism, it is in drafting machines and locomotive coupling rods, and the formula cannot see why.

constraint · Mobility Three verdicts, and only one instrument can give all three. Every 10-link graph that satisfies Grübler's count, split by what is actually true of it. 230 are mechanisms with 10 links. 1,165 carry a subchain whose own count is exactly nought — and neither of the two standing routes can see them: the count returns one and the rank returns one, and both are right, because a rigid subchain removes exactly the freedoms it is supposed to. What is false is the description. 483 carry a subchain whose count is below nought, and those the rank does catch: the surplus pins repeat a constraint already imposed, the Jacobian loses rank, and the measured mobility comes out above the count. The third instrument — a count run over every subset of the links — is the only one that answers the question at all.

What a count cannot see

At ten links, 1,878 graphs satisfy Grübler's rule and 230 are mechanisms. The other 1,648 contain a subchain that is already a structure — and on 1,165 of them the count says one degree of freedom, the rank of the constraint Jacobian says one degree of freedom, and both are right about a mechanism that does not have ten links.

topology · Topology Bennett's four-bar at 40°. Four bars, four revolute joints, and axes that are not parallel — a spatial four-bar, which Kutzbach counts at -2 degrees of freedom. Bennett's condition, sin α / a = sin β / b, makes the screw system rank 3 instead of 4, so the mechanism has 1. Driving the first joint through a full turn, 48 of 48 positions assemble. Orthographic projection, viewed from 40° azimuth and 24° elevation; dashed stubs mark the joint axes.

Bennett, and the condition that moves it

A spatial four-bar is immobile by every count there is, and generically it cannot even be assembled at more than isolated configurations. Bennett found the one relation between four lengths and two twists that makes it turn through a full revolution — and break the relation by two parts in a thousand and most of the travel is gone.

spatial · Spatial How many wiggles it takes. The growth vector: how many independent directions are available after one bracket, two, three. The first number is what the constraints leave and the last is the dimension of the configuration space, so the length of the row is how deep the manoeuvring has to go. A car needs one bracket more than a trolley and a car with a trailer one more again — and the ball changes by one depending only on whether it may be twisted.

How many wiggles

A bracket of two permitted directions may point somewhere new; the bracket of that with a permitted direction may point somewhere newer still. How deep the process goes before it stops is an integer — 2·3 for a wheel, 2·3·4 for a car, 2·3·4·5 for a car and trailer — and the same integer turns up as the exponent of a manoeuvre nobody told it about.

rolling · Rolling Six sizes, one freedom, and a count going the other way. Creases and constraints both grow as the square of the sheet's side, and they grow at different rates: two per panel against three per interior vertex. So the counted column runs (n − 1)(3 − n) and is positive at two, nought at three and increasingly negative after that, while the measured mobility is one on every row. The redundant column is the difference and it is exactly (n − 2)² — one at three, four at four, nine at five, thirty-six at eight. A twelve-by-twelve sheet of a hundred and forty-four panels is counted at minus ninety-nine and has a hundred repeated constraints, and it is the same mechanism as the smallest one on this table.

The freedom that survives repetition

A Miura sheet has one freedom at four panels and one at a hundred and forty-four, and the count runs the other way: plus one, then nought, then minus three, minus fifteen, minus ninety-nine. The gap between them is exactly (n − 2) squared, which is a hundred repeated constraints on a sheet with one degree of freedom.

networks · Network 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.

applied · Mobility How far inside the hull the origin actually is. The same seven arrangements with their margins drawn rather than tabulated, because the shape of this chart is the argument: the quantity is not a probability and not a percentage, it is a distance — how far the origin sits from the nearest face of the hull of the contact rows, with every row a unit vector so the number is comparable across arrangements. The two that hold come in at 0.211 and 0.091; the five that do not come in at exactly nought, and they are drawn at nought rather than left off. A margin that falls smoothly to nothing is what makes this a measurement: an arrangement approaching one that lets go says so before it does.

The test is a program, not a rank

Three independent routes to one yes-or-no: enumerate the escape cone's extreme rays by cross products, take the convex hull of the contact rows and ask where the origin is, or hand the whole thing to a simplex. They agree on every arrangement — and the first version of the third one reported a disc as held, which is the one part in the field that no number of contacts holds.

holding · Restraint 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.

pairs · Pair elbow arm at a posture. elbow arm, drawn from 6 joint values through a product of 6 exponentials — no equation is solved anywhere in this picture, because an open chain has none to solve. The thin lines are the joint axes at this configuration, which are also the columns of the arm's Jacobian: the tool's velocity is a sum of turns about exactly those lines. Here the smallest singular value of that Jacobian is 0.3674 and the largest is 2.407, so the arm is comfortably away from a configuration where a direction of motion is lost. Drag θ₃ elbow.

Two routes to a Jacobian

An open chain's Jacobian is a list of its joint axes, drawn as lines in the same picture as the arm. A finite difference of its own forward kinematics is a completely different computation and has to agree — and when the two disagreed by 5 × 10⁻⁵, the fault was in a function six phases old that every spatial loop on this site had been using.

serial · Serial 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.

constraint · Mobility How often a set of screws is a group. Take a subspace of the twists at random and ask whether it is closed under the Lie bracket — whether doing two of its motions in one order and undoing them in the other leaves you inside it. Every one-dimensional subspace is, trivially and importantly: a single screw always generates a one-parameter subgroup, which is the same statement as every screw is a joint somebody could build. Above one dimension, not one of eighty thousand is a group, and every one of them generates the whole of the rigid displacements at the first bracket. So a mechanism whose motion lies inside a proper subgroup is not merely unusual; it is a coincidence of measure zero — and it is the coincidence every planar mechanism, every spherical one and every Sarrus linkage on this site is built on.

Almost nothing is a group

Eighty thousand subspaces of the twists were drawn at random and closed under the Lie bracket. Above one dimension, not one of them was already closed, and every single one generated the whole of the rigid displacements at the first bracket. Two pins with parallel axes close at three; move one axis a hair and they close at six.

pairs · Pair Where a tolerance stack-up stops meaning anything. The ratio between the first-order tolerance estimate and the band measured by building every corner linkage, for two four-bars at ±0.002 on each length. The crank-rocker's ratio is 1 at all 180 positions — a stack-up is exactly right for it, everywhere. The parallelogram is a change-point linkage, where all four bars can lie on one line, and at that position the estimate exceeds the measurement by a factor of 4.8e+5. The difference is not in the arithmetic, which is identical; it is that a derivative describes a map that can be inverted, and at a change point the map cannot.

Where a stack-up stops working

On a crank-rocker the first-order tolerance estimate matches the measured band at every one of 180 positions, to eight parts in ten thousand. On a parallelogram it exceeds it by a factor of 475,512. Same arithmetic, same tolerance, two linkages that differ only in their proportions — and nothing in the calculation says which one it is being run on.

practice · Tolerance What the count is right about. Freedoms minus dependencies, against what the count predicts, on seven assemblies from three different representations. The difference is exact every time and it is exact for a reason that has nothing to do with mechanisms: the count is unknowns minus constraints, the rank is a number no larger than either, and the two nullities are what each of them has left over. So a count is not wrong in the way a mismeasurement is wrong. It is a statement about a difference being read as a statement about one of the terms — and on four of these seven rows both terms are large and the difference is nearly meaningless.

A constraint that has been said already

Every constraint matrix leaves two null spaces, and a mechanism only lives in one of them. The other is the set of combinations of constraints that come to nothing — and its dimension is exactly the amount by which the count is wrong, on a deployable ring, a Miura sheet and a framework with twelve bars and six joints.

networks · Network What each loop's constraint system is made of. For each mechanism: the order of the screw system its joints span, the order of the reciprocal system — the wrenches it carries without moving, which is always six minus the first — and what those wrenches are. A planar four-bar carries one force and two couples; a mechanism whose motion lies in no subgroup carries screws of finite pitch instead, and 2 of these 6 do.

What a mechanism cannot do

A mechanism's freedoms are a subspace of screw space. Everything orthogonal to that subspace under the reciprocal product is a force the mechanism carries without moving — so the constraints are not a separate thing to be worked out, they are what is left, and one matrix gives both.

spatial · Spatial elbow arm at a posture. elbow arm, drawn from 6 joint values through a product of 6 exponentials — no equation is solved anywhere in this picture, because an open chain has none to solve. The thin lines are the joint axes at this configuration, which are also the columns of the arm's Jacobian: the tool's velocity is a sum of turns about exactly those lines. Here the smallest singular value of that Jacobian is 0.3674 and the largest is 2.407, so the arm is comfortably away from a configuration where a direction of motion is lost. Drag θ₅ pitch.

The wrist is three joints and one point

Three axes meeting at a point is what makes a six-joint arm's inverse problem solvable in closed form, and it is why every industrial arm is built that way. Move one of those axes by ten millimetres and the construction goes on returning eight confident answers, every one of them out by three and a half.

serial · Serial Free to spin, and it cannot turn at all. An ellipse of semi-axes 1.4 and 0.9 inside four flat walls that touch it at the ends of its own axes. Every normal points at the centre, so every row's moment is nought and the four rows span two dimensions rather than three: the cone of permitted twists is the whole spin axis, a line through the origin, and the first-order answer is that the part is free to turn either way. Drag the angle and watch what happens. The ellipse's reach in the direction of the top and bottom walls is √(a²sin²θ + b²cos²θ), which is smallest at θ = 0 and grows from there, so any rotation whatever drives it into both of them — by 0.016 at this angle. A nullity is a candidate and not a motion, and this is the shape of case the fields before this one could not produce: not a mechanism at a singularity, but an ordinary part in an ordinary pocket. positioned by solving, not by drawing.

Free to turn and unable to

An ellipse in a pocket the size of its own bounding box has four contacts whose rows span two dimensions, so the cone of permitted twists is a whole line and the first-order answer is that it spins both ways. It cannot turn by any amount whatever: the penetration grows as the square of the angle, with a fitted exponent of 1.9944, and a circle in the same pocket turns for ever.

holding · Restraint Kutzbach, plus the constraints counted twice. Each row is 6(n − j − 1) + Σf, then the redundant constraints ν measured from the two legs' constraint systems, then the mobility measured from the rank of the whole loop's screw system. The first column is wrong for 5 of 6 of these mechanisms; adding ν repairs every one. The catch is that ν is not a property of the joint graph — Bennett's linkage and a spatial four-bar with one twist changed have the same graph and different ν — so the corrected formula needs the measurement it was supposed to replace.

The formula is repaired by the thing it replaced

Kutzbach's count is wrong about most of the mechanisms worth building, and every textbook gives the same repair — add back the constraints that were imposed twice. The repair works on every loop this site has. It is also not a formula, because the number it adds cannot be read off the joint graph.

constraint · Spatial Whose constraints stay put. The largest principal angle between a mechanism's screw system at the start of its motion and at each later position. The 3 mechanisms whose motion lies in a subgroup of the rigid displacements — planar, spherical, translational — never leave the same subspace, and read between 1.9e-6 and 3.0e-6 degrees, which is the precision of an arccosine near one rather than a movement. The paradoxical ones turn through 49° and 57°. Every curve runs over its own range of motion, because Bricard's linkage assembles over 120 degrees and a shared axis would hide it.

Two ways to be overconstrained

A planar four-bar and Bennett's four-bar report the same redundancy, the same rank and the same wrong count. One of them is overconstrained at every set of link lengths; the other at exactly one ratio and nowhere near it. The difference is not in any of the numbers so far — but it is measurable, and the measurement is an angle.

spatial · Spatial Three singularities, and where each one is. The elbow singularity is at the edge of the workspace, where the arm is straight and cannot reach further — nothing is lost that could have been used. The shoulder singularity is where the two ways of facing the target merge, on a cylinder of radius 0.18 m about the base axis which is also the boundary of a hole the arm cannot reach into at all. The wrist singularity is the one that stops real machines: axes four and six in line, one rotation gone, in the middle of an ordinary working volume with nothing about the tool's position to suggest it. The rank falls by exactly one in all three — and what is lost differs. The elbow and the shoulder are constrained by a pure force, along the arm and across it; the wrist is constrained by a screw of pitch −0.629, which is a force and a couple together. Those are reciprocal screws rather than singular vectors, because a singular vector's direction depends on whether the arm was written in metres or millimetres and a screw does not.

Where the arm loses a direction

An arm has three singularities and they are three different events. Two of them are inside an ordinary working volume, all three drop the rank by exactly one, and what each one takes away is a screw — a pure force along the arm at the elbow, a pure force across it at the shoulder, and at the wrist a screw of pitch −0.629 that is a force and a couple together.

serial · Serial Four joints that give a group, and four that do not. Two chains of four revolute-and-slide joints, each drawn at its home position with its joint axes dashed, and each with a cloud of the tool positions it reaches. The counts are identical: four joints, four freedoms, the same Jacobian rank everywhere off a singularity. On the left the three pins are parallel and the slide is along them, and the displacement set is the Schoenflies group — every translation and one rotation direction, four dimensions, closed. On the right the axes are at random and the set is four-dimensional too, and it is inside no group smaller than all the rigid displacements. The instrument is in the caption of each panel: take the logarithms of the displacements the chain reaches and count the dimensions they occupy. Four means a group. Six means there is nothing to be inside.

Four joints that give a group, and four that do not

Two chains of four joints. Same joint types, same count, same mobility, same Jacobian rank, same everything this site has measured for twenty-two fields. One of them reaches a four-dimensional set of displacements that closes under composition; the other reaches a four-dimensional set whose logarithms fill all six dimensions. The difference is six hundredths of a radian in where two axes point.

pairs · Pair Two numbers, and where they differ. Every mechanism in this field, with what its constraints leave and what its brackets fill. On every mechanism without a rolling contact the two columns are the same number, which is why nobody had to say which one mobility meant. Here only the rail agrees with itself — and the rail is the one mechanism in the table that cannot go anywhere new.

The count that counts the wrong thing

Mobility has meant one number for six fields, because until a wheel appeared no mechanism could tell two questions apart. A rolling wheel has two velocity freedoms and a three-dimensional reachable set, and the formula that gives 2 is not wrong — it is answering the question about instants when the question anybody asks is about intervals.

constraint · Mobility The smallest mechanism that is not one. Two bars from a free joint to two pinned ones, with the three points in line. The constraint matrix has two rows and two columns, both rows are horizontal, and its rank is 1: one freedom left over, pointing straight up, and one dependency among the two bars. Move the joint up and neither bar changes length to first order, which is what the freedom says. To second order both bars get longer, by the same amount and in the same direction, and there is nothing to trade off against — which is what the dependency says. The obstruction is the dependency applied to the second-order stretch and comes to 1.414214; anything but nought there and the freedom is not the beginning of a motion. Lifted by 0.34 the bars are 0.05622 longer, which is the whole argument drawn to scale.

It moves to first order and not at all

Two bars from one joint to two pinned ones, all three in line: the rank leaves a freedom pointing straight up, and lifting the joint stretches both bars. The obstruction is 1.414214, the walk travels a millionth of what it is asked to, and how far it gets is a property of the tolerance rather than of the mechanism.

networks · Network What each instrument returns, on each kind of graph. The 8-link census, three rows, and the same three questions asked of every graph in it. Grübler returns 1 in every row — it has to, because that is what the census selected on. The rank returns 1 in the first two rows and 2 in the third. Only the third column changes across all three rows, and it is the one this site did not have before this field: a mobility computed for every subset of the links rather than for the whole. Read down the middle two columns and the site's standing pair of routes is unanimous about 62 graphs, of which only 16 are what it says they are.

The count was right and the name was wrong

The constraint field has checked Grübler's count against a Jacobian rank since the foundation, and the two disagree only where the geometry is special. Here is an assembly where they agree, where both are correct, and where the mechanism does not have the number of links it is described as having.

constraint · Mobility One tool pose, a curve of postures. 41 postures of S-R-S arm, every one of them holding the tool at the same pose. The tool moved 1.0e-13 of a metre over the whole sweep, which is the arithmetic's noise floor rather than a tolerance. The joint that moves most is the elbow, and it runs on a circle of radius 0.192 m about the line from shoulder to wrist — a curve computed here by walking the Jacobian's null space, and known independently as the intersection of two spheres. The two agree to 10⁻¹³.

The freedom that does nothing

Give an arm a seventh joint and one tool pose stops having eight answers and starts having infinitely many. The arm changes shape while the tool stands still — measured at 3.3 × 10⁻¹² of a metre over forty-one postures — and the elbow runs on a circle that two entirely different computations agree about to a tenth of a picometre.

serial · Serial 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.

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.

constraint · Mobility How much each set of wheels forbids. Every wheel contributes the same row, and whether it is a constraint or a drive is one factor of sin γ in it — γ being the angle the rollers make with the wheel's own axle. At γ = 0 the row says the body may not move across the wheel and the wheel's speed drops out of the statement; at γ = 45° the row says nothing about the body at all and fixes the wheel's speed instead. The whole difference between a machine that shuffles and one that slides sideways is in that factor.

The wheel that forbids nothing

Every wheel contributes the same row to the same matrix, and whether that row is a constraint on the vehicle or a statement about the wheel's own speed is decided by one factor of sin γ. At γ = 0 the vehicle may not move across the wheel; at 45° the row says nothing about the vehicle at all, and sideways costs exactly what forwards costs — to the last digit, and at no other angle.

rolling · Rolling The framework Maxwell's count calls a structure. Six joints and twelve bars in space. Three coordinates each gives eighteen unknowns, six rigid motions come off, and twelve bars is exactly twelve constraints — Maxwell's count is 6 against six rigid motions, which is the definition of isostatic: no mechanism, no redundancy, every bar carrying its own share and nothing spare. The rank is 11, not twelve. There is one dependency among the bars and one freedom left over, and the freedom is a genuine finite motion: walked here with every bar held to 4.4e-16 of its own length. The reason is a symmetry — three pairs of joints exchanged by a half turn about one line — and it is built into the coordinates rather than asserted about the result. positioned by solving, not by drawing.

Twelve bars and a symmetry

Six joints and twelve bars in space is Maxwell's count exactly: no mechanism, no redundancy, nothing spare. Place three pairs of the joints so that a half turn about one line exchanges them and it moves — a finite motion, walked with every bar held to five ten-thousand-billionths of its own length, on a framework the arithmetic calls a structure.

spatial · Spatial Sarrus, as two planes meeting in a line. Each arm of Sarrus's linkage is three pins with parallel axes, so each arm holds the platform inside a planar group — the one whose normal is that arm's axis direction. The platform has to satisfy both, so what it may do is the intersection, and the intersection of two planar groups whose normals are not parallel is the one-dimensional group of translations along their common perpendicular. The platform goes up and down and does nothing else, and that is the exact straight line the spatial field measured to 10⁻¹⁶ of its span — arrived at here with no mechanism solved and no tolerance anywhere. Drag the arms towards each other: the answer is a translation at every angle but zero, where the two groups become one group and the intersection jumps to three dimensions. That is the linkage built flat, and it is the configuration in which it stops being a straight-line mechanism.

Two planes meeting in a line

Sarrus's linkage draws an exact straight line out of six pin joints, and the spatial field proved it by solving the mechanism sixty times and measuring a departure of 9.8 × 10⁻¹⁶. Here the same fact comes out of two planes and a cross product, with no mechanism solved anywhere — and the two routes are not redundant, because only one of them can tell you the linkage as built delivers it.

pairs · Pair Two branches, meeting where the sheet is flat. The same vertex's two folding modes, plotted as one crease's fold angle against another's. Both curves pass through the origin, which is the flat sheet, and they cross there and nowhere else. That crossing is what a rank cannot see: at the origin the tangent directions of both branches are available to the constraint matrix at once, so the nullity there counts them all and the mechanism has only one of them once it has left. Every branch argument this site has made — a four-bar's assembly configurations, an arm's eight postures, the components a solve's paths turned out to run between — is this picture with different axes.

Where the branches meet

A flat sheet is the one configuration every folding of a pattern passes through, and it is the one configuration where the rank is wrong about all of them. Three of a three-by-three Miura sheet's four apparent freedoms are not motions — and a grid that folds to no angle at all reports exactly the same four.

networks · Network 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.

Compose two positions and see where you land

Take two configurations a mechanism actually reaches, compose the displacements that got it there, and ask what kind of thing the result is. A planar four-bar lands inside planar motion, to 4 × 10⁻¹⁶. Sarrus lands on its own line. Bennett's linkage lands three tenths of a radian outside the four dimensions its own displacements occupy — a one-freedom motion that generates all six.

pairs · Pair Every axis through one point. Four wheels on one rigid body, each rolling without sliding. Each turns about some point on its own axle line, and a rigid body has one such point, so every axle line has to pass through it. That is the whole of steering geometry, and it is a rank condition rather than a formula: here the four rows have rank 2 of 3, leaving a one-dimensional family of twists, and the centre they agree on is 12.000 m to the side. The scrub is 3.2e-17 m per metre — zero, to the last digit.

Every axis through one point

Bolt several rolling wheels to one rigid body and they impose one condition between them: every axle line must pass through a single point. The familiar steering formula falls out of it as a consequence rather than being quoted — cot δₒ − cot δᵢ = 0.574074 at a turn of six metres, of eight, of twelve and of twenty, on a track of 1.55 m and a wheelbase of 2.7.

rolling · Rolling Arms with their tools pinned down. Pin an arm's tool to the ground and the open chain is a closed loop, which the first field of this site knows how to count. Kutzbach gives 6(n − 1) − 5n = n − 6 for a loop of n revolutes, and the measurement is n minus the rank of its screw system — the columns of the arm's own Jacobian, read as constraints rather than as velocities. The two agree on every row but one, and the one is the arm at a wrist singularity: the formula says the pinned arm is a structure and the mechanism has a freedom. That is the finding this site opened with, arrived at from the far end of its subject.

Pin the tool and it is a loop

Hold an arm's tool still and the open chain becomes a closed one, which this site has known how to count since its first field. Kutzbach's criterion says a pinned six-joint arm is a structure. At each of its three singularities the measurement says it can still move — the site's founding finding, arrived at from the far end of its own subject.

serial · Spatial Six assemblies, one routine, three disagreements and one accident. Every row is the same three steps: write down the constraint Jacobian, take its rank, and subtract it from the number of unknowns. The representations differ — bars between points, bodies joined by pins, panels joined by creases, one cell of a pattern that repeats for ever — and the routine does not. The counted column is the arithmetic on the numbers of bodies and joints; the measured column is the nullity of the matrix. They agree on the lazy tong and on the kagome cell and disagree on the other four, most sharply on the deployable ring, which the count declares immobile and which is sold as a mechanism that opens. The right-hand column is the reason: constraints that repeat what another constraint has already said, which the count has no way of seeing and the rank cannot help seeing. The fourth row is worth reading twice: the count says nothing can move and nothing can, so the two agree — and they agree for the wrong reason, because that pattern's flat state shows four freedoms and not one of them is a motion.

Six things a network is not

A count that is right about a difference and read as an answer, a nullity taken for a mobility, a flat state that cannot tell a mechanism from a structure, a scissor ring that closes nowhere, a vertex that folds while its sheet does not, and a null space computed with an instrument whose floor is above the answer. Six claims, each with the number that kills it.

wrong · Misconception What the repeated constraints cost the drawing. Move an interior vertex of the flat pattern and the folded state generally stops existing. It survives if the change to the vertex closures can be absorbed by a change in the fold angles — and the part that cannot be absorbed is exactly the part that lies along a dependency, because a dependency is a direction in residual space the fold angles cannot reach. So the number of conditions a pattern's shape has to satisfy is at most the number of dependencies among its constraints, and on the Miura family it is exactly that: one at three by three, four at four, nine at five, measured by taking the rank of the obstruction. A twelve-by-twelve sheet has a hundred conditions on where its vertices may be. That is why a grid whose vertices are anywhere at all does not fold, and it is the same number, read the other way round, as the amount by which the count is wrong.

What a pattern has to satisfy

Move an interior vertex of a crease pattern and the folded state generally stops existing. How many conditions the drawing has to meet is not a matter of taste: it is exactly the number of dependencies among the constraints, measured at one, four and nine on three sizes of sheet, and a hundred on a sheet of a hundred and forty-four panels.

networks · Network 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.

Six things a hold is not

A rank read as a restraint, a count read as an answer, four contacts placed the wrong way round, a nullity taken for a spin, a part free in every direction and unable to leave, and a tolerance offered as a cure for an arrangement that was never a hold. Six claims, each with the number that kills it.

wrong · Misconception A network with no boundary at all. The kagome lattice, drawn out to 5 cells across and continuing for ever. The measurement is made on one cell: 3 joints, 6 bars, and a bar that leaves the cell comes back into it, written against the far end's position in the neighbouring cell. There is no boundary anywhere in the arithmetic, and the size of the network has gone from being a parameter to not existing. The highlighted triangle is the cell; every other line on the page is a copy of one of its six bars. positioned by solving, not by drawing.

The cell that repeats for ever

Take the size of a network to infinity and it stops being a parameter. What is left is one cell, six bars, and a question nobody has to ask about a finite assembly: does the pattern's period count as a body? A square grid is rigid if it does not and shears if it does, and so does the kagome.

networks · Network Why this measurement is not a derivative. The honest failure mode of the field's instrument, drawn rather than hidden. Every set looks like its own tangent space near the identity — that is what a tangent space is — so a chain sampled over a thousandth of a radian reports the dimension of its velocities, which is the number the screw system already gives. Four pins at random has four joints, and sampled over 10⁻⁹ radians its displacements occupy four dimensions; sampled over two radians they occupy six. The step is at 10⁻⁶, which is where the departure from the tangent space falls below the rank tolerance — so the position of the step is a fact about arithmetic and the two plateaux are facts about the mechanism. A group is a statement about displacements you could compose, and no derivative can make it.

The instrument that is not a derivative

Sample a chain of four random pins over a millionth of a radian and its displacements occupy four dimensions, exactly as a SCARA arm's do. Sample the same chain over two radians and they occupy six. The step is at 10⁻⁶, and where it sits is a fact about arithmetic while the two plateaux are facts about the mechanism.

pairs · Pair 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.

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.

holding · Mobility What a calibration can and cannot see. The singular values of elbow arm's identification Jacobian — the matrix of how the tool pose moves when each model parameter is nudged, over 11 postures. There are 36 parameters and only 30 of them can be found: the last 6 directions come out at 2.3e-8, which is the difference noise, against 6.3e-4 for the weakest real one — a gap of 3e+4. And 30 is exactly 4R + 2P + 6 for this arm's 6 turning and 0 sliding joints, which is a count from the literature meeting a rank measured from the arm's own arithmetic.

What a calibration cannot see

A six-joint arm's model has thirty-six parameters and a measurement can find thirty of them. The other six are not hard to measure — they are combinations that move the tool by exactly nothing, at every posture, and no instrument ever built will separate them. The count is 4R + 2P + 6, and it comes out of a rank on four different arms.

serial · Serial 5 braces, and it is rigid. A 3×3 grid of squares with 5 of its cells braced by a diagonal, and no freedom left. The bipartite graph on the 3 columns and 3 rows, with one edge per braced cell, has 1 component — and the number of freedoms is one less than that, at every bracing there is. Nothing in the rank computation knows about columns, rows or graphs.

Which diagonal rigidifies a grid

A three-by-three grid of squares needs five diagonals and eighty-one of the hundred and twenty-six ways of placing five will do. Which ones is not a rank question at all: it is whether a graph on the grid's columns and rows is connected, and eighty-one is the number of that graph's spanning trees.

networks · Network What each machine is sold with. Every mechanism in this field arrives with a number. The middle column is that number; the right-hand column is what the mechanism was measured to have, by the library named under the row. The coloured bar and the word under each machine are the verdict: exact means the mechanism has the number at every position, mean that it is the average of something that varies within every cycle, bounded that it is false by a stated and negligible amount, point that it is the value at one position, and quoted that the mechanism has no such quantity at all. The lit row is the one this essay is about. Of the 14 rows, 6 are quoted and 3 are exact.

Which numbers survived

Fourteen machines were measured against the numbers they are sold with. Three survived exactly, and all three are counts. One is false by a stated bound, two are exact means of things that vary, three are honest values at a stated position, and six name quantities their mechanisms do not have. The pattern is not about honesty; it is about what kind of thing a number is.

applied · Misconception The same patch twisted 17.2°: 35 mechanisms, all at the edge. A rhombus of 8 × 8 kagome cells — 192 joints, 346 bars — with every up-pointing triangle turned by 17.2° about its own centre. Each joint is drawn with an area proportional to its weight: its share of the diagonal of the projector onto the patch's mechanisms, which does not depend on how the mechanisms are written down and adds up over the joints to the number of mechanisms, 35. That number is Maxwell's count, 2 × 192 − 346 − 3, and the rank agrees with no redundant bar. The mean weight is 0.341 on the outermost ring of cells and 0.015 on the innermost. The twist kinks every line of bars at every joint, and the mechanisms fall away from the edge 23-fold in 3 cells.

The count says how many and not where

A kagome lattice has three joints and six bars in every cell and counts to exactly nothing, so a patch cut from it has as many mechanisms as its edge has lost bars: 5L − 5 for a rhombus of L cells a side, which the rank confirms at every size with no bar redundant. Straight or twisted, the number is the same. Where the mechanisms are is not: a straight patch keeps nearly half its edge weight in the middle, and a patch whose triangles are turned by 17° keeps a twentieth.

networks · Network

Named alongside it

The objects these essays reach for when they reach for this one.

MobilityConstraintRedundant constraintDegrees of freedomOverconstraintGrübler's criterionJacobianConstraint jacobianNull spaceNetworkScrew systemDisplacement subgroup

All concepts