Machine Kinematics
solved, then drawn
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The thread: The singularities are the point

Dead centres, toggle positions and the configurations where a mechanism locks or changes branch are treated as the subject rather than as edge cases. They are where the mechanical advantage goes to infinity and where the machine stops working.
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. Contacts that only push

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.

 A rolling wheel, where it was driven to. The mechanism at a configuration nothing wrote down: it was reached by integrating permitted velocities from the start of the trail, and there is no equation here whose root it is. The barred line at each wheel is the direction that wheel forbids — the subject of the whole field, and the one thing a photograph of a car cannot show. The constraint residual along the drawn history is 0.0e+0. Wheels, and where they may not go

A constraint that takes nothing away

A rolling wheel forbids one direction of motion and removes no coordinate from the mechanism's description. It cannot slide sideways and it can still be brought to any position at any heading — and the gap between those two sentences is the whole of this field, because in every mechanism built of pins and slides the two agree.

 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. Out of the plane

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.

 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 chain before the lengths

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.

 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. Several legs, one platform

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.

 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. Drawn wrongly

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.

 What one contact forbids, drawn as a place. A single contact on one edge of a square, and the whole plane coloured by what it permits. A rotation about a point is a twist, and a twist is permitted when it does not drive the part into the obstacle; because a rotation about (x, y) is affine in the point, the condition is a half-plane and the boundary is a straight line — the line through the contact along its own surface. On one side of it only anticlockwise rotations are permitted, on the other only clockwise, and the two together are the whole plane bar the line itself. So one contact rules out exactly half of what the part could do and leaves the other half untouched, which is why the count of contacts a hold needs is one more than the dimension rather than equal to it: the first 1 of them cannot leave nothing over. The picture is exact — the regions are clipped polygons, not a sampled grid. Contacts that only push

What one contact forbids

A rotation about a point is a twist, and a twist is affine in the point — so what a single contact permits is a half-plane of centres, with the boundary being the contact surface's own line. Reuleaux drew it in 1875 and it is exact rather than sampled, which is why every figure in this field is a picture of the plane rather than of a cone.

 Of 1176 exactly correct syntheses, 176 could be built. Every pair of points on a 7×7 grid over the moving body, each pair synthesised into a four-bar and each four-bar verified by the forward solver as reaching all three prescribed poses. 1176 of them do, exactly. Then each is swept from the first pose in both directions, carrying the branch the way a built mechanism must: 810 cannot reach all three without being taken apart and reassembled, and 190 reach them in the wrong order. 176 — 15% — are mechanisms rather than theorems. Nothing in the construction distinguishes them. Drawn wrongly

Exactly right, and unbuildable

A linkage synthesised through three prescribed positions reaches all three. That is a theorem and it holds exactly. Whether it reaches them in one piece, without being taken apart, and in the order asked for, are separate questions the construction says nothing about — and of 1,176 exactly correct solutions, 176 could be built.

 A 6-slot Geneva wheel. The driving pin enters a radial slot, carries the wheel through 60°, and leaves. The centre distance is not free: it must be crank ÷ sin(180°/6) = 60.00 so that the pin enters along the slot, with the crank and slot perpendicular. That is the mechanism's one design requirement and it is what makes the driven wheel start and stop from rest — measured here at 1.5e-3 against a peak of 1.000. What it does not fix is the acceleration, which peaks at 1.35 and is why film sprocket holes tear. Prescribed motion

Stopping thirty times a second

A Geneva wheel turns continuous rotation into steps, and its one design requirement is that the pin enters the slot along the slot so the driven wheel starts and stops from rest. That fixes every dimension from the slot count. What it does not fix is the acceleration, which is why film sprocket holes tear.

 The circle of points going straight, at 66°. Every point on this circle is, at this instant, travelling in a straight line: its path has zero curvature there. The circle passes through the pole — where the point is not moving at all — and its diameter is 47.64, which is the pole's own speed divided by the plane's angular rate. Nothing here was assumed to be a circle. The locus is the zero set of a quadratic whose |w|² coefficient is φ′³, a real number with no cross term and no difference between its two square terms, and a general conic fitted to the sampled locus returns those coefficients at 8.1e-15 and 6.4e-15. At this position the coupler is close to translating, the pole has run off the canvas and the circle with it — the figure is the size of the mechanism, and δ here is 13.6 coupler lengths. positioned by solving, not by drawing. The motion, not the mechanism

The circle of points going straight

At any instant some points of a moving plane are travelling in a straight line. They form a circle — not nearly a circle, a circle — and the reason is one real coefficient in a quadratic. A coupler curve has an inflection exactly when that circle sweeps over the tracing point, which turns out to be rare.

 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. 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.

 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. What can move

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.

 The transmission angle through one turn. μ is the angle at B between coupler and rocker, computed from each solved position rather than from a formula. It runs from 54.3° to 100.3° for these lengths. The shaded band is the usual design rule — keep μ between 40° and 140° — and this linkage stays inside it throughout. The rule is about geometry alone: nothing here knows about friction, and a mechanism with a comfortable μ can still be a poor machine. Linkages

The transmission angle

The angle at which the coupler meets the rocker decides how much of an applied force becomes useful output torque and how much goes into the bearings. It is pure geometry, it is computed here from every solved position rather than from a formula, and it is the number a linkage is judged by after Grashof has said it turns.

 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. The chain before the lengths

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.

 Burmester's curves, contoured rather than drawn. With three poses, every point of the moving body works: three images, one circumcircle. With four, a point's four images are concyclic only if it lies on a particular cubic — the circle-point curve — and the fixed pivots those points want lie on a second cubic, the centre-point curve. Both are drawn here as contours of a measured quantity: at each point of a 150×150 grid, how far the fourth image misses the circle through the other three, contoured at zero. Points refined onto the contour are concyclic to 3.8e-15; points 0.47 away from it miss by at least 2.0e-1. The two curves are keyed in the legend and the four prescribed poses are drawn faintly for scale. Three poses leave a designer the whole plane; four leave a curve. The four prescribed poses are outlined faintly for scale, and both curves are clipped to the frame: the centre-point curve is a cubic with unbounded branches that reach 360 units on a mechanism three units across, and the part worth looking at is the part near the machine. The problem backwards

What the fourth position costs

With three prescribed positions every point of the coupler will do, and a designer is spoilt for choice. Add a fourth and the whole plane collapses to a curve — only points on a particular cubic have four images that lie on a circle, and the cubic is Burmester's.

 6 ways to assemble the same three actuator angles. The actuators are at 216°, 48°, 144° in every panel, so the three elbows are at the same three points throughout and only the platform differs. Each pose satisfies all three legs to 6.7e-16. Found by reducing the problem to one equation in the platform angle and scanning it at 0.100° — exhaustive to that resolution and no further, which is the honest thing to say about a root count. Over a survey of 6750 actuator triples this mechanism ranges from none to 6. Several legs, one platform

One command, six answers

Lock the three motors of a planar platform and the platform can be in as many as six different poses, every one of them satisfying every leg exactly. Which one it is in was decided by how it was assembled and where it has been since — and the number of answers is not a property of the mechanism but of where the motors happen to be.

 The gap is a function, and it has corners. The smallest gap over every tested pair, at each of 360 solved positions of the crank. Two things are visible that a check at the ends could not report. The minimum, 0.1041, is in the middle of the travel and not at either end. And the curve has 4 corners, each one a change in which pair is closest — the colours — so the function is piecewise smooth rather than smooth, and its minimum is not where a derivative vanishes. Refining off the sample grid by golden section moves the answer by 1.4e-5, which is what a corner rather than a smooth minimum looks like. Links with a width

A gap with corners in it

The clearance between two parts is a function of the crank angle, and it is not a smooth one. It has a corner wherever the closest pair of features changes hands, so its minimum is not where a derivative vanishes and is not at either end of the travel.

 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. Out of the plane

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.

 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. Wheels, and where they may not go

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.

 The count that moves, and the one that does not. 676 sets of leg lengths for one 3-RPR platform, the third leg held at 2.2. Every one of them has exactly 6 complex solutions. The number that is real runs 0, 2 — 141 cells at 0, 535 cells at 2 — and that number is what a machine shop would call the assembly modes. How many answers

The count that does not move

A mechanism does not have a number of assembly modes. Its family has a complex solution count that never changes, and each member has a real count that does — 676 sets of leg lengths for one platform, all with six complex solutions, and nought, two or four of them real. The number a machine shop cares about is the one that is not a property of the machine.

 A defect that draws nothing wrong. The same four-bar swept 360 times with the pre-correction Jacobian and with the corrected one, at eight coupler-point offsets. Corrected, every position is reached at every offset. Uncorrected: 5 of the offsets lose nothing at all, and then it loses 58, 159, 267 of 360. The picture was never wrong — a refused position is simply not drawn — so the only symptom was a sweep with fewer frames in it than it asked for. Drawn wrongly

The solver was refusing a quarter of the sweep

Four numbers in this site's Jacobian had the wrong sign, from the foundation phase until now. Every picture it ever drew was correct, because a wrong derivative does not move a converged answer — it just makes Newton crawl, until the stall rule declares the position unreachable. The symptom was a sweep quietly returning fewer frames than it asked for, and no gate in the fleet has a rule against that.

 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. One path to the tool

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.

 Where the coupler is pivoting, at 70°. At any instant the coupler is turning about one point — not a pin, and usually not on the mechanism at all. Kennedy's theorem finds it: the crank and coupler share the pin at A, the coupler and rocker share B, so the coupler's centre relative to the frame must lie on both O₂A extended and O₄B extended, and it is where they cross. The dashed lines are that construction. The cross is a completely different route to the same point — the place where the coupler's solved velocity field is zero, computed from the Jacobian and knowing nothing about Kennedy. Across 119 positions the two agree to 2.7e-15. At this instant the centre lies outside the frame — the two construction lines are nearly parallel, the coupler is close to translating, and the pivot has run off rather than gone missing. The paths points trace

Where the coupler is turning

At every instant the coupler of a four-bar is rotating about a single point — not a pin, and usually not on the mechanism at all. Track that point in two different frames and you get two curves which, rolled on each other without slipping, reproduce the coupler's motion exactly. The bars are one way of producing it and not the motion itself.

 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. What a joint is

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.

 A cam profile is an envelope, and its curvature obeys the same law. The dashed curve is the pitch curve — where the roller's centre travels — and the solid one is the surface that has to be cut, which is the pitch curve offset inward by the roller radius. That offset is the conjugate law of this field with one centre of curvature sent to infinity, and it says ρ_cut = ρ_pitch − r. Measured off the drawn polyline at six angles, the worst departure is 4.8e-7: at 95° the pitch curve has radius 29.52 and the cut profile 21.52, against 21.52 predicted. It is also why undercutting is a curvature condition rather than an accident: where ρ_pitch falls below the roller radius the offset turns itself inside out. Prescribed motion

A profile is an envelope

A cam's surface is not a curve somebody drew. It is the envelope of the roller as the roller runs round the pitch curve, and its curvature is the pitch curve's less the roller radius — a law that this site's cam library was breaking for six months, in the one curve that gets manufactured and the only one no check looked at.

 4 contacts, and the centres they still allow. The same four, placed pinwheel. The same square, the same four edges, the same distance along each — and taken the same way round rather than alternately. Every row's moment then has the same sign, so no positive combination can cancel it, and the part turns. Each contact contributes one half-plane of permitted centres per sense, and the shaded regions are what survives all 4 of them: the darker one is where an anticlockwise rotation is still permitted and the lighter one where a clockwise one is. What is left is the escape, and it is a region rather than a direction: any point inside it will do as a centre. The enumeration finds 4 extreme rays, of which 4 are rotations and the rest are translations — the corners of the region and its unbounded directions respectively. positioned by solving, not by drawing. Contacts that only push

The escape is a place

A part that is not held escapes, and the useful thing is not that it escapes but where. The extreme rays of the cone are the corners of a region of the plane and its unbounded directions are translations — so the answer to 'this does not hold' is a picture with a shape, and the shape says where the next contact has to go.

 30 teeth and two pallets. An escape wheel of 30 teeth and a pair of pallets spanning 4 tooth pitches. The heavier line at each pallet is the locking face, here an arc about the arbor; the lighter one is the impulse face the tooth slides along once it is let go. The wheel is drawn where the contact puts it, not where it looks well: at this pallet angle the tooth in play sits on the lock face and the wheel is -0.0323° from it. Dragging the pallet through its whole engagement moves the wheel by 0.107° of recoil. Of the 6.0° the wheel turns each beat, -37.3% is drop and does nothing. Drawn wrongly

The escapement that could not alternate

Four drawings of intermittent mechanisms that appear everywhere and would not work: pallets spanning a whole number of teeth, a pawl whose pivot is on the wrong side of the tooth face, a Geneva at the wrong centre distance, and an intermittent gear with no locking arc. Each one is put through the library that draws the working version, and each returns a number.

 A quantity that is not there, refusing to converge. A second central difference of a function with a bounded second derivative settles as the step is halved; across a step in the first derivative it doubles, every time. The rising line is a mutilated gear at the instant its teeth engage, where the driven wheel goes from stopped to full pitch-line speed: its successive estimates grow by a factor of 2.000000, which is the signature and not an accident of the step. The flat line is a six-slot Geneva at the same point in its cycle, whose pin enters along the slot and whose acceleration is a number. This is the only way to report an acceleration that does not exist: not by quoting a large one, but by showing the measurement refuse. Motion that stops

The gear with its teeth cut away

Leave teeth on part of a gear's circumference and take the rest off, and the output turns for part of the input's revolution and stops for the rest. It is the cheapest intermittent drive there is and it engages at full speed, so its output's velocity has a step and its acceleration is not a large number — it is not a number, and the way to report that is to watch a difference quotient refuse to converge.

 Where a point held by three strands may be. Three anchors, three strands of 130, 130, 120 mm, and a point tied to all three. A rigid link of those lengths would leave nothing to decide — three distance equations in two unknowns have no solution at all — and three strands leave a region, because each of them says no further than rather than exactly. The region is the intersection of the three discs; its area here is 2721.0 mm² and it has 3 corners. Inside it nothing is taut and the point has both its freedoms; on an arc one strand is taut and it has one; at a corner two are taut and it has none. positioned by solving, not by drawing. Members that pull

The strand that is slack

A rigid link removes a freedom wherever the mechanism stands. A strand removes one only where it is taut — so a point held by three of them has two freedoms in the middle of its region, one on an arc, none at a corner, and no single mobility count describes it at all.

 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. As built

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.

 A pin in a hole is a short link. Left: a pin of radius 0.86 in a hole of radius 1, so the clearance is 0.14. The pin's centre may sit anywhere within that of the hole's centre. Right: the same joint as it enters the kinematics — a binary link of fixed length 0.14 and free direction, with a revolute at each end. That is not an analogy. It is the same set of relative positions, so every count, every Jacobian and every solve on this site applies to it unchanged, and a four-bar with play at each pin is a mechanism with eight links and eight joints. As built

A clearance is a link

A pin in a hole is not a joint at a point. Its centre may sit anywhere within the difference of the two radii, so the two links it joins are connected by a body of fixed length and free direction — a binary link with a revolute at each end. That is not an analogy, and taking it literally makes a four-bar a mechanism with eight links, eight joints and five degrees of freedom.

 The roll centre through the travel — double wishbone. The roll centre is a construction: the instantaneous centre of the upright, joined to the contact patch, extended to the car's centreline. It is quoted as a height. Over 160 mm of travel it moves 54 mm — 52 mm to 106 mm — The number in a specification is the value at one position of a curve, and the curve is steeper than the thing it is a property of. Machines you have met

A roll centre is not a point

The roll centre is a construction on the instantaneous centre of the wheel's upright, and every step of it is exact. What it is not is a height: over eighty millimetres of bump and droop it moves 54 mm on a wishbone and 131 mm on a strut, and on the strut it goes below the road.

 A lower bound that happened to be tight. The site's own Gough platform at its home pose. The search starts Newton from a spread of guesses and reports what it lands on: 16 from 400, 16 from 1200, 16 from 4000. Tracking every one of the 1458 Bézout paths says there are 28 poses in the complex numbers and 16 of them are real. The search was right. Nothing available to the search could have said so. How many answers

The search that was right

This site has reported sixteen assemblies for its Gough platform and labelled the number a lower bound found by search, everywhere it appears. Tracking every path says there are twenty-eight poses and sixteen of them are real. The lower bound was tight. Nothing available to the search could have said so, and a second method that was supposed to settle it turns out to have the same defect one level up.

 Four pulleys, one idler, and one equation. A closed strand over four fixed pulleys and an idler carried on an arm. The idler sits outside the loop the other four make and bulges the strand out to reach it, so its wrap is signed the other way — -25.34° against the four positive ones — and the five still add to exactly one turn. The arm angle is not a free choice: the strand has a length, and matching it is one scalar equation, whatever the number of pulleys. At this position the run is 1018.046 mm and the idler takes up 0.4387 mm of strand for every millimetre it moves along the bisector of its two spans. positioned by solving, not by drawing. Members that pull

The tensioner is the unknown

A linkage closes when a vector comes back to where it started: two equations, two unknowns. A strand closes when a number does — its length — however many bodies it runs over. So a run with one free body is determined, a run with two is not, and the arm angle that takes up 1,020 mm of belt is the root of one scalar equation solved to 1·10⁻¹³ mm.

 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. One path to the tool

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.

 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. What a joint is

What a point sees

A group of displacements has no shape, so the only picture of one is what it does to something. Fix a point and the six lower pairs draw a line, a circle, a helix, a cylinder, a sphere and a plane — the six surfaces the pairs are made of. And two of the twelve sweep the same surface and are still different groups, which is the honest caption on the whole method.

 Where the curvature is standing still, at 66°. The cubic of stationary curvature: every point on it traces a path whose curvature has stopped changing at this instant, so its osculating circle fits for one order longer than an ordinary point's. It passes through the pole — twice, with a double point there — and through both moving pins, which is the plainest case there is, since a pin traces an exact circle and a constant curvature is certainly a stationary one. The expression whose zero set this is looks like a quartic and its fourth-degree terms cancel to 1.2e-14 of the third-degree ones. positioned by solving, not by drawing. The motion, not the mechanism

Where the curvature stands still

One derivative further on there is a second locus: the points whose path curvature has momentarily stopped changing. The expression whose zero set it is looks like a quartic, its fourth-degree terms cancel, and the cancellation is measured at ten to the minus fourteen rather than assumed.

 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. Contacts that only push

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.

 Where the platform stops being controllable, at 0°. Every point is a position of the platform's centre at a fixed orientation of 0°, shaded by how far it is from a direct singularity — pale is near. Unshaded means unreachable, which is the workspace boundary and the ordinary kind of singularity. The line is det A = 0, traced through the field rather than tested for: it runs through the middle of the reachable region in 102 segments, and on it the platform can move with all three actuators locked. 3312 of 6561 sampled positions are reachable. Several legs, one platform

The workspace is not a shape you choose

A serial arm's reach is roughly a sphere and can be quoted as a number. A parallel mechanism's is the intersection of three reachability conditions, changes with every degree of orientation, and has a surface of uncontrollable poses cutting through the middle of it. There is no formula. There is a map, and it has to be computed.

 How far the crank turns before the rocker does. With a clearance of 0.01 at each pin, the crank must be turned this far on reversal before the rocker moves at all. Through most of the cycle it is about 2.8°, and at its best 1.44°. At the two positions where the rocker reverses — marked — it is unbounded: the output velocity passes through zero there, so no amount of crank rotation moves the rocker out of its clearance band. The peak in a plot like this is therefore a property of the sampling and not of the mechanism; it reads 402° at 96 samples and grows without limit as the sampling is refined. The number worth quoting is the plateau. As built

How far the crank turns first

Reverse the input of a four-bar with a hundredth of clearance at each pin and the output does not move for about 2.8° of crank rotation. At the two positions where the rocker reverses it does not move at all, however far the crank is turned — the lost motion is unbounded there, and the peak in any plot of it is a property of the sampling rather than of the mechanism.

 What each of them can reach. Nine hundred control histories of four legs each, from the same starting configuration, with the resulting position plotted. The wheel's cloud is two-dimensional and fills the region; the trolley's is one-dimensional and lies exactly on its rail — the same number of coordinates, the same number of constraints, the same count of freedoms, and a reachable set of a different dimension. Nothing here is a matter of degree. Drawn wrongly

Not unreachable, only expensive

The sentence is false of every wheeled mechanism in this field and true of exactly one — the trolley bolted to a rail. A rolling constraint forbids a direction and reaches everywhere; the mistake is reading a statement about instants as a statement about intervals, and it is made in both directions.

 One member of the family is a pair of wheels. The same demand — one plus 0.4 sin φ — scaled by a constant, and for each the angle the driven wheel is out by after one turn of the input. It has to be zero, or the teeth do not line up with themselves and there is no wheel. Only the unscaled member closes; five per cent either way leaves the output 18° out, which is a third of a tooth on a thirty-tooth wheel and a mechanism that seizes on its second turn. Non-circular gearing is a search rather than a drawing for exactly this reason: the closure condition is one equation on a whole function, and almost no function satisfies it. The shape is the unknown

The demand that cannot be met

Ask for an output rate and the two pitch curves follow with no design step in between — so the interesting question is not how to draw them but which demands admit any pair of wheels at all. The answer is one equation on a whole function, it is about the demand's mean and nothing else, and five per cent of error leaves the output eighteen degrees out after a turn.

 Two circles each, so two to the power of the dyads. A dyad has two solutions, and a chain whose groups are all dyads is solved one dyad at a time, so the number of ways it can be assembled at a given input angle is 2 raised to the number of them. That is a prediction made from the graph about a count of configurations, and it is checked here against a count: the same chain solved from 240 random seeds, with the distinct converged configurations counted. The two agree in every row. Watt's did not at first — it came back at eight — and every one of the four extra answers had its ternary link mirrored: three distances fix a triangle only up to reflection, so the distance equations admit a part that has been turned inside out. A reflected link is a different part rather than a different pose, and the solver refuses those frames now. How many answers

Two to the power of the dyads

How many ways a mechanism can be assembled at a given input angle is a count of configurations, and it is predicted here by a graph: two circles per pair of links, so two to the power of the number of pairs. The prediction came back four for Watt's chain and the count came back eight, and the four extra had a link turned inside out.

 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. Out of the plane

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.

 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. One path to the tool

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.

 Three double points, and the one that is real is never visited. The coupler curve of a four-bar with ground 4, crank 1, coupler 3.5, rocker 3, coupler point at u = 0.45, v = 0.50, both ovals solved. The dashed circle is where the coupler's orientation can fail to be fixed by the point it carries; it passes through both fixed pivots and through the third pivot of the cognate construction, centre (2.000, 0.010), radius 2.0000. The curve's three finite double points are on it. The one that is real is isolated — a point of the curve no oval passes through, at (3.964, −0.368). The other two are a complex-conjugate pair and have no place in the plane. The paths points trace

A point the machine never reaches

Every coupler curve has three finite double points, and an odd number of them are real, so no coupler curve has none. On the standard crank-rocker the only real one is a point of the curve that neither assembly ever visits, that no contour plot can find, and that sits on the circle through the three pivots of Roberts's cognates.

 Six words, and the shortest of them. Every path a car that may not reverse and may not turn tighter than R can take between two placements is one of six shapes: three arcs, or two arcs with a straight between. All six are drawn; the shortest is LSR at 2.2557 R and the longest is RSR at 14.788 R. A numerical shooting solve that shares no line of code with the closed forms returns 2.2557 R, which agrees to 4e-16. Wheels, and where they may not go

A circle for the first millimetre

The shortest path for a car that may not reverse, from here to a point one millimetre to the side at the same heading, is 31.417 m for a five-metre turning radius. The shortest path to a point twenty metres to the side is 31.416 m. The cost of going sideways is not monotonic in how far sideways, and below a crossover at 2.956 R it is exactly 2πR + δ.

 How far a joint-space move bows off the line. Every joint runs from its start value to its end value at a constant rate — the simplest possible move, and the one that can never be refused, because every point along it is a set of joint values and every set of joint values is a pose. The tool does not travel in a straight line while it happens. It bows away by 402 mm over a move of 1223 mm, which is 32.9% of the distance travelled and enough to hit something that the straight line would have missed. One path to the tool

A straight line at constant speed

Run every joint from its start value to its end value and the tool bows 402 mm off the line between them. Insist on the line instead and the arm will follow it — until the path passes near a singularity, where the joint rates a metre a second demands grow as one over the distance, measured at an exponent of −1.010.

 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. What a joint is

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.

 The same error, twice, in two directions. A Sarrus linkage with one axis of one chain tilted off true, and the motion range that survives. Tilted within the plane the chain works in, it does not care: at 0.2 radians — eleven and a half degrees, which is not a manufacturing error by any standard — it still drives through a full turn. Tilted out of that plane, 0.001 radians stops it dead. Two hundred times the error, in the other direction, for no cost at all. What separates them is whether the perturbation lies in the screw system the mechanism leaves unconstrained — so "an overconstrained mechanism must be exact" is not merely crude, it is wrong about the case it is usually said of. As built

Fragility has a direction

Tilt one axis of a Sarrus linkage out of true by a thousandth of a radian and it stops dead. Tilt the same axis of the same mechanism by two hundred times as much, in the other direction, and it drives through a full turn with nothing measurably wrong. Three orders of magnitude between two errors of the same size — and the direction that matters is the one the reciprocal screw system names.

 Free in every direction, and it cannot get out. A disc of radius 1 among 3 point obstacles on a circle of radius 1.100. The shaded discs are the obstacles grown by the part's own radius, which is what the part's centre may not enter — the configuration space, and for a round part it is the plane itself. The part is caged when those grown discs overlap enough to close a ring around it, which happens below R = 1/sin(π/3) = 1.154701, and here it does. At every configuration inside the cage the part is free. The three normals all point at its centre, the rank of its rows is two, the escape cone is a whole line, and none of that has anything to do with whether it can leave. A hold is a statement about velocities at one configuration; a cage is a statement about where a finite motion can go, and the second does not follow from the first in either direction. positioned by solving, not by drawing. Contacts that only push

Free at every instant and going nowhere

Three points on a circle of 1.1 radii around a unit disc leave it free in every direction at every configuration — rank two, margin nought, the whole plane of centres shaded — and it cannot get out. The threshold is 1/sin(π/n), which is 1.154701 for three, and a flood fill of the free space agrees with the formula at every radius sampled.

 The slider turns round at the dead centre. The slider's position against the crank angle, through the dead centre. The curve has a maximum there — that is what a dead centre is — so the slider retreats on both sides of it, and the latch works because a mechanism resting past the top of this curve has to be pushed back up it before anything can move. Set 4° past, the slider is 71 µm below the peak, and because the curve is quadratic there, doubling the setting quadruples the depth. Machines you have met

Locked on purpose

A toggle clamp, a landing-gear downlock and the catch on a folding table are all the same mechanism parked a few degrees past its dead centre, where the slider's motion is second order in the crank's. Seventy-one microns of slider travel undoes a latch set four degrees over — and setting it eight degrees over does not double that, it quadruples it.

 One angle, read twice. Draw against the angle the locking face is tilted by. It is not a separate design quantity from recoil: the pallet's moment per unit of wheel torque is minus the rate at which the wheel is driven backwards, which is virtual work and holds to nine figures. So an escapement cannot be made to hold its own lock without also being made to push its train back — and at δ = 0 the arc has neither, which is a lock that any disturbance opens. The flat face starts above zero because the tooth rests below the corner, where a straight face is no longer tangent to anything. Motion that stops

The angle that holds the lock

A locking face cut exactly concentric with the pallet arbor has no tendency to hold itself: the tooth's push aims straight at the pivot and its moment is zero, so the smallest disturbance opens the lock. Tilt the face and the moment becomes ρ sin δ — and the same tilt, by virtual work, is exactly the rate at which the wheel is driven backwards. Draw and recoil are one angle read twice.

 The error between the precision points. chebyshev: worst 0.2232°, RMS 0.1475°; least-squares: worst 0.3900°, RMS 0.1203°; minimax: worst 0.2105°, RMS 0.1463°. The error is zero at each precision point by construction and nowhere else. minimax has the smallest maximum here, and which curve is best depends entirely on which measure is asked for. The problem backwards

The linkage that is only nearly right

Stop demanding that a linkage pass exactly through three points, and ask instead that it be close everywhere. Three linkages result, none of them passing exactly through anything, and each is the best by a different measure — the least-squares fit beats the interpolant on average error and loses to it on the worst case. An optimiser gives exactly what it was asked for and nothing else.

 What a platform's own sensors settle. One three-legged platform, measured at 75 poses, with three different sets of sensors on it. Reading every joint angle determines 12 of the 15 numbers that describe the machine and leaves three undetermined, and one of the three is the scaling of the whole machine — every reading is dimensionless, so multiplying every length by a constant changes nothing any sensor sees. Laying a rule across two base pivots once recovers exactly one more, and the one it recovers is the size. Legs that report their own extension need no rule: a reading with a length in it breaks the scaling direction outright, and 7 of 9 come back. What none of the three determines is where the frame's origin sits, which is a convention rather than a defect. Several legs, one platform

A platform that measures itself

A three-legged platform with every joint read can be calibrated from its own sensors with no instrument in the room. Left to itself it shrinks the machine to a fiftieth of a per cent of its size — and reports a residual five orders smaller than the right answer's for doing it.

 A parallelogram's sextic is a circle and a quartic. The parallelogram with ground 4, crank 1.5, coupler 4, rocker 1.5, coupler point at u = 0.45, v = 0.50. Its eliminated sextic divided by the circle of radius 1.5000 about (1.800, 2.000) leaves a remainder of 6.3 × 10⁻¹⁶ of its largest coefficient, and the quotient is a quartic whose leading form is exactly a multiple of (x² + y²)². Of the configurations solved across a full turn on both assemblies, 720 are on the circle — the machine as a parallelogram, coupler parallel to the ground — and 720 on the quartic, the machine crossed. The two factors meet at four finite points: 2 are change points, marked solid, where one configuration belongs to both; the other 2 are places the two drawings merely cross. The paths points trace

A sextic that comes apart

A parallelogram chain's coupler curve is not one curve. Its sextic divides exactly by a circle, leaving a quartic, and each factor is one of the two things the machine can do. The division leaves rounding and nothing else, a coupler one millionth too long leaves a remainder a million times larger than rounding, and the two factors meet at the two places where the machine has to choose.

 The four eight-link chains a compass cannot position. Every one of the twenty ways of choosing a frame and a driven link, on each of these four chains, leaves a group of four links or more that has to be solved as one system. There is no order in which they come apart two at a time, so there is no ruler-and-compass construction for any of them and no closed form for their positions. They are numbers 1, 3, 4, 10 of the sixteen, and they do not share an assortment: 4×2 + 4×3 and 5×2 + 2×3 + 1×4 both appear. Three of the four are among the most symmetric chains in the census — automorphism groups of 16, 8, 8 against a median of three across the sixteen — which is the direction one would guess, since a symmetric chain has few genuinely different places to attach a driven link. The fourth has an automorphism group of 2, so symmetry is a tendency here and not the reason. The chain before the lengths

Four that a compass cannot reach

Twelve of the sixteen eight-link chains can be positioned two links at a time, from at least one choice of frame and input. Four cannot be positioned that way from any of their twenty choices — and at ten links ninety of the two hundred and thirty are in the same position.

 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. What can move

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.

 One linkage, two curves. The same bars, the same lengths, the same driving angle — assembled two ways. One trace is where the polynomial vanishes and the other is not: the worst value of xy − 0.5 along the second is 8.7e-1, against 4.7e-14 along the first. Every position on both was solved to 9.5e-14. Nothing about the second linkage is defective; one of its parallelograms is a crossed one, so a direction is being carried wrongly, and the machine is faithfully computing a different function. The curve as an equation

The proof drew more than the curve

Sixteen ways to assemble one linkage. Eight of them close. Four put the tracing point on the curve and four put it somewhere else — at a closure residual of 9.6 × 10⁻¹⁵, which is the same floor the right ones reach. No tolerance on the closure could ever have told them apart.

 A corner in a curve, and the reason for it. This coupler point traces a curve with a cusp — a corner, where the curve stops and reverses rather than turning. The reason is that a cusp happens where the tracing point is momentarily still, and the only point of a moving plane that is momentarily still is the pole. So the cusps of a coupler curve are the instants when the pole passes through the tracing point: they are the crossings of the tracing point by the moving centrode, which is a statement about two curves in the coupler's own plane with the fixed plane not involved at all. This point was chosen by taking the pole's body coordinate at one instant, so it is on the moving centrode by construction; at the corner its speed is 9.9e-15 against 2.87 a fifth of a radian later. positioned by solving, not by drawing. The motion, not the mechanism

Where a curve has a corner

Some coupler curves have corners in them — points where the curve stops, turns round and comes back. A corner happens where the tracing point is momentarily still, the only point of a moving plane that is momentarily still is the pole, and so the corners of a coupler curve are decided entirely inside the coupler's own plane.

 Every pin position, and what it costs — 3 mm between the doors. One cell per candidate pin position: the door is swung through 95° from each and the worst clearance recorded. Light cells clear; dark cells mean the door would pass through the partition or through the door beside it, by up to 18 mm. The horizontal line is the door's own front face. With the doors touching, the boundary sits exactly on it — because the front corner's sideways rate is the pin's distance behind it, and that is zero only there. Each millimetre of gap buys a few millimetres of depth, and at 3 mm the deepest pin that still clears is 10 mm behind the face. Machines you have met

Where a hinge pin can go

A cabinet door's front corner moves sideways as it opens at a rate equal to how far the pin sits behind it, so with the doors touching, no pin behind the door's face can open one without going through the next. Three millimetres of gap buys nine and a half millimetres of depth, and that is the whole reason a concealed hinge has four bars instead of a pin.

 The error between the precision points. chebyshev: worst 0.2232°, RMS 0.1475°; least-squares: worst 0.3900°, RMS 0.1203°. The error is zero at each precision point by construction and nowhere else. chebyshev has the smallest maximum here, and which curve is best depends entirely on which measure is asked for. The problem backwards

Where an optimiser starts

An approximate synthesis is a local search on an objective that is non-convex, disconnected and not everywhere defined, so the answer depends on where the search began. Nothing in the optimisation supplies that. What supplies it is the exact constructions the field spent four rungs on, and an atlas of coupler curves — which is why a method superseded by computers is still the thing that feeds them.

 A joint with no way out in the plane it is drawn in. A dovetail. The tail is wider at its far end than at the mouth it went in through, so every direction out of the mouth is blocked by a slanted face and every direction further in is blocked by the floor. In the plane of this drawing the joint cannot be taken apart at all, and the direction it does come apart in is the one the drawing does not show. The moving part touches the rest at 3 faces, each contributing one inequality on the direction it may be translated in — the direction must not have a negative component along that face's inward normal — and the set of directions that satisfy all of them is a cone in two dimensions rather than three, because a translation has no moment term. That is why a removal cone can be drawn as an angle where a mobility cone cannot. Here it is empty: every direction is refused by one face or another, so the part cannot be taken out by any translation and cannot have been put in by one either. That is a statement about the assembly and not about the part, and the direction the joint does come apart in is perpendicular to this drawing. positioned by solving, not by drawing. Contacts that only push

Which way it comes out

Drop the rotation from the inequalities and the cone lives in two dimensions rather than three, so it can be drawn as an angle: a block in a vee has ninety degrees of directions out, a key in a slot has exactly one and no arc around it, and a dovetail has none at all. Three answers, and each of them is a different kind of joint.

 20,000 four-bars, and not one with three circuits. 20,000 four-bars with the ground at one and the other three lengths drawn from 0.05 to 3, each counted exactly. crank-rocker: 3,031, of which 3,031 have two circuits, 0 have one and 0 cannot be assembled; double crank: 3,086, of which 3,086 have two circuits, 0 have one and 0 cannot be assembled; Grashof double rocker: 1,497, of which 1,497 have two circuits, 0 have one and 0 cannot be assembled; triple rocker (non-Grashof): 12,386, of which 0 have two circuits, 9,528 have one and 2,858 cannot be assembled. The most circuits any linkage has is 2. How many answers

Never three circuits

A four-bar has one circuit or two, and twenty thousand random four-bars counted exactly contain no exception. The reason is a count of four points where the two assemblies merge, which makes the configuration curve a curve of genus one, and Harnack's theorem allows a real curve of genus one two pieces and no more. A six-bar's curve has genus five or seven, and its circuits go to four and six.

 Where the assembly count changes, and a loop round a cusp. With the first motor held at 216°, each point of the square is a setting of the second and third motors, shaded by how many assemblies the platform has there, sampled on a 41 by 41 grid: 2 assemblies at 1,261, 4 assemblies at 347, 6 assemblies at 73. Each edge between two shades is a curve of direct singularities, where two assemblies merge and vanish, and two such edges meet in a sharp point. The 2 marked points are the cusps in this window, each found as a triple root of the closure equation. The dashed circle, of radius 8°, is the loop the motors are driven round; it encloses 1 cusp, the one at (101.35°, 185.32°), and starts at the open marker. Several legs, one platform

Round a cusp into another assembly

A parallel platform's assembly mode was supposed to change only through a direct singularity. Driven round a small loop of motor angles that encloses a cusp of the singular curve, the standard three-legged platform leaves one assembly and arrives in another, turned 52° from where it started, and at no point on the way is it nearer than 0.0716 to singular.

 Neither one comes out, and the two of them do. Two congruent Z-shaped parts in a tray that is open at the top. Each has a step that lies over the other's, so part A's four contacts with part B have normals at all four points of the compass and leave it no free direction at all — and the same is true of B, for the same reason and by symmetry. The blocking is mutual and there is no order in which the two can be taken out one at a time. Together they have 6 contacts, all of them with the tray, and exactly one direction out: straight up. So the removal cone of a set of parts is not built from the removal cones of its members, and which part comes out first is a question with no answer here. positioned by solving, not by drawing. Contacts that only push

Neither part comes out first

Two congruent Z-shaped pieces in a tray open at the top. Each has four contacts with the other, with normals at all four points of the compass, so each alone is blocked in every direction there is — and the pair lifts straight out. The removal cone of a set of parts is not built from the removal cones of its members, and *which part comes out first* is a question with no answer.

 One angle, read twice. Draw against the angle the locking face is tilted by. It is not a separate design quantity from recoil: the pallet's moment per unit of wheel torque is minus the rate at which the wheel is driven backwards, which is virtual work and holds to nine figures. So an escapement cannot be made to hold its own lock without also being made to push its train back — and at δ = 0 the arc has neither, which is a lock that any disturbance opens. The flat face starts above zero because the tooth rests below the corner, where a straight face is no longer tangent to anything. Motion that stops

One test, three mechanisms

Whether a pawl holds, whether an escapement's lock draws itself deeper, and how much a four-bar's coupler can do for its rocker are the same question asked three times: on which side of a pivot does a contact normal pass? All three are one cross product, none evaluates a force, and the three answers are used for completely different things.

 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. 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.

 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. Out of the plane

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.

 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. Many of one thing

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.

 How much of Bennett's turn survives a bar being wrong. The same four bars and the same four twists, with one bar's length changed by the amount on the left and nothing else touched. The bar shows the fraction of 48 sampled positions of the first joint at which the loop closes to within 10⁻⁹. Two parts in a thousand already costs most of the travel. This is what it means for a mechanism to work only on a condition rather than approximately near one — and it is why Bennett's linkage was a curiosity for eighty years before anyone could machine to it. Out of the plane

Bennett's condition is a ratio

A spatial loop's parameters are lengths and angles together, so a scaling touches only half of them. Bennett's condition — a over sine alpha equals b over sine beta — is a relation between the two halves, and what it demands of a machine is a relation between its lengths and its twists rather than a property of either.

 Where a calibration measures. The four-bar drawn at the 6 poses a selection chose, one over another, with the tracing point marked at each. The largest gap between consecutive chosen poses is 150°. They are spread because rows of the identification Jacobian at nearby poses are nearly the same row, which is a statement about the matrix and reads here as a picture of a machine in visibly different configurations. Numbers that were measured

The pose the machine cannot reach

A measurement plan is drawn against the nominal machine and executed on the real one, and the real one does not go quite where the drawing says. A pose that falls outside the travel returns no reading at all — which is not an error, not a failure of the instrument, and not nothing: it is a measurement of the limit position.

 Two motors, two circles, two places for the hand. A planar five-bar with its motors 1.0 apart, arms 1 and distal links 1.25, at motor angles 100° and 60°. Each distal link holds the hand on a circle of radius 1.25 about its elbow, and the elbows are 1.678 apart, less than the 2.50 at which the circles would only touch, so they meet twice. The hand drawn solid is at (0.229, 1.850) with det A 0.995; the other assembly, dashed, is at (0.098, 0.001) with det A -0.995 — the same size and the opposite sign. Several legs, one platform

The smallest parallel robot

Two motors, two arms, and two links meeting at a hand: a planar five-bar is the smallest parallel robot there is. Its forward problem is two circles, so the hand has two places to be, and each is named by the sign of one determinant. That is the whole reason it cannot do what the three-legged platform does, and cannot change assembly without passing through the one configuration where the two meet.

 Eight four-bars, one from each region the three signed sums cut. One linkage from each of the eight sign patterns of T₁ = g + c − a − b, T₂ = g + b − a − c and T₃ = b + c − a − g, each solved at a crank angle in the middle of its range. The thick arc round the left pivot is where the input's pin can go and the arc round the right pivot is where the output's can: crank-rocker + + +, input 100% of a turn and output 22%; double crank − − +, input 100% of a turn and output 100%; rocker-crank − + −, input 22% of a turn and output 100%; double rocker + − −, input 23% of a turn and output 21%; 0–π rocker + + −, input 76% of a turn and output 72%; π–π rocker + − +, input 84% of a turn and output 51%; π–0 rocker − + +, input 84% of a turn and output 85%; 0–0 rocker − − −, input 67% of a turn and output 91%. Grashof's condition names four of these and calls the other four one thing; the arcs show that the four triple rockers differ in which way along the ground line each rocker swings through. Linkages

Eight kinds of four-bar

Grashof's condition gives a four-bar one of four names and calls every linkage that fails it a triple rocker. Three signed sums of the lengths give eight, and a census of four thousand random linkages finds every one moving exactly as its signs say — because the planes where those sums vanish are the only places a four-bar's motion can change its kind.

 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. What a joint is

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.

 11 assortments are arithmetically possible and 7 contain a mechanism. The 10-link census organised the way every published table organises it: by how many links carry two pins, three, four and more. The assortments themselves are a small piece of arithmetic — the degrees must sum to twice the pin count and none may be below two — and it admits 11 of them. 4 contain no chain at all. Each of those 4 needs a link carrying six, seven or eight pins, and a link with that many pins in a chain this small always drags a structure in with it: the graphs exist, they satisfy Grübler exactly, and every one of them has a rigid subchain. That is a result the arithmetic cannot reach, because the arithmetic never looks at where a pin goes. The chain before the lengths

Eleven assortments and four that are empty

How many links carry two pins, how many carry three, how many carry four: two lines of arithmetic admit eleven answers at ten links. Seventy-eight graphs have degrees the last four of them describe, every one of those graphs satisfies Grübler's rule exactly, and not one of them is a mechanism.

 Free space, in pieces. The driving angle round the circle, with the arcs at which the machine is both assembled and clear drawn heavy. One stud in the way takes a bite out of the turn and leaves 1 arc: the crank can still reach every remaining angle by going the other way. Two studs leave 2, covering 74% of the turn — and every configuration in both arcs is a perfectly good solution of the same constraint equations, on the same assembly branch, at the same mobility. Nothing a solver computes distinguishes an angle in one arc from an angle in the other; what separates them is that the machine cannot be driven from one to the other. Links with a width

Free space comes in pieces

Every arc on this site has ended at a configuration the mechanism cannot reach. Put two studs in a four-bar's way and its drive falls into two arcs whose ends are configurations it reaches perfectly well and cannot occupy — and no quantity the solver computes tells one arc from the other.

 In space the arithmetic allows almost nothing. A body in space has six freedoms and a revolute joint takes five, so a mobility of one needs (6n − 7)/5 joints — and that is an integer only when the link count leaves a remainder of two on division by five. The whole table is this: 7, 12, 17, 22 links, and nothing else. At seven links the degrees must sum to fourteen across seven links with none below two, so every link is binary and the graph is a single seven-cycle: there is exactly one spatial chain, and it is a loop. That is the census explanation for something the spatial field has lived with since it was written — every spatial mechanism on this site is one closed loop — and it had never been stated as a count. The next admissible size is twelve links and thirteen joints, where two assortments are arithmetically possible, 157 candidates give 33 graphs, and 5 of them are chains — every one with ten binary links and two ternary, so the assortment with a quaternary link is empty exactly as four of the planar ones are. Out of the plane

In space there is one chain

A body in space has six freedoms and a revolute joint takes five, so a mobility of one needs (6n−7)/5 joints — an integer only when the link count leaves a remainder of two on division by five. At seven links every link is binary, the graph is a single seven-cycle, and there is exactly one spatial chain.

 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. One path to the tool

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.

 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. Drawn wrongly

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.

 Two Burmester points, or none, depending where the crank is. A Burmester point's path stays on one circle to fifth order. A planar motion has at most four of them; two of this mechanism's are always its own moving pins, whose paths are exact circles and satisfy every order at once. The other two are real for 67 per cent of the turn and complex for the rest, and the count changes without anything about the mechanism changing. The window matters and is stated: points beyond a hundred coupler lengths from the pole are not counted, and widening the window from six to four hundred moves the count of positions-with-two from 191 to 245 out of 360. A silent cap here would read as an absence. The motion, not the mechanism

The circle a point stays on longest

One condition further on are the points whose path holds a circle to fifth order. A planar motion has at most four; two of them are always the mechanism's own pins, and the other two are real for two thirds of a turn and complex for the rest — a count that changes while nothing about the mechanism does.

 What is left of a 12-tooth flank. The flank the rack generated on a 12-tooth wheel, with every point tested against the cutter at every other position of the cut. The pale dots survive; the dark ones are inside the cutter at some later instant and are not on the finished tooth — 39 of 203 of them, the deepest by 5.19e-2 mm. Undercutting is not a shape, it is a removal: the same corner that leaves the fillet comes back for flank the straight edge had already generated, and what is lost is the part of the involute nearest the base circle, which is exactly the part the mating wheel's tip needs. Drag to change the tooth count. positioned by solving, not by drawing. The shape is the unknown

The cutter takes back the tooth

Undercutting is usually explained as a shape: a tooth with a waist in it. It is better understood as an event — the corner that leaves the fillet comes back through flank the straight edge has already generated — and seen that way the threshold at seventeen teeth is a comparison of two measured points that passes through zero.

 Everything one planetary can do, and the gap in the middle. A single epicyclic has three shafts, so there are six ways of choosing which is held, which is driven and which comes out. Each gives a band of reductions as the tooth counts run over every design that can be cut, assembled with three planets and kept clear of undercutting. The bands above 1 are drawn; between them is a gap running from 1.6304 to 2.5862 that no single planetary reaches in any configuration — and a reduction of exactly 2, which is the most ordinary thing anybody asks a gearbox for, is inside it. The gap's width as a factor is exactly the smallest achievable ring-over-sun ratio, 1.5862, which is a statement about how small a planet may be and how large a sun may be. More than one input

The reductions a planetary cannot give

One epicyclic offers six ratios, and the formula for each of them suggests the whole positive line is available. Sweep every design that can actually be cut and assembled and the reachable set has a hole in it running from 1.630 to 2.586 — the width of which has a closed form — and a reduction of exactly 2, the most ordinary thing anybody asks a gearbox for, sits in the middle of it.

 Two taut paths, and no way between them. A strand from one point to another past a peg. Each side gives a path that is taut — straight where it can be, on the surface where it must be, leaving at a right angle — and each is the shortest path on its own side: 219.165 mm on one and 200.643 mm on the other, against 200 mm of open air the strand cannot use. Neither can turn into the other without passing through the peg, which is a different kind of non-uniqueness from the assembly branches of a linkage: those are separate roots of one equation, and these are separate classes of one minimisation. positioned by solving, not by drawing. Members that pull

The taut path has more than one answer

A strand from one point to another past a peg has two taut paths — 200.643 mm on one side and 219.165 on the other, against 200 mm of open air it cannot use. Both are shortest. Neither can become the other without passing through the peg, and no computation recovers which one was threaded.

 A period, in the space the mechanism lives in. One whole period of an escapement, plotted as wheel angle against pallet angle. Give a four-bar its crank angle and its coupler is somewhere definite; give this its pallet angle and the wheel may be in any of 3 places, because what settles it is which face of which pallet a tooth is against. The path crosses itself and no amount of solving removes the crossing. The vertical jumps are the drops, drawn at the pallet angle of the release because during a drop nothing is touching anything and where the pallet is by then is a question about torque. Over the period the wheel advances 12.000000°, which is one tooth exactly. Motion that stops

Where the input stops deciding

Give a four-bar its crank angle and its coupler is somewhere definite. Give an escapement its pallet angle and the wheel may be in any of three places, because what settles it is which face of which pallet a tooth is against. The path crosses itself and no amount of solving removes the crossing — the state of these mechanisms has a discrete part, and that is what makes them a different kind of object.

 What a singularity does, and what it does not do. The reflector driven straight through the configuration at which its two placements merge — here θ = 0.800, where the rhombus flattens onto its own mirror. Two numbers are plotted. The closure residual is how well the bars are satisfied, and it does not move: 8.3e-14 on both sides. The departure is how far the output is from the angle the gadget is supposed to produce, and it goes from the floor to order one at 0.800. Nothing breaks. The gadget goes on being a perfectly good linkage and stops being the function it was built to be. The curve as an equation

Where the machine stops being the function

Drive a reflector through the angle at which its rhombus flattens and it comes out computing something else. Nothing breaks: every bar is the length it was, the closure residual stays at 8 × 10⁻¹⁴, and the machine goes on turning. That is why every compiled machine in this field works over an arc and not a turn — the quintic's over a tenth of a radian.

 Three legs and four, at 0°. The same slice of positions at a platform angle of 0°, shaded by how well the platform is held — pale is near singular. Left, three legs: 3,312 reachable samples and a singular curve through them in 102 segments. Right, the same three legs and a fourth: 3,198 reachable, because the fourth leg must reach too, and no curve. What is left of the singular set in this slice is 1 isolated point, at (-1.449, -0.811), where all four lines meet. Positions held above 0.1 go from 2,989 to 3,178, and at no sampled position is the four-legged platform held less well than the three-legged one. Several legs, one platform

What a fourth leg buys

Three leg lines fail to hold a platform when they meet at a point, which is one condition, so in every slice of the workspace the failures form a curve. Four lines fail only when all four meet at a point, which is two conditions, so the curve becomes isolated points. The fourth leg buys that and more, and it costs a machine that can no longer be assembled from any four motor angles.

 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. Out of the plane

A name for each overconstraint

The spatial field separated subgroup overconstraint from paradoxical by measuring how far a mechanism's screw system turns: 2 × 10⁻⁶ degrees against 89. That is a verdict without a name. Closing the logarithms of the reached displacements under the bracket gives the same verdict and says which group — planar, spherical, a translation — and for Bennett's linkage it says six.

 One piece, and still not reachable. The state of a linear ratchet — a cable tie — is how far in it is pulled, and its free space is the whole interval: every state is connected to every other, with no barrier anywhere. What it does not have is a way back. From the marked state the reachable set is everything forward and only as far back as the tooth the pawl has already dropped into, so 52.1% of ordered pairs are reachable and 4.2% are reachable both ways. Connectivity is symmetric; reachability is an order, and a component count answers the first question and cannot be asked the second. Motion that stops

One piece, and still not reachable

The site decides whether two configurations can be joined by asking whether they are in the same connected component, and that relation is symmetric because a path run backwards is a path. A one-way mechanism breaks the symmetry and leaves the connectivity alone: its free space is a single interval with no barrier anywhere in it, and about half of the ordered pairs of states cannot be joined by any admissible motion.

 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. Drawn wrongly

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.

 Forwards it settles, backwards it runs away. A trailer starting a hundredth of a radian out of line, with the steering held straight. Driving forwards the angle decays as e^(−s/d); reversing, the same equation runs the other way and it doubles every 4.16 m. Nothing about forces is involved and nothing about the driver: it is the sign of one exponent, and the length scale is the trailer's own length. The curve flattens at the top because the sine that generates it saturates — the runaway is exponential only while the angle is small. Wheels, and where they may not go

The angle that doubles

A trailer a hundredth of a radian out of line decays back into line as e^(−s/d) driving forwards and grows as e^(+s/d) reversing — doubling every 4.16 m for a six-metre trailer. And a jackknifed rig is not a rig that has lost anything: its growth vector is 2·3·4·5 at a hitch angle of zero, of ninety degrees and of a hundred and eighty.

 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. Many of one thing

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.

 What a tensioner can take up, and where it stops being one. The run's length as the arm swings, over the whole interval in which the idler is actually touching the strand: 4.123 to 4.774 radians, and 1013.37 to 1030.99 mm — a range of 17.62 mm on a strand of a metre. Outside that interval the geometry still returns tangent lines and a length; what it returns is not a strand, because the path it describes cuts through the pulleys. The curve is flat at the left-hand end and steep at the right, which is the whole of a tensioner's design problem: its authority is proportional to the sine of half its own wrap, so an idler set near the edge of contact travels a long way and takes up nothing. Members that pull

Where a strand stops touching

A body joining a run costs length as the square of how far it intrudes — exponent 2.0000, measured over four decades — so at the moment contact begins the strand's length is stationary. That is why a tensioner set at the edge of its own contact takes up 0.000245 mm of belt per millimetre it travels.

 The ratio survives; the continuity does not. An involute pair holds its ratio at any centre distance, and that is not the same as working at any centre distance. The contact ratio — the length of the contact path divided by the base pitch, which counts how many pairs of teeth are engaged at once — starts at 1.647 for this 24 : 36 pair and falls as the shafts move apart, because the useful part of the line of action is bounded by the two tip circles. It reaches one at 2.82 mm, and below one a pair of teeth lets go before the next has picked up: the drive stops being continuous and becomes a series of arrivals. That is the real limit on the involute's indifference, and it is a limit on the teeth rather than on the tooth form. The shape is the unknown

Where two shapes stop touching

A conjugate pair is exact at every instant it has a contact. It does not have one for ever: a profile is an arc rather than a curve, and both ends of that arc are somebody's decision — which is why the useful question about a pair of shapes is not whether they mesh but for how long.

 A joint with no way out in the plane it is drawn in. A dovetail. The tail is wider at its far end than at the mouth it went in through, so every direction out of the mouth is blocked by a slanted face and every direction further in is blocked by the floor. In the plane of this drawing the joint cannot be taken apart at all, and the direction it does come apart in is the one the drawing does not show. The moving part touches the rest at 3 faces, each contributing one inequality on the direction it may be translated in — the direction must not have a negative component along that face's inward normal — and the set of directions that satisfy all of them is a cone in two dimensions rather than three, because a translation has no moment term. That is why a removal cone can be drawn as an angle where a mobility cone cannot. Here it is empty: every direction is refused by one face or another, so the part cannot be taken out by any translation and cannot have been put in by one either. That is a statement about the assembly and not about the part, and the direction the joint does come apart in is perpendicular to this drawing. positioned by solving, not by drawing. Contacts that only push

A cone has no size

What a set of contacts permits is a cone of twists, and a cone is closed under positive scaling by definition — so nothing about it changes when the part it holds is made bigger. Except that a twist is a screw, a screw has a pitch, and a pitch is a length.

 A rolling wheel, where it was driven to. The mechanism at a configuration nothing wrote down: it was reached by integrating permitted velocities from the start of the trail, and there is no equation here whose root it is. The barred line at each wheel is the direction that wheel forbids — the subject of the whole field, and the one thing a photograph of a car cannot show. The constraint residual along the drawn history is 0.0e+0. Wheels, and where they may not go

A wheel that cannot report its radius

Rolling relates a wheel's turning to a vehicle's travelling, and the relation has a length in it. So a rolling constraint is the one place on this site where an angle measurement does carry a size — and the size it carries is the one thing a vehicle's own odometry can never separate from its wheelbase.

 Every position loses its hold at -150° and 30°. The smallest singular value of the six leg lines of the Gough–Stewart platform, held level, as it is turned through a whole revolution at 4 positions: (0, 0, 2.4), (0.4, -0.3, 2.4), (0, 0, 3.2), (-0.5, 0.2, 1.8). At -150° and 30° all 4 curves reach zero together — the largest of them is 9.7e-9 — and 1° either side the least is 1.28e-3. Nowhere else does any of them touch zero. Several legs, one platform

A yaw that is singular everywhere

Turn the standard hexapod platform 30° about the vertical, hold it level, and it is singular: not at one pose, but at every position it can be put in, with one screw motion that none of its six locked legs can resist. The angle is not a property of the dimensions. It comes out of one line of trigonometry that no spread of the anchor points can change.

 Four offset slider-cranks, one from each region the limit leaves. One slider-crank from each region of T₂ = b − a + e and T₃ = b − a − e, with the slide the dashed vertical line a distance e from the crank's pivot and the ground line dashed across. The thick arc round the pivot is where the crank's pin can go and the thick stretch of the slide is where the slider can: crank-rocker + +, crank 1, rod 3.5, offset 0.8, the crank reaching 100% of a turn and the slider between 2.37 and 4.43 on each side of it; double rocker − −, crank 3.5, rod 1.2, offset 0.6, the crank reaching 22% of a turn and the slider between 2.22 and 4.66 on each side of it; 0–π rocker + −, crank 2, rod 2.5, offset 1.6, the crank reaching 65% of a turn and the slider running from −4.21 to 4.21 through the ground line; π–π rocker − +, crank 2, rod 2.5, offset −1.6, the crank reaching 65% of a turn and the slider running from −4.21 to 4.21 through the ground line. These are the four of the eight four-bar kinds in which T₁ is positive. Linkages

Four kinds of slider-crank

An offset slider-crank is a four-bar whose output bar and ground have grown without bound, and in that limit the three signed sums that sort four-bars into eight kinds lose one of their signs. Four kinds survive. A census of four thousand finds every slider-crank moving as its region predicts, the textbook condition for a full crank turn turns out to be one region exactly, and each of the four kinds that vanish is carried, at a length that can be written down, into the survivor that shares its other two signs.

 How many times each curve passes through the circular points. For each body of five machines, the degree of the curve a point on it draws, from a random complex line, and the number of finite points where that curve meets a line through the circular point I, x + iy = c, and one through J, x − iy = c. The difference is how many times the curve passes through that circular point. Four-bar, crank: degree 2, 1 and 1 finite, circularity 1; four-bar, coupler: degree 6, 3 and 3 finite, circularity 3; four-bar, rocker: degree 2, 1 and 1 finite, circularity 1; Watt six-bar, coupler: degree 6, 3 and 3 finite, circularity 3; Watt six-bar, second coupler: degree 6, 3 and 3 finite, circularity 3; Watt six-bar, output: degree 2, 1 and 1 finite, circularity 1; Stephenson six-bar, coupler: degree 6, 3 and 3 finite, circularity 3; Stephenson six-bar, arm: degree 18, 9 and 9 finite, circularity 9; Stephenson six-bar, output: degree 2, 1 and 1 finite, circularity 1; slider-crank, crank: degree 2, 1 and 1 finite, circularity 1; slider-crank, coupler: degree 4, 3 and 3 finite, circularity 1; elliptic trammel, rod: degree 2, 2 and 2 finite, circularity 0. Every body of the three machines built from pins alone has circularity exactly half its degree; the slider-crank's coupler curve and the trammel's ellipse do not. How many answers

Nine times through each circular point

A line through a circular point meets a curve at that point once for every time the curve passes through it, so counting its finite meetings measures how often, with no equation. The Stephenson six-bar's arm curve, degree eighteen, meets such lines in nine finite points: it passes nine times through each circular point. Every curve drawn by a body of a machine built from pins alone does this at exactly half its degree; a slide breaks it.

 Two permitted motions, and a composite that lifts off. A 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. What a joint is

A higher pair has no group

A disc resting on a straight edge may slide along it and may turn about its own centre. Both keep the contact exactly. Do one and then the other and the contact lifts off the edge by |t sin φ| — up to 1.2 radii over an ordinary range — so the two freedoms are real and the pair of them is not closed. The count is still right and there is nothing for it to be the dimension of.

 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. Drawn wrongly

Six things a chain is not

A count read as a verdict, a rank trusted where it is blind, a fingerprint used as a proof, a list of five taken for a complete one, a solver treated as a convenience, and a census read as a catalogue of machines. Six claims, each with the number that kills it.

 A parallelogram at its flat position, exact and built slightly wrong. Left, a parallelogram with ground 3, cranks 2 and coupler 3 lying flat, where its two assemblies meet; faintly, the two ways it can go on, drawn at 30°. Middle, the same linkage with its coupler 0.05 too long, drawn at the closest it can come to the flat position: 7.49° away on either side, so the input's circle is thick where the input can go and red across the 15.0° it can never enter. Right, the input crank 0.05 too short, at the flat position: its two assemblies put the output crank 43.76° apart, and they do not meet at any input angle. What can move

A parallelogram a micron wrong

A parallelogram linkage sits exactly where two kinds of four-bar meet, so a parallelogram that has actually been made is always one of four other machines. Make one bar a micron wrong on a 300 mm frame and the input stops a tenth of a degree short of lying flat, or the output turns round there with an acceleration that grows as one over the square root of the error. A third bar turns the square root back into a misfit of one micron.

 Two identical rotors in mesh, and the one place they touch. Two rotors, each with 2 cycloidal lobes on a pitch circle of radius 50, on centres 100 apart and turning at the same speed in opposite senses, drawn with the first turned −20°. Each rotor's roots were computed from its tips, and the mate is the same rotor turned. At this position they touch at one point, on the first rotor's tip, and the common normal there misses the pitch point by 2.2 × 10⁻¹⁰. The contact sits on the describing circle tangent to both pitch circles, within 8.5 × 10⁻¹⁰, so the normal is the chord from the pitch point to it. The normal's moment arm about the mate's shaft is 32.14, positive when the contact turns the mate forward. The shape is the unknown

Rotors that mesh and cannot drive each other

Two identical lobed rotors on shafts turning one to one are each other's conjugate: give half of a lobe and the meshing equation computes the other half so exactly that the rotor is its own mate. The pair holds its ratio at every instant and still cannot drive itself, because the one contact between them pushes the driven rotor backwards for exactly half of every turn.

 A 3-RPR with similar triangles at 0°, its leg lines meeting wherever it is put. A planar platform with three extending legs, its base pivots on a circle of radius 2 and its platform points on one of radius 0.6 at the same angles, so the two triangles are similar, turned to 0° and drawn at two positions. Each leg's line is continued past its ends. With the platform's centre at (0.4, −0.3) the three lines meet at (0.5714, −0.4286), within 1 × 10⁻¹⁶ of every line, which is where a scale factor of 0.3 about that point carries the base triangle onto the platform; the smallest singular value of the three legs is 3 × 10⁻⁹. With the platform's centre at (−0.55, 0.3) the three lines meet at (−0.7857, 0.4286), within 2 × 10⁻¹⁶ of every line, which is where a scale factor of 0.3 about that point carries the base triangle onto the platform; the smallest singular value of the three legs is 0. The meeting point moves with the platform, and the three lines meet wherever it is. Several legs, one platform

Two orientations no position can rescue

A planar platform on three extending legs whose platform triangle is a scaled copy of its base is singular at every position it can be put in, at exactly two orientations: its three leg lines meet at one point wherever it is. The two orientations are read off the attachment points, a platform a few per cent from similar is held at its worst orientation only in proportion to how far from similar it is, and the revolute-legged 3-RRR, built on the same similar triangles, does not inherit any of it.

 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. Wheels, and where they may not go

One bracket, two subjects

A wheel can be parked sideways because the forbidden direction is the bracket of two permitted ones. A mechanism's motion is a group when the brackets of its permitted twists are already permitted. Same operation, same two plateaux, and the two fields want opposite answers — which is why the two are named for different things here, with each one saying beside itself that the other exists.

 What one point can and cannot see of a group. For each of the twelve: the group's dimension, the dimension of one point's orbit under it, and the difference — the stabiliser, the motions that leave that particular point exactly where it is. The orbit is the only picture a group has, and this table is the honest caption on it. Planar motion and spherical motion are both three-dimensional and both sweep a point over a two-dimensional surface, so each leaves one motion doing nothing at all: a turn about the plane's normal in one case, a turn about the radius in the other. A point does not see the whole group, and no drawing of one trajectory can be a complete picture of what a joint permits. Drawn wrongly

Six things a joint is not

A freedom count read as a description, a screw system read as a group, a pair list read as a convention, a trajectory read as a determination, a nominal alignment read as a delivered one, and a higher pair read as a larger joint. Six claims, each of them what a careful person would say, each answered with a number.

 The free configurations, with joint limits. Every point is a pair of joint angles for a two-link arm; the pale region is the configurations at which neither link touches an obstacle, and the dark one is where something is in the way. The free space is in 2 pieces. The arm's joints cannot turn all the way round, so the edges of the square are edges — and now the barrier separates. The two crosses put the tool at exactly the same point, and the arm cannot get from one to the other at all. One path to the tool

The space of configurations

A two-joint arm's configurations are a torus, and drawing one on a page turns it into a square whose opposite edges are secretly the same line. Count the free space on the square and get three pieces; count it on the torus and get one. Eighty single-obstacle arrangements were tried and not one of them cut the torus in two — what does that is a pair of hard stops.

 Nine bars joining two sets of three joints, at three placements of one motion. Joints B₁, B₂ and B₃ lie on the horizontal line at -2, 1, 3, joints W₁, W₂ and W₃ on the vertical line through the same point at -1.5, 1, 2.5, and every joint of one set is barred to every joint of the other. Counted, nine bars on six joints leave no freedom. Drawn here at three placements, the joints have slid along their lines — B₂ at 0.632 and W₂ at 1.265; B₂ at 1.000 and W₂ at 1.000; B₂ at 1.265 and W₂ at 0.632 — and every one of the nine bars has the same length in all three, to 4.4e-16. What can move

Nine bars that ought to be rigid

Join each of three joints to each of three others and the nine bars leave no freedom, by the count and by the rank, wherever the joints are. Put one set on a line and the other on a line at right angles and the framework moves, all the way round a loop, with no bar repeating any other: take away any one of the nine and the motion is unchanged, take away any two and it gains a freedom. Tilt the lines by a degree and it still has a freedom by rank and cannot move at all.

 A fourth leg through the meeting point, and one beside it. The platform with similar triangles at its singular orientation, at one position, with a fourth base pivot at (0, -2). Left: the fourth platform point placed where the similarity puts it, so the fourth leg's line passes 2.8e-17 from the point the other three meet at, and the smallest singular value of all four is 2.59e-9 — nothing has changed. Right: the same base pivot with the platform point moved 0.28 round the platform, so the fourth line misses the meeting point by 0.3993 and the four legs hold at 2.914e-1. The ringed point is where the three original lines meet, and it moves with the platform. Several legs, one platform

One placement of every placement

A planar platform whose two triangles are similar is singular at every position it can be put in, at two orientations. A fourth leg removes both — unless its own pair of attachment points is related by the same similarity, and then it changes nothing at all. Swept right round the platform, exactly one placement of the fourth attachment fails, the similarity names it in advance, and the holding a placement buys is proportional to how far it sits from that one point.

 The body points whose three images are in a line. The three prescribed poses, faint, and a curve through them. At every point of a 110×110 grid over the moving body, the point is placed in all three poses and the signed height of its image triangle is measured; the curve is where that length is zero. Those are the points whose three images lie on a line, so the dyad they want is a slide rather than a crank. The curve is a circle. Refined onto the contour, its points lie on the circle through the three image poles to 2e-15, where the pole triangle's own circle misses them by up to 1.98. Its radius is 8.767, so inside a window three units across it reads as a gentle arc; 2 of the three image poles are in the frame and the third, P₁₃, is 6.8 units away. The problem backwards

Where a pin becomes a slide

Three-position synthesis gives every point of the moving body a fixed pivot, except the points whose three images fall in a line. Those want a slide, and they are not scattered: they lie on one circle, the circle through the three image poles, which a single line of algebra predicts and a contour of a measured length draws to 10⁻¹⁵.

 Every way the same bars can be put together. The machine compiled from a rectangular hyperbola has 4 parallelograms, and a parallelogram's four bars also close as an antiparallelogram — so there are 16 ways to assemble it. One mark per way. 4 of them put the tracing point on the curve, 4 put it somewhere else, and 8 do not close at all. The ones that are wrong are not broken: they satisfy every bar to 9.6e-15 while the polynomial at their tracing point reads 1.5e-1. This is the gap in Kempe's original argument, and no tolerance on the closure could ever have found it. Drawn wrongly

Six things a compiled linkage is not

A closure residual read as a verdict, a theorem read as a design, an exact answer read as an accurate one, a degree read as a cost, a construction read as a search, and a neighbourhood read as a turn. Six claims, each of them what a careful person would say, each answered with a number.

 One set of lengths, two machines. Three measured input–output pairs, marked, and the linkage Freudenstein's relation returns from them — which is the truth's four lengths to fourteen figures. The relation is a statement about the two angles and it holds on both assembly branches, because it was derived by squaring and that is the step that forgets which one the mechanism is on. So the identified linkage assembled the way the data was taken passes through every reading, to 2.53e-14 radians, and assembled the other way misses them by up to 268° — at the first precision point it reads -111.6° where 98.8° was wanted. That is not a near miss and not a failure either. It is the other answer. Numbers that were measured

One set of lengths, two machines

Three measured input–output pairs return a four-bar's four lengths to fourteen figures. Assembled the way the data was taken, that linkage reproduces every reading to 2.5 × 10⁻¹⁴ radians. Assembled the other way — which the same four lengths permit — it misses them by 268°, and no equation in the identification knows the difference.

 The configuration space of a crank rocker. Every point of the square is a pair of angles — the crank's and the rocker's — and the curve is where the coupler is exactly the right length to join them. That curve is the mechanism: one equation in two angles leaves one freedom, which is the mobility. It has 2 components, and that is the two assembly branches. A built linkage cannot cross between them, because there is no path in the set to cross by — and each goes all the way round, which is what makes the crank a crank. The square's left and right edges are the same line, and so are its top and bottom. One path to the tool

Branches were components all along

Seven words have been used for one thing. An assembly branch, a circuit, an assembly mode, a working mode, a posture and a branch defect are all statements about the connected components of a mechanism's configuration space — and once that is said, a four-bar's two circles, a platform's six modes and a synthesis defect stop being three subjects.

 The two circuits at a flat position, and the play that joins them. The input and output angles of a parallelogram whose input is short by 1e-3, near the flat position where the exact parallelogram's two motions would cross. With no play the linkage's configurations are two curves, an upper and a lower, 0.1095 radians apart in output angle at the flat input angle — the square root of the error, not the error. Each shaded band is what a radial play of a stated fraction of the error makes reachable, and the innermost boundary is the play-free pair. At a play equal to the error the bands meet and the linkage can pass from one circuit to the other. What can move

A length error is undone by its own size

A parallelogram built a thousandth wrong loses its change point, and the two motions it could have chosen between end up a tenth of a radian apart — the square root of the error rather than the error. The radial play that joins them again is a thousandth exactly: not of that order, that number. It is the same number a third crank charges the same linkage in misfit, and no pin is worth more of it than any other.

 36% shared, and they never touch. The region left arm visits over a whole drive, the region right arm visits, and — in the third colour — the part of the plane both of them visit. The shared area is 4.652 square units, 35.9 per cent of the smaller of the two regions. The two parts are drawn at the configuration where they come closest, and at that configuration the gap between them is 0.3799 — more than twice a link's width, and positive everywhere else on the drive. A test that asks whether the regions intersect has reported a collision between two parts that are never in the same place at the same time. Links with a width

The regions overlap and the parts never meet

A swept region is a projection along time, and a projection cannot be undone. Chebyshev's two arms share thirty-six per cent of the ground the smaller of them covers and never come within twice a link's width of each other — a false alarm the region test cannot avoid, and one it cannot make at all against anything that stands still.

 A paired platform with its pairs rotated 60°, turned through a revolution. The paired Gough platform — base anchors in pairs 25° apart on a radius of 2.2, platform anchors in pairs 40° apart on 1.1 — with the platform's pairs centred 60° round from the base's, held level and turned through a revolution at four positions. All four curves reach nought together at 30° and −150°, which is 90° − ρ and 180° from it. At the centred position the platform is held at 0.0363 at a yaw of 0°, and it can turn ±17° before that falls to half. Dragging the rotation carries the two dead yaws across the revolution together. Several legs, one platform

The dead yaw is a design choice

A paired hexapod held level is singular everywhere at a yaw of 30° because its platform pairs sit 60° round from its base pairs. Rotate them by ρ instead and the dead yaws move to 90° − ρ and 180° from it, exactly, at every rotation from 0° to 120°. The furthest they can be from home is a quarter-turn each way, at ρ = 0 — where the platform is also best held at home and can turn ±75.5° before its holding halves, against ±17° for the usual 60°.

 Six coupler points of one slider-crank: three at other distances from the crank pin, three exactly a rod away. A slider-crank with a crank of 1 and a rod of 3 on a slide through the crank's pivot, drawn at one position with the closed curves six points of its rod trace over both assemblies. A point is given as (u, v) in the rod's own units — u along the rod from the crank pin, v across it. On the left (0.4, 0.5), (−0.3, 0.2), (1.2, −0.6): their distances from the crank pin are 0.64, 0.36, 1.34 rods. On the right (0.6, 0.8), (0, 1), (−0.8, 0.6), each exactly one rod from the crank pin, as far as the slider pin is. All six are quartics, all six are bounded — no real branch runs off the page — and nothing in the drawing tells the two panels apart. The difference is at infinity. How many answers

Where a slide puts the rest of the degree

A slider-crank's connecting rod draws a quartic that passes once through each circular point, which leaves two of its four meetings with the line at infinity unaccounted for. They are not along the slide. A point u along the rod and v across it sends them to the complex slopes [2v ± i(1 − u² − v²)] / [(1 + u)² + v²], whatever the crank, rod or offset — confirmed by slicing and by the fitted equation — and they are real only for points exactly a rod's length from the crank pin, where they merge into one direction at half the point's angle.

 Three machines, one curve. The coupler curve of a four-bar, drawn three times by three different linkages. Roberts's theorem gives every coupler curve exactly three four-bars that trace it, and their proportions are not close: the cranks here are 1.600, 2.214, 2.558, a spread of 60%. Read as an identification problem this is a least-squares objective with three separate exact minima and nothing between them — so an instrument that records only where the tracing point went has three answers however good it is, and no amount of data chooses. What chooses is knowing where the ground pivots are, which is a different measurement rather than a better one. Numbers that were measured

Three machines, one curve

Roberts's theorem says every four-bar coupler curve is drawn by exactly three different four-bars. Read as an identification problem that is a least-squares objective with three separate exact minima, whose cranks differ by sixty per cent — so an instrument that records only where the tracing point went has three answers, and no amount of data chooses between them.

 A framework built off square, and the play it asks for to move at all. The nine bars with their two lines 4° from perpendicular, as built and after the first joint has been pushed 0.2 along its line. The framework has a freedom by rank and no motion, so the push cannot be taken with the bars at their lengths. With every bar allowed to be wrong by 1.509e-4 — which is a radial clearance of 7.547e-5 at each end — a placement exists, and the worst bar in it is out by 1.509e-4. The same push on the perpendicular framework needs no allowance at all, because there it is a motion. What can move

The right angle as a tolerance

Dixon's nine bars move only when their two lines are exactly perpendicular, and a framework built a degree off square has a freedom by rank and no motion at all. Give its joints clearance and it moves a bounded distance: the play each bar needs is proportional to the tilt and to the square of the travel, one constant serves every tilt, and all nine bars end up at that play exactly. Then the framework reaches its first crossing and the law is left three hundred times behind.

 24 teeth inside 72. An internal pair, drawn from the same involute the external pairs are drawn from. Three things are different and they are one difference. The centre distance is 24.0 — the difference of the pitch radii rather than their sum. Both base tangency points lie on the same side of the line of action, so the pitch point falls outside the segment between them rather than inside it. And the contact ratio is 1.931 against the external pair's 1.707, because the annulus's tip circle cuts the line on the far side of its own tangency and lengthens the contact path instead of shortening it. The pinion turns the same way as the annulus, which no external pair ever does. Teeth

The mesh with one curvature reversed

Turn an annulus's teeth inward and the same involute law produces a different machine: a centre distance that is a difference, two base tangencies on one side of the line of action, more contact than an external pair carries, and three separate floors on the tooth counts, all of them the same statement about where an involute stops existing.

 One link, five bodies. Five bars with the same two pins, offset by -0.24, -0.12, 0, 0.12, 0.24 of the link's length, drawn to a common scale. Every one of them holds its two pins exactly the same distance apart, so every one of them is the same link: put any of them into a mechanism and the mechanism solves to the same joint positions at every configuration. Nothing in the kinematics of this collection — no loop equation, no velocity, no coupler curve, no mobility count — can tell them apart. What they do not have in common is which ground they occupy on the way from one pin to the other. Links with a width

A link may be bent

A link is two pins at a fixed distance and the metal between them is a free choice. Bending it moves no joint of the mechanism by more than 10⁻¹³ and moves the clearance by a tenth of a link length — enough to build a machine that a straight bar refuses, and worth exactly nothing against a bearing pedestal the link sweeps over.

 Eight contacts at once, and the sense each turns the disc. The 12-pin drive at 40° of its eccentric. 8 pins are in contact with the disc at this instant, and each one's line is the common normal, which passes through the pitch point on the pin circle. A pin can only push, so the sense in which it turns the disc is decided by which side of the disc's own centre its normal passes: 4 of the contacts turn it one way and 4 the other, with the largest arm in each sense 42.2 and 45.0 on a pitch offset of 55.0. A pair with one contact has no such choice, which is the whole of why two identical rotors cannot drive each other. The shape is the unknown

What a second contact is for

Two identical rotors that are exactly each other's conjugates cannot drive each other, and the reason has nothing to do with conjugacy. A ring of pins and the disc they generate is just as exactly conjugate, has eight to eleven contacts at once instead of one, and never loses more than thirty per cent of the arm its geometry allows.

 Where a tilted platform is still singular. A slice of the workspace at height 2.4, at the dead yaw and 3° of tilt, with each position shaded by how well the platform is held there — dark where the six legs are nearly dependent and pale where they are not. Level, this whole square would be uniformly dark. Tilted, the dark places are a curve through it, which is what an ordinary direct singularity looks like on a slice. Driving a search downhill from forty-eight starts reaches a smallest singular value of 0.00e+0, so the locus is a singularity rather than a shallow valley. Several legs, one platform

Tilted, near the dead yaw

A paired Gough platform held level is singular at one yaw wherever it stands. Tilt it and that stops being true — the six moments are no longer equal and the home position is held. It is a poor rescue: the rise is quadratic in the tilt, so a degree buys a sixty-fourth of what eight degrees buys, and what the tilt actually does is not remove the singularity but turn it into a surface through the workspace.

 Reachable, and dexterous. A 3-link planar arm with links 1.6, 1.2, 0.7. The outer region is everywhere the tool can be put: an annulus from 0.00 to 3.50. The inner region is everywhere it can be put at every tool angle — from 1.10 to 2.10, which is 26% of the area. Both are measured by counting cells on a 260 × 260 grid and both agree with the area computed from the radii to 0.06%, which is what makes the picture a measurement. One path to the tool

An arm's parameters and its poses

A three-link planar arm has three lengths and a tool position that carries a length, so nothing about it is invisible to a measurement — and it is nevertheless the mechanism on this site where a calibration is hardest, because its parameter count is high, its poses are three-dimensional and its Jacobian is singular where a designer likes to work.

 Nine parameters, two of them invisible. A Watt six-bar has seven lengths, a fraction that says where a point rides on its rocker, and a third ground pivot with two coordinates. Reading its output link with a protractor over 28 poses gives a matrix of rank 7: two directions are invisible, at 8.24e-10 and 5.06e-10 against a largest of 7.49e+0. One is scaling the whole machine, with a zero against the fraction, because a fraction is not a length. The other is scaling the second loop alone about O₄ — that loop is a four-bar in its own right and its own size does not reach the output angle. The two wrong guesses a reader would try, scaling those five parameters about O₂ or scaling the first loop alone, are refused at 1.5e-1 and 2.0e-1. Numbers that were measured

Nine parameters, two of them invisible

A Watt six-bar has seven lengths, a fraction and a ground pivot's two coordinates. Read by a protractor on its output link, its identification Jacobian has rank seven — and the second missing direction is not a scaling of the machine at all. It is a scaling of the second loop alone, about the pivot the two loops share.

 A four-bar at 1°, solved. Ground 4, crank 1, coupler 3.5, rocker 3. Every joint position here is the output of a Newton–Raphson solve on the loop-closure equations, converged to 3.3e-14 — not a placement that looked right. Grashof's condition classifies these lengths as a crank rocker, and sweeping the crank through 360° confirms it: 120 of 120 positions assemble. The transmission angle at this instant is 54.3°. Drawn wrongly

Six things a measurement cannot tell you

A calibration with a perfect residual whose fourth number is a starting guess, a rank that says nothing about a second answer, an improvement that proves nothing about a parameter, a class with no margin, a plan scored on poses that were refused, and a model missing something no data can find. Six claims, each with the number that kills it.

 Two circuits at an ordinary change point, and how far apart a length error leaves them. A four-bar on the Grashof boundary g + a = b + c — ground 4, crank 1, coupler 2.50, output 2.50 — with its crank short by 1e-3, near the one input angle at which its two assemblies would meet. Built exactly, the two curves would cross at the origin. Built with the error they pass each other 0.0400 radians apart at the flat input angle, against the law 2√(2bδ/c(g + a)) = 0.0400, and the pin clearance that rejoins them is 1.0000e-3: the error itself. Dragging the coupler's share of b + c moves the separation and leaves the clearance where it is. What can move

Every change point lies flat

A parallelogram a thousandth wrong is rejoined by a pin clearance of exactly a thousandth, at any of its four bearings. The obvious guess is that an ordinary change point — a linkage on one Grashof boundary with no equal bars — would need a clearance with a constant in front and would reveal which bearing is loose. It does neither, because every change point has its four joints on one line. What does acquire a constant is the angle: 2√(2bδ/c(g + a)) when the circuits separate, and a stall constant with no coupler or output in it at all.

 The common normal, and where it is. Two joint axes 1.0 unit apart, nominally parallel, 0.50° apart in one plane and 0.05° in the other. The Denavit–Hartenberg convention takes all four of its numbers from the one line that meets both at right angles, and for these two axes that line crosses the first 11.3 units from the joint — off this page by a factor of about 11, which is why the tilt here is drawn at 26° and not at a fraction of one. Along it, the "link length" of a link 1.0 unit long reads 0.9950, and the angle round the first axis reads 5.71°. Nothing has moved by more than 0.50°. Numbers that were measured

The common normal, and where it is

The Denavit–Hartenberg convention reads all four of its numbers off one line: the common normal between two joint axes. Two parallel axes do not have one — every perpendicular meets both at right angles — and two nearly parallel axes have one that is somewhere else entirely.

 The machine is fine; the description is not. Above: the offset the Denavit–Hartenberg chart assigns to a pair of nominally parallel axes, over three decades of twist, running from 0.0030 to 1102 link lengths. Below: the condition number of the identification Jacobian in a chart that describes the second axis by two small rotations from the first and never asks for a common normal, over the same range. It is 7.5501 at every one of them, flat to 2.4e-9. The machine is the same machine in both rows and it is perfectly well behaved. What breaks is a convention that locates its parameters on a line which, for two parallel axes, does not exist. Numbers that were measured

The chart breaks, the machine does not

The same two axes, over the same three and a half decades of twist. In one description a parameter runs from 0.003 to 1,102; in another the condition number is 7.5501 and does not move in the fifth figure. A quantity that diverges in one chart and is constant in another is a property of the chart.

 Which numbers have a size. Twelve quantities from eight fields of mechanism kinematics, each scaled by taking every length in its mechanism up and down together and fitting the power its value follows. A shape lands on zero, a length on one, a curvature on minus one, an area on two, and every row lands on an integer to 3.3e-11. The sorting is not a units argument — it is measured from each field's own library, by asking that library the same question. What it says is that most of what kinematics computes is a shape: a mobility, a Grashof class, a transmission angle, a contact ratio and a velocity ratio are all unchanged by making the machine bigger, so all of them are recoverable from a measurement that cannot recover a single length. The row worth stopping at is the tolerance band: held to a fixed ±0.01 it lands on minus one, because ±0.01 is a length and a bigger machine held to the same absolute tolerance is a better machine. Written as a percentage the same band lands on zero. One quantity, two drawing conventions, two different kinds of number. Numbers that were measured

A count is neither

Every quantity in the scaling survey lands on an integer power — zero for a shape, one for a length, two for an area. A mobility lands nowhere. It has no dimension at all, it does not move under any perturbation, and the probe that sorts the rest of the site's numbers returns nothing for it.

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