Machine Kinematics
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The thread: The singularities are the point, part 2 of 2

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.

180 essays in 2 parts: part 1 · part 2

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.

 Nine contacts at once, and the sense each turns the disc. The 12-pin drive at 40° of its eccentric. 9 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: 5 of the contacts turn it one way and 4 the other, with the largest arm in each sense 49.9 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 nine or ten contacts at once instead of one, and never loses more than a fifth 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.

 A strip of distorted kagome with its mechanisms drawn by weight: one edge, the other, or both. A strip of kagome lattice 10 cells wide, periodic along its length (6 of its 12 cells drawn), with each up-triangle's three joints moved along the triangle's sides by 0.1, 0.1 and −0.1. Each joint is drawn with an area proportional to its share of the strip's mechanisms, summed over every wavevector along the strip. The cut leaves each long edge the same number of missing bars, so the count says the two edges share the mechanisms equally. They do when the three moves have one sign. With the third reversed, the top half carries 19.99 of the 24 and the bottom 4.01. The bulk's windings, measured from the twisted kagome's, are (1, −1). Many of one thing

An edge that knows which side it is on

Cut a strip from a kagome lattice and each long edge loses the same bars, so the count says the two edges share the strip's mechanisms equally. Move each triangle's three joints by the same signed amount and they do. Reverse one move and every mechanism that is not a translation moves to one edge. The shift is a whole number per cell, set by the winding of one six-by-six determinant round the lattice's zone. It changes only where one family of bars lines up straight.

 Four points of one connecting rod: two crossings, none, two isolated points, two crossings again. The closed curves drawn over both assemblies by four points of the connecting rod of a slider-crank with a crank of 1 and a rod of 3 on a slide through the crank's pivot, which is marked in each panel. Each curve is a quartic and owes two double points. (0.12, 0.15), 0.19 rods from the crank pin: two crossings, at (±0.65, ±0.36), ringed; (0.4, 0.5), 0.64 rods from the crank pin: a complex pair, so the curve crosses itself nowhere; (0.75, 0.94), 1.20 rods from the crank pin: two isolated points at (±4.06, ±3.00), dotted, off the drawn curve; (1.1, 1.0), 1.49 rods from the crank pin: two crossings, at (±2.66, ±2.95), ringed. Each panel has its own scale. The second curve has no visible double point at all, and the third has two that no configuration of the machine ever reaches. How many answers

The double points that leave for infinity

A slider-crank's connecting rod draws a quartic that owes two double points. Close to the crank pin they are two real crossings. Between a crank's length and a rod's length from it they are a complex pair and the curve crosses itself nowhere. At exactly one rod both run off to the same real point at infinity and meet there as a tacnode whose branches are complex. Past that they come back real. Offsetting the slide changes the two circles by (R² − a²)(R² − b²) + e²R² = 0 and sends only one of the pair to infinity.

 Two legs that share a line of force. Two legs of three revolute joints each under one platform, drawn about their common centre: the first leg's joints in one colour and the second's in another. The second leg was built so that each of its joints passes through a point of one line, the dark one, which also crosses all three joints of the first leg — the six marked points, each met to 1e-16. A force along that line does no work on any joint of either leg, so both legs resist it: it is a load path the platform has twice over. The dashed line is what comes with it: the axis of the one twist both legs permit, a screw of pitch -0.049 that passes 0.032 from the shared line at 57.2° to it. Dragging moves the shared line along the first leg's first joint and builds the second leg afresh through each position. Out of the plane

A shared load is a shared motion

Put two legs of three revolute joints under one platform and let them share a line of force, so that both carry the same load. The platform does not merely gain a redundant load path; it gains a motion in the same act, because three constraints and three make six. The motion is a screw about a different line, tied to the shared one by h cos φ = d sin φ on two hundred constructed pairs to 4 × 10⁻¹⁵. Pushed along it, the two legs part as the square of the push: the platform shakes and does not move.

 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 in this collection 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.

 A seam that holds two mechanisms. Ten rows of a strip of kagome lattice sixty cells wide, centred on the line where two differently distorted lattices meet. Below it every up-triangle's joints are moved along its sides by 0.3, 0.3, −0.3, polarization −1 in the component that counts along the seam; above it by −0.3, −0.3, 0.3, polarization +1. No bar is missing anywhere along the seam. Each joint is drawn with an area proportional to its share of the strip's two mechanisms at one wavevector. They sit on the seam and die away from it into both lattices, and the strip's outer edges, out of the picture, have none. The wavevector along the seam is 3/12 of a turn per cell. Many of one thing

A seam with no bar missing

Join two distorted kagome lattices of opposite polarization along a line. Every joint on the line keeps all four of its bars, so Maxwell's count says the line holds nothing. It holds two per wavevector: two mechanisms one way up, two states of self-stress the other way up, exactly the difference of the two polarizations. The self-stresses are tensions the lattice carries with no load on it, along a line where no bar is spare. In a strip sixty cells wide the prediction holds at every wavevector a quarter of the zone or more from its centre.

 A disc cut undersize, turned until a pin touches. A twelve-pin cycloidal drive whose disc was cut as if its pins were 1 larger than they are, drawn at 10° of the eccentric with the disc turned a further 1.082° about its own centre — the turn at which the first pin touches. One pin touches, pin 0, marked in its own colour. The pins within half the clearance of touching are drawn in a second colour and the rest plainly: 5 of the twelve are within half the clearance and 7 further off. The clearance is exaggerated so that it can be seen; at every clearance measured, one pin touches first. The shape is the unknown

A clearance leaves one contact

A cycloidal drive's disc, exactly as its pins generate it, touches nine or ten of its twelve pins at once, and that is why it can drive where two identical rotors cannot. Cut the disc two hundredths undersize, as a real drive is cut, and a rigid disc turned until it touches meets one pin, at every angle of the eccentric and in both directions. The arm that pin gives is never worse than the exact disc's best. What a clearance takes away is load sharing, and deflection gives it back: contacts that may give by the whole clearance put six or seven pins within reach.

 The disc, the wheel, and what has to be cut out of it. A 6-slot Geneva at 20° of driver, with the driver's locking disc of radius 13.5 drawn about its axis and the wheel drawn as the material it actually has — inside its rim, short of the 6 flat chords cut across it, one between each pair of slots, and clear of the 6 slots. The disc and the wheel share the region near the line of centres, so the disc has to be cut away wherever the wheel is ever there while the pin is driving. Swept over the whole index that cut-away spans 126.8°, against an index sweep of 120° — it is wider, because the rim is still swinging through the disc's circle after the pin has left the slot. Motion that stops

A flat holds the wheel to second order

A Geneva wheel is held between indexes by a disc on its driver sitting in a concave arc, and that disc is so large that one pin is all a wheel of four slots or more can carry. Hold the wheel on a flat chord instead and the disc must shrink to the crank times sin(π/n). Its cut-away falls from 236° to 127° on six slots, and a second pin fits at every count from four to twelve. The cost is the hold itself. An arc lets the locked wheel rock in proportion to the running clearance and a flat as its square root: 0.022° against 1.48° at a clearance of 0.02 on a crank of 30.

 The one real double point a four-bar owes is the one that leaves when its rocker becomes a slide. Left: a four-bar with a crank of 1 pivoted at the origin, a coupler of 3 and a rocker of 5 pivoted at (3, −5), so that the rocker's circle touches the line y = 0 where a slider-crank's slide would run. The coupler point is (0.4, 0.5) in the coupler's own units, and the curve it draws over both assemblies is shown with the circle through the three cognate pivots, on which all three finite double points lie. Two of them are a complex pair near the crank; the third, the only real one, is an isolated point at (0.92, −5.32) that no configuration reaches. Right: the same picture divided by the rocker length for rockers of 4, 10, 20, 50, 200, 100,000, with the drawn rocker's point ringed. The machine shrinks to the origin, the pivot circle settles onto a fixed circle, and the isolated point settles onto one point of it, (−0.484, −0.580), marked with a cross: the value d(|w|² − 1)/(w̄ − 1) for w = 0.4 + 0.5i and d the direction from the slide to the rocker's pivot. How many answers

The double point a slide cannot keep

A four-bar's coupler curve owes three finite double points and a slider-crank's owes two, so a four-bar whose rocker grows into a slide has to lose one. Parity decides which: the one that leaves is always real, and always an isolated point no configuration reaches. It runs off along d(|w|² − 1)/(w̄ − 1) times the rocker's length, a law that knows nothing of the crank, the rod, the offset or where the rocker touches the slide. On the circle one coupler length from the crank pin the law is zero, and all three double points leave together like the cube root of the rocker.

 Where in the workspace each yaw is singular, once the platform tilts. A horizontal slice of the workspace at height 2.4, for the paired platform with its pairs rotated 0° and tilted 1° about an axis 0° from x (the arrow). Each square is shaded by the yaw, measured from the dead yaw 90°, at which that position is singular — found as the root of the six leg screws' determinant — blue where it is below the dead yaw and orange above, deepest at ±0.208°. Level, every position would be singular at the dead yaw exactly. Tilted, the singular yaw is a plane over the slice: over the whole workspace, the yaw times the height fits a·x + b·y to 0.7% of its spread, and the dashed line is where it is nought. The plane's slope turns with the tilt axis. Dragging turns the axis. Several legs, one platform

The dead yaw opens into a wedge

A paired Gough platform held level is singular everywhere at one yaw and nowhere a quarter of a degree either side of it. Tilt it and each position of the workspace becomes singular at its own yaw, displaced from the dead one in proportion to the tilt and to the position's horizontal offset over its height. So the orientations that put a singularity somewhere in the workspace are not a surface through the dead yaw but a wedge standing on it, about a quarter of a degree of yaw wide on each side for every degree of tilt — and the rotation between the pairs, which chooses where the dead yaw is, barely changes how much room it needs.

 A third leg that keeps the platform's shake. The two legs of the earlier construction — the second built through a line of force of the first, the dark line — share one twist, whose axis is the dashed line: a screw of pitch -0.049. A third leg is drawn in a third colour, built so that its three joints span a family of motions containing that twist: two joints chosen freely and the third solved for. The platform on all three legs keeps its one freedom. The dotted line is the axis of the one wrench the third leg shares with the first: a screw of pitch 0.201, not a line of force, and 0.244 from the pair's shared line. Dragging picks another third leg from the same family. Out of the plane

A third leg that keeps the shake

Two three-joint legs that share a line of force share one twist as well, and a third leg in general position takes it away. A third leg whose own freedom contains that twist does not, and building one is three conditions on twelve numbers: two joints anywhere and the third solved for. Such a leg must share a load with each of the first two — but not the line they share. What it shares with each is a screw with a pitch of its own, reciprocal to the twist through the sum of their two pitches, and only a third leg built through the pair's line shares the line itself. With any of them the platform still shakes rather than moves.

 A stop on the coupler, and the one direction in which it stops nothing. A four-bar — frame 4, crank 1.2, coupler 3.4, rocker 2.8, crank at 70° — whose coupler is stopped by a contact with the frame at (1.59, 2.16). The dashed line through the stop is its normal; the ringed point is the coupler's pole, where the crank's and rocker's lines meet, at (2.505, 6.883). With the normal 10° off the line to the pole the stop removes the four-bar's one freedom; through the pole it removes nothing, because the contact point is already moving along the stop's face. The graph of the stopped four-bar is the same in every case. The chain before the lengths

A contact turns and slides at once

A cam on its follower, a gear tooth or a pin in a slot takes one freedom away where a pin or a slide takes two, and it turns as well as slides. Give every joint of the four-bar, Watt's and Stephenson's chains a third label and there are 1,317 different chains. The rule that read slides off the graph extends to all of them, agreeing with the constraint rank on every one, and the two obvious shortcuts miss 36 and 769. What the graph cannot record is which way a contact's normal points — and on these chains that never matters, while a single stop added to a four-bar holds everywhere except through one point.

 A cam cleat holds while its pivot leans inside the friction cone. A rope along the bottom, pulled to the right, pressed against a fixed backstop by a cam on a frictionless pivot. The cam can push on the rope only along the line from the contact to its pivot, because a body on one pin with one other force is a two-force member. That line leans 15° from the contact's normal; the shaded wedge is the friction cone, 21.80° either side of the normal for a friction coefficient of 0.4. The line is inside the cone, so the cam's force can resist the pull, and it squeezes the rope with 3.73 times the pull. Dragging moves the pivot. Contacts that only push

A hold the load squeezes shut

A cam cleat bites harder the harder the rope pulls, and a wedge dragged by a bar into a narrowing gap squeezes the bar harder the harder it is drawn. Neither needs a screw: the load supplies its own squeeze, if the geometry lets it. A cam on a free pivot can push only along the line to its pivot, so it holds exactly while that line leans from the contact's normal by no more than the friction angle. A wedge's squeeze passes through two contacts, and it jams while its angle is no more than the sum of their friction angles — twice the figure usually quoted. And the set of loads either one holds does not shrink towards its threshold. It is lost there, all at once.

 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.

 A six-bar followed round one circuit, with the links that turn against the frame coloured. A Watt six-bar — a four-bar of ground 4, crank 3.17, coupler 3.10 and rocker 1.90, with a dyad of 3.83 and 3.03 hung from its rocker to a third ground pivot — followed once round one of its 4 circuits. Links drawn in ink come back to where they started; links drawn in colour make a whole turn against the frame. On this circuit that is rocker + dyad link + output, and they turn together, so every one of the fifteen relative rotations between pairs of links is either nought or that one turn. The pins' paths are dashed. Linkages

Fifteen pairs and one cut

A six-bar has fifteen pairs of links, and each pair has a relative rotation. Followed round every circuit of more than four thousand random Watt and Stephenson six-bars, those fifteen rotations come from six whole-number windings. Almost always they take two values, so the links split into a group that turns once against the rest. How many inversions can be driven is then the number of links touching that cut: nought, three, four or five, never one, two or six. One chain in ten can have different cuts on different circuits, and a Stephenson chain can turn its rocker twice against its dyad.

 The third-order pole goes where the drive puts it. The four-bar of this field at a crank angle of 60°, with its inflection circle. The cross is the velocity pole and the ring Ball's point, where the circle meets the cubic of stationary curvature. The square is the third-order pole — the coupler point whose jerk is nought — when the crank is driven with an angular acceleration of 1 and a jerk of 6, each per the right power of its speed. It is at (0.492, 10.663). The faint curve is where it goes as the jerk alone is varied: a circle through the velocity pole. Every point off the inflection circle is the third-order pole of exactly one drive; on the circle only the velocity pole and Ball's point can be. Dragging changes the drive. The motion, not the mechanism

The third pole goes anywhere but the circle

The velocity pole is fixed by the geometry and the acceleration pole by the geometry up to one number, its place on the inflection circle. The next one, the point of a moving plane whose jerk is nought, depends on two numbers of the drive — and two numbers are enough to put it anywhere. Every point of the plane off the inflection circle is the third-order pole of exactly one drive, because a body point's jerk is affine in the crank's acceleration and jerk. On the circle only two points qualify: the velocity pole, and Ball's point, where the path is straight to third order.

 A slider-crank whose crank becomes a slide loses both its double points. Left: a slider-crank with a crank of 5 pivoted a crank's length before the point where its crank pin crosses the slide, a rod of 3 and the slide along y = 0, drawn with the slides' eventual crossing at the origin. The rod point (0.3, 0.2) in the rod's own units draws the curve shown, and its two double points are the dots: isolated points, at (−7.90, 2.47) and (−2.10, −2.47). Right: the same double points divided by the crank, with the crank's pivot at the origin and the crossing at (1, 0), for cranks of 5, 10, 30, 100, 300, 1000. The pair settles symmetrically onto the crosses at ±(0.595, −0.505) — the law ± i(|w| − e^(i·arg w))/√(1 − |w|²) for w = 0.3 + 0.2i — and leaves in proportion to the crank. How many answers

The trammel takes both double points

Grow a slider-crank's crank without bound and its crank pin's circle flattens into a second slide: the machine becomes the elliptic trammel, whose curves are conics with no double points at all. Both of the slider-crank's double points have to go, and parity cannot say how. They go together, symmetric through the crank's pivot, at ±i(|w| − e^(i·arg w))/√(1 − |w|²) crank lengths — a law that knows nothing of the rod or the slide's offset. Inside the one-rod circle they are two real isolated points, outside it a complex pair. And on the circle with the rod as its diameter, where the trammel draws straight lines, one of them does not leave at all: it settles on the crossing of the two slides.

 A pin with friction lets the cam's force miss the pivot, by the friction circle. A rope pulled to the right, gripped by a cam whose pivot pin has a radius of 0.16 (drawn larger) and friction 0.5, so the pin's force on the cam may lie anywhere tangent to or inside its friction circle, radius 0.072. The dashed lines from the contact touch that circle; the cam's force on the rope can lie anywhere between them, where a frictionless pin would allow only the line through the centre. The shaded wedge is the contact's friction cone, 21.80° either side of the normal for μ = 0.4. At a lean of 23° the cleat holds a forward pull (it holds up to 24.14°, against 21.80° with a frictionless pin) and lets go when pulled back (it locks backwards below 2.56°). Contacts that only push

A stiff pin buys grip with release

A cam cleat on a frictionless pin holds a rope while the line from the contact to the pivot leans no more than the contact's friction angle. Give the pin friction and the cam's force may miss the pivot by the pin's friction circle, so the cleat holds to a lean steeper by exactly the angle that circle subtends: tan γ ≤ μ + (ρ/h)√(1 + μ²). But the same friction makes a cleat leaning less than arctan(ρ/h) lock against a backward pull as well, and refuse to let go. Both edges of the working window rise, the lower at a slope of one and the upper at √(1 + μ²): the grip gained is almost all paid for in release.

 A seat round the stop's nose holds the four-bar if it is tighter than the nose centre's path allows. The four-bar of frame 4, crank 1.2, coupler 3.4 and rocker 2.8 at a crank angle of 70°, its coupler carrying a round nose of radius 0.3 centred at (1.593, 2.160). The nose sits in a hollow seat on the frame of radius 0.8, touching it at (1.650, 2.454), on the line from the nose's centre to the coupler's pole at (2.505, 6.883), off the top of the drawing. The dashed circle is where the nose's centre may go without the nose entering the seat: a disc of radius 0.50. The curve through the nose's centre is the path the four-bar gives it, with a radius of curvature of 0.893. The seat's disc is tighter than the path, so the path leaves it either way and the four-bar cannot turn: the stop holds. The threshold is a seat of radius 1.193, the path's radius plus the nose's. The chain before the lengths

A stop aimed at its pole holds by curvature

A stop whose normal passes through its pole forbids nothing at first order, and whether it holds at all is decided by curvature. For a round nose the deciding curve is not the one it touches: the seat that just holds is the path of the nose's centre moved out by the nose, 1.193 on a four-bar where the contact point's own circle says 0.776. Past it, one contact holds both ways — loosely, as the square root of its clearance.

 The singular set cuts the workspace's poses in two, and every path across it meets it. A slice of the paired Gough platform's poses, tilted 2° about x: the yaw's offset from the dead yaw across, and the position along the workspace's low diagonal, from one corner at height 1.8 to the other, up. Each cell is shaded by the sign of the six leg screws' determinant; the line between the two shades is where it is nought, the singular set. Tilted, it leans: each position is singular at its own yaw, the low corners about 0.48° either side. Three paths from −2.5° to +2.5° are drawn, one held still, one moving, one wandering, and each meets the line: held at the middle, 1 crossing; moving corner to corner, 1 crossing; wandering, 1 crossing. No path from one shade to the other can avoid it. Several legs, one platform

No path across the dead yaw misses it

A paired Gough platform that has to turn from one side of its dead yaw to the other cannot be steered round the singularity by tilting, by moving, or by turning the long way. The six legs' determinant has one sign everywhere on one side and the other everywhere on the other, so every path passes a singular pose — two hundred random ones do, an odd number of times each. Tilt chooses only where the crossing happens and how briskly: a quarter quicker at best, and threefold slower if it is done by moving instead of turning.

 Turning one joint about the shared line tunes the shake's first coefficient to nought. Two legs of three revolute joints each under one platform, sharing a line of force, the dark line, that crosses every joint of both — so they share one twist and the platform shakes along it. The second leg's first joint is turned about the shared line by 0°, which keeps it crossing the line and keeps the shared twist. The shake's second-order coefficient, the reciprocal product of the shared wrench with half the difference of the two legs' bracket terms, is 2.62e-2 here. It passes through nought at 36.488° and again at 109.612°; at those turns the platform still shakes, one order higher. Out of the plane

A shake has an order

Two legs of three joints that share a line of force let a platform shake: pushed along the one twist they share, the legs part as the square of the push. The square's coefficient is a bracket of the legs' own joints, computable without pushing at all, and it can be tuned to nought by turning one joint. The shake does not become a motion. It moves up an order, to the cube, and a second tuned joint moves it to the fourth power. Each order is one more condition, and only legs whose freedoms are groups meet them all.

 Dixon's nine bars with every joint somewhere inside its clearance. Dixon's framework — joints at -2, 1, 3 on one line and -1.5, 1, 2.5 on a line square to it, every joint of one set barred to every joint of the other — with each joint moved to a seeded point inside a disc about its nominal place. The discs are drawn with a radius of 0.12 so that they can be seen; the framework measured has joint errors about a hundred times smaller. The nominal joints lie on two lines, which is one conic, and that is what lets nine bars on six joints flex. The joints as built lie on no conic: at an error of ρ, seed 2's smallest singular value is 0.1746ρ where the nominal framework's is nought, so by rank it is a structure. What can move

The clearance that makes the framework generic

A clearance lets Dixon's nine bars move, and it also moves their joints off the two lines that let them move at all. Six joints off a conic make the framework rigid by rank, with a smallest singular value proportional to its distance from a conic. The same clearance that did the damage pays for it fourteen times over: a framework built anywhere inside its clearance discs is pushed to its first crossing using at most 7% of the allowance those discs give. The tilt of the lines decides how far it goes; its own joint errors change that by about 1%.

 A curved slot bends at its mouth with the crank's circle, or against it. A six-slot Geneva wheel whose slots are curved: mouth curvature 0.5 of the crank's. The driver's pin runs round the dashed circle and enters each slot along its mouth, which is radial so that the wheel starts from rest. At entry the pin's acceleration is square to the slot, and the part of it the slot's own bend takes up does not turn the wheel. Solved over the index, the wheel starts with an acceleration of 0.289 and stops with a deceleration of 0.866, in units of the crank's speed squared, against 0.577 at both ends for a straight slot; the two always add to 1.155. Its peak is 1.649 against the straight slot's 1.350. Motion that stops

A curved slot moves the step

A Geneva's pin arrives with an acceleration of tan(π/n) of the crank's speed squared, and no choice of entry direction removes it. A choice of slot shape can: bend the slot's mouth with the crank's own circle and the slot absorbs the pin's acceleration instead of handing it to the wheel. But the pin leaves by the mouth it came in by, and there the same bend adds to the step. Solved at four slot counts and seven curvatures, the entry and exit steps are tan(π/n)(1 ∓ κ̂) and always sum to 2 tan(π/n). A curved slot moves the step from one end of the index to the other, every bend raises the peak, and no slot tried beat the straight one.

 With a seventh leg the dead yaw is not a dip at all. The paired Gough platform held level at home, turned through sixty degrees of yaw centred on its dead yaw: the smallest singular value of its six leg screws, which falls to 8.9e-9 at the dead yaw, and of the seven screws with a seventh leg from the base circle to the platform circle at base 60°, platform 255°. With it the smallest singular value is 0.0288 at the dead yaw and no less than 0.0287 anywhere in the sweep, against the six legs' 0.0141 ten degrees off the dead yaw and 0.0320 thirty degrees off. The best placement of the 576 tried is the first the slider offers. Several legs, one platform

A seventh leg opens the dead yaw

Six legs can never carry a paired Gough platform across its dead yaw without passing a singular pose, because their determinant changes sign there and every path crosses nought. Seven legs make no square matrix and have no determinant to change sign. They fail only where the twist the six leave free also does no work on the seventh, a second condition that paths miss. Placed well, a seventh leg holds the platform's smallest singular value near 0.029 through the dead yaw, above the six legs' own value ten degrees off it, on every one of a hundred random paths across. What it cannot do is lift the platform further than the six's next-weakest direction allows.

 A pair moves until its gap reaches what its joints' play can take up. Two legs of three revolute joints each, sharing one twist, pushed along it: the gap left between their nearest poses, along the one wrench they share, for the pair built through the shared line, the pair with one joint tuned and the pair with two. The horizontal line is the gap the six joints' play can take up, 4.52 times a radial play of 10⁻⁶. Each pair travels until its curve meets the line: built through the shared line, 1.31e-2; one joint tuned, 6.63e-2; two joints tuned, 1.39e-1. Out of the plane

A shake travels as a root of the play

Two three-revolute legs that share a twist let their platform shake along it: the gap between their nearest poses grows as the square of the push, or the cube once one joint is tuned, or the fourth power once two are. With play in the joints, the pair moves until that gap uses up what the play can take, and that is a root: the square, cube or fourth root of the play. At a play of a millionth the three pairs travel 0.013, 0.066 and 0.139 along their twist. A tuned joint is worth more the tighter the joints are, five times the travel at a millionth and eleven at a hundred-millionth.

 The seat at the pole, and the friction cone at its contact. The four-bar of frame 4, crank 1.2, coupler 3.4 and rocker 2.8 at 70° of crank, its coupler's round nose of radius 0.3 in a seat of radius 0.8 on the frame, aimed through the coupler's pole. At the contact the shaded wedge is the friction cone for a coefficient of 0.1: every force the seat can press with lies within 5.7° of the normal. The normal itself runs through the pole, so it has no lever about it; the cone's edges do. Under a moment of 0.001 on the crank against a contact of stiffness 10⁶ the crank turns 1.70e-4 radians, against 1.48e-3 with no friction; released, it stays at 1.70e-4. The chain before the lengths

A seat with friction stays where it is pushed

A stop aimed through its pole holds a four-bar by curvature, because its normal has no lever about the pole. Friction has one, and the obvious guess is that it lets such a stop hold at first order. It does something narrower and stranger. It cannot hold what curvature releases, since a gap that opens carries no force for friction to be a share of. It makes a seat that holds up to twenty-six times stiffer, turning its cube-root give into a square root. And it makes any stop whose pole lies inside its friction cone stay where a load pushed it: released, the seat at the pole does not come back.

 The crank's acceleration fixes its jerk, so the two numbers a point needs cannot both be met. For the coupler point (0.45, 0.5) of the standard four-bar, over the crank's stroke from 30° to 150°: the second schedule number λ = θ⃛/θ̇³ that, with the first κ = θ̈/θ̇², makes the point's jerk nought at each angle; and the λ that a crank actually following that κ has, κ′ + 2κ², since both numbers come from one function, the crank's speed against its angle. Their difference, the mismatch, reaches 1.77 and is nought only at 39.8°. Wherever it is not nought no crank holds the point's jerk at nought. The motion, not the mechanism

No drive holds the third pole

At one crank angle a drive's acceleration and jerk can put the coupler's third-order pole on almost any point, making that point's jerk nought. Over a stretch of the stroke they cannot. A crank's schedule is one function, its speed against its angle, and its jerk number is fixed by its acceleration number: λ = κ′ + 2κ². So the two numbers a point needs at every angle must satisfy that relation, and on the standard four-bar they miss it by as much as the numbers themselves, agreeing only at isolated angles. The best schedule a crank can follow takes a fifth to a third off a coupler point's jerk, and no more.

 Taken in one step, the limit is a family of laws. A four-bar with its crank a pivoted a crank's length to the left of the point where its two pins' circles cross, and its rocker c pivoted a rocker's length below it, both growing with c = λa, so that near that point it becomes the elliptic trammel. The coupler point (0.3, 0.2) in the rod's units has three double points, all leaving in proportion to the machine; plotted divided by a + c, with the crossing at the origin. Faint dots: the real ones at 49 ratios from a thousandth to a thousand. Large dots: at λ = 1, at (-0.110, -0.161), (-0.827, -0.176), (-0.059, -0.692), all three real. At either end one point settles where the one-step limit taken first would put it — the four-bar's law with the rocker grown first, its mirror with the crank grown first — and the other two collapse onto the crossing at this scale. In between, all three are spread. How many answers

A four-bar becomes a trammel along a family of laws

A four-bar becomes the elliptic trammel when its crank and its rocker both grow without bound, their pivots receding at right angles, and its three double points must all leave. Taken one link at a time the limit sends one point off along the four-bar's law and then the other two along the trammel's. Taken in one step it does something neither order sees: all three leave together, in proportion to the machine, at positions set by the coupler point and by λ, the rocker's length over the crank's. The two one-step orders are the family's two ends, λ → ∞ and λ → 0, and at equal rates the coupler point (0.3, 0.2) has its three double points up to 0.86 crank lengths from where either order puts them.

 A cleat's lever, and the squeeze the load has left it. A cam cleat at a lean of 10°, its pivot pin with friction 0.5 and a radius a tenth of the pivot's height (drawn larger, with its friction circle), and a release lever on the cam. Along a load path the rope's pull rises to one, falls to nought and rises again, and the squeeze follows it between the band's edges, 4.50 and 7.62 times the pull. Here the pull is 1, the squeeze 4.50, and the lever must turn the cam with a moment of 0.412 times the pivot's height to free the rope. Contacts that only push

A cleat set hard lets go easiest

A cam cleat on a frictionless pin is freed by a lever with no effort at all. With pin friction the lever must beat h(T − N tan γ) + ρ√(T² + N²), and since a stiff pin lets the squeeze N sit anywhere in a band, the effort is set by where in the band it sits: nought at the top, twice the pin's friction moment at the bottom. A rising load drags the cam in to the bottom, so a cleat holding the load it was set by is the hardest to free — 0.412 times the pull times the pivot's height at a lean of 10° — while one jammed hard by hand, or one whose load has eased, frees itself. Below the backward-lock lean nothing jumps in the effort. What changes is the state: the slack cleat keeps the squeeze its largest pull left.

 Past the crossing, one push names two placements, one in each family. Dixon's framework, joints at −2, 1, 3 on one line and −1.5, 1, 2.5 on a line square to it, held by B₁ and the direction of B₁W₁, with B₂ pushed 0.5 along its line. Solid: the placement on the family the framework starts in (++), reached before the crossing. Dashed: the placement at the same push on the family beyond it (+−), reached after W₂ has passed the other line at a push of 0.591 — the faint framework — and the push has begun to fall again. Every bar has its drawn length in both. A framework with no errors moves from one to the other through the crossing; a framework as built has no motion and must be carried there by its play. What can move

The price of the other family

At its crossing, Dixon's framework passes from one family of placements to another, and past it every push names two placements, one in each. A framework built with its joints anywhere inside a clearance ρ has no motion to follow, so its play has to carry it. The price of going on is proportional to ρ, from 0.002ρ to 0.20ρ across ten frameworks, exactly as a parallelogram's change point is — and reversing every joint error leaves it unchanged. Eight of the ten find the far family dearer than their own and two find it cheaper, and all of them find it far inside the 2ρ their clearance gives.

 A pin steered off its circle carries the wheel on any law. A 4-slot Geneva wheel and its driver. The pin rides in a radial guide on the driver and a stationary cam sets its radius, so its path through the index (solid) is not the crank's circle (dashed). Here it is cut for the modified trapezoid law over the circle's own window of 90° of driver: the wheel starts with an acceleration of nought and stops with nought, against tan(π/4) = 1.000 at both ends for the crank pin, and peaks at 3.112 against the crank pin's 5.407, in units of the driver's speed squared. The pin's radius moves between 0.667 and 0.708 on a crank of 0.707. Motion that stops

A steered pin keeps only the window

A Geneva's step at entry and exit belongs to its pin's circle, not to its slots. Carry the pin in a radial guide on the driver and set its radius with a stationary cam, and the wheel can be given any law at all — the pin is simply where the driver's ray meets the slot's line. Cut for a modified trapezoid, a six-slot wheel starts and stops with no step and peaks 14% below the crank pin, while the cam moves the pin by 1.2% of the crank. What the circle cannot hand over is its window: no ray before the crank pin's entry meets a resting slot, so the index can never be longer, and every degree spent holding the wheel in its slot comes out of it as the square.

 The disc, the wheel, and what has to be cut out of it. A 6-slot Geneva at -40° of driver, with the driver's locking disc of radius 13.5 drawn about its axis and the wheel drawn as the material it actually has — inside its rim, short of the 6 flat chords cut across it, one between each pair of slots, and clear of the 6 slots. The disc and the wheel share the region near the line of centres, so the disc has to be cut away wherever the wheel is ever there while the pin is driving. Swept over the whole index that cut-away spans 126.8°, against an index sweep of 120° — it is wider, because the rim is still swinging through the disc's circle after the pin has left the slot. Drawn wrongly

Six things an indexer is not

More pins make a Geneva smoother, a finer ratchet is a more precise one, a locked wheel is a still wheel, a curved slot gives a smooth start, a cam indexer is always gentler than a Geneva, and a mechanism whose states are all in one piece can go from any of them to any other. Six things said about indexing mechanisms, each the natural reading of something true, each answered by a number the timing essays measured. They fail the same way: each judges the mechanism by its motion, and an indexer is decided at its switches — the instants a pin enters, a pawl drops or a lock takes over.

 A V of two faces, each turned off the line to the pole. The four-bar of frame 4, crank 1.2, coupler 3.4 and rocker 2.8 at 70° of crank, its coupler's round nose of radius 0.3 resting in a V of two flat faces on the frame. Each face's normal is turned off the line from the nose to the coupler's pole so that at the contact it misses the pole by 12.79°, one each way, so that each face holds the crank one way at first order. The shaded wedges are the friction cones for a coefficient of 0.2, half-angle 11.3°: the pole lies outside each, so released from either side the crank springs back. Each face's moment per radian, through a contact of stiffness 10⁶, is 72 thousand. The chain before the lengths

Return and stiffness come from the same aim

A stop with friction looked like a trade: aim it near its pole and friction multiplies its stiffness seventeen times, aim it outside the friction cone and it springs back when released. The trade is an artefact of how the gain was counted. Friction's factor multiplies a stiffness that vanishes at the pole, and measured outright a stop aimed further away is stiffer at every friction — 271 per radian at 1.1° and 104,000 at 21.3° with none, 4,631 and 184,000 at a coefficient of 0.3. Return and stiffness are bought by the same aim. What the aim costs is the other side: a face aimed off its pole holds nothing the other way, so a stop that must return both ways is a V of two faces, and across a friction band from 0.1 to 0.2 a V aimed 12.3° each way returns from both and is 8,800 times stiffer than the seat at the pole.

 Near its threshold, the disc's flank sharpens to a point under the pin. One root of a twelve-pin cycloidal disc, pins of radius 5 on a ring of 60, cut with the eccentric at 0.97 of the threshold 0.9740 R/N. The line is the disc's edge; the circle is a pin touching it at the flank's sharpest point, 31.0° of the eccentric from the root. There the edge bulges towards the pin with a radius of 2.23, the flank's bend 7.23 less the pin's 5 (small circle, centre marked), so the contact is convex against convex, and a line contact's pressure per unit load there is 1.80 times a pin on a flat. At the root itself the edge is a hollow round the pin. The shape is the unknown

A disc cut at its threshold grows a point

A cycloidal disc's roots stay untrimmed up to an eccentric where the pin path's flank bends with the pin's own radius, and it looked as though a disc cut exactly there would have its most conformal contact, the pin and the flank sharing a curvature. It has its least. The disc's edge is the path offset inwards by the pin's radius, so where the path bends round a centre on the disc's side the edge bulges towards the pin with the path's radius less the pin's: at the threshold, a point. A line contact's pressure per unit load there rises as the inverse square root of the distance to the threshold, and the pins passing it are carrying load. Each of four drives has an eccentric of least contact stress at 0.67 to 0.80 of its threshold; cut at 0.97, the peak pressure is 9% to 80% higher for a lever 21% to 46% longer.

 Where on the coupler a scheduled crank is smoothest. The standard four-bar's coupler plane, A at the origin and B at (1, 0), shaded darker where a point's jerk per unit of its own travel over the stroke from 30° to 150° is smaller, with the crank's speed scheduled to make that point's jerk least: from 1.34 to 5.12, measured in time over a stroke of fixed duration. The smoothest scheduled point is (−0.02, −0.80), at 1.263 against 4.555 with a steady crank; the smoothest with a steady crank is (−1.00, 1.25), at 3.397. Ball's point at the stroke's middle lies at (1.55, 3.27), off the map, where it scores 1.702 scheduled. The motion, not the mechanism

The smoothest coupler point is chosen in time

A crank's speed can be scheduled to cut a coupler point's jerk, though never to nought. Which point should a designer free to choose it pick? Measured as the schedule essays measured jerk, per crank speed cubed, the answer is a point whose schedule swings the crank's speed four to one, and it is a cheat: that measure divides by the speed it changes. Measured in time over a stroke of fixed duration, per unit of the point's own travel, the smoothest point of the standard four-bar is (−0.02, −0.80) in the coupler's frame. Its schedule cuts its jerk by 72% with a speed swing under two to one, and it is 63% smoother than the best a steady crank finds anywhere. Ball's point, the instant's straightest, lies in a second, shallower valley four coupler lengths away.

 A seventh leg does no work on the free twist only when it leans as the six do. At home, level at the dead yaw, the seventh leg's signed work on the twist the six legs cannot resist, against how far round the axis its platform attachment sits from its base attachment. It is nought at exactly two leans, 82.5° and 97.5°, the two leans of the six legs themselves (ticks), positive between them and negative everywhere else, most negative at 278°. The best leg of the earlier placement map leans 285° and does work −0.506, near that extreme of −0.507. Several legs, one platform

A seventh leg must lean its own way

Seven legs on a Gough platform fail only where the twist the six leave free does no work on the seventh. Near the paired platform's dead yaw that set is decided by one angle: how far round the axis a leg's platform attachment sits from its base attachment. The six legs all lean 82.5° or 97.5°. A seventh at either lean joins them and is singular at every level position; a seventh at any other lean is singular at none. Tilted, the failing leans spread out from the six's, to 74.5°–107° at four degrees of tilt and 44.5°–137.5° at eight, and never reach the far side of the axis, where the best leg leans.

 Every boundary touches the circle on the rod. The coupler's plane in the rod's units, crank pin at (0, 0) and rocker pin at (1, 0). Shaded: the coupler points whose three departing double points are all real at a ratio of rocker to crank of 1, the region where R = λ⁴A + λ²B + C is negative; 8.1% of a window from −1.4 to 2.4 along the rod and ±1.5 across it. Dotted: the circles one rod from each pin, the boundary at the family's two ends. Dashed: the circle with the rod as its diameter. Every boundary in the family touches that circle and never crosses it; at this ratio it touches at (0.50, ±0.50), the points where −B/2A = λ². Inside it all three are real at every ratio. How many answers

Three discs decide the real double points

A four-bar growing into the elliptic trammel loses three double points, and how many of them are real depends on the coupler point and on the ratio λ of rocker to crank. The boundary of the region where all three are real is one polynomial, R = λ⁴A + λ²B + C, quadratic in λ² and short when written in the squared distances from the two pins. Its sign is the count at every one of 23,100 cells tried. As a quadratic in λ² its discriminant is the circle with the rod as diameter times a figure of eight, so that circle is touched by every boundary and crossed by none, and a coupler point's whole history along the family is settled by three discs.

 Where two coefficients vanish, a curve; where three do, two points. The second leg's three joints each turned about the shared line by ψ₁, ψ₂ and ψ₃. Every point of the curve makes the shake's second- and third-order coefficients nought, σ₂ = σ₃ = 0, found by continuation from the earlier two-joint tuning (black) both ways, over an arc of 7.25 rad, until it reaches configurations where the two legs come close to sharing a second twist (crosses), where every coefficient runs off to infinity. Left: ψ₂ against ψ₁; right: ψ₃ against ψ₁. The curve is drawn in one colour where σ₄ is negative and another where it is positive; it changes sign at the two dots, (106.31°, 140.35°, 24.66°) and (136.39°, 57.24°, 18.34°), where σ₄ is nought too. Marked: 50% of the way along, at (107.0°, 141.4°, 25.2°), where σ₄ is positive. Out of the plane

Three tuned joints stop at the fifth order

Two legs of three revolute joints that share a line of force let their platform shake, and every joint of the second leg turned about that line clears one more order: the square of the push, then the cube, then the fourth power. A third tuned joint was expected to buy the fifth. Read off the two legs' own Taylor series instead of a push test, the set where the second- and third-order coefficients vanish is a curve in the three tuning angles, and the fourth-order coefficient crosses nought on it twice. At both points the gap falls as the fifth power of the push, confirmed independently with slopes of 4.97 and 4.89. The fifth-order coefficient is −0.0042 at one and 0.109 at the other, and with three angles spent there is nothing left to clear it: three tuned joints stop at the fifth order. And a higher order is not always more travel: in a millionth of play the second of them moves less than the fourth-order pair.

 A straight move ends where its posture does. The six-joint arm with its stops on, moving its tool in a straight line at a fixed orientation in the posture it starts in (back · elbow up · wrist flipped). Faint: the arm at the start. Solid: where the move ends, 66.0 cm along, stopped by the base joint at its stop. Dashed: the same line continued, which this posture cannot follow. Of the arm's usable postures at the start, the best would have gone 125.5 cm. One path to the tool

A straight move ends at a stop

A six-joint arm asked to move its tool in a straight line at a fixed orientation keeps the posture it starts in, because changing posture means passing a singularity. Followed from three hundred random poses inside the arm's joint stops, in random directions, the starting posture carries the tool a median of 19.5 cm. Half the moves end at a joint stop, two in five where the arm stretches straight and its posture runs out, and three in a hundred near the wrist singularity, the one the literature on straight-line moves is about. The best of the arm's other usable postures would have gone a median of 34.5 cm, and in a third of the moves it would have gone further. On the way, an earlier count of how many postures an arm can hold was found to be reading its base angle a turn out.

 A cam cleat whose pin swings on a sprung link. A cam cleat at a lean of 2°, below its pin's lock lean of 2.56°, its pin carried on a link that swings about a frictionless pivot 2.2 pin heights above the rope, held by a spring. The link's lever is chosen so that, with the spring a quarter as stiff as the rope at that lever, cam and link turning together squeeze the rope 12.53 times the pull, the pin's own bottom edge. Along a path the pull rises to one, falls to nought and rises again. At this step the pull is 0: a fixed pin squeezes 12.53, this link 0.00, and a link at half that ratio 6.27. Contacts that only push

A cleat on a swinging pin has two leans

Carry a cam cleat's pin on a sprung link and, while the pin holds, cam and link turn together about the link's pivot: a second cam, frictionless, whose squeeze per unit pull is the cot of its own lean softened by the spring against the rope. A rising load is met at the pin's edge and a falling one at the link's, so a slack cleat keeps only what the link falls short of the edge — 12.5 times the pull on a fixed pin at 2°, 6.3 on a link at half the edge, nothing on a link at the edge itself. Nothing in it depends on the load's size: every force doubles with the pull.

 A rod on three blocks, on three slides through one point, and it moves. The double slider of slides at 70° and a rod of 2, with a third block on the rod at a point of its rolling circle (radius 1.064, drawn at this instant) and a third slide along the line that point draws, through the same crossing. Five bodies and six joints of one freedom each: the count gives 3·4 − 2·6 = 0. The rod turns anyway, all the way round, every pin on its slide, because each point of the rolling circle runs on a line through the crossing and the third slide is that line. The paths points trace

A third slide moves only through the crossing

Every point of a double slider's rolling circle runs on a line through the slides' crossing, so a rod can ride a third block on a third slide there, and a fourth and a fifth. The count gives nought, then minus one, then minus two; the rod turns freely every time, because each slide past the second is implied by the first two. Build the third slide slightly wrong and the count wins, in a way that depends on the error's kind: moved parallel, the rod turns freely inside the block's clearance and does not assemble outside it; turned by ε about the crossing, it moves only in two windows a turn, of 2·arcsin(c/(2r sin ε)) each, where the third pin passes through the crossing.

 Each family carries one self-stress, and the price is how hard the errors load it. Dixon's framework pushed 0.5 along B₂'s line, on the family it starts in. Pinned and pushed it has eight free coordinates against nine bars, so it carries exactly one self-stress: bar forces that balance at every joint. Each bar is drawn with a width in proportion to its share of that stress, tension solid and compression dashed: B₁W₁ -0.126, B₁W₂ 0.210, B₁W₃ -0.140, B₂W₁ 0.270, B₂W₂ -0.395, B₂W₃ 0.350, B₃W₁ -0.333, B₃W₂ 0.587, B₃W₃ -0.337. A framework as built with joint errors d pays, per unit ρ, |ω·e| / Σ|ωᵢ|, where e is the bar-length error its joints make. Framework 4 pays 0.0293 on the near family and 0.0267 on the far one; framework 7 pays 0.0131 and 0.0023. What can move

The price is one self-stress

A Dixon framework built with its joints a little out of place has to be carried past its crossing by play, and the play it needs was measured by bisection, framework by framework. It has a closed form. Pinned and pushed, the framework carries exactly one self-stress on each family, and the least play is the errors' bar misfits weighted by that stress, over the stress's size. The formula matches every measured price to 0.6%, says why two frameworks of ten prefer the far family, and finds that one in twelve does.

 Half a turn on, the six legs lean alike again, and the seventh still the other way. The paired Gough platform from above, level, turned half a turn from its dead yaw: the base circle of radius 2.2, the platform circle of radius 1.1 (dashed), and the six legs, each now leaning 262.5° or 277.5° round the axis, two angles with the same sine, so the six are singular here at every position. The seventh leg (heavy) leans 105°, with the opposite sine, and does work 0.373 to 0.560 in size on the six legs' free twist across the level workspace: never nought. Several legs, one platform

A seventh leg has no dead yaw of its own

Seven legs can only be singular where six already are, since a 7 × 6 matrix that loses rank takes every 6 × 6 minor with it. Level, over a whole turn of yaw and the whole workspace, the paired Gough platform's six legs are singular at two yaws only, the dead yaw and the yaw half a turn on, and at both a seventh leg fails exactly when it leans as the six do. The best seventh leg leans the other way at both, so it has no dead yaw of its own. Its singular set exists, and it hugs the six legs' dead yaws, but it comes in only at tilts of 15° and more.

 A crank that must come round: slow through the stroke, quick through the return. The crank's speed over a whole turn, as a share of its mean, under periodic schedules that leave it at the speed it started with and give the working stroke, 30° to 150° (shaded), a third of the cycle: smoothest stroke point, return nearly free (−0.02, −0.80), from 0.66 to 1.87; smoothest stroke point, return weighted a thousandth (−0.02, −0.80), from 0.83 to 1.46; the turn's own point, return weighted a thousandth (1.44, 1.89), from 0.81 to 1.35. Highlighted: the turn's own point, return weighted a thousandth. The motion, not the mechanism

A crank that must come round picks another point

A crank scheduled over one stroke can cut the smoothest coupler point's jerk by 72%. A crank running continuously must come back to the speed it started at, and its working stroke must keep its share of the cycle. Held to both, the same point keeps two-thirds of its gain only by putting five and a half times a steady crank's jerk into the return, and no schedule beats a steady crank on both strokes. Choose the point for the whole turn instead and a gentle schedule, its speed swinging only 1.66 to one, cuts the working stroke's jerk by 74% and the return's by 18% together. That point is three coupler lengths from the first.

 A V carried on a slide and pressed onto the nose. The four-bar of frame 4, crank 1.2, coupler 3.4 and rocker 2.8 at 70° of crank, its coupler's round nose of radius 0.3 held in a V of two flat faces. The V rides on a slide along its own axis and a spring of force F presses it onto the nose, so it can be pushed back as the crank turns. Each face's normal is turned 45° off the V's axis, and misses the line to the coupler's pole at its contact by 47.6°. The shaded wedges are the friction cones for a coefficient of 0.2. Turning the crank past the V takes a moment of 1.39 F; released, the V pushes the crank back with 0.64 F. The chain before the lengths

A V pressed onto its nose has a window

A V of two faces holds a four-bar's coupler both ways and returns it, as long as each face is aimed off the coupler's pole by more than the friction angle. Squeeze the nose between both faces and two different things can happen. A rigid V cut a hundred-thousandth tight cannot lock, because the crank moves the nose across the V and never into it; it only stops short of home, inside a dead band that is μ/tan ψ of the play an equally loose fit would leave. A V pressed on by a spring, as a detent is, rides back along its own axis, and that is the direction a wedge locks in. It returns from an aim of arctan μ at the contact and locks at 90° less arctan μ about its axis, so its aim has a window, and the V that is most nearly reversible inside it is a right angle.

 A seventh joint steers the base roll off its stop. The S-R-S arm with stops on every joint, moving its tool in a straight line at fixed orientation from the same start (faint). With its upper-arm roll locked, a six-joint arm, the move ends 27.0 cm along at the base roll at its stop (dashed arm). With all seven joints and the self-motion steered at every step to move the joint nearest its stop away from it, it ends 61.5 cm along at the base roll at its stop (solid arm). Shown 60.0 cm along, with the six-joint arm stopped where its move ended. One path to the tool

A seventh joint cannot move the elbow

A seven-joint arm can change its shape while its tool stands still, and on a straight move that freedom can be spent keeping its joints off their stops. Spent that way, it lengthens the median move from 23.3 cm to 33.3 cm, against the same arm with its extra joint locked. But the gain is all or nothing. Half the moves a stop used to end gain under a tenth, and a quarter gain more than two and a half times. One stop it never gets round: the elbow's, whose angle is set by the distance from shoulder to wrist and is the same, to 10⁻¹⁵ rad, whatever the other joints do. Nor does it ever reach past the edge of the arm's reach.

 Three steered pins hand a six-slot wheel from slot to slot. A 6-slot Geneva wheel with no locking disc and a driver carrying 3 pins, each in a radial guide whose radius a stationary cam sets. Each pin is in some slot for 120° of every driver turn: it runs in along a resting slot, indexes the wheel on a modified sine law over 40°, and runs out along the slot it moved. The solid curve is the path every pin follows while it is in a slot; the dashed circle is the crank's. At the moment shown pin 1 is running in, pin 2 is clear of the wheel, pin 3 is clear of the wheel; the wheel is held by one slot. Motion that stops

A wheel held by its slots needs three pins

A steered pin can hold a Geneva's wheel by running along a resting slot, but every pin is in some slot for exactly the crank pin's window, 180° less 360°/n of every turn, because it can leave a slot only by the mouth it will reach. Two windows of less than half a turn can never cover a turn, so two pins cannot hold a wheel without its disc at any slot count. Three can, on six slots and more, and four on four. At the Geneva's own share of index to dwell the index is no harder than the crank pin's, and the slots hold the wheel tighter than any lock in the field. The disc's work moves to the steering cam: on a run the cam's pressure angle is the run's own angle, and the six-slot drive's runs are 40°.

 A growing four-bar walks away from its own double points. A four-bar with a rod of 3, its crank and rocker both of length t, pivoted so that both pivot circles pass through the slides' crossing, with the coupler point at (0.3, 0.2) in the rod's units. Drawn in units of t: the two pivot circles (dotted), their second meeting (ringed), the real coupler curve the machine draws (solid), and its real double points, filled where two real configurations meet and hollow where none reaches. At size t = 7 the curve's real double points are isolated (2.02), isolated (3.32), crossing (0.93); the number in brackets is how far the line of rod orientations through each passes from the unit circle's centre, a crossing below one and isolated above. As t grows the curve shrinks onto the crossing and the second meeting and the double points stay where they are. How many answers

A growing four-bar reaches none of its departing double points

A four-bar growing into the elliptic trammel loses three double points, and wherever they are real the question is whether the machine passes through them. In the limit it passes through none: a configuration that reaches a point a crank's length away needs its rod along an isotropic direction, which no real rod has. How fast each point leaves the machine's reach is a ratio of four distances, √(1 + λ²)·|Z|·|Z − Zₛ| / (2b·|Z + 1|·|Z + iλ|), and it vanishes only at the pivot circles' second meeting. Every crossing is lost at a finite size: through a cusp, when the law predicts it, or by meeting another crossing and leaving as a complex pair, when it does not.

 Either side of the crossing, only W₂ is somewhere else. Dixon's framework, pinned at B₁ with the bar B₁W₁ held in direction, a distance s = 0.3 along its motion before its first crossing (solid) and the same distance beyond it (dashed, W₂'s three bars only). The motion is Dixon's closed form with t = 1 − s²: every joint but W₂ is a function of s², so it is in the same place on both sides — here to the last digit — and W₂ is at s and −s, on opposite sides of the line through the black joints. The push on B₂ is the same at s and −s and is largest at the crossing, a fold; W₂'s side of the black line passes through nought. What can move

A push turns back at the crossing

Dixon's framework as built has play enough to go on past its first crossing onto either family, and the question left open was which one a push on B₂ chooses: the family whose self-stress is cheaper, or something else. Neither. Dixon's own formula says that near the crossing every joint but W₂ moves as the square of the distance along the motion, so the push is at a fold there on both sides, and no push can carry the framework through. Ten frameworks as built, pushed to their turning point and pulled back, all retrace, including the two that are cheaper beyond. Driven instead by W₂'s side of the black line, all ten go through, needing exactly the worst price on their path, and the price is continuous through the crossing.

 The pin opposite the third runs on the perpendicular slide, a quarter turn behind. The double slider of slides at 70° and a rod of 2, with a third block at a point of its rolling circle (radius 1.064) and a fourth at the opposite point, each on the slide its pin draws through the crossing. The two extra pins are the ends of a diameter of the rolling circle, so their slides are at right angles and their positions along them are 2r cos(φ − φ₃) and −2r sin(φ − φ₃): here 1.864 and 1.027. Four blocks and the count gives minus one; the rod turns, and the fourth pin passes the crossing a quarter turn after the third. The paths points trace

A fourth block can only take freedom away

A rod on a double slider can carry a third block on a third slide through the crossing and still turn, and a slide turned slightly wrong leaves it free only in two windows a turn. The hope was that a fourth block, at the point of the rolling circle opposite the third, would travel in opposition and cancel the error. It cannot, for two reasons. The opposite pin is the other end of a diameter: it runs on the perpendicular slide, a quarter of a turn out of phase, so its windows fall exactly between the third's. And whatever its place, a block only adds a condition the rod must also meet, so the rod's freedom is an intersection and can only shrink. With the double slider's own blocks exact, the third and the opposite fourth turned 0.0066 rad each lock the rod, where the third alone still moves at any error.

 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.

 B₂ and W₂ go round like the ends of a trammel bar. Dixon's framework in its own frame: the black joints on the horizontal line, the white on the vertical, nine bars joining every black to every white. Round the whole motion, all four families joined at their crossings, B₂ is at √2 cos ψ on the black line and W₂ at √2 sin ψ on the white, so the bar between them (heavy) slides with its ends on two perpendicular lines and turns once as the framework goes once round. The corner of the rectangle on B₂ and W₂ (small dot) is at angle ψ on the dotted circle of radius √2, and goes round it once. Here ψ = 60°: B₂ at 0.707, W₂ at 1.225. The arc at W₂ is the motor's angle, from the bar W₂B₂ to the bar W₂B₃, 37.2°. What can move

A drive that goes round is an angle

Dixon's framework moves round a loop through four crossings, and a push on B₂ folds at two of them while a drive on W₂'s side folds at the other two. A lever moving both was proposed as the drive that would go all the way round. It cannot, and no drive that is one number read off the joints can: round the whole loop B₂ and W₂ sit at √2 cos ψ and √2 sin ψ on their two lines, the ends of a trammel bar, so the loop is a circle and every such number has a top and a bottom on it. A lever only moves its two folds off the crossings. What goes round is an angle. Four of the framework's eighteen joint angles turn once as it goes once round, never stopping, and a motor at W₂ takes ten frameworks as built all the way round at the worst price on the loop — 1.00 to 1.46 times their worst crossing — after which every turn repeats the first.

 Moved apart, the nearest joints still go round a circle. A Dixon framework with B₂ starting 1 from the crossing point on the black line and W₂ starting 1.4 on the white, every other joint where Dixon put it: blacks at −2 and 3, whites at −1.5 and 2.5. Round its whole motion B₂ and W₂ stay on a circle of radius √(1 + 1.4²) = 1.720 (dotted), because the bar between them (heavy) keeps its length. Here ψ = 250°: B₂ at −0.588, W₂ at −1.617, W₁ at −1.704. The angle at B₂ between the bars to W₂ and W₁ turns once round the loop with its speed between 0.046 and 1.954 per unit of ψ, slowest as B₂ crosses the white line. What can move

Put the motor where the rocking bar is longest

Dixon's framework goes round its loop with its two nearest joints on the ends of a trammel bar, and a motor at a joint angle with that bar for a side carries it round with no fold. With the nearest joints starting at different distances it looked as though the loop would become an ellipse and some angle would nearly stop. The loop stays a circle at every ratio, because the bar between the two joints is a bar. Each of the four turning angles runs at one radian per unit of the loop, plus or minus the rate of its other side, which rocks fastest at a crossing and there turns at the trammel bar's length over its own. That gives each angle's speed ratio in closed form, says none of them ever stops, and puts the motor at whichever joint has the longest rocking bar.

 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.

180 essays in 2 parts: part 1 · part 2

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