Machine Kinematics
solved, then drawn
Concept

Assembly branch — where it appears

One of the ways a mechanism can be put together for the same input, and a connected component of its configuration space. A mechanism has to be taken apart to change branch, so which branch a machine was assembled in is a permanent property of that machine rather than a state it visits.

Named by 33 essays across 12 fields — each of them below, with the objects they name alongside it.

A four-bar at 60°, 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 0.0e+0 — 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 66.9°.

Four bars and four pins

The smallest interesting machine there is. Four lengths decide everything about it — which link can turn all the way round, how hard it pushes, where it stops and whether it can be assembled at all — and every one of those is a number that falls out of a solve rather than a judgement about a drawing.

linkages · Fourbar Two circles, four answers, two of them nowhere. A four-bar with its crank held at 52° is two circles: the coupler pin is 3.5 from the crank pin and 3 from the far ground pivot. Two quadratics in two unknowns, so Bézout's number is four — and the tracker finds two. The other two paths run off to infinity, and they do so for every pair of circles ever drawn: two circles meet the line at infinity in the same two points, and those are what the fourth and third answers are.

Two circles, four answers

A four-bar with its crank held still is two circles, and two circles meet twice. Bézout's theorem says four. The two missing answers are not a rounding error and are not special to these link lengths — they are the same two points for every pair of circles ever drawn, and they are the beginning of a way of counting that has so far been done by hand.

algebra · Algebra 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.

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.

wrong · Misconception the reflector, solved. A rhombus whose far vertex is held on a line through the pivot. One side is the input, the line is the mirror, and the other side comes out reflected in it — which is where negation, doubling and addition all come from. The relation it satisfies is (μ, θ) ↦ 2μ − θ, and across a sweep of 41 positions the worst departure from it is 1.3e-13 radians. Every joint here is the output of a Newton–Raphson solve on the bar lengths; nothing is placed by the formula the picture is about.

Four bars that add two angles

A rhombus on two links from one pivot points along the bisector of their angle, exactly, because a rhombus has equal sides. Hold its far vertex on a line and it reflects instead. From those two facts come negation, doubling and addition — and every whole-number combination of two angles a compiled machine needs.

computing · Compute 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.

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.

algebra · Algebra How much of a compiled machine is computing anything. Each machine's bars split two ways: the ones that build an angle — reflectors, means, rigid offsets, the arm — and the ones that carry a direction from where it was computed to where it is needed. On the smallest machines the arithmetic is nearly all of it. By the quintic the carrying is 80 per cent, and it goes on rising, because the arithmetic grows with the number of terms and the carrying grows with the number of pairs of them. That is the answer to why a universality construction is enormous, and it is not about the algebra being hard.

A parallelogram carries an angle, and only so far

A direction computed at the frame is no use where it is needed. A parallelogram carries one from point to point — but only between two points a fixed distance apart, and that single proviso is what makes a compiled machine quadratic in the number of terms and turns most of it into transport.

computing · Compute 8 postures, one tool pose. The tool is at one place, held one way. Eight different sets of joint values put it there, and this is all of them at once: two ways for the base joint to face the target, two for the elbow, and two for the wrist — two by two by two. Every posture here came out of a closed form and was checked by running the arm forward again; the worst disagreement is below 10⁻¹⁴ of a metre. Which of them a machine can actually use is a separate question, and it is answered by the joint limits rather than by the geometry.

Eight ways to hold the same tool

A six-joint arm asked to put its tool at one place, held one way, has eight answers. Not approximately eight and not eight found by looking — two for the base, two for the elbow, two for the wrist, each exact to a hundredth of a femtometre, and a search from six hundred starting postures finds those eight and no ninth.

serial · Serial A vertex is a spherical linkage, and the sectors are its link lengths. The four creases of one folded vertex, drawn as directions from the vertex itself, with the great-circle arcs between consecutive ones. Those arcs are the sector angles of the flat pattern — 80°, 60°, 100°, 120° — and they are those angles at every fold, to 8.9e-16 radians. That is the whole of the claim in the title: four axes through a point at fixed arcs from each other is a spherical four-bar, the object this site's spatial field built two phases ago, and a crease pattern's vertex is one of them with the arcs printed on the paper. The dihedral angle of the sheet at each crease is a half turn less that crease's fold angle, which here run 68.8°, 14.4°, 68.8°, 14.4°. positioned by solving, not by drawing.

Every vertex is a spherical linkage

Four creases through a point at fixed arcs from one another is a spherical four-bar — the object the spatial field is built on — with its link lengths printed on the paper as sector angles. The arcs hold to four parts in ten thousand million million at every fold, and on a flat-foldable vertex the half-angle tangents keep a ratio constant to nine figures.

networks · Network 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.

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.

algebra · Algebra A four-bar that computes log₁₀. The output rocker's angle, mapped back into y, against the input crank's angle mapped back into x. The pale curve is log₁₀ x and the solid one is what the linkage does. They agree exactly at the 3 precision points and nowhere else; the largest disagreement over the range is 0.2232° of rocker, which is 7.47e-4 in y. The chebyshev spacing put the precision points at 1.0670, 1.5000, 1.9330.

Three problems called synthesis

Prescribing a path, prescribing a whole pose, and prescribing a relation between two angles are three different problems with three different counts, and the word synthesis covers all of them. The third has a property the others do not — eliminate the coupler angle and the design equation becomes linear, so a four-bar that computes a logarithm falls out of a 3 × 3 solve with no iteration at all.

synthesis · Synthesis Two ovals of one sextic. A four-bar with ground 4, crank 1, coupler 3.5, rocker 3, its coupler point solved at 720 crank angles on each assembly. Each assembly closes on its own oval through a full turn of the crank, and the two ovals never meet: they come no closer than 1.712. Every solved point satisfies the one eliminated sextic to 6.6 × 10⁻¹⁶ of its largest term. The machine drawn solid and the one drawn faint are the same four bars at the same crank angle of 60°, and taking a pin out is the only way from one oval to the other.

The curve the other assembly draws

A crank-rocker's two assemblies do not share a coupler curve. Each draws a whole closed oval of its own through a full turn of the crank, the two ovals never meet, and both are the zero set of one sextic, so the equation a machine's own motion determines also describes a second machine it can never become.

curves · Coupler 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.

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.

algebra · Algebra 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.

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.

curves · Coupler A dyad is two circles, and that is why it has two answers. The whole of what makes a dyad easy. Two links, three pins: one pin onto something already placed at each end, one pin between them. The free pin is at a fixed distance from each of the placed ones, so it lies on both circles — and two circles meet in two points, in nought, or in one. That is a quadratic with a closed form, and its two roots are the two assembly branches every four-bar on this site has. A group of four links has no such picture: its unknowns do not separate into one circle each, the system does not factor, and what replaces the compass is Newton's method from a seed. positioned by solving, not by drawing.

What has to be solved together

Hold a link, turn a neighbour, and the rest of a mechanism comes apart into the smallest sets that can be positioned one after another. Every set of two links is two circles meeting — a quadratic, two branches, no solver. A set of four is a system, and this site's Newton solve stops being a convenience.

topology · Topology 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.

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.

curves · Coupler 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.

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.

topology · Topology 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 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.

computing · Compute 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.

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.

algebra · Algebra Crank angle θ and crank angle −θ, joined. The coupler curve alone, with 12 chords, each joining the point the tracing point reaches at a crank angle θ between 0° and 180° to the point it reaches at −θ on the same assembly. Every chord is square to the dashed axis, to 7.5 × 10⁻¹⁶ in the cosine, and every midpoint lies on it. Measured at 48 pairs, the reflection of one point misses the other by at most 4.2 × 10⁻¹⁵. The crank's own angle is the pairing: the two places the curve crosses its axis are crank angles 0° and 180°, the only angles equal to their own negatives.

A symmetric curve from a lopsided machine

A four-bar with a crank of 1, a ground of 3 and a rocker of 2.5 has no symmetry anywhere in it. Make the rocker, the coupler and the arm from the rocker pin to the tracing point one length, and the curve it draws is its own mirror image to 4 × 10⁻¹⁵, about a line through the rocker pivot turned from the ground line by exactly half the coupler's angle at that pin.

curves · Coupler A brace is one redundant equation, on purpose. The compiled machine, counted and measured, with and without 4 braces. The count says the braced machine has -3 degrees of freedom — it cannot move — and the rank of the constraint Jacobian says it has 1, the same as before. Every brace contributes exactly one equation the others already imply, which is what overconstraint is, and here it is being added deliberately: the redundancy is what removes the assemblies the count knows nothing about. This is Grübler being wrong for the useful reason rather than the embarrassing one.

A bar between two midpoints

In a parallelogram the midpoints of two opposite sides are exactly one side apart, and in the crossed assembly they are not. One bar between them admits the first and refuses the second — and it is one redundant equation per parallelogram, added on purpose, on a site whose constraint field is otherwise about overconstraint arriving by accident.

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

Where the branches meet

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

networks · Network One crank, four machines, one area. Four crank-rockers sharing only a crank of length 1, with grounds, couplers and rockers all different, each tracing the point 30% of the way from the crank pin to the rocker pin. The curves have different shapes and sizes and different places in the plane, and the area each one encloses is 2.199115 = (1 − 0.3)·π·1² — on both assemblies of every one, with a worst difference of 3.2 × 10⁻¹⁴.

The area a coupler point encloses

Trace a point on the line through a crank-rocker's two moving pins and the region its curve encloses has area (1 − u)πa²: the crank pin's own circle, scaled by how far along the line the point sits. No ground, coupler or rocker length appears in it. Four machines that share only a crank enclose 2.199115 each, to 3 × 10⁻¹⁴.

curves · Coupler 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.

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.

bodies · Body 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.

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.

timing · Intermittent 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.

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.

computing · Compute 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.

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.

timing · Intermittent How far the driven link turns is a fact about the lengths, not the chain. Each of the sixteen eight-link chains, given the arbitrary placement its own layout produces, driven from its first available choice of frame and input, and swept until a frame stops closing. 10 of the sixteen reach every angle and the rest rock through between 107° and 244°. Nothing in this chart is a property of the chains. Change the placement and the bars change; the census above them does not. It is here because it is the sharpest way to say what this field does and does not decide, and because the temptation to read a topology census as a catalogue of machines is exactly the mistake it prevents.

The chain has no lengths

Every number in this field survives multiplying every link by a different scale factor, because there are no lengths to scale. Which is also the statement of what a census cannot decide — and the sixteen eight-link chains, each given one arbitrary set of dimensions and driven, produce a chart in which nothing belongs to the chains.

topology · Topology 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.

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.

constraint · Mobility 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.

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.

serial · Serial 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.

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.

wrong · Misconception 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.

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.

serial · Algebra 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.

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.

constraint · Mobility 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.

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.

constraint · Mobility

Named alongside it

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

Configuration spaceGrashof's conditionLoop closureToleranceConnected componentCoupler curveKinematic chainMobilityChange pointCircuitRedundant constraintAssur group

All concepts