What is refuted here, and by what

A part touched on all four sides cannot move.

What decides it: Four contacts, one on each edge of a square, each a third of the way along, taken the same way round: every row's moment about the centre has the same sign, no positive combination of them can come to nothing, and the part rotates clockwise about any of a large region of centres. Moving two of the four to the other ends of their own edges — the same four edges, the same distance along — takes the margin from nought to 0.211.

Tested in A constraint that only pushes · the restraint sequence.

Which joints exist is a matter of engineering convention — somebody decided that pins, slides, screws, ball joints and flat ways were the useful ones.

What decides it: The permitted set of a surface contact is the surface's own symmetry group, which is computable from the surface's normals with no convention in it: one row per sampled point saying that the velocity a twist gives that point is tangent, and the answer is a null space. Eleven surfaces — plane, sphere, circular cylinder, prism, thread, cone, torus, spheroid, shaft with collars and two with no symmetry at all — give six distinct groups and nothing else, with the weakest rank decision separated by a factor of 6.6 × 10¹⁵.

Tested in A joint is a surface that slides on itself · the pair sequence.

A graph with the right number of links and the right number of pins is a mechanism with that many links.

What decides it: At ten links and thirteen pins, 1,878 graphs satisfy Grübler's rule with every link carrying at least two pins, and 230 of them are mechanisms. The other 1,648 contain a subchain whose own count is zero or less — 1,501 of them a plain triangle — so they move as mechanisms with fewer links, one of which is welded out of several pieces.

Tested in The mechanism is the graph · the topology sequence.

The involute is one of several workable tooth profiles, chosen because it is easy to cut.

What decides it: Requiring the velocity ratio to stay constant while the contact point slides forces the profile: the common normal at contact must pass through a fixed point, and the curve whose normal does that is the involute of the base circle. The test here is run on the drawn polyline rather than on the parameterisation — an earlier version compared each flank to itself and announced the answer it had assumed.

Tested in Why a tooth is an involute · the tooth sequence.

Two mechanisms with the same numbers of links, joints and link types are the same mechanism.

What decides it: Watt's chain and Stephenson's both have six links, seven pins and an assortment of four binary and two ternary links. No count made on either separates them. They are not isomorphic — their characteristic polynomials differ at the λ² term — they give two mechanisms and three respectively, and their shortest independent loops involve eight link lengths against nine.

Tested in Same links, same pins, different machines · the topology sequence.

A universal joint couples two shafts, so the output turns at the speed of the input.

What decides it: The output angle from the closed form and from the solved spatial loop agree, and both run fast then slow twice per revolution, between cos β and 1/cos β for a shaft angle β. At 30° the output is a third faster at some points of the turn than at others, and no choice of joint quality changes it — the variation is fixed by the angle alone.

Tested in The joint that is not constant velocity · the spatial sequence.

Watt's linkage draws a straight line.

What decides it: The traced coupler path is measured against the best-fit line over the full stroke. Watt's deviates by about 9% of its travel and Chebyshev's by 12%; Peaucellier's cell, drawn by the same machinery, is straight to 9.8 × 10⁻¹⁶ of its span. Watt knew this and said so; the approximation was the point, and it is the later retelling that dropped the word.

Tested in The straight-line problem · the coupler sequence.

A check that compares a program against itself is still a check, as long as the two calculations look different.

What decides it: This site's velocity check obtains the analytic velocity by solving with the same Jacobian it is testing, so it agreed to 10⁻¹³ while four of that Jacobian's terms carried the wrong sign. The sensitivity comparison here does not share the route: one side rebuilds the mechanism and solves from scratch, the other differentiates the equations. They agree to 2 × 10⁻⁸ relative, and a sign error cannot survive both.

Tested in Two routes to a sensitivity · the tolerance sequence.

A four-bar jams at its toggle — the toggle and the collapsed transmission angle are two names for the same configuration.

What decides it: Computed on one linkage, the two are 222° apart. At the toggle the mechanical advantage reaches 1,673 and the mechanism is at its strongest; at the worst transmission angle the coupler's force goes into the output bearing and it is at its weakest. Opposite situations, opposite design responses.

Tested in Two things called jamming · the misconception sequence.

A higher-degree curve always costs more to build than a lower-degree one.

What decides it: A lemniscate is degree four and expands to five cosine terms; a general cubic is degree three and expands to eight. Compiled, the lemniscate is fifty bars and the cubic a hundred and one. The lemniscate's symmetry cancels ten of the fifteen frequency pairs its degree allows, and cancellation is a property of the coefficients, not of the degree.

Tested in A circle costs one term · the compute sequence.

Six contacts remove a body's six freedoms, so a part on six pads is fixed.

What decides it: The 3-2-1 scheme's six contacts are six independent constraints — rank six, and a bilateral version of them would fix the box exactly — and the box lifts straight off. Six vectors can span six dimensions and cannot positively span them, so the minimum is seven; and across four thousand random six-contact arrangements on a box, the number that held is nought.

Tested in Four in the plane and seven in space · the restraint sequence.

Two mechanisms with the same joint counts and the same mobility are kinematically equivalent.

What decides it: A four-bar and a slider-crank have four links, four one-freedom pairs and mobility one; so does a four-bar with one pin replaced by a screw. All three agree on every count and on the rank of the constraint Jacobian, and their coupler points trace a sextic, a different sextic and a curve that does not close at all. The count is the dimension of the permitted set and the permitted sets are three different groups.

Tested in The count cannot tell a pin from a slide · the pair sequence.

A count and a rank agreeing is a mechanism confirmed.

What decides it: Across the 71 eight-link graphs that pass Grübler's rule, 46 contain a subchain whose own mobility is exactly nought. Every one of those 46 measures a Jacobian mobility of 1, which is what the count says, and both are right — the assembly does move with one degree of freedom. It is not an eight-link mechanism: three or more of its links never move relative to one another. Neither instrument distinguishes it from the 16 that are.

Tested in What a count cannot see · the topology sequence.

Two kinematic chains with the same characteristic polynomial are the same chain.

What decides it: Among the 230 ten-link chains there are two cospectral pairs. The first pair does not even share an assortment — one chain has two ternary and two quaternary links, the other has six ternary — so the cheapest invariant in the subject separates them and the spectrum does not. Their polynomials agree in all eleven coefficients. There are no such pairs at six or eight links, which is why the test survived.

Tested in Right until the size nobody checked · the topology sequence.

An extra support makes a location more secure.

What decides it: A seventh contact on an exactly constrained part adds no rank, so it removes no freedom. What it does is make one contact of the set unable to touch: with contact errors of ten microns, the added pad under a Kelvin clamp is left with a gap of 3.5 µm. Nothing about the part's position has been made more secure, and one of the seven has become a rattle.

Tested in The seventh contact · the mobility sequence.

A figure that draws correctly was computed correctly.

What decides it: Four terms in this site's analytic Jacobian carried the wrong sign from the foundation phase onward. Every picture it ever drew was right, because a wrong derivative is still usually a descent direction — it costs steps, not answers. What it did instead was quietly refuse positions: 0 of 360 crank angles at zero coupler offset, 267 of 360 at an offset of 1.2. A sweep with missing frames is a sweep, and a published count was wrong by two thirds.

Tested in The solver was refusing a quarter of the sweep · the misconception sequence.

A mechanism can be given any number of freedoms from one to six by choosing the right single joint or the right compound one, so a five-freedom joint is a design option.

What decides it: There is no five-dimensional subgroup of the rigid displacements. Twenty thousand random five-dimensional subspaces of the twists were tested for closure under the Lie bracket and every one of them generated the whole six-dimensional algebra; no subalgebra of dimension five appears anywhere in this field — not from a surface, not from a chain, not from a loop, not from an intersection. A joint with five freedoms permits a set that is not closed under composition, so a body reached by two of its displacements in succession is somewhere the joint does not allow.

Tested in Twelve kinds of freedom · the pair sequence.

A chain drawn one way and the same chain drawn another way can be told apart by looking.

What decides it: A chain relabelled at random forty times produces forty different edge lists, forty different adjacency matrices and one canonical form. The layout every figure in this field uses reads only that canonical form, so the forty relabellings produce byte-identical drawings — asserted by relabelling and requiring the closest pair of links in the picture to be unchanged to 10⁻⁹.

Tested in Deciding that two chains are one · the topology sequence.

An escapement's pallets span a whole number of tooth pitches.

What decides it: Then both pallets meet the teeth in step: they lock together and release together, and the wheel is arrested by both or by neither. The beat budget for such a mechanism sums to −0.000000000° — the impulse takes the wheel forward 2.267° and the drop takes it back 2.235° — against half a pitch of 6° for the correct span. It is not a badly proportioned escapement; it is a mechanism that cannot escape.

Tested in The escapement that could not alternate · the misconception sequence.

A pin joint is a revolute pair: two links sharing an axis, with one relative freedom between them.

What decides it: Only if the pin and the hole have the same diameter, in which case it does not turn. A real pin of radius r in a hole of radius R has a centre free anywhere within c = R − r, which is kinematically a binary link of length c with a revolute at each end. Grübler's own arithmetic then gives a four-bar with play at all four pins five degrees of freedom rather than one, and four of them are a hundredth of a millimetre wide.

Tested in A clearance is a link · the clearance sequence.

Backlash is slop — wear, or a pair that was made badly. A good gear pair has none.

What decides it: A pair with none cannot be assembled: at the centre distance where the two teeth exactly fill the circular pitch the mesh is at its limit, and any closer the teeth interfere and the gears will not go together. Backlash is bought with a centre distance — 0.03 of centre-distance error buys 0.022 of it on a 20-and-40-tooth pair — and it is specified on the drawing for the same reason a running fit is.

Tested in Backlash is an allowance · the tooth sequence.

Multiplying an angle by n with linkages costs about n gadgets, so high-degree curves are expensive because their angle arithmetic is expensive.

What decides it: Doubling is one reflector and adding is two gadgets, so n is reached by its binary expansion: eight costs three gadgets and seven costs six, for a smaller number. Across a whole compiled machine the arithmetic never exceeds a few dozen bars, while the transport reaches a hundred and fifty-four parallelograms on the same machine.

Tested in Doubling is cheaper than adding · the compute sequence.

A search that has stopped finding new solutions has found them all.

What decides it: This site reported sixteen assemblies of its platform from a random search — sixteen from 400 starting points and still sixteen from 4,000, which looks like convergence. Tracking every path says twenty-eight poses exist and sixteen of them are real at that pose. The search was exactly right and could not have known: a lower bound that happens to be tight is the one outcome that leaves no evidence.

Tested in The search that was right · the algebra sequence.

Watching where a mechanism's points go tells you what group its motion is in.

What decides it: A point's orbit has the dimension of the group minus the dimension of its stabiliser, and two of the twelve groups have identical orbit dimensions from different dimensions: planar motion and spherical motion are both three-dimensional and both sweep a chosen point over a two-dimensional surface. A point sees the orbit and never the stabiliser, so a trajectory is evidence about a group and not a determination of one.

Tested in What a point sees · the pair sequence.

A chain of n links gives n mechanisms, one for each link that can be grounded.

What decides it: The four-link chain gives one mechanism, not four: every link is carried to every other by a symmetry of the chain, so grounding any of them gives the same machine drawn from a different angle. Across the census the average is 1.00, 2.50, 4.44 and 7.97 mechanisms per chain at four, six, eight and ten links — always below the link count, and approaching it only because large chains mostly have no symmetry at all.

Tested in Which link to bolt down · the topology sequence.

A part whose contact rows leave a null direction can move along it.

What decides it: An ellipse of semi-axes 1.4 and 0.9 in a pocket touching it at the ends of its axes has rank two and a null direction that is a pure spin. Rotating it by θ makes its reach toward the top and bottom walls grow as √(a²sin²θ + b²cos²θ), so it penetrates both by (a²−b²)θ²/2b — measured at 6.4 × 10⁻⁵ for a hundredth of a radian, with a fitted exponent of 1.9944 against 2. A circle in the same pocket penetrates by exactly nothing at every angle.

Tested in Free to turn and unable to · the restraint sequence.

A four-bar's coupler curve is one closed curve, which the mechanism covers in two pieces, one per assembly, meeting at the limit positions.

What decides it: On the standard crank-rocker each assembly draws a complete closed oval through a full turn of the crank, 720 solved positions apiece, and the two ovals never come closer than 1.712. The two assemblies' rocker pins are never less than 4.132 apart at any whole degree. The description fits a triple rocker, whose assemblies do meet at its dead centres, and not the linkage it was written about.

Tested in The curve the other assembly draws · the coupler sequence.

A chain with j pins offers 2j different ways of being driven.

What decides it: The four-link chain has four pins and eight ordered frame-and-input pairs, and every one of them gives the same driven mechanism: a single orbit. Watt's chain has fourteen pairs and four orbits, Stephenson's fourteen and five. Across the eight-link census, 320 listed pairs are 153 genuinely different driven mechanisms, and the ratio is not constant — it depends on how symmetric each chain is.

Tested in Eight ways to drive it, and one machine · the topology sequence.

A mechanism with fewer controls than coordinates cannot reach every configuration — a car with two controls and three pose coordinates is missing a direction, and no sequence of manoeuvres can supply it.

What decides it: The forbidden direction is the Lie bracket of two permitted ones, so it is reached by performing them in the wrong order. The four-leg manoeuvre is flown at seven amplitudes and the net displacement lies along the computed bracket, with a fitted exponent of 1.997 against a predicted 2. What the constraint costs is a power of the amplitude, not the direction: a mechanism with 0.5 m of room gains 100 mm a cycle and one with 0.25 m gains 25 mm, and both get there.

Tested in Not unreachable, only expensive · the misconception sequence.

An extra contact makes a hold more secure.

What decides it: On a hexagon held by five contacts, one of the five can be removed with the margin unchanged at 0.0914 to every figure. It is constraining nothing. What it does instead is add a second condition the contact errors must satisfy before the part goes in at all — the arrangement's fit conditions go from one to two — and become the one contact in the set that can be left with a gap.

Tested in The contact that is free not to touch · the restraint sequence.

Every solution of a linkage's distance equations is a configuration of the linkage.

What decides it: Watt's chain decomposes into two dyads, so it has 2² = 4 assemblies at a given input angle. Solving its pin coordinates from 240 random seeds returns eight distinct converged solutions, all with residuals below 10⁻¹². Four of the eight have the ternary link mirrored: three distances fix a triangle only up to reflection, and a reflected link is a different part rather than a different pose.

Tested in Two to the power of the dyads · the algebra sequence.

A coupler curve that never crosses itself has no double points, so every point of the equation is a point the coupler passes through.

What decides it: The standard crank-rocker draws two ovals that never cross and never touch, and its sextic still has a real double point, at (3.964, −0.368), 0.628 from the nearer oval. It is isolated: both orientations the coupler would need there are complex, so no assembly reaches it. Across 400 coupler curves the number of real double points is one or three, never none, because the other two come as a complex-conjugate pair.

Tested in A point the machine never reaches · the coupler sequence.

A machine whose axes are nominally parallel behaves like one whose axes are exactly parallel, to within the alignment error.

What decides it: The group is not a continuous function of the alignment. Three pins at exactly parallel axes have a displacement span of three; tilt two of them by 0.06 radians and the span is six, with no intermediate value at any tilt — the dimension jumps at zero. What varies continuously with the tilt is the bracket defect, which is the right quantity to quote for a machine made to a tolerance, and it is a different number from the dimension.

Tested in Four joints that give a group, and four that do not · the pair sequence.

A part that is free to move in every direction is not restrained.

What decides it: A unit disc among three points at radius 1.1 has rank two, a margin of nought and a uniformly shaded plane of permitted rotation centres — free at every instant, in every direction, at every configuration it can reach. The area its centre can reach is 0.060, bounded, and it cannot leave. Push the obstacles out to 1.25 radii and the reachable area is 26.9 and unbounded. The threshold is exactly 1/sin(π/3) = 1.154701.

Tested in Free at every instant and going nowhere · the restraint sequence.

Any linkage can be positioned by drawing circles, two links at a time.

What decides it: Of the sixteen eight-link chains, four have no choice of frame and input at all for which the remaining links come apart two at a time — 320 frame-and-input pairs between the sixteen, and on those four chains none of their twenty each works. Their smallest indecomposable group is four links, and a group of four does not separate into one circle per unknown.

Tested in What has to be solved together · the topology sequence.

A self-calibration that reaches a small residual has found the machine.

What decides it: An angle-sensed platform fitted to its own readings with no reference length reaches a residual of 1.5e-10 and returns a machine 0.24% of the true size. The correctly anchored fit, on the same readings, settles at 1.05e-4 — seven hundred thousand times larger, and right. Every residual here is a difference of lengths, so shrinking the machine shrinks the residual, and the objective's global minimum is a platform of no size at all.

Tested in A platform that measures itself · the parallel sequence.

A linkage that cannot be positioned in closed form has been driven from the wrong link.

What decides it: Four of the sixteen eight-link chains have twenty frame-and-input choices each, and none of the eighty is all dyads. Changing the driving link on those chains changes which four- or six-link group has to be solved as a system; it never removes one. At ten links the same is true of ninety chains of two hundred and thirty.

Tested in Four that a compass cannot reach · the topology sequence.

Grübler's count fails only when the link lengths are special.

What decides it: At ten links, 1,648 of the 1,878 graphs satisfying the count contain a subchain that is already a structure, at every set of dimensions. On 1,165 of them the count returns one, the rank of the constraint Jacobian returns one, and both are right about an assembly that has fewer links than it is described as having. No choice of lengths repairs it, because a triangle does not bend.

Tested in The count was right and the name was wrong · the mobility sequence.

A linkage built to a correct construction draws what the construction says it draws.

What decides it: The construction fixes the bar lengths and not the assembly. The machine compiled from xy = 0.5 has four parallelograms and therefore sixteen assemblies; eight close, and four of those eight trace a curve on which the polynomial reaches 0.15. Their closure residual is 9.6 × 10⁻¹⁵, identical to the correct ones, because a crossed parallelogram satisfies all four of its bars.

Tested in The proof drew more than the curve · the compute sequence.

A parallel platform can change from one assembly mode to another only by passing through a direct singularity.

What decides it: With the first motor held at 216°, the other two are driven round a circle of 8° radius about (98.15°, 185.32°), which encloses a cusp of the singular curve at (101.35°, 185.32°). The assembly that starts with the platform at −79.4° completes the loop continuously and arrives at the assembly at −27.3°, another pose of the same three motor angles. Along the whole loop det A and every leg's entry in B keep their signs, and the smallest singular value of the leg lines never falls below 0.0716.

Tested in Round a cusp into another assembly · the parallel sequence.

A Panhard rod locates the axle.

What decides it: It locates the axle on an arc. Over ±80 mm of travel the axle moves 3.56 mm sideways, always in the same direction, so a car on a rough road is shifted sideways twice per bounce. A Watt's linkage over the same travel moves it 34 µm — and the two errors grow as the square and the fifth power of the travel respectively, so the ratio between them is not a constant either.

Tested in Holding an axle still · the coupler sequence.

An assembly that can be taken apart can be taken apart one piece at a time.

What decides it: Two congruent Z-shaped pieces in a tray: each has four contacts with the other whose normals point in all four axis directions, so neither has a single free translation. The pair, whose six contacts are all with the tray, has exactly one — straight up. The assembly comes apart and no piece of it is the first.

Tested in Neither part comes out first · the restraint sequence.

Enumerating chains is a matter of generating the graphs and removing duplicates.

What decides it: Generating every labelled graph with the right degree sequence and deduplicating by canonical form reaches 8,494 complete graphs at eight links to produce 71, and does not finish at ten within ten minutes. The same enumeration with one pruning rule reaches 92 and 3,000 respectively, and finishes ten links in about half a second — a difference of shape rather than of speed.

Tested in The candidates a search throws away · the topology sequence.

An overconstrained mechanism is one whose joints all lie in a proper subgroup of the rigid displacements, so finding the subgroup explains every such mechanism.

What decides it: Four of the six overconstrained loops here are of that kind and two are not. Bennett's linkage and a Bricard six-bar have the same counts, the same ranks and the same redundancy as a planar four-bar, and the logarithms of the displacements their moving link reaches span four dimensions and close under the bracket at six. There is no proper subgroup containing their motion, and the composite of two reached displacements lies 0.31 and 0.20 of its own length outside the four dimensions the reached ones occupy.

Tested in Compose two positions and see where you land · the pair sequence.

Any distribution of pins among links that satisfies the mobility arithmetic can be built.

What decides it: Eleven distributions are admissible at ten links. The four requiring a link with six, seven or eight pins produce seventy-eight graphs between them, every one connected, simple, with no link below two pins and a Grübler count of exactly one — and every one of the seventy-eight contains a subchain that is already a structure. The arithmetic admits them and the adjacency refuses them, with no exceptions.

Tested in Eleven assortments and four that are empty · the topology sequence.

The spatial mechanisms on this site are single loops because single loops are what got built.

What decides it: A spatial all-revolute chain of one degree of freedom needs (6n−7)/5 joints, so only 7, 12 and 17 links are admissible. At seven links the degrees must sum to fourteen over seven links with none below two, so every link is binary and the graph is a seven-cycle: exactly one chain, and it is a loop. The next admissible size is twelve links, where 33 graphs pass the count and five are chains.

Tested in In space there is one chain · the spatial sequence.

Grübler's count tells you whether an assembly can move.

What decides it: It tells you the difference between the freedoms and the dependencies among the constraints, exactly, on every mechanism there is. A deployable ring of eight angulated pairs counts at nought and has four freedoms and four dependencies; a Miura sheet of thirty-six panels counts at minus fifteen and has one freedom and sixteen dependencies. The count is right about the difference in both cases and useless about the answer.

Tested in Six things a network is not · the misconception sequence.

Bracing a compiled linkage makes it draw its curve for a whole turn of its input.

What decides it: Bracing removes discrete alternative assemblies. It does not remove the configurations at which two placements of a joint merge, and a machine driven through one comes out on the other placement with every bar still satisfied. The quintic's machine works over 0.17 radians of driving angle, braced or not, and what ends the arc is a reflector reaching the angle at which its mirror lies along its input.

Tested in Where the machine stops being the function · the compute sequence.

A linkage's accuracy improves gradually with the number of links, so an exact one is the limit of a sequence of better approximations.

What decides it: Measured over their own working arcs: Watt four bars at 9.0 per cent of the stroke, Chebyshev four bars at 12.4 per cent, Peaucellier seven bars at 4 × 10⁻¹⁶, a compiled machine five bars at 4.8 × 10⁻¹⁴. Fourteen decades separate the two groups and no mechanism sits between them, because exactness comes from an algebraic identity holding rather than from an error being made small.

Tested in Exact costs more than close · the compute sequence.

Tightening the tolerances on a locating scheme eventually pins the part down.

What decides it: The permitted poses are { p : A p ≥ −c }, a polyhedron whose every dimension scales linearly with the clearance c — measured at seven clearances four orders of magnitude apart with the ratios constant to 1e-9. It is bounded exactly when the arrangement holds. A square in a vee has an unbounded pose set at every clearance, so shrinking c narrows the box in two coordinates and never closes the direction the part slides out along.

Tested in Held is not located · the restraint sequence.

Six independent constraints leave a body no freedom.

What decides it: True of equations and false of inequalities. Every exact-constraint coupling on this site — Kelvin, Maxwell, three-two-one — has rank six and an escape, and a Maxwell coupling's escape is a pure vertical translation with nothing else in it. Six vectors can span six dimensions and can never positively span them, so no arrangement of six frictionless contacts holds anything, and across four thousand random six-contact arrangements on a box the number that held is nought.

Tested in Six things a hold is not · the misconception sequence.

A parallel platform's singularities are isolated poses that a planner can steer round, so moving the platform somewhere else always leaves a singular pose behind.

What decides it: Held level and turned 30° about the vertical, the site's Gough–Stewart platform is singular at every position tried: the smallest singular value of its six leg lines is 9.7 × 10⁻⁹ or less at four positions spread through the workspace, 2.1 × 10⁻⁸ or less over a whole horizontal square, and 1.28 × 10⁻³ or more one degree either side. The same holds at −150°. No change of position leaves those yaws singular-free.

Tested in A yaw that is singular everywhere · the parallel sequence.

A higher pair is a joint with more freedoms than a lower pair, so its permitted motions are a larger group.

What decides it: A disc on a straight edge permits a slide and a turn, and the composite of the two is not permitted: it carries the disc's centre off the line by exactly |t₂ sin φ₁|, which agrees with the composition to 4 × 10⁻¹⁶ and reaches 1.2 radii over an ordinary range. The permitted set is not closed under composition, so it is not a group of any size, and the freedom count that says two is the dimension of a set rather than of a group.

Tested in A higher pair has no group · the pair sequence.

A graph with the right number of links and pins, in which every link carries at least two pins, is a mechanism with that many links.

What decides it: At ten links, 1,878 graphs satisfy every one of those conditions and 230 are mechanisms. The other 1,648 contain a subchain whose own mobility count is zero or less — 1,501 of them a plain triangle — and on 1,165 of them Grübler's count returns one, the rank of the constraint Jacobian returns one, and both are right about an assembly that does not have ten links.

Tested in Six things a chain is not · the misconception sequence.

A census of chains is a catalogue of mechanisms.

What decides it: The sixteen eight-link chains, each given the arbitrary placement its own layout produces and driven from its first available frame-and-input pair, turn through arcs from 134° to a full revolution — ten of the sixteen reaching every angle and six not. Change the placement and every bar in that chart changes. The census above it does not.

Tested in The chain has no lengths · the topology sequence.

Two joints with the same number of freedoms are interchangeable in a kinematic model.

What decides it: A revolute, a prismatic and a helical pair each have one freedom, each take five away in space, and each give the same mobility for any mechanism they are put in. Their permitted sets are three different groups: one sends a point round a circle, one along a line, and one along a helix at a ratio the joint fixes. Swapping a pin for a screw in a four-bar leaves every count unchanged and gives a loop whose motion does not close after a full turn.

Tested in Six things a joint is not · the misconception sequence.

A fixture's contact tolerances can be allocated equally, because every contact does the same job.

What decides it: The weights in the fit condition are the coefficients of the positive combination that cancels. On a square held by four contacts they are a quarter each, and on a hexagon held by five one of the two conditions runs from 0.144 to 0.424 — a spread of 3.47. A micron at the heaviest contact spends three and a half times as much of the budget as a micron at the lightest, and nothing on the drawing says so.

Tested in Which contact to make accurately · the tolerance sequence.

A mechanism whose loop closes to 10⁻¹⁴ is doing what its construction says it does.

What decides it: Sixteen assemblies of one twenty-bar compiled machine: eight close, four put the tracing point on the curve and four put it somewhere else, with the polynomial reaching 0.15 where zero was wanted. All eight close to 9.6 × 10⁻¹⁵. The two populations are not separated in that quantity at all, so no tolerance on it can tell them apart.

Tested in Six things a compiled linkage is not · the misconception sequence.

The zero-pitch screws of a three-system form a cone through its centre, so every line of that cone is a joint axis that would leave a leg the same freedom.

What decides it: The directions form that cone and the lines do not. A three-system with no pure translation holds exactly one screw with each direction, and on the leg measured here each zero-pitch axis misses the centre by between 0.165 and 0.551. A revolute joint placed on the cone's own line through the centre lies outside the system by at least 0.109 of the screw; one placed on the true axis lies inside it to 10⁻¹⁵. The true axes rule a hyperboloid, and the cone is its asymptotic cone.

Tested in The lines a leg turns about and the lines it is pushed along · the spatial sequence.

Two frameworks with the same joints, the same bars and the same count of mechanisms have the same mechanisms.

What decides it: An eight-by-eight kagome patch has 192 joints, 346 bars and 35 mechanisms whether its triangles are straight or turned by 17.2°, and the rank agrees both times with no redundant bar. The mean share of the mechanism space at a joint three cells in from the edge is 0.118 on the straight patch and 0.015 on the twisted one, and the single mechanism with the most movement in the middle puts 59% of it there on the straight patch and 16% on the twisted — less than the middle's 25% share of the joints.

Tested in The count says how many and not where · the network sequence.

A slide sends a coupler curve to infinity along the slide's own direction.

What decides it: Lines parallel to the slide meet the slider-crank's rod quartic four times, the full degree, for every coupler point tried, so the curve does not pass through the slide's point at infinity at all. Lines of slope [2v ± i(1 − u² − v²)] / [(1 + u)² + v²] meet it three times, or twice when u² + v² = 1, and the quartic fitted to the drawn curve has exactly those slopes left after dividing its top part by x² + y². Only the slider pin itself, u = 1 and v = 0, points along the slide.

Tested in Where a slide puts the rest of the degree · the algebra sequence.

A fixture should press a part near the middles of its faces, where the contacts are well away from the edges.

What decides it: On a regular octagon, sliding one of the four best contacts from its corner to the middle of its edge takes the hold margin from 0.284 to 0.158, and on to nothing at the far corner. From six sides up the best four contacts all sit at the ends of edges, and the four contacts of the octagon's best placement moved to the middles of their edges hold nothing at all: every one of them then pushes through the centre, and no combination resists a turn.

Tested in The hold is in the corners · the restraint sequence.

If two links cross on the drawing, they interfere.

What decides it: Chebyshev's linkage draws with its two long arms crossing and clears by 0.380 over its whole working arc. A drawing of a motion has thrown the time away: interference is a condition on configurations, and two bodies in the same place at different instants are two bodies that never met.

Tested in Six things a body is not · the misconception sequence.

In a ball bearing with its outer race held, the balls and cage go round at half the shaft's speed.

What decides it: Half is the limit of vanishingly small balls. Two rolling constraints put the cage at (1 − d/D)/2 of the shaft speed: 0.475 for balls a twentieth of the pitch diameter, 0.400 for a fifth, 0.325 for 0.35. The gear-train solver, given a planetary whose sun and ring have D − d and D + d teeth, returns the same fractions exactly — 19/40, 2/5, 13/40 — because a ball rolling between two races is a planet meshing with a sun and a ring.

Tested in A bearing is a planetary with no teeth · the rolling sequence.

Replacing a pin of a one-freedom chain by a slide leaves a one-freedom mechanism, since Grübler's count gives both joints the same weight.

What decides it: True for one or two slides on the four-bar, Watt and Stephenson chains, and false from three. Of the twelve different Stephenson chains with three slides, two carry pins that cannot turn; with slides on the three joints of the path 0–4–5–3, all four pins of the other loop are locked although no loop holds a single pin. With four slides in a loop the four-bar has two freedoms, not one. Graph and Jacobian rank agree on every one of the 102 placements.

Tested in A slide turns nothing · the topology sequence.

A compiled machine's cost rises with the curve it has to draw, so a curve with more components in it costs more terms than one of those components alone.

What decides it: The circles r² = 1.44 and r² = 2.56 have a product equation of one term, which is exactly what either of them costs alone; six concentric circles in three such pairs cost three terms against six for the same circles moved off the law. The machine for the pair also runs over 5.200 radians against the single circle's 1.560.

Tested in Two circles for the price of one · the compute sequence.

A calibration whose residual falls to the instrument's noise has determined the machine.

What decides it: One on this site's four-bar fits thirty readings to 1.8 × 10⁻¹⁶ radians and returns four lengths, none of which is the machine's — they are the machine's multiplied by 0.99229, every one of them, to fifteen figures. The residual is as good as a residual gets and one of the four numbers is the starting guess.

Tested in Six things a measurement cannot tell you · the misconception sequence.

At an ordinary change point, unlike a parallelogram's, the clearance that restores the change point carries a geometric constant and the four bearings are not equally effective.

What decides it: Measured on linkages along g + a = b + c with ground 4, crank 1 and couplers from 1.5 to 3.5, for all eight single-length errors at 10⁻³ and 10⁻⁶, the clearance that rejoins the circuits or lets the crank through is the error to within 1.4 × 10⁻¹⁰ of itself. Each of the four pins given a play of 4 × 10⁻⁴ leaves the loop 6.0000 × 10⁻⁴ from closing. The constant belongs to the angle instead: the circuits separate by 2√(2bδ/c(g + a)), 0.0400 radians at δ = 10⁻³ on the linkage with coupler and output both 2.5.

Tested in Every change point lies flat · the mobility sequence.

Two mechanisms with the same links and the same joints have the same tolerance problem.

What decides it: All sixteen eight-link chains have three independent loops, forced by ten pins and eight links. The total length of the shortest independent set of loops runs from twelve links to fifteen across the sixteen, so the smallest number of link dimensions any stack-up on them can involve differs by three. Watt's six-bar needs eight and Stephenson's needs nine, on identical link and pin counts.

Tested in Where the shortest loops are · the tolerance sequence.

A bearing in a mechanism is a revolute joint, so a mechanism's mobility can be counted from its bill of materials.

What decides it: Which pair a bearing gives is decided by its surfaces, not by its name. A plain journal with no thrust faces is a cylindrical pair — two freedoms — and a mechanism counted as though it were a revolute is short by one freedom per bearing. A deep-groove ball bearing locates axially and is a revolute; an angular-contact pair mounted back to back is a revolute; a needle roller without collars is cylindrical; and a spherical plain bearing is a spherical pair with three.

Tested in The pair a catalogue sells · the clearance sequence.

A hinge with more knuckles is more overconstrained, so it needs more play — a piano hinge of forty knuckles is a far harder thing to make than a door hinge of three.

What decides it: The play needed is the Chebyshev fit of a line to the bore centres, and it is 0.409 of the bore tolerance at three knuckles and 0.966 at forty — two and a half times, not thirteen. It saturates at one, because the nominal axis already misses every bore by at most the largest error, and it is held by two or three bores whatever the count. A two-knuckle hinge needs nought.

Tested in A piano hinge is not forty door hinges · the overconstraint sequence.

A linkage can only approximate a curve given to it; exactness is for special cases like Peaucellier's straight line.

What decides it: A polynomial in the two angles of a two-link arm is a finite sum of cosines of whole-number combinations of them, each of which a handful of bars produces exactly. Compiled that way, a rectangular hyperbola comes out as twenty bars whose tracing point satisfies xy − 0.5 to 1.3 × 10⁻¹⁴ over the whole of its working arc, and a quintic as four hundred and thirteen. Exactness is general; what is special about Peaucellier is that it is small.

Tested in A demand that is an equation · the compute sequence.

A robot arm is a mechanism like any other, so the hard part is working out where it can be.

What decides it: For a closed loop the forward problem is a Newton solve that can fail and the inverse problem is often a division. For an open chain the forward problem is a product of exponentials that cannot fail — 400 joint settings sampled out to twelve radians, every one giving a rigid pose to 1.1 × 10⁻¹⁵ — and the inverse problem has eight answers. The difficulty is not smaller; it is at the other end.

Tested in The chain that does not close · the serial sequence.

Bézout's theorem gives the number of solutions of a polynomial system.

What decides it: It gives four for the four-bar's two circles, which meet twice; the other two are the circular points at infinity, and they are there for every pair of circles rather than for these lengths. On the Gough platform the same bound is 1,458 against 80 finite solutions. Bézout counts solutions in complex projective space with multiplicity, and every word of that is load-bearing.

Tested in Two circles, four answers · the algebra sequence.

A mechanism's dimensions are inputs. You choose them, you make the parts to them, and thereafter the analysis knows them.

What decides it: Read a four-bar's output angle at twenty-four positions and ask which of its four lengths the readings determine. Three. The fourth is annihilated by every row of the matrix that relates readings to parameters, to 7.6 × 10⁻¹⁵, and no instrument that reads an angle will ever recover it.

Tested in A dimension is a measurement · the parameter sequence.

Two shapes with no crossing edges are apart, so the smallest distance between their edges is the gap between them.

What decides it: A block sitting entirely inside a bell crank's convex hull crosses none of its edges, and the smallest edge-to-edge distance is 0.18. The block is inside. One point of each polygon settles it, and the version without that test reported a comfortable clearance for a part buried in another part.

Tested in A gap is a number · the body sequence.

A roller running on a flat surface is a prismatic pair.

What decides it: A prismatic pair fixes the relative angle of the two bodies as well as one of their relative positions: two constraints. A roller that can turn in its track fixes only the position: one. On a three-stage scissor lift the difference is exactly two degrees of freedom — the formula gives 1 with the rollers counted as pins in slots and −1 with them counted as sliders, and the Jacobian measures 1.

Tested in A roller is not a slider · the mobility sequence.

Writing a polynomial as a sum of cosines is a Fourier expansion, so it is an approximation and it is infinite.

What decides it: It is neither. Squaring and multiplying two-term Laurent polynomials can only produce finitely many powers, so the sum terminates: a general quintic gives eighteen terms and stops. Evaluated against the polynomial itself at three hundred random pairs of angles the worst disagreement is 1.8 × 10⁻¹⁴, which is rounding rather than truncation — there is nothing left over to truncate.

Tested in Every curve is a sum of cosines · the compute sequence.

Denavit–Hartenberg parameters are just a convention for writing an arm down, so which description you use is a matter of taste.

What decides it: Tilt one axis of a SCARA arm by 10⁻⁶ radians. The joint screws move by 10⁻⁶, because they are the axes. The table's d parameter moves from undefined to −349,969 — a factor of 3.5 × 10¹¹ times the nudge — and only when the tilt is in one particular direction. A description that changes by order one when the machine does not change at all is not a matter of taste when it is what a calibration fits.

Tested in Four numbers or a screw · the serial sequence.

A mechanism goes singular at the edge of its workspace, where it has run out of reach.

What decides it: That is the serial arm's singularity and it is real. A parallel mechanism has a second kind with no serial counterpart, where the platform gains a freedom the locked motors cannot resist — it moves with every actuator held still. The workspace map here computes both, and the second lies inside the workspace rather than on its boundary, which is why it cannot be designed around by staying away from the edge.

Tested in Locked, and still moving · the parallel sequence.

There are six lower pairs because six kinds of surface are easy to machine accurately.

What decides it: Manufacturability decides which pairs are common and not which exist. The census here contains a cone, a torus and a scalene ellipsoid, none of them a standard bearing surface; the first two give the revolute pair and the third gives no joint at all, and the reason is the dimension of a symmetry group rather than the cost of grinding one.

Tested in Six, and no others · the pair sequence.

Constant-acceleration motion is the best cam law, because it gives the lowest peak acceleration.

What decides it: It does give the lowest peak, by 57% against cycloidal, and that is the whole of the claim's support. Differentiating the drawn follower displacement a third time shows what it costs: the jerk is impulsive at every junction. The comparison is run here on all the laws at once, and each wins only on the measure it was given.

Tested in The law that costs least is not the smoothest · the cam sequence.

Using a part's convex hull as its body is a safe approximation: it can only report interference that is not there, and a design that passes on the hull certainly passes.

What decides it: The hull of a right-angled bell crank is 105% more material than the crank — more than double — and all of it is in the notch, which is exactly where a designer puts something. A block in that notch clears the real part by 0.020 and is 0.120 inside the hull. The direction of the error is safe; its size makes the check useless.

Tested in A shape with a dent in it · the body sequence.

A four-bar synthesised through three prescribed positions will reach all three.

What decides it: It will, and the construction is exact — the residual at each pose is at the level of arithmetic. Whether it reaches them in one piece is a different question, and sweeping all 1,176 exactly-correct syntheses on this site leaves 176 that a motor can actually drive through the three poses: 810 have a branch or circuit defect and 190 an order defect. Nothing in the construction distinguishes them.

Tested in Exactly right, and unbuildable · the misconception sequence.

Exact straight-line motion from pin joints alone needs Peaucellier's eight-bar inversion cell.

What decides it: It needs eight bars in the plane. Sarrus's linkage does it with six, exactly, and was published eleven years earlier — by leaving the plane. Its closure is solved here as a spatial loop, and the output translates with no rotation at all: the residual on the rotation part of the closure transform stays at the level of arithmetic through the whole travel.

Tested in Sarrus, and the straight line that is exact · the spatial sequence.

Grübler's criterion tells you whether a mechanism moves.

What decides it: Five links and six pins: the formula returns zero, meaning a structure. The rank of the constraint Jacobian measures one freedom, and a 360° sweep drives the mechanism through 59 of 60 sampled positions. The criterion is a theorem about a generic assembly, and the mechanisms worth building are the ones that are not generic.

Tested in The mechanism Grübler says cannot move · the mobility sequence.

A ratchet's lost motion is a matter of how accurately it is made.

What decides it: A perfectly made ratchet of twenty-four teeth loses up to 15° of input before its output moves, because the pawl has nowhere to drop except a tooth. Clearance in the joints adds to that and does not cause it; the design term survives every tolerance going to zero.

Tested in The resolution is the pitch · the intermittent sequence.

Check a mechanism's clearance at the ends of its travel and at its dead centres; those are where the extremes are.

What decides it: The worst clearance of a four-bar against a post bolted to its frame is 0.104 at a crank angle of 2.11 radians — neither end, and no dead centre. Both ends read a comfortable 0.5 and above. The cabinet door in the applied field is the same shape of answer: its closest approach to the carcase is a quarter of the way open.

Tested in A gap with corners in it · the body sequence.

Adding two angles with links requires a gear pair or a special mechanism; bars alone cannot do arithmetic.

What decides it: A rhombus on two equal links from a common pivot has its far vertex at their vector sum, so its diagonal bisects them exactly. Reflecting the frame direction in that bisector gives the sum of the two angles. Four bars, no gears, and the output matches its closed form to 2.7 × 10⁻¹³ radians across ninety-seven solved positions.

Tested in Four bars that add two angles · the compute sequence.

A dwell is a mechanism briefly having nothing to do.

What decides it: A Geneva during its dwell has one degree of freedom, exactly as it does while the pin is in the slot; Grübler counts 3(3−1) − 2(2) − 1 = 1 either way. What is zero is not a freedom but the derivative of the transmission function, over an interval — and producing a zero over an interval rather than at a point takes a specific curve.

Tested in The arc that is concentric with the pivot · the intermittent sequence.

A singularity is where the Jacobian loses rank, so gimbal lock and a wrist singularity are the same kind of event.

What decides it: At joint values (0.30, 0.04, 1.56, 0, −1.60, 0.20) the arm's Jacobian has rank 6 and its smallest singular value is 0.502 — as far from singular as this arm ever gets. The z-y-z Euler rate matrix at the same posture has rank 2. One of those matrices belongs to the mechanism and the other to a choice of three numbers for describing an orientation.

Tested in Two routes to a Jacobian · the serial sequence.

Adding tolerances statistically rather than in the worst case halves the stack-up, so use root-sum-square.

What decides it: The ratio between the two rules is bounded above by the square root of the number of contributions — exactly 2 for four lengths, and reached only when all four contribute equally. Measured through this linkage's turn it runs from 1.42 to 1.96, never 2. And the whole saving is bought with an assumption that the four errors are independent, which is false the moment one fixture locates two of the features.

Tested in Worst case and the square root · the tolerance sequence.

A mechanism's screw system tells you which group its motion is in.

What decides it: A screw system is a subspace of twists and a group's algebra is a subspace closed under the Lie bracket, and the second condition is almost never satisfied: of eighty thousand random subspaces of dimensions two to five, none is closed, and all eighty thousand generate the whole of the algebra at the first bracket. A screw system is evidence about a group only when the closure is checked, and the check fails far more often than it passes.

Tested in Almost nothing is a group · the pair sequence.

A direction can be copied anywhere in a linkage with a parallelogram, so transporting angles is cheap.

What decides it: A parallelogram copying a direction from P to R needs bars PR and the copy's far side, so |PR| must be fixed. Partial sums along a summing chain are not at fixed distances from the pivot, so each term's direction must ripple down the chain one fixed-length bar at a time: N(N−1)/2 parallelograms for N terms. On the quintic that is 154 of the machine's 242 joints.

Tested in A parallelogram carries an angle, and only so far · the compute sequence.

Inverse kinematics is solved numerically: you iterate until the tool is where you want it.

What decides it: Iteration finds *a* posture — whichever one the seed was nearest. From 600 scattered seeds, 522 converge and they land on 8 distinct postures. The construction returns all eight in one pass, exact to 1.8 × 10⁻¹⁵, and tells you which is which: front or back, elbow up or down, wrist or wrist flipped. A machine has to choose among them, and the choice is made on joint limits and cable routing rather than on residuals.

Tested in Eight ways to hold the same tool · the serial sequence.

The contact patch is a point of the upright, so its sideways motion is the scrub.

What decides it: A wheel is a disc: the point of it that touches the road is the lowest point of that disc, and it moves across the tread as the wheel cambers. Tracking a point fixed to the upright instead disagrees by up to 14.7 mm on this suspension against a total scrub of 12.2 mm — and at full bump the two have opposite signs, so the shortcut is not a small error but a different answer.

Tested in The wheel is the coupler · the fourbar sequence.

A tolerance stack-up is a standard calculation: sum the sensitivities times the tolerances and read off the band.

What decides it: Run on a crank-rocker it agrees with the measured band to 0.09% at all 180 positions. Run on a parallelogram at its change point it exceeds the measured band by 4.8 × 10⁵. The calculation is a first-order expansion and it needs the constraint Jacobian to be invertible; at a change point it is not, and nothing in the arithmetic notices.

Tested in Where a stack-up stops working · the tolerance sequence.

Adding a second instrument to a calibration is redundant once the first one determines every parameter.

What decides it: A coordinate machine on the tracing point recovers all six parameters at a condition number of 162. Adding a protractor on the output link recovers the same six at 26 — six times better, with no parameter gained. The two instruments carry information in different directions of the same space, so neither is a subset of the other.

Tested in A ruler and a protractor · the parameter sequence.

Two linkages draw the same curve if their traced points lie close together everywhere.

What decides it: Closeness of two sampled traces is a statement about the sampling. Fitting each trace independently for the degree-six polynomial that vanishes on it, on a common normalisation, gives twenty-eight coefficients per linkage that agree to 5 × 10⁻⁷ across all three cognates — a statement about the curves rather than about where either mechanism happened to be sampled.

Tested in Three linkages, one equation · the coupler sequence.

A wrist whose axes nearly intersect is near enough: the closed-form solution will be nearly right.

What decides it: Offset one wrist axis by 10 mm. The construction still returns eight postures, labelled front and back and elbow up and down as usual, and every one of them puts the tool somewhere else — the best is 3.6 mm out, which is 0.36 of the offset. Nothing inside the construction can notice, because every step of it is arithmetic about a wrist centre that has stopped standing still.

Tested in The wrist is three joints and one point · the serial sequence.

A planar mechanism lies in a plane; the third dimension is a detail of manufacture with nothing kinematic in it.

What decides it: Not one of the nine machines in this catalogue fits in a single plane, and the four-bars need three rather than two. The offsets are indeed not kinematic — every length, angle and position is unchanged — but which link goes in which plane is a decision the plane does not make, and a machine assembled without making it is a machine whose parts are in the same place.

Tested in Two bars that have to cross · the body sequence.

Drop is a clearance the designer sets.

What decides it: The wheel advances exactly half a tooth pitch per beat, by the counting, and that half pitch is spent on the impulse, the drop and the lock-in run. Change the lift and the drop changes by exactly minus as much, to twelve figures: over lifts from 2.6° to 5.0° the impulse runs 1.64° to 2.88° and the drop 4.36° to 3.12°, and the sum never moves.

Tested in Where the tooth lets go · the intermittent sequence.

A calibration is more accurate the more positions it measures, so measure as many as time allows.

What decides it: The observability of a four-bar's worst-recovered parameter rises from 0.118 at three poses to 0.379 at twenty, and two thirds of that rise has happened by ten. What the extra poses buy after that is averaging of noise, which improves as the square root and is a different gain — and eight well-chosen poses beat twelve evenly spaced ones outright.

Tested in How many poses are enough · the parameter sequence.

A serial chain of n joints has an n-dimensional displacement set, so n numbers describe where its tool can be.

What decides it: The set is n-dimensional and the numbers needed to describe it are the dimension of the smallest subgroup containing it, which is not the same thing. Four pins at random reach a four-dimensional set of displacements whose logarithms occupy all six dimensions; a SCARA arm's four joints reach a four-dimensional set whose logarithms occupy four. Same joint count, same mobility, same rank — and one motion is a group and the other is a slice of everything.

Tested in A chain multiplies · the pair sequence.

A flat follower's face has to be a little wider than the cam's lobe, and the lift is what sets it.

What decides it: The lift sets nothing. The contact sits at ds/dθ from the follower's axis, so the face spans from the least value of the velocity to the greatest — 19.10 for a cycloidal rise and return of lift 10, and 15.00 for a harmonic one of the same lift over the same spans. Same lift, same base circle, faces differing by a quarter.

Tested in A follower needs a face · the cam sequence.

With one wheel on ice a differential sends all the drive to the wheel with no grip, and that is a failure of the mechanism.

What decides it: The mechanism's own condition — that the cage turns at the mean of the two wheels — is satisfied exactly throughout, and the null space returns the spinning wheel at precisely twice the cage. Which of the permitted motions occurs is decided by what is pushing, and no geometry decides it. The kinematic statement that is available, and is the one worth having, is the opposite one: a locked axle in a turn has no permitted motion at all, and scrubs 9.74 m per circle to get one.

Tested in One wheel on ice · the transmission sequence.

A mechanism that satisfies its constraints to 10⁻¹⁴ is doing what it was designed to do.

What decides it: The compiled machine's closure residual sits at 10⁻¹⁴ on every one of its sixteen assemblies, including the four whose tracing point is nowhere near the curve — the worst of those reads 0.15 against a polynomial that should be zero. Closure is a statement about the bars and the specification is a statement about the point, and no tolerance on the first can decide the second.

Tested in The machine, compiled · the compute sequence.

A deadbeat escapement's locking faces are flat surfaces set at the right angle.

What decides it: A flat face tangent to the concentric arc has no draw and no recoil at the corner and grows both in proportion to how far below the corner the tooth rests. Its recoil is quadratic in the supplementary arc — 0.0022°, 0.0088°, 0.0358°, 0.1464° at half, one, two and four degrees — where the arc's is identically zero at every one.

Tested in The wheel that goes backwards · the intermittent sequence.

A parallel platform's motion has to be worked out from the closure equations of its loops, so it depends on the leg lengths and the attachment geometry.

What decides it: The motion type does not depend on any length. Each leg's joints, if they share a group, confine the platform to that group; the platform's permitted set is the intersection; and an intersection of groups is a group computed from the legs' axis directions alone. Two planar arms with perpendicular normals give a one-dimensional translation whatever their links measure — checked against the mechanism's own solved motion, which reports a closure of one. Only the degenerate arrangements, where two legs' groups coincide, change the answer.

Tested in Three legs and one plane · the parallel sequence.

At a singularity the arm cannot move in the direction of the Jacobian's smallest singular vector.

What decides it: That vector is a mixture of radians and metres and it moves when the units do. At the wrist singularity it is (−0.35, −0.47, −0.15 | 0.05, 0.71, 0.37) with the arm written in metres and (0.58, 0.78, 0.24 | 0.00, 0.00, 0.00) with the same arm written in millimetres. The reciprocal screw is the same line in both, with a pitch of −0.6285 m and −628.5 mm.

Tested in Where the arm loses a direction · the serial sequence.

The direction a bicycle was ridden can be told from its tracks by which one is deeper, or by the way the mud is thrown, or by which track crosses over the other.

What decides it: The tracks decide it geometrically and none of those readings is needed. The rear wheel is towed, so at every instant it points at the front wheel: the tangent to the rear track, extended forward by the wheelbase, lands on the front track. Measured over a sinuous run, that construction misses by a mean of 0.13 mm the right way round and 447 mm the wrong way — a factor of 3,554 — and the same measurement returns the wheelbase as 1.0503 m against a true 1.0500 m without being told it.

Tested in Which way did the bicycle go · the rolling sequence.

A universality construction is enormous because the algebra of a high-degree curve is complicated.

What decides it: Split a compiled machine's bars by what they are for. On the quintic the angle arithmetic — every reflector, mean and rigid offset that turns two arm angles into eighteen integer combinations — is about a quarter of the machine, and three hundred and eight of its four hundred and thirteen bars are parallelograms whose only job is to carry a direction from where it was made to where it is used.

Tested in Five bars for a line, four hundred for a quintic · the compute sequence.

Draw and recoil are two separate defects a good escapement minimises together.

What decides it: They are one quantity. The pallet's moment per unit of wheel torque equals minus the rate at which the wheel is driven back as the pallet goes deeper, and the two agree to nine figures at every tilt tested. An escapement cannot be given a lock that holds itself without being given a wheel that is pushed backwards.

Tested in The angle that holds the lock · the intermittent sequence.

The region a mechanism occupies is the region its coupler curve and its links' centrelines cover, thickened by the link width.

What decides it: Thickening a curve gives the region a point sweeps, not the region a body sweeps: a bar's far corner reaches further than its pin does, by the boss radius on every side and by more at the ends. On the crank alone the two differ by the whole of the boss: the pin sweeps a circle of radius 1 and the material sweeps a disc of radius 1.163, an area 35% larger.

Tested in The room a machine sweeps · the body sequence.

A cam's motion law is chosen for the accelerations it produces, so the choice depends on the cam's size.

What decides it: A law is a dimensionless function of a dimensionless argument. Its peak acceleration coefficient, its jerk behaviour and its ranking against other laws are the same for a cam of any lift and any base circle — which is why a table of laws can be printed once and used everywhere, and why the numbers in it are coefficients rather than accelerations.

Tested in A lift is a size and a law is a shape · the cam sequence.

Doubling the number of measurements halves the error, so a long calibration is a better calibration.

What decides it: It halves the part of the error that comes from noise, which is a factor of 1.41 per doubling and nothing else. It does not improve the conditioning: the condition number of this four-bar's identification Jacobian is 6.5 at three poses, 5.2 at five, and 5.2 at every count after that, flat to the printed digits.

Tested in What another measurement is worth · the parameter sequence.

A four-bar's two assembly branches, as a solver labels them, are its two motions.

What decides it: On a parallelogram chain, ground 4, crank 1.5, coupler 4, rocker 1.5, each of the solver's two branches spends 360 of its 720 solved configurations on the circle factor of the coupler sextic and 360 on the quartic factor, swapping at 0° and 180° of crank angle. The factors are the motions: one keeps the coupler parallel to the ground and the other crosses it. A branch is a label on which intersection of two circles was taken.

Tested in A sextic that comes apart · the coupler sequence.

A least-squares fit always returns an answer, so fitting a polynomial to a curve cannot establish its degree.

What decides it: It can, because the quantity that decides is not the residual but the gap between the smallest singular value of the design matrix and the next. On a coupler curve, degree five leaves a smallest singular value of 1.0 × 10⁻³ against a next of 1.9 × 10⁻³ — no gap, no answer — and degree six leaves 3.3 × 10⁻⁸ against 9.3 × 10⁻⁵, a factor of 2.8 × 10³.

Tested in The equation a four-bar satisfies · the algebra sequence.

A gear train can be geared to any ratio by choosing the wheels carefully enough.

What decides it: The ratio between a sidereal day and a mean solar one is 1.0027379, and among all 12,321 pairs of wheels between 40 and 150 teeth the closest is one to one — no gearing at all beats every single pair. Two pairs get to 0.157 seconds a day and none of the 1,413,721 candidates is exact.

Tested in A clock is a factorisation · the tooth sequence.

A parallel mechanism's mobility is what the count says it is, so a platform with three legs of five joints each has three freedoms and that is the design.

What decides it: The count gives the dimension and the intersection gives the group. Three legs whose groups are three planar groups with parallel normals leave the platform the whole planar group — three freedoms, and specifically planar motion, which no count can distinguish from three freedoms of any other kind. Change one leg's normal and the intersection falls to a translation: same joint counts, mobility three by arithmetic, and one freedom in fact.

Tested in Legs intersect · the pair sequence.

Overconstrained mechanisms are paradoxical curiosities: they need exact geometry, so they are not manufacturable.

What decides it: The misfit a perturbation creates in Bennett's linkage is 0.507 times the perturbation, measured over four decades. Shared among four joints that is 0.127 δ of play per pin — sixteen microns for a link of 1.6 cm machined to a part in a thousand, which is an ordinary running fit. The mechanism is unbuildable only in a model where the joints are points, and the model that says so is the same one that says a planar four-bar has mobility −2.

Tested in Why a hinge works · the overconstraint sequence.

Observability indices measure the same thing and any of them will do for comparing pose sets.

What decides it: Normalise four of them to their own value at eighteen poses and compare at eight: two read 1.000 — finished — and two read 0.667 and 0.817. A factor of 1.5 between the extremes, and a disagreement about whether any further pose is worth taking.

Tested in Four indices, four answers · the parameter sequence.

A four-bar has at most two circuits because it has two assemblies at each crank angle.

What decides it: A Stephenson six-bar also has at most four assemblies at each crank angle, exactly as a Watt six-bar does, and a search finds one with six circuits and a Watt six-bar with four. The number of assemblies over a crank angle does not bound the number of circuits. What holds a four-bar at two is that its two assemblies merge at only four points in the complex numbers — genus one — and a real curve of genus one has at most two components.

Tested in Never three circuits · the algebra sequence.

A coupler curve with a mirror symmetry must come from a linkage with a mirror symmetry.

What decides it: The linkage here has a crank of 1, a ground of 3 and every other length 2.5, and its curve is symmetric to 4.2 × 10⁻¹⁵ over 48 mirrored pairs on both assemblies. The symmetry lives in a circle about the rocker pin that three of the machine's points share, not in the machine — and one of the curve's Roberts cognates draws the identical symmetric curve with no three lengths equal and no pivot on the axis at all.

Tested in A symmetric curve from a lopsided machine · the coupler sequence.

Adding constraints to remove unwanted assemblies makes the mechanism less able to move.

What decides it: Braced, the machine compiled from a quintic has 1,249 equations in 1,096 unknowns and a Grübler count of −153 — by the count it cannot move at all. Its Jacobian has rank 1,095, so its mobility is one, exactly as it was unbraced. Every brace is one equation the others already imply, and it removes assemblies rather than freedoms.

Tested in A bar between two midpoints · the compute sequence.

Working out whether a pawl holds, or whether a lock draws, is statics.

What decides it: Both are the sign of a cross product of two directions with a length in it. A contact transmits along one direction and that direction is a property of the surface; the moment's sense follows from where the pivot is. No magnitude appears in either calculation, and neither answer changes if the load is doubled.

Tested in One test, three mechanisms · the intermittent sequence.

A redundant arm has a spare joint, so the inverse problem has one more solution than before.

What decides it: It has infinitely many. The solution set stops being a list of eight postures and becomes a curve: forty-one postures were walked along it here and the tool moved 3.3 × 10⁻¹² m in total. A count and a continuum are not the same kind of answer, and every algorithm that assumed a finite list has to be rewritten.

Tested in The freedom that does nothing · the serial sequence.

Grübler's rule fails only when a mechanism's geometry is special, so a count applied to generic geometry is reliable.

What decides it: Special geometry is one of three failure modes and it is the mildest. The rule also cannot see a rigid subchain, which is a fact about the graph rather than the geometry and costs it eight graphs per mechanism at ten links; and it cannot see which group a joint's freedom belongs to, so it reports the same number for three joints that are not interchangeable in any mechanism. All three are structural rather than numerical, and none of them is repaired by generic geometry.

Tested in The freedom that is a set · the mobility sequence.

Sarrus's linkage draws a straight line because its two arms are identical and symmetry forces the platform to go straight up.

What decides it: Symmetry is not required and does not appear in the argument. The two arms need only have non-parallel axis directions; their link lengths, their pivot positions and their proportions may all differ. What produces the straight line is that each arm confines the platform to a planar group and two planar groups with non-parallel normals meet in a one-dimensional translation group — a statement about two normals and a cross product, in which no length appears.

Tested in Two planes meeting in a line · the pair sequence.

A gear pair's ratio is a property of its geometry that a measurement of the mesh recovers.

What decides it: The ratio is two integers divided. Scaling the module from 2 to 4 doubles the centre distance from 60 to 120, doubles the base pitch from 5.904 to 11.809, and leaves the ratio at exactly 2 and the contact ratio at 1.63519 — unchanged to every digit, because neither contains a length at all.

Tested in The module is a size, the ratio is a shape · the tooth sequence.

A flat-faced follower has no pressure angle, so its cam can be made as small as the lift allows.

What decides it: The pressure angle is zero, and the size limit moves rather than vanishing. The profile's radius of curvature at the contact is R₀ + s + s″, and it must stay positive or the face bridges a hollow and the follower is held above its programme. For a cycloidal rise and return of 10 over 120° the smallest base circle is 5.332, found by the formula and by bisecting on the drawn curve to five decimal places; on a base of 2 the face is held 0.446 high.

Tested in A flat face asks for a convex cam · the cam sequence.

Every parallel mechanism can change assembly mode without a singularity by going round a cusp, as the three-legged planar platform can.

What decides it: A planar five-bar with motors 1.0 apart, arms 1 and distal links 1.25 has two assemblies, and their det A are equal and opposite. At 1,026 random settings of the motors the assembly label and the sign of det A agreed every time, and of 305 closed loops of the motors whose tracks completed, none ended in the other assembly. A cusp needs three assemblies merging, and two circles only ever have two answers.

Tested in The smallest parallel robot · the parallel sequence.

The degree of a mechanism's curve can only be had from its equation, either eliminated symbolically or fitted to traced points.

What decides it: Neither is needed. The coupler point is required to lie on a random complex line, the loop closure and the line are solved completely by continuation, and the distinct solutions are counted: six on each of four random lines, from eight tracked paths, with no polynomial in the plane's coordinates computed anywhere. A trace test confirms that the six are all of them — their sum along a pencil of lines is affine to 1.3 × 10⁻¹⁵ and bends by at least 3.6 × 10⁻⁴ when any one is left out.

Tested in A degree counted on a line · the algebra sequence.

A four-bar that fails Grashof's condition is a triple rocker, and there is nothing more to say about which kind.

What decides it: Three signed sums of the lengths split the failing linkages into four regions whose rockers swing through different positions. Of 4,000 random four-bars, the 1,771 that fail Grashof's condition fall into groups of 459, 428, 456 and 428, and every one of them swings through the positions its signs predict — measured by asking, at 360 angles of each grounded link, whether the rest of the chain can close.

Tested in Eight kinds of four-bar · the fourbar sequence.

The area a coupler curve encloses depends on all four link lengths, because the shape of the curve does.

What decides it: The shape does and the area need not. For a point on the coupler line of a crank-rocker the area is (1 − u)πa², which contains the crank and nothing else: four crank-rockers with a crank of 1 and every other length different enclose 2.199115 each, on both assemblies, with a worst difference of 3.2 × 10⁻¹⁴ between the traced polygon and the formula.

Tested in The area a coupler point encloses · the coupler sequence.

A six-bar gives more design freedom than a four-bar because it has more links.

What decides it: The number of independent dimensional parameters is two per pin less four — four for the four-bar, ten for either six-bar, sixteen for any eight-link chain. It depends on the pin count and on nothing else, so Watt's chain and Stephenson's have exactly the same number of free parameters and pose different synthesis problems. More links buys freedom through the pin count, and which chain is chosen buys something the count cannot express.

Tested in Choosing the chain before the lengths · the synthesis sequence.

Two configurations of a one-freedom mechanism on the same assembly branch can be driven between; a branch is a connected set.

What decides it: It is connected in the solution set of the constraint equations and need not be connected in what the machine can occupy. With two studs bolted to its frame, a four-bar's drive is two arcs covering 74% of the turn; every angle in both solves to 10⁻¹⁴ on the same branch, at the same mobility, and the crank cannot be driven from one arc to the other.

Tested in Free space comes in pieces · the body sequence.

A mobility count that comes out negative means the mechanism is overconstrained by accident and something is wrong.

What decides it: Here it means the opposite. Every one of the 154 surplus equations in the braced machine is a brace, added on purpose to remove assemblies that draw the wrong curve, and every one of them is implied by the constraints already present. The count reads −153 and the rank reads a mobility of one, unchanged from the unbraced machine, at every size measured.

Tested in One freedom and four hundred links · the mobility sequence.

A six-joint arm with its tool clamped is a structure — six joints, six constraints, nothing left.

What decides it: At an ordinary posture, yes: Kutzbach counts zero and the rank measures zero. At each of the arm's three singularities the screw system's rank falls to five and the pinned arm has a freedom left — it can still move, with its tool bolted down, and the formula that counted it a structure knows nothing about where the axes are.

Tested in Pin the tool and it is a loop · the spatial sequence.

Enumerate the extreme combinations and take the spread. It is the exact answer and computers are fast.

What decides it: It is 2ⁿ mechanisms at every position. Sixteen for a four-bar and 128 for a Watt six-bar are both cheap; twenty parameters, which is an ordinary spatial mechanism, is 1,048,576 against twenty linear solves for the derivative route. The two agree to 0.01% on the six-bar, so the cheap route is not an approximation of the expensive one in any sense that matters — it is the same answer arrived at by a method whose cost grows linearly.

Tested in Seven lengths and a hundred corners · the tolerance sequence.

A mechanism's output is a function of its input, possibly a multi-valued one with a branch to be chosen.

What decides it: For an escapement it is neither. One pallet angle occurs at three different wheel angles in a single period, and the three are not branches of one solve — they are different contacts, each with its own equation. Choosing a branch does not disambiguate them, because a branch is a choice within one system and these are three systems.

Tested in Where the input stops deciding · the intermittent sequence.

A calibration that fits every reading has found the machine.

What decides it: This one fits thirty readings to 1.8 × 10⁻¹⁶ radians and returns g = 3.96916 for a machine whose ground length is 4.00000. Every other length is wrong by the same factor. The fit is exact, the shape is exact, and the four numbers it prints are not measurements of anything.

Tested in Every length wrong, every reading right · the parameter sequence.

Given enough traced points, a least-squares fit will find the degree of any mechanism's curve, as it does for a four-bar's.

What decides it: On 1,600 traced points of a Stephenson six-bar's arm curve, with the fit given its best chance (a QR factorisation in a Chebyshev basis rather than the normal equations), the ratio of the second-smallest singular value to the smallest is 7.45 at degree six, 14.07 at fourteen, 1.44 at seventeen and 1.36 at eighteen. No degree is distinguished. The four-bar's sextic, by the same fit, is decided by a ratio of 4.3 × 10¹⁰. The arm curve's degree, eighteen, comes from slicing: eighteen distinct witness points on each of four random lines, from thirty-two tracked paths.

Tested in The curve nobody eliminates · the algebra sequence.

A redundant leg removes a parallel platform's direct singularities.

What decides it: It changes their dimension rather than removing them. At a platform angle of 0° the three-legged platform's singular set is a curve through the slice, 102 segments long; with a fourth leg attached at 180° it is one isolated point, at (−1.449, −0.811), where all four leg lines meet. And a fourth leg placed along a line one of the three already supplies removes nothing: the singular curve stays, 74 segments of it, unmoved.

Tested in What a fourth leg buys · the parallel sequence.

Any cam can be made to drive its follower both ways by closing a yoke round it.

What decides it: A yoke's two faces are a fixed distance apart, so the cam's breadth along the follower's line must equal that distance at every angle, which is s(θ) + s(θ + 180°) constant. The cam field's standing programme — a rise and a return of 20 over 120° with dwells of 30° and 90° — varies on that sum by 9.502, and measured across the drawn cam it varies by the same 9.502. A yoke wide enough to fit it anywhere leaves 9.502 of lost motion at 45°; one that closes that gap cannot be assembled at 120°.

Tested in A cam that holds its follower both ways · the cam sequence.

A plain crank-rocker cannot get past a time ratio of about 1.2.

What decides it: The limit depends on the swing. Taking, for each swing and ratio, the member of the whole family of crank-rockers with the best worst transmission angle, the 40° rule is kept up to a ratio of 1.322 for a 30° swing, 1.207 for a 60° swing and 1.025 for a 90° swing, and at no ratio at all for a swing beyond 100°. The remark is exact for one swing and wrong in both directions elsewhere.

Tested in A swing and a time ratio · the fourbar sequence.

A four-bar that Grashof's rule names a crank-rocker has a crank that turns and a rocker that swings.

What decides it: One of the two machines Roberts's construction gives for the standard crank-rocker is also named a crank-rocker by the rule, because its shortest bar is next to the frame. Its input pin's path encloses nothing and its output pin's encloses 1.7357, which is its whole circle: the input swings and the output turns. An area formula that took the name at its word left out 1.727876 of the 2.116354 that machine encloses.

Tested in Where three machines keep one area · the coupler sequence.

A screw system that does not turn through a mechanism's motion tells you the mechanism is planar, spherical or translational.

What decides it: It tells you the motion lies in some proper subgroup and cannot say which. Sarrus's linkage and a planar four-bar both report a screw system that stands still to the noise floor; their motions are a one-dimensional translation group and a three-dimensional planar group, which are different in dimension, in type and in what the mechanism is for. The closure dimension is one and three respectively, and the classifier names both.

Tested in A name for each overconstraint · the spatial sequence.

A revolute is a point in the plane; giving it a size is a detail of manufacture with no geometry in it.

What decides it: Two pins on one link cannot be closer than the material round them, and a construction that returns points is free to put them anywhere. The shortest link in the synthesis field's three-position survey is 0.103 units on a mechanism whose poses span 2.2, which is a link shorter than the boss a modest pin needs — and a tenth of the survey's exact solutions have their closest pair of pins inside 0.41.

Tested in A pin is not a point · the body sequence.

Prescribing more points is a step towards prescribing a whole curve, so a fine enough sampling amounts to an exact specification.

What decides it: It does not converge to it. A four-bar can meet nine points exactly and no more; a tenth makes the system over-determined and the answer becomes an optimiser's best compromise, which is a different kind of object. Prescribing the curve as a polynomial skips the sequence entirely — the compiled machine is exact at every point of the curve at once, and costs hundreds of bars rather than four.

Tested in Prescribing a curve rather than points · the synthesis sequence.

A jackknifed articulated vehicle is stuck because the mechanism has reached a singular configuration in which it can no longer be manoeuvred.

What decides it: The growth vector of a car and trailer is (2, 3, 4, 5) at a hitch angle of zero, forty-five degrees, ninety, a hundred and thirty-five and a hundred and eighty. Every configuration remains reachable from every other, and nothing about the mechanism's reachability changes as it folds. What a jackknifed rig has run out of is room to drive forwards, which is where the angle is undone — the driving is the resource, not the geometry.

Tested in The angle that doubles · the rolling sequence.

A mechanism's screw system is a subspace of twists, so measuring it tells you the subspace its motion lies in.

What decides it: A screw system is the tangent space to the motion at one configuration, and every set equals its tangent space to first order. Four pins at random, sampled over 10⁻⁹ radians, report a four-dimensional set of displacements — the same answer a SCARA arm gives, whose motion really is inside a four-dimensional group. The two are told apart only by sampling wide enough for the set to curve away from its own tangent space, which happens above 10⁻⁶ radians and is where every measurement in this field is made.

Tested in The instrument that is not a derivative · the pair sequence.

A finer escape wheel is a better escapement, since it wastes less of its travel in drop.

What decides it: The drop falls with the tooth count and the tolerance stack does not: 0.382° at fifteen teeth, 0.392° at thirty, 0.388° at forty-eight. So the margin collapses from 26 to 4.5 across that range and reaches one at about seventy-two teeth, past which the mechanism is not buildable to these tolerances at all.

Tested in What a drop cannot be smaller than · the tolerance sequence.

An arm's accuracy is set by its joints, so the best encoders should go where the finest motion is needed — at the wrist.

What decides it: A microradian at the shoulder moves the tool 1.557 µm and a microradian at the fifth joint moves it 0.220 µm; the sixth joint's axis passes through the tool and its error moves the tool not at all. Every one of those numbers is the perpendicular distance from the tool to that joint's axis, and none of them is a property of the joint.

Tested in Where an error at the shoulder ends up · the tolerance sequence.

An angle sensor cannot recover a length, so a vehicle's odometry cannot recover its own dimensions.

What decides it: A wheel encoder is an angle sensor whose reading is tied to a distance by the wheel's radius, which is a length. Odometry recovers the wheelbase in units of the wheel radius exactly, and neither in metres — one dimensionless combination from a pair of dimensionless readings, which is the standing rule producing a different-looking answer.

Tested in A wheel that cannot report its radius · the rolling sequence.

A mechanism with one degree of freedom goes through the same cycle of positions over and over.

What decides it: A geared five-bar has one degree of freedom at every gear ratio, by Grübler's count and by the rank of its constraint Jacobian, with the smallest singular value never below 0.418. At a ratio of 3 to 2 it is back where it started after two input turns. At the golden ratio it is never back: the second crank's distance from home sets new records at 1, 2, 3, 5, 8, 13, 21, 34, 55 and 89 turns and never reaches nought.

Tested in One freedom, and a motion that never repeats · the mobility sequence.

The best dwell comes from the highest order of contact between the coupler point's path and the dwell link's circle.

What decides it: At an angle of 120.8024° between coupler and arm the symmetric curve holds its osculating circle to sixth order, and a six-bar built there stays inside 0.1% of its output swing for 58.7° of crank and inside 1% for 90.5°. At 118.6°, where the contact is only fourth order, it stays inside 0.1% for 80.1°; at 116.1° it stays inside 1% for 122.9°. The longest dwell at a stated tolerance is found by giving some order back.

Tested in The flattest dwell is not the longest · the coupler sequence.

An oscillating roller follower is an offset follower whose offset is fixed by where the pivot is placed.

What decides it: On an arm of 50 balanced on a prime circle of 40, the distance of the roller's line of motion from the cam's centre runs from −7.45 on the low dwell to 12.39 at full lift of 20, so the offset is a ramp and not a number. On the standing programme, whose strokes are mirror images, the best such arm peaks at 23.20° where a centred slide peaks at 21.22°, a premium that is 5.40°, 1.98°, 0.78°, 0.22° and 0.031° at arms of 30, 50, 80, 150 and 400 — within 3% of 5,000/L² at every length — and it leans the follower 10.73° on the low dwell. On the quick-rise programme a pivot on the wrong side, rebalanced by its own angle, costs 0.80° at an arm of 50; a sliding offset moved to the wrong side costs 11.04°.

Tested in An arm is an offset that grows with the lift · the cam sequence.

An offset slider-crank's crank turns fully when its connecting rod is longer than its crank, as a centred one's does.

What decides it: Of 4,000 random offset slider-cranks, crank and rod between 0.2 and 5 and offset between −5 and 5, a rod longer than the crank misjudges 1,275. A rod longer than the crank plus the offset decides all 4,000, and it is exactly the region where both surviving signed sums, b − a + e and b − a − e, are positive — the crank-rocker's region of the eight kinds of four-bar, carried to the limit.

Tested in Four kinds of slider-crank · the fourbar sequence.

Tricircularity is a special property of the four-bar's coupler sextic.

What decides it: Sliced by lines through each circular point, every body of the four-bar, the Watt six-bar and the Stephenson six-bar draws a curve that passes through each circular point exactly half its degree times: the circles once, the sextics three times, and the Stephenson arm curve, of degree eighteen and with no known equation, nine times, from 32 tracked paths and on five lines through each point. A slider-crank's connecting-rod quartic passes through each once, not twice, and an elliptic trammel's ellipse not at all. Three is half of six; the property belongs to machines made of pins.

Tested in Nine times through each circular point · the algebra sequence.

A scissor lift stows flat: its height is proportional to the sine of its opening angle, and that goes to zero.

What decides it: It goes to zero for lines. For bars of half-width 0.035 with bosses of 0.047, the fold stops at φ = 0.0951 radians, where two bars sharing a plane touch — leaving 0.0949 per stage, which is twice the boss radius to four figures. A six-stage stack stows at 0.570 rather than at nothing, and the floor is per stage, so it grows with the count.

Tested in A stack that has to fit · the network sequence.

A better measuring machine would let you calibrate every parameter of a robot's model.

What decides it: Six of this arm's thirty-six model parameters lie in directions along which the tool pose does not change at all — not slightly, exactly. The identification matrix's singular values fall from 6.3 × 10⁻⁴ to 2.3 × 10⁻⁸ between the thirtieth and the thirty-first, a gap of 2.7 × 10⁴, and the thirty-first is the difference noise. No instrument helps, because there is nothing there to measure.

Tested in What a calibration cannot see · the serial sequence.

Kempe's theorem means any curve you want can be drawn by a linkage, so linkage design for curve tracing is a solved problem.

What decides it: The theorem says a linkage exists and is silent about its size and about how much of the curve it covers. Built and measured: a general quintic costs four hundred and thirteen bars, of which three quarters carry angles rather than compute them, and it traces its curve over 0.17 radians of driving angle. A solved problem would have to say something about those two numbers, and the theorem says nothing about either.

Tested in What universality is worth · the compute sequence.

A belt drive's ratio is set by its pulley diameters, so measuring the ratio measures the pulleys.

What decides it: It measures their ratio. Scale both pulleys and the centre distance together and every wrap angle, every tangent direction and the velocity ratio are unchanged — so a drive's motion determines its proportions completely and its size not at all, which is the standing result arriving at a mechanism made of rope.

Tested in A drum is a size, a wrap is a shape · the strand sequence.

A small residual means a good calibration.

What decides it: A model missing a parameter this machine has settles at a residual of 4.2 × 10⁻³ and reduces the error over the measured range by a factor of thirty-three. Every practical test says it worked, and the rocker it reports is eight per cent short. Meanwhile a fit whose parameters are exactly right to fourteen figures can carry a residual of 268°, because it assembled the machine the other way.

Tested in Reading a residual · the parameter sequence.

A linkage made to within a small tolerance moves like its drawing, to within something of the size of that tolerance.

What decides it: A parallelogram with a ground of 3, cranks of 2 and a coupler of 3 is the corner of four regions of length space. With its coupler longer by δ its input cannot come within the square root of δ/3 radians of the flat position: 5.7735 × 10⁻⁶ at δ = 10⁻¹⁰ and 1.8257 × 10⁻³ at 10⁻⁵, by bisection on the exact closure, against the law to seven figures. Scaled to a 300 mm frame, a micron of error costs 0.365 mm of crank-pin travel. The same error in a three-crank chain leaves a misfit of exactly one micron.

Tested in A parallelogram a micron wrong · the mobility sequence.

When an implicit fit leaves more than one polynomial vanishing on the traced points, it was given too few points.

What decides it: The circle motion of a parallelogram chain, ground 4, crank 1.5, coupler 4 and rocker 1.5, traced at 1,438 points, leaves a degree-six fit a null space of exactly fifteen behind a drop of 3.7 × 10¹³, and the circle times each of four random quartics lies in it to 4.7 × 10⁻¹⁵ or better. One oval of the standard crank-rocker, traced at 2,880 points, leaves a null space of fourteen at degree eight behind a drop of 57, where the sextic's multiples number six; both ovals give six behind 1.8 × 10⁹. A large null space with a drop to rounding is algebra however few the points; a count with a drop of two decades is arithmetic however many.

Tested in A null space of fifteen is not noise · the coupler sequence.

A roller in a groove changes walls where the programme's acceleration changes sign, so a cycloidal programme has a fixed number of crossovers.

What decides it: It changes walls where the follower's mass times its acceleration plus the load changes sign, which depends on speed. For a follower of 0.5 kg pressed toward the cam by 20 N, the quick-rise cycloidal programme has no crossovers below 267.6 rpm, two up to 505.5 rpm and four above; at 300 rpm its rise's two fall at 58.18° and 76.82°, and at 3,000 rpm at 45.11° and 89.89°. Under the constant-acceleration law they are pinned at 45.00° and 90.0° and arrive as steps of m·2A·ω² and m·A·ω²: 3,200 N and 1,600 N at 3,000 rpm.

Tested in A roller in a groove changes walls with the speed · the cam sequence.

Two rotors whose profiles are exactly conjugate can drive each other through a whole turn, since conjugate action guarantees continuous contact and a constant ratio.

What decides it: Two identical two-lobe cycloidal rotors on pitch circles of radius 50 keep one contact at all but the two hand-over positions of 361 sampled through a lobe pitch, with the normal through the pitch point to 10⁻¹⁰. Its moment arm about the driven shaft is −50·sin 2φ to 8.4 × 10⁻⁹: zero whenever a tip lies on the line of centres, and negative, turning the driven rotor backwards, at half the positions. Circular tips 20 high change sign at the same three angles.

Tested in Rotors that mesh and cannot drive each other · the meshing sequence.

A planar parallel platform's direct singularities form a curve in each orientation slice, so at any orientation there is a position at which it can be controlled.

What decides it: A 3-RPR platform with base pivots on a circle of radius 2 and platform points on one of radius 0.6 at the same angles is singular at every one of 625 positions across a square 1.8 wide when turned to 0° or to 180°: the largest smallest singular value of its three legs there is 1.6 × 10⁻⁸. Its three leg lines meet at the point about which the base triangle scales onto the platform, to 3.3 × 10⁻¹⁶, at every position tried. Turn its platform points 40° in their own frame and the two orientations move to −40° and 140°.

Tested in Two orientations no position can rescue · the parallel sequence.

A linkage built from crank-rockers cannot give a quick return near 2 without a transmission angle far below 40°.

What decides it: A single crank-rocker with a 60° swing keeps at best 19.47° at a time ratio of 2. A drag link of ground 1, crank 2.5, coupler 2.25 and output 1.5, whose least transmission angle is 41.41°, driving the crank of the crank-rocker whose own ratio is 1.2 and least angle 40.32°, gives the whole machine a ratio of 2.7108 at a phase of 284.8° between the stages — by closed form and, within 0.001, by the six-bar swept in the solver with the swing held at 60.00°. Of 821 drag links on a grid that keep 40°, 147 reach 2 with that crank-rocker.

Tested in A drag link ahead of a crank-rocker · the fourbar sequence.

A one-freedom linkage draws an algebraic curve whose degree is fixed by the links it is built from, as a four-bar's coupler draws a sextic.

What decides it: One geared five-bar, cranks 1 and 1 on pivots 3 apart and couplers 3 and 2.5, draws a pin curve of degree 6 at gear ratio 1 : 1, 10 at 2 : 1, 16 at 3 : 2, 26 at 5 : 3, 42 at 8 : 5, 68 at 13 : 8, 110 at 21 : 13 and 178 at 34 : 21, which is 4·max(p, q) + 2·min(p, q), counted as the roots of one Laurent polynomial in w = exp(iφ) with every root satisfying the machine to 4 × 10⁻¹³, and confirmed by homotopy continuation to 3 : 2 and by an implicit fit at 6, 10 and 16. At the golden ratio no fit up to degree fourteen has a drop of even one decade. The same links draw curves of every degree the ratio asks for.

Tested in Every rational gear ratio has a degree · the algebra sequence.

A Geneva wheel with several pins has to satisfy two separate conditions: enough of the turn left over for a dwell, and pins spaced far enough apart that two are never in slots at once.

What decides it: They are one inequality written twice. The engagement arc is 180° − 360°/n and the spacing is 360°/p, so the arc fits between the pins exactly when p(n − 2)/2n is under one. Checked over fifty-six slot-and-pin pairs, from 3 to 37 slots and 1 to 9 pins, the two tests give the same verdict in every row.

Tested in Two pins and no dwell at all · the intermittent sequence.

A mechanism with hundreds of links is structurally complicated and has to be solved as a whole.

What decides it: Walk the joints of a compiled linkage, placing any whose position is fixed by two already-placed neighbours. On a machine of two hundred and forty-two joints the walk finishes with two hundred and thirty-eight placed alone and one dyad — the same decomposition a four-bar of four links does not have. Size and structural depth are independent.

Tested in A machine with one dyad in it · the topology sequence.

The Lie bracket in kinematics is one tool, so a result about brackets in one field carries over to another.

What decides it: The two brackets on this site act on different objects. The rolling field's is a bracket of vector fields on a configuration manifold, computed by finite differences and varying from point to point; the pairs field's is an exact bilinear map on a six-dimensional algebra with constant structure. A distribution can be integrable at one configuration and not at another; a subspace of twists is closed or is not, everywhere. The two fields also want opposite answers — non-closure is why a car can park and why a mechanism has no motion type.

Tested in One bracket, two subjects · the rolling sequence.

Put an obstacle in an arm's way and it can no longer get from one side of it to the other.

What decides it: Not for an arm whose joints turn all the way round. Eighty single-disc arrangements were tested and every one of them left the free configurations in a single connected piece; a thirteen-disc fence did too. The arm goes round the back, by folding — and the folded configurations are off the top of the drawn square and back on at the bottom, where the picture cannot show them. Add joint limits and the same fence cuts the space in two.

Tested in The space of configurations · the serial sequence.

The mechanism has four lengths, so tolerance the four lengths.

What decides it: No drawing tolerances a link length; it tolerances the two holes the length is between, and the length is derived. Whether the two errors are independent decides the answer in both directions — with the frame holes located separately the lengths-only band is optimistic by 11%, and with both bored in one setup it is pessimistic by 34%. The sensitivity is geometry and this site computes it; which features are independent is a fact about a factory and it does not.

Tested in Tolerancing the holes · the tolerance sequence.

A rocker geometry is set up by making the pad square to the valve with the valve shut.

What decides it: Squaring it there puts the whole lift on one side of the geometry's symmetry: the pad sweeps 1.35 mm across the valve tip and the peak lift is 0.51% short. Squaring it at mid-lift instead — a shim under the stud — gives 0.34 mm of sweep and a peak lift within 0.0003% of the arm ratio's promise. The rule is not about being square; it is about where the symmetry is centred.

Tested in Where the pad touches · the cam sequence.

If a calibration makes the machine more accurate, the parameters it found are closer to the truth.

What decides it: This one makes the machine thirty-three times more accurate over the range it measured and returns a rocker of 2.7526 for a machine whose rocker is 3.0000. The improvement is real and repeatable; the parameter is out by eight per cent, and no diagnostic in the fit says so.

Tested in A parameter the model has not got · the parameter sequence.

A mechanism that moves when its count says it cannot has a redundant constraint, and removing that constraint leaves the same mechanism.

What decides it: Dixon's framework of nine bars moves with one freedom where the count gives none, and no one of its bars is the redundant one. Removing any single bar leaves one freedom and exactly the same motion, the removed bar's length drifting by at most 7.1 × 10⁻¹⁵ as the other eight are followed; removing any of the thirty-six pairs leaves two freedoms. The redundancy belongs to the nine together.

Tested in Nine bars that ought to be rigid · the mobility sequence.

Each of the three four-bars that draw one coupler curve keeps the closed-form part of its enclosed area in a different term of the area formula.

What decides it: It is a fact about one of the four sets of three. A double crank's three machines each occupy all three closed-form columns and carry the same three numbers — 17.6934, 16.3426 and 0.1018 — permuted between them. A double rocker's two cognates occupy the same column with the same value, −0.007854. In both triple-rocker sets no machine has a closed-form term at all. The claim is refused in three of the four.

Tested in The kind is decided before the lengths are · the coupler sequence.

Each inversion of the slider-crank chain has its own conditions for a continuously rotating input, so a mechanism must be classified after its frame is chosen.

What decides it: The chain has three relative rotations and its four inversions read two of the three each. Over 1,500 chains every circuit's winding triple is one of exactly three — (1, 0, −1), (0, 1, 1) and (0, 0, 0) — fixed by the signs of b − a + e and b − a − e, which are the same two quantities for every choice of frame, up to a sign and a change of slot. The count of inversions with a turning input is three, three, none and none across the four regions.

Tested in Three rotations, and four benches · the fourbar sequence.

A rotor pair can be given a running clearance either by cutting the rotors back or by setting the shafts a little further apart, and the two amount to the same thing.

What decides it: Cutting both rotors back by 1 leaves a gap between 1.9998 and 2.0000 over a whole lobe pitch. Opening the centres by 1 leaves a gap between 0.1255 and 1.0000, a ratio of eight to one, and at an opening of 0.1 the ratio is 23.7 to one with the tightest place 4.2% of the nominal. A seal is specified at its tightest, so the second buys a twenty-fourth of what it costs.

Tested in The clearance that is the seal · the meshing sequence.

Adding a redundant leg to a parallel platform removes its singularities, so a fourth leg is a cure for an architecture that is singular by design.

What decides it: On a 3-RPR whose base and platform triangles are similar, a fourth leg whose own pair of attachment points is related by that same similarity leaves the platform singular at every position at both of the two orientations, exactly as the three-legged one was — measured at 2.6 × 10⁻⁹, which is the solver's floor. Swept round the platform, that is the one placement in three hundred and sixty that fails, and it is named by the similarity before the sweep is run.

Tested in One placement of every placement · the parallel sequence.

Putting a cam follower on a swinging arm instead of a slide is a way of improving the cam, so a flat face on an arm should work at least as well as one on a slide.

What decides it: It is worse at every arm length and there is no arm at which it is not. The smallest base circle that leaves the cam convex is 10.6640 for a sliding face and 10.7096, 10.8483, 11.6079, 14.0644, 18.4640 and 28.6789 for pivots at 20000, 5000, 1000, 300, 150 and 90. An arm pivoted 60 away has no convex cam at any base circle at all.

Tested in A flat face on an arm is worse · the cam sequence.

Two mechanisms that draw curves of the same degree are drawing curves of the same kind, so a gear ratio's sign is a fact about the machine and not about the curve.

What decides it: At every ratio tried the two senses give one degree — 6, 10, 14, 16, 24, 26 and 42 — and different circularities. The internal mesh passes through each circular point exactly half its degree at all seven; the external mesh passes through twice the larger tooth count, which is 33.3% of the degree at 1 : 1 and 42.9% at 3 : 1. One is at the ceiling for its degree and the other is not.

Tested in The mesh inside keeps the half · the algebra sequence.

A power-split transmission's geared-neutral point is a ratio the machine can be set to, so the output can be held at rest with the engine running.

What decides it: It is a boundary, not a setting. The output ratio's sensitivity to the variator is (1 + K)/(K − v)², so it is 15 at a gap of 0.4 and 24,000 at a gap of 0.01. Put the variator's own uncertainty — 1.6% from a belt, a centre distance and a sheave each out by the tolerance table's amounts — through that, and the output ratio is uncertain by 110% at a gap of 0.02, which is more than the ratio itself.

Tested in A bounded ratio made unbounded · the transmission sequence.

A one-way clutch with several rollers has a finer resolution than one with a single roller, as a ratchet with several pawls has a finer resolution than one with a single pawl.

What decides it: A ratchet's pawls are staggered against its teeth, so m of them divide the lost motion by m: 15°, 7.50°, 5.00° and 3.75° at 24 teeth. A clutch's rollers all sit at the same angle on identical flats, so they reach their seats at the same instant — measured across three, four, five, eight and twelve faces the lost motions differ by nought, and the number of rollers is a question about load rather than about resolution.

Tested in A ratchet with no teeth · the intermittent sequence.

Every question about a mechanism is ultimately a question about numbers you could measure.

What decides it: A chain has no numbers. Sixteen eight-link chains exist, the count is exact, and no scaling, tolerance or instrument bears on it — the field's evidence is a canonical form and an enumeration, and its errors are of a kind no measurement would ever find.

Tested in A graph has no numbers at all · the topology sequence.

A four-bar's two assembly branches are two ways of building the same linkage, so they are the same mechanism described twice.

What decides it: For a Grashof linkage they are two disjoint circles in configuration space, each spanning the full 360° of crank angle, and no motion joins them. For a non-Grashof linkage they are one circle: the two rocker solutions converge as the crank approaches its limit — 120° apart at the start, 3.3° apart at 117°, and gone at 117.5° — and the linkage passes from one to the other by turning back. Same construction, different topology, and the difference is Grashof's condition.

Tested in Branches were components all along · the algebra sequence.

Which bearing of a four-bar carries the play decides how much of a change point it can restore, so the loose one is the one to find.

What decides it: Each of the four pins was given a radial play of 4 × 10⁻⁴ and its displacement searched over every radius and direction of its disc. All four leave the loop 6.0000 × 10⁻⁴ from closing — the shortfall less the play — and they agree with each other to nought. Every clearance enters the loop sum as one more displacement vector, so only the total matters and a linkage cannot tell which of its bearings is loose.

Tested in A length error is undone by its own size · the mobility sequence.

A Bowden cable's lost motion depends on how tightly its sheath is bent, so routing it on generous radii keeps the free play down.

What decides it: The play depends on the angle turned and not on the radius. A single 90° bend with a clearance of 0.25 leaves the pulled inner 0.392282 short of the sheath on a bend radius of 15 and 0.392283 on a radius of 140; the law c·φ gives 0.392699, and the remainder comes from the straights at the ends. Over six routes the shortfall is the clearance times the total turning to within 0.05% at a clearance of 0.01, and an S-bend whose net turning is nought loses as much as a U-bend.

Tested in A strand in a tube · the strand sequence.

Oldham's coupling transmits constant velocity between offset shafts because its two slides are at right angles.

What decides it: The slides at 60° give the same ratio of exactly one: the relative motion of the hubs is the translation group T2 whenever the two slides are not parallel, and the output's angle stays on the input's to rounding. The right angle sets how fast the slides run: their peak speed is the offset times the shaft speed divided by sin β — 1.000 at 90°, 1.155 at 60°, 2.000 at 30° — and the disc's centre runs twice a turn round a circle of diameter e/sin β.

Tested in A coupling that only translates · the pair sequence.

Twisting a pair of rotors along their shafts averages out a profile that seals unevenly, so a helical pair can be given a decent running clearance by shapes that would not hold one flat.

What decides it: A wrap of one lobe pitch holds the seal's open area constant to 2.4 × 10⁻⁴ against a flat swing of 4.757 — but it holds it at −0.4085, the flat pair's own average, which the wrap cannot move because a moving average moves no average. The tightest section anywhere on the rotor is the profile's own tightest, −2.420, at every shaft angle rather than once a turn.

Tested in A twist steadies what it cannot tighten · the meshing sequence.

A compiled machine's size is a property of the curve it draws, so five bars for a line and four hundred and thirteen for a quintic measure how hard those curves are to draw with bars.

What decides it: The same line, written as its own equation multiplied by 1 + x² + y², compiles to fifty bars instead of five, and its machine traces the same line to 1.2 × 10⁻¹². The factor is at least one at every real point, so not one point of the curve has moved. The penalty for that factor is 10.00 on the line, 3.75 on a hyperbola, 2.45 on a circle and 2.07 on a cubic, so it is not a property of the factor either.

Tested in The price is on the equation · the compute sequence.

A rank deficiency is the way a calibration fails to determine its answer.

What decides it: It is one way. Three cognate four-bars trace one coupler curve with cranks of 1.600, 2.214 and 2.558 — sixty per cent apart — and the identification Jacobian at each of the three is of full rank and perfectly ordinary. No derivative detects an alternative that is not nearby, and there are two of them.

Tested in Three machines, one curve · the parameter sequence.

How much clearance a framework needs to absorb a misfit can be read off the residual a least-squares fit leaves.

What decides it: A least-squares fit minimises the sum of nine squares and leaves the nine bars unequal, from 4.35 × 10⁻⁵ to 2.28 × 10⁻⁴ on a framework 4° off square pushed 0.2. Its worst bar asks for 1.508 times the clearance that is needed. The placement that minimises the largest of the nine puts all nine at 1.5095 × 10⁻⁴ exactly, to 3 × 10⁻⁹, because every bar can be traded against every other.

Tested in The right angle as a tolerance · the mobility sequence.

A pair of rotors that are each other's conjugates cannot drive itself, so conjugacy between identical shapes is what puts the dead points there.

What decides it: A ring of twelve pins and the eleven-lobed disc they generate are exactly as conjugate, by the same routine from the same equation, and have between eight and eleven contacts at every angle of the input. The largest arm available never falls below 70.7% of the pitch offset in either sense, against the identical pair's nought. Six pin counts from six to twenty all keep more than one contact and all keep an arm.

Tested in What a second contact is for · the meshing sequence.

A paired Gough platform's dead yaw is a property of the level slice, so a tilting task can pass near it by tilting.

What decides it: The rise is second order: the smallest singular value at the home position goes from 8.9 × 10⁻⁹ level to 4.4 × 10⁻⁷ at a quarter of a degree, with a fitted exponent of 1.987. And the singularity is not removed. At three degrees of tilt the workspace spans 3.4 × 10⁻⁷ to 1.0 × 10⁻³, a factor of three thousand, and a descent from forty-eight starts reaches exactly nought — at one degree, at three and at six.

Tested in Tilted, near the dead yaw · the parallel sequence.

A four-axis arm is a six-axis arm with two axes left off, so it can do a subset of what the six-axis one does and is otherwise the same kind of machine.

What decides it: A SCARA arm's reachable displacements are a group and a six-axis arm's are not — four dimensions that close under the bracket against six that have nothing to be inside. The consequence is a guarantee rather than a restriction: everything true of the SCARA near one pose is true near every other, transported by an element of the group, so its tool face is level at every reachable configuration by construction. Four random axes give a four-dimensional set whose logarithms span all six and which guarantees nothing.

Tested in The arm that is a group · the serial sequence.

A serial arm is harder to calibrate than a linkage because more of its parameters are unrecoverable.

What decides it: Fewer are. An arm measured by tool position has readings that carry a length, so nothing is invisible and the rank is full. What makes it hard is the parameter count, a pose space of dimension three rather than one, and a Jacobian that is singular exactly where the reach is greatest.

Tested in An arm's parameters and its poses · the serial sequence.

A link length has a tolerance, and a tolerance analysis takes the lengths as its variables.

What decides it: A link is two holes and the length between them is derived. With the holes located independently the derived length's band is √2 times what a lengths-only analysis assumes; with their errors entirely common it is zero however badly the pair is placed. The two analyses agree at exactly one value of a number that is a fact about a factory.

Tested in Where a length comes from · the parameter sequence.

A tolerance analysis on link lengths is conservative — it may be a little pessimistic but it will not under-state the band.

What decides it: It under-states it by exactly √2 whenever the holes of a part are located independently, which is the normal case for a frame too large for one setup. The two analyses agree at a shared fraction of one half and the lengths-only one is optimistic below it.

Tested in Where the two analyses cross · the parameter sequence.

The parts of a mechanism that most need making accurately are the ones there are most of.

What decides it: A compiled machine's bars are three quarters parallelograms carrying directions and a quarter angle arithmetic. Lengthening each bar in turn by 10⁻⁴ and reading the polynomial at the tracing point, the worst offenders are consistently reflectors: 9.1 × 10⁻³ against the translators' 2.0 × 10⁻⁴ on the same machine. Size and fragility live in different parts.

Tested in Exactness a micron destroys · the tolerance sequence.

A part's tolerances describe the part, so two drawings of the same part with the same tolerances describe the same thing.

What decides it: Three holes at 0, 1.4 and 3.5, each located to ±0.01. Chained, the far gap is ±0.010 and the span is ±0.0141. From a baseline, the far gap is ±0.0141 and the span is ±0.010. Same part, same process, two analyses, exactly √2 apart in opposite places.

Tested in The same part, dimensioned twice · the parameter sequence.

Denavit and Hartenberg's four parameters describe any pair of consecutive joint axes.

What decides it: They describe any pair with a unique common normal. Two parallel axes have infinitely many, so d and θ are supplied by convention rather than measured; and two axes half a degree from parallel have exactly one, sitting 11.3 link lengths from the joint. Nothing has moved by as much as a degree.

Tested in The common normal, and where it is · the parameter sequence.

A mechanism whose parameters are ill-conditioned is an ill-conditioned mechanism.

What decides it: Two nearly parallel axes have a DH offset that runs to 1,102 link lengths and an identification Jacobian, in a chart with two small rotations instead of a common normal, whose condition number is 7.5501 at thirty degrees of twist and 7.5501 at a hundredth. Same machine, same range, one number diverging and one not moving.

Tested in The chart breaks, the machine does not · the parameter sequence.

A model with more parameters describes a machine more completely, so a calibration with more parameters is more thorough.

What decides it: A six-revolute arm modelled at six numbers per joint transform plus two frames has forty-eight parameters and thirty distinguishable directions. The other eighteen are a family of exactly equivalent models; a fit moves along them arbitrarily, the residual is unaffected, and every number it returns along those directions is a fact about the damping.

Tested in Six per joint is two too many · the parameter sequence.

A tolerance band is a property of the mechanism's proportions, so scaling the machine leaves it unchanged.

What decides it: A band computed from a fixed ±0.01 on each length scales as the reciprocal of the machine's size: the fitted exponent is −1.000. A bigger machine held to the same absolute tolerance is a proportionally better machine. Expressed as a percentage instead, the same band lands on zero. One quantity, two drawing conventions, two kinds of number.

Tested in Which numbers have a size · the parameter sequence.

Everything a scaling leaves unchanged is a shape, so a mobility is a shape.

What decides it: A shape is a quantity that a scaling does not move and something else does. A mobility is moved by nothing continuous at all: it is an integer, it is constant on an open set of parameters, and the probe's control column — which requires a quantity to respond to a five per cent change in one length — returns zero for it.

Tested in A count is neither · the parameter sequence.

A calibration's parameter list comes with the machine's model, so choosing it is not a decision.

What decides it: Fit four lengths to a machine whose tracing point is 0.198 units off and the rocker comes back eight per cent short. Fit four lengths to angle data and one of the four is a starting guess. The same machine and the same readings support both failures, and which one occurs is decided entirely by the column list.

Tested in What a model is allowed to change · the parameter sequence.

A mechanism that solves, sweeps and passes its mobility count is a mechanism that can be built.

What decides it: Give the site's own crank rocker — ground 4, crank 1, coupler 3.5, rocker 3 — links a fifth of a unit wide and sweep it. Every position solves to 10⁻¹⁴, the mobility is one by formula and by rank, and the coupler is inside the frame by 0.432 for the whole turn. Nothing in the closure equations mentions it, because a distance constraint has no width.

Tested in A link that takes up room · the body sequence.

A pawl holds when the line of the tooth face, extended, passes on the far side of its pivot.

What decides it: That is one of two boundaries. The other is the contact normal extended, on which the push has no moment about the pivot at all, and the two are perpendicular — so the holding region is a pair of opposite quadrants rather than a half-plane. Over a square of pivot positions centred on the contact, the one-line rule gets 53.1% of them wrong.

Tested in A joint that works one way · the intermittent sequence.

An epicyclic needs its own formula, because its planet axes move.

What decides it: It needs one extra label. Write the mesh relation relative to the body the two axes are stationary in and the same row covers both cases — an ordinary train is the one where that body is always the frame. The same eight lines return 4 for a planetary with its ring held, 2176/106 for a compound epicyclic, −1/100 for a harmonic drive and 2 for a differential with one wheel stopped, and every one of them is a fraction rather than a rounded decimal.

Tested in A ratio is a null space · the transmission sequence.

Six contact points constrain a rigid body.

What decides it: Six contacts whose normals are not independent as wrenches do not. Three V-grooves cut parallel to each other give six contacts, rank five, and leave the part free to translate along the grooves — measured here as a screw of infinite pitch along (1, 0, 0). The condition is not six contacts; it is six independent ones, and independence is a rank.

Tested in Six points and no more · the mobility sequence.

Tighten the tolerances until the mechanism is accurate enough.

What decides it: The four lengths of this site's four-bar contribute 38%, 26%, 25% and 11% of the output band. Tightening all four equally buys three quarters of its accuracy from the two that matter and pays for four; tightening only the coupler buys 38% of the improvement for a quarter of the cost. The ranking is computed from the same sensitivities the band is, and it is not the ranking the drawing suggests.

Tested in The four lengths do not matter equally · the tolerance sequence.

The lever diagram is a handy way to remember epicyclic ratios.

What decides it: It is an exact statement and it is available for a stated reason. A gearset whose frame carries no teeth admits the motion in which every member turns at the same speed; that motion is one direction of a two-dimensional space, so each member has one remaining coordinate, and taking that coordinate to be its lever position makes the speeds the heights of a straight line by construction. Checked in exact rationals across four gearsets and four motions each, the residual is 0/1. Two trains here have no lever, and the same test says why.

Tested in The lever that is the gearset · the transmission sequence.

Measure a mechanism's motion carefully enough and for long enough and you will know what it is made of.

What decides it: Three four-bars at 0.6×, 1× and 1.5× put their output links at the same angle at every crank position, to 1.4 × 10⁻¹⁶ radians over ninety-six sampled positions. No length of measurement and no quality of protractor separates them, because the quantity being measured does not contain the answer.

Tested in The direction no protractor can see · the parameter sequence.

Make the base circle big enough to keep the pressure angle down and the cam will work.

What decides it: Pressure angle is one of two limits and not the binding one. The curvature of the profile is computed here from the drawn curve, and where the radius of curvature falls below the roller's the cutter removes material the follower needed — the cam cannot be made, at any pressure angle. Both limits fall as the base circle grows, which is why one number appears to fix both and does not.

Tested in The cam that cannot be cut · the cam sequence.

A set of contacts holds a part when some positive combination of their rows comes to nothing.

What decides it: A disc on eight contacts satisfies that exactly: its rows lie in a plane through the origin and the eight of them cancel with equal weights, so the program returns a share of 0.125 and reports the arrangement held. It holds nothing at any number of contacts. Positively spanning a subspace is not positively spanning the space, and the program alone cannot tell the two apart — it needs the rank as a guard.

Tested in The test is a program, not a rank · the restraint sequence.

A four-bar function generator has four design parameters.

What decides it: It has three. Scale one by any factor and every input-output pair it produces is unchanged, so the fourth number is a choice of units. Freudenstein's derivation says so on its face — his relation has three coefficients, and this site's own implementation fixes the ground length at one and has never said that is why.

Tested in The coordinates the site already had · the parameter sequence.

Sweep the mechanism finely enough and a clearance check is conclusive.

What decides it: Finely enough is not a property a sweep can report about itself. Twelve samples of a crank passing a stud return a smallest gap of +0.007 and the machine is 0.010 inside the stud. What makes a sweep conclusive is a bound on the gap between samples: at V = 1.256 per radian and Δθ = 0.571, the least the gap can be between two adjacent samples is −0.281, and the certificate refuses.

Tested in A sweep that missed nothing · the body sequence.

A planetary gearset works if the ring's teeth equal the sun's plus twice the planet's.

What decides it: That condition only says a planet fits between them. Whether n equally spaced planets can be assembled is a separate condition on the same integers — the sun and ring counts must sum to a multiple of n — and it fails silently: the first planet drops in anywhere, and the second is out of mesh by a fraction of a tooth that no clearance absorbs. Of 1,822 sun-and-ring pairs whose difference is even, only 33.4% will take three equally spaced planets.

Tested in The gearset that could not be assembled · the misconception sequence.

A gearbox's ratios are chosen to give equal steps.

What decides it: There are not enough numbers to choose them with. A Simpson three-speed's whole ladder is a function of one proportion and a Ravigneaux four-speed's of two, so once first gear and reverse are picked the rest follow. Equal steps is then an equation rather than a target: K² − K − 1 = 0, whose root is φ. It is achievable — a Simpson of 55 in 89 has steps equal to 0.0024%, and it assembles — and no catalogue gearbox uses it, the ones measured here being 7.5% and 10.8% away.

Tested in The steps are not free · the transmission sequence.

Which plane each link goes in is a graph colouring of the conflicts, so any proper colouring with the fewest colours will do.

What decides it: A pin joining planes 1 and 3 passes through plane 2, and a link in plane 2 whose material covers that pin is pierced. Of the six proper three-plane colourings of a crank rocker, four put a pin through a link and two can be built. Of Peaucellier's 192, exactly 96 can. Permuting colours preserves a colouring and destroys betweenness, so no colouring argument can see it.

Tested in A plane is a colour · the body sequence.

Kutzbach's criterion counts a spatial mechanism's degrees of freedom from its joint graph.

What decides it: The count is compared against the number of joints minus the rank of the joint screws, for every closed loop on this site, on the mechanism ledger. It is wrong about most of them. The repair every textbook gives — add back the constraints imposed more than once — is right, and the missing term is measured here from the two legs' reciprocal systems rather than assumed.

Tested in The formula is repaired by the thing it replaced · the spatial sequence.

Poses for a calibration should be spread evenly through the machine's travel.

What decides it: Even spacing is a good default and it is not the best. Eight poses chosen to maximise the smallest singular value reach 0.2957 where twelve evenly spaced ones reach 0.2935 — the same answer from a third fewer measurements. The chosen set is not evenly spaced: on a pool of thirty-six its eight poses leave gaps running from 10° to 140°.

Tested in Where a calibration should measure · the parameter sequence.

A variator's two sheaves move equal and opposite amounts, so the sum of the running radii is constant.

What decides it: The belt's length is πS + 2d·asin(d/C) + 2√(C² − d²) with S the sum and d the difference of the radii, so dS/dd = −(2/π)·asin(d/C), which is zero only at d = 0. The sum is constant to first order and departs at second — about d²/πC — falling from 110.00 mm to 107.21 mm across the travel here. Using the constant-sum assumption to work out the ratio is 2.9% wrong at one end of the shift and 7.4% wrong at the other.

Tested in A ratio with no steps in it · the transmission sequence.

An overconstrained mechanism only works if its geometry is exact, which is why paradoxical linkages are curiosities rather than machines.

What decides it: The Sarrus linkage tolerates a 0.2 radian tilt of one axis within the plane its chain works in — eleven and a half degrees — and is destroyed by a 0.001 radian tilt of the same axis out of that plane. The sentence is true of one direction and false of the other, and which is which is decided by whether the perturbation lies in the screw system the mechanism leaves unconstrained.

Tested in Fragility has a direction · the overconstraint sequence.

An over-centre latch is locked.

What decides it: It is held by a distance rather than by a constraint. The latch measured here sits 71 µm below its dead-centre position when set four degrees over; a load that moves the slider that far unlocks it, and nothing geometric prevents it. What makes it a good latch is that the same 71 µm is worth 4° of crank rotation — the motion ratio there is eight times the mid-stroke value — so it is hard to reach by accident and easy to reach on purpose.

Tested in Locked on purpose · the fourbar sequence.

A compound epicyclic's reduction is exact, because it is a ratio of tooth counts.

What decides it: The ratio is exact and the mechanism is not free. The two meshes of the drive published here ask for centre distances of 19 and 18.5 modules from one planet shaft — half a tooth apart — so one of them has to run away from the centre it was cut at, or the two stages have to be cut at modules 2.7% apart. The exactness is bought, and this is the invoice.

Tested in A hundred to one from a difference of one · the transmission sequence.

The space a mechanism needs is the extent of its joint positions over its motion.

What decides it: It is the extent of its material, which is larger by the widths, by the bosses, and by the ends that overhang their pins. Across nine machines the material's box is 13% to 36% larger in area than the joints', on links a twentieth of their length wide — and the machine fills between 50% and 78% of the box it needs.

Tested in The hole the machine needs · the body sequence.

A door swings clear if its hinge is on the edge it turns about.

What decides it: A door 18 mm thick, hinged at its back corner and opening 95° next to a door 3 mm away, passes 16.6 mm into its neighbour. Sweeping the pin over a grid of positions shows the region that clears is bounded by the door's own front face: with the doors touching, every pin behind that plane fouls, and the boundary is exactly where the front corner's sideways velocity changes sign.

Tested in Where a hinge pin can go · the coupler sequence.

Grashof's condition tells you whether a four-bar's crank turns all the way round.

What decides it: It tells you that and not how nearly it fails. This site's own four-bar has s + l = 5 against p + q = 6.5, a margin of 1.5 units — comfortable. A four-bar at 4, 1, 3.4, 1.65 has a margin of 0.05, and a twentieth of a unit of manufacturing error on any one link changes what kind of machine it is.

Tested in Grashof is a shape test · the fourbar sequence.

Offsetting a cam's follower reduces its pressure angle.

What decides it: It reduces it on one stroke. The offset is subtracted from the follower's velocity inside the pressure angle, so the stroke whose velocity has the offset's sign gains what the other loses. On a programme whose rise and return are mirror images the best offset is zero to 6 × 10⁻¹². On one that rises in 90° and returns over 170° it is 5.88, and there the rise and the return peak at the same 21.81°.

Tested in An offset trades the rise for the return · the cam sequence.

A four-part block and tackle has a velocity ratio of four.

What decides it: Differentiating the strand's own length gives 3.927 with the blocks 220 mm apart and 3.617 at 90 mm, against 3.99999 at twenty metres. The integer is the limit as the parts of line become parallel, and a tackle is used at close quarters. The same number comes independently from the sum of the parts' cosines, to every figure printed.

Tested in Six things a strand is not · the misconception sequence.

Grashof's condition says whether a four-bar's crank turns all the way round, and that is the whole of what the four lengths decide about whether it will run.

What decides it: With a bearing pedestal at each ground pivot, every fully rotating four-bar tested sweeps a link straight over a pivot — the coupler over the driving pivot on all four crank rockers, the crank over the far pivot on the drag link — so its closest approach is exactly zero and no positive link width is admissible. Not one of the rockers does, and they take widths up to 0.198 of their shortest link.

Tested in The crank that cannot turn all the way · the body sequence.

A mechanism's mechanical advantage and its sensitivity to manufacturing error are separate design questions.

What decides it: In a difference mechanism they are the same number to within a unit convention. The gain and the conditioning number differ by one for the compound epicyclic, by a factor of two for the differential pulley, and not at all for the differential screw here — and all three conditioning numbers are |a/(a−b)| for that mechanism's own two nearly equal quantities. Measured by perturbing each mechanism by one part in ten million: 10.000000, 20.000000, 19.528340, against 10, 20 and 19.528302 predicted.

Tested in Three mechanisms, one subtraction · the transmission sequence.

A kinematic analysis tells you whether a mechanism will work.

What decides it: It tells you where the mechanism can be and how fast. Whether the pin survives, whether the follower stays on the cam, whether a poor transmission angle becomes an actual lock, and whether the mechanism is still accurate in ten years are four separate questions, none of which any figure on this site answers, and each of which has decided the fate of real machines.

Tested in What is still outside · the practice sequence.

In a running gear mesh every tooth eventually meets every tooth.

What decides it: Only if the counts are coprime. A tooth on a 20-tooth pinion in a 40-tooth wheel meets exactly two of the wheel's forty teeth and meets them for ever; add one tooth to the pinion and it meets all forty. Walking the mesh and counting gives the same number as z₂/gcd(z₁,z₂) on every pair tried. Of the 506 feasible planetary gearsets in this site's catalogue, 47.0% are coprime at both meshes and 53.0% are not — and the site's own default gearset has a planet tooth that meets exactly one sun tooth.

Tested in Which tooth meets which · the tooth sequence.

Scaling a mechanism scales all of its parameters.

What decides it: It scales its lengths. A spatial loop's parameters are lengths and twist angles together, and a twist is dimensionless — so a scaling acts on a proper subset of the parameter space, and a condition relating the two halves is neither scale-free nor scale-covariant. Bennett's a/sin α = b/sin β scales on the left and does not on the right.

Tested in Bennett's condition is a ratio · the spatial sequence.

A calibration measures the mechanism at the poses it was planned for.

What decides it: It measures it at the poses the mechanism has. A double rocker asked for twenty-four evenly spaced crank angles returns seven readings; the other seventeen are configurations that do not exist. The rank is still three, because seven rows are more than three columns — but the plan and the measurement are different objects and only one of them was scored.

Tested in The pose the machine cannot reach · the parameter sequence.

A 53/11 chain drive turns the wheel 4.818 times per turn of the cranks.

What decides it: It does, to five decimal places, per whole turn — and at no instant within one. The instantaneous ratio swings from 4.756 to 4.948, a fluctuation of 4.0%, because the effective radius of each sprocket varies within every tooth. The mean here is measured by integrating the velocity law and comes out at 4.8182 against a tooth ratio of 4.81818, which the integration is never given.

Tested in The chain is a polygon · the tooth sequence.

A planetary gear set can be given any reduction you like by choosing the tooth counts.

What decides it: Its reduction with the ring held is 1 + ring/sun, and a ring must exceed its sun by two whole planets, so the reduction cannot be less than 2 on any design whatever. Over the catalogue of designs that can actually be cut, assembled with three equally spaced planets and kept clear of undercutting, the floor is 2.5862 and the ceiling 9.1765; across all six configurations of the same gearset nothing at all is reachable between 1.6304 and 2.5862, and the gap's width as a factor is exactly the smallest achievable ring-over-sun ratio.

Tested in The reductions a planetary cannot give · the transmission sequence.

A chuck centres a section exactly when a turn of 360°/j is one of the section's own symmetries.

What decides it: That is sufficient and not necessary. A six-jaw chuck centres a square bar at every orientation, to fourteen figures, and 60° is not a symmetry of a square. The condition the offset's own formula gives is gcd(n, j) > 1, because a sum over j equally spaced directions picks out only those harmonics of the support function whose index is ±1 modulo j. The two rules agree whenever the jaw count is prime, and three is prime, which is why the wrong one has never caused trouble.

Tested in Where the jaws put it · the restraint sequence.

Every manufacturing error on a four-bar's four lengths moves its output.

What decides it: A quarter of a per cent added to all four moves it by exactly zero — 0.000° at a crank angle of one radian, against 0.168° for the same error on the ground length alone. One direction of the four-dimensional tolerance box is harmless, and with equal tolerances it is exactly a quarter of the box's mean squared error.

Tested in A band with a direction in it · the tolerance sequence.

A coupler curve's curvature is a property of the linkage's proportions.

What decides it: It is a property of its proportions divided by its size. The fitted exponent of a coupler path's curvature under a uniform scaling is −1.000000, so a machine at twice the scale traces a path of half the curvature at every corresponding point — the same curve, drawn bigger, and every radius doubled.

Tested in A curvature is a size with a minus sign · the curvature sequence.

A mechanism's force-transmission behaviour is a property of the machine, so measuring it needs the machine.

What decides it: It needs the machine's proportions and nothing else. The transmission angle at a crank angle of one radian is 65.91° on this four-bar and 65.91° on the same four-bar at a tenth of the size, to the solver's floor. A protractor that cannot recover one length recovers the whole curve.

Tested in The transmission angle has no size · the transmission sequence.

Choosing a mechanism type is a matter of judgement rather than of search.

What decides it: Every requirement in this essay's filter is a statement about the graph and none mentions a length: at least one link with four pins, a link none of whose neighbours is binary, and a driving choice that comes apart into pairs. Applied in turn to the 230 ten-link chains they leave a specific, countable subset — and each is checkable in microseconds, before a single dimension is chosen.

Tested in A catalogue is a search space · the topology sequence.

An exact number is a robust number.

What decides it: The compound reduction here is 20.528301886…:1, exactly 2176/106, and no measurement can disagree with it. Change one ring from 69 teeth to 70 and it becomes 28.63:1 — a 39% change from a part that is 1.4% different. Exactness is a statement about a number's relation to its mechanism; robustness is a statement about its relation to the mechanism's parameters, and they are unrelated properties.

Tested in A ratio that is a count · the tooth sequence.

Two configurations of a mechanism can be joined by a motion exactly when they are in the same connected component of its free space.

What decides it: A cable tie's free space is a single interval — every state connected to every other, no barrier anywhere. Sampling it at 400 points, 52.1% of the 160,000 ordered pairs are reachable and 4.2% are reachable both ways. Connectivity is symmetric and reachability is an order, and the first cannot see the second.

Tested in One piece, and still not reachable · the intermittent sequence.

A two-stage gear train can be given any ratio, and making the shafts coaxial is a matter of arranging the layout.

What decides it: Coaxial shafts mean the two stages share one centre distance, so z₁ + z₂ = z₃ + z₄ on top of the ratio condition — two equations in four integers. Enumerated exhaustively with wheels between 14 and 120 teeth, twelve to one has no solution until a wheel reaches 63 teeth, sixteen to one manages at 56, and no ratio that is not the square of a rational admits two identical stages at all.

Tested in Two shafts that must be in line · the transmission sequence.

Put the best bearing where the load is highest.

What decides it: That is where the wear will be, which is a different question from where the error will be. Ranked by what its clearance costs the output, the crank-to-coupler pin contributes 32% and the rocker-to-frame pin 21% — and the rocker pin is the one carrying the most load on this linkage. The two rankings are computed from different quantities and they do not agree.

Tested in Which pin to buy · the clearance sequence.

A cone of permitted motions is a scale-free object, so nothing in the restraint field has a size.

What decides it: A cone is closed under positive scaling, so its own description is scale-free. Its rays are screws, and a screw's pitch is a length — the ratio of a translation to a rotation. Scale the fixture and every pitch in the cone scales with it, so the cone's shape is a shape and the pitches labelling its rays are not.

Tested in A cone has no size · the restraint sequence.

A calibration is as accurate as the instrument that fed it.

What decides it: It is the instrument's accuracy divided by the smallest singular value of the identification Jacobian. On this four-bar with thirty poses that is a factor of 1.90 measured and 2.16 bounded — so a protractor good to a milliradian gives a shape good to about two. Change the pose set and the factor changes with nothing else moving.

Tested in The instrument's error, multiplied · the parameter sequence.

A synthesis returns the linkage that meets a demand, so the demand determines the answer.

What decides it: It determines it up to a similarity, and the similarity is supplied by the demand's own coordinates rather than by the construction. Scale three prescribed positions and every circle, every centre and every returned linkage scales exactly with them — the construction cannot produce a size the demand did not contain.

Tested in What a synthesis assumes it knows · the synthesis sequence.

The site's boundary was settled when the practice field drew it.

What decides it: It named feature-based tolerancing as the first thing to do next and predicted it would be a straightforward extension of the tolerance machinery. It was straightforward and it was not an extension: it is the same subject as identification, because a length derived from holes and a parameter derived from readings are one question asked at two ends of a machine's life.

Tested in Where the boundary moved again · the practice sequence.

A three-position synthesis that is exact, drivable and in the right order is a linkage that can be built.

What decides it: Of 1,176 exactly correct syntheses, 176 pass all three kinematic tests. At a pin boss of 0.2 on poses spanning 2.2 units, 116 of the 1,176 have two pins closer than one boss diameter — and eight of them are among the 176. At 0.4 it is 313 and 27. The shortest link the construction returns is 0.103 units long.

Tested in A defect that is not kinematic · the synthesis sequence.

An exactly constrained part has no freedoms left.

What decides it: Read the same six contacts as inequalities rather than equations and every coupling on this site has an escape. A Maxwell coupling's is a pure translation along its axis, a Kelvin clamp's is a twist dominated by the same lift, and a Kelvin clamp with a seventh pad added has exactly the same escape as one without it. The rank is six in every case, which is the number the claim is made from.

Tested in Six hold nothing · the mobility sequence.

Reduce the clearance to reduce the lost motion.

What decides it: It works, in proportion, and it runs into a floor: a joint with no clearance when cold has none to spare when the machine warms up. Fixing the clearance's direction instead removes the lost motion entirely — exactly zero, because there is nothing left to take up on reversal — and converts what remains into a repeatable offset whose average calibrates out. Measured on this site's four-bar, 2.80° of play becomes 0.343° of offset, and 0.136° of that is a constant.

Tested in Taking up the play · the clearance sequence.

A measurement of a mechanism's motion tells you about its geometry.

What decides it: It tells you about its proportions. Every quantity in this field is a length, a reciprocal length or an area, and the field's central object — a clearance between two parts — is a length compared against a half-width chosen from a catalogue. An angle-only measurement recovers none of it, and the site's other fields it recovers almost entirely.

Tested in A body is all size · the body sequence.

A prismatic pair takes away two freedoms and leaves one, and that is what it costs.

What decides it: That is what the pair costs. What the parts that realise it cost is a guide longer than the stroke by the whole length of the block: a block of 0.6 on a crank of 1 needs a guide of 2.6 rather than 2.0, measured by shortening the guide until the block's corners leave the rails, and agreeing with the arithmetic to 6 × 10⁻⁵. Nearly a quarter of that guide is never stroke.

Tested in The block in the guide has a length · the pair sequence.

Scaling a compiled linkage leaves the equation it computes unchanged, because the linkage's behaviour is unchanged.

What decides it: The linkage traces the same curve at a different size, and the polynomial it evaluates is multiplied by the scale factor. Those are compatible — a polynomial and its multiple vanish on the same set — and they are not the same statement: the curve is preserved and the function is not.

Tested in A compiled machine and its own scale · the compute sequence.

Three-position synthesis works for any point of the moving body: its three images have a circumcentre, and that is the fixed pivot.

What decides it: It works for every point except those on one circle, and near that circle it works only in a sense no machine can use. A point a thousandth of a unit off the circle is exact and asks for a crank 855 units long on a body two units across, because the crank grows as one over the miss — measured over four decades with the product settling to 0.855. The points on the circle want a slide, and a slider-crank built from one reaches all three poses to 2 × 10⁻¹⁵.

Tested in Where a pin becomes a slide · the synthesis sequence.

Every dimension of a mechanism carries a tolerance, so every quantity it produces does.

What decides it: A gear ratio does not, and not because the tolerance is small. There is no length in it: the ratio is a quotient of two integers and the module, the centre distance and every diameter are absent from the relation entirely. A variator's ratio, computed by the same site from the same kind of mechanism, has a three-term stack that reaches 2.99% worst case and 2.04% in quadrature at the tall end of its travel.

Tested in The ratio that has a tolerance · the tolerance sequence.

Specifications should be more accurate.

What decides it: Accuracy is not the failing. Six of the fourteen numbers measured here name a quantity that does not exist at any position — a roll centre height, a percentage of a cotangent condition, a rise-per-ram ratio — and no amount of accuracy helps a quantity that has no value. What would help is a different kind of statement: a range, an order, or a stated position. Three of the fourteen already do that and are the least troublesome rows in the table.

Tested in Which numbers survived · the misconception sequence.

Identifying a mechanism's dimensions identifies the mechanism.

What decides it: It identifies the parts list. A four-bar with these four lengths can be put together two ways, both satisfy the relation the identification solved, and one of them reaches none of the readings it was identified from — reading −111.6° at a pose where 98.8° was measured. Which assembly a machine is in is not in the equations and is not recoverable from them.

Tested in One set of lengths, two machines · the parameter sequence.

Where accelerations matter, a cam indexer beats a Geneva, because a cam can be cut to a better motion law.

What decides it: It depends on the slot count and on which law. At the Geneva's own index angle and step, a cycloidal cam has the lower peak acceleration only below 5.19 slots, a modified sine below 6.23 and simple harmonic motion below 8.06; a six-slot Geneva peaks at 5.653 in step-over-angle-squared units against the cycloidal cam's 6.283. Only constant acceleration beats it at every count. What the Geneva cannot match is the entry: its acceleration is tan(π/n) the moment the pin enters — 0.577 of the input speed squared at six slots — where a cycloidal or modified-sine cam starts from nought.

Tested in When the index law becomes a choice · the intermittent sequence.

A paired Gough platform's platform anchors should sit midway between its base anchors, rotated 60°, so that the legs triangulate.

What decides it: That rotation puts the orientation at which the platform is singular everywhere 30° from home, and at the centred position the platform's holding halves after ±17° of yaw. Rotated 0°, with each platform pair inside its own base pair, the dead yaws move to ±90°, the holding at home rises from 0.0363 to 0.0411, and the platform turns ±75.5° before it halves. At a rotation of 90° the home orientation is itself dead. The dead yaw is 90° − ρ at every rotation measured, to 7 × 10⁻⁶ of a degree.

Tested in The dead yaw is a design choice · the parallel sequence.

A power-split transmission should place its geared neutral at the edge of the variator's travel, since the travel near the pole cannot be used.

What decides it: That is the best placement only if the variator's tolerance is the same everywhere. With one tolerance of 1.6% and a trim at 10% output uncertainty, the forward span peaks at 5.55 where the travel's top meets the trim and falls to under 0.7 of that as the pole moves in. With the variator's own tolerance, which rises from 0.98% to 2.24% along its travel, the forward span is 4.81 at best and stays within a tenth of it from a fixed ratio of 0.6 to 1.2 — while the reverse span grows from nothing to 3.9.

Tested in Sliding the travel across the pole · the transmission sequence.

An open chain of n links is one thing: a sequence of joints from base to tool.

What decides it: An open chain is a tree, and there are 1, 2, 3, 6, 11, 23, 47 and 106 distinct trees on three to ten links. Exactly one tree at each size is a path. The other 105 at ten links branch, and every one of them is a mechanism with nine degrees of freedom and no loop — the same count, a different machine, and the serial field has drawn only the path.

Tested in An arm is a tree · the serial sequence.

A grooved cam's roller changes walls at every sign change of the force the groove must supply, so the count of sign changes is the count of impacts.

What decides it: A sign change starts a flight across the clearance, and the flight need not arrive. On the quick-rise cycloidal cam carrying 0.5 kg against 20 N with 0.05 mm of clearance, the force first changes sign at 267.6 rpm; at 270 rpm the roller lifts 4.4 µm and falls back onto the wall it left, and it first reaches the far wall at 275.8 rpm. Where it does arrive, its landing speed grows as the 0.664 power of the clearance under a cycloidal law and the 0.5000 power under constant acceleration — and a flight that ends in a dwell lands at √(2cF/m) = 63.2 mm/s at every speed.

Tested in The time a crossover takes · the cam sequence.

The inflection circle belongs to the mechanism: it is where the points of the moving body travel in straight lines.

What decides it: It belongs to one direction of the motion. Held the other way — the coupler still and the frame moving — the frame's points that travel straight lie on a different circle of the same diameter, the first reflected through the pole. At a crank angle of 225° on the four-bar with ground 4, crank 1, coupler 3.5 and rocker 3, frame points taken round that reflected circle have path curvature nought to 3 × 10⁻⁹ in coupler-length units, while round the ordinary circle it is at least 0.8 away from the pole.

Tested in The frame seen from the coupler · the curvature sequence.

A pulled Bowden inner is short of its sheath by the clearance times the total turning, so a route's lost motion follows from its bend angles alone.

What decides it: On two 8° bends of radius 40 the inner holds 95% of each bend at a clearance of 0.5 and loses 97% of what the law predicts; at a clearance of 10 it holds none of either and loses 43% of it. What it loses flattens onto 1.204, the route's length minus the straight line between its ferrules, which has no clearance in it at all — and the law passes that ceiling at a clearance of about 4.3 and goes on rising.

Tested in Which walls a strand is held by · the strand sequence.

How many pins a Geneva driver can carry is decided by the slots: the wheel indexes p times a turn and the indexes must not overlap, so p is bounded by 2n/(n−2).

What decides it: The locking disc's cut-away, measured by sweeping the wheel into the driver's frame, is 119° on a three-slot wheel against a 60° index and 236° on a six-slot wheel against 120°. Two cut-aways of 236° do not fit in a turn. The disc allows 2, 1, 1, 1, 1, 1, 1 pins at 3, 4, 5, 6, 8, 10, 12 slots where the slot arithmetic allows 5, 3, 3, 2, 2, 2, 2 — it is the binding condition at every count.

Tested in The disc decides the pin count · the intermittent sequence.

A power split's forward span is set by where its travel sits against the pole, so the best placement puts the top of the travel at the tolerance trim and the span is 1 + τ(1−r)/p.

What decides it: The variator's branch carries v/(K−v) of the engine's power and reaches the engine's own at v = K/2, which for K = 1.4 is 0.700. The tolerance trim is at 1.207. A variator rated for the engine's power is therefore stopped at 0.700 and reaches a span of 1.70 against the 5.375 the kinematic arithmetic promises; the tolerance only becomes the binding condition at a rating of τ/p = 6.25.

Tested in The power that goes round twice · the transmission sequence.

A mechanism read by an angle sensor has exactly one unrecoverable direction, which is its overall size.

What decides it: A Watt six-bar has two. The second scales its output loop about O₄ and leaves the four-bar alone, and it is invisible at 2.1 × 10⁻¹⁰ — while the same five parameters scaled about O₂ instead are plainly visible at 1.5 × 10⁻¹, a difference of nine orders. The centre of the scaling is what decides it.

Tested in Nine parameters, two of them invisible · the parameter sequence.

For a dwell mechanism, the best design is the one with the highest order of contact at the dwell point.

What decides it: Only in the limit of a band of nought. At every finite band the longest dwell comes from detuning the sixth-order angle by an amount that grows as the cube root of the band — 2.26° at a tenth of a per cent of the swing, 0.49° at ten parts per million — and it dwells (27/4)^(1/6) = 1.3747 times as long as the sixth-order machine, measured at 1.3755 and 1.3728. The detuned output dips by the whole band and returns to its vertex value at the ends of the dwell; the sixth-order output never uses the lower half of its band at all.

Tested in The dip that buys the dwell · the coupler sequence.

The best drag link to put ahead of a shaper's crank-rocker is the one that gives the largest time ratio at an acceptable transmission angle.

What decides it: That design reaches 2.71 on a crank-rocker with a 60° swing and a ratio of 1.2 of its own, and drives the working stroke so unevenly that the rocker's fastest speed over the central 80% of its swing is 2.77 times its slowest. A different drag link at a different phase reaches 2.03 with a ratio of 1.21 — more even than the same crank-rocker driven directly at constant speed, which gives 1.91 — with every stage still at 40°.

Tested in A quick return that cuts evenly · the fourbar sequence.

An open chain has no closure equation, so every set of joint angles is a configuration it has.

What decides it: Every set of joint angles solves, and 21.4% of them put the arm's first and third links inside each other. The forbidden set is a two-dimensional region in the relative angles and does not depend on the base joint at all, which is checked at forty pairs of relative angles with zero disagreements.

Tested in The arm that hits itself · the serial sequence.

Calibrating a mechanism requires an external measuring instrument and a datum.

What decides it: Two encoders on a four-bar recover its shape to 8.6 × 10⁻¹⁵ with no fixture, no datum and nothing outside the machine. What they cannot recover is its size, ever — the same three numbers describe this machine and one a quarter of it, and an encoder reads an angle.

Tested in A machine that measures itself · the parameter sequence.

The gap between links was a small omission: a body on each link and a collision test, and the site's inventory gains one line.

What decides it: It gains a field. A body and a test are half a day; what they open is a clearance that is a function with corners, a sweep that needs a Lipschitz certificate, a layer assignment that is a colouring plus an ordering, a free space in pieces, a fourth kind of synthesis defect, and a four-bar that turns all the way round exactly when it cannot be built with bearings at both pivots.

Tested in Where the boundary moved · the practice sequence.

Tolerancing features rather than lengths changes which dimension should be held tightest.

What decides it: Only when the parts are made differently. Under one process for the whole machine the two allocations are identical to the last digit, because the transmission factor is then common and divides out. With a frame jig-bored at ninety per cent shared beside a crank drilled twice at ten, the tightest tolerance moves from the rocker to the crank and the frame's loosens by 1.8.

Tested in Which feature to hold tight · the parameter sequence.

A tolerance turns every computed quantity into an interval, and the interval is read the same way the number was.

What decides it: Not when the quantity has a sign. A four-bar clearing a post by 0.104 as drawn clears by 0.083 at the worst corner of a ±0.02 band on its four lengths — a band a fifth of the nominal clearance wide, from tolerances an eighth of a link width. Read as a magnitude that is a modest spread; read as a sign it is a fifth of the way to a machine that does not assemble.

Tested in A clearance inside a tolerance box · the tolerance sequence.

A small error in a machine's geometry produces a small error in its parameters.

What decides it: Half a degree of twist and a twentieth of a degree of tilt on a pair of nominally parallel axes one unit apart gives a DH offset of −11.35 units. At a hundredth of a degree of twist it is −1,102. The geometry has not moved by as much as a degree and the parameter has moved by three orders of magnitude.

Tested in A number that runs away · the parameter sequence.

Synthesis and calibration are different problems that happen to use similar mathematics.

What decides it: For a four-bar read by angle they are the same 3 × 3 linear system with the pairs prescribed instead of measured. Three rows is a solve and thirty is a least squares, and every question this field asks — which parameters are recoverable, how well, from which poses — exists only because there are more rows than columns.

Tested in A calibration is a synthesis with more equations · the parameter sequence.

A calibration's accuracy is one number, so two measurement plans can be ranked by comparing it.

What decides it: A coordinate machine on the tracing point recovers the coupler-point coordinates an order better than the four lengths; a protractor recovers the three invariants at 1.4, 1.6 and 1.9. The single amplification is the worst of the list, and two plans whose worst entries are equal can have entirely different lists.

Tested in Two instruments disagree about the worst · the parameter sequence.

A calibration with a small residual, a full-rank Jacobian and a well-conditioned answer has determined the machine.

What decides it: It has determined the machine among the models it was given, near the answer it found. A missing parameter whose column is nearly parallel to one present is absorbed with the residual at the noise, the rank full and the conditioning fine, and nothing in the data distinguishes that case from a correct model.

Tested in What this field cannot measure · the parameter sequence.

A mechanism's accuracy is a number: this linkage is accurate to a tenth of a degree.

What decides it: Building all sixteen extreme combinations of four toleranced lengths and solving each, at every crank angle, gives a band that is 0.73° wide at 30° of crank rotation and 0.36° wide at 120°. It is a curve with a maximum, not a value, and quoting one figure for it quotes whichever position the mechanism happened to be measured at.

Tested in A length is a range · the tolerance sequence.

A Geneva wheel with six slots is driven for a sixth of the driver's turn.

What decides it: The pin is in a slot while the driver sweeps 180° − 360°/n, so the driven fraction is (n − 2)/2n: a third of the turn at six slots, not a sixth. The two agree only at four slots, which is the case a handbook is most likely to have drawn.

Tested in The mechanism that waits · the intermittent sequence.

A specification is either right or wrong.

What decides it: Of fourteen quoted numbers measured here, six name a quantity the mechanism has no instance of — a roll centre height, a percentage of Ackermann, a lift ratio — and being wrong is not available to them. Three are exact at every position. The other five are right in a stated and different sense each: a mean over a cycle, a bound on an error, or a value at one position. Right and wrong is two verdicts for a question that needs five.

Tested in The number on the box · the misconception sequence.

A four-bar linkage has a velocity ratio, the way a gear pair does.

What decides it: Solving the velocity problem at every position of one turn gives an output-to-input ratio running from −0.29 to 0.51 and changing sign on the way. A gear pair's ratio is a number; this is a curve through zero, and the single figure usually quoted is its average, which the mechanism holds at two instants of the cycle.

Tested in The ratio that is not a number · the misconception sequence.

A gearbox has a set of ratios.

What decides it: The gearset has a plane of permitted motions and no ratio at all. A ratio appears only when something is engaged, and each engagement is one more linear condition that cuts the plane down to a line. The Ravigneaux gearset drawn here offers seven distinct ratios; the transmission built on it is sold with five, and the two left over are not defects — they are directions in the plane nobody bought the parts to reach.

Tested in Two inputs and one output · the transmission sequence.

Grashof's condition is a rule of thumb for whether a crank will turn all the way round.

What decides it: It is a theorem, and the site checks it the only way that means anything: the condition predicts which link rotates fully from the four lengths alone, and a 360° sweep of the solver measures which one does. They agree on every linkage tested, including the ones at the boundary, where the prediction is that the mechanism passes through a change point rather than that it nearly fails.

Tested in Grashof, predicted and then swept · the fourbar sequence.

A gear needs at least seventeen teeth or the cutter will undercut it.

What decides it: The threshold computed from the pressure angle is 17.097 at 20°, so seventeen teeth undercut slightly and the rule is a rounding of a real number in the unsafe direction. The threshold also moves with the pressure angle — 11.5 at 25° — and profile shift removes it altogether, which is measured here on the drawn flank rather than quoted.

Tested in Undercutting, and the seventeen-tooth rule · the tooth sequence.

An arm's reach is the number in its catalogue: 1.6 metres means it works within 1.6 metres.

What decides it: The reach is the radius of a sphere the arm touches at exactly one posture, and this arm gets to 99.08% of its own triangle-inequality bound. Inside that sphere is a hole it cannot enter at all, and the region where the tool can be held at *any* orientation is 26.1% of the reachable area for one set of link lengths and 0% for the same three lengths in a different order.

Tested in Where the hand can go · the serial sequence.

An epicyclic gear train has a ratio.

What decides it: It has three shafts and one equation relating them, so fixing a different member gives a different ratio from the same gears. Every ratio on this site is computed twice, by Willis's equation and by the tabular method, and quoting one of the three as the train's ratio is choosing a boundary condition without saying so.

Tested in Epicyclic ratios, two ways · the tooth sequence.

A three-speed automatic has three gears because it has three sets of gears.

What decides it: It has one gearset with two degrees of freedom and three ways of taking one of them away. The Simpson gearset here has one sun, two rings and two carriers, and eighteen brake-and-clutch combinations across two possible inputs; they give five distinct ratios — 2.4595, 1.4595, 1.0000, −2.1765 and an unused 3.1765 — and the transmission built on it is sold with the first four.

Tested in Holding a member chooses the ratio · the transmission sequence.

A mechanism has a number of assembly modes.

What decides it: Its family has a complex solution count that does not change, and each set of link lengths has a real count that does. Sweeping 676 sets of leg lengths for one platform, the complex count is constant and the real count moves between 0, 2 and 4. Quoting one number for 'the' number of assemblies picks a member of the family and calls it the family.

Tested in The count that does not move · the algebra sequence.

A four-speed automatic's ratios are four numbers a designer chose.

What decides it: They are two. The Ravigneaux's seven exact ratios are 1+K₂, 1+K₂/K₁, 1, 1−1/K₁, −(K₁−1), 1+K₁/K₂ and 1+1/K₂, where K₁ and K₂ are ring-over-sun for the two suns; the short planet's tooth count appears in none of them, and doubling every count in the gearset returns the identical seven fractions. Choosing first gear and reverse fixes second, third and the overdrive with nothing left to spend.

Tested in Four speeds from two numbers · the transmission sequence.

An intermittent gear's acceleration at engagement is very large.

What decides it: It is not a number. A second central difference of the output angle at engagement grows by a factor of 2.000000 with every halving of the step and goes on growing; quoting a value would be quoting the step it was taken at. A six-slot Geneva measured the same way converges, because it enters along its slot.

Tested in The gear with its teeth cut away · the intermittent sequence.

A Gough–Stewart platform has forty assembly configurations for a given set of leg lengths.

What decides it: Forty is the count for a general platform, and homotopy continuation on this site's own platform returns twenty-eight finite solutions from 1,458 tracked paths. Its six base anchors are three symmetric pairs — a special architecture — and the twelve missing poses are at infinity. Perturbing the anchors by 0.05 brings them back at coordinates of order 10³.

Tested in Twenty-eight, not forty · the algebra sequence.

This suspension has a roll centre 73 mm above the ground.

What decides it: It has that roll centre at one position. Swept through ±80 mm of wheel travel the construction returns every value from 52 mm to 106 mm on the wishbone here, and from −10 mm to 121 mm on the strut — so the quantity being quoted takes 54 and 131 different values respectively over the travel a road wheel uses in ordinary driving.

Tested in A roll centre is not a point · the fourbar sequence.

This mechanism has 3° of lost motion.

What decides it: Computed at every position of a turn, the lost motion of a four-bar with 0.01 of clearance at each pin is 1.4° at its best, about 2.8° through most of the cycle, and unbounded at the two positions where the output velocity passes through zero. A plot of it reads 38° at 48 samples and 402° at 192 — the peak is a pole, so the largest value in any table is a statement about the table.

Tested in How far the crank turns first · the clearance sequence.

This car has 100% Ackermann steering.

What decides it: The correct steering condition is cot δ_outer − cot δ_inner = w/l, which is not a rational function of a crank angle; a four-bar's output is. The trapezoid with the textbook arm angle meets the condition at straight ahead and at no other angle in the lock range, and is 2.06° out at 35° of inner lock. Choosing the arm angle to minimise the worst error gets one more exact angle and 0.34°, and there is no fourth.

Tested in The steering that is never right · the fourbar sequence.

Moving an arm's joints smoothly from one posture to another moves the tool smoothly from one place to the other.

What decides it: It moves it smoothly and not straight. Between two ordinary postures of this arm the tool bows 402 mm off the straight line joining them, on a move of 1,223 mm — 32.9% of the distance travelled, which is enough to hit whatever the straight line would have missed.

Tested in A straight line at constant speed · the serial sequence.

A watch's lift angle of 44° is a convention chosen by the industry.

What decides it: It is a consequence. The pallets fix how far the lever must swing — their lock plus their lift — and the sine rule then fixes the balance's swing from the ratio of the roller radius to the fork's: sin A / sin B = r / f, to twelve figures. At the proportion watches are built to, r/f = 0.15, the lift angle comes out at 44.41° from geometry that was never told what the answer should be.

Tested in Detached, and safe while detached · the intermittent sequence.

The shortest path between two poses is the straight line between them.

What decides it: Between the same two poses of this arm, the tool travels 1.223 m if the position is interpolated straight, 1.443 m along the single screw motion Chasles's theorem guarantees, and 1.550 m if the joints are interpolated. All three are somebody's definition of shortest, and there is no fact of the matter, because a pose is a rotation and a translation and adding a radian to a metre requires a length nobody wrote down.

Tested in The distance between two poses · the serial sequence.

A 1.6 rocker gives 1.6 times the cam's lift at the valve.

What decides it: It gives 1.605 at the position the ratio was measured at — the valve shut — and less everywhere else, because both moment arms swing as the rocker turns. Solved through a full cam event, the instantaneous ratio falls to 1.588 at peak lift and the valve reaches 12.78 mm against the 12.84 the multiplication promises. The 0.51% shortfall is a property of the linkage's setup, not of the cam.

Tested in The cam is not the valve · the cam sequence.