Field

What can move

Before a mechanism does anything it must be able to. Mobility is countable, and the count can be wrong — which is how some of the most useful mechanisms ever built were nearly ruled out.

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

What decides whether it moves

Mobility as the difference between a structure and a mechanism, counted by Grübler's criterion and measured from the rank of the constraint Jacobian, and why the two are worth computing separately.

Mobility, counted and measured. Grübler's criterion counts links and joints and knows nothing about the dimensions; the rank of the constraint Jacobian measures the dimensions and knows nothing about the topology. They agree for four of these five. The parallelogram with a redundant third bar is the exception: the formula declares it a structure with zero degrees of freedom, and it moves. The formula is the one that is wrong, because it cannot see that the third bar's constraint equations are already implied by the other two.

Counting and measuring mobility

The two independent routes to a mechanism's mobility, what each one can and cannot see, why the topology has to be declared rather than inferred, and how a rank threshold is chosen.

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

The mechanism Grübler says cannot move

The over-constrained parallel linkage, why Grübler's criterion fails on it, what a redundant constraint is, and why the redundancy is deliberate rather than a mistake.

What each kind of joint takes away. Grübler's formula is M = 3(n − 1) − 2j₁ − j₂, and the 2 and the 1 in it are not conventions. A lower pair — a pin or a slide — holds two bodies together over a surface and leaves one relative freedom, so it costs 2. A higher pair — a cam against a follower, a wheel on a rail — touches at a point, the contact travels along both surfaces, and it costs 1. Five chains, each built and each measured from the rank of its constraint Jacobian, which has never heard of the formula. The last row is the one worth having: count that cam contact as a pin, as is very easily done, and the formula returns 0 where the mechanism has 1. The Jacobian does not move.

What each joint takes away

The lower and higher pairs, why a pin costs two freedoms and a rolling contact costs one, and five chains whose counted mobility is checked against the rank of their constraint Jacobians.

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

The formula is repaired by the thing it replaced

The modified mobility criterion tested on nine spatial loops, with the redundant constraints measured from the legs' reciprocal systems rather than assumed, and the sense in which the corrected formula is a measurement in a formula's clothes.

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

The count that counts the wrong thing

What the mobility count means when a mechanism's constraints are on velocities: the two numbers, when they agree, and the arithmetic that has to change on a mechanism with a wheel in it.

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

The count was right and the name was wrong

A second failure of the mobility count, at generic dimensions: what a rigid subchain does to a description, why neither of the field's two routes can see it, and the third instrument that can.

What it takes to build each of the twelve. The same twelve, read as a bill of materials. Six of them are one joint, because a lower pair permits the whole symmetry group of its surface and those six groups are exactly the symmetry groups surfaces have. The other five with a dimension take a chain: two slides for planar translation, three for Cartesian motion, a thread and two slides for the screw-in-a-plane group, and three parallel pins with a slide along them for Schoenflies motion — which is a SCARA arm, and is why a pick-and-place machine has four joints and not one. The group each chain produces is measured from four hundred sampled poses rather than declared, and every row agrees.

The freedom that is a set

The mobility count re-read as arithmetic on the dimensions of displacement groups, with what the arithmetic keeps and what it discards.

One freedom, whatever the count says. Every one of these machines has exactly one degree of freedom, measured as the number of unknowns minus the rank of the constraint Jacobian. Unbraced, the count agrees. Braced, the count says the largest machine has -153 — that it cannot move, by a wide margin — and the rank says it still turns exactly as it did. The gap is one equation per brace and every one of those equations is implied by the others. This is the constraint field's oldest example, at a scale nobody would try by hand: a count that is wrong by a hundred and fifty-three about a mechanism that works.

One freedom and four hundred links

The constraint field's oldest comparison at a scale nobody would attempt by hand: a count that is wrong by 154 about a mechanism whose motion is not in doubt.

A geared five-bar at 3 : 2, its coupler pin traced over 2 input turns. Two cranks of length 1 on pivots 3 apart, meshed through gears whose pitch circles, dashed, have radii 1.800 and 1.200, so the second crank turns 1.5000 times for every turn of the first and the other way. Coupler bars of 3 and 2.5 join them at the pin C, and the thin curve is where C goes over 2 turns of the input. The two crank pins are always between 1 and 5 apart and the bars can span 0.5 to 5.5, so the chain never meets a dead position. After 2 turns the machine is exactly where it began and the curve is closed.

One freedom, and a motion that never repeats

A geared five-bar whose mobility is one at every gear ratio and whose motion closes only when the ratio is rational: the count and the rank against the torus of crank angles and the continued fraction.

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

A parallelogram a micron wrong

The parallelogram four-bar as the corner where four regions of length space meet: every single-length error sorted into a gap or a stall at each flat position, both square-root laws measured against the exact closure from 10⁻¹⁰ to 10⁻¹, the output's reversal and its universal shape, and why a redundant third bar makes the cost of an error linear.

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

Nine bars that ought to be rigid

Dixon's flexible framework of nine bars measured three ways: the count, the rank of its constraint Jacobian, and a forced move. The motion followed on the bar lengths alone round a closed loop through four crossings, every bar removable and no pair, and the tilted placements that flex to first order and miss a forced move by its square.

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

A length error is undone by its own size

Radial clearance at a parallelogram's pins, measured as a budget on one residual: the play that rejoins two circuits separated by a square root is the length error itself, for gaps and for stalls alike, over six decades; the gap left below the threshold closes as another square root; a search over each pin's disc finds all four worth the same; and the threshold equals the misfit a third crank imposes.

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

The right angle as a tolerance

A clearance tolerance for Dixon's flexible framework, measured by bisection on whether a placement exists with all nine bars inside it: the square-in-travel, linear-in-tilt law over four decades, the least allowance putting every bar at it while a least-squares fit asks for half as much again, two searches from different starting points agreeing exactly, and the cliff at the crossing point.

Every change point lies flat

The pin clearance that restores a change point, measured on four-bars along the Grashof boundary g + a = b + c rather than at the parallelogram corner; why the flat configuration of every change point is collinear and the threshold is therefore the length error exactly, at every bearing alike; the closed forms for the separation and the stall derived from a law of cosines and checked to 10⁻⁷; and the parallelogram as the worst linkage on its wall for lost choice.

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

The clearance that makes the framework generic

Dixon's flexible K₃,₃ framework with every joint displaced at random within a clearance disc of radius ρ: rank nine and a smallest singular value σ₉ proportional to the six-point conic determinant, the allowance needed to push it collapsing onto one curve per framework in units of ρ, its first-order rate 0.41 σ₉, a worst case of 0.14ρ against the 2ρ a clearance of ρ provides, and the tilt of the lines deciding the travel.

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

The price of the other family

Dixon's framework as built, pushed past its first crossing: the far family's placement at the same push, the allowance to reach it measured by bisection for ten frameworks at three sizes of error, the price proportional to ρ to 4.2%, blind to the signs of the errors to 1.3%, and the play at which a framework can only come back the way it came.

The price is one self-stress

Dixon's framework as built, pushed to 0.5 on either family: the framework drawn by its self-stress on each family; the measured price against |ω·e| / Σ|ωᵢ| for ten frameworks; the two families' stresses bar by bar; four thousand frameworks priced on both families; the price along the passage from the stress; and the ten frameworks' unadjusted bar errors, prices and preferences.

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

A push turns back at the crossing

Dixon's framework at its first crossing: the exact placements a distance s before and beyond it, differing only in W₂; the push and W₂'s side against s at both kinds of crossing, on log axes; ten frameworks as built pushed to their turning point and pulled back; the same driven by W₂'s side across the black line; the first-order price along the motion with the side drive's least play; and a ledger of prices and outcomes.

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

A drive that goes round is an angle

Dixon's framework driven round its whole loop: the joints in Dixon's own frame with the trammel bar B₂W₂ and the motor's angle; the push, the side and two levers round the loop with their folds; all eighteen joint angles' turns and rates; the motor's rate round the loop; the play ten frameworks as built need to hold the motor's angle at every point; and three of them driven round three times.

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

Put the motor where the rocking bar is longest

Dixon frameworks whose two nearest joints start at different distances from the crossing point: the framework with W₂ moved out towards W₁; B₂ and W₂ round the loop for three starting distances, against the ellipse that was expected; the four turning angles' speeds round one loop; the speed ratios against the starting distance with the white line scaled and with W₂ alone moved, closed form and measured; and a ledger.

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