Machine Kinematics
solved, then drawn
Concept

Deployable — where it appears

A mechanism designed to change size or shape between a stowed configuration and a working one, built from many copies of one unit. Its distinguishing feature is arithmetic rather than function: the constraints of a deployable repeat one another, so the count declares it a structure, and the repetition has to be arranged by a symmetry or by a single parameter chosen once for every unit.

Named by 7 essays across 3 fields — each of them below, with the objects they name alongside it.

Six assemblies, one routine, three disagreements and one accident. Every row is the same three steps: write down the constraint Jacobian, take its rank, and subtract it from the number of unknowns. The representations differ — bars between points, bodies joined by pins, panels joined by creases, one cell of a pattern that repeats for ever — and the routine does not. The counted column is the arithmetic on the numbers of bodies and joints; the measured column is the nullity of the matrix. They agree on the lazy tong and on the kagome cell and disagree on the other four, most sharply on the deployable ring, which the count declares immobile and which is sold as a mechanism that opens. The right-hand column is the reason: constraints that repeat what another constraint has already said, which the count has no way of seeing and the rank cannot help seeing. The fourth row is worth reading twice: the count says nothing can move and nothing can, so the two agree — and they agree for the wrong reason, because that pattern's flat state shows four freedoms and not one of them is a motion.

Many loops, one freedom

A scissor lift, a folded sheet and a deployable ring are one small unit repeated thirty times, and three things change at once: the count of bodies becomes a parameter, mobility becomes the rank of a matrix, and a unit that moves can be rigid the moment it is joined to another of itself.

networks · Network What the count is right about. Freedoms minus dependencies, against what the count predicts, on seven assemblies from three different representations. The difference is exact every time and it is exact for a reason that has nothing to do with mechanisms: the count is unknowns minus constraints, the rank is a number no larger than either, and the two nullities are what each of them has left over. So a count is not wrong in the way a mismeasurement is wrong. It is a statement about a difference being read as a statement about one of the terms — and on four of these seven rows both terms are large and the difference is nearly meaningless.

A constraint that has been said already

Every constraint matrix leaves two null spaces, and a mechanism only lives in one of them. The other is the set of combinations of constraints that come to nothing — and its dimension is exactly the amount by which the count is wrong, on a deployable ring, a Miura sheet and a framework with twelve bars and six joints.

networks · Network The ring a count says cannot move. 8 pairs of angulated elements, each pair two mirror-image bent bars pinned at their kinks, with each pair joined to the next at two pins on a common radius. 16 bodies and 24 pins give Grübler's 3(n − 1) − 2j = 0: no freedom at all, a structure. The rank of the constraint Jacobian is 44 of 48, which leaves 4 — the three rigid motions of the whole ring and one deployment — and 4 constraints that repeat what the others have already said. The kink angle is not a style: it is 135.0000°, a half turn less the 45.0000° the ring subtends per pair, and any other value gives a ring that will not deploy. Inner radius 0.6876, outer 2.2688. positioned by solving, not by drawing.

The ring that closes at every size

Two bent bars pinned at their kinks hold the angle between their connection lines at 135.000000° whatever you do to them, and two straight ones hold it at nothing. That is the whole difference between a scissor chain that grows in a line and a ring of eight that opens and shuts — and the count says the ring cannot move.

networks · Network The network where counting works. 5 scissor units, each two bars pinned at their middles, with consecutive units sharing their end pins. Sixteen bodies and twenty-two pins at eight units, and Grübler's 3(n − 1) − 2j gives 4 — three rigid motions and one internal freedom — against a measured nullity of 4 and no redundant constraints at all. That holds at one unit and at sixteen. The tong is here to make the field's point in the direction nobody expects it: a network is not a place where counting fails, it is a place where counting stops being checkable by hand, and one of the two assemblies that spans this plate is counted perfectly. Span 8.1388 mm at this opening, which is 5 times one unit's 1.6278. positioned by solving, not by drawing.

One input at one end

A lazy tong's reach is 2nL cos θ — exactly, to a part in a million million, over eight sizes and sixty openings — so it multiplies its input by the number of units. It multiplies everything else by the same number, including the part of the drawing nobody wanted multiplied: a unit cut a hundredth of a radian out puts a tong of thirty-two units 0.374 out at the far end.

networks · Network Six assemblies, one routine, three disagreements and one accident. Every row is the same three steps: write down the constraint Jacobian, take its rank, and subtract it from the number of unknowns. The representations differ — bars between points, bodies joined by pins, panels joined by creases, one cell of a pattern that repeats for ever — and the routine does not. The counted column is the arithmetic on the numbers of bodies and joints; the measured column is the nullity of the matrix. They agree on the lazy tong and on the kagome cell and disagree on the other four, most sharply on the deployable ring, which the count declares immobile and which is sold as a mechanism that opens. The right-hand column is the reason: constraints that repeat what another constraint has already said, which the count has no way of seeing and the rank cannot help seeing. The fourth row is worth reading twice: the count says nothing can move and nothing can, so the two agree — and they agree for the wrong reason, because that pattern's flat state shows four freedoms and not one of them is a motion.

Six things a network is not

A count that is right about a difference and read as an answer, a nullity taken for a mobility, a flat state that cannot tell a mechanism from a structure, a scissor ring that closes nowhere, a vertex that folds while its sheet does not, and a null space computed with an instrument whose floor is above the answer. Six claims, each with the number that kills it.

wrong · Misconception A 4-stage scissor, open and as flat as it goes. On paper a scissor stack's height is n·L·sin φ and goes to nothing as the bars lie down. The bars are made of something: the two colours are the two planes the stack needs — two, at any number of stages, which is why a scissor folds at all — and the bars that share a plane are the ones two stages apart. They meet at φ = 0.0951, leaving each stage 0.0949 high, which is exactly the two bosses that touch: 2 × 0.0473. The stowed height of the whole stack is 0.380 rather than zero, and it grows with the stage count in the way a stowed lift's does.

A stack that has to fit

A scissor stack's height is n·L·sin φ and goes to nothing as the bars lie down — on paper. The bars are made of something, and what stops the fold is two bosses meeting: every stage keeps 0.0949 whatever the stack does, which is exactly twice the boss radius.

networks · Network The same multiplier, applied to the error. A tong whose units are cut to an angle 0.01 radians away from the drawing. If one unit is out, the span is out by that unit's share and nothing more; if every unit is out the same way — which is what a machine setting or a worn tool produces — the error is multiplied by the unit count, exactly, to 7.6e-14. The third column is what would happen if the errors were independent and equally likely either way: the accumulation goes as the square root of the count instead, and the difference between the two columns at thirty-two units is a factor of 5.66. Which column applies is a question about how the parts were made, not about the mechanism.

The error that is repeated

Thirty-two units cut on one setting of one machine are thirty-two copies of one error, not thirty-two draws from a distribution — so a tong's span is out by thirty-two times a unit's, not by the square root of thirty-two times it. The two estimates differ by a factor of 5.66, and the second one is the comforting one.

practice · Tolerance

Named alongside it

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

NetworkGrübler's criterionMobilityRedundant constraintScissor linkageNull spaceRankConstraint jacobianDesign ruleError accumulationFlat stateTolerance

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