Machine Kinematics
solved, then drawn
Concept

Exact mechanism — where it appears

A mechanism whose output satisfies an algebraic identity rather than approximating a demand. Exactness is a property of the relation the lengths satisfy, so it does not improve gradually with the number of links and does not survive a tolerance unchanged.

Named by 6 essays across 3 fields — each of them below, with the objects they name alongside it.

What the machine draws, against where the polynomial vanishes. Two objects, found two ways. The thin line is the set where x^4 + 2x^2y^2 + y^4 − 1.2x^2 + 1.2y^2 is zero, walked over a grid with no mechanism involved. The marks are where the compiled machine's tracing point went, one per converged solve, over the 147 positions of its working arc. The machine's constraint set never mentions the polynomial, so evaluating it at each traced point is an independent check: the worst value over the whole arc is 3.7e-13. The arc is 1.30 radians of the driving angle and not the whole turn, and past that arc it draws something else.

A demand that is an equation

Every field on this site is handed its demand geometrically — three positions, a sampled path, a ratio at each angle — and hands back a mechanism that is right at those places and approximately right between them. This one is handed a polynomial, and the mechanism that comes back satisfies it everywhere it moves.

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

Four bars that add two angles

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

computing · Compute What it costs to multiply an angle. Multiplying an angle by n is done by doubling and adding — 2k is one reflector applied to k, and 2k+1 is one addition of k's result and the original — so the cost follows the binary expansion of n and not n itself. Eight costs three doublings; seven costs six gadgets, twice as much, for a smaller number. The other line is what a chain of n−1 additions would cost. The difference matters because the highest multiple a curve of degree d needs is d, so the arithmetic in a compiled machine grows like d log d while the carrying grows like d⁴ — which is why the arithmetic is not what makes these machines large.

Doubling is cheaper than adding

Multiplying an angle by eight costs three gadgets and multiplying it by seven costs six. The cost of an integer multiple follows the binary expansion of the integer and not its size — which is why the arithmetic in a compiled machine grows like d log d while everything else grows like the fourth power.

computing · Compute Exactness is not bought with links. Five straight-line mechanisms, each measured over its own working arc — walked out to its dead centres and back a tenth — and each plotted at its own bar count. Watt's four bars are wrong by 9.0 per cent of the stroke and Chebyshev's by 12.4; Peaucellier's seven are exact. There is nothing in between, and adding bars to an approximation does not walk down the axis: the compiled machine is exact for the same reason Peaucellier is — an exact algebraic relation — and its extra bars buy generality rather than accuracy.

Exact costs more than close

Watt's four bars are wrong by nine per cent of their stroke and Chebyshev's by twelve. Peaucellier's seven are exact to 4 × 10⁻¹⁶, and a compiled machine is exact to 4.8 × 10⁻¹⁴ in five. There is nothing in between — adding bars to an approximation does not walk down the axis, and the four-bar in every beam engine ever built is on the wrong end of it.

computing · Compute Every way the same bars can be put together. The machine compiled from a rectangular hyperbola has 4 parallelograms, and a parallelogram's four bars also close as an antiparallelogram — so there are 16 ways to assemble it. One mark per way. 4 of them put the tracing point on the curve, 4 put it somewhere else, and 8 do not close at all. The ones that are wrong are not broken: they satisfy every bar to 9.6e-15 while the polynomial at their tracing point reads 1.5e-1. This is the gap in Kempe's original argument, and no tolerance on the closure could ever have found it.

Six things a compiled linkage is not

A closure residual read as a verdict, a theorem read as a design, an exact answer read as an accurate one, a degree read as a cost, a construction read as a search, and a neighbourhood read as a turn. Six claims, each of them what a careful person would say, each answered with a number.

wrong · Misconception One bar made 0.0001 too long, one bar at a time. Every bar of the machine compiled from a lemniscate lengthened by 0.0001 in turn, the machine re-solved, and the polynomial read at the tracing point. It is no longer zero anywhere. The worst bar takes it to 9.1e-3 — an amplification of 91 — and the median bar to 2.8e-4. The bars that matter are the reflectors, which are the cheapest part of the machine; the translators, which are most of it, barely move the answer at all. Size and fragility live in different parts.

Exactness a micron destroys

Lengthen one bar of a compiled machine by a ten-thousandth and its tracing point leaves the curve. On the smallest machine the error comes out smaller than it went in; on a fifty-bar one it comes out ninety times larger — and the bars that matter are the reflectors, which are the cheapest part of the machine.

practice · Tolerance

Named alongside it

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

Compiled linkageAssembly branchReflectionRhombusToleranceUniversalityWorking arcAlgebraic curveAngle bisectorAngle multiplierApproximationBinary expansion

All concepts