Machine Kinematics
solved, then drawn
Concept

Sextic — where it appears

A curve of degree six, which is what a general four-bar coupler point traces. The degree is why choosing a linkage to draw a wanted curve is hard, and why the curves were catalogued rather than computed.

Named by 6 essays across 2 fields — each of them below, with the objects they name alongside it.

Five points on one coupler. The same four-bar, with a tracing point rigidly attached to the coupler at five different places. Each curve is a sextic — degree six — and moving the attachment point a little changes it a great deal. That sensitivity is the reason coupler-curve synthesis was done with atlases of printed curves for most of the twentieth century: there is no simple inverse, so the practical method was to look one up.

What a coupler point draws

A point rigidly attached to the coupler of a four-bar traces a curve of degree six. Move the attachment a little and the curve changes a great deal. For most of the twentieth century the practical way to find the linkage that draws a wanted curve was to look it up in a book of printed atlases.

curves · Coupler 3 four-bars, one coupler curve. Three different four-bars, with different ground pivots, different link lengths and different proportions — 0.692 and 0.799 times the size of the first. Every one of them draws this same curve. Roberts's theorem says there are always exactly three, and the construction is one complex multiplication: write the coupler point as λ = (P − A)/(B − A), put the third fixed pivot at O₂ + λ(O₄ − O₂), and the other two linkages fall out with their bars' roles permuted — what is a coupler in one is a crank in another. The curves here were traced separately, each from its own solver runs, and agree to 1.3e-5 against a sampling resolution of 1.4e-5 — which is to say, as closely as the comparison can tell.

Three linkages, one curve

Every coupler curve is drawn by three different four-bars, not one. The other two can be constructed from the first with a single complex multiplication, they have different proportions and different ground pivots, and the roles of their bars are permuted — what is a coupler in one is a crank in another.

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

The curve the other assembly draws

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

curves · Coupler Three double points, and the one that is real is never visited. The coupler curve of a four-bar with ground 4, crank 1, coupler 3.5, rocker 3, coupler point at u = 0.45, v = 0.50, both ovals solved. The dashed circle is where the coupler's orientation can fail to be fixed by the point it carries; it passes through both fixed pivots and through the third pivot of the cognate construction, centre (2.000, 0.010), radius 2.0000. The curve's three finite double points are on it. The one that is real is isolated — a point of the curve no oval passes through, at (3.964, −0.368). The other two are a complex-conjugate pair and have no place in the plane.

A point the machine never reaches

Every coupler curve has three finite double points, and an odd number of them are real, so no coupler curve has none. On the standard crank-rocker the only real one is a point of the curve that neither assembly ever visits, that no contour plot can find, and that sits on the circle through the three pivots of Roberts's cognates.

curves · Coupler A parallelogram's sextic is a circle and a quartic. The parallelogram with ground 4, crank 1.5, coupler 4, rocker 1.5, coupler point at u = 0.45, v = 0.50. Its eliminated sextic divided by the circle of radius 1.5000 about (1.800, 2.000) leaves a remainder of 6.3 × 10⁻¹⁶ of its largest coefficient, and the quotient is a quartic whose leading form is exactly a multiple of (x² + y²)². Of the configurations solved across a full turn on both assemblies, 720 are on the circle — the machine as a parallelogram, coupler parallel to the ground — and 720 on the quartic, the machine crossed. The two factors meet at four finite points: 2 are change points, marked solid, where one configuration belongs to both; the other 2 are places the two drawings merely cross.

A sextic that comes apart

A parallelogram chain's coupler curve is not one curve. Its sextic divides exactly by a circle, leaving a quartic, and each factor is one of the two things the machine can do. The division leaves rounding and nothing else, a coupler one millionth too long leaves a remainder a million times larger than rounding, and the two factors meet at the two places where the machine has to choose.

curves · Coupler Where a fit's singular values fall away, on each motion of a parallelogram. The 28 singular values of the degree-six fit, largest first and relative to the largest, for points traced on the parallelogram's circle motion, on its quartic motion, and on both. On the circle motion the last 15 lie below a drop of 3.7 × 10¹³, from 0.232 to 6.2 × 10⁻¹⁵; on the quartic motion the last 6 lie below a drop of 3.0 × 10¹⁰, from 6.3 × 10⁻⁵ to 2.1 × 10⁻¹⁵; on both motions the last one lies below a drop of 2.5 × 10¹², from 3.1 × 10⁻⁴ to 1.2 × 10⁻¹⁶. A drop of ten decades or more is a null space that is exactly there: fifteen sextics vanish on a circle, six on a quartic, and one on both.

A null space of fifteen is not noise

Points traced on one motion of a parallelogram four-bar leave a degree-six fit with fifteen polynomials that vanish on them, behind a drop of thirteen decades: the circle times every quartic. Half an oval of an ordinary coupler curve leaves two to five, behind drops of two. The count is the same kind of number in both cases, and only the drop beside it says which one is algebra.

curves · Coupler

Named alongside it

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

Coupler curveAssembly branchDouble pointImplicit equationParallelogramChebyshev's linkageCircuitCognate linkageSynthesisAtlasChange pointConstraint

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