Machine Kinematics
solved, then drawn
Concept

Solutions at infinity — where it appears

Roots that a polynomial system acquires from the way it was written rather than from the mechanism, sitting where a coordinate runs away. They inflate the Bézout bound and have to be discounted before a count of solutions means anything about a machine.

Named by 8 essays across one field — each of them below, with the objects they name alongside it.

What became of Bézout's paths. four-bar coupler pin: 2 of 4 paths arrived at a solution and 2 went to infinity; 3-RPR platform: 6 of 16 paths arrived at a solution and 10 went to infinity; Gough, generic: 80 of 1458 paths arrived at a solution and 1378 went to infinity. The surplus is not merely wasted — it is cheap: a path on its way to infinity is abandoned in a handful of steps, while every path that arrives is tracked in full.

The paths that leave

Bézout's number over-counts, and the over-count is enormous — 1,458 tracked paths for 80 solutions. The obvious response is to find a method that tracks only the paths that arrive. That method exists, it was built, and it is four times slower, because the surplus paths are not merely surplus. They are cheap.

algebra · Algebra The twelve that were at infinity, coming back. Every solution of the platform's direct kinematics, plotted by how far from the origin it sits, as the six anchors are jittered. At no jitter the site's own platform has 56 solutions and the largest is at 25.8. At a jitter of 0.2 there are 80, and the extra ones arrive from far out — they were never missing, they were at infinity.

Twenty-eight, not forty

The general six-legged platform has forty poses for a given set of leg lengths, and that number has been quoted beside a picture of a platform that has twenty-eight. Its anchors are arranged symmetrically, which makes it a special architecture, and the missing twelve poses are not missing. They are at infinity, and perturbing the anchors brings them back.

algebra · Algebra How many times each curve passes through the circular points. For each body of five machines, the degree of the curve a point on it draws, from a random complex line, and the number of finite points where that curve meets a line through the circular point I, x + iy = c, and one through J, x − iy = c. The difference is how many times the curve passes through that circular point. Four-bar, crank: degree 2, 1 and 1 finite, circularity 1; four-bar, coupler: degree 6, 3 and 3 finite, circularity 3; four-bar, rocker: degree 2, 1 and 1 finite, circularity 1; Watt six-bar, coupler: degree 6, 3 and 3 finite, circularity 3; Watt six-bar, second coupler: degree 6, 3 and 3 finite, circularity 3; Watt six-bar, output: degree 2, 1 and 1 finite, circularity 1; Stephenson six-bar, coupler: degree 6, 3 and 3 finite, circularity 3; Stephenson six-bar, arm: degree 18, 9 and 9 finite, circularity 9; Stephenson six-bar, output: degree 2, 1 and 1 finite, circularity 1; slider-crank, crank: degree 2, 1 and 1 finite, circularity 1; slider-crank, coupler: degree 4, 3 and 3 finite, circularity 1; elliptic trammel, rod: degree 2, 2 and 2 finite, circularity 0. Every body of the three machines built from pins alone has circularity exactly half its degree; the slider-crank's coupler curve and the trammel's ellipse do not.

Nine times through each circular point

A line through a circular point meets a curve at that point once for every time the curve passes through it, so counting its finite meetings measures how often, with no equation. The Stephenson six-bar's arm curve, degree eighteen, meets such lines in nine finite points: it passes nine times through each circular point. Every curve drawn by a body of a machine built from pins alone does this at exactly half its degree; a slide breaks it.

algebra · Algebra Six coupler points of one slider-crank: three at other distances from the crank pin, three exactly a rod away. A slider-crank with a crank of 1 and a rod of 3 on a slide through the crank's pivot, drawn at one position with the closed curves six points of its rod trace over both assemblies. A point is given as (u, v) in the rod's own units — u along the rod from the crank pin, v across it. On the left (0.4, 0.5), (−0.3, 0.2), (1.2, −0.6): their distances from the crank pin are 0.64, 0.36, 1.34 rods. On the right (0.6, 0.8), (0, 1), (−0.8, 0.6), each exactly one rod from the crank pin, as far as the slider pin is. All six are quartics, all six are bounded — no real branch runs off the page — and nothing in the drawing tells the two panels apart. The difference is at infinity.

Where a slide puts the rest of the degree

A slider-crank's connecting rod draws a quartic that passes once through each circular point, which leaves two of its four meetings with the line at infinity unaccounted for. They are not along the slide. A point u along the rod and v across it sends them to the complex slopes [2v ± i(1 − u² − v²)] / [(1 + u)² + v²], whatever the crank, rod or offset — confirmed by slicing and by the fitted equation — and they are real only for points exactly a rod's length from the crank pin, where they merge into one direction at half the point's angle.

algebra · Algebra Four points of one connecting rod: two crossings, none, two isolated points, two crossings again. The closed curves drawn over both assemblies by four points of the connecting rod of a slider-crank with a crank of 1 and a rod of 3 on a slide through the crank's pivot, which is marked in each panel. Each curve is a quartic and owes two double points. (0.12, 0.15), 0.19 rods from the crank pin: two crossings, at (±0.65, ±0.36), ringed; (0.4, 0.5), 0.64 rods from the crank pin: a complex pair, so the curve crosses itself nowhere; (0.75, 0.94), 1.20 rods from the crank pin: two isolated points at (±4.06, ±3.00), dotted, off the drawn curve; (1.1, 1.0), 1.49 rods from the crank pin: two crossings, at (±2.66, ±2.95), ringed. Each panel has its own scale. The second curve has no visible double point at all, and the third has two that no configuration of the machine ever reaches.

The double points that leave for infinity

A slider-crank's connecting rod draws a quartic that owes two double points. Close to the crank pin they are two real crossings. Between a crank's length and a rod's length from it they are a complex pair and the curve crosses itself nowhere. At exactly one rod both run off to the same real point at infinity and meet there as a tacnode whose branches are complex. Past that they come back real. Offsetting the slide changes the two circles by (R² − a²)(R² − b²) + e²R² = 0 and sends only one of the pair to infinity.

algebra · Algebra The one real double point a four-bar owes is the one that leaves when its rocker becomes a slide. Left: a four-bar with a crank of 1 pivoted at the origin, a coupler of 3 and a rocker of 5 pivoted at (3, −5), so that the rocker's circle touches the line y = 0 where a slider-crank's slide would run. The coupler point is (0.4, 0.5) in the coupler's own units, and the curve it draws over both assemblies is shown with the circle through the three cognate pivots, on which all three finite double points lie. Two of them are a complex pair near the crank; the third, the only real one, is an isolated point at (0.92, −5.32) that no configuration reaches. Right: the same picture divided by the rocker length for rockers of 4, 10, 20, 50, 200, 100,000, with the drawn rocker's point ringed. The machine shrinks to the origin, the pivot circle settles onto a fixed circle, and the isolated point settles onto one point of it, (−0.484, −0.580), marked with a cross: the value d(|w|² − 1)/(w̄ − 1) for w = 0.4 + 0.5i and d the direction from the slide to the rocker's pivot.

The double point a slide cannot keep

A four-bar's coupler curve owes three finite double points and a slider-crank's owes two, so a four-bar whose rocker grows into a slide has to lose one. Parity decides which: the one that leaves is always real, and always an isolated point no configuration reaches. It runs off along d(|w|² − 1)/(w̄ − 1) times the rocker's length, a law that knows nothing of the crank, the rod, the offset or where the rocker touches the slide. On the circle one coupler length from the crank pin the law is zero, and all three double points leave together like the cube root of the rocker.

algebra · Algebra A slider-crank whose crank becomes a slide loses both its double points. Left: a slider-crank with a crank of 5 pivoted a crank's length before the point where its crank pin crosses the slide, a rod of 3 and the slide along y = 0, drawn with the slides' eventual crossing at the origin. The rod point (0.3, 0.2) in the rod's own units draws the curve shown, and its two double points are the dots: isolated points, at (−7.90, 2.47) and (−2.10, −2.47). Right: the same double points divided by the crank, with the crank's pivot at the origin and the crossing at (1, 0), for cranks of 5, 10, 30, 100, 300, 1000. The pair settles symmetrically onto the crosses at ±(0.595, −0.505) — the law ± i(|w| − e^(i·arg w))/√(1 − |w|²) for w = 0.3 + 0.2i — and leaves in proportion to the crank.

The trammel takes both double points

Grow a slider-crank's crank without bound and its crank pin's circle flattens into a second slide: the machine becomes the elliptic trammel, whose curves are conics with no double points at all. Both of the slider-crank's double points have to go, and parity cannot say how. They go together, symmetric through the crank's pivot, at ±i(|w| − e^(i·arg w))/√(1 − |w|²) crank lengths — a law that knows nothing of the rod or the slide's offset. Inside the one-rod circle they are two real isolated points, outside it a complex pair. And on the circle with the rod as its diameter, where the trammel draws straight lines, one of them does not leave at all: it settles on the crossing of the two slides.

algebra · Algebra Taken in one step, the limit is a family of laws. A four-bar with its crank a pivoted a crank's length to the left of the point where its two pins' circles cross, and its rocker c pivoted a rocker's length below it, both growing with c = λa, so that near that point it becomes the elliptic trammel. The coupler point (0.3, 0.2) in the rod's units has three double points, all leaving in proportion to the machine; plotted divided by a + c, with the crossing at the origin. Faint dots: the real ones at 49 ratios from a thousandth to a thousand. Large dots: at λ = 1, at (-0.110, -0.161), (-0.827, -0.176), (-0.059, -0.692), all three real. At either end one point settles where the one-step limit taken first would put it — the four-bar's law with the rocker grown first, its mirror with the crank grown first — and the other two collapse onto the crossing at this scale. In between, all three are spread.

A four-bar becomes a trammel along a family of laws

A four-bar becomes the elliptic trammel when its crank and its rocker both grow without bound, their pivots receding at right angles, and its three double points must all leave. Taken one link at a time the limit sends one point off along the four-bar's law and then the other two along the trammel's. Taken in one step it does something neither order sees: all three leave together, in proportion to the machine, at positions set by the coupler point and by λ, the rocker's length over the crank's. The two one-step orders are the family's two ends, λ → ∞ and λ → 0, and at equal rates the coupler point (0.3, 0.2) has its three double points up to 0.86 crank lengths from where either order puts them.

algebra · Algebra

Named alongside it

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

Double pointCircular pointsSlider-crankGenusBezoutDegreeElliptic trammelGough platformPolynomial systemWitness setAlgebraic curveCentrode

All concepts