Machine Kinematics
solved, then drawn
Field

How many answers

Newton finds a configuration. It cannot tell you how many there are, and a search that has stopped finding new ones is not a proof that there are no more. Written as polynomials, a mechanism's closure conditions have a number of solutions that is a property of its shape — and this is where the site stops counting by looking.
Two circles, four answers, two of them nowhere. A four-bar with its crank held at 52° is two circles: the coupler pin is 3.5 from the crank pin and 3 from the far ground pivot. Two quadratics in two unknowns, so Bézout's number is four — and the tracker finds two. The other two paths run off to infinity, and they do so for every pair of circles ever drawn: two circles meet the line at infinity in the same two points, and those are what the fourth and third answers are.

Two circles, four answers

Bézout's theorem applied to the smallest mechanism on the site — why the count of a mechanism's assemblies is a property of the shape of its equations rather than of its dimensions, and where the missing solutions go.

 Every path, in the plane of one unknown. The 16 tracked paths of the 3-RPR platform, projected onto the complex plane of x. Each curve starts at a solution of the start system and ends at a solution of the target or leaves the frame on its way to infinity. This run is γ random, start constants complex, and it found 6 solutions.

Following a root from a problem already solved

How homotopy continuation works, why the deformation has to be complex, and a measurement of where the genericity actually has to come from.

 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

Why Bézout's bound is so loose for mechanism equations, where the missing solutions go, and a measurement showing that eliminating the waste costs more than the waste.

 The count that moves, and the one that does not. 676 sets of leg lengths for one 3-RPR platform, the third leg held at 2.2. Every one of them has exactly 6 complex solutions. The number that is real runs 0, 2 — 141 cells at 0, 535 cells at 2 — and that number is what a machine shop would call the assembly modes.

The count that does not move

The difference between a mechanism family's complex solution count and a particular mechanism's assembly modes, measured over a grid of leg lengths, and what a singularity is in that language.

 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 Gough platform's classical count of forty, computed rather than quoted, and the discovery that the symmetric platform drawn here is not a general one.

 A lower bound that happened to be tight. The site's own Gough platform at its home pose. The search starts Newton from a spread of guesses and reports what it lands on: 16 from 400, 16 from 1200, 16 from 4000. Tracking every one of the 1458 Bézout paths says there are 28 poses in the complex numbers and 16 of them are real. The search was right. Nothing available to the search could have said so.

The search that was right

A search-based assembly count checked against a complete one, and what monodromy does when it has found 78 of 80.

 Two circles each, so two to the power of the dyads. A dyad has two solutions, and a chain whose groups are all dyads is solved one dyad at a time, so the number of ways it can be assembled at a given input angle is 2 raised to the number of them. That is a prediction made from the graph about a count of configurations, and it is checked here against a count: the same chain solved from 240 random seeds, with the distinct converged configurations counted. The two agree in every row. Watt's did not at first — it came back at eight — and every one of the four extra answers had its ternary link mirrored: three distances fix a triangle only up to reflection, so the distance equations admit a part that has been turned inside out. A reflected link is a different part rather than a different pose, and the solver refuses those frames now.

Two to the power of the dyads

An assembly count predicted from a topology and checked against a solve: the dyad decomposition, why the branches multiply, and the reflection that satisfies every equation and is not a pose.

 What degree a coupler curve actually is. A four-bar's coupler point traced at 602 solved positions, then asked which polynomials vanish on those points. For each degree the design matrix's smallest singular direction is taken and its worst residual plotted. Degrees four and five leave nothing near zero; degree six drops by ten decades and degree seven buys nothing further. The textbook sentence — a coupler curve is a sextic — comes back as a measurement, and the decision is made by the gap between the smallest singular value and the next, which here is a factor of 2.8e+3 — at 15, 21, 28 and 36 monomials respectively.

The equation a four-bar satisfies

Reading the implicit equation of a four-bar's coupler curve off its traced points, and why the singular-value gap rather than the residual is what decides the degree.

 20,000 four-bars, and not one with three circuits. 20,000 four-bars with the ground at one and the other three lengths drawn from 0.05 to 3, each counted exactly. crank-rocker: 3,031, of which 3,031 have two circuits, 0 have one and 0 cannot be assembled; double crank: 3,086, of which 3,086 have two circuits, 0 have one and 0 cannot be assembled; Grashof double rocker: 1,497, of which 1,497 have two circuits, 0 have one and 0 cannot be assembled; triple rocker (non-Grashof): 12,386, of which 0 have two circuits, 9,528 have one and 2,858 cannot be assembled. The most circuits any linkage has is 2.

Never three circuits

Why a four-bar has at most two circuits: an exact census, the four branch points of its configuration curve, Riemann–Hurwitz and Harnack's bound, and the six-bars whose higher genus lets them pass it.

 A line across the four-bar's coupler curve. The curve traced by a point on the four-bar's coupler, over every real configuration, with the machine drawn faintly at one of them. A real line crosses it at 4 marked points. Written as polynomials, the machine and the line have 8 paths to track; 6 arrive, 2 leave for infinity, and the arrivals draw 6 distinct points — 4 real and 2 complex. A random complex line gives 6 as well, which is the curve's degree.

A degree counted on a line

The degree of a four-bar's coupler curve measured by slicing it with random complex lines and counting the witness points, with a trace test for completeness and monodromy for irreducibility.

 Every body of the Stephenson six-bar, sliced by one line. One random complex line, and a point on each body of the Stephenson six-bar required to lie on it. Every system has Bézout number 32. crank: 8 configurations drawing 2 distinct points, each 4 times, so degree 2; ternary coupler: 12 configurations drawing 6 distinct points, each 2 times, so degree 6; rocker: 8 configurations drawing 2 distinct points, each 4 times, so degree 2; arm: 18 configurations drawing 18 distinct points, each 1 time, so degree 18; output: 12 configurations drawing 2 distinct points, each 6 times, so degree 2. The configurations are what the tracker counts; the points are what the curve has.

The curve nobody eliminates

The degree of a Stephenson six-bar's arm curve measured by witness points where elimination is not attempted and the implicit fit fails; why each body of a six-bar draws its points a different number of times; and the Watt chain whose second coupler draws a sextic twice.

 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

The circularity of curves drawn by planar linkages, measured by slicing each machine with lines through the circular points: half the degree for every body of the four-bar, Watt and Stephenson six-bars, but not for the slider-crank's connecting rod or the elliptic trammel; the four-bar sextic written in z and z̄; and the double points the circular points leave room for.

 The closed path of a geared five-bar's pin at four gear ratios. A five-bar with two cranks of length 1 on pivots 3 apart, couplers 3 and 2.5, and a gear on each crank, drawn over one whole cycle at ratios 1 : 1, which closes after 1 input turn; 2 : 1, which closes after 1 input turn; 3 : 2, which closes after 2 input turns; 5 : 3, which closes after 3 input turns. Both assembly branches of the coupler pin are drawn, one in each colour. The curves they make together have degree 6, 10, 16, 26.

Every rational gear ratio has a degree

The degree of a geared five-bar's pin curve at rational gear ratios, counted by homotopy continuation and by the roots of one Laurent polynomial left after eliminating the pin; why its span is 4·max(p, q) + 2·min(p, q); the real roots as the drawn crossings; what an implicit fit can find; and why a gear passes each circular point 2·max(p, q) times rather than half the degree.

 The same five-bar with its gears meshed outside and inside. A geared five-bar drawn over a whole cycle at four ratios, with its two cranks turning against each other in the top row and together in the bottom one — an external mesh and an internal one, which is the same machine with the ratio's sign changed. Both assembly branches are drawn. The curves have the same degree in both rows — 1 : 1 at 6, 2 : 1 at 10, 3 : 2 at 16, 5 : 3 at 26 — and they are not the same curves: the lower ones are the maximally circular members of their degree and the upper ones are not, which is a difference nothing about the drawing shows.

The mesh inside keeps the half

The internally geared five-bar's coupler curve: the same degree law as the external mesh at every convergent, a circularity of exactly half the degree rather than twice the larger tooth count, the double-point budget that leaves it a perfect square in the finite plane, and a real line counted on the polynomial and on the drawn polyline.

 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

The two non-circular points at infinity of a slider-crank's coupler quartic: predicted from the complex configurations where the crank pin runs to infinity, checked by counting finite meetings with lines of the predicted slopes and by dividing the fitted quartic's top part by x² + y², independent of crank, rod and offset, and real exactly on the circle one rod from the crank pin, at half the point's angle.

 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

The two double points of a slider-crank's coupler quartic, found by two copies of the machine and by the fitted equation: real crossings inside one crank length of the crank pin, a complex pair out to one rod, a tacnode at the real point at infinity on the one-rod circle with branch ratio cos²(ψ/2), real again beyond with the rod's moving centrode dividing crossings from isolated points, and the offset machine's two circles in closed form.

 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

The four-bar's three finite double points followed as its rocker lengthens into a slider-crank's slide: the real, isolated one that leaves, derived in isotropic coordinates as q = d(|w|² − 1)/(w̄ − 1), checked to 10⁻³ over 240 machines with the miss falling as one over the rocker, the two that stay converging on the slider-crank's pair at the same rate, and the cube-root departure of all three on the one-coupler circle.

 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

A slider-crank whose crank grows into a second slide: the curve and its two double points at one crank, and the pair divided by the crank settling on the law; which way the pair leaves for every rod point, real inside the one-rod circle and complex outside; five rod points counted at a crank of 1 and of 1,000; the gap to the law closing as the square of one over the crank; 120 machines with random rods and offsets against the law; and the rod-diameter circle, where one double point stays at the slides' crossing.

 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 whose crank and rocker grow together into the elliptic trammel: its three double points, divided by the sum of the two, as the ratio of rocker to crank runs from a thousandth to a thousand; their gap to the law against the machine's size at equal rates; the family's ends against the two one-step laws; how many of the three are real over the coupler's plane at three ratios; the double point that stays at the slides' crossing on the circle with the rod as diameter; and ninety random machines against the law.

 Every boundary touches the circle on the rod. The coupler's plane in the rod's units, crank pin at (0, 0) and rocker pin at (1, 0). Shaded: the coupler points whose three departing double points are all real at a ratio of rocker to crank of 1, the region where R = λ⁴A + λ²B + C is negative; 8.1% of a window from −1.4 to 2.4 along the rod and ±1.5 across it. Dotted: the circles one rod from each pin, the boundary at the family's two ends. Dashed: the circle with the rod as its diameter. Every boundary in the family touches that circle and never crosses it; at this ratio it touches at (0.50, ±0.50), the points where −B/2A = λ². Inside it all three are real at every ratio.

Three discs decide the real double points

The boundary of the region where a four-bar growing into a trammel has three real departing double points, found as a discriminant: the three curves it is mixed from, its sign checked against the law's own count cell by cell and ray by ray, the ratio at which it touches each point of the circle on the rod, the four histories a coupler point can have, and the window of ratios in which a point of the lens has only one real.

 A growing four-bar walks away from its own double points. A four-bar with a rod of 3, its crank and rocker both of length t, pivoted so that both pivot circles pass through the slides' crossing, with the coupler point at (0.3, 0.2) in the rod's units. Drawn in units of t: the two pivot circles (dotted), their second meeting (ringed), the real coupler curve the machine draws (solid), and its real double points, filled where two real configurations meet and hollow where none reaches. At size t = 7 the curve's real double points are isolated (2.02), isolated (3.32), crossing (0.93); the number in brackets is how far the line of rod orientations through each passes from the unit circle's centre, a crossing below one and isolated above. As t grows the curve shrinks onto the crossing and the second meeting and the double points stay where they are.

A growing four-bar reaches none of its departing double points

A four-bar whose crank and rocker grow together into the elliptic trammel, and whether the machine reaches its departing real double points: the real curve and the points in units of the crank as the machine grows; each point's line of rod orientations moving off the unit circle in proportion to size; the size beyond which no crossing is left, mapped over the coupler's plane; that size against the rate law, by how the last crossing leaves; and three routes classifying every point.

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