Machine Kinematics
solved, then drawn
Concept

Dyad — where it appears

Two links joined by one joint, with a joint at each far end onto something already placed. It is the smallest Assur group and the reason most mechanisms have closed forms: the free joint lies on a circle about each placed one, which is a quadratic with two roots and no solver.

Named by 8 essays across 3 fields — each of them below, with the objects they name alongside it.

One moving pin, and the pivot it turns about. Choose any point of the moving body — this one at (-0.55, 0.5) in the body's own frame. In the three prescribed poses it lands in three places, and three points that are not in a line lie on exactly one circle. That circle's centre is where the fixed pivot has to be and its radius is how long the link has to be: here 1.0860, and all three images sit at that distance to within 10⁻¹². There is no iteration and no tolerance in the construction, because three points determine a circle exactly. The freedom is entirely in which point of the body to pick.

Three positions, and a circumcentre

The whole of three-position synthesis is one observation: a moving point occupies three places, three points that are not in a line lie on exactly one circle, and that circle's centre is where the fixed pivot has to be. No iteration, no tolerance, and every point of the coupler is a candidate.

synthesis · Synthesis Burmester's curves, contoured rather than drawn. With three poses, every point of the moving body works: three images, one circumcircle. With four, a point's four images are concyclic only if it lies on a particular cubic — the circle-point curve — and the fixed pivots those points want lie on a second cubic, the centre-point curve. Both are drawn here as contours of a measured quantity: at each point of a 150×150 grid, how far the fourth image misses the circle through the other three, contoured at zero. Points refined onto the contour are concyclic to 3.8e-15; points 0.47 away from it miss by at least 2.0e-1. The two curves are keyed in the legend and the four prescribed poses are drawn faintly for scale. Three poses leave a designer the whole plane; four leave a curve. The four prescribed poses are outlined faintly for scale, and both curves are clipped to the frame: the centre-point curve is a cubic with unbounded branches that reach 360 units on a mechanism three units across, and the part worth looking at is the part near the machine.

What the fourth position costs

With three prescribed positions every point of the coupler will do, and a designer is spoilt for choice. Add a fourth and the whole plane collapses to a curve — only points on a particular cubic have four images that lie on a circle, and the cubic is Burmester's.

synthesis · Synthesis Five positions, and what is left of the curve. Five prescribed poses of a moving body. With four of them, every point of the pale curve is a usable fixed pivot — a one-parameter family. The fifth pose is one more equation, and it leaves 4 points. Bézout's number for the system is 16; 4 paths arrive; 4 of those are real. Every pair of the 4 is a four-bar, so there are 6 candidate linkages and 2 of them reach all five poses in one piece and in order. 1 of the 4 pivots is too far away to draw in frame and is marked at the edge with its true distance — which is why some of the linkages have a bar twenty times the size of the body.

Five positions, and what is left

Three prescribed poses leave a whole plane of choices. Four leave a curve. Five leave four points, and finding them is the first thing in this site's synthesis field that a compass cannot do — it needs two cubics intersected, which is algebra rather than construction. Four points give six four-bars, and two of them can be built.

synthesis · Synthesis 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

How many ways a mechanism can be assembled at a given input angle is a count of configurations, and it is predicted here by a graph: two circles per pair of links, so two to the power of the number of pairs. The prediction came back four for Watt's chain and the count came back eight, and the four extra had a link turned inside out.

algebra · Algebra A dyad is two circles, and that is why it has two answers. The whole of what makes a dyad easy. Two links, three pins: one pin onto something already placed at each end, one pin between them. The free pin is at a fixed distance from each of the placed ones, so it lies on both circles — and two circles meet in two points, in nought, or in one. That is a quadratic with a closed form, and its two roots are the two assembly branches every four-bar on this site has. A group of four links has no such picture: its unknowns do not separate into one circle each, the system does not factor, and what replaces the compass is Newton's method from a seed. positioned by solving, not by drawing.

What has to be solved together

Hold a link, turn a neighbour, and the rest of a mechanism comes apart into the smallest sets that can be positioned one after another. Every set of two links is two circles meeting — a quadratic, two branches, no solver. A set of four is a system, and this site's Newton solve stops being a convenience.

topology · Topology The four eight-link chains a compass cannot position. Every one of the twenty ways of choosing a frame and a driven link, on each of these four chains, leaves a group of four links or more that has to be solved as one system. There is no order in which they come apart two at a time, so there is no ruler-and-compass construction for any of them and no closed form for their positions. They are numbers 1, 3, 4, 10 of the sixteen, and they do not share an assortment: 4×2 + 4×3 and 5×2 + 2×3 + 1×4 both appear. Three of the four are among the most symmetric chains in the census — automorphism groups of 16, 8, 8 against a median of three across the sixteen — which is the direction one would guess, since a symmetric chain has few genuinely different places to attach a driven link. The fourth has an automorphism group of 2, so symmetry is a tendency here and not the reason.

Four that a compass cannot reach

Twelve of the sixteen eight-link chains can be positioned two links at a time, from at least one choice of frame and input. Four cannot be positioned that way from any of their twenty choices — and at ten links ninety of the two hundred and thirty are in the same position.

topology · Topology A machine with one thing in it that must be solved at once. Each compiled machine's joints, walked in the order they can be placed: a joint goes down as soon as two things already placed decide where it is. Every one of these machines unwinds completely, and each contains exactly one pair that has to be solved together — the arm and the parallelogram that carries its second angle back to the pivot. Nothing larger than a dyad appears in a machine of two hundred and forty joints. The topology field's four-bar, at four, has no such decomposition at all; a compiled linkage is enormous and structurally trivial, which are not the same axis.

A machine with one dyad in it

Two hundred and forty joints, and two hundred and thirty-eight of them can be placed one at a time from parts already positioned. The whole of what has to be solved simultaneously is a single pair — the arm and the parallelogram carrying its second angle home. Size and structural depth are different axes, and a compiled machine is extreme on one and trivial on the other.

topology · Topology The body points whose three images are in a line. The three prescribed poses, faint, and a curve through them. At every point of a 110×110 grid over the moving body, the point is placed in all three poses and the signed height of its image triangle is measured; the curve is where that length is zero. Those are the points whose three images lie on a line, so the dyad they want is a slide rather than a crank. The curve is a circle. Refined onto the contour, its points lie on the circle through the three image poles to 2e-15, where the pole triangle's own circle misses them by up to 1.98. Its radius is 8.767, so inside a window three units across it reads as a gentle arc; 2 of the three image poles are in the frame and the third, P₁₃, is 6.8 units away.

Where a pin becomes a slide

Three-position synthesis gives every point of the moving body a fixed pivot, except the points whose three images fall in a line. Those want a slide, and they are not scattered: they lie on one circle, the circle through the three image poles, which a single line of algebra predicts and a contour of a measured length draws to 10⁻¹⁵.

synthesis · Synthesis

Named alongside it

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

Assur groupKinematic chainAssembly branchCircumcentreLoop closurePosition analysisPrecision positionSynthesisBurmester theoryCanonical formConstraintType synthesis

All concepts