Machine Kinematics
solved, then drawn
Concept

Kinematic synthesis — where it appears

Designing the mechanism from the motion rather than deriving the motion from the mechanism. It is the whole subject run backwards, and it is where atlases, precision points and defect tests come from.

Named by 12 essays across 4 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 Three prescribed positions of a rigid body. The whole of the design problem, before any mechanism exists. A body has to occupy these three positions — each one a place and an angle, three numbers — and what carries it between them is not yet decided. A forward analysis starts from link lengths and finds the motion. This starts from the motion, and the lengths are what has to be found. The marked points are the poles: any planar displacement is a rotation about one point, so each pair of poses has one, and the arcs show the turn each represents through the body's own origin. A pole is a property of the displacement and not a mechanism — nothing has been chosen yet. 1 of the 3 poles lies outside this frame and is not drawn; near-parallel displacements push their pole a long way off.

The problem the other way round

Every essay before this one starts from link lengths and finds the motion. That is the reader's problem, because lengths are what a drawing shows. It is not the designer's problem, which is the reverse — and the reverse is hard enough that for a century the practical method was to look the answer up in a book.

synthesis · Synthesis Of 1176 exactly correct syntheses, 176 could be built. Every pair of points on a 7×7 grid over the moving body, each pair synthesised into a four-bar and each four-bar verified by the forward solver as reaching all three prescribed poses. 1176 of them do, exactly. Then each is swept from the first pose in both directions, carrying the branch the way a built mechanism must: 810 cannot reach all three without being taken apart and reassembled, and 190 reach them in the wrong order. 176 — 15% — are mechanisms rather than theorems. Nothing in the construction distinguishes them.

Exactly right, and unbuildable

A linkage synthesised through three prescribed positions reaches all three. That is a theorem and it holds exactly. Whether it reaches them in one piece, without being taken apart, and in the order asked for, are separate questions the construction says nothing about — and of 1,176 exactly correct solutions, 176 could be built.

wrong · Misconception 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 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 A four-bar that computes log₁₀. The output rocker's angle, mapped back into y, against the input crank's angle mapped back into x. The pale curve is log₁₀ x and the solid one is what the linkage does. They agree exactly at the 3 precision points and nowhere else; the largest disagreement over the range is 0.2232° of rocker, which is 7.47e-4 in y. The chebyshev spacing put the precision points at 1.0670, 1.5000, 1.9330.

Three problems called synthesis

Prescribing a path, prescribing a whole pose, and prescribing a relation between two angles are three different problems with three different counts, and the word synthesis covers all of them. The third has a property the others do not — eliminate the coupler angle and the design equation becomes linear, so a four-bar that computes a logarithm falls out of a 3 × 3 solve with no iteration at all.

synthesis · Synthesis The error between the precision points. chebyshev: worst 0.2232°, RMS 0.1475°; least-squares: worst 0.3900°, RMS 0.1203°. The error is zero at each precision point by construction and nowhere else. chebyshev has the smallest maximum here, and which curve is best depends entirely on which measure is asked for.

Where an optimiser starts

An approximate synthesis is a local search on an objective that is non-convex, disconnected and not everywhere defined, so the answer depends on where the search began. Nothing in the optimisation supplies that. What supplies it is the exact constructions the field spent four rungs on, and an atlas of coupler curves — which is why a method superseded by computers is still the thing that feeds them.

synthesis · Synthesis How many points a linkage will take. function generation: 5 free numbers, 1 net constraint per point, 5 precision points; motion generation: 4 free numbers, 1 net constraint per point, 5 precision points; path generation: 9 free numbers, 1 net constraint per point, 9 precision points. Every count in the literature that differs from these differs about what is being counted as free, not about the geometry.

How many points may be prescribed

Five poses, five angle pairs, nine points — three numbers that get quoted as properties of a four-bar and are properties of what somebody decided to count as free. Derive them instead, and the fifth precision point turns a linear solve into a system with 128 paths, twelve finite solutions, four real ones, and exactly one linkage anybody could build.

synthesis · Synthesis Every pivot on these arcs gives a 60° swing at a time ratio of 1.2. The rocker's pivot O₄ is fixed and its two limit positions B₁ and B₂ are 60° apart. At a limit the crank and coupler are in line, so the crank pivot sees the chord B₁B₂ at the angle δ = 180°(Q − 1)/(Q + 1) = 16.36°, and the points that see a chord at a fixed angle are arcs of two circles. The thick stretches are the 828 sampled pivots that give a crank-rocker with exactly this swing and ratio; the rest of each circle gives a linkage of another kind. The linkage drawn is the member whose worst transmission angle is largest, at both limits: ground 1.222, crank 0.478, coupler 1.130 and rocker 1, with its worst transmission angle 40.32°.

A swing and a time ratio

A shaper's specification gives the rocker's swing and how much quicker the return must be than the cut, and those two numbers do not fix a linkage. They leave a one-parameter family of crank-rockers on the arcs of two circles, every member exactly right, and the transmission angle chooses between them — which is also what decides that a 60° swing cannot return more than 1.207 times as fast and keep 40°.

linkages · Fourbar What each curve costs, in cosines. Every polynomial in the two arm angles is a constant plus a sum of terms A cos(mα + nβ + φ) with whole-number m and n, and the number of those terms is what a machine has to build. The count is not the degree and not the monomial count: a circle costs one term, a general line two, and a lemniscate — degree four — costs five, fewer than the cubic above it, because its symmetry cancels frequency pairs the cubic keeps. Each row's expansion was checked against a direct evaluation of its own polynomial at random angles, worst disagreement 1.8e-14.

Prescribing a curve rather than points

A four-bar can be made to pass through nine prescribed points and no more; past nine the problem is over-determined and the answer is an optimiser's. Prescribe the whole curve as an equation instead and there is no counting to do — but the mechanism that comes back has four hundred bars where the four-bar had four.

synthesis · Synthesis Bars against terms, over the whole catalogue. One mark per compiled machine. The bar count rises much faster than the term count, and the reason is the summing chain: term k has to have its direction carried to the k−1th vertex of the chain, one parallelogram per hop, so the carrying costs a translator for every pair of terms. Nine curves, from five bars to four hundred and thirteen, on a term count that goes from one to eighteen.

What universality is worth

The linkage exists, it is four hundred and thirteen bars, and it draws ten degrees of its curve. All three are true and only the first is in the theorem — which is the ordinary shape of a result about what exists, and the reason it was worth building one to find out.

computing · Compute A drag link driving a crank-rocker at a phase of 284.8°. A drag link, ground 1, crank 2.5, coupler 2.25, output 1.5, whose output crank carries the crank of a crank-rocker with a 60° swing, ground 1.2223, crank 0.4783, coupler 1.1298, output 1, turned 284.8° ahead of it, drawn at an input angle of 40°. The two cranks on the middle pivot are one rigid part. The thick arc at the right is the rocker's swing. At this phase the whole machine returns 2.71 times as fast as it works; the crank-rocker alone, driven at constant speed, returns 1.2 times as fast.

A drag link ahead of a crank-rocker

A crank-rocker with a 60° swing keeps a transmission angle of 40° only up to a time ratio of 1.207, and at a ratio of 2 no crank-rocker keeps even 20°. Drive its crank from the output of a drag link, whose cranks both turn but not at the same speed, and a pair in which each stage keeps 40° returns 2.71 times as fast as it works. The phase between the two stages decides almost all of it: the same two linkages give anything from 1.003 to 2.71.

linkages · Fourbar

Named alongside it

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

Coupler curveBranch defectFunction generationPath generationPrecision positionSynthesisUniversalityAlgebraic curveApproximate synthesisAssembly branchAtlasBezout

All concepts