Machine Kinematics
solved, then drawn
Concept

Layer assignment — where it appears

A map from a mechanism's links to parallel planes with no two conflicting parts sharing one and no pin passing through a link. It is a colouring plus an ordering: permuting the planes preserves the first condition and destroys the second, so most proper colourings are not buildable.

Named by 9 essays across 4 fields — each of them below, with the objects they name alongside it.

crank rocker: the closest pair at one position. The site's standard four-bar: ground 4, crank 1, coupler 3.5, rocker 3. Every joint is where the solver put it, exactly as in the linkage field; the material is the only thing added. The heavy segment joins the two closest points over every pair of parts that is tested — which excludes pairs sharing a pin, since their material surrounds that pin by construction — and its length is the gap: 0.5438 here, between coupler · frame. A negative value is a penetration depth, the distance the pair would have to be moved apart, and it is drawn in the warning colour.

Two bars that have to cross

The site's own four-bar has its coupler inside its frame by a full link width for the whole of a turn. It is not an impossible mechanism; it is a mechanism that cannot be built in one plane — and the plane it has been drawn in for twenty-three fields was a convenience nobody had to pay for.

bodies · Body A colouring is not an assignment. A pin joining links in planes 1 and 3 has to pass through plane 2, and anything in plane 2 whose material covers that pin is pierced by it. That condition involves three links at once where a colouring's conditions involve two, and it is invisible to a colouring because permuting colours preserves a colouring and destroys betweenness. It does not change the number of planes on any machine here. What it removes is arrangements: of the 6 three-plane colourings of a crank rocker, 2 can be built, and of Peaucellier's 192 exactly 96 can.

A plane is a colour

Assigning links to parallel planes so that no two conflicting parts share one is a graph colouring, and the answer for a four-bar is three. Then the pins have to get through, and the problem stops being a colouring: of the six proper three-plane colourings, two can be built.

bodies · Body How close a synthesis puts two pins. The distribution of the smallest pin-to-pin distance on any one link, over all 1,176 exact syntheses. Burmester's construction returns points, and points can be arbitrarily close together: the shortest here is 0.103, on a mechanism whose poses span more than two units. The shaded band is what a boss of radius 0.25 forbids — 148 of them, 12.6%. The link that is worst is most often the crank, which is not where a designer looks: the frame's two ground pivots are the pair everybody checks by eye.

A pin is not a point

A joint in the fields before this one is a name and two coordinates. A pin is a cylinder with material round it, a length through the stack of plates, and a head — and every one of those turns some construction that returns points into a construction that may return nothing buildable.

bodies · Body A 4-stage scissor, open and as flat as it goes. On paper a scissor stack's height is n·L·sin φ and goes to nothing as the bars lie down. The bars are made of something: the two colours are the two planes the stack needs — two, at any number of stages, which is why a scissor folds at all — and the bars that share a plane are the ones two stages apart. They meet at φ = 0.0951, leaving each stage 0.0949 high, which is exactly the two bosses that touch: 2 × 0.0473. The stowed height of the whole stack is 0.380 rather than zero, and it grows with the stage count in the way a stowed lift's does.

A stack that has to fit

A scissor stack's height is n·L·sin φ and goes to nothing as the bars lie down — on paper. The bars are made of something, and what stops the fold is two bosses meeting: every stage keeps 0.0949 whatever the stack does, which is exactly twice the boss radius.

networks · Network What a sweep at the wrong resolution reports. One machine — a crank passing a stud a tenth of a unit across — swept at six sample counts. At eight samples the smallest gap found is 0.0809 and the machine reads as clear; at twelve it is 0.0067 and still clear; from sixteen on it is negative and the crank is inside the stud by 0.0099. Nothing about the twelve-sample answer looks wrong: the curve it draws is smooth, its minimum is interior, its margin is small and positive. A sweep cannot report what it did not look at, and the repair is not more samples but a bound on what happens between them.

Six things a body is not

A verdict read as a measurement, a hull read as a part, a sweep read as a proof, a drawing read as a configuration, a geometry read as a force, and a plane read as a place. Six claims, each of them what a careful person would say, each answered with a number.

wrong · Misconception 6 pairs, 16 cuts if each is cut alone, 4 for all of them together. Peaucellier's cell's pairs of parts that never touch but whose whole-drive regions share ground, each in its own lane: its intervals between visits to shared cells (thin, one in several drawn) and the fewest cuts that pair needs alone (solid ticks). long arm A / rhombus QB: 5 arcs; long arm A / rhombus PB: 3 arcs; long arm B / rhombus AQ: 5 arcs; long arm B / rhombus AP: 3 arcs; rhombus AQ / rhombus PB: 3 arcs; rhombus QB / rhombus AP: 3 arcs. Cut separately they need 16 cuts. The dashed lines are the fewest cuts for every pair at once, 4, at −1.132, −0.290, 0.658, 1.360 radians — as few as the worst pair alone needs — and the intervals they were placed at do not overlap, so no partition does with fewer.

Four cuts serve the whole cell

A region test between moving parts needs its drive cut where each pair's two visits to shared ground separate, and one pair of Peaucellier's arms needs four cuts. The cell has six such pairs, and cut one at a time they would need sixteen. Cut together they need four, the same four moments, because every pair crosses its shared ground on either side of them. Built from a whole-drive region test the cell would take six planes; the four cuts give back the true four, where equal arcs need eight to do it.

bodies · Body 3 handovers, and 3 cuts where the shared ground stands empty. Four in turn: each tool's lift against the cam angle, from its own cam, with a thick bar under it while its front is inside the shared ground. The dashed lines are the fewest cuts that clear every pair of tools at once, 3, at 114.5°, 204.5°, 294.5°, and the shaded bands are the windows each may move in, 65° to 115°, 155° to 205°, 245° to 295°: the times when no tool is inside the shared ground. The pairs need 1 cut at worst and 6 between them; the station changes hands 3 times after the turn's seam.

A cam machine is cut at its handovers

In one symmetric linkage, every pair of parts crosses its shared ground on either side of the same few moments, so the whole machine needs no more cuts of its drive than its worst pair. Four tools driven by four cams into one work station do not share their moments. Each pair needs one cut, the pairs need six between them, and the machine needs three: one for each time the station changes hands. Each cut can move only within the time the station stands empty, which is the cam diagram's own timing margin, and when a schedule is squeezed until that margin closes, the tools collide.

bodies · Body Two stations, each shared by its own tools, and an arm that visits both. A cell with two work stations 3 apart. Station A is fed from the left and pressed from above; station B is pressed from above and emptied to the right; an arm pivoted below and between them picks a part up at A, swings it to B and parks midway. Every tool is 0.3 wide and reaches 0.15 past its station's centre, so the shaded squares are the only ground any two tools share, each square shared by its own station's tools and the arm. At 230° station A holds nothing and station B nothing.

Two stations share a cut when they hand over together

A cam machine's region test needs one cut of its drive for every time its shared ground changes hands. A cell with two work stations has two pieces of shared ground, each with its own handovers, and a single cut can serve both if it falls when both stations stand empty. So the cell needs somewhere between its busier station's count and the two stations' sum, and its cams decide where. Out of step, a press-and-transfer cell needs four cuts; with the same tools and strokes put in step, two. The transfer arm that carries the part between stations makes one shared cut for free, because while it swings it is at neither.

bodies · Body a crank rocker with a post: the closest pair at one position. The same four-bar with a post bolted to the frame, just clear of the coupler's path. Every joint is where the solver put it, exactly as in the linkage field; the material is the only thing added. The heavy segment joins the two closest points over every pair of parts that is tested — which excludes pairs sharing a pin, since their material surrounds that pin by construction — and its length is the gap: 0.1809 here, between coupler · post. A negative value is a penetration depth, the distance the pair would have to be moved apart, and it is drawn in the warning colour.

Where the boundary moved

An earlier essay drew a line around what the essays here compute and listed one thing on the far side as a gap rather than a boundary: interference between links, which needed no new physics, only a body and a test. Here is what that turned out to cost and what it turned out to open.

practice · Practice

Named alongside it

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

Link bodyInterferenceClearanceBossConfiguration spaceConflict graphSigned clearanceSwept volumeChromatic numberFree spaceKinematic chainLower pair

All concepts