Machine Kinematics
solved, then drawn
Concept

Swept region — where it appears

Every point any part of a machine occupies at some position of its drive. It is a region rather than a curve, because a body's far corner reaches places its pin never goes, and it is the shape a guard has to enclose and the set an obstacle has to stay out of.

Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.

The room crank rocker needs. The shaded region is every point any part of this machine occupies at some position of its drive, computed as an occupancy grid over 150 solved configurations and outlined by marching squares — so the outline drawn is the boundary of the set the area was counted from rather than a second object that agrees with it. Its area is 12.78 square units, filling 54% of the box that contains it. The dashed rectangle is the box the joints need, which is what every figure that treats links as lines could have told you; the material needs a box 20% larger in area.

The room a machine sweeps

Every point any part of a machine occupies at some position of its drive. It is a region rather than a curve, its area is an integral computed two ways, and the one shape in the field with a closed form is what the grid is calibrated against.

bodies · Body The box the joints need, and the box the machine needs. Every machine in the catalogue, measured twice over the same drive: the extent of its joint positions, which is what a site drawing links as lines can report, and the extent of its material, which is what has to fit in something. The ratio runs from 1.13 to 1.36 — between thirteen and 36 per cent more area than the skeleton suggests, on machines whose links are a twentieth of their length wide. The last column is how much of the material's box the swept region actually fills, and it is where the packaging argument really is: a machine at 50% is a machine with a great deal of room inside its own envelope that nothing may be put in.

The hole the machine needs

A four-bar's joints fit in a box five units by four. Its material needs a box twenty per cent larger in area, and fills barely half of it. Both numbers are design quantities, and until a link had a width neither could be stated.

bodies · Body A link, as a distance and as a body. The same two links every field before this one has drawn as lines, drawn as the material they are made of. A bar is a rectangle with its ends rounded off to the bosses that surround its pins, and it is convex; a bell crank is two arms meeting at a shared pin, and it is not — its inner corner turns the wrong way by 0.52, which is the single fact that puts it outside every separating-axis test here. The pins are marked because they are what has not changed: the constraint equations are the same, the solve is the same, and the positions are the same. What is new is everything between the pins.

A body is all size

Twenty-five of the fields before this one compute quantities that are mostly shapes, recoverable from an angle sensor and transferable between machines of any size. This one computes clearances, footprints and swept areas, and not one of them is a shape — which makes it the only field whose whole output needs a ruler.

bodies · Body 36% shared, and they never touch. The region left arm visits over a whole drive, the region right arm visits, and — in the third colour — the part of the plane both of them visit. The shared area is 4.652 square units, 35.9 per cent of the smaller of the two regions. The two parts are drawn at the configuration where they come closest, and at that configuration the gap between them is 0.3799 — more than twice a link's width, and positive everywhere else on the drive. A test that asks whether the regions intersect has reported a collision between two parts that are never in the same place at the same time.

The regions overlap and the parts never meet

A swept region is a projection along time, and a projection cannot be undone. Chebyshev's two arms share thirty-six per cent of the ground the smaller of them covers and never come within twice a link's width of each other — a false alarm the region test cannot avoid, and one it cannot make at all against anything that stands still.

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

Three phases ago this site drew a line around what it computes 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.

InterferenceLink bodyOccupancy gridFootprintSigned clearanceBossConfiguration spaceConserved quantityDesign ruleFree spaceIdentifiableLayer assignment

All concepts