Machine Kinematics
solved, then drawn
Concept

Footprint — where it appears

The extent of a machine's material over its whole drive, as against the extent of its joint positions. The second is what a site drawing links as lines can report; the first is what has to fit in something, and it is larger by the widths, the bosses and the ends that overhang their pins.

Named by 3 essays across one field — 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

Named alongside it

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

Link bodyOccupancy gridSwept regionBossInterferenceIdentifiableScale invarianceSigned clearanceSwept pathSwept volumeWorkspace

All concepts