Machine Kinematics
solved, then drawn
Concept

Rigid body — where it appears

A piece of a mechanism whose points keep fixed distances from one another. A point attached to one is at a fixed angle and a fixed fraction of its length, which is how a compiled machine applies a phase and a length at no cost in freedoms.

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

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: 0.2561 here, between rocker · 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.

A link that takes up room

For twenty-three fields a link on this site has been a distance between two points, and a distance cannot collide with anything, because it is not anywhere. Give every link a body and a question arrives that none of the constraint equations can ask.

bodies · Body How much of a compiled machine is computing anything. Each machine's bars split two ways: the ones that build an angle — reflectors, means, rigid offsets, the arm — and the ones that carry a direction from where it was computed to where it is needed. On the smallest machines the arithmetic is nearly all of it. By the quintic the carrying is 80 per cent, and it goes on rising, because the arithmetic grows with the number of terms and the carrying grows with the number of pairs of them. That is the answer to why a universality construction is enormous, and it is not about the algebra being hard.

A parallelogram carries an angle, and only so far

A direction computed at the frame is no use where it is needed. A parallelogram carries one from point to point — but only between two points a fixed distance apart, and that single proviso is what makes a compiled machine quadratic in the number of terms and turns most of it into transport.

computing · Compute Every pin position, and what it costs — 3 mm between the doors. One cell per candidate pin position: the door is swung through 95° from each and the worst clearance recorded. Light cells clear; dark cells mean the door would pass through the partition or through the door beside it, by up to 18 mm. The horizontal line is the door's own front face. With the doors touching, the boundary sits exactly on it — because the front corner's sideways rate is the pin's distance behind it, and that is zero only there. Each millimetre of gap buys a few millimetres of depth, and at 3 mm the deepest pin that still clears is 10 mm behind the face.

Where a hinge pin can go

A cabinet door's front corner moves sideways as it opens at a rate equal to how far the pin sits behind it, so with the doors touching, no pin behind the door's face can open one without going through the next. Three millimetres of gap buys nine and a half millimetres of depth, and that is the whole reason a concealed hinge has four bars instead of a pin.

applied · Coupler

Named alongside it

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

ClearanceConstraintInterferenceAssembly branchCouplerHingeInstantaneous centreLink bodyLinkageMobilityOverconstraintParallelogram linkage

All concepts