Machine Kinematics
solved, then drawn
Concept

Homotopy class — where it appears

The set of paths that can be deformed into one another without passing through an obstacle, which labels a strand's route. It is a discrete input rather than something computed — two strands with the same endpoints in different classes are the same mechanism configuration and different mechanisms.

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

One tendon over two joints. A two-link chain with a strand anchored off to the left, running over an idler at each joint and terminating on the far link. Each idler is centred on its joint's axis, which is the one arrangement that makes the strand's length a linear function of the joint angles: the coupling measured here is 14.000000 and 10.000000 mm per radian, against radii of 14 and 10. Total strand 232.845 mm. positioned by solving, not by drawing.

One strand over many joints

Route a tendon over an idler centred on a joint's axis and the strand's length becomes an exactly linear function of the joint angle — 8·10⁻¹⁴ mm of departure over 203 degrees of travel. Move that idler 6 mm off the axis and the same drive's arm swings from 8.00 to 16.52 mm per radian.

strands · Strand Two taut paths, and no way between them. A strand from one point to another past a peg. Each side gives a path that is taut — straight where it can be, on the surface where it must be, leaving at a right angle — and each is the shortest path on its own side: 219.165 mm on one and 200.643 mm on the other, against 200 mm of open air the strand cannot use. Neither can turn into the other without passing through the peg, which is a different kind of non-uniqueness from the assembly branches of a linkage: those are separate roots of one equation, and these are separate classes of one minimisation. positioned by solving, not by drawing.

The taut path has more than one answer

A strand from one point to another past a peg has two taut paths — 200.643 mm on one side and 219.165 on the other, against 200 mm of open air it cannot use. Both are shortest. Neither can become the other without passing through the peg, and no computation recovers which one was threaded.

strands · Strand

Named alongside it

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

Design ruleIdlerStrandBranchConfiguration spaceEccentricLever armSerial chainShortest pathTangencyTautTendon

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