Machine Kinematics
solved, then drawn
Concept

Chasles's theorem — where it appears

The statement that any rigid displacement is one rotation about a line together with a translation along that same line. It is what makes the screw the natural object for spatial kinematics, since it says every displacement already is one whether or not it was described that way.

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

A screw of pitch 0.25. A single screw drawn from its own decomposition. The six numbers of the twist go into a decomposition that returns a direction, a point on the axis and a pitch of 0.2500; the helix is then the path of a point at radius 0.5 about that axis, advancing 0.250 along it per radian turned. Over 1.15 turns the point advances 1.806 — the pitch times the angle, which is what pitch means. This screw is a screw.

Every motion is a screw

Chasles showed that any rigid displacement whatever is a turn about some line together with a slide along that same line. Not approximately, and not usually — always, with the line and the amount of slide computable from the motion. It is the fact that makes spatial kinematics a subject rather than a pile of special cases.

spatial · Spatial Three shortest paths. The tool's route between the same two poses, under three interpolations. The straight one is position interpolated along a line with the rotation carried separately, which is what most controllers do. The screw path is the single turn about a single axis that Chasles's theorem says takes any pose to any other — the only one of the three that mentions no coordinate system — and it is 18.0% longer. The joint-space path is what the arm does when nobody asks for anything in particular, and it is 26.8% longer again. The screw axis itself is drawn: pitch -0.096 m per radian.

The distance between two poses

Ask an arm to take the shortest route between two poses and three reasonable definitions give three different paths, of lengths 1.223, 1.443 and 1.550 metres. The disagreement is not numerical. There is no distance between two rigid poses until somebody chooses a length to measure a radian in, and on this arm the choice changes which of two poses is nearer at 74.5 millimetres.

serial · Serial

Named alongside it

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

ScrewTwistConditioningConstraintCoupleInstant centreInterpolationMetricPitchPoseRigid displacementScrew axis

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