Machine Kinematics
solved, then drawn
Concept

Self-calibration — where it appears

A mechanism identifying its own dimensions from redundant sensing of its own motion, with no instrument outside it. A closed loop with a sensor at each end gives one equation per pose, which is enough for the machine's shape and never enough for its size.

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

What a platform's own sensors settle. One three-legged platform, measured at 75 poses, with three different sets of sensors on it. Reading every joint angle determines 12 of the 15 numbers that describe the machine and leaves three undetermined, and one of the three is the scaling of the whole machine — every reading is dimensionless, so multiplying every length by a constant changes nothing any sensor sees. Laying a rule across two base pivots once recovers exactly one more, and the one it recovers is the size. Legs that report their own extension need no rule: a reading with a length in it breaks the scaling direction outright, and 7 of 9 come back. What none of the three determines is where the frame's origin sits, which is a convention rather than a defect.

A platform that measures itself

A three-legged platform with every joint read can be calibrated from its own sensors with no instrument in the room. Left to itself it shrinks the machine to a fiftieth of a per cent of its size — and reports a residual five orders smaller than the right answer's for doing it.

parallel · Parallel A rolling wheel, where it was driven to. The mechanism at a configuration nothing wrote down: it was reached by integrating permitted velocities from the start of the trail, and there is no equation here whose root it is. The barred line at each wheel is the direction that wheel forbids — the subject of the whole field, and the one thing a photograph of a car cannot show. The constraint residual along the drawn history is 0.0e+0.

A wheel that cannot report its radius

Rolling relates a wheel's turning to a vehicle's travelling, and the relation has a length in it. So a rolling constraint is the one place on this site where an angle measurement does carry a size — and the size it carries is the one thing a vehicle's own odometry can never separate from its wheelbase.

rolling · Rolling A machine measuring its own shape. A four-bar with an encoder at each end of its one freedom. Every pose gives one scalar equation — Freudenstein's, which is linear in the three invariants — so 40 poses make a three-column least squares with a condition number of 8.758 and no instrument outside the machine anywhere in it. With perfect encoders the invariants come back to 8.64e-15; with encoders good to a milliradian they come back to 5.40e-3, an amplification of about 6.26. What comes back is a shape and only a shape: the same three numbers describe this machine and one a quarter of the size, and nothing an encoder can read separates them.

A machine that measures itself

Put an encoder at each end of a one-freedom loop and every pose gives one scalar equation. The equation is Freudenstein's, it is linear in three unknowns, forty poses make a three-column least squares at a condition number of 8.76 — and no instrument outside the machine is involved anywhere.

metrology · Parameter

Named alongside it

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

IdentifiableCalibrationScale invarianceAssembly-modeClosureFreudenstein equationFreudenstein invariantsHolonomyIdentification jacobianLeast-squaresNoise amplificationNonholonomic

All concepts