Machine Kinematics
solved, then drawn
Concept

Cognate ambiguity — where it appears

The fact that a coupler curve is traced by three different four-bars, read as a statement about measurement. An identification that sees only the traced path has three exact answers with nothing between them, so no amount of data and no better instrument chooses among them.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

Three machines, one curve. The coupler curve of a four-bar, drawn three times by three different linkages. Roberts's theorem gives every coupler curve exactly three four-bars that trace it, and their proportions are not close: the cranks here are 1.600, 2.214, 2.558, a spread of 60%. Read as an identification problem this is a least-squares objective with three separate exact minima and nothing between them — so an instrument that records only where the tracing point went has three answers however good it is, and no amount of data chooses. What chooses is knowing where the ground pivots are, which is a different measurement rather than a better one.

Three machines, one curve

Roberts's theorem says every four-bar coupler curve is drawn by exactly three different four-bars. Read as an identification problem that is a least-squares objective with three separate exact minima, whose cranks differ by sixty per cent — so an instrument that records only where the tracing point went has three answers, and no amount of data chooses between them.

metrology · Parameter A four-bar at 1°, solved. Ground 4, crank 1, coupler 3.5, rocker 3. Every joint position here is the output of a Newton–Raphson solve on the loop-closure equations, converged to 3.3e-14 — not a placement that looked right. Grashof's condition classifies these lengths as a crank rocker, and sweeping the crank through 360° confirms it: 120 of 120 positions assemble. The transmission angle at this instant is 54.3°.

Six things a measurement cannot tell you

A calibration with a perfect residual whose fourth number is a starting guess, a rank that says nothing about a second answer, an improvement that proves nothing about a parameter, a class with no margin, a plan scored on poses that were refused, and a model missing something no data can find. Six claims, each with the number that kills it.

wrong · Misconception What each instrument recovers. The same twenty poses of the same four-bar, read three ways. A protractor on the output link recovers 3 of the four lengths and leaves the fourth exactly invisible, because its readings are dimensionless in the lengths and scaling the machine does not move them. A coordinate machine on the tracing point recovers all six parameters — the four lengths and the two that say where the tracer sits — at a condition number of 162.3. Using both recovers the same six at 26.1, 6.2 times better, which is the case for putting two instruments on one machine: not more parameters, better-conditioned ones.

What this field cannot measure

Every field on this site has a boundary and this one has three: a direction the readings cannot span, an alternative no derivative detects, and a model nobody thought of. The first is computable exactly, the second needs a search, and the third is not detectable from data by any method at all.

metrology · Parameter

Named alongside it

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

CalibrationIdentifiableIdentification jacobianMeasurement residualUnidentifiable directionUnmodelled parameterBranch ambiguityCognate linkageCoupler curveLeast-squaresObservability indexthe Roberts–Chebyshev theorem

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