Machine Kinematics
solved, then drawn
Concept

Minimal parameterisation — where it appears

A description of a mechanism with no two parameter vectors giving the same motion. A model that is complete but not minimal has a set of exactly equivalent answers, so a fit returns whichever the damping prefers and every number it reports is arbitrary.

Named by 4 essays across one field — each of them below, with the objects they name alongside it.

The coordinates that do not move. Above: the four link lengths as the whole machine is scaled from 0.4× to 2.5×, four straight lines through the origin. Below: Freudenstein's K₁ = g/a, K₂ = g/c and K₃ = (g² + a² + c² − b²)/2ac over the same range, three horizontal lines whose total variation is 1.8e-15. Each of them is homogeneous of degree zero in the lengths, so the scale ray is a level set of all three at once — and the map from a four-bar's shape to its three K's is invertible, so they are not merely invariant but complete. The identifiable quotient of a four-bar's parameter space is three-dimensional, and this site has had its coordinates since its first essay on synthesis.

The coordinates the site already had

Three of a four-bar's four parameters are recoverable, so there are three recoverable quantities. They are Freudenstein's K's, which this site has used to design function generators for as long as it has synthesised anything — and the same 3 × 3 linear system, read backwards, identifies a machine from three measured angle pairs.

metrology · Parameter Nine parameters, two of them invisible. A Watt six-bar has seven lengths, a fraction that says where a point rides on its rocker, and a third ground pivot with two coordinates. Reading its output link with a protractor over 28 poses gives a matrix of rank 7: two directions are invisible, at 8.24e-10 and 5.06e-10 against a largest of 7.49e+0. One is scaling the whole machine, with a zero against the fraction, because a fraction is not a length. The other is scaling the second loop alone about O₄ — that loop is a four-bar in its own right and its own size does not reach the output angle. The two wrong guesses a reader would try, scaling those five parameters about O₂ or scaling the first loop alone, are refused at 1.5e-1 and 2.0e-1.

Nine parameters, two of them invisible

A Watt six-bar has seven lengths, a fraction and a ground pivot's two coordinates. Read by a protractor on its output link, its identification Jacobian has rank seven — and the second missing direction is not a scaling of the machine at all. It is a scaling of the second loop alone, about the pivot the two loops share.

metrology · Parameter Six per joint is two too many. The number of parameters a serial chain's model carries, against the number of revolute joints. The upper bar of each pair counts six for every joint transform and six for each of the base and tool frames, which is what a reader expects; the lower bar is the number a measurement can distinguish. At 6 joints they are 48 and 30. The difference is not a saving, it is a warning: a model with 48 parameters fitted to any amount of data has an 18-dimensional set of exactly equivalent answers, so the fit returns whichever one the damping happens to prefer and every one of its numbers is arbitrary. Four per revolute because a rotation about the joint's own axis changes nothing and a translation along it is the joint variable; six rather than twelve for the two frames for the same reason, one level out.

Six per joint is two too many

A joint transform is six numbers, and a six-joint arm with a base and tool frame is forty-eight. A measurement can distinguish thirty. The difference is not a saving — it is an eighteen-dimensional set of exactly equivalent answers, and a fit returns whichever member of it the damping prefers.

metrology · Parameter The identification Jacobian of a four-bar, read by coordinate machine. One row for every number the instrument reads and one column for every parameter that might be wrong. Each cell is the derivative of that reading with respect to that parameter, drawn to the right of its centre line when positive and to the left when negative, with the largest entry in the whole matrix at 3.14e+0. 24 rows against 6 columns: far more equations than unknowns, which is what makes an identification a least-squares problem rather than a solve, and what makes the question of which combinations of columns cancel a real one. These are the same derivatives the tolerance field computes one at a time — the same matrix read down instead of across.

What a model is allowed to change

Before a calibration runs, somebody decides which numbers it may move. Leave one out and the fit absorbs it into the others; put one in that the instrument cannot see and the fit returns whatever the damping preferred. Both decisions are made before any measurement, both are checkable in advance, and neither is usually checked.

metrology · Parameter

Named alongside it

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

IdentifiableStructural identifiabilityCalibrationIdentification jacobianRedundant parameterScale invarianceSingular valueUnidentifiable directionDH parametersFreudenstein equationFreudenstein invariantsFunction generation

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