Machine Kinematics
solved, then drawn
Concept

Noise amplification — where it appears

The factor by which the instrument's error is multiplied on its way into the recovered parameters. It is bounded by the reciprocal of the smallest singular value of the identification Jacobian, so it belongs to the mechanism and the poses rather than to the instrument.

Named by 10 essays across one field — each of them below, with the objects they name alongside it.

What another measurement is worth. The smallest singular value of the identification Jacobian — the observability of the worst-recovered parameter — against how many poses were measured. The lower curve takes poses evenly round the crank; the upper one chooses each next pose to make this number as large as it can. Both rise steeply and then flatten: the 13 evenly-spaced poses already give four fifths of what 20 give. Chosen poses reach at 4 what even spacing needs 5 to reach. The flattening is not diminishing returns on accuracy — it is the information in a pose being a direction in a three-dimensional space, which a handful of well-spread poses already spans. Everything after that is averaging noise, which is a different gain and improves as the square root.

How many poses are enough

Three readings determine a four-bar's shape and the thousandth adds almost nothing. The rank is reached at three because there are three parameters, and everything after that is conditioning — which is a different quantity, improves for a different reason, and stops improving much sooner than anybody expects.

metrology · Parameter The order the poses are chosen in. Each next pose chosen to make the worst-recovered parameter as observable as possible, numbered in the order it was taken, on the crank's own dial. The first three land 150° apart at the widest — they have to, since three parameters need three independent rows and nearby poses give nearly the same row — and every one after that bisects a gap. Nothing told the routine to spread them; it maximises a singular value and spreading is what that turns out to mean. The eighth pose is worth 7.4% more observability than the seventh.

Where a calibration should measure

Choose each next pose to make the worst-recovered parameter as observable as it can be, and something happens that nobody asked for: the first three land as far apart as they can get, and every one after that bisects a gap. Nothing told the routine to spread them. It maximises a singular value, and spreading is what that turns out to mean.

metrology · Parameter What another measurement is worth. The smallest singular value of the identification Jacobian — the observability of the worst-recovered parameter — against how many poses were measured. The lower curve takes poses evenly round the crank; the upper one chooses each next pose to make this number as large as it can. Both rise steeply and then flatten: the 13 evenly-spaced poses already give four fifths of what 20 give. Chosen poses reach at 4 what even spacing needs 5 to reach. The flattening is not diminishing returns on accuracy — it is the information in a pose being a direction in a three-dimensional space, which a handful of well-spread poses already spans. Everything after that is averaging noise, which is a different gain and improves as the square root.

What another measurement is worth

The observability of a four-bar's worst-recovered parameter goes 0.118, 0.162, 0.188, 0.208 — and then keeps going up by less and less until adding a pose changes the fourth decimal place. The flattening is not diminishing returns on accuracy. It is a space of fixed dimension being filled.

metrology · Parameter Four indices, four answers. Four of the five observability indices in use, each divided by its own value at 18 poses so their shapes can be compared: their absolute sizes differ by ten orders and a shared axis would draw three flat lines. They are five different questions about one list of singular values — the geometric mean, the smallest alone, the reciprocal condition number, and two normalisations of the smallest — and at eight poses they disagree by a factor of 1.50 about how much of the job is done. A pose set chosen to maximise one is not the set that maximises another, and the literature quotes the choice as a preference.

Four indices, four answers

Five numbers are in use for scoring how well a set of poses determines a mechanism's parameters. They are five different questions about one list of singular values, they rank pose sets differently, and the literature quotes the choice between them as a matter of preference. It is a matter of what the report has to carry.

metrology · Parameter The instrument's error, multiplied. The error in the recovered shape against the error in each reading, over four decades, each point the mean of six independent calibrations and the open marks the worst of the six. The slope is 0.9994 — the error is linear in the noise, with no threshold and no saturation — and the constant is 1.90. So a protractor good to a milliradian gives a shape good to about 1.9 milliradians' worth, and the factor belongs to the mechanism and the poses rather than to the instrument. The bound from the smallest singular value is 2.16, which the measurement sits under, as it must.

The instrument's error, multiplied

Repeat a whole calibration on independently noised readings at four levels three decades apart and the error in the recovered shape is linear in the noise, with a fitted slope of 0.9994 and a constant of 1.90. That constant belongs to the mechanism and the poses, not to the instrument — and it is bounded by one over the smallest singular value.

metrology · Parameter A fit converging onto noise. The sum of squared residuals through a calibration of a four-bar built 3% long on the coupler and 1% short on the rocker, started from the nominal dimensions. It falls by a factor of 1.2e+3 in 19 steps and then stops, at 4.05e-5 — which is the noise, not the machine. The readings were given 1.00e-3 radians of error each, and no fit can go below what its data contains. The last two steps fall faster than the ones before them, which is what a Gauss–Newton descent does when the Jacobian has full rank on the directions it is allowed to move in.

Reading a residual

A residual that falls to the instrument's noise and stops means the model is right. A residual that stops above it means something is missing, and which something can be read off how the leftover is distributed over the poses — as a constant, as a pattern in the crank angle, or as one bad reading.

metrology · Parameter A calibration that improves the machine and reports the wrong one. The truth's tracing point is 0.198 units from where the model says it is, and the model has only four lengths with which to say so. Fitted over half a turn, it reduces the error there by a factor of 33 and over the other half by a factor of 21, so every practical test says the calibration worked. It got there by moving the rocker by -0.2474 — 8.2% — and the coupler by -0.0317. The residual it cannot drive away, 4.18e-3, is the only signal that anything is missing, and it is the signal a practitioner is most likely to read as instrument noise.

A parameter the model has not got

The machine's tracing point is 0.198 units from where the model says it is, and the model has only four lengths with which to say so. It absorbs the discrepancy: the error over the measured half-turn falls by a factor of thirty-three, the error over the other half falls by twenty-one, and the rocker comes back eight per cent short.

metrology · Parameter 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 A number that runs away. The Denavit–Hartenberg offset of a pair of nominally parallel axes, against how far from parallel they actually are, for a fixed out-of-plane tilt of 0.05°. The marks are extracted from the geometry by finding the common normal; the line is the closed form −A cos α cos β sin β / (sin²α cos²β + sin²β), and the two agree to 2.4e-16 relative over three decades. At 30° of twist the offset is 0.0030 of a link length; at 0.01° it is 1102. The dashed line is the worst case over the tilt, which sits at β = α and is exactly A/2α. Nothing about the machine has changed by as much as a degree.

A number that runs away

A hundredth of a degree of unintended twist on a nominally parallel pair of joint axes puts the Denavit–Hartenberg offset at −1,102 link lengths. The extraction from the geometry and the closed form agree to 5 × 10⁻¹⁶ over three decades, and the worst case over the tilt is exactly A/2α.

metrology · Parameter 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.

Two instruments disagree about the worst

A protractor recovers three of a four-bar's parameters at a condition number of 5.2. A coordinate machine recovers six at 162. Neither number says which parameter is worst recovered, and when both are asked, they name different ones — because a condition number is a summary of a list and the list is what a report needs.

metrology · Parameter

Named alongside it

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

CalibrationIdentification jacobianLeast-squaresIdentifiableObservability indexPose selectionSingular valueMeasurement residualUnmodelled parameterBranch ambiguityCommon normalCoupler point

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