Machine Kinematics
solved, then drawn
Concept

Identification jacobian — where it appears

The matrix of derivatives of every measured reading with respect to every parameter of a mechanism's model, stacked over all the poses measured. Its rank says how many of those parameters the measurement determines and its null space says which combinations no measurement moves.

Named by 22 essays across 4 fields — each of them below, with the objects they name alongside it.

The identification Jacobian of a four-bar, read by protractor. 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 4.09e-1. 14 rows against 4 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.

A dimension is a measurement

Every library on this site takes the numbers on the drawing as given: chosen by a designer, cut by a machinist, and thereafter known. They are not known. This field runs the same kinematics with the parameters as the unknowns and the motion as the data, and the first thing that appears is a question with an exact answer — which of them can be recovered at all.

metrology · Parameter The identification Jacobian of a four-bar, read by protractor. 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 4.07e-1. 12 rows against 4 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.

The matrix a calibration inverts

One row for every number an instrument reads, one column for every parameter that might be wrong. Every entry is a derivative the tolerance field has been computing since its first essay — so this field's central object arrived already built, and what is new is which way it is read.

metrology · Parameter Three machines a protractor cannot tell apart. The same four-bar at 0.60×, 1.00×, 1.50×, drawn one inside another at the same crank angle. Every one of them puts its output link at 102.914064°, and the three readings differ by 2.8e-14° — which is the solver's floor rather than a difference. A protractor on the output link is reading a function of the ratios of the lengths, so it is the same function for every member of this family, at every crank angle, exactly. Whatever such an instrument recovers, it is not the size of the machine.

The direction no protractor can see

A four-bar's output angle depends only on the ratios of its lengths. That is a sentence anybody would agree to, and it has a consequence with a number attached: the vector of the four lengths is annihilated by every row of the machine's own identification Jacobian, to 7.6 × 10⁻¹⁵, at every pose, for ever.

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.

A ruler and a protractor

What a measurement recovers is decided by the units of its readings. An angle is dimensionless and cannot see a size; a position is not and can. And putting both instruments on one machine recovers no more parameters than the better of them alone — it recovers the same ones six times better conditioned.

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.

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 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 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 Where a calibration measures. The four-bar drawn at the 6 poses a selection chose, one over another, with the tracing point marked at each. The largest gap between consecutive chosen poses is 150°. They are spread because rows of the identification Jacobian at nearby poses are nearly the same row, which is a statement about the matrix and reads here as a picture of a machine in visibly different configurations.

The pose the machine cannot reach

A measurement plan is drawn against the nominal machine and executed on the real one, and the real one does not go quite where the drawing says. A pose that falls outside the travel returns no reading at all — which is not an error, not a failure of the instrument, and not nothing: it is a measurement of the limit position.

metrology · Parameter Every length wrong, every reading right. A four-bar was built to the dimensions in the upper bar of each pair and its output angle read at 30 positions. A calibration started from the nominal dimensions returns the lower bar. It reproduces every one of those readings to 1.81e-16 radians and not one of its four numbers is the machine's: they are the machine's multiplied by 0.992289, every one of them, to 3.0e-16. The shape is recovered exactly — the distance in Freudenstein's three invariants is 6.3e-16 — and the size is a free parameter the damping happened to leave near where it started. A machinist handed these numbers would build a machine that works and is not this one.

Every length wrong, every reading right

A four-bar was built out of true and measured at thirty positions. A calibration started from the nominal dimensions reproduces every reading to 1.8 × 10⁻¹⁶ radians and returns four lengths, not one of which is the machine's. They are the machine's, multiplied by 0.99229 — every one of them, to fifteen figures.

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 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 Reachable, and dexterous. A 3-link planar arm with links 1.6, 1.2, 0.7. The outer region is everywhere the tool can be put: an annulus from 0.00 to 3.50. The inner region is everywhere it can be put at every tool angle — from 1.10 to 2.10, which is 26% of the area. Both are measured by counting cells on a 260 × 260 grid and both agree with the area computed from the radii to 0.06%, which is what makes the picture a measurement.

An arm's parameters and its poses

A three-link planar arm has three lengths and a tool position that carries a length, so nothing about it is invisible to a measurement — and it is nevertheless the mechanism on this site where a calibration is hardest, because its parameter count is high, its poses are three-dimensional and its Jacobian is singular where a designer likes to work.

serial · Serial 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 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 The machine is fine; the description is not. Above: the offset the Denavit–Hartenberg chart assigns to a pair of nominally parallel axes, over three decades of twist, running from 0.0030 to 1102 link lengths. Below: the condition number of the identification Jacobian in a chart that describes the second axis by two small rotations from the first and never asks for a common normal, over the same range. It is 7.5501 at every one of them, flat to 2.4e-9. The machine is the same machine in both rows and it is perfectly well behaved. What breaks is a convention that locates its parameters on a line which, for two parallel axes, does not exist.

The chart breaks, the machine does not

The same two axes, over the same three and a half decades of twist. In one description a parameter runs from 0.003 to 1,102; in another the condition number is 7.5501 and does not move in the fifth figure. A quantity that diverges in one chart and is constant in another is a property of the chart.

metrology · Parameter The identification Jacobian of a four-bar, read by protractor. 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 4.09e-1. 14 rows against 4 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.

A calibration is a synthesis with more equations

The site's second field prescribes three input–output pairs and solves a 3 × 3 linear system for a linkage. This one measures thirty pairs and solves the same system in the least-squares sense. Same matrix, same coefficients, same closed form — and the only structural difference produces every question this field is about.

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 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.

CalibrationIdentifiableSingular valueLeast-squaresMeasurement residualNoise amplificationObservability indexPose selectionUnidentifiable directionScale invarianceUnmodelled parameterGrashof's condition

All concepts