Machine Kinematics
solved, then drawn
Series

Parameter — the series

32 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 1 · metrology
  2. 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.

    part 2 · metrology
  3. 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.

    part 2 · metrology
  4. 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.

    part 3 · metrology
  5. 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.

    part 3 · metrology
  6. 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.

    part 4 · metrology
  7. Where a length comes from. A coupler 3.5 units long is a part with two holes in it, and neither hole's position is the length. Each carries an error of 0.010, of which 60% is common to both because they were bored in one setup — the whole pattern shifts by that much and the distance between the holes does not change. What survives is the independent part, 0.0063 at each hole, combining to 0.0089 on the length. A drawing that tolerances the length at ±0.010 is describing a part nobody makes, and it is out by a factor of 0.894 — optimistic below a shared fraction of one half and pessimistic above it.

    Where a length comes from

    A coupler 3.5 units long is a part with two holes in it, and neither hole's position is the length. What reaches the length is only the part of each hole's error the two do not share — so a drawing that tolerances the length is describing a part nobody makes, and is out by a factor of √2 in one direction or by everything in the other.

    part 4 · metrology
  8. The common normal, and where it is. Two joint axes 1.0 unit apart, nominally parallel, 0.50° apart in one plane and 0.05° in the other. The Denavit–Hartenberg convention takes all four of its numbers from the one line that meets both at right angles, and for these two axes that line crosses the first 11.3 units from the joint — off this page by a factor of about 11, which is why the tilt here is drawn at 26° and not at a fraction of one. Along it, the "link length" of a link 1.0 unit long reads 0.9950, and the angle round the first axis reads 5.71°. Nothing has moved by more than 0.50°.

    The common normal, and where it is

    The Denavit–Hartenberg convention reads all four of its numbers off one line: the common normal between two joint axes. Two parallel axes do not have one — every perpendicular meets both at right angles — and two nearly parallel axes have one that is somewhere else entirely.

    part 4 · metrology
  9. 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.

    part 5 · metrology
  10. 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.

    part 5 · metrology
  11. 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.

    part 5 · metrology
  12. 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.

    part 5 · metrology
  13. Where the two analyses cross. The band on the output angle at a crank angle of 57°, with every hole on every part given a position error of 0.010, against how much of that error each pair of holes shares. The flat line is what an analysis on the four lengths gives, which is the same number whatever the answer to that question. They cross at 0.53 and nowhere else: below it the lengths-only answer is optimistic, reaching 1.414× at holes located independently, and above it pessimistic, reaching zero when the error is entirely common and the distance between two holes is perfect however badly the pair is placed. The crossing is at one half because two holes contribute √2 and the surviving fraction is √(1 − shared).

    Where the two analyses cross

    Tolerance the lengths and you get one number whatever the shop does. Tolerance the holes and you get a curve, running from 1.414 times that number when nothing is shared to zero when everything is. They meet at a shared fraction of exactly one half, and the crossing does not depend on the machine, the tolerance or which length is being asked about.

    part 5 · metrology
  14. The same part, dimensioned two ways. Three holes at 0.00, 1.40, 3.50, each located to ±0.010 by the process. Dimensioned as a chain, each hole is placed from the last, so the two gaps are ±0.010 each and the errors accumulate over the whole span to ±0.0141. Dimensioned from a baseline, every hole is placed from H0, so the span is ±0.010 and the gap between H1 and H2 — which touches the datum at neither end — is ±0.0141. The two schemes are exactly √2 apart in opposite places. Nothing about the part decides this and nothing about the process does; the drawing does, and a tolerance analysis that starts from the lengths has already thrown the information away.

    The same part, dimensioned twice

    Three holes, one process, two drawings. Dimensioned as a chain the errors accumulate and the span is ±0.0141; dimensioned from a baseline the span is ±0.0100 and the gap between the second and third holes is ±0.0141. Exactly √2 apart, in opposite places, and nothing about the part or the process decides which — the drawing does.

    part 5 · metrology
  15. 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.

    part 5 · metrology
  16. 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.

    part 5 · metrology
  17. 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.

    part 6 · metrology
  18. 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.

    part 6 · metrology
  19. 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.

    part 6 · metrology
  20. 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.

    part 6 · metrology
  21. 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.

    part 6 · metrology
  22. Which feature to hold tight. A budget of 0.020 divided between the four lengths in inverse proportion to their sensitivities, and the same budget divided between the features those lengths are derived from — a frame jig-bored in one setup at 0.90 shared, a crank drilled twice at 0.10, a coupler at 0.60 and a rocker at 0.30. The tightest tolerance moves from b to a. And the null result matters as much: with one process for the whole machine the two allocations are identical to the last digit, because the transmission factor is then common and divides out. Feature-based tolerancing changes the answer when the parts are made differently and not merely when they are made.

    Which feature to hold tight

    A budget divided between four lengths in inverse proportion to their sensitivities gives one answer. The same budget divided between the features those lengths are derived from gives the same answer exactly — unless the parts are made differently, in which case the tightest tolerance moves from the rocker to the crank and the frame's loosens by a factor of two.

    part 6 · metrology
  23. 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.

    part 6 · metrology
  24. 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.

    part 7 · metrology
  25. 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.

    part 7 · metrology
  26. One set of lengths, two machines. Three measured input–output pairs, marked, and the linkage Freudenstein's relation returns from them — which is the truth's four lengths to fourteen figures. The relation is a statement about the two angles and it holds on both assembly branches, because it was derived by squaring and that is the step that forgets which one the mechanism is on. So the identified linkage assembled the way the data was taken passes through every reading, to 2.53e-14 radians, and assembled the other way misses them by up to 268° — at the first precision point it reads -111.6° where 98.8° was wanted. That is not a near miss and not a failure either. It is the other answer.

    One set of lengths, two machines

    Three measured input–output pairs return a four-bar's four lengths to fourteen figures. Assembled the way the data was taken, that linkage reproduces every reading to 2.5 × 10⁻¹⁴ radians. Assembled the other way — which the same four lengths permit — it misses them by 268°, and no equation in the identification knows the difference.

    part 7 · metrology
  27. Which numbers have a size. Twelve quantities from eight fields of mechanism kinematics, each scaled by taking every length in its mechanism up and down together and fitting the power its value follows. A shape lands on zero, a length on one, a curvature on minus one, an area on two, and every row lands on an integer to 3.3e-11. The sorting is not a units argument — it is measured from each field's own library, by asking that library the same question. What it says is that most of what kinematics computes is a shape: a mobility, a Grashof class, a transmission angle, a contact ratio and a velocity ratio are all unchanged by making the machine bigger, so all of them are recoverable from a measurement that cannot recover a single length. The row worth stopping at is the tolerance band: held to a fixed ±0.01 it lands on minus one, because ±0.01 is a length and a bigger machine held to the same absolute tolerance is a better machine. Written as a percentage the same band lands on zero. One quantity, two drawing conventions, two different kinds of number.

    Which numbers have a size

    Take every length in a mechanism up and down together and fit the power each computed quantity follows. A transmission angle lands on zero, a coupler point's speed on one, a path curvature on minus one, an enclosed area on two — and a tolerance band held to a fixed ±0.01 lands on minus one, which nobody would guess.

    part 7 · metrology
  28. 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.

    part 7 · metrology
  29. 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.

    part 7 · metrology
  30. Which numbers have a size. Twelve quantities from eight fields of mechanism kinematics, each scaled by taking every length in its mechanism up and down together and fitting the power its value follows. A shape lands on zero, a length on one, a curvature on minus one, an area on two, and every row lands on an integer to 3.3e-11. The sorting is not a units argument — it is measured from each field's own library, by asking that library the same question. What it says is that most of what kinematics computes is a shape: a mobility, a Grashof class, a transmission angle, a contact ratio and a velocity ratio are all unchanged by making the machine bigger, so all of them are recoverable from a measurement that cannot recover a single length. The row worth stopping at is the tolerance band: held to a fixed ±0.01 it lands on minus one, because ±0.01 is a length and a bigger machine held to the same absolute tolerance is a better machine. Written as a percentage the same band lands on zero. One quantity, two drawing conventions, two different kinds of number.

    A count is neither

    Every quantity in the scaling survey lands on an integer power — zero for a shape, one for a length, two for an area. A mobility lands nowhere. It has no dimension at all, it does not move under any perturbation, and the probe that sorts the rest of the site's numbers returns nothing for it.

    part 8 · metrology
  31. 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.

    part 8 · metrology
  32. 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.

    part 8 · metrology

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