Machine Kinematics
solved, then drawn
Concept

Least-squares — where it appears

Finding the answer that makes a set of residuals as small as possible in the sum-of-squares sense. Fitting a polynomial to a sampled curve is the version with no right-hand side, where the answer is the design matrix's smallest singular direction.

Named by 22 essays across 7 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 calculation in these essays 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 linkages, one polynomial. Roberts's theorem says three different four-bars draw the same coupler curve. Here each one is traced, and each trace is fitted separately for the sextic that vanishes on it — on a common normalisation, or the comparison would be between three polynomials in three coordinate systems. The twenty-eight coefficients agree across all three to 5.0e-7. The three traces are drawn on top of one another and the curve is the same object each time; the test shares nothing with the construction that produced the cognates, which is why it is a test.

Three linkages, one equation

Roberts's theorem says three different four-bars draw the same coupler curve. Fitted separately for the sextic that vanishes on each trace, on a common normalisation, the twenty-eight coefficients agree across all three to 5 × 10⁻⁷ — a test of the theorem that shares nothing with the construction the cognates came from.

curves · Coupler 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 error between the precision points. chebyshev: worst 0.2232°, RMS 0.1475°; least-squares: worst 0.3900°, RMS 0.1203°; minimax: worst 0.2105°, RMS 0.1463°. The error is zero at each precision point by construction and nowhere else. minimax has the smallest maximum here, and which curve is best depends entirely on which measure is asked for.

The linkage that is only nearly right

Stop demanding that a linkage pass exactly through three points, and ask instead that it be close everywhere. Three linkages result, none of them passing exactly through anything, and each is the best by a different measure — the least-squares fit beats the interpolant on average error and loses to it on the worst case. An optimiser gives exactly what it was asked for and nothing else.

synthesis · Synthesis 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 What degree a coupler curve actually is. A four-bar's coupler point traced at 602 solved positions, then asked which polynomials vanish on those points. For each degree the design matrix's smallest singular direction is taken and its worst residual plotted. Degrees four and five leave nothing near zero; degree six drops by ten decades and degree seven buys nothing further. The textbook sentence — a coupler curve is a sextic — comes back as a measurement, and the decision is made by the gap between the smallest singular value and the next, which here is a factor of 2.8e+3 — at 15, 21, 28 and 36 monomials respectively.

The equation a four-bar satisfies

Every textbook says a coupler curve is a sextic. Traced at six hundred solved positions and fitted at degrees four through seven, the answer comes back as a measurement: nothing vanishes below six, degree six drops by ten decades, and degree seven buys nothing — with the decision made by a gap of 2.8 × 10³ rather than by a residual.

algebra · Algebra 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 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 Five poses, five four-pose slides, and the compromise between them. Left, the moving body's frame with the slider point of each four-pose subproblem — the point whose four images are in a line when one pose is left out — and the compromise point whose five images all miss one guide by the same amount. Right, the fixed frame: the body in its five poses and the images of the chosen point, with its guide. Leaving out pose 1 to 5 misses that pose by 0.5304, 0.0836, 0.0383, 0.0422, 0.1288 units; the compromise misses every pose by 0.0140, on alternate sides. The body is about two units long.

The slide five poses almost have

Five prescribed poses leave no slide: a body point would need five images in a line, three conditions on two coordinates. But each of the five four-pose subproblems has exactly one slider point, and each misses the pose it left out by a distance a pair of compasses can find. Those five misses price the best compromise without solving for it. The slide that misses every pose by the same amount misses by one over the sum of their reciprocals — the misses add like resistances in parallel — and giving up the best single pose instead is always between two and four times worse.

synthesis · Synthesis 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 Along one line the nearly free curve stays nearly free. The arm's pivot is fixed at the centre of its reach (dashed); the quintic x⁵ + 0.4x²y − y = 0 is drawn with its own point — where the catalogue put the pivot — moved 0.2 of a link along the line x = 0, which is the same as moving the pivot the other way. Heavy: the arc priced, the part of the quintic within 1.4 of its own point. Thin: the nearest curve built from at most four cosines of the arm's angles, found by trying every set of four: 6.0% of the reach away, the line through the pivot bent by the square of the radius. The equation itself costs 22 cosines here.

The quintic's cheapness runs along a line

A quintic whose equation costs a compiled machine eighteen cosines is drawn to a few per cent of the arm's reach by four of them, and its exact count jumps to 22 or 30 the moment the arm's pivot moves. Priced by the nearest curve instead, the two directions part completely. Along the line x = 0 the nearest four-term curve stays within 7.3% of the reach out to 0.4 of a link and the nearest six-term within 3.4%. Across it the four-term price goes from 3.2% to 18% between a twentieth and a tenth of a link, and from 0.15 on four terms buy nothing over three. The move along adds four cosines of amplitude 0.2 and 0.4 times the move; the move across adds twelve, sixty to ninety times larger in total.

computing · Compute Re-chosen attachment points buy back a third of the tilt at four times the springs. The tilting-head lamp with one parallelogram and the smallest springs of the earlier essay. Every spring 2 cm free and every attachment point re-chosen for the least worst moment, all springs still pulling: 0.0611 N·m, a third less, with peak pulls 29.6, 0.0, 41.6, 195.1 N (lower arm, upper arm, elbow bracket to head, wrist), 266.3 N in all. The upper arm's spring is dotted where it has no stiffness.

A free length is absorbed where the springs attach

Give every spring of the smallest balanced desk lamp 2 cm of free length and refit the stiffnesses, and the lamp is left with 0.093 N·m of unbalance, growing in exact proportion to the free length. The spare fourth spring that ideal springs had no use for recovers under three per cent of it, and moving the head spring beyond the threshold only makes it worse. What absorbs a free length is a spring whose length barely changes. Re-chosen attachment points buy back a third of the unbalance that way, and pay for it with four times the pull.

applied · Fourbar The smallest springs that balance the lamp, and the one that is not needed. The tilting-head desk lamp with one parallelogram, its springs of no free length placed to make the total of their peak pulls over the working range least, inside an outline of brackets 0.15 m high, attachment points 0.08 m past the wrist and along the head's 0.15 m. Peak pulls 22.7, 0.0, 23.2, 17.5 N (lower arm, upper arm, elbow bracket to head, wrist), 63.4 N in all and 23.2 at most. The head spring meets the head at 0.0615 m, the threshold, and the upper arm's spring has no stiffness at all: it is left out.

Six things a balancing spring is not

More springs share the load, a spring that must push wants a compression spring, a stiffer spring balances better, a refitted stiffness repairs a free length, least squares finds the balance, and the best spring is one of no free length. Six things said about springs that balance a mechanism, each the natural reading of something true, each answered by a number the essays on the desk lamp and the rocking slide measured. They fail the same way. Each treats the spring as the thing to be sized, and a balance is decided first by where the spring is hung and how its error is judged.

wrong · Misconception The cheap pivots lie in a valley tilted off the line x = 0. The price of the quintic's arc drawn by the line through the pivot bent by the square of the radius, four cosines, as a share of the arm's reach, at pivots a two-hundredth of a link apart across and a twentieth along, within 0.16 and 0.45 of the catalogue's point. Darkest: within 5%; then within 10% and 20%. The cheapest pivot at each height (dots) lies on a line at dx/dy = -0.172, 9.8° off the line x = 0 (dashed), with prices from 1.4% to 2.7%; on x = 0 itself the price reaches 7.3%.

The quintic's cheap pivots lie off the line

Four cosines draw the quintic within a few per cent of the arm's reach from its catalogue pivot, and the earlier measurement found that cheapness surviving along the line x = 0 and dying across it. Mapped over a grid of pivots, it does neither exactly. The cheap pivots form a narrow valley tilted ten degrees off that line, about a twentieth of a link wide. In the valley four cosines hold the arc to 1.4–2.5% at every height out to 0.4 of a link; on the line itself, at the same heights, they miss by 6–7%. At two valley pivots every set of four cosines and every set of six was tried, and none beat the bent line. The amplitudes of the cosines a move adds predict a tilt a seventh as large, so they do not explain it.

computing · Compute Allowed 100 newtons, the lamp spends them on its attachment points. The tilting-head lamp with one parallelogram, every spring 2 cm free, its attachment points chosen inside the earlier outline for the least worst moment on any joint with the springs' peak pulls capped at 100 N in total. Worst moment 0.0620 N·m; peak pulls 22.5, 28.7, 13.9, 34.9 N (lower arm, upper arm, elbow bracket to head, wrist), 100.0 N in all. Each spring's zigzag is drawn heavier the harder it pulls, and dotted where it pulls less than half a newton.

The first pull goes to the spring the lamp did without

Give every spring of the balanced desk lamp 2 cm of free length and let it have more pull, and the worst moment it is left with falls from 0.086 N·m at 62 N to 0.052 at 227 N, the most any attachment points inside the outline can do. Half of that improvement costs the first 18 newtons; the second half costs the next 147. Spent on stiffness alone, with the attachment points held, extra pull buys nothing at all. And the first place it goes is the upper arm's spring, the spare that ideal springs had no use for, hung lower on its bracket while the head spring moves out past the threshold. The wrist spring, which carried almost all of the earlier re-attached design's cost, takes the largest share only after 140 N.

applied · Fourbar 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

Named alongside it

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

CalibrationIdentification jacobianSingular valueIdentifiableNoise amplificationMeasurement residualMinimaxObservability indexPose selectionCoupler curveFreudenstein invariantsParallelogram

All concepts