Machine Kinematics
solved, then drawn
Concept

Singular value — where it appears

One of the numbers saying how much a matrix stretches along each of its principal directions. The smallest one decides a rank, and the ratio between the smallest kept and the largest discarded is what makes that decision a measurement rather than an assertion.

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 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 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 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 body of the Stephenson six-bar, sliced by one line. One random complex line, and a point on each body of the Stephenson six-bar required to lie on it. Every system has Bézout number 32. crank: 8 configurations drawing 2 distinct points, each 4 times, so degree 2; ternary coupler: 12 configurations drawing 6 distinct points, each 2 times, so degree 6; rocker: 8 configurations drawing 2 distinct points, each 4 times, so degree 2; arm: 18 configurations drawing 18 distinct points, each 1 time, so degree 18; output: 12 configurations drawing 2 distinct points, each 6 times, so degree 2. The configurations are what the tracker counts; the points are what the curve has.

The curve nobody eliminates

A point on the arm of the site's dwell six-bar draws a curve whose equation nobody writes down and no fit can find: at degree eighteen a fit needs a hundred and ninety coefficients, and its singular values have no gap to decide by. Sliced by a random line, the same machine has thirty-two paths to track and eighteen arrive, on every line tried, all eighteen distinct and all on one curve.

algebra · Algebra Three legs and four, at 0°. The same slice of positions at a platform angle of 0°, shaded by how well the platform is held — pale is near singular. Left, three legs: 3,312 reachable samples and a singular curve through them in 102 segments. Right, the same three legs and a fourth: 3,198 reachable, because the fourth leg must reach too, and no curve. What is left of the singular set in this slice is 1 isolated point, at (-1.449, -0.811), where all four lines meet. Positions held above 0.1 go from 2,989 to 3,178, and at no sampled position is the four-legged platform held less well than the three-legged one.

What a fourth leg buys

Three leg lines fail to hold a platform when they meet at a point, which is one condition, so in every slice of the workspace the failures form a curve. Four lines fail only when all four meet at a point, which is two conditions, so the curve becomes isolated points. The fourth leg buys that and more, and it costs a machine that can no longer be assembled from any four motor angles.

parallel · Parallel 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 Every position loses its hold at -150° and 30°. The smallest singular value of the six leg lines of the Gough–Stewart platform, held level, as it is turned through a whole revolution at 4 positions: (0, 0, 2.4), (0.4, -0.3, 2.4), (0, 0, 3.2), (-0.5, 0.2, 1.8). At -150° and 30° all 4 curves reach zero together — the largest of them is 9.7e-9 — and 1° either side the least is 1.28e-3. Nowhere else does any of them touch zero.

A yaw that is singular everywhere

Turn the standard hexapod platform 30° about the vertical, hold it level, and it is singular: not at one pose, but at every position it can be put in, with one screw motion that none of its six locked legs can resist. The angle is not a property of the dimensions. It comes out of one line of trigonometry that no spread of the anchor points can change.

parallel · Parallel Where a fit's singular values fall away, on each motion of a parallelogram. The 28 singular values of the degree-six fit, largest first and relative to the largest, for points traced on the parallelogram's circle motion, on its quartic motion, and on both. On the circle motion the last 15 lie below a drop of 3.7 × 10¹³, from 0.232 to 6.2 × 10⁻¹⁵; on the quartic motion the last 6 lie below a drop of 3.0 × 10¹⁰, from 6.3 × 10⁻⁵ to 2.1 × 10⁻¹⁵; on both motions the last one lies below a drop of 2.5 × 10¹², from 3.1 × 10⁻⁴ to 1.2 × 10⁻¹⁶. A drop of ten decades or more is a null space that is exactly there: fifteen sextics vanish on a circle, six on a quartic, and one on both.

A null space of fifteen is not noise

Points traced on one motion of a parallelogram four-bar leave a degree-six fit with fifteen polynomials that vanish on them, behind a drop of thirteen decades: the circle times every quartic. Half an oval of an ordinary coupler curve leaves two to five, behind drops of two. The count is the same kind of number in both cases, and only the drop beside it says which one is algebra.

curves · Coupler A 3-RPR with similar triangles at 0°, its leg lines meeting wherever it is put. A planar platform with three extending legs, its base pivots on a circle of radius 2 and its platform points on one of radius 0.6 at the same angles, so the two triangles are similar, turned to 0° and drawn at two positions. Each leg's line is continued past its ends. With the platform's centre at (0.4, −0.3) the three lines meet at (0.5714, −0.4286), within 1 × 10⁻¹⁶ of every line, which is where a scale factor of 0.3 about that point carries the base triangle onto the platform; the smallest singular value of the three legs is 3 × 10⁻⁹. With the platform's centre at (−0.55, 0.3) the three lines meet at (−0.7857, 0.4286), within 2 × 10⁻¹⁶ of every line, which is where a scale factor of 0.3 about that point carries the base triangle onto the platform; the smallest singular value of the three legs is 0. The meeting point moves with the platform, and the three lines meet wherever it is.

Two orientations no position can rescue

A planar platform on three extending legs whose platform triangle is a scaled copy of its base is singular at every position it can be put in, at exactly two orientations: its three leg lines meet at one point wherever it is. The two orientations are read off the attachment points, a platform a few per cent from similar is held at its worst orientation only in proportion to how far from similar it is, and the revolute-legged 3-RRR, built on the same similar triangles, does not inherit any of it.

parallel · Parallel A paired platform with its pairs rotated 60°, turned through a revolution. The paired Gough platform — base anchors in pairs 25° apart on a radius of 2.2, platform anchors in pairs 40° apart on 1.1 — with the platform's pairs centred 60° round from the base's, held level and turned through a revolution at four positions. All four curves reach nought together at 30° and −150°, which is 90° − ρ and 180° from it. At the centred position the platform is held at 0.0363 at a yaw of 0°, and it can turn ±17° before that falls to half. Dragging the rotation carries the two dead yaws across the revolution together.

The dead yaw is a design choice

A paired hexapod held level is singular everywhere at a yaw of 30° because its platform pairs sit 60° round from its base pairs. Rotate them by ρ instead and the dead yaws move to 90° − ρ and 180° from it, exactly, at every rotation from 0° to 120°. The furthest they can be from home is a quarter-turn each way, at ρ = 0 — where the platform is also best held at home and can turn ±75.5° before its holding halves, against ±17° for the usual 60°.

parallel · Parallel Where a tilted platform is still singular. A slice of the workspace at height 2.4, at the dead yaw and 3° of tilt, with each position shaded by how well the platform is held there — dark where the six legs are nearly dependent and pale where they are not. Level, this whole square would be uniformly dark. Tilted, the dark places are a curve through it, which is what an ordinary direct singularity looks like on a slice. Driving a search downhill from forty-eight starts reaches a smallest singular value of 0.00e+0, so the locus is a singularity rather than a shallow valley.

Tilted, near the dead yaw

A paired Gough platform held level is singular at one yaw wherever it stands. Tilt it and that stops being true — the six moments are no longer equal and the home position is held. It is a poor rescue: the rise is quadratic in the tilt, so a degree buys a sixty-fourth of what eight degrees buys, and what the tilt actually does is not remove the singularity but turn it into a surface through the workspace.

parallel · Parallel 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 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 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.

Identification jacobianIdentifiableCalibrationLeast-squaresObservability indexNoise amplificationPose selectionArchitecture singularityImplicit equationSingularityUnidentifiable directionAlgebraic curve

All concepts