Calibration — where it appears
Named by 33 essays across 7 fields — each of them below, with the objects they name alongside it.
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.
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.
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.
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.
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.
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.
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.
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.
Grashof is a shape test
The oldest classification in the subject compares sums of lengths, so it is unchanged by making the machine bigger — which means a protractor recovers it exactly without recovering a single length. What it does not recover is the margin, and the margin is what says whether the classification is safe.
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.
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.
The transmission angle has no size
The geometric half of force transmission is an angle in a triangle whose three sides scale together, so it is the same at every size — 54.31° at its worst on this machine, whatever units the drawing is in. Which means a measurement that cannot recover a single length recovers the whole of what this field computes.
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.
Where the boundary moved again
The practice field's inventory ended by naming what the work after it should do first: take the feature positions as the variables and derive the lengths. That is done, and it turned out not to be an extension of the tolerance field but half of a different one — because where a length comes from and what a measurement determines are the same question.
What a calibration cannot see
A six-joint arm's model has thirty-six parameters and a measurement can find thirty of them. The other six are not hard to measure — they are combinations that move the tool by exactly nothing, at every posture, and no instrument ever built will separate them. The count is 4R + 2P + 6, and it comes out of a rank on four different arms.
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.
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.
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.
The error that is an integral
A tolerance on a link length moves an output by a bounded amount. A tolerance on a wheel radius moves a vehicle by an amount that grows with how far it has driven: one per cent of mismatch between two wheels bends a commanded straight line onto a 30 m radius, and a four-metre square comes back 1.60 m from where the machine thinks it is.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
Named alongside it
The objects these essays reach for when they reach for this one.
IdentifiableIdentification jacobianLeast-squaresMeasurement residualSingular valueNoise amplificationObservability indexPose selectionToleranceUnidentifiable directionScale invarianceUnmodelled parameter