Machine Kinematics
solved, then drawn
Concept

Kinematic solve — where it appears

Finding a configuration that satisfies a mechanism's closure conditions, which on this site means Newton–Raphson against an analytic Jacobian. Nothing is drawn that was not solved, and the residual is judged at a tolerance four orders looser than the one the solver drives to.

Named by 5 essays across 3 fields — each of them below, with the objects they name alongside it.

Two circles, four answers, two of them nowhere. A four-bar with its crank held at 52° is two circles: the coupler pin is 3.5 from the crank pin and 3 from the far ground pivot. Two quadratics in two unknowns, so Bézout's number is four — and the tracker finds two. The other two paths run off to infinity, and they do so for every pair of circles ever drawn: two circles meet the line at infinity in the same two points, and those are what the fourth and third answers are.

Two circles, four answers

A four-bar with its crank held still is two circles, and two circles meet twice. Bézout's theorem says four. The two missing answers are not a rounding error and are not special to these link lengths — they are the same two points for every pair of circles ever drawn, and they are the beginning of a way of counting that has so far been done by hand.

algebra · Algebra 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 A defect that draws nothing wrong. The same four-bar swept 360 times with the pre-correction Jacobian and with the corrected one, at eight coupler-point offsets. Corrected, every position is reached at every offset. Uncorrected: 5 of the offsets lose nothing at all, and then it loses 58, 159, 267 of 360. The picture was never wrong — a refused position is simply not drawn — so the only symptom was a sweep with fewer frames in it than it asked for.

The solver was refusing a quarter of the sweep

Four numbers in this site's Jacobian had the wrong sign, from the foundation phase until now. Every picture it ever drew was correct, because a wrong derivative does not move a converged answer — it just makes Newton crawl, until the stall rule declares the position unreachable. The symptom was a sweep quietly returning fewer frames than it asked for, and no gate in the fleet has a rule against that.

wrong · Misconception 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.

metrology · Parameter

Named alongside it

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

CalibrationIdentifiableMeasurement residualIdentification jacobianLeast-squaresLoop closureSingular valueAssembly branchAssembly-modeBezoutBranch ambiguityCircular points

All concepts