Machine Kinematics
solved, then drawn
Concept

Freudenstein invariants — where it appears

The three coefficients of Freudenstein's relation, each a ratio of link lengths and so unchanged by scaling the machine. They are coordinates on the identifiable part of a four-bar's parameter space: three measured angle pairs determine them by a linear solve, and they determine the linkage's shape.

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

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.

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 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 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.

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 4.09e-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

Named alongside it

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

Scale invarianceCalibrationFreudenstein equationIdentifiableLeast-squaresGrashof's conditionIdentification jacobianMeasurement residualPrecision positionFunction generationMinimal parameterisationNoise amplification

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