Machine Kinematics
solved, then drawn
Concept

Root-sum-square — where it appears

Combining tolerances in quadrature, on the assumption that they are independent and not all at their limits at once. It gives a band roughly a third narrower than adding the magnitudes, and whether that assumption is safe is a question about how the parts were made rather than about the mechanism.

Named by 7 essays across 4 fields — each of them below, with the objects they name alongside it.

Two routes to the same derivative. How much the output angle moves when the coupler length moves, through one turn, computed twice. One route rebuilds the mechanism at b ± 10⁻⁶ and solves both from scratch; the other differentiates the constraint equations and solves one linear system against the analytic Jacobian. They lie on top of each other — the strip beneath plots the difference on a four-decade log scale, and its largest value anywhere in the turn is 2.9e-10 against a sensitivity of order 0.41. That is the only independent check there is of the Jacobian itself, whose coupler rows carried four wrong signs from the foundation phase to 2026-08-12 without ever drawing anything wrong.

A band with a direction in it

One whole direction of a four-bar's tolerance box does nothing. A machine made a quarter of a per cent too big all over has an output error of exactly zero — and an aluminium four-bar heated by a hundred degrees has an output error of exactly zero, while one with a steel frame has 0.076°.

practice · Tolerance Half a tooth, spent three ways. Each bar is one beat of the escape wheel: exactly half a tooth pitch, 6.0° on 30 teeth, whatever the faces are cut like. The dark part is the impulse, which is the only part that does anything to the pendulum; the pale part is the drop, in which nothing is touching anything; the short tail is the lock-in run, in which the arriving tooth drags the wheel backwards as it settles. On the arc with no draw that tail is exactly zero and the budget has two terms. On every other face it is not, and the three still sum to the half pitch to twelve figures — which is the check, since the three are computed from three different contacts.

What a drop cannot be smaller than

Two thirds of an escape wheel's travel is drop, and drop does nothing. The obvious economy is to cut it down, and it cannot be cut down, because every dimension it is made of has a tolerance and a drop smaller than the accumulated error is a tooth that does not clear the pallet it is leaving. The stack is 0.39°, and it barely moves when the tooth count triples.

practice · Tolerance Every joint's lever arm. The thin lines run from the tool to each joint's axis, meeting it square. Their lengths are what a radian of error at each joint costs the tool in metres — not a rule of thumb but the Jacobian column, which is ω × r and therefore that perpendicular exactly. The shortest of them belongs to the joint nearest the work and the longest to the joint furthest from it, which is why an arm's accuracy is decided at the shoulder and its resolution at the wrist.

Where an error at the shoulder ends up

The same angular error at every joint of an arm, and the tool is out by 0.156 mm because of the shoulder, 0.017 mm because of the wrist roll and exactly nothing because of the last joint. The numbers are not properties of the joints. Each one is the distance from the tool to that joint's axis, measurable off the drawing with a ruler.

serial · Tolerance Near its mirror, a double Cardan joint's lag is an ellipse in its two errors. A W double Cardan joint whose working angles average 15°, with a phasing error δ between its middle yokes across and a difference ε between its two working angles up, both in degrees. The curves are where the predicted lag spread over a turn, tan a·√((sin a·δ)² + ε²), is 0.05°, 0.1°, 0.15°, 0.2°. The dots are 35 joints solved from their pins, labelled with their measured spread in degrees; they meet the ellipse to 0.06% at worst. The ellipse is 3.86 times as wide as it is tall: at 15° a degree of phasing error does what 0.259° of mismatch between the angles does.

Two errors, one ellipse

A double Cardan joint keeps step only while a mirror fixes its motion, and any real one is a little off. Split each error by what the mirror does to it and only the half it reverses costs anything: a phasing error spreads the lag at sin a · tan a per degree, a difference between the working angles at tan a, and raising both angles together costs nothing at all. The two costs leave their marks an eighth of a turn apart, so they add as the sides of a right triangle — a root sum of squares that no assumption of independence was needed for.

pairs · Pair Six balls, two grooves each, and a cage. A Rzeppa joint bent 20°, input shaft along z and output shaft turned about x. Each ball lies where the plane of its inner-race groove, through the input's axis, meets the plane of its outer-race groove, through the output's axis; at equal input and output angles every ball is in the plane bisecting the two shafts, and the cage holds them there. Left, from the side: the two shafts and the bisecting plane edge-on, the balls on it. Right, along the cage's normal: the six balls round their circle, the plane of the bend drawn across it. At an input angle of 0° a cage clearance of c lets the output lag or lead by up to 0.350 c radians, c in ball-circle radii.

A Rzeppa cage adds its clearances

A Rzeppa joint keeps constant velocity because its cage holds every ball in the plane that bisects the two shafts. Give the cage's windows a clearance c and the output can lag or lead the input. Six balls might be expected to limit that lag through the worst of them, or as a root sum of squares. They do neither. Opposite balls are one constraint, the cage's two tilts spend two of the three, and what is left adds the three pairs' allowances like clearances in series: 0.350 c at a 20° bend, twice what the worst ball alone would allow. The lag falls to nought each time a ball crosses the plane of the bend, grows with the bend itself, and a four-ball joint cannot hold it at all.

pairs · Pair Where the two analyses cross. The band on the output angle at a crank angle of 57°, with every hole on every part given a position error of 0.010, against how much of that error each pair of holes shares. The flat line is what an analysis on the four lengths gives, which is the same number whatever the answer to that question. They cross at 0.53 and nowhere else: below it the lengths-only answer is optimistic, reaching 1.414× at holes located independently, and above it pessimistic, reaching zero when the error is entirely common and the distance between two holes is perfect however badly the pair is placed. The crossing is at one half because two holes contribute √2 and the surviving fraction is √(1 − shared).

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.

metrology · Parameter The same part, dimensioned two ways. Three holes at 0.00, 1.40, 3.50, each located to ±0.010 by the process. Dimensioned as a chain, each hole is placed from the last, so the two gaps are ±0.010 each and the errors accumulate over the whole span to ±0.0141. Dimensioned from a baseline, every hole is placed from H0, so the span is ±0.010 and the gap between H1 and H2 — which touches the datum at neither end — is ±0.0141. The two schemes are exactly √2 apart in opposite places. Nothing about the part decides this and nothing about the process does; the drawing does, and a tolerance analysis that starts from the lengths has already thrown the information away.

The same part, dimensioned twice

Three holes, one process, two drawings. Dimensioned as a chain the errors accumulate and the span is ±0.0141; dimensioned from a baseline the span is ±0.0100 and the gap between the second and third holes is ±0.0141. Exactly √2 apart, in opposite places, and nothing about the part or the process decides which — the drawing does.

metrology · Parameter

Named alongside it

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

ToleranceWorst caseSensitivityClearanceConstant velocityDerived lengthFeature toleranceSetup errorSymmetryBacklashBeatCalibration

All concepts