Machine Kinematics
solved, then drawn
Concept

Stack-up — where it appears

The combined effect of several dimensions' tolerances on one output quantity of the assembly they belong to. Adding the worst cases overstates it and adding the variances understates it, so the site computes the output band by solving the mechanism at the corners as well.

Named by 6 essays across 3 fields — each of them below, with the objects they name alongside it.

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 Whether the part goes in is one inequality. Two of the four contacts are moved and the other two left where the drawing says; the horizontal and vertical axes are those two errors, inward positive. The shaded region is where the part still goes in and the unshaded region is where it does not fit at all — not fits badly, not is located wrongly: there is no position and no orientation the part can take. The boundary is the straight line 0.250·e₁ + 0.250·e₂ = 0, whose coefficients are the shares from the previous figure. Four probe points are marked, each checked twice — once by the inequality and once by a linear program that looks for a pose and reports the program infeasible when there is none — and the two agree at every one. A hold turns a set of tolerances into a single condition, and the weights in it are what say which contact is worth making accurately.

Which contact to make accurately

A hold turns a set of contact tolerances into one linear inequality, and the weights in it are the coefficients of the combination that cancels — a quarter each on a square held by four, and 0.144 to 0.424 on a hexagon held by five. Above that line the part goes in and below it there is no pose it can take at all: not badly located, not out of position, no fit.

holding · Tolerance What a belt ratio's tolerance is made of. The three ways a variator's ratio can be wrong when the parts are wrong, at a nominal ratio of 1.848: a belt 0.3% long, a centre distance 0.2 mm out, and a sheave 0.15 mm from where it was commanded. Worst case they add to 2.98% and combined in quadrature they are 2.04% — the two conventions the practice field already argues about. The last row is a gear train's ratio under the same treatment, and it is empty: there is no length in a tooth count for a tolerance to be on.

The ratio that has a tolerance

Every dimension in this collection has been given a range at some point, and the ratios never were, because a gear ratio is a count and a count has no tolerance. A belt ratio is a quotient of two solved lengths, so every length in the mechanism is in it — and the amplification from belt length to ratio runs from 2.7 to 6.4 across the travel, which puts a whole per cent on a mechanism whose gearbox equivalent has none at all.

practice · Tolerance The same links and pins, and between 12 and 15 link lengths in the stack-up. Every closed loop in a mechanism is one equation a tolerance analysis has to satisfy, and the equation involves every link the loop passes through. All 16 chains here have the same number of independent loops — 3, which is pins minus links plus one and is fixed by the two totals — but not the same shortest set of them. The bars are the total length of a minimum cycle basis, and they run from 12 to 15. So the smallest number of link dimensions that any stack-up on this mechanism can involve is decided by the graph, before a single dimension has been chosen, and two topologies a count cannot tell apart differ by 3 of them.

Where the shortest loops are

A tolerance stack-up goes round a loop, and every link the loop passes through is a dimension in it. Two eight-link chains with the same links, the same pins and the same number of loops can need twelve link lengths in their shortest independent set or fifteen — decided by the graph, before any dimension is chosen.

practice · Tolerance One bar made 0.0001 too long, one bar at a time. Every bar of the machine compiled from a lemniscate lengthened by 0.0001 in turn, the machine re-solved, and the polynomial read at the tracing point. It is no longer zero anywhere. The worst bar takes it to 9.1e-3 — an amplification of 91 — and the median bar to 2.8e-4. The bars that matter are the reflectors, which are the cheapest part of the machine; the translators, which are most of it, barely move the answer at all. Size and fragility live in different parts.

Exactness a micron destroys

Lengthen one bar of a compiled machine by a ten-thousandth and its tracing point leaves the curve. On the smallest machine the error comes out smaller than it went in; on a fifty-bar one it comes out ninety times larger — and the bars that matter are the reflectors, which are the cheapest part of the machine.

practice · Tolerance A clearance that has a sign, over a tolerance box. The same machine at each of the sixteen corners of a ±0.02 band on its four lengths, with the smallest gap over a whole drive computed at each. The nominal machine clears by 0.1041; the worst corner clears by 0.0763 and the best by 0.1321, a band 0.0558 wide from a tolerance of 0.02 on each length. Every other quantity computed from the lengths becomes an interval when the lengths do. This one has a sign, and an interval that reaches zero is not a wider answer to the same question — it is a different answer, because on that side of it the parts do not go together. 24 random draws inside the box beat no corner, which is the check that the extremes are where they are assumed to be.

A clearance inside a tolerance box

Every quantity derived from the lengths becomes an interval when the lengths become ranges. This one has a sign, and an interval that reaches zero is not a wider answer to the same question — it is a different answer, because on that side of it the parts do not go together.

practice · Tolerance

Named alongside it

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

SensitivityToleranceWorst caseClearanceBelt driveCanonical formCompiled linkageConditioningContinuously variableCycle basisError accumulationError amplification

All concepts