Machine Kinematics
solved, then drawn
Concept

Worst case — where it appears

Combining tolerances by adding their magnitudes, which assumes every one is at its limit and pulling the same way. It always bounds the real spread and usually overstates it badly, so this site reports it beside the band the mechanism actually produces rather than instead of it.

Named by 18 essays across 6 fields — each of them below, with the objects they name alongside it.

The network where counting works. 5 scissor units, each two bars pinned at their middles, with consecutive units sharing their end pins. Sixteen bodies and twenty-two pins at eight units, and Grübler's 3(n − 1) − 2j gives 4 — three rigid motions and one internal freedom — against a measured nullity of 4 and no redundant constraints at all. That holds at one unit and at sixteen. The tong is here to make the field's point in the direction nobody expects it: a network is not a place where counting fails, it is a place where counting stops being checkable by hand, and one of the two assemblies that spans this plate is counted perfectly. Span 8.1388 mm at this opening, which is 5 times one unit's 1.6278. positioned by solving, not by drawing.

One input at one end

A lazy tong's reach is 2nL cos θ — exactly, to a part in a million million, over eight sizes and sixty openings — so it multiplies its input by the number of units. It multiplies everything else by the same number, including the part of the drawing nobody wanted multiplied: a unit cut a hundredth of a radian out puts a tong of thirty-two units 0.374 out at the far end.

networks · Network 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 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 square a robot thinks it drove, and the two it drove. A differential drive with wheels one per cent apart in radius, sent round a four-metre square twice — once clockwise and once anticlockwise. What it believes is the square; what it did closes 1.60 m out one way and 1.40 m out the other, and the errors point different ways. A wrong track width instead gives 0.177 m and 0.177 m — the same both ways round. That difference is why the test is run in both directions: one run cannot tell the two errors apart and two runs can.

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.

practice · 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 The same multiplier, applied to the error. A tong whose units are cut to an angle 0.01 radians away from the drawing. If one unit is out, the span is out by that unit's share and nothing more; if every unit is out the same way — which is what a machine setting or a worn tool produces — the error is multiplied by the unit count, exactly, to 7.6e-14. The third column is what would happen if the errors were independent and equally likely either way: the accumulation goes as the square root of the count instead, and the difference between the two columns at thirty-two units is a factor of 5.66. Which column applies is a question about how the parts were made, not about the mechanism.

The error that is repeated

Thirty-two units cut on one setting of one machine are thirty-two copies of one error, not thirty-two draws from a distribution — so a tong's span is out by thirty-two times a unit's, not by the square root of thirty-two times it. The two estimates differ by a factor of 5.66, and the second one is the comforting one.

practice · Tolerance 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 a length comes from. A coupler 3.5 units long is a part with two holes in it, and neither hole's position is the length. Each carries an error of 0.010, of which 60% is common to both because they were bored in one setup — the whole pattern shifts by that much and the distance between the holes does not change. What survives is the independent part, 0.0063 at each hole, combining to 0.0089 on the length. A drawing that tolerances the length at ±0.010 is describing a part nobody makes, and it is out by a factor of 0.894 — optimistic below a shared fraction of one half and pessimistic above it.

Where a length comes from

A coupler 3.5 units long is a part with two holes in it, and neither hole's position is the length. What reaches the length is only the part of each hole's error the two do not share — so a drawing that tolerances the length is describing a part nobody makes, and is out by a factor of √2 in one direction or by everything in the other.

metrology · Parameter A slide spends one ball of every pair. A 6-ball Rzeppa joint bent 20°, at the symmetric position, with its cage slid along its own normal by 0 of the window clearance c and then tilted to allow the largest lag. Each column is one ball's window, from −c to +c, and the dot is where the ball sits in it at that lag: −1.00, 1.00, −1.00, 1.00, −1.00, 1.00 c. The lag is 0.3500 c. Opposite balls lie on one line through the joint's centre, so a lag or a tilt moves them opposite ways and a slide moves them the same way: whatever the slide gives one ball of a pair it takes from the other, and the lag falls as 1 − s/c.

A floating cage frees three balls and not six

A Rzeppa cage that may slide along its own normal as well as tilt has a third freedom, and counting constraints against freedoms says a six-ball joint should then lose its hold on the lag altogether. It loses nothing: a slide moves both balls of an opposite pair the same way, so it spends one ball's clearance for every one it frees, and the six-ball lag stays at 0.350 c. A three-ball joint, whose balls have no opposites, loses its bound outright, because three ball centres always lie in some plane. Offset grooves then add a second limit, set by the worst ball alone, which takes over from the cage only past an offset of about 0.35 times the tracks' play over the windows'.

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 Past the play, a caged joint shares its load and a grooved one leans on one ball. Two six-ball Rzeppa joints bent 20°, at an input angle of 15°, each twisted 0.03 of a window clearance past its free lag, with every contact a spring of the same stiffness and the cage and balls settled where the stored energy is least. Bars: each ball's share of the total contact load. Left, grooves offset 0.1, where the cage sets the play: 0.17, 0.20, 0.19, 0.16, 0.14, 0.15, 5.91 balls' worth. Right, grooves offset 0.7, where the grooves do: 0.00, 1.00, 0.00, 0.00, 0.00, 0.00, 1.00 balls' worth. Dotted: an even sixth.

A tight groove leans on one ball

A six-ball Rzeppa joint's free play is set by the tighter of its cage and its offset grooves, and the two limits look alike from outside. They are not alike once the play is taken up. With every contact made a spring of the same stiffness and the joint twisted past its free lag, a joint whose cage binds spreads the first load across four to six balls' worth and, a third of a window clearance on, gives every ball between 0.17 and 0.19 of it. A joint whose grooves bind puts the first load on one ball alone over most of a ball pitch, and two clearances on it still spreads it over fewer than six. At the same torque its hardest-pressed ball carries 1.5 to 3.0 times the caged joint's. Tightening the tracks to cut the play buys the play with uneven loading.

pairs · Pair 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 Tighten the windows and the cage's limit drops below the grooves'. The six-ball Rzeppa joint bent 20° with its grooves offset 0.7 and a track play of one: across a ball pitch, the free lag the grooves alone allow (solid) and the cage alone allows with windows of clearance 0.5 (dashed). The joint's play is the smaller of the two. The cage is the limit wherever its curve is the lower, here at every input angle; it becomes the limit everywhere once the windows' clearance is below 0.559 of the track play. The largest play over the pitch is 0.163.

A tight cage spreads the load until the tracks join in

A Rzeppa joint whose offset grooves set its play loads one ball first, and at equal torque its hardest-pressed ball carries up to three times a joint whose cage sets the play. Tighten the cage's windows below 0.559 of the track play and the cage becomes the limit at every input angle: the first load spreads over 3.6 balls' worth or more everywhere, the play falls, and at low torque the worst ball carries no more than a caged joint's. The cage does not stay in charge. Once the twist reaches the gap between the two limits the tracks engage, unevenly, and the worst ball climbs back towards the untightened joint's. A tighter cage widens the gap and pushes that torque up, so the windows' clearance is a choice about the torque at which even loading should end.

pairs · Pair Which feature to hold tight. A budget of 0.020 divided between the four lengths in inverse proportion to their sensitivities, and the same budget divided between the features those lengths are derived from — a frame jig-bored in one setup at 0.90 shared, a crank drilled twice at 0.10, a coupler at 0.60 and a rocker at 0.30. The tightest tolerance moves from b to a. And the null result matters as much: with one process for the whole machine the two allocations are identical to the last digit, because the transmission factor is then common and divides out. Feature-based tolerancing changes the answer when the parts are made differently and not merely when they are made.

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.

metrology · Parameter 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.

ToleranceSensitivityClearanceRoot-sum-squareConstant velocityDerived lengthFeature toleranceSetup errorStack-upSymmetryBacklashCalibration

All concepts