Machine Kinematics
solved, then drawn
Concept

Parallelogram — where it appears

A four-bar whose opposite links are equal, so that its coupler translates without turning. It is the standard example of an overconstrained mechanism and of one that changes branch at a configuration where all four links line up.

Named by 17 essays across 9 fields — each of them below, with the objects they name alongside it.

Grashof's classification, predicted and then swept. Four sets of link lengths. For each, Grashof's condition predicts from the lengths alone whether the input can rotate a full turn, and the solver then attempts all 180 positions and reports how many assembled. The prediction and the measurement agree in every case, which is what licenses quoting the classification for a mechanism nobody has swept.

Grashof, predicted and then swept

Add the shortest link to the longest. If the total does not exceed the other two, some link can turn a full revolution. It is a sentence about four numbers, it was published in 1883, and it is the kind of claim these essays refuse to print without measuring — so every linkage here is also asked for all 360 positions and required to agree.

linkages · Fourbar Three parallel bars, and a formula that says this cannot move. Five links and six pin joints, so Grübler's criterion gives 3(5−1) − 2(6) = 0 and calls it a structure. The Jacobian has rank 5 against 6 free coordinates, so it measures one degree of freedom — and the sweep assembles 59 of 60 positions, which settles the matter. The third bar removes no freedom because its constraint is already implied by the other two, and a formula that counts joints cannot notice that they happen to be parallel. Mechanisms of exactly this kind carry drafting machines, anglepoise lamps and locomotive coupling rods, where the redundant bar is there for load sharing and for keeping the linkage out of its change point.

The mechanism Grübler says cannot move

Three parallel bars between two frames. Five links, six pins, and the criterion every engineering course teaches gives zero degrees of freedom — a structure. It is a mechanism, it is in drafting machines and locomotive coupling rods, and the formula cannot see why.

constraint · Mobility 3 four-bars, one coupler curve. Three different four-bars, with different ground pivots, different link lengths and different proportions — 0.692 and 0.799 times the size of the first. Every one of them draws this same curve. Roberts's theorem says there are always exactly three, and the construction is one complex multiplication: write the coupler point as λ = (P − A)/(B − A), put the third fixed pivot at O₂ + λ(O₄ − O₂), and the other two linkages fall out with their bars' roles permuted — what is a coupler in one is a crank in another. The curves here were traced separately, each from its own solver runs, and agree to 1.3e-5 against a sampling resolution of 1.4e-5 — which is to say, as closely as the comparison can tell.

Three linkages, one curve

Every coupler curve is drawn by three different four-bars, not one. The other two can be constructed from the first with a single complex multiplication, they have different proportions and different ground pivots, and the roles of their bars are permuted — what is a coupler in one is a crank in another.

synthesis · Synthesis Where a tolerance stack-up stops meaning anything. The ratio between the first-order tolerance estimate and the band measured by building every corner linkage, for two four-bars at ±0.002 on each length. The crank-rocker's ratio is 1 at all 180 positions — a stack-up is exactly right for it, everywhere. The parallelogram is a change-point linkage, where all four bars can lie on one line, and at that position the estimate exceeds the measurement by a factor of 4.8e+5. The difference is not in the arithmetic, which is identical; it is that a derivative describes a map that can be inverted, and at a change point the map cannot.

Where a stack-up stops working

On a crank-rocker the first-order tolerance estimate matches the measured band at every one of 180 positions, to eight parts in ten thousand. On a parallelogram it exceeds it by a factor of 475,512. Same arithmetic, same tolerance, two linkages that differ only in their proportions — and nothing in the calculation says which one it is being run on.

practice · Tolerance Three bars, and no rotation left. Each of a delta robot's legs ends in a parallelogram, which keeps the bar on the platform parallel to the bar on the arm. That leg therefore permits the platform no turn about either direction perpendicular to its bar: it imposes two couple constraints, drawn here as rings about the directions they act on. Three legs impose 6; together they span only 3, so 3 are redundant. What is reciprocal to them is three couples — three pure translations, and the platform cannot turn at all.

Why the platform stays flat

A delta robot has three legs and three freedoms, and there is no obvious reason those freedoms should be the three translations rather than some mixture. The reason is a parallelogram in each leg, and the argument from there to "the platform cannot turn at all" is a constraint computation that takes six wrenches and a rank.

parallel · Parallel A parallelogram's sextic is a circle and a quartic. The parallelogram with ground 4, crank 1.5, coupler 4, rocker 1.5, coupler point at u = 0.45, v = 0.50. Its eliminated sextic divided by the circle of radius 1.5000 about (1.800, 2.000) leaves a remainder of 6.3 × 10⁻¹⁶ of its largest coefficient, and the quotient is a quartic whose leading form is exactly a multiple of (x² + y²)². Of the configurations solved across a full turn on both assemblies, 720 are on the circle — the machine as a parallelogram, coupler parallel to the ground — and 720 on the quartic, the machine crossed. The two factors meet at four finite points: 2 are change points, marked solid, where one configuration belongs to both; the other 2 are places the two drawings merely cross.

A sextic that comes apart

A parallelogram chain's coupler curve is not one curve. Its sextic divides exactly by a circle, leaving a quartic, and each factor is one of the two things the machine can do. The division leaves rounding and nothing else, a coupler one millionth too long leaves a remainder a million times larger than rounding, and the two factors meet at the two places where the machine has to choose.

curves · Coupler What the clearance has to swallow. Bennett's linkage with its second length multiplied by 1 + δ, and the closure error the solver drives down to and then cannot improve on. The loop does not close at any δ tried, including one part in a million. But the gap is exactly proportional to δ — the ratio varies by 0.07% across four decades — with a measured constant of 0.507. Shared over 4 joints that is 0.127 δ of play per pin, so a linkage machined to one part in a thousand needs about 0.20 mm of clearance in a link of 1.6 m, or a hundredth of a millimetre in a link of 1.6 cm. That is an ordinary running fit, and it is why a mechanism that cannot be built is in every folding table.

Why a hinge works

A door hinge with three knuckles is overconstrained — three axes imposed where one would do, and exactly parallel is a condition no bored hole has ever met. It works because the misfit is 0.507 times the error and the play in each knuckle is larger than that. The mechanisms these essays called unbuildable are built every day, and the thing that builds them is the clearance that was already there.

practice · Overconstraint A parallelogram at its flat position, exact and built slightly wrong. Left, a parallelogram with ground 3, cranks 2 and coupler 3 lying flat, where its two assemblies meet; faintly, the two ways it can go on, drawn at 30°. Middle, the same linkage with its coupler 0.05 too long, drawn at the closest it can come to the flat position: 7.49° away on either side, so the input's circle is thick where the input can go and red across the 15.0° it can never enter. Right, the input crank 0.05 too short, at the flat position: its two assemblies put the output crank 43.76° apart, and they do not meet at any input angle.

A parallelogram a micron wrong

A parallelogram linkage sits exactly where two kinds of four-bar meet, so a parallelogram that has actually been made is always one of four other machines. Make one bar a micron wrong on a 300 mm frame and the input stops a tenth of a degree short of lying flat, or the output turns round there with an acceleration that grows as one over the square root of the error. A third bar turns the square root back into a misfit of one micron.

constraint · Mobility Where a fit's singular values fall away, on each motion of a parallelogram. The 28 singular values of the degree-six fit, largest first and relative to the largest, for points traced on the parallelogram's circle motion, on its quartic motion, and on both. On the circle motion the last 15 lie below a drop of 3.7 × 10¹³, from 0.232 to 6.2 × 10⁻¹⁵; on the quartic motion the last 6 lie below a drop of 3.0 × 10¹⁰, from 6.3 × 10⁻⁵ to 2.1 × 10⁻¹⁵; on both motions the last one lies below a drop of 2.5 × 10¹², from 3.1 × 10⁻⁴ to 1.2 × 10⁻¹⁶. A drop of ten decades or more is a null space that is exactly there: fifteen sextics vanish on a circle, six on a quartic, and one on both.

A null space of fifteen is not noise

Points traced on one motion of a parallelogram four-bar leave a degree-six fit with fifteen polynomials that vanish on them, behind a drop of thirteen decades: the circle times every quartic. Half an oval of an ordinary coupler curve leaves two to five, behind drops of two. The count is the same kind of number in both cases, and only the drop beside it says which one is algebra.

curves · Coupler Oldham's coupling: two slides between two offset shafts. An input hub on one axis and an output hub on a parallel axis 0.6 away, joined by a disc that slides in a slot across the input hub and carries a tongue across the output hub at 90° to the slot. At an input angle of 35° the disc has slid 0.491 along the slot and the output hub -0.344 along the tongue, and the output hub has turned through exactly the input's angle, because neither slide can change an orientation. The disc's centre, the dot, runs round the dashed circle of diameter 0.600 — the offset divided by sin 90° — and goes round it 2 times for each turn of the shafts.

A coupling that only translates

A coupling between two parallel, offset shafts turns its output at exactly the input's speed when, and only when, the relative motion of its two hubs contains no rotation — when it lies in the translation group. Oldham's two slides give that group by construction and so do two equal parallel cranks; a four-bar that is not a parallelogram gives the whole planar group and its output wanders by more than a radian. And Oldham's right angle is not what makes the ratio one: it is what makes the slides slide least.

pairs · Pair A two-arm lamp balanced at every pose. A two-arm desk lamp: a lower arm 0.4 m long pivoted on its base, an upper arm carrying a 0.8 kg head 0.35 m out. A parallelogram — the lower arm, a short crank at the base, a link parallel to the lower arm, and the upper arm's first few centimetres — keeps the base crank parallel to the upper arm, so the upper arm's spring can run from a post above the base pivot to that crank and see the upper arm's angle alone. The lower arm's spring runs from the same post to the lower arm. Both springs have no free length and stiffnesses matched to their arms' weights. At every pose the lamp's total potential energy is the same: here it differs from the pose with both arms upright by 3e-16 of the head's moment, and the joint torques are 0 and 0 N·m — rounding. Dragging swings the lamp.

The lamp that stays where it is put

A desk lamp that stays wherever it is left is balanced, not held: its springs store exactly the energy its weight gives up, at every pose. For one arm that takes a spring with no free length, fixed above the pivot, whose energy has a cos θ term that cancels the weight's to 10⁻¹⁵. A spring with a free length L₀ leaves an unbalance of exactly L₀/s of the weight's torque, and a stiffness a fraction ε off leaves 2ε of its moment. For two arms it takes a parallelogram: without one, the upper spring sees the angle between the arms and the lamp's energy varies by twice the head's moment; with one, both springs sit at the base and the lamp is flat everywhere.

applied · Fourbar A tilting head balanced with one parallelogram and two springs. A desk lamp with a lower arm 0.4 m long, an upper arm of 0.35 m and a 0.5 kg head that tilts on the upper arm's end, its weight 0.08 m from that wrist. A parallelogram in the lower arm keeps a bracket at the elbow upright. Four springs of no free length: the lower arm's from a post on the base, the upper arm's from the elbow's bracket, and two for the head — one from the top of the elbow's bracket to a point on the head, which sees the head's angle from the vertical, and one across the wrist from the upper arm, which cancels the term the first brings in the angle between upper arm and head. At this pose, lower arm 0°, upper arm 80°, head 130° from the upward vertical, the lamp's energy differs from the first pose's by 5e-15 of the head's moment, and the torque at the wrist is 9e-10 N·m. Dragging moves all three joints.

Two springs stand in for a parallelogram

A lamp's head tilts on the end of its arm, and its weight's moment is a cosine of the head's angle from the vertical, so a spring can balance it only if the spring sees that angle. A second parallelogram carries the vertical out to the wrist and one spring does it. So do two springs with one parallelogram, or six with none: each spring that reaches across a joint brings a term in the joint's angle, and another spring across that joint takes it away. A spring across the wrist alone balances the head at one arm angle and leaves sin(Δ/2) of its moment over a range Δ.

applied · Fourbar Where the head spring hangs decides whether the arm's spring must push. The tilting-head desk lamp with one parallelogram: lower arm 0.4 m, upper arm 0.35 m, a 0.5 kg head with its weight 0.08 m beyond the wrist, and four springs of no free length. The spring from the top of the elbow's bracket to the head is hung 0.03 m along the head (the slider). Solved for an exact balance, its stiffness is 131 N/m, the wrist spring's 687, the lower arm's 561, and the upper arm's −434, negative: a spring that would have to push (drawn red). The upper arm's spring pulls exactly when the head attachment point is beyond 0.0615 m, 76.9% of the way to the head's centre of mass.

A spring that must push is hung in the wrong place

Balancing a lamp with a tilting head exactly, with springs of no free length, sometimes asks one spring for a negative stiffness, a spring that would have to push. It looked like a property of the lamp: a light upper arm under a heavy head might admit no design in which every spring pulls. It is a property of one attachment point. The upper arm's spring pulls exactly when the head spring meets the head beyond a point that is always short of the head's centre of mass, 77% of the way on this lamp, and at most 99% even with a head fifty times the arm's mass. Hang the head spring at its centre of mass and every spring pulls on every lamp.

applied · Fourbar The smallest springs that balance the lamp, and the one that is not needed. The tilting-head desk lamp with one parallelogram, its springs of no free length placed to make the total of their peak pulls over the working range least, inside an outline of brackets 0.15 m high, attachment points 0.08 m past the wrist and along the head's 0.15 m. Peak pulls 22.7, 0.0, 23.2, 17.5 N (lower arm, upper arm, elbow bracket to head, wrist), 63.4 N in all and 23.2 at most. The head spring meets the head at 0.0615 m, the threshold, and the upper arm's spring has no stiffness at all: it is left out.

The lamp needs three springs

A desk lamp with a tilting head, balanced exactly with springs of no free length, was built with four springs whose peak pulls add to 162 N. Searched over every attachment point inside the same outline, the least total is 63.4 N, and it is reached with three: the head spring hung at the threshold where the upper arm's spring stops being needed, 77% of the way to the head's centre of mass. Beyond the threshold a second spring is the first one split in two, and splitting a spring never makes it smaller. A design with no parallelogram and six springs is never close: 213 N at its best.

applied · Fourbar Re-chosen attachment points buy back a third of the tilt at four times the springs. The tilting-head lamp with one parallelogram and the smallest springs of the earlier essay. Every spring 2 cm free and every attachment point re-chosen for the least worst moment, all springs still pulling: 0.0611 N·m, a third less, with peak pulls 29.6, 0.0, 41.6, 195.1 N (lower arm, upper arm, elbow bracket to head, wrist), 266.3 N in all. The upper arm's spring is dotted where it has no stiffness.

A free length is absorbed where the springs attach

Give every spring of the smallest balanced desk lamp 2 cm of free length and refit the stiffnesses, and the lamp is left with 0.093 N·m of unbalance, growing in exact proportion to the free length. The spare fourth spring that ideal springs had no use for recovers under three per cent of it, and moving the head spring beyond the threshold only makes it worse. What absorbs a free length is a spring whose length barely changes. Re-chosen attachment points buy back a third of the unbalance that way, and pay for it with four times the pull.

applied · Fourbar The smallest springs that balance the lamp, and the one that is not needed. The tilting-head desk lamp with one parallelogram, its springs of no free length placed to make the total of their peak pulls over the working range least, inside an outline of brackets 0.15 m high, attachment points 0.08 m past the wrist and along the head's 0.15 m. Peak pulls 22.7, 0.0, 23.2, 17.5 N (lower arm, upper arm, elbow bracket to head, wrist), 63.4 N in all and 23.2 at most. The head spring meets the head at 0.0615 m, the threshold, and the upper arm's spring has no stiffness at all: it is left out.

Six things a balancing spring is not

More springs share the load, a spring that must push wants a compression spring, a stiffer spring balances better, a refitted stiffness repairs a free length, least squares finds the balance, and the best spring is one of no free length. Six things said about springs that balance a mechanism, each the natural reading of something true, each answered by a number the essays on the desk lamp and the rocking slide measured. They fail the same way. Each treats the spring as the thing to be sized, and a balance is decided first by where the spring is hung and how its error is judged.

wrong · Misconception Allowed 100 newtons, the lamp spends them on its attachment points. The tilting-head lamp with one parallelogram, every spring 2 cm free, its attachment points chosen inside the earlier outline for the least worst moment on any joint with the springs' peak pulls capped at 100 N in total. Worst moment 0.0620 N·m; peak pulls 22.5, 28.7, 13.9, 34.9 N (lower arm, upper arm, elbow bracket to head, wrist), 100.0 N in all. Each spring's zigzag is drawn heavier the harder it pulls, and dotted where it pulls less than half a newton.

The first pull goes to the spring the lamp did without

Give every spring of the balanced desk lamp 2 cm of free length and let it have more pull, and the worst moment it is left with falls from 0.086 N·m at 62 N to 0.052 at 227 N, the most any attachment points inside the outline can do. Half of that improvement costs the first 18 newtons; the second half costs the next 147. Spent on stiffness alone, with the attachment points held, extra pull buys nothing at all. And the first place it goes is the upper arm's spring, the spare that ideal springs had no use for, hung lower on its bracket while the head spring moves out past the threshold. The wrist spring, which carried almost all of the earlier re-attached design's cost, takes the largest share only after 140 N.

applied · Fourbar

Named alongside it

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

Four-barChange pointDesign ruleParallelogram linkageRedundant constraintThresholdToleranceApproximationConstraintCoupler curveGrashof's conditionJacobian

All concepts