A shake travels as a root of the play

A shared load is a shared motion found that two legs of three revolute joints each, holding one platform, share a twist exactly when their joint axes all cross one line, the line of a wrench both legs can carry. Pushed along that twist, the platform moves at first order and no further: the two legs’ nearest poses drift apart in proportion to the square of the push. That is a shake. A shake has an order found what the square is made of, a Lie bracket of the legs’ joint screws read against the shared wrench. Turning one joint of the second leg about the shared line until that number is nought raises the gap to the cube of the push. Turning a second joint raises it to the fourth power.

That essay measured the gap and stopped at its exponent. Real joints have play, and a gap smaller than the play is a gap the joints simply absorb. So a shaking platform in real joints is a mechanism over some range, and the question the essay left is how large a range. Its natural guess was that a shake of order $k$, in joints with play $c$, moves about $c^{1/k}$ before the play runs out.

The guess is right. What it hides is a constant, and how much the exponent is worth.

Three pairs, one shared line #

The three pairs measured are the earlier essay’s. The first leg has its three joints at fixed places, and the shared line is a transversal crossing all three. The second leg is built through the same line, so the two share one wrench along it and one twist. Left as built, the pair shakes at second order. Turning the second leg’s first joint about the shared line by 36.49° keeps it crossing the line and so keeps the shared twist, and at that angle the bracket that sets the square is nought: the pair shakes at third order. Turning the first joint to 50.77° and the second to 23.62° nulls the cube’s coefficient as well: fourth order.

Nothing about the three pairs looks different. Each has six joint axes crossing one line, and the tuned pairs differ from the untuned one only in two angles that no drawing would single out. That is why the earlier essay could find the tuning only by computing a bracket, and why the difference in travel below is invisible in the hardware. The pairs differ in the order of their shake, and only measurement of how far they move shows it.

The gap lies along one wrench #

The push test is the same as the earlier essays’. One joint of the first leg is held at a push $\varepsilon$ along the shared twist, the other five joint angles of both legs are solved by Gauss–Newton to bring the two legs’ platform poses as close as they will go, and the difference left is the gap. The gap is a small displacement of the platform, six numbers in general. Here it is effectively one.

The five free joint angles can close any small displacement except one kind, the kind the two legs together cannot produce, which is a motion along the one wrench they share. So the gap that survives the solve lies along that wrench. Measured on all three pairs at every push up to a tenth, its component along the shared wrench is its whole size to two parts in a thousand. That is what makes the play’s job simple to state. The play does not have to absorb a six-dimensional error, only a single number, the gap’s component along the shared wrench.

What the joints’ play can take #

A revolute joint with radial play $c$ lets its outer link shift by up to $c$ in any direction square to its axis. Why a hinge works treated exactly this play as the hinge’s own small extra freedoms. A shift $\delta$ does work on the shared wrench equal to the part of the wrench’s force along $\delta$. The most any one joint can contribute is $c$ times the part of the force square to its axis, and all six joints together can close a gap of $c\,G$ along the wrench, with

$$G = \sum_j |\mathbf f \times \mathbf u_j|,$$

$\mathbf f$ the shared wrench’s force direction and $\mathbf u_j$ the joints’ axes.

On these legs the shares are between 0.42 and 0.91, and $G$ is 4.52. The capacity is the same for all three pairs to the last digit. That is not a coincidence of the numbers. Tuning turns joints about the shared line, the wrench’s force runs along that line, and turning an axis about a line keeps its angle to it. A joint whose axis ran along the shared line would contribute nothing, and none of these does, since every joint crosses the line at an angle.

The travel is a root #

With the gap’s component along the wrench $a\,\varepsilon^k$ for a pair of order $k$, the pair moves until that equals what the play can close:

$$a\,\varepsilon^k = c\,G \quad\Longrightarrow\quad \varepsilon = \left(\frac{cG}{a}\right)^{1/k}.$$

That is the earlier essay’s guess with its constant. The constant $a$ is the gap’s coefficient, read from the smallest pushes: 0.026 for the untuned pair, 0.015 with one joint tuned and 0.012 with two.

The law was checked against the thing it claims to predict. For each pair and each play from $10^{-8}$ to $10^{-4}$, the travel was found by bisection on the push, every trial a full solve of the two legs and a fresh measurement of the gap. Wherever the travel stays below a tenth, the law matches the bisection to 3%. The slopes on a logarithmic plot are 0.501, 0.332 and 0.243 against the law’s half, third and quarter. The fourth-order pair drifts a little from its line at the largest plays, where its travel reaches four-tenths and the gap’s higher terms join in.

The numbers are not small. With a play of a millionth in every joint, where the legs are of order one, the untuned pair travels 0.013 along its twist, thirteen thousand times the play. The pair with one tuned joint travels 0.066 and the pair with two 0.139. A platform that the first-order count calls rigid and the second-order count calls a shake is, in real joints, a mechanism with a definite range, and the range is set by a root of the play rather than by the play.

The constant comes from a drawing #

The constant $a$ for the untuned pair need not be measured at all. A shake has an order found the gap’s second-order coefficient to be the reciprocal product of the shared wrench with half the difference of the two legs’ bracket terms, computed from the joint screws alone with no push. On this pair that bracket is 0.02621, and the push test’s coefficient is 0.02621. So for a pair shaking at second order the whole travel law can be read off its joint axes:

$$\varepsilon = \sqrt{\frac{c\,G}{|\sigma_2|}},$$

with $G$ from the axes’ angles to the shared line and $\sigma_2$ from their brackets. A designer handed two legs and a play can say how far the platform will move without solving anything. For the tuned pairs the bracket is nought by construction, and their cube and fourth-power coefficients were measured from the push test. The earlier essay found no closed form for them.

A tuned joint is worth more in tighter joints #

Tuning lowers the gap’s coefficient a little, from 0.026 to 0.012, and raises its exponent, and the exponent is what matters. Because the travel is a root of the play, the advantage of a higher order is a ratio of roots, and it grows as the play shrinks.

At a play of $10^{-4}$, one tuned joint buys 2.3 times the untuned pair’s travel and two buy 3.2 times. At $10^{-6}$ the factors are 5.0 and 10.6. At $10^{-8}$ they are 10.9 and 34. In loose joints the three pairs move by similar amounts, and it hardly matters which order the shake is. In tight joints the untuned pair barely moves while the doubly tuned one moves almost as if it were a mechanism.

That reverses the obvious intuition about precision. It might seem that a precisely made pair, with very small play, is the one that behaves most like its first-order model, rigid but for its shake. It is, for an untuned pair, whose travel falls as the square root of the play. For a pair tuned to third or fourth order the travel falls much more slowly, so in precise joints the tuning is what decides whether the pair moves at all.

What the numbers mean in a machine #

The push is measured as the size of the shared twist, and for these legs the twist is mostly a turn. Of a unit twist, 0.87 is rotation about its screw axis and the rest a slide along and about it. So a travel of 0.013 is a turn of about two-thirds of a degree, with a slide that goes with it. The legs are about one and a half units long, so a play of a millionth is a micrometre on legs of a metre and a half, the play of a good plain bearing.

Read that way, the untuned pair with micrometre joints turns freely through two-thirds of a degree about its shaking screw. The pair with one tuned joint turns through more than three degrees, the pair with two through nearly seven. With joints ten times tighter, a tenth of a micrometre over the same legs, the untuned pair’s turn falls to a fifth of a degree while the doubly tuned pair’s falls only to under four degrees. None of these would be called rigid by anyone who picked the platform up.

Where the travel goes #

The travel is measured along the shared twist, the screw the two legs’ freedoms have in common. For the legs here that twist is a screw motion of the platform about an axis crossing the shared line. A third leg that keeps the shake found that a third leg crossing the same line keeps the same twist and the same square. So a three-legged platform built that way travels as the untuned pair does, and tuning it means tuning two of its three leg pairs at once.

What the travel is not is the platform’s motion in any other direction. Every other direction is held to first order by the joints, and the play lets the platform rattle in those directions only by about the play itself. The shaking direction is the only one in which a millionth of play becomes a hundredth of motion. That is the practical signature of a shake. A platform that seems tight in every direction but one, and loose in that one by far more than its joints should allow, is shaking at some order. The ratio of its looseness to its joints’ play says which. A platform loose by the square root of its play, measured at two plays, is shaking at second order; one whose looseness barely falls as the joints are tightened is shaking at a higher order, and the slope of looseness against play on a logarithmic plot is one over the order. That makes the order a measurable property of a built platform, not only of its drawing.

The same root elsewhere on the site #

The square root is not peculiar to legs. It moves to first order and not at all found the simplest object with a shake, a joint between two bars in line, which the rank says can move straight up and which stretches both bars when it does. There, too, the walk gets as far as the tolerance allows and no further, and the distance is the square root of the tolerance over the obstruction’s size. That is this essay’s law at $k = 2$, with the bars’ stretch in place of the joints’ play. The same essay named a case its own field did not reach: a flex whose obstruction vanishes at second order and is blocked at third, which would walk further than a tolerance-limited crawl and much less than it was asked. The pair with one tuned joint is that case, built, and its travel is the cube root the earlier essay’s argument implies.

A piano hinge is not forty door hinges is the opposite case, where play absorbs a misfit that is fixed rather than one that grows with the motion. A hinge’s redundant knuckles bind unless their bore errors are smaller than the play in their pins, and the comparison there is linear, error against play. A shake’s misfit is not fixed. It starts at nothing and grows as a power of how far the platform has gone, so the play is used up gradually, and a high power uses it up late.

That is the whole difference between the cases, and it explains why order matters so much more than the coefficient. Halving the coefficient of an untuned pair’s gap buys travel in proportion to the square root of two. Raising the order by one changes the root, and in tight joints that is worth a factor of five or ten.

What the play does not settle #

Play is not only radial. A real revolute joint also lets its outer link tilt, by the radial play over the bearing’s length, and slide along its axis by an end play. Tilts add capacity in proportion to the joints’ distances from the wrench’s line, and end play adds it along the axes. Both would raise $G$, and neither is included.

Elastic legs. The links are rigid. Elastic links absorb a gap too, in proportion to the force that closing it needs, and a stiff platform with play behaves between the play’s root law and the elastic links’ linear one.

What happens at the end of the travel. The law says where the play runs out, not what the platform does there. Past that push the gap has to be closed by forcing the joints and links, and the platform’s resistance rises steeply but from nothing, as a stop’s does when a stop aimed at its pole holds by curvature. How stiff the platform is just past its travel, and how that stiffness depends on the order, is not measured.

Which way the play is taken up. The capacity assumes every joint’s play is free to shift whichever way closes the gap. A loaded platform has its joints pressed one way already, and only the slack side of each joint’s play is available. That could halve the travel, or more, and depends on the load.

Still open: a third tuned joint #

Each tuned joint raised the order by one and cost one condition. The pair with two tuned joints shakes at fourth order, its coefficient is still about 0.012, and its travel at a millionth of play is already 0.14.

The distinct argument would be the fifth order and what it costs. A third joint turned about the shared line gives a third free angle, and the fourth-order coefficient could be driven to nought with it. The gap would then be the fifth power of the push and the travel the fifth root of the play, which at a millionth is $10^{-6/5}$ times a constant, sixty times the square root of a millionth. The measurement would find that third angle, confirm the fifth-power gap, and ask whether two legs of three joints can be tuned to every order. If they can, they are a mechanism, which is what two wrists are. If the conditions run out at some order, that order is the most a pair of three-joint legs can be made to move without being redesigned.