A third leg that keeps the shake

A shared load is a shared motion put two legs of three revolute joints under one platform and let them share a line of force, so that both legs carry the same load. It found that the platform gains a motion in the same act — a twist both legs permit, a screw about a different line — because the twists two such legs share and the wrenches they share are equal in number: three constraints and three make six. Pushed along the shared twist, the two legs part as the square of the push, so the platform shakes rather than moves. And a third leg in general position removes the shake, since its freedom does not contain the twist, while keeping the redundancy.

That essay ended on the case it had not built: a third leg whose freedom does contain the shared twist. Its question was which legs those are, whether such a leg must also share the first two legs’ line of force, and whether the fourth redundant wrench the count predicts is a pure force like the first. This essay builds those legs and measures what they share.

Building a leg around a given twist #

A leg of three revolute joints permits exactly the motions spanned by its three joint axes, each a line — a screw of pitch nought — and resists the reciprocal system of wrenches that do no work on any of them. What a leg of three joints leaves free drew that span as a three-system with its own principal screws. For the leg’s freedom to contain a given screw $S$, $S$ must be a combination of the three joint lines:

$$S = a\,L_1 + b\,L_2 + L_3.$$

That is a construction, not only a test. Choose two joint lines $L_1$ and $L_2$ anywhere, and a number $a$; then $L_3 = S - aL_1 - bL_2$ is a line exactly when its pitch form — the dot product of its angular and linear parts — is nought. That form is quadratic in $a$ and $b$ together, but $L_1$ and $L_2$ are lines, so their own pitch forms vanish and it is linear in $b$ once $a$ is fixed:

$$Q(S) - a\,\langle S, L_1\rangle - b\,\langle S, L_2\rangle + a b\,\langle L_1, L_2\rangle = 0,$$

with $\langle\,,\rangle$ the reciprocal product. One division gives $b$, and the third joint follows. So every choice of two lines and one number gives a third leg containing $S$. A free leg has twelve numbers in it, four for each joint line; this family has nine. Containing a given screw is three conditions, which is the count a three-dimensional freedom in a six-dimensional space should give.

The construction is checked on what it produces. For 200 third legs built this way from random joint lines, the distance from $S$ to the span of the leg’s three joint screws is at most $9 \times 10^{-15}$ of $S$’s length — the twist is in the freedom to rounding.

One freedom kept, four redundancies #

Put a third leg built this way under the platform with the first two. The platform’s freedoms are the twists all three legs permit, and its redundancies are how many of the nine wrenches the legs resist are more than six can span.

For every one of the 200 legs the platform keeps exactly one freedom — the shared twist — and has four redundant wrenches. A third leg placed anywhere else leaves the platform rigid with three: nine wrenches spanning six dimensions. The one extra redundancy is the price of the kept freedom, and it is forced by the same count as before: the nine wrenches span only five dimensions when one twist is reciprocal to all of them.

The count that cannot tell the difference #

The mobility count has no way to see any of this, and it is worth saying exactly why. A three-legged platform on three-revolute legs has eight links counting the base — the platform, two in each leg and the ground — and nine one-freedom joints, so the Kutzbach count in space gives $6(8 - 1 - 9) + 9 = -3$. The number Kutzbach is missing is the correction: the true mobility is the count plus the number of redundant constraints, and the count is really the platform’s freedoms minus its redundancies.

With a third leg in general position that is $0 - 3 = -3$. With a third leg that keeps the shake it is $1 - 4 = -3$. The count is identical because the kept freedom and the extra redundancy arrive together, which is the same equality of shared twists and shared wrenches that the first essay found for two legs, now appearing across all three. No formula built from how many links and joints a platform has can distinguish the two platforms; only the screws can.

That is also the sense in which the shake is one of two ways to be overconstrained. Both platforms are overconstrained by the count. One is overconstrained and rigid; the other is overconstrained and movable to first order, with the extra constraint the price of the extra freedom — and in both, the thing left over after the counting is done is a screw whose pitch says what kind of object it is.

A third leg must share a load with each of the first two #

The first essay’s counting applies to any two three-joint legs, not only to the first pair: the twists two legs share and the wrenches they share are equal in number. A third leg that contains $S$ shares $S$ with the first leg and with the second. So it must share a wrench with each.

It does, every time: in all 200 cases the third leg shares exactly one wrench with the first leg and exactly one with the second. The first essay’s question was whether those shared wrenches are the pair’s shared line of force — whether a third leg that keeps the shake automatically joins the load path the first two already share.

They are not. Not one of the 400 shared wrenches is the pair’s line, and not one is a line at all. Their pitches run from −0.83 to 2.45; the smallest in size is $1.2 \times 10^{-3}$; every one of their axes is at least 0.032 from the shared line. And no wrench is shared by all three legs. What a third leg containing the twist shares with each of the first two is a screw — a force along an axis together with a couple about it — and a different screw with each.

A pitched wrench is a real load path, but a different kind from a line of force. A force along a line passes through a joint axis it meets without doing work on it, which is why a leg whose three joints all meet a line resists a force along it. A wrench of pitch $h$ is a force along its axis together with a couple of $h$ times that force about the same axis, and a leg resists it only if the force’s moment and the couple cancel about every joint axis — which is what reciprocity says. The third leg and the first leg can pass such a wrench between them through the platform in either proportion, and the platform feels nothing. It is redundancy in exactly the sense the first essay counted, carried by a screw rather than a line.

So the answer to the first essay’s question is no. The fourth redundancy is not a pure force, and a leg that keeps the platform’s motion is not, as a rule, a leg that joins its shared load path.

Unless the third leg is built through the line #

There is one way to make the third leg share the line, and the construction above shows why it works. Build the third leg’s first two joints through points of the pair’s shared line $\ell$. Two lines that meet are reciprocal, so $\langle L_1, \ell\rangle = \langle L_2, \ell\rangle = 0$; and $S$, being a twist both legs permit, is reciprocal to every wrench either resists, $\ell$ among them. Then $\langle L_3, \ell\rangle = \langle S,\ell\rangle - a\langle L_1,\ell\rangle - b\langle L_2, \ell\rangle = 0$, and the solved third joint meets $\ell$ too — without being asked to.

So a third leg with two joints through $\ell$ has all three through it, $\ell$ does no work on any of its joints, and the leg resists a force along $\ell$. For 60 legs built that way, both shared wrenches are $\ell$ itself — pitch nought to $4 \times 10^{-13}$, axis on $\ell$ to $2 \times 10^{-6}$ — and all three legs share it. The redundancy count is still four; what has changed is that one line of force now runs through all three legs.

That is the construction the first essay’s second leg used, applied to the third: a leg through the shared line. What this essay adds is that it is the only way to get the shared line into the third leg, and that part needs no census: a leg resists a force along $\ell$ exactly when $\ell$ is reciprocal to all three of its joint lines, and two lines are reciprocal exactly when they meet or are parallel. So a third leg shares $\ell$ if and only if every one of its joints meets it. Containing the twist is not enough.

The law the shared wrench obeys #

The first essay found the shared twist tied to the shared line by $h\cos\phi = d\sin\phi$: a pure force does no work on a screw motion exactly when the motion’s pitch times the cosine of the angle between the axes equals the distance between them times its sine. With a wrench that carries a pitch of its own, the general law of reciprocity takes both pitches:

$$(h_t + h_w)\cos\phi = d\sin\phi.$$

Every one of the 200 pairs of twist and shared wrench lies on it, the reciprocal product at most $2 \times 10^{-15}$. The first essay’s law is the case $h_w = 0$. The third leg’s shared wrenches are free to have any pitch the law allows, and the law is then a relation between two pitches, a distance and an angle rather than a constraint on the twist alone. Which is to say the wrench a third leg shares can sit anywhere the reciprocity allows — anywhere on a large family of screws reciprocal to $S$ — and the particular one is set by where the third leg’s joints happen to be.

Still a shake #

Keeping the shared twist in the third leg’s freedom keeps it as a first-order freedom of the platform. Whether it becomes a motion — whether the platform can actually move along it — is the question the first essay’s push test answered for two legs, and it is answered the same way here.

Push the platform a distance $\varepsilon$ along the twist, hold one joint of the first leg where that push puts it, and solve the other eight joints of the three legs to bring their platform poses as close together as they will come. The gap left falls as $\varepsilon^{2}$ — a log–log slope of 2.00 — both for a third leg built in general and for one built through the line. A finite motion would leave nothing; the platform is movable to first order and no further, exactly as with two legs. A third leg placed anywhere refuses the push at slope 1.00, in proportion to it, which is what a rigid platform does.

So the third leg can keep the shake but cannot turn it into a motion. The contrast is with the linkages the spatial field has met that do move against their count: Bennett’s four-bar and Sarrus’s linkage are overconstrained by the count and move through a finite range, because their geometry satisfies a condition at every position of the motion rather than only at the start. That is not surprising — the second order was already refusing the motion with two legs, and a third leg adds constraints rather than removing any — but it rules out the hope that a well-chosen third leg might supply the missing second-order freedom.

How near a third leg is to keeping it #

The first essay measured how near two legs are to a freedom: move one joint of the second leg off the shared line by $\delta$ and the smallest singular value of the six wrenches grows in proportion to $\delta$. The same instrument applies one leg further on.

Move the third leg’s solved joint off its line by $\delta$, square to it, and the platform is rigid at every $\delta$ tried — the nine wrenches span six dimensions. But the smallest singular value of their stack grows with slope 1.00 on log axes, at about 0.45 times the miss. A third leg’s nearness to keeping the shake is a length, as the second leg’s nearness to creating it was, and it is measured in the units of the geometry.

What it means for a designer #

The chain of results through these essays is now long enough to state as a rule about three-legged platforms built from three-revolute legs.

A shared line of force in two legs brings a shared twist, a shake of the platform, whatever else is done (the earlier essay).

A third leg removes the shake unless its freedom contains the twist, and containing a given twist is three conditions — so a third leg chosen for other reasons almost never does, and one that does was built to.

A third leg that keeps the shake shares a pitched wrench with each of the first two, not the line they share. Only a third leg built through that line shares it, and then the line is a load path through all three legs.

For a designer who wants the redundancy without the shake, the second statement is the reassuring one: the bad case needs three coincidences. For one who wants a platform to be deliberately compliant along a twist — a flexure-like mechanism built from rigid joints — the third statement says what the third leg will carry while it cooperates.

What these legs do not cover #

Three revolute joints each. Every leg here is a chain of three pins. Legs with a slide, or with more joints, have freedoms of other dimensions, and the equal counting of shared twists and shared wrenches holds only for a pair whose constraint dimensions sum to six.

The family is sampled, not mapped. The nine-parameter family of third legs was sampled at 200 members from one distribution of joint lines. That none of them shares a pure force with either first leg is a finding about those 200; that none shares the pair’s own line unless all three joints meet it is proved above.

One pair. Every third leg in the census was built around the same first pair and the same shared twist. The pitches of the shared wrenches are a property of that twist as well as of the third leg, and a pair with a twist of different pitch would give a different spread.

First and second order. The push test establishes that the kept freedom is a shake. It does not measure how stiff the shake is in any mechanical sense, since the gap between two legs’ reachable poses is a geometric residual, not a stiffness.

Still open: four legs and the second order #

With three legs the second order still refuses the shared twist. A fourth leg could not supply a missing second-order freedom either — constraints only add — but it can be asked a sharper question: how many coincidences does it take for the second order to stop refusing?

The distinct argument would be the condition for the shared twist to be a finite motion rather than a shake: for two wrists it is — the first essay found two wrists turning about the line through their centres for as far as they are pushed — and for two legs through a shared line it is not. Somewhere between those constructions is the condition that makes the second-order residual vanish. The measurement would take the push test’s second-order coefficient as a function of the legs’ geometry, find the families of legs on which it is nought, and ask whether those families are exactly the ones in which every leg’s freedom contains the whole one-parameter motion rather than only its tangent — which would make the difference between a shake and a motion a statement about a curve of screws rather than about one.