A shared load is a shared motion

A leg of three revolute joints lets the platform on its end move by a three-dimensional family of twists and resists a three-dimensional family of wrenches. The lines a leg turns about and the lines it is pushed along put a picture on both. The revolute axes the leg permits and the lines of force it holds are the two rulings of one hyperboloid, and every line of one ruling meets every line of the other. A force whose line crosses all three hinge axes loads the leg without turning any of them.

That essay stopped at two legs. A platform held by two such legs can move only by what both permit, which legs intersect established for groups and which holds for screw systems in the same form: the platform’s instantaneous freedom is the intersection of the two legs’ freedoms. It carries a load along two paths at once exactly when both legs resist it. The question left open was what a coincidence between the two legs’ surfaces means. If the legs share a line, is it a load path both carry, or an axis the platform can turn about?

For two legs of three joints each, the answer is that it is always both, and the reason has no geometry in it.

Three and three make six #

Stack the six wrenches the two legs resist. The platform can move by a twist exactly when that twist does no work on any of them, so the number of independent twists both legs permit is six less the rank of the stack. The number of wrenches the two legs share, the loads either leg could carry alone, is how far the rank falls short of the six wrenches stacked. Both counts are “six minus the rank”, so they are equal. For two three-joint legs, the number of shared loads equals the number of freedoms, whatever the legs look like.

This is the index theorem a constraint that has been said already found on frameworks, written for a platform: freedoms minus redundancies equals the count, and the count here is 6 − 3 − 3 = 0. A single leg of three joints, or a pair whose counts do not balance, would not tie the two numbers together. A pair that does balance cannot gain one without the other.

The same count governs a platform that everyone has met. A Gough–Stewart platform stands on six legs, each a strut with a ball joint at either end, and each strut can push on the platform only along its own line. That is six wrenches, one per leg, in six dimensions, so its count also balances. Six legs and a square root found its forward problem hard and its inverse problem easy. The index says what happens at its singular poses. When the six leg lines fall into a special arrangement and their six forces lose a rank, the platform gains a motion that no leg can resist, and in the same act the leg forces become indeterminate, because a set of loads in the six legs then balances with no external load at all. A flight simulator’s singularity and a self-stress are one event, for the reason two three-joint legs share their loads and their freedoms.

Two legs in general position share nothing. Across 2,000 pairs of legs placed at random, not one had a shared wrench or a shared twist. The smallest singular value of any stack, the measure of how nearly six wrenches fail to span six dimensions, was 1.2 × 10⁻⁴, and the typical value is of order a tenth. A shared line is therefore something a designer does deliberately or by the accident of a symmetric layout. Random placement does not produce one.

There is one arrangement in which sharing is automatic, and it is the familiar one. A wrist is a leg whose three joints pass through one point. Every line of force it holds passes through that point, because a force through the centre has no moment about any of the three axes. Two wrists at different centres therefore share exactly one line of force, the line through both centres. By the count they share exactly one twist, and it is a turn about that same line. A platform held at two ball joints spins about the line between them, as a door turns about the line through its two hinge pins. The census row for two wrists finds the shared wrench and the shared twist on the line through the centres to 2 × 10⁻¹⁵, the twist with pitch nought.

Building a pair that shares a line of force #

The construction follows from the earlier essay’s test. Take the leg the earlier essays’ three-system figures are drawn about and pick one of its lines of force: a line crossing all three of its joint axes, found as the intersection of the plane through a point of the first axis and the second axis with the plane through that point and the third. Then build a second leg whose three joints each pass through a point of that line, in directions of their own. A force along the line now passes through all six hinges and turns none of them, so both legs resist it.

The measurement confirms that the stack of six wrenches loses exactly one rank. The one wrench both legs resist, extracted independently from the reciprocal of both legs’ freedoms, is the constructed line with pitch nought. It agrees to 1 × 10⁻¹⁵ in position and direction. That is the load path.

The platform also gains exactly one freedom. The twist both legs permit is not the line of force, and it is not a pure turn. It is a screw of pitch −0.049 about an axis passing 0.032 from the shared line at 57° to it: the dashed line in the opening figure.

The two are related, and the relation is the same reciprocity that built the rulings. The shared twist does no work on anything either leg resists, including the shared line of force. A force along a line does no work on a screw motion of pitch h exactly when h cos φ = d sin φ, where d is the distance between the two lines and φ the angle between them. The moment of the force about the screw’s axis must cancel the force’s component along the slide.

Two hundred constructed pairs, each with a random first leg and a random line of its force, gained exactly one freedom every time. Every one satisfies the relation to 4 × 10⁻¹⁵ across six decades of d sin φ. None of the shared twists is a pure turn. Their pitches run from −1.31 to +1.51, and the nearest to nought is 2.6 × 10⁻⁵. A zero pitch would have needed the twist’s axis to meet the shared line, and nothing in the construction asks for that.

So a designer who routes a load through both legs of a two-legged platform gives the platform a screw motion, and can say where it is before computing anything else. Its axis lies near the shared line or is nearly parallel to it, and its pitch is what the relation fixes.

Sharing an axis instead #

The construction runs the other way. Choose a line the first leg can turn about, a member of its freedom’s ruling with pitch nought, and not one of its own joints. Then build the second leg with that line as one of its joints. Both legs now permit a turn about it, and the platform has a freedom that is a pure rotation about a known axis.

The count forces the matching redundancy, and the measurement finds it: one wrench both legs resist, of pitch 0.112, on an axis 0.298 from the shared axis at 20°. The shared axis is recovered from the two legs’ freedoms to 5 × 10⁻¹⁶, and the stack of wrenches again has rank five.

So the question has no either-or answer. A shared line of force brings a screw motion, and a shared axis brings a shared load that is not a pure force: it is a wrench with a couple about its own line. Each is the other’s reciprocal partner, placed by the same relation. A designer who wanted only one of them, the redundant load path for stiffness or the pivot for motion, gets both.

First order and no further #

The platform can move along the shared twist, but only to first order. A twist both legs permit is an instantaneous motion. As the platform moves, each leg’s freedom changes with its joint angles, and a finite motion needs the two freedoms to go on sharing a twist all the way along. It moves to first order and not at all met the planar version: two bars in line leave a vertical freedom in the rank, and lifting the joint stretches both bars.

The test here is direct. Push the platform a distance ε along the shared twist by holding one joint of the first leg where that push puts it, then solve the other five joints — two of the first leg and all three of the second — so that the two legs’ platform poses agree as nearly as they can. A finite motion would leave a gap of nothing. A rigid platform, pushed along a twist the legs do not share, leaves a gap in proportion to ε, because the legs part at once. A first-order motion should leave a gap in proportion to $\varepsilon^2$.

Over three decades of push the gap for the shared line of force falls with slope 1.98, reaching 7.6 × 10⁻⁹ at a push of 3 × 10⁻⁴. The shared axis gives 1.98 as well. The refusal, two legs in general position pushed along one of the first leg’s joints, gives 1.01. The shared twist is a real first-order freedom, and the second order takes it away. The platform can be moved by a small amount against no stiffness from the linkage, and it returns: its stiffness along that twist comes from the second-order term and whatever the joints and links add by their own compliance, not from the geometry at first order.

The two wrists are the contrast that makes the slope mean something. Their shared twist is a finite motion. Pushed along it by any amount, they leave a gap of no more than 1.7 × 10⁻¹⁶, which is rounding, because each wrist permits every rotation about its own centre and a rotation about the line through both centres keeps both centres where they are. The difference between the wrists and the constructed pair is not in the count or in the first-order picture, where both have one shared line and one shared twist. It is that the wrists keep sharing as they move. Their shared line is the line through two fixed points, and it stays where it was. The constructed pair’s shared line of force was a coincidence of six hinge positions, and the first push moves the hinges off it.

That is worse, in one respect, than either reading the earlier essay offered. A mechanism has a motion that can be driven and is honest about it. A redundant structure has a load path and stays rigid. A two-legged platform with a shared line of force is a structure along every direction but one, and along that one it is a shake: a direction in which its stiffness is set by elastic deformation and not by geometry. Twelve bars and a symmetry showed what it takes to make such a freedom finite, a symmetry holding the whole framework on the special set. Nothing in this construction supplies one.

How near a pair is to a freedom #

A shared line is an exact coincidence, and a built platform has none exactly. What it has is a pair of legs whose six wrenches nearly fail to span six dimensions. The earlier essay proposed measuring that by the distance between the two legs’ rulings, and the cleanest version is the smallest singular value of the stacked wrenches.

Moving one of the second leg’s joints off the shared line by a distance δ restores the full rank, and the platform is rigid again. The smallest singular value of the stack grows in proportion, with slope 0.994 over five and a half decades and a coefficient of about 0.23. So the conditioning number has a picture attached to it: how far one hinge misses the line the other leg’s hinges all cross. A platform whose legs are laid out so that one line nearly crosses all six hinges is nearly shaky along the reciprocal twist, and the nearness is a length.

The practical reading goes further than the singular value. If every constraint were equally stiff, the platform’s stiffness along the nearly shared twist would be proportional to the square of that smallest singular value, because a stiffness is a quadratic form in the constraint directions. A hinge that misses the shared line by a tenth of the leg’s size gives a singular value of about 0.023, and a platform about a two-thousandth as stiff along that twist as along a direction whose singular value is near one. The stiffness was not computed here; the scaling is what the singular value implies and is stated as such.

A third leg takes the motion and keeps the loads #

Add a third leg in general position to the pair that shares a line of force. Nine wrenches now stand in six dimensions. The stack regains full rank, so the platform loses its freedom, and three of the nine wrenches are redundant, as three legs of three joints must have whatever their arrangement.

The count no longer ties redundancy to freedom, because nine is more than six. The index is 6 − 9 = −3, so three legs carry three redundant load paths as a matter of course, and a freedom now needs a coincidence of a different order. A twist must be permitted by all three legs at once. The two-legged pair already provides a candidate, its one shared twist, and a third leg makes the platform shaky exactly when its freedom also contains that twist.

The difference in how these arrangements are designed follows from this. Two-legged platforms are rare because they are balanced on the edge the count describes: every coincidence a designer adds is a load path and a freedom together. Three-legged platforms, the kind the problem swaps ends describes, can be given shared load lines freely, because the third leg absorbs the freedom each one would bring, and they become mechanisms only by the stronger coincidence.

What is not settled #

The shared twist’s second-order behaviour is measured on the pairs drawn, and the slope of two is the generic case, not a theorem about every pair. A pair built with more symmetry, such as a second leg that is the first leg’s mirror image in a plane through the shared line, could share the twist to a higher order or all the way. The census here drew its second legs at random through the line and saw nothing of the kind.

The stiffness claim is a scaling from the singular value, not a stiffness computed with compliant joints, and its constant is not given.

Still open: when a third leg’s freedom holds the shared twist #

With three legs, the first coincidence, two legs sharing a line of force, costs nothing: the third leg covers it. The platform becomes shaky only if the third leg’s freedom also contains the one twist the first two share. That twist is a screw of known pitch and axis, and a three-joint leg’s freedom contains a screw of non-zero pitch only in a particular arrangement of its joints.

The distinct argument there would be that arrangement: given the first pair’s shared screw, which three-joint legs contain it, and how large a family of them there is. It would also check whether a third leg built so will always share a line of force with each of the first two as well. The count says that a platform with one freedom on three such legs carries four redundant wrenches instead of three. Whether the extra one is a pure force along a line, as the pair’s own was, is not known. The measurement would be the same census with a third leg, the three pairwise intersections and the triple one, and the push test on the result.