The lines a leg turns about and the lines it is pushed along

A leg of three revolute joints between a base and a platform lets the platform move in a three-dimensional family of ways, and what a leg of three joints leaves free found that family’s normal form: three principal screws, mutually perpendicular, meeting at one centre, with pitches $h_1 \le h_2 \le h_3$ that price every other screw of the family by its direction alone.

The same leg also holds things. Any load on the platform that does no work on any of the motions the leg permits is carried by the leg’s structure without a joint turning, and those loads form a family of their own — the reciprocal three-system, the constraint that is the other half of every freedom. Its order has been computed wherever a leg or a loop has been analysed. Its principal screws have not, and the essay that found the freedom’s normal form closed by naming that debt and a second one beside it: the surface that the screws of a given pitch sweep.

The two debts turn out to be one object. The freedom and the constraint share a frame, their pitches are negatives of each other, and at pitch nought both families of lines lie on a single surface — a hyperboloid of one sheet whose two rulings are the axes the leg turns about and the lines along which it can be pushed. Measuring that also exposes a claim the earlier essay made about where a leg’s alternative joint axes lie, which was wrong, and the correction is the last section here.

Two computations that land on one frame #

The reciprocal system is computed from the leg’s joint screws with no reference to the freedom’s principal screws at all. Each joint’s twist is swapped end for end, so that the reciprocal product becomes an ordinary dot product, and the orthogonal complement of the three swapped twists is taken; that complement is a basis of three wrenches. That basis is then handed to the same generalised eigenproblem that found the freedom’s normal form — $\det(A - hM) = 0$, with $A$ the reciprocal Gram matrix of the basis and $M$ the angular one — and it returns three principal wrenches with three pitches.

The leg drawn above has three joint axes chosen to look like nothing in particular: one vertical, one tilted mostly along $x$, one mostly along $y$, at three different heights. Its freedom’s principal pitches are −0.3766, −0.0338 and +0.8075.

The constraint’s come out at +0.3766, +0.0338 and −0.8075.

That is not a relabelling. The wrench system’s centre is $5 \times 10^{-16}$ from the twist system’s, each wrench axis is parallel to one twist axis to rounding, and each wrench pitch is the negative of that twist axis’s pitch to $10^{-15}$. The check is written so it could have failed in a nearby way: if the reciprocal merely had the same three pitches in a different order, the pairs would disagree by as much as 1.6, twice the largest pitch.

Why the sign flips #

The reason is short, and it is worth having because it says the result is not a property of this leg.

The reciprocal product of a twist and a wrench is the rate at which the wrench does work on the motion. Refer both to the common principal frame. A twist along principal axis $i$ — a turn about a line with a slide along it, in the proportion its pitch sets — has angular part $e_i$ and linear part $h_i e_i$; a wrench along the same axis with pitch $k$ has force $e_i$ and moment $k e_i$. Their reciprocal product is $h_i + k$ — the moment working against the rotation plus the force working against the slide. For the wrench to do no work on that twist, $k = -h_i$.

That settles the three principal axes one at a time, and it also shows the pairing extends to every direction. A three-system with no pure translation contains exactly one screw with a given angular part, because the map from a screw of the system to its angular part is linear and invertible. So along every direction in space there is one twist the leg permits and one wrench it resists, and the pitch law $h = h_1 l^2 + h_2 m^2 + h_3 n^2$ read with every $h_i$ negated prices the wrench.

Two hundred random directions confirm it directly, with nothing borrowed from the principal frame: for each, the twist is solved out of the freedom’s basis and the wrench out of the constraint’s, each is taken apart into an axis and a pitch as a spatial body’s six freedoms allow any screw to be, and the two pitches sum to nought within $10^{-15}$.

So the six numbers that describe what a three-joint leg leaves free describe what it holds as well. A designer who has the frame and three pitches of a leg’s freedom has its constraint without computing anything further.

A force the leg holds is a line that meets all three joints #

The most useful members of the constraint are its zero-pitch wrenches — pure forces, loads with a line of action and no couple about it. The wrench pitch law vanishes on the same cone of directions as the twist pitch law does, because negating every $h_i$ does not move the set where their weighted sum is nought.

A pure force along a line does no work on a rotation about an axis exactly when the two lines are coplanar: when they meet, or are parallel. A force through a hinge has no moment about the hinge. So a force the leg can hold without any joint turning is a force whose line meets all three joint axes — and a line meeting three given skew lines is one of the oldest objects in geometry.

Taken from the reciprocal system and not constructed as transversals, eight such lines each pass within $10^{-15}$ of every joint axis, and none does more than $7 \times 10^{-16}$ of work on a unit rotation of any joint. The contrast row is the case that must fail: a line drawn through one point of the first joint and one point of the second meets those two exactly and passes 0.199 from the third, on which a unit force along it does 0.169 of work. The third joint turns, and the leg does not hold it.

That is also a practical instrument that needs no screw theory to use. To know whether a three-revolute leg can carry a given load line rigidly, draw the line and ask whether it crosses all three hinge axes. If it misses one, that hinge turns.

The two rulings of one hyperboloid #

Three mutually skew lines have a one-parameter family of lines meeting all three, and those transversals sweep a quadric surface. The three original lines also lie on it, and the surface carries a second one-parameter family of lines — the lines meeting every transversal — of which the three originals are members. A surface ruled twice by straight lines in this way is a hyperboloid of one sheet, and its two families are its rulings.

The leg makes both of those families concrete. One ruling is the revolute axes of its freedom, the zero-pitch twists; the other is the lines of force of its constraint, the zero-pitch wrenches. That is a stronger statement than “the forces meet the leg’s three joints”: it says every force line meets every possible alternative joint axis, and both families lie on one surface whose equation in the principal frame is

$$h_1 x^2 + h_2 y^2 + h_3 z^2 + h_1 h_2 h_3 = 0.$$

Sixteen lines of each family are solved from the two bases independently, and every line of the first comes within $2 \times 10^{-15}$ of every line of the second. Within a family the lines never meet: the closest drawn pair of revolute axes stays 0.012 apart — neighbours on a ruling sampled sixteen times round, so the number falls as more lines are drawn and never reaches nought. Every line of both families satisfies the surface equation at three points along it to $10^{-15}$. The dark line in the figure, dragged round its ruling, crosses the three grey joints at every stop, and the work it does on them stays at the rounding floor throughout.

The hyperboloid is the picture the three-system was missing. A two-system is a surface — its screws sweep a cylindroid, one line per pitch at each point along a nodal axis — and a three-system’s zero-pitch lines are one ruling of a hyperboloid whose other ruling is its constraint. Order two gives one surface per system; order three gives one surface per pitch, and at pitch nought freedom and constraint share it.

Every pitch has its quadric, and the constraint shares each one #

The equation above is the $p = 0$ member of a family. The axes of the twists of pitch $p$ in the system lie on

$$(h_1 - p)x^2 + (h_2 - p)y^2 + (h_3 - p)z^2 + (h_1 - p)(h_2 - p)(h_3 - p) = 0,$$

the pitch quadric. The classical form is not taken on trust here: sixty screws are taken at random from the leg’s freedom, each is decomposed into an axis and a pitch, and the quadric at that screw’s own pitch vanishes at three points along its axis to $2 \times 10^{-15}$. The $p = 0$ quadric, evaluated on the same axes, misses each of them by at least 0.79 of its pitch — a line crosses a wrong surface once, but it does not lie on it.

Negating every $h_i$ and $p$ together leaves that equation unchanged up to an overall sign. So the wrenches of pitch $-p$ in the constraint lie on the same surface as the twists of pitch $p$ in the freedom, for every $p$ at once.

The dial runs the pitch between the leg’s two extreme principal pitches. In that interval the cone $(h_1 - p)l^2 + (h_2 - p)m^2 + (h_3 - p)n^2 = 0$ is real, the surface is a hyperboloid of one sheet, and it is ruled by a family of twists and a family of wrenches; every drawn line sits on it to $10^{-16}$. Near either end the hyperboloid’s throat narrows and the lines crowd onto one principal axis, which is the only screw of the system with that extreme pitch. Past the ends the cone has no real directions, the system has no screw of that pitch, and the surface has no lines on it. The middle pitch is where the quadric passes from being ruled about one principal axis to being ruled about another, and the parameterisation of its lines has to change axis there too; one that does not draws only half of each family above it.

So the answer to the earlier essay’s question — what surface does a three-system sweep? — is a one-parameter family of quadrics, one per pitch, nested about one frame, and the constraint sweeps the same family with the labels reversed.

Where a leg’s alternative joints actually are #

The earlier essay stated that a designer could move any joint of a leg along the zero-pitch cone through the system’s centre and leave the platform’s freedom unchanged, and drew the cone’s generators as though they were those alternative joints. It also counted them as a two-parameter family.

Both statements were wrong, and the argument above shows why. The zero-pitch directions form that cone. But a system with no pure translation has exactly one screw with each direction, so each direction on the cone belongs to exactly one revolute axis, at one position — and those positions lie on the hyperboloid, not on the cone. The family of alternative joint axes is one-parameter, it is one ruling of the hyperboloid, and the cone through the centre is the surface’s asymptotic cone: the shape the rulings approach far from the centre and never reach.

The size of the mistake is measurable. On the leg drawn here each zero-pitch axis misses the centre by between 0.165 and 0.551, against joint separations of about one. A revolute screw placed on the cone’s own line through the centre lies outside the leg’s freedom by at least 0.109, as a fraction of the screw’s size; the same direction placed on the ruling lies inside it to $10^{-15}$. A joint moved “along the cone” as the earlier essay described would have changed the leg’s freedom, and the platform with it.

The earlier essay has been corrected, and the negative it stated survives intact: a direction off the cone cannot be a joint axis of the leg anywhere. It is the positive — where on a line of that direction the joint must go — that needed the hyperboloid.

It is worth recording how the error lasted. Every check in that essay was about directions: the principal axes perpendicular, the pitch law on directions, the leg’s own joint directions at pitch nought. None placed a revolute joint at a position and asked whether its screw belonged to the span, which is the one test that distinguishes a line from a direction. The figure drew directions carried to the centre and captioned them as lines. That is the shape of a defect the curvature field has met before — a construction whose every measured property is right and whose drawn object is not the object named.

What this changes for someone building a platform #

The constraint of a three-joint leg is read off its freedom. Same centre, same axes, pitches negated. A table of a leg’s three principal pitches is a table of what it holds.

A load line is held exactly when it crosses all three hinges. That test is exact, needs no screw arithmetic, and is how a designer checks a support path against a leg: a strut, a cable or a contact whose line of action passes through all three hinge axes loads the leg without turning any of them.

A joint may be moved to another line of its ruling. The family is one-parameter, the lines are skew to one another, and they are fixed by the three joints the leg started with: three skew lines determine the hyperboloid. Moving one joint to another line of the same ruling keeps the surface, keeps both systems, and keeps every line of force the leg held.

Two legs on one platform constrain it by the intersection of their freedoms. Legs intersect makes that statement about groups; for screw systems it is a statement about two hyperboloids. A platform on two three-joint legs is rigid unless the two freedoms share a screw, and a shared zero-pitch screw is a line lying on both surfaces — which, for two quadrics in general position, it does not. The problem swaps ends at the platform, and the rulings are where it swaps.

Degenerate legs are the manufactured ones, and the surface degenerates with them. Three joints through one point — a wrist — give three zero pitches, so $h_1 h_2 h_3 = 0$ and the hyperboloid collapses onto its own asymptotic cone: the wrist’s axes really do all pass through the centre, and there the earlier essay’s picture is right. Three parallel joints contain a pure translation and have no finite centre, and neither construction applies.

What is not settled #

The pitch quadrics are established on one leg in general position, by measurement on random screws and on sampled lines, together with a derivation for the principal axes. The classical statement is general; the checks here are one leg’s worth of evidence for it and one leg’s worth of evidence that the computations are right.

Hunt’s classification of three-systems — the cases by the signs and coincidences of the three pitches — is still not built. The rulings sharpen what it would add: in each case the surface family changes type, and the cases where a hyperboloid of one sheet becomes a cone, a pair of planes or nothing real are the cases in which a leg’s alternative joints change character.

Still open: the leg that is also a structure #

A three-joint leg’s constraint is a three-system with its own zero-pitch ruling, and a platform held by three such legs is rigid when the three constraints together span all six dimensions. A platform arranged so that one line of force is common to two legs’ constraints has a redundancy, and one arranged so that one revolute axis is common to all three legs’ freedoms has a motion.

Its distinct argument would be those coincidences read as incidences on the hyperboloids: when two legs’ surfaces share a line, what the shared line is for — a common load path that both legs carry, or a common axis the platform can turn about — and how many independent coincidences a three-legged platform can be given before it becomes a mechanism. Two things would come out of it. A condition a designer can draw rather than compute, stated as lines on surfaces; and the measured distance between two legs’ rulings as the platform’s nearness to gaining a freedom, which would be a conditioning number with a picture attached to it.