What a leg of three joints leaves free

The two-system is a surface, and that essay ends by naming what it did not build:

The one that matters most is the three-system, because a leg of three joints spans one and almost every parallel mechanism in the next field has legs of three or more. What a three-system’s screws sweep is a quadric rather than a cylindroid, and the classification of which quadric is what tells a designer what a leg leaves free. The order of those systems and their reciprocals is computed here, and neither is drawn.

That was five essays ago. This draws one.

The object #

Three revolute joints in series between a base and a platform. Each contributes one screw — a zero-pitch screw about its own axis — and the platform’s freedom is everything in their span: a three-system, a three-dimensional subspace of the six-dimensional space of screws.

What the platform cannot do is everything reciprocal to that span, which is another three-system, of wrenches. One matrix gives both, and this field established that four essays ago. What it did not establish is what either of them looks like.

Six numbers, and then it is finished #

A three-system has a normal form, and the classical statement of it is short.

There are three principal screws. Their axes are mutually perpendicular and meet at one point — the centre of the system. Their pitches $h_1 \le h_2 \le h_3$ are the roots of $\det(A - hM) = 0$, where $A$ is the reciprocal Gram matrix of any basis and $M$ the angular one. And then every screw of the system is priced by its direction alone:

$$h = h_1 l^2 + h_2 m^2 + h_3 n^2,$$

with $(l, m, n)$ the direction cosines of its axis in the principal frame.

Six numbers — three pitches and a frame — stand in for a two-parameter family of screws with a pitch attached to each. That is what a normal form is for, and it is worth stating why this one matters: a designer asking what a leg leaves free is asking for those six numbers, and nothing else about the three joints enters.

Measuring it, and what the measurement had to avoid #

The pitches are a generalised eigenproblem and they are solved as one, through $M$’s Cholesky factor and a symmetric eigendecomposition, rather than by expanding a cubic. A cubic’s roots are a well-conditioned problem made ill-conditioned by hand, and there is a standing finding about exactly that: conditioning beats truncation.

The claims are then checked on the leg drawn above, and there are four of them.

The three principal axes are mutually perpendicular. Worst cosine between any pair: $1.1 \times 10^{-16}$.

They meet at one point. The three lines are fitted to a common point by least squares and the worst distance from it is $2.6 \times 10^{-16}$. Three lines in space generically miss each other entirely, so this is a real claim rather than a formality.

The pitch law holds. And this is the one that makes the whole thing a normal form rather than a change of basis that happens to look tidy.

The eigenproblem was handed an orthonormalised basis of the span. It has no idea which three screws generated it, and no idea that any of the two hundred and forty test screws exist. That the law prices all of them is the statement being tested.

And the frame is basis-independent. Re-running the whole computation from a randomly recombined basis of the same span gives the same three pitches, which is the claim that the six numbers belong to the system rather than to how it was written down.

What a pitch is, and why one is negative #

The pitches quoted above run from −0.377 to +0.808 on the drawn leg, and a negative pitch deserves a sentence because it is not a small quantity or an error.

A screw’s pitch is the ratio of the translation to the rotation in the motion it generates: turn by an angle and slide by pitch times that angle, along the same axis. It has the units of length and it carries a sign, which says whether the slide is along the rotation’s own direction or against it — right-handed or left-handed, in the language of a thread.

So a three-system with pitches of mixed sign contains right-handed screws, left-handed ones, and the pure rotations between them. That is the generic case and it is why the zero-pitch cone below is non-empty: a family containing both handednesses must contain the boundary.

A system with three pitches of the same sign would be a family of screws that all thread the same way and contain no revolute axis at all. Such systems exist, and no leg of three revolute joints can span one, for the reason the cone section gives: the leg’s own axes are zero-pitch members of its own span.

The lines the system contains, and why a designer wants them #

A screw of pitch nought is a pure rotation about a line. So the zero-pitch members of a system are the revolute axes it contains — and the pitch law makes them trivial to find:

$$h_1 l^2 + h_2 m^2 + h_3 n^2 = 0$$

is a quadratic cone of directions, and it is non-empty exactly when the extreme pitches straddle zero.

They always do, for a leg of revolutes. The system was spanned by three zero-pitch screws, so it contains them, so the pitch law must vanish somewhere — and the measurement confirms it from the other direction: the leg’s own three joint axes come out with pitch $1.4 \times 10^{-16}$ in the system they generate.

That cone is a useful outcome, and it is a design statement rather than a classification one — with a limit that has to be stated exactly, because an earlier version of this essay stated it wrongly.

A direction off the cone cannot be a joint axis of this leg at any position. That is the negative, and it holds without further computation: a designer with a packaging constraint that forces a joint’s direction can check it against the cone before working out anything else.

The positive is narrower than the picture suggests. A three-system with no pure translation holds exactly one screw with each angular part, so each direction on the cone belongs to one revolute axis at one position — and that axis does not in general pass through the centre. The lines drawn above are directions carried to the centre for the eye, not joint positions: a revolute joint placed on one of those lines lies outside the system. The alternative joint axes are therefore a one-parameter family, one line per direction, and the surface those lines rule is the hyperboloid a leg’s freedom and its constraint share, whose asymptotic cone is the cone drawn here.

The size of the difference is not small. On the leg drawn here the true zero-pitch axes miss the system’s centre by between 0.165 and 0.551, on a leg whose joints are about one unit apart, and a revolute joint placed on one of the cone’s lines through the centre lies outside the leg’s freedom by at least 0.109 of the screw — so a joint moved there would change what the platform can do. The distinction between a direction and a line is exactly the one the direction-only checks above cannot see: every one of them tests an angle, and none places a joint and asks whether its screw is in the span.

So a designer may move any joint of this leg to another line of that family and the platform’s freedom is unchanged — same three-system, same principal pitches, same reciprocal wrench system, same constraint, a different machine. A rank says a leg leaves three things free; the family says which other legs leave the same three. And because the family is one-parameter, the choice is a single number per joint — where along the ruling to put it — which is a design space a drawing can show and a designer can search by hand.

Two degeneracies, and they are the mechanisms people build #

The normal form has cases where it collapses, and in this subject the collapsed cases are not edge cases — they are the arrangements that get manufactured.

Three axes through one point. All three principal pitches come out exactly zero. That is a spherical three-system: every screw in it is a pure rotation about a line through the point, so the platform can turn about anything through that point and translate about nothing. It is a wrist, and the reason every wrist is built that way is that its freedom is a whole three-system of rotations rather than three rotations.

Three parallel axes. The computation is refused, by name: the system contains a pure translation, so $M$ is singular, one principal pitch is infinite and there is no finite centre. That is a planar three-system — two translations and a rotation — and it is the freedom of every planar mechanism here. The refusal is the right answer, because the normal form above genuinely does not apply: a system with an infinite pitch has a different one.

Both are worth having as refusals rather than as special cases handled quietly. A function that returned a large number for the parallel case would have been reporting a finite centre for a system that has none — and this field has a recorded finding about exactly that shape of failure one order down, where a two-system of two intersecting axes produced eight screws with a pitch of NaN and an axis at the origin, and was found only because somebody wrote a test whose “generic” example happened to be degenerate.

The third degeneracy is the one that is not a mechanism, and it is worth naming so the list is complete. Two of the three pitches equal gives a system with an axis of symmetry: the pitch law depends only on the angle to one direction, so the screws of a given pitch lie on a cone of revolution rather than on a general quadric. It happens at isolated arrangements of the three axes, it is not refused because nothing about it is singular, and it is the case a classification has to name separately.

What this says about a parallel platform #

The reason the two-system essay named the three-system as the one that matters is the field next door, where a platform is carried by several legs at once and its freedom is the intersection of what its legs leave free.

That intersection is now computable in the normal form rather than only as a rank. Two three-systems in six-dimensional space generically meet in nothing at all — three plus three is six — so a platform on two three-joint legs is generically rigid, and the platforms that move do so because their legs’ systems meet in a subspace they generically would not. Which is the same statement two ways to be overconstrained makes about loops, arriving from the leg’s side.

Three legs of three joints each leave nine screws’ worth of freedom and the platform gets the intersection of three three-systems, which is generically empty and in practice is not — a Stewart platform’s six legs leave six freedoms because each leg is six joints rather than three, and a three-legged platform with three joints a leg is a structure that somebody arranged to move.

What the cone adds to that is the design lever. If a platform’s legs’ systems must intersect in a stated subspace, each leg’s own joints can be moved to any line of its own family of alternative axes without disturbing the intersection at all. The condition is on the systems and the joints are free inside them, which is a much larger design space than moving joints and re-checking a rank.

That is a claim about what the instrument is for rather than a measurement, and the platform-side computation that would use it is not built. Naming it is the honest form of an essay that has produced a tool and not yet the thing the tool is for.

Where the classification would go #

Hunt’s enumeration of the three-systems is the substantial classical result here, and it is not built here. What it builds is the instrument the enumeration is a statement about, so it is worth saying what the enumeration adds.

The three pitches, up to an overall scale and a relabelling, are two numbers — and the classification is a partition of that two-dimensional space into cases: all three pitches distinct and of mixed sign, two equal, one zero, one infinite, and so on. Each case has a name and a characteristic set of mechanisms. What a designer gets from it is a lookup: measure the three pitches of a candidate leg, find the case, and read off what family the leg belongs to.

The lookup is worth more than it sounds because it is finite. What two bodies can be free to do, in the sense a three-joint leg decides, is not a continuum of possibilities but a short list of types with continuous parameters inside each. That is the same shape of result as the enumeration of lower pairs — six and no others — one order up, and it is the reason type synthesis is a subject with answers.

What this argument cost, and what it reused #

It is worth recording what had to be built, because the answer is less than it looks, and the reason is what was already to hand.

The generalised eigenproblem is new: a Cholesky factor of the angular Gram matrix, a similarity transform, and a symmetric eigendecomposition. The concurrency of three lines is new, as a least-squares fit to a common point with the residual reported. The zero-pitch cone is new.

Everything else was already here. The screw of a revolute joint, the reciprocal product, the span of a set of screws, the axis and pitch of a screw taken apart and rebuilt, the order of a system read off a rank with a gap in it — all of that has been in place since one of this field’s early essays and all of it is used unchanged.

One piece was nearly worked out twice. The eigendecomposition was first done by hand, because the two-system version of the same problem handles order two only and a three-by-three case looked like a different problem. It is not: the symmetric eigendecomposition every rank in these essays is read off does the three-by-three case unchanged, and doing it twice would have been two answers to one question, agreeing until the day they did not.

That is the ordinary experience of extending a field that has been built carefully: the new mathematics is a page and the temptation to re-implement the old mathematics is the expensive part.

Still open: the quadric, and the reciprocal’s own frame #

Two things are named here and not drawn, and both are one step away.

The pitch quadric. The screws of a given pitch $p$ in a three-system have axes lying on a quadric surface — a hyperboloid for most pitches, degenerating at the three principal values — and that surface is the three-system’s answer to the cylindroid. The pitch law gives it immediately as $h_1 l^2 + h_2 m^2 + h_3 n^2 = p$ on the direction sphere, but a surface needs the axes’ positions as well as their directions, and the position of the axis of a given screw of the system is a second computation that is not done here. The zero-pitch cone above is not the $p = 0$ member of that family but its asymptotic cone: the axes of pitch nought do not pass through the centre, and the surface they rule is a hyperboloid that approaches the cone far from it.

The reciprocal system’s own normal form. A three-system’s reciprocal is a three-system of wrenches, and it has its own three principal screws with their own pitches. The classical relation between the two sets is clean — the axes coincide and the pitches change sign — and the reciprocal system’s order is computed at every loop here while its principal screws never are. That is the same debt this essay has just paid, one column over in the same table.