When the link lengths are angles

The universal joint has been on this site since the expansion phase, described as a closed loop of four revolute joints whose axes all pass through one point. Kutzbach counts it at minus two; it turns; and its output angle runs ahead of and behind its input in a way that has been quietly costing driveline engineers effort for two centuries.

It was introduced as a curiosity of the spatial count. It is not a curiosity. It is a member of a family — arguably the family, since it is the only one of the classical spatial linkages that requires no special condition at all — and this essay is about the family.

What is fixed and what is free #

Take four revolute axes and make them all pass through one point. Every joint then rotates about a line through that point; every link is a rigid body one of whose points is pinned to the centre; and the whole mechanism is confined to spherical motion.

That is a subgroup of the rigid displacements, and by the previous essays’ argument the consequences follow without further work. The screw system is the three-dimensional space of rotations about the centre, at every position, for every set of dimensions. The reciprocal is three pure forces through the centre. The redundancy is three, the mobility is one, and the count is wrong by exactly as much as the planar four-bar’s is.

The interesting part is what “link length” means now.

A link joins two joints. In the plane it is a bar of some length, and the length is the distance between the pins. Here the two joints are two lines through the centre, so what separates them is not a distance — it is an angle, the arc between the two axis directions measured on the unit sphere.

So a spherical four-bar has four arcs where a planar four-bar has four lengths, and everything downstream follows the same pattern with an angle in place of a length. That is not an analogy; it is a substitution that works, and the reason it works is that the sphere is a metric space in which the arcs play exactly the role lengths play in the plane.

Grashof, with sines #

The planar Grashof condition says that a four-bar has a link that rotates fully exactly when

$$s + l \le p + q$$

— the shortest plus the longest, against the other two. On this site that condition is not quoted; it is predicted from the four lengths and then measured by sweeping the crank through a full turn and asking whether every position assembled.

The spherical form is the same inequality on the arcs. Sorted:

$$s + l \le p + q, \qquad \text{arcs in degrees}$$

and the site does the same two things with it. The hero figure’s arcs sorted are 40, 80, 90, 100, so $s + l = 140$ against $p + q = 170$: the condition holds, the shortest arc should turn fully, and sweeping the input through 36 positions reaches all 36.

Change the arcs to 80, 60, 95, 70 — sorted 60, 70, 80, 95 — and $s + l = 155$ against $p + q = 150$. The condition fails. Swept, the input reaches 33 of 36 positions and refuses three of them, because there is no configuration of those four arcs with the input there.

spherical-linkage asserts the agreement of the two before it draws anything: the prediction from the arcs must match whether the sweep reached every position. It is the same assertion the planar field has carried since the foundation, moved to the sphere.

The one thing that must not drift #

A spherical linkage is defined by every axis passing through one point, and nothing in the solver knows that.

SpatialLoop closes a loop by Newton–Raphson on the logarithm of its closure transform. It has no concept of a sphere, a centre, or concurrency. It is handed four axes and told to close the loop, and if the solve wandered off the sphere it would return joint angles that closed a loop perfectly well while drawing a mechanism that is not the one asked for.

The figure would look fine. Four bars, four pins, moving plausibly. This is the failure mode the whole site is built against, arriving in a new place.

So assertSphericalStaysOnItsSphere measures it: over 36 positions, the perpendicular distance from the centre to every joint axis, worst case. It comes out at exactly zero — not near zero — because the loop is constructed from axes through the origin and every transform in the walk is a rotation about an axis through the origin, so the property is preserved by construction rather than by convergence.

That is the right outcome and it is worth saying why the check is kept anyway. The claim being tested is not “the arithmetic is accurate” but “nothing in the construction lets this slip”, and the second is a claim about the code that a future change could falsify without touching the arithmetic. A check that currently returns exactly zero is a check that will report the day it stops.

The arcs are the lengths, and they hold #

The second thing the figure checks is the one that makes it a figure about a linkage rather than about four lines.

At any position, the angle between consecutive axis directions is the arc of the link between them. Those four angles are the mechanism’s dimensions and they cannot change — a bar does not stretch. So the figure measures them from the drawn directions and compares against what the placement asked for, at every drag frame.

This catches an ordering mistake that is easy to make and invisible once made. The construction takes the arcs ground-first, and the walk round the loop starts at the crank, so the arc measured at step i is arcs[i + 1] rather than arcs[i]. The first version of the check compared them in the wrong order and failed on a correct figure — reporting 60 where 80 was expected, which is exactly the off-by-one it was. Had the shift gone the other way, or had the four arcs happened to be more nearly equal, it would have passed while comparing the wrong pairs.

The universal joint, placed #

A Hooke joint is a spherical four-bar with two of its four arcs at exactly 90 degrees.

The input shaft and the first pin of the cross are perpendicular; the second pin and the output shaft are perpendicular; the two pins of the cross are perpendicular to one another. Three right angles, and the fourth arc — between the two shafts — is the shaft angle β, the one quantity a driveline designer actually chooses.

That places every property of the joint the earlier essay measured. Its arcs sorted are β, 90, 90, 90, and for any β below 90 the condition $s + l \le p + q$ reads $\beta + 90 \le 180$, which holds. So the input turns fully, always, at every shaft angle — which is a thing everybody knows about universal joints and which arrives here as a special case of a general condition rather than as a fact about universal joints.

And the non-constant velocity ratio has the same standing. A spherical four-bar’s output angle is a nonlinear function of its input angle in general, exactly as a planar four-bar’s is; the universal joint’s particular relation $\tan\phi = \tan\theta / \cos\beta$ is that general nonlinearity with three right angles substituted in. The joint is not special for being non-constant. It would have been special for being constant, and the reason people expect constancy is that the mechanism looks symmetric, which it is — just not in the way that would help.

The two branches, and what they mean here #

The $\pm$ in the construction above is not a detail of the arithmetic. It is the same two-branch structure the planar four-bar has, and it means the same thing.

For most input angles a planar four-bar has two assembly configurations: the coupler and rocker reflected about the diagonal. A built mechanism is in one of them and cannot reach the other without being taken apart, which is why every sweep on this site carries the previous solution forward rather than re-solving from scratch — re-solving would let a figure jump branches mid-animation and show a machine that dismantled itself between frames.

On the sphere the two branches are the two intersections of the two cones, reflected in the plane through the two ground axes. Everything that is true of the planar branches is true of these: a physical linkage sits on one, the change from one to the other happens only where the two coincide, and the place they coincide is a dead centre — a configuration where the coupler and rocker arcs lie along one great circle and the mechanism momentarily has an extra instantaneous freedom.

That is the same list of consequences the planar field spends several essays on, and it is worth stating that the transport is not partial. Dead centres, branch defects, the transmission angle and its worst value all have spherical forms obtained by the same substitution. What this site has built is the linkage, the condition and the check that the arcs hold; the rest of the transported theory is named here and not drawn.

Why the family matters beyond the joint #

Three reasons, and the third is the one that made this worth a rung of its own.

It is a whole planar theory, transported. Coupler curves, transmission angles, dead centres, cognates, the branch and circuit defects of a synthesis: every one has a spherical form, obtained by the same substitution of arcs for lengths. This site has not built them, and saying so is more useful than implying otherwise — what it has built is the linkage and the condition, which is the entrance.

It is the mechanism for redirecting a shaft. Any application that needs rotation about one axis converted to rotation about another, with the two axes meeting, is a spherical linkage problem. That includes the universal joint, the constant-velocity joints designed to fix its ratio, most steering couplings, and the wrist of a great many robot arms.

It is the second subgroup. Planar and spherical are the two three-dimensional subgroups of rigid motion that contain rotations, and between them they account for almost every overconstrained mechanism anyone builds on purpose. The overconstraint essay makes the general argument; this field supplies half of its evidence.

What the construction had to solve #

Building the linkage is a small piece of spherical trigonometry and it is worth writing down, because it is where the substitution stops being a slogan.

The planar construction places the crank pin on a circle and finds the coupler pin by intersecting two circles — one of coupler radius about the crank pin, one of rocker radius about the fixed pivot. Two intersections, which are the two assembly branches.

The spherical construction places the crank axis on a cone about the ground axis and finds the coupler axis by intersecting two cones — one of the coupler arc about the crank axis, one of the rocker arc about the far ground axis. Two intersections, which are the two assembly branches.

Concretely: the wanted direction $z_2$ satisfies $z_2 \cdot z_1 = \cos b$ and $z_2 \cdot z_3 = \cos c$, two linear equations in three unknowns with $|z_2| = 1$ closing the system. Solving gives

$$z_2 = k_1 z_1 + k_2 z_3 \pm k_3 (z_1 \times z_3)$$

with the $k$s from the two cosines and the angle between $z_1$ and $z_3$. The $\pm$ is the branch. If the coefficient under the square root is negative the four arcs do not close at that input angle, and the construction says so rather than returning a complex direction — the same refusal the planar circle intersection makes.

This is the whole reason the substitution works: the sphere has a law of cosines, so the constructions transport. It also shows where it stops. The sphere is bounded, so an arc of 200 degrees is an arc of 160 degrees the other way round, and a “long” link is not a meaningful idea past 180.

The plane is the small-arc limit #

The substitution a length becomes an angle and a sine appears runs one way in this essay: planar results are carried up to the sphere. It runs the other way too, and reading it in that direction says which of the two subjects is the general one.

Shrink a spherical four-bar. Make its four arcs small — a little mechanism on a very large sphere — and $\sin a \to a$, so the spherical Grashof condition $\sin s + \sin l \le \sin p + \sin q$ becomes $s + l \le p + q$, which is the planar condition on the arcs read as lengths. Every other result goes the same way, because every one of them is the planar statement with sines in it.

So planar kinematics is the small-arc limit of spherical kinematics, not a separate theory that happens to have an analogue. The plane is what a sphere looks like locally, the four bars are what four arcs look like when they are short, and the reason every planar result has a spherical counterpart is that the planar result was the limiting case all along.

The departure is computable, which turns the observation into something usable. $\sin a$ differs from $a$ by $a^3/6$, so the relative error in treating an arc as a length is about $a^2/6$: at arcs of 10° that is 0.5%, at 30° it is 4.6%, and at 90° the two quantities are 1 and 1.571 and there is nothing left of the approximation. A spherical mechanism with short arcs can be designed with planar tools and a correction; one with arcs near a right angle cannot be designed with them at all.

Which places the universal joint exactly where its behaviour says it should be. Two of its four arcs are at 90°, which is as far from the planar limit as the sphere goes, so it is the least planar member of the family — and its famous departure from a constant velocity ratio is the same statement. A Hooke joint at a small operating angle is nearly planar and nearly constant-velocity; the two nearlies are the same nearly.

It also settles a question the substitution invites. If the plane is the limit of the sphere, is the sphere the limit of something else? It is not, in this direction: planar and spherical are the two three-dimensional subgroups of the rigid displacements and neither contains the other. A spherical mechanism degenerates to a planar one as its arcs shrink, and a planar one does not degenerate to anything — which is the sense in which the sphere is the richer of the two and the plane is where the subject started for reasons of drawing rather than of generality.

What is not here #

The solve. The construction above places the four axes in an assembled configuration, and then the loop’s own solver takes over — the site’s standing rule is that a drawn position is the output of a solve, not of a construction, and the construction is a seed.

That distinction has bitten this site before. The expansion phase recorded a check that seeded a solver with a closed form and then agreed with it, having never run: Newton moved the seeded angles by exactly zero at every position and the assertion passed on a solver that could have been broken. The figure here avoids that by construction rather than by care, because the drag frames are not seeded from the construction at all — each one is driven from the previous position, in small steps, exactly as a shaft turns.

That mattering is not hypothetical either. A spherical four-bar assembles near one input angle and the solve is asked for others; jumping straight to 180 degrees hands Newton a seed on the wrong side of the configuration space, and the loop that sweeps 36 of 36 positions perfectly well fails to close when asked for one of them directly. driveTo walks there in increments and carries each solution forward as the next guess — which is not a numerical convenience, it is what the machine does.