The wrist is three joints and one point

The last three joints of an industrial arm meet at a point. Not nearly: exactly, to whatever the machining allows, and the design goes to some trouble to make it so — the motors are pushed back down the forearm and the drive is taken forward through concentric shafts, which is more expensive than putting each motor at its own joint.

The reason is not mechanical. It is that three axes through one point make the inverse problem solvable in closed form, and a controller solving it a thousand times a second cannot afford anything else.

The property, measured #

The claim is that the last three joints do not move the wrist centre. It is one line to check, and the site’s own gate checks it: drive joints four, five and six through twenty-four combinations spanning their ranges, and watch the point.

It moves by $6.0 \times 10^{-16}$ metres. That is the floating-point noise of a product of six transforms, and it is what “the axes meet” means operationally.

Everything in the previous essay rests on that number. The construction’s first step is to find the wrist centre by backing off from the target along the tool axis, and its second is to solve three joints for that point as though the last three did not exist. Both steps are licensed by this measurement and by nothing else.

Pieper’s condition #

The general statement is Pieper’s condition, from 1968: a six-joint arm has a closed-form inverse solution if three consecutive axes intersect at a point, or if three consecutive axes are parallel.

Both cases work for the same reason. The three special joints contribute a motion that leaves something invariant — a point, for the intersecting case; a direction, for the parallel one — and that invariant is enough to split six coupled equations into two sets of three. Without it, the six equations stay coupled and the problem is a piece of elimination theory that was open until the late 1980s and has sixteen solutions.

It is worth pausing on what kind of statement that is. A geometric coincidence in a machine is being arranged deliberately, so that an equation will decouple. The arm is not better at reaching things for having a spherical wrist. It is better at being told where to reach, which is a property of the arm and its controller together rather than of the arm.

This site has collected several designs of that shape. A Watt linkage’s tracer is placed where it is because the error’s fifth-order term vanishes there. A gear’s involute flank is the curve it is because that is the one that keeps a ratio constant. A Bennett linkage’s link lengths satisfy a relation because otherwise it does not move at all. The spherical wrist joins them: a shape chosen for what it makes computable.

The wrist is a spherical mechanism #

Three revolute axes through a point is not a new object on this site. Every point of the moving body stays on a sphere about that point, and the mechanism is a spherical one — the same class as the universal joint and the spherical four-bar whose link lengths are angles.

That connection pays off immediately, because it says what the wrist can do without any calculation. A spherical mechanism’s configuration is a rotation; three independent axes through the point span the whole rotation group; therefore a spherical wrist with three non-coplanar axes can produce every orientation. Not most, not a large subset: all of them.

It also says where the trouble is. A spherical three-joint chain is a set of Euler angles by another name, and reading three angles off a rotation is the classical problem with the classical degeneracy: when the middle angle is zero, the first and third axes have become the same line and only their sum is determined. The wrist has lost a freedom.

The important thing about that row is what it is not. It is not gimbal lock. Gimbal lock is a rank drop in the map from three chosen numbers to an angular velocity, and it goes away if different numbers are chosen. This is a rank drop in the arm’s Jacobian: two of the six columns have become the same line, and no change of description puts them back. Two routes to a Jacobian separates the two by exhibiting a posture where the coordinates are singular and the arm is not.

Reading three angles off a rotation #

The wrist half of the construction is a z-y-z Euler decomposition, and it is worth writing out because its two branches are two of the eight and its degeneracy is the singularity.

Whatever the first three joints did, the wrist has to supply the rest: a rotation $R$, expressed in the frame the forearm has arrived at. With axes $z$, $y$, $z$ the three angles come out as

$$\beta = \operatorname{atan2}\!\left(\pm\sqrt{r_{13}^2 + r_{23}^2},\ r_{33}\right), \qquad \alpha = \operatorname{atan2}(\pm r_{23}, \pm r_{13}), \qquad \gamma = \operatorname{atan2}(\pm r_{32}, \mp r_{31})$$

with one sign choice running through all three. Take it positive and the wrist folds one way; take it negative and $\beta$ changes sign while $\alpha$ and $\gamma$ each shift by $\pi$, which is the wrist flipped over with the tool pointing exactly where it was.

When $\beta = 0$ the square root vanishes and neither $\alpha$ nor $\gamma$ is determined — only their sum is, because the first and third axes have become the same line. There is no right answer to return there, and what this site’s code does is a convention: it puts the whole sum into the first angle and zero into the third. That is a choice, it is stated in the function, and it is the honest way to handle a degeneracy — the alternative, quietly returning whatever the arctangents produce from a matrix full of zeros, is how a controller ends up commanding a joint to swing 180° because a rounding error picked a branch.

The singularity is everywhere and nowhere #

An obvious defence against the wrist singularity would be to stay away from wherever it is. It does not work, and the reason is one this field keeps arriving at: a singularity is a set of configurations, not a set of places.

Sample the wrist-singular set — the fifth joint at zero, every other joint anywhere within its limits — and ask where the tool ends up. Over 1,680 such postures the tool lands at radii from 0.495 to 2.580 m, against a workspace that reaches 2.578. The singular configurations cover essentially the entire working volume.

So there is no region of space to avoid. At almost every point the arm can reach, there are postures that are singular and postures that are not, and which of the two the arm is in depends on how it got there. That is exactly the distinction the workspace essay drew between a singular configuration and a boundary, and it is why avoiding singularities is a question about paths rather than about geometry — which is the next essay but one.

What happens when the axes miss #

Now the measurement this essay exists for.

Real wrists do not always intersect. A hollow wrist, which several manufacturers build so that cables and hoses can run through the arm rather than flapping outside it, has axes that miss each other by a few millimetres. So does any wrist with a bearing arrangement that will not fit through a point. The datasheets describe these as compact, not as defective, and they are correct.

Take the arm above, move its fifth axis sideways by 10 mm, and run the same construction.

It returns eight postures. They are labelled front and back, elbow up and elbow down, wrist and wrist flipped, exactly as before. Each of them is a set of six joint values that looks entirely ordinary. And every one of them puts the tool somewhere other than where it was asked: the best is 3.6 mm out, the worst 7.7 mm, on a 10 mm offset.

Nothing inside the construction can notice. Every step of it is arithmetic about a wrist centre, and on this arm the wrist centre is not a fixed point of the last three joints — driving them moves it by 6.9 mm. The construction is answering a question about an arm that does not exist, and it has no way to find that out.

What finds it out is the round trip: run each answer forward through the product of exponentials and compare with the pose asked for. That check is why every inverse solution in this field is run forward before it is drawn, and this is the case it exists for.

The miss scales with the offset almost exactly linearly — 0.74 mm at a 2 mm offset, 3.6 at 10, 16 at 50 — which is a useful thing to know and a dangerous thing to rely on. It says a nearly-intersecting wrist gives a nearly-right answer, which is true; it does not say that the answer is good enough, because 3.6 mm is enormous for an arm whose repeatability is quoted in hundredths of a millimetre.

What machines with offset wrists actually do #

They do not use this construction. There are three ways out and all three are in service.

Iterate. Damped least squares from the previous posture, which converges in a few steps because the previous posture is close, and which is what the search route does. The cost is determinism: the answer depends on where the iteration started, and near a singularity it can jump branch without saying so.

Iterate from the closed form. Use the decoupled construction as a seed — it is 3.6 mm out, which is an excellent starting guess — and take two Newton steps. This is the common answer, and it keeps the branch labelling: the eight seeds converge to eight solutions that inherit their names.

Solve the general case. The sixteen-solution elimination exists and can be implemented. It is expensive and it is done, in offline planners rather than in servo loops.

The third is worth a note on this site’s boundaries. Its algebra field has homotopy continuation and root counting, and a general 6R inverse problem is exactly the kind of polynomial system that machinery is for. It has not been done here, and saying “the machinery exists” is not the same as having done it — which is the same distinction that field made about Bézout numbers.

Three parallel axes, for the same reason #

Pieper’s other case deserves its own sentence, because it is what the machines in the other half of the market are built on.

A palletiser, a SCARA and most excavators have three consecutive parallel axes, and their inverse problems are also closed-form. The invariant there is a direction rather than a point: three parallel revolutes cannot change the direction of anything perpendicular to them, so the orientation about that axis is a sum of three angles and the position is a planar two-link problem. It is the planar arm’s solution wearing a different hat.

And the two cases have opposite relationships to the parameterisation essay’s finding. Intersecting axes are fine for a DH table and awkward to build; parallel axes are easy to build and are exactly where the DH table has no answer. A designer picking Pieper’s parallel case for its stiffness gets, for free, a description of the arm whose parameters jump by order one when an axis is tilted by a millionth of a radian.

The wrist that is not there #

One design avoids all of it by having no wrist at all: a three-joint arm carrying a tool that does not need orienting — a drill on a Cartesian gantry, a pick-and-place head, a sucker. Three joints, three coordinates, one closed-form solution and no wrist singularity anywhere in the workspace, because the machine cannot be asked for an orientation it might lose.

That is not a compromise, it is a specification. A large fraction of the arms in the world are of this kind, and they are cheaper, faster and more accurate than a six-joint arm for the jobs they can do. What they cannot do is present a tool at an angle, and the moment a job requires that — welding a fillet, laying a bead round a corner, deburring an edge — the three extra joints arrive and so does everything in this field.

What the concentric shafts cost #

The mechanical price of putting three axes through one point is worth stating, because it is not small and it explains why some arms decline to pay it.

The three wrist motors cannot sit at the wrist. There is no room — three motors and three gearboxes will not fit inside a sphere the size of a fist, and if they did, their mass at the end of the forearm would be the dominant load on the shoulder. So they sit at the back of the forearm and drive the wrist through concentric shafts or through a bevel train running the length of it.

That has three consequences, all kinematic and all measurable elsewhere on this site:

• Coupling between the wrist joints. With a shared shaft train, turning one wrist joint turns another unless the transmission compensates. Some arms compensate mechanically, some in software, and the ones that compensate in software have a wrist whose joint angles are not the numbers the motors read. That is a gear-train ratio problem sitting inside a kinematics problem.
• Backlash where it is worst. Three trains in series, each with a mesh, at the end of the longest lever in the machine. The backlash essay counts what a mesh has to give away; here it is given away at the joint whose errors are amplified least, which is the one saving grace.
• A longer forearm. The motors have to go somewhere, and where they go makes the link they are in longer — which moves the wrist further from the shoulder, which is exactly the lever arm the error budget says the shoulder’s error is multiplied by.

None of that changes what this essay’s construction does. All of it changes whether the arm the construction describes is the arm on the floor, and that is the subject of the two essays about calibration and error.

What the decoupling buys, restated #

Three axes through a point costs concentric shafts, a longer forearm, and a singularity in the middle of the working volume that will never go away.

It buys a construction: eight exact postures from a target pose, in a few dozen operations, each labelled by which way the arm is folded. Every industrial arm on the planet has made that trade, and the trade was made against an equation rather than against a load. It is the clearest case this site has of a mechanism shaped by what somebody needed to be able to compute.

The linearity of the miss with the offset is what makes the second remedy work, and it is worth saying why. A seed whose error is bounded and known in advance is a much better seed than one that is merely usually good: at a 10 mm offset the construction lands 3.6 mm out, and 3.6 mm is a distance a damped least-squares iteration crosses in two or three steps from a starting point with no singularity between it and the answer. So the iteration count is bounded by the offset rather than by the pose, and a controller can guarantee a solve time — which is the property a real-time system actually needs and which an unseeded iteration cannot promise. That is the practical value of measuring the error’s scaling rather than merely noting that the construction is wrong: an error of unknown size is a reason to abandon a method, and an error known to be linear in a manufacturing dimension is a reason to keep it and budget for it.