Machine Kinematics
solved, then drawn
Concept

Wrist singularity — where it appears

The posture where two of an arm's wrist axes line up, costing one rotational freedom in the middle of an otherwise ordinary working volume. It is a genuine loss of freedom rather than an artefact of the coordinates, and it is why a straight-line move through the middle of a workspace can demand unbounded joint rates.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

elbow arm at a posture. elbow arm, drawn from 6 joint values through a product of 6 exponentials — no equation is solved anywhere in this picture, because an open chain has none to solve. The thin lines are the joint axes at this configuration, which are also the columns of the arm's Jacobian: the tool's velocity is a sum of turns about exactly those lines. Here the smallest singular value of that Jacobian is 0.3674 and the largest is 2.407, so the arm is comfortably away from a configuration where a direction of motion is lost. Drag θ₅ pitch.

The wrist is three joints and one point

Three axes meeting at a point is what makes a six-joint arm's inverse problem solvable in closed form, and it is why every industrial arm is built that way. Move one of those axes by ten millimetres and the construction goes on returning eight confident answers, every one of them out by three and a half.

serial · Serial Three singularities, and where each one is. The elbow singularity is at the edge of the workspace, where the arm is straight and cannot reach further — nothing is lost that could have been used. The shoulder singularity is where the two ways of facing the target merge, on a cylinder of radius 0.18 m about the base axis which is also the boundary of a hole the arm cannot reach into at all. The wrist singularity is the one that stops real machines: axes four and six in line, one rotation gone, in the middle of an ordinary working volume with nothing about the tool's position to suggest it. The rank falls by exactly one in all three — and what is lost differs. The elbow and the shoulder are constrained by a pure force, along the arm and across it; the wrist is constrained by a screw of pitch −0.629, which is a force and a couple together. Those are reciprocal screws rather than singular vectors, because a singular vector's direction depends on whether the arm was written in metres or millimetres and a screw does not.

Where the arm loses a direction

An arm has three singularities and they are three different events. Two of them are inside an ordinary working volume, all three drop the rank by exactly one, and what each one takes away is a screw — a pure force along the arm at the elbow, a pure force across it at the shoulder, and at the wrist a screw of pitch −0.629 that is a force and a couple together.

serial · Serial A SCARA arm, and where it can put its tool. A SCARA arm at its home position, with the axes dashed and a cloud of the tool positions reached over random joint values. The displacement set of an open chain is the product of its joints' groups, one factor per joint, and the question this field asks of it is whether the product is itself a group. Here the logarithms of the reached displacements occupy 4 dimensions, so the motion lies inside Schoenflies motion and composing two of its displacements gives another one. The cloud is a fact about the reach and not about the group: a chain of finite links covers a bounded piece of its group and never the whole of it, which is a separate question and a different field's.

The arm that is a group

A SCARA arm has four joints and a six-axis robot has six, and the usual explanation is that four is enough for the job. The better one is that the job is a four-dimensional group of displacements which is not the symmetry group of any surface — so it cannot be one joint, and four is what it costs. The arm's tool face is level everywhere it can reach, and the reason is not that anybody checked.

serial · Serial

Named alongside it

The objects these essays reach for when they reach for this one.

RankSerial manipulatorSingularityWorkspaceClosed formConditioningDegrees of freedomDisplacement subgroupEuler anglesInverse kinematicsJacobianLie bracket

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