Machine Kinematics
solved, then drawn
Concept

Pieper condition — where it appears

The geometry that makes a six-joint arm's inverse problem solvable in closed form: three consecutive axes meeting at a point, or three parallel. It is why almost every industrial arm has a spherical wrist: the condition is a design constraint accepted in order to make the inverse problem solvable in closed form.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

8 postures, one tool pose. The tool is at one place, held one way. Eight different sets of joint values put it there, and this is all of them at once: two ways for the base joint to face the target, two for the elbow, and two for the wrist — two by two by two. Every posture here came out of a closed form and was checked by running the arm forward again; the worst disagreement is below 10⁻¹⁴ of a metre. Which of them a machine can actually use is a separate question, and it is answered by the joint limits rather than by the geometry.

Eight ways to hold the same tool

A six-joint arm asked to put its tool at one place, held one way, has eight answers. Not approximately eight and not eight found by looking — two for the base, two for the elbow, two for the wrist, each exact to a hundredth of a femtometre, and a search from six hundred starting postures finds those eight and no ninth.

serial · Serial 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

Named alongside it

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

Closed formInverse kinematicsSerial manipulatorAssembly branchEuler anglesJoint limitsNewton–RaphsonOpen chainRankRoot countSingularitySpherical four-bar

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