Machine Kinematics
solved, then drawn
Concept

Joint limits — where it appears

The hard stops on a joint's travel, which turn a configuration space from a torus into a box and decide which solutions a machine can use. They decide which of an inverse problem's several solutions a real machine can actually use, and they are why a posture that exists on paper may be unavailable.

Named by 4 essays across one field — each of them below, with the objects they name alongside it.

Reachable, and dexterous. A 3-link planar arm with links 1.6, 1.2, 0.7. The outer region is everywhere the tool can be put: an annulus from 0.00 to 3.50. The inner region is everywhere it can be put at every tool angle — from 1.10 to 2.10, which is 26% of the area. Both are measured by counting cells on a 300 × 300 grid and both agree with the area computed from the radii to 0.09%, which is what makes the picture a measurement.

Where the hand can go

A robot is sold on its reach, which is one number and describes a sphere the arm touches at one posture. The set the tool can actually be put in is an annulus with a hole; the set it can be put in at every orientation is a quarter of that; and reordering the same three links leaves the first unchanged and destroys the second.

serial · Serial 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 One tool pose, a curve of postures. 41 postures of S-R-S arm, every one of them holding the tool at the same pose. The tool moved 1.0e-13 of a metre over the whole sweep, which is the arithmetic's noise floor rather than a tolerance. The joint that moves most is the elbow, and it runs on a circle of radius 0.192 m about the line from shoulder to wrist — a curve computed here by walking the Jacobian's null space, and known independently as the intersection of two spheres. The two agree to 10⁻¹³.

The freedom that does nothing

Give an arm a seventh joint and one tool pose stops having eight answers and starts having infinitely many. The arm changes shape while the tool stands still — measured at 3.3 × 10⁻¹² of a metre over forty-one postures — and the elbow runs on a circle that two entirely different computations agree about to a tenth of a picometre.

serial · Serial The free configurations, with joint limits. Every point is a pair of joint angles for a two-link arm; the pale region is the configurations at which neither link touches an obstacle, and the dark one is where something is in the way. The free space is in 2 pieces. The arm's joints cannot turn all the way round, so the edges of the square are edges — and now the barrier separates. The two crosses put the tool at exactly the same point, and the arm cannot get from one to the other at all.

The space of configurations

A two-joint arm's configurations are a torus, and drawing one on a page turns it into a square whose opposite edges are secretly the same line. Count the free space on the square and get three pieces; count it on the torus and get one. Eighty single-obstacle arrangements were tried and not one of them cut the torus in two — what does that is a pair of hard stops.

serial · Serial

Named alongside it

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

Inverse kinematicsOpen chainSerial manipulatorWorkspaceAssembly branchRedundancyAnnulusClosed formConfiguration spaceConnected componentDexterous workspaceForward kinematics

All concepts