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 6 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 A straight move ends where its posture does. The six-joint arm with its stops on, moving its tool in a straight line at a fixed orientation in the posture it starts in (back · elbow up · wrist flipped). Faint: the arm at the start. Solid: where the move ends, 66.0 cm along, stopped by the base joint at its stop. Dashed: the same line continued, which this posture cannot follow. Of the arm's usable postures at the start, the best would have gone 125.5 cm.

A straight move ends at a stop

A six-joint arm asked to move its tool in a straight line at a fixed orientation keeps the posture it starts in, because changing posture means passing a singularity. Followed from three hundred random poses inside the arm's joint stops, in random directions, the starting posture carries the tool a median of 19.5 cm. Half the moves end at a joint stop, two in five where the arm stretches straight and its posture runs out, and three in a hundred near the wrist singularity, the one the literature on straight-line moves is about. The best of the arm's other usable postures would have gone a median of 34.5 cm, and in a third of the moves it would have gone further. On the way, an earlier count of how many postures an arm can hold was found to be reading its base angle a turn out.

serial · Serial A seventh joint steers the base roll off its stop. The S-R-S arm with stops on every joint, moving its tool in a straight line at fixed orientation from the same start (faint). With its upper-arm roll locked, a six-joint arm, the move ends 27.0 cm along at the base roll at its stop (dashed arm). With all seven joints and the self-motion steered at every step to move the joint nearest its stop away from it, it ends 61.5 cm along at the base roll at its stop (solid arm). Shown 60.0 cm along, with the six-joint arm stopped where its move ended.

A seventh joint cannot move the elbow

A seven-joint arm can change its shape while its tool stands still, and on a straight move that freedom can be spent keeping its joints off their stops. Spent that way, it lengthens the median move from 23.3 cm to 33.3 cm, against the same arm with its extra joint locked. But the gain is all or nothing. Half the moves a stop used to end gain under a tenth, and a quarter gain more than two and a half times. One stop it never gets round: the elbow's, whose angle is set by the distance from shoulder to wrist and is the same, to 10⁻¹⁵ rad, whatever the other joints do. Nor does it ever reach past the edge of the arm's reach.

serial · Serial

Named alongside it

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

Inverse kinematicsWorkspaceAssembly branchOpen chainRedundancySerial manipulatorNull spaceSingularityAnnulusClosed formConfiguration spaceConnected component

All concepts