Machine Kinematics
solved, then drawn
Concept

Open chain — where it appears

A kinematic chain with no loop in it, so every joint setting is admissible and its pose is a product of transforms rather than a solve. Because there is nothing to close, its forward problem is a multiplication that cannot fail — and the whole difficulty moves to the inverse problem, where one tool pose has several answers.

Named by 7 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 θ₂ shoulder.

The chain that does not close

Every mechanism on this site so far has been a loop, and a loop is why a configuration here is a solve. An arm has no loop. Its pose is a product of six transforms, evaluated, with nothing to converge and nothing to refuse — and the difficulty does not disappear, it moves to the other end of the problem.

serial · Serial SCARA, as four numbers a joint. A Denavit–Hartenberg table describes each joint by the common perpendicular between its axis and the next one: how long it is, how much the axes twist across it, where along the first axis it meets, and at what angle. 3 of these 3 rows have no answer for the last two, because the axes they describe are parallel and two parallel lines have infinitely many common perpendiculars, all the same length, at every point along them. The arm is perfectly ordinary; it is the description that has run out.

Four numbers or a screw

An arm can be written down as four numbers a joint or as a line in space with a pitch on it. Both are minimal, both describe the same machine to the last bit, and one of them jumps by three hundred and fifty thousand when an axis is tilted by a millionth of a radian.

serial · Serial 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 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 θ₃ elbow.

Two routes to a Jacobian

An open chain's Jacobian is a list of its joint axes, drawn as lines in the same picture as the arm. A finite difference of its own forward kinematics is a completely different computation and has to agree — and when the two disagreed by 5 × 10⁻⁵, the fault was in a function six phases old that every spatial loop on this site had been using.

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 Arms with their tools pinned down. Pin an arm's tool to the ground and the open chain is a closed loop, which the first field of this site knows how to count. Kutzbach gives 6(n − 1) − 5n = n − 6 for a loop of n revolutes, and the measurement is n minus the rank of its screw system — the columns of the arm's own Jacobian, read as constraints rather than as velocities. The two agree on every row but one, and the one is the arm at a wrist singularity: the formula says the pinned arm is a structure and the mechanism has a freedom. That is the finding this site opened with, arrived at from the far end of its subject.

Pin the tool and it is a loop

Hold an arm's tool still and the open chain becomes a closed one, which this site has known how to count since its first field. Kutzbach's criterion says a pinned six-joint arm is a structure. At each of its three singularities the measurement says it can still move — the site's founding finding, arrived at from the far end of its own subject.

serial · Spatial 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.

Serial manipulatorForward kinematicsInverse kinematicsJoint limitsScrewWorkspaceAssembly branchJacobianKutzbach's criterionLoop closureMobilityProduct of exponentials

All concepts