Machine Kinematics
solved, then drawn
Concept

Forward kinematics — where it appears

The map from a mechanism's joint values to the pose of its tool, which for an open chain is a product of exponentials and cannot fail. It is the easy direction: no iteration, no branch, no possibility of refusal, which is exactly what a loop-closure equation does not offer.

Named by 8 essays across 2 fields — 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 Three legs, one platform. A 3-RRR planar parallel mechanism at (0.20, -0.15) turned 11.5°, elbows up/up/up. The three actuator angles were computed one leg at a time and independently, which is what makes this direction cheap. The dashed lines extend each leg's second link: those are the three forces the legs can transmit to the platform, and the mechanism is controllable exactly while they stay independent. Here they miss one another by 0.651, and the smallest singular value of the three is 0.9550. At this position the platform can be turned through 206° in all before a leg runs out of reach.

The easy problem and the hard one change places

For a robot arm, working out where the hand is takes a walk down a chain and working out what the joints must be to put it somewhere is the hard part. Connect the platform by three legs instead of one and both statements reverse — and the reversal is not a matter of degree.

parallel · Parallel 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 6 ways to assemble the same three actuator angles. The actuators are at 216°, 48°, 144° in every panel, so the three elbows are at the same three points throughout and only the platform differs. Each pose satisfies all three legs to 6.7e-16. Found by reducing the problem to one equation in the platform angle and scanning it at 0.100° — exhaustive to that resolution and no further, which is the honest thing to say about a root count. Over a survey of 6750 actuator triples this mechanism ranges from none to 6.

One command, six answers

Lock the three motors of a planar platform and the platform can be in as many as six different poses, every one of them satisfying every leg exactly. Which one it is in was decided by how it was assembled and where it has been since — and the number of answers is not a property of the mechanism but of where the motors happen to be.

parallel · Parallel Six legs, six numbers. A Gough–Stewart platform at (0.00, 0.00, 2.40) with a rotation vector of (0.00, 0.00, 0.15). The six leg lengths run from 3.077 to 3.263, and every one of them was computed from the pose by a single subtraction and a square root, with no leg consulting another. Going the other way — six lengths in, one pose out — takes an iterative solve and does not have one answer. The six leg lines span a screw system of order 6; the smallest singular value of the six is 0.0316, which is how far this pose is from a configuration where they become dependent and the platform stops being controllable.

Six legs and a square root

A Gough–Stewart platform's inverse problem is one subtraction and one square root per leg, computed six times without any leg consulting another. Its forward problem has forty solutions. That gap is the whole design, and it is why flight simulators are built this way and robot arms are not.

parallel · Parallel Where the assembly count changes, and a loop round a cusp. With the first motor held at 216°, each point of the square is a setting of the second and third motors, shaded by how many assemblies the platform has there, sampled on a 41 by 41 grid: 2 assemblies at 1,261, 4 assemblies at 347, 6 assemblies at 73. Each edge between two shades is a curve of direct singularities, where two assemblies merge and vanish, and two such edges meet in a sharp point. The 2 marked points are the cusps in this window, each found as a triple root of the closure equation. The dashed circle, of radius 8°, is the loop the motors are driven round; it encloses 1 cusp, the one at (101.35°, 185.32°), and starts at the open marker.

Round a cusp into another assembly

A parallel platform's assembly mode was supposed to change only through a direct singularity. Driven round a small loop of motor angles that encloses a cusp of the singular curve, the standard three-legged platform leaves one assembly and arrives in another, turned 52° from where it started, and at no point on the way is it nearer than 0.0716 to singular.

parallel · Parallel Two motors, two circles, two places for the hand. A planar five-bar with its motors 1.0 apart, arms 1 and distal links 1.25, at motor angles 100° and 60°. Each distal link holds the hand on a circle of radius 1.25 about its elbow, and the elbows are 1.678 apart, less than the 2.50 at which the circles would only touch, so they meet twice. The hand drawn solid is at (0.229, 1.850) with det A 0.995; the other assembly, dashed, is at (0.098, 0.001) with det A -0.995 — the same size and the opposite sign.

The smallest parallel robot

Two motors, two arms, and two links meeting at a hand: a planar five-bar is the smallest parallel robot there is. Its forward problem is two circles, so the hand has two places to be, and each is named by the sign of one determinant. That is the whole reason it cannot do what the three-legged platform does, and cannot change assembly without passing through the one configuration where the two meet.

parallel · Parallel

Named alongside it

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

Assembly-modeDirect singularityParallel mechanismWorking modeInverse kinematicsOpen chainPlatformScrewSerial manipulatorWorkspaceConditioningCuspidal mechanism

All concepts