Machine Kinematics
solved, then drawn
Concept

Workspace — where it appears

The set of places a mechanism's tool can be put, which for an arm is a region with a hole in it rather than the sphere its reach suggests. Its boundary is where the arm is fully stretched or fully folded, and the hole in the middle is the volume the arm cannot reach because its own links are in the way.

Named by 23 essays across 5 fields — each of them below, with the objects they name alongside it.

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 Three bars and four bars. On the left, two bars to a common point: three links, three joints, and Grübler gives 3(3−1) − 2(3) = 0. The Jacobian agrees — two free coordinates, rank 2, nothing left over — and the shape cannot change without a bar changing length. On the right, one more bar and one more joint gives mobility 1, and the whole of this site follows from that difference. The triangle is why bridges are triangulated and the quadrilateral is why machines are not.

What decides whether it moves

Before a mechanism does anything it has to be able to. Two bars pinned to a point cannot move; three can. The count that separates them is one subtraction, it is the first thing anybody computes about a machine, and it can be wrong in a way that no amount of care with the arithmetic will catch.

constraint · Mobility Losing a freedom and gaining one. Left, the nearest inverse-kinematic singularity: a leg is straight to 1.1e-5 and cannot reach further, so the platform has lost a freedom. That is the workspace boundary and it is a serial arm's singularity. Right, the nearest direct-kinematic singularity: the three leg lines pass through one point to 0.0001, so the three forces the legs transmit are no longer independent and the platform can turn about that point with every actuator locked. It has gained a freedom, and it is 1.97 from the other configuration and nowhere near the edge of the reach.

Locked, and still moving

A serial arm goes singular at the edge of its reach, where it loses a freedom, and the failure is visible as an arm gone straight. A parallel mechanism has a second kind with no serial counterpart — it gains a freedom, in the middle of the workspace, at poses nothing about the legs' reach marks out.

parallel · Parallel 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 Where the platform stops being controllable, at 0°. Every point is a position of the platform's centre at a fixed orientation of 0°, shaded by how far it is from a direct singularity — pale is near. Unshaded means unreachable, which is the workspace boundary and the ordinary kind of singularity. The line is det A = 0, traced through the field rather than tested for: it runs through the middle of the reachable region in 102 segments, and on it the platform can move with all three actuators locked. 3312 of 6561 sampled positions are reachable.

The workspace is not a shape you choose

A serial arm's reach is roughly a sphere and can be quoted as a number. A parallel mechanism's is the intersection of three reachability conditions, changes with every degree of orientation, and has a surface of uncontrollable poses cutting through the middle of it. There is no formula. There is a map, and it has to be computed.

parallel · Parallel How many dimensions each chain's displacements occupy. Every chain in the field, with the dimension its reached displacements' logarithms occupy. A chain of n joints always has n freedoms; what varies is whether those freedoms compose. Where the bar equals the joint count the motion is inside a group and the group is named; where it reaches six there is no proper group containing the motion, and the two chains that do are the ones whose axes were chosen at random. Nothing about the joints themselves differs — three pins are three pins, and the two rows differ only in where the axes point.

A chain multiplies

An open chain's displacements are the product of its joints' groups, one factor per joint, in order. Sometimes the product is a group — three parallel pins and a slide along them give Schoenflies motion, which is a SCARA arm and is why it has four joints. Usually it is not, and then the chain's poses are a four-parameter set that needs six numbers to describe.

pairs · Pair What every pair of groups meets in. The intersection of two subgroups is always a subgroup — that needs no computation — and which one is the useful part. This is the design rule behind every parallel machine in these essays: choose legs whose groups meet in the motion the platform is wanted to have, and it has that motion whatever the leg lengths are, with no synthesis and no tolerance. The row and column are built about different axes, at right angles, because an intersection is a statement about particular subgroups rather than about their kinds: two planar groups with the same normal meet in the whole of themselves, and two with different normals meet in a line.

Three legs and one plane

The parallel field argued that a platform stays flat from the structure of its legs, one leg at a time. The same fact is one line of linear algebra: each leg confines the platform to a group, the platform gets the intersection, and an intersection of groups is a group whatever the leg lengths are. The motion type is decided before a single dimension is chosen.

parallel · Parallel Three singularities, and where each one is. The elbow singularity is at the edge of the workspace, where the arm is straight and cannot reach further — nothing is lost that could have been used. The shoulder singularity is where the two ways of facing the target merge, on a cylinder of radius 0.18 m about the base axis which is also the boundary of a hole the arm cannot reach into at all. The wrist singularity is the one that stops real machines: axes four and six in line, one rotation gone, in the middle of an ordinary working volume with nothing about the tool's position to suggest it. The rank falls by exactly one in all three — and what is lost differs. The elbow and the shoulder are constrained by a pure force, along the arm and across it; the wrist is constrained by a screw of pitch −0.629, which is a force and a couple together. Those are reciprocal screws rather than singular vectors, because a singular vector's direction depends on whether the arm was written in metres or millimetres and a screw does not.

Where the arm loses a direction

An arm has three singularities and they are three different events. Two of them are inside an ordinary working volume, all three drop the rank by exactly one, and what each one takes away is a screw — a pure force along the arm at the elbow, a pure force across it at the shoulder, and at the wrist a screw of pitch −0.629 that is a force and a couple together.

serial · Serial Four joints that give a group, and four that do not. Two chains of four revolute-and-slide joints, each drawn at its home position with its joint axes dashed, and each with a cloud of the tool positions it reaches. The counts are identical: four joints, four freedoms, the same Jacobian rank everywhere off a singularity. On the left the three pins are parallel and the slide is along them, and the displacement set is the Schoenflies group — every translation and one rotation direction, four dimensions, closed. On the right the axes are at random and the set is four-dimensional too, and it is inside no group smaller than all the rigid displacements. The instrument is in the caption of each panel: take the logarithms of the displacements the chain reaches and count the dimensions they occupy. Four means a group. Six means there is nothing to be inside.

Four joints that give a group, and four that do not

Two chains of four joints. Same joint types, same count, same mobility, same Jacobian rank, same everything measured here across twenty-two fields. One of them reaches a four-dimensional set of displacements that closes under composition; the other reaches a four-dimensional set whose logarithms fill all six dimensions. The difference is six hundredths of a radian in where two axes point.

pairs · Pair The box the joints need, and the box the machine needs. Every machine in the catalogue, measured twice over the same drive: the extent of its joint positions, which is what a site drawing links as lines can report, and the extent of its material, which is what has to fit in something. The ratio runs from 1.13 to 1.36 — between thirteen and 36 per cent more area than the skeleton suggests, on machines whose links are a twentieth of their length wide. The last column is how much of the material's box the swept region actually fills, and it is where the packaging argument really is: a machine at 50% is a machine with a great deal of room inside its own envelope that nothing may be put in.

The hole the machine needs

A four-bar's joints fit in a box five units by four. Its material needs a box twenty per cent larger in area, and fills barely half of it. Both numbers are design quantities, and until a link had a width neither could be stated.

bodies · Body 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 A 3-RPR with similar triangles at 0°, its leg lines meeting wherever it is put. A planar platform with three extending legs, its base pivots on a circle of radius 2 and its platform points on one of radius 0.6 at the same angles, so the two triangles are similar, turned to 0° and drawn at two positions. Each leg's line is continued past its ends. With the platform's centre at (0.4, −0.3) the three lines meet at (0.5714, −0.4286), within 1 × 10⁻¹⁶ of every line, which is where a scale factor of 0.3 about that point carries the base triangle onto the platform; the smallest singular value of the three legs is 3 × 10⁻⁹. With the platform's centre at (−0.55, 0.3) the three lines meet at (−0.7857, 0.4286), within 2 × 10⁻¹⁶ of every line, which is where a scale factor of 0.3 about that point carries the base triangle onto the platform; the smallest singular value of the three legs is 0. The meeting point moves with the platform, and the three lines meet wherever it is.

Two orientations no position can rescue

A planar platform on three extending legs whose platform triangle is a scaled copy of its base is singular at every position it can be put in, at exactly two orientations: its three leg lines meet at one point wherever it is. The two orientations are read off the attachment points, a platform a few per cent from similar is held at its worst orientation only in proportion to how far from similar it is, and the revolute-legged 3-RRR, built on the same similar triangles, does not inherit any of it.

parallel · Parallel 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 paired platform with its pairs rotated 60°, turned through a revolution. The paired Gough platform — base anchors in pairs 25° apart on a radius of 2.2, platform anchors in pairs 40° apart on 1.1 — with the platform's pairs centred 60° round from the base's, held level and turned through a revolution at four positions. All four curves reach nought together at 30° and −150°, which is 90° − ρ and 180° from it. At the centred position the platform is held at 0.0363 at a yaw of 0°, and it can turn ±17° before that falls to half. Dragging the rotation carries the two dead yaws across the revolution together.

The dead yaw is a design choice

A paired hexapod held level is singular everywhere at a yaw of 30° because its platform pairs sit 60° round from its base pairs. Rotate them by ρ instead and the dead yaws move to 90° − ρ and 180° from it, exactly, at every rotation from 0° to 120°. The furthest they can be from home is a quarter-turn each way, at ρ = 0 — where the platform is also best held at home and can turn ±75.5° before its holding halves, against ±17° for the usual 60°.

parallel · Parallel Where a tilted platform is still singular. A slice of the workspace at height 2.4, at the dead yaw and 3° of tilt, with each position shaded by how well the platform is held there — dark where the six legs are nearly dependent and pale where they are not. Level, this whole square would be uniformly dark. Tilted, the dark places are a curve through it, which is what an ordinary direct singularity looks like on a slice. Driving a search downhill from forty-eight starts reaches a smallest singular value of 0.00e+0, so the locus is a singularity rather than a shallow valley.

Tilted, near the dead yaw

A paired Gough platform held level is singular at one yaw wherever it stands. Tilt it and that stops being true — the six moments are no longer equal and the home position is held. It is a poor rescue: the rise is quadratic in the tilt, so a degree buys a sixty-fourth of what eight degrees buys, and what the tilt actually does is not remove the singularity but turn it into a surface through the workspace.

parallel · Parallel A SCARA arm, and where it can put its tool. A SCARA arm at its home position, with the axes dashed and a cloud of the tool positions reached over random joint values. The displacement set of an open chain is the product of its joints' groups, one factor per joint, and the question this field asks of it is whether the product is itself a group. Here the logarithms of the reached displacements occupy 4 dimensions, so the motion lies inside Schoenflies motion and composing two of its displacements gives another one. The cloud is a fact about the reach and not about the group: a chain of finite links covers a bounded piece of its group and never the whole of it, which is a separate question and a different field's.

The arm that is a group

A SCARA arm has four joints and a six-axis robot has six, and the usual explanation is that four is enough for the job. The better one is that the job is a four-dimensional group of displacements which is not the symmetry group of any surface — so it cannot be one joint, and four is what it costs. The arm's tool face is level everywhere it can reach, and the reason is not that anybody checked.

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 260 × 260 grid and both agree with the area computed from the radii to 0.06%, which is what makes the picture a measurement.

An arm's parameters and its poses

A three-link planar arm has three lengths and a tool position that carries a length, so nothing about it is invisible to a measurement — and it is nevertheless the mechanism in this collection where a calibration is hardest, because its parameter count is high, its poses are three-dimensional and its Jacobian is singular where a designer likes to work.

serial · Serial Where in the workspace each yaw is singular, once the platform tilts. A horizontal slice of the workspace at height 2.4, for the paired platform with its pairs rotated 0° and tilted 1° about an axis 0° from x (the arrow). Each square is shaded by the yaw, measured from the dead yaw 90°, at which that position is singular — found as the root of the six leg screws' determinant — blue where it is below the dead yaw and orange above, deepest at ±0.208°. Level, every position would be singular at the dead yaw exactly. Tilted, the singular yaw is a plane over the slice: over the whole workspace, the yaw times the height fits a·x + b·y to 0.7% of its spread, and the dashed line is where it is nought. The plane's slope turns with the tilt axis. Dragging turns the axis.

The dead yaw opens into a wedge

A paired Gough platform held level is singular everywhere at one yaw and nowhere a quarter of a degree either side of it. Tilt it and each position of the workspace becomes singular at its own yaw, displaced from the dead one in proportion to the tilt and to the position's horizontal offset over its height. So the orientations that put a singularity somewhere in the workspace are not a surface through the dead yaw but a wedge standing on it, about a quarter of a degree of yaw wide on each side for every degree of tilt — and the rotation between the pairs, which chooses where the dead yaw is, barely changes how much room it needs.

parallel · Parallel The joint space an arm's own material forbids. A three-link planar arm, at every pair of relative joint angles on a 101 × 101 grid, with the pairs at which its first and third links are inside each other shaded. It is 21.4% of the arm's own joint space, and it is a two-dimensional picture of a three-dimensional space because the first joint does not enter it: turning the whole arm about its base carries every link with it, so a collision at one base angle is a collision at all of them — checked at forty pairs of relative angles with 0 disagreements. Only the first and third links can meet; consecutive links share a pin and are excluded, exactly as in the closed-loop machines.

The arm that hits itself

A three-link arm's joint space is a cube of angles and it may not use a fifth of it, because its own material is in the way. The forbidden set does not depend on where the arm is pointing — which is why it can be drawn as a picture rather than described as a volume.

serial · Serial The singular set cuts the workspace's poses in two, and every path across it meets it. A slice of the paired Gough platform's poses, tilted 2° about x: the yaw's offset from the dead yaw across, and the position along the workspace's low diagonal, from one corner at height 1.8 to the other, up. Each cell is shaded by the sign of the six leg screws' determinant; the line between the two shades is where it is nought, the singular set. Tilted, it leans: each position is singular at its own yaw, the low corners about 0.48° either side. Three paths from −2.5° to +2.5° are drawn, one held still, one moving, one wandering, and each meets the line: held at the middle, 1 crossing; moving corner to corner, 1 crossing; wandering, 1 crossing. No path from one shade to the other can avoid it.

No path across the dead yaw misses it

A paired Gough platform that has to turn from one side of its dead yaw to the other cannot be steered round the singularity by tilting, by moving, or by turning the long way. The six legs' determinant has one sign everywhere on one side and the other everywhere on the other, so every path passes a singular pose — two hundred random ones do, an odd number of times each. Tilt chooses only where the crossing happens and how briskly: a quarter quicker at best, and threefold slower if it is done by moving instead of turning.

parallel · Parallel 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.

SingularityInverse kinematicsArchitecture singularityJoint limitsParallel mechanismSerial chainSingular valueConstraintDesign ruleDirect singularityDisplacement subgroupPlatform

All concepts