Machine Kinematics
solved, then drawn
Concept

Serial chain — where it appears

A chain of links joined end to end with no loop, so that its pose is a product of transforms rather than the solution of a closure equation. Nothing about a serial chain has to converge, and the difficulty moves to the inverse problem, where one tool pose can have eight answers.

Named by 10 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 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 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 this site has measured for 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 One tendon over two joints. A two-link chain with a strand anchored off to the left, running over an idler at each joint and terminating on the far link. Each idler is centred on its joint's axis, which is the one arrangement that makes the strand's length a linear function of the joint angles: the coupling measured here is 14.000000 and 10.000000 mm per radian, against radii of 14 and 10. Total strand 232.845 mm. positioned by solving, not by drawing.

One strand over many joints

Route a tendon over an idler centred on a joint's axis and the strand's length becomes an exactly linear function of the joint angle — 8·10⁻¹⁴ mm of departure over 203 degrees of travel. Move that idler 6 mm off the axis and the same drive's arm swings from 8.00 to 16.52 mm per radian.

strands · Strand The chains that never close, counted by the same routine. An open chain has one joint fewer than it has links, so its graph is a tree and its mobility is the joint count rather than one. Feeding the same enumerator that produced the closed-chain census — with the minimum-degree condition relaxed to one, because an arm's base and its end each carry a single joint — gives 1, 2, 3, 6, 11, 23, 47, 106 for three links up to ten. That is the number of unlabelled trees, a sequence anybody can look up, and reproducing it is the strongest check the enumerator gets: it was written for a different problem, tested against three mechanism counts, and asked here for a number from a different subject entirely.

An arm is a tree

The serial field's chains are the ones that never close, and as graphs they are trees. There are 106 distinct arrangements of ten links joined that way, and exactly one of them is the straight arm every essay in the field has drawn — the other 105 branch.

serial · Serial 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 on this site 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 The common normal, and where it is. Two joint axes 1.0 unit apart, nominally parallel, 0.50° apart in one plane and 0.05° in the other. The Denavit–Hartenberg convention takes all four of its numbers from the one line that meets both at right angles, and for these two axes that line crosses the first 11.3 units from the joint — off this page by a factor of about 11, which is why the tilt here is drawn at 26° and not at a fraction of one. Along it, the "link length" of a link 1.0 unit long reads 0.9950, and the angle round the first axis reads 5.71°. Nothing has moved by more than 0.50°.

The common normal, and where it is

The Denavit–Hartenberg convention reads all four of its numbers off one line: the common normal between two joint axes. Two parallel axes do not have one — every perpendicular meets both at right angles — and two nearly parallel axes have one that is somewhere else entirely.

metrology · Parameter A number that runs away. The Denavit–Hartenberg offset of a pair of nominally parallel axes, against how far from parallel they actually are, for a fixed out-of-plane tilt of 0.05°. The marks are extracted from the geometry by finding the common normal; the line is the closed form −A cos α cos β sin β / (sin²α cos²β + sin²β), and the two agree to 2.4e-16 relative over three decades. At 30° of twist the offset is 0.0030 of a link length; at 0.01° it is 1102. The dashed line is the worst case over the tilt, which sits at β = α and is exactly A/2α. Nothing about the machine has changed by as much as a degree.

A number that runs away

A hundredth of a degree of unintended twist on a nominally parallel pair of joint axes puts the Denavit–Hartenberg offset at −1,102 link lengths. The extraction from the geometry and the closed form agree to 5 × 10⁻¹⁶ over three decades, and the worst case over the tilt is exactly A/2α.

metrology · Parameter Six per joint is two too many. The number of parameters a serial chain's model carries, against the number of revolute joints. The upper bar of each pair counts six for every joint transform and six for each of the base and tool frames, which is what a reader expects; the lower bar is the number a measurement can distinguish. At 6 joints they are 48 and 30. The difference is not a saving, it is a warning: a model with 48 parameters fitted to any amount of data has an 18-dimensional set of exactly equivalent answers, so the fit returns whichever one the damping happens to prefer and every one of its numbers is arbitrary. Four per revolute because a rotation about the joint's own axis changes nothing and a translation along it is the joint variable; six rather than twelve for the two frames for the same reason, one level out.

Six per joint is two too many

A joint transform is six numbers, and a six-joint arm with a base and tool frame is forty-eight. A measurement can distinguish thirty. The difference is not a saving — it is an eighteen-dimensional set of exactly equivalent answers, and a fit returns whichever member of it the damping prefers.

metrology · Parameter

Named alongside it

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

WorkspaceCalibrationIdentifiableDegrees of freedomDH parametersDisplacement subgroupLie bracketSchoenflies motionSubalgebraCommon normalMobilityParallel axis defect

All concepts