Machine Kinematics
solved, then drawn
Concept

Subalgebra — where it appears

A subspace of the twists that is closed under the Lie bracket, so that doing two of its motions in one order and undoing them in the other lands inside it again. It is what makes a set of screws the algebra of a group of displacements, and among subspaces picked at random above one dimension it never happens.

Named by 13 essays across 5 fields — each of them below, with the objects they name alongside it.

The twelve kinds of freedom, and which are joints. Every connected group of rigid displacements, up to where its axis points and where its origin sits. There are twelve, the height on the page is the dimension, and a line means the lower one is contained in the upper — computed by asking whether each generator of the smaller lies in the span of the larger, with all twelve built about a common axis. Filled discs are joints: six of the twelve are the symmetry group of a surface and can be a single pair, and six are not and have to be built out of a chain. There is nothing at dimension five, which is not obvious and is checked rather than assumed: twenty thousand random five-dimensional subspaces of the twists were closed under the bracket, and every one generated the whole of the six.

Twelve kinds of freedom

Every set of displacements that is closed under composition is one of twelve, up to where its axis points. Six of them are joints somebody sells. Four are motions a designer may perfectly well want and cannot buy at any price. And there is nothing at all of dimension five — checked here on twenty thousand random subspaces, every one of which generated the whole of the six.

pairs · Pair How often a set of screws is a group. Take a subspace of the twists at random and ask whether it is closed under the Lie bracket — whether doing two of its motions in one order and undoing them in the other leaves you inside it. Every one-dimensional subspace is, trivially and importantly: a single screw always generates a one-parameter subgroup, which is the same statement as every screw is a joint somebody could build. Above one dimension, not one of eighty thousand is a group, and every one of them generates the whole of the rigid displacements at the first bracket. So a mechanism whose motion lies inside a proper subgroup is not merely unusual; it is a coincidence of measure zero — and it is the coincidence every planar mechanism, every spherical one and every Sarrus linkage on this site is built on.

Almost nothing is a group

Eighty thousand subspaces of the twists were drawn at random and closed under the Lie bracket. Above one dimension, not one of them was already closed, and every single one generated the whole of the rigid displacements at the first bracket. Two pins with parallel axes close at three; move one axis a hair and they close at six.

pairs · Pair 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 on this site: 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 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 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 on this site: 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.

Legs intersect

A serial chain multiplies its joints' groups and the product is almost never a group. A parallel machine's platform gets the intersection of what its legs permit — and an intersection of groups is a group, always, with no coincidence required. That is the only construction in this field that produces closure for free, and it is why a platform can be designed for a motion type instead of discovered to have one.

pairs · Pair Sarrus, as two planes meeting in a line. Each arm of Sarrus's linkage is three pins with parallel axes, so each arm holds the platform inside a planar group — the one whose normal is that arm's axis direction. The platform has to satisfy both, so what it may do is the intersection, and the intersection of two planar groups whose normals are not parallel is the one-dimensional group of translations along their common perpendicular. The platform goes up and down and does nothing else, and that is the exact straight line the spatial field measured to 10⁻¹⁶ of its span — arrived at here with no mechanism solved and no tolerance anywhere. Drag the arms towards each other: the answer is a translation at every angle but zero, where the two groups become one group and the intersection jumps to three dimensions. That is the linkage built flat, and it is the configuration in which it stops being a straight-line mechanism.

Two planes meeting in a line

Sarrus's linkage draws an exact straight line out of six pin joints, and the spatial field proved it by solving the mechanism sixty times and measuring a departure of 9.8 × 10⁻¹⁶. Here the same fact comes out of two planes and a cross product, with no mechanism solved anywhere — and the two routes are not redundant, because only one of them can tell you the linkage as built delivers it.

pairs · Pair Four instruments, and only the last one names the group. Every instrument this site has for an overconstrained loop, on the same six mechanisms. Kutzbach's count gives −2 for a planar four-bar and −2 for Bennett's. The rank of the constraint Jacobian gives three and three. Both are right and neither separates them. The last two columns are this field's: the span is how many dimensions the logarithms of the displacements the moving link actually reaches occupy, and closes at is the dimension after those are closed under the bracket. A planar four-bar closes at three and the three are planar motion; Sarrus closes at one, a translation, which is the exact straight line the spatial field measured by solving the mechanism sixty times. Bennett closes at six: its displacements occupy four dimensions and no group smaller than all of them contains those four. That is what "paradoxical" has meant on this site for six phases, stated as an integer.

Compose two positions and see where you land

Take two configurations a mechanism actually reaches, compose the displacements that got it there, and ask what kind of thing the result is. A planar four-bar lands inside planar motion, to 4 × 10⁻¹⁶. Sarrus lands on its own line. Bennett's linkage lands three tenths of a radian outside the four dimensions its own displacements occupy — a one-freedom motion that generates all six.

pairs · Pair Four instruments, and only the last one names the group. Every instrument this site has for an overconstrained loop, on the same six mechanisms. Kutzbach's count gives −2 for a planar four-bar and −2 for Bennett's. The rank of the constraint Jacobian gives three and three. Both are right and neither separates them. The last two columns are this field's: the span is how many dimensions the logarithms of the displacements the moving link actually reaches occupy, and closes at is the dimension after those are closed under the bracket. A planar four-bar closes at three and the three are planar motion; Sarrus closes at one, a translation, which is the exact straight line the spatial field measured by solving the mechanism sixty times. Bennett closes at six: its displacements occupy four dimensions and no group smaller than all of them contains those four. That is what "paradoxical" has meant on this site for six phases, stated as an integer.

A name for each overconstraint

The spatial field separated subgroup overconstraint from paradoxical by measuring how far a mechanism's screw system turns: 2 × 10⁻⁶ degrees against 89. That is a verdict without a name. Closing the logarithms of the reached displacements under the bracket gives the same verdict and says which group — planar, spherical, a translation — and for Bennett's linkage it says six.

spatial · Spatial Why this measurement is not a derivative. The honest failure mode of the field's instrument, drawn rather than hidden. Every set looks like its own tangent space near the identity — that is what a tangent space is — so a chain sampled over a thousandth of a radian reports the dimension of its velocities, which is the number the screw system already gives. Four pins at random has four joints, and sampled over 10⁻⁹ radians its displacements occupy four dimensions; sampled over two radians they occupy six. The step is at 10⁻⁶, which is where the departure from the tangent space falls below the rank tolerance — so the position of the step is a fact about arithmetic and the two plateaux are facts about the mechanism. A group is a statement about displacements you could compose, and no derivative can make it.

The instrument that is not a derivative

Sample a chain of four random pins over a millionth of a radian and its displacements occupy four dimensions, exactly as a SCARA arm's do. Sample the same chain over two radians and they occupy six. The step is at 10⁻⁶, and where it sits is a fact about arithmetic while the two plateaux are facts about the mechanism.

pairs · Pair Two pins, and what their bracket costs. The bracket, on the smallest case there is. Two revolute joints span a two-dimensional set of twists whichever way they are arranged, and no count on this site can tell the two arrangements apart. Their brackets can. Two parallel pins bracket to a translation, which was not in the span, and the span closes at three — planar motion, which is the group the pair of them lives in. Two skew pins bracket to something that closes at six: nothing smaller than the whole of the rigid displacements contains them. The defect column is how far the bracket lies outside the original span as a fraction of its own length, and in both cases it is of order one — which is the ordinary case, and is why a mechanism confined to a subgroup is the exception.

One bracket, two subjects

A wheel can be parked sideways because the forbidden direction is the bracket of two permitted ones. A mechanism's motion is a group when the brackets of its permitted twists are already permitted. Same operation, same two plateaux, and the two fields want opposite answers — which is why the two are named for different things here, with each one saying beside itself that the other exists.

rolling · Rolling Oldham's coupling: two slides between two offset shafts. An input hub on one axis and an output hub on a parallel axis 0.6 away, joined by a disc that slides in a slot across the input hub and carries a tongue across the output hub at 90° to the slot. At an input angle of 35° the disc has slid 0.491 along the slot and the output hub -0.344 along the tongue, and the output hub has turned through exactly the input's angle, because neither slide can change an orientation. The disc's centre, the dot, runs round the dashed circle of diameter 0.600 — the offset divided by sin 90° — and goes round it 2 times for each turn of the shafts.

A coupling that only translates

A coupling between two parallel, offset shafts turns its output at exactly the input's speed when, and only when, the relative motion of its two hubs contains no rotation — when it lies in the translation group. Oldham's two slides give that group by construction and so do two equal parallel cranks; a four-bar that is not a parallelogram gives the whole planar group and its output wanders by more than a radian. And Oldham's right angle is not what makes the ratio one: it is what makes the slides slide least.

pairs · Pair 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

Named alongside it

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

Displacement subgroupLie bracketConstraintMobilityRankSchoenflies motionTwistOrbitOverconstraintScrew systemWorkspaceDegrees of freedom

All concepts