Machine Kinematics
solved, then drawn
What a joint is

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.

Assumes A chain multiplies and Almost nothing is a group.

15 min read 8 figures The singularities are the pointTwo routes to the same number

Take two arms.

The first has three revolute joints with parallel axes and a prismatic joint sliding along the same direction. That is a SCARA arm, and there are tens of thousands of them in factories.

The second has four revolute joints whose axes point in four unrelated directions. That is an ordinary four-axis arm and there is nothing wrong with it.

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.
Fig. 1 The two chains at their home positions, with the joint axes dashed and a cloud of the tool positions each reaches. Every count anybody would apply to the two is identical.

Now put every instrument this site owns on them.

Link count: four moving links each. Joint count: four one-freedom pairs each. Mobility by Grübler and Kutzbach: four each — an open chain’s mobility is its joint count, which the serial field established as the reason an arm has no closure equation to solve. Rank of the Jacobian: four each, away from singularities. Number of assembly configurations: not applicable to either, since there is no loop. Workspace dimension: both reach a four-dimensional set of tool poses.

Every one of those numbers agrees, and the two machines are not the same kind of object.

The one number that differs

Sample each chain over random joint values, take the logarithm of every pose reached, and count the dimensions those logarithms occupy.

The SCARA’s occupy four. The random arm’s occupy six.

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.
Fig. 2 Every chain in this field. The rows where the bar equals the joint count are the ones whose motion lies inside a group; the two rows that overshoot to six are the ones whose axes were chosen without a condition.

Four means the reached displacements are inside a four-dimensional group, and the classifier names it: Schoenflies motion, every translation together with a rotation about the axis direction. Composing two of the arm’s displacements gives another displacement of the same kind, and every pose the arm reaches has its tool face at the same attitude about two of the three axes.

Six means there is nothing to be inside. The random arm’s reached set is four-dimensional too — it has four joints and four freedoms and its tool poses form a four-parameter family — but describing that family takes six numbers, because it is a curved slice of the whole displacement group rather than a piece of a smaller one.

A four-dimensional set is not the same thing as a four-dimensional group, and until this field the site had no way to say which one a mechanism had.

The gap is worth making concrete rather than leaving as a difference of two integers. Suppose the SCARA arm is at one pose and the question is which poses are reachable from there by the same machine. Because the reached set is a group, the answer is the same set again, translated: whatever was reachable from the home pose is reachable from this one, carried over by the displacement that arrived here. Nothing has to be recomputed.

Do the same with the random arm and there is no such statement. What is reachable from a given pose depends on which pose it is, in a way that has to be worked out configuration by configuration. That is not an inconvenience; it is the ordinary situation for a mechanism, and the SCARA’s exemption from it is the whole reason its geometry was chosen.

Six hundredths of a radian

The comparison above uses two obviously different arms, which makes it easy to believe the difference is about something else — link geometry, or how far apart the axes are. So here is the same finding with the difference made as small as possible.

Three revolute joints with axes exactly parallel: span three, and the group is planar motion.

The same three joints with two of the axes tilted by 0.060.06 radians — three and a half degrees, which is a bad alignment and not an absurd one: span six.

Nothing else changes. The joints are in the same places, the links are the same lengths, the counts are identical, and the rank of the Jacobian is three in both cases at every configuration away from a singularity.

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.
Fig. 3 The mechanism of the jump, on two joints instead of three. Two pins with parallel axes bracket to a translation and close at three; two skew pins bracket to something whose closure is six. The bracket lies a full hundred per cent outside the original span in both cases.

The dimension is not a continuous function of the alignment. It is three at exactly zero tilt and six at every other tilt, with nothing in between at any angle, however small. That is what a measure-zero condition looks like on approach: not a gradual loss, but a cliff at exactly one value.

What varies continuously, and is the number to quote

A machine is made to a tolerance. Its axes are parallel to a few arcminutes, never exactly. So a dimension that is three at exactly zero and six everywhere else is the wrong quantity to hand an engineer, and this field would be dishonest if it stopped there.

The right quantity is the defect: how far the brackets lie outside the span, as a fraction of their own length. It is a continuous function of the alignment, it goes to zero as the tilt does, and it is what the closure test actually measures — the dimension is a threshold applied to it.

A perfectly aligned chain reports a defect at the floor of double precision. The chain tilted by 0.060.06 radians reports a defect of order one. In between, the defect scales with the tilt, and what it says is not the mechanism is not planar but the mechanism is this far from being planar.

That is the same reading the practice field gives every exact statement on this site. Peaucellier’s cell is exactly straight and a real one is straight to its clearances; a planar four-bar is exactly planar and a real one is planar to its alignment. The exact statement is about the geometry and the tolerance is a separate measurement, and confusing the two produces either false precision or false alarm.

Compose two positions and see where you land. The group axiom run as an experiment. For each loop, take two displacements the moving link actually reaches, compose them, and measure how far the result lies outside the set the reached displacements span. The four trivial loops come back at the floor — and the floor here is the solve, not the arithmetic, because a configuration is a root found to 10⁻¹³ and its logarithm inherits that. Bennett's linkage and the Bricard six-bar come back at two to three tenths. There is no threshold between the two answers; there are twelve orders of magnitude.
Fig. 4 The defect as a quantity rather than a threshold, on the field’s six loops. The trivial ones sit at the solver’s own floor and the paradoxical ones at two to three tenths — twelve orders of magnitude apart, with nothing between.

Reading the two clouds

The clouds of reached tool positions in the opening figure are worth a paragraph, because they look like the answer and are not.

Both clouds are three-dimensional blobs, and they are roughly the same size, because both arms have links of about the same length. Nothing about their shapes says which arm is in a group. That is not a failure of the drawing: a position is three numbers and a displacement is six, so a cloud of positions has thrown away the attitude before the reader ever sees it, and the attitude is where the difference lives.

The SCARA arm’s tool face is level at every point of its cloud. The random arm’s tool face points somewhere different at every point of its cloud. A picture of positions cannot show that, and this is the same limitation the orbit rung states in general: a point sees the orbit and never the stabiliser, so a cloud of one point’s positions is evidence about a group and never a determination of one.

Which is why the caption under each panel carries a number the picture cannot. The span is computed from full displacements — logarithms of four-by-four transforms — and it is the only thing on the page that distinguishes the two machines.

Why anyone builds the aligned one

If the aligned chain is a coincidence and the misaligned one is generic, the obvious question is why the coincidence is worth engineering for.

The answer is that being in a group is a guarantee that transports. If a machine’s displacements are inside Schoenflies motion, then its tool face is level at every reachable configuration — not at the configurations somebody tested, and not to within an interpolation between them. The statement holds for the whole set at once, because a group looks the same from every one of its elements.

That is a strong property and it is exactly what a pick-and-place machine is sold on. Nobody checks a SCARA arm’s levelness pose by pose; the geometry guarantees it, and the guarantee is a group-theoretic one whether or not anybody says the word.

The random arm has no such guarantee, and does not need one: it is built to reach general poses and the tool’s attitude is commanded rather than constrained.

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.
Fig. 5 The SCARA’s reach. Every pose in the cloud has the tool face level, and the reason is not that the cloud was checked — it is that the four joints’ groups all lie inside one four-dimensional group.

What it costs to lose it

Two consequences of the cliff, both of them practical.

Alignment is a functional requirement rather than a quality one. A SCARA arm whose axes are out of parallel does not have slightly worse Schoenflies motion; it has no Schoenflies motion, and its tool face tilts by an amount proportional to the misalignment as the arm sweeps. The specification that keeps the machine in its group is the parallelism of three axes, and it is not obvious from any mobility count that this is the specification that matters.

A model that assumes the group is exact will not fit a real machine. What a calibration cannot see is a parameter that does not move the tool, and a calibration that fits a SCARA arm within the Schoenflies model has no parameter for axis tilt at all — so the tilt shows up as unexplained residual and is attributed to whatever parameter is nearest. That is a real failure mode of a real procedure, and its cause is that the model was chosen from a group and the machine is not exactly in one.

The near-miss is the honest example

Of the two comparisons in this essay, the second is the one that should be remembered, and it is worth saying why.

A SCARA arm against a four-axis arm is a comparison between two designs. Somebody chose the axes, and the choice is visible in the drawing: one machine obviously has parallel joints and the other obviously does not. A reader can be forgiven for thinking the group-theoretic apparatus is a formal way of noticing something that was already obvious.

Three parallel pins against three nearly-parallel pins is a comparison between one design and the same design built badly. The drawings are indistinguishable at any reasonable scale. Every count is identical. And the answer changes completely.

Nor is the badly-built version hypothetical. Three parallel bores in a casting are parallel to the accuracy of a jig; a robot’s link is assembled from parts with their own tolerances; a machine that has been dropped is not the machine that left the factory. The nominal geometry is in a group and the delivered geometry is not, always, on every machine of this kind ever built.

That is the case the instrument is for, and it is also the case where every other instrument on the site is silent. The count reads two integers and cannot see an angle at all. The rank reads a matrix of positions and returns three in both cases, because three is the mobility and the mobility is unaffected. The screw system reads the twists available at one configuration, and at any single configuration the two chains’ screw systems are three-dimensional and nearly identical — the difference is that one of them turns as the mechanism moves and the other does not, which is the spatial field’s drift measurement and is a derivative rather than a statement about composing displacements.

The same story with a loop in it

The comparison generalises to closed mechanisms, and there the stakes are higher, because a closed loop with the wrong alignment does not merely behave differently — it does not assemble at all.

The parallelogram with a third parallel bar is the site’s oldest example: Grübler counts zero degrees of freedom and it moves, because the three bars being parallel and equal is an exact condition. Make one of them a hair long and the mechanism is rigid. That is the same cliff, in a loop, where the consequence is not a tilted tool but a machine that will not go together.

A planar four-bar, and the group it moves inFour pins with parallel axes. Kutzbach counts −2 and it moves. The dashed stubs are the joint axes, the solid marker is one point of link 1 and the faint curve is everywhere that point goes. Every frame is a **solve**: the joint angles are the unknowns, the closure of the loop is the equation, and a frame is drawn only where the residual comes below 10⁻⁹. What this field adds to the picture is one number. Take the displacements this link reaches, take their logarithms, and close them under the bracket: the answer is **3**, so the motion lies inside planar motion and composing two of its displacements gives a third one it also reaches, to 4.4e-16. positioned by solving, not by drawing.4 joints · a planar four-barinside G
Fig. 6 A planar four-bar built as a spatial loop. Kutzbach counts −2 and the mechanism moves; the reason is that all four axes are parallel, which is exactly the condition that puts every joint’s group inside one planar group.

And Bennett’s linkage is the same cliff at a condition that is not an alignment at all — a relation between four lengths and four twist angles — and there the group that appears at the bottom of the cliff turns out to be no group. That is the field’s central result and this rung is its warm-up: the machinery that tells a SCARA arm from a four-axis arm is the machinery that tells Sarrus from Bennett.

How wide the sampling has to be, again

One methodological point has to be repeated here because this essay is where somebody would try to reproduce the measurement and get the wrong answer.

The span is computed over joint values sampled across ±2.2 radians. Sample the same random arm over a millionth of a radian and it reports a span of four — indistinguishable from the SCARA — because over a range that small every set is its own tangent space and the tangent space has the dimension of the joint count.

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.
Fig. 7 The four random pins, measured at twelve sampling ranges. Four on the left and six on the right, with the step at 10⁻⁶ where the departure from the tangent space falls under the rank tolerance.

So the number in this essay is not a property of the arm alone; it is a property of the arm measured over a stated range, and the range is part of the claim. That is a rung of its own and it is the sharpest statement of why this field is not a restatement of screw theory: the screw system is the left-hand plateau, and everything this field measures lives on the right-hand one.

The tolerance belongs to the relation, not to the parts

The cliff makes alignment a functional requirement, and it is worth following that into how such a machine is specified and made, because the consequence is a rule about datums rather than about accuracy.

Every one of the twelve groups is picked out by a relation between joint axes: parallel, intersecting, perpendicular, coaxial. Not one of them is a property any single joint has. A bearing bore is round or it is not, and its roundness says nothing about whether it is parallel to the bore next to it — so the quantity that decides which group a chain reaches is not a property of any part on the drawing.

That means the tolerance cannot be written on a part. Three bearings each made to a micron, assembled into a casting whose bores were bored in three separate setups, gives three axes whose parallelism is the accumulation of three setups’ errors — and the chain is generic, with a defect that has nothing to do with how good the bearings were. Specifying the parts tightly buys nothing at all against the failure that matters.

What buys it is a datum strategy: bore all three in one setup, on one machine, without releasing the work. Then the parallelism is the machine’s own slideway rather than the sum of three fixturings, and it is one error instead of three. That is why a SCARA arm’s shoulder and elbow housings are made as one casting and machined together, and why the manufacturing drawing carries a parallelism callout between features rather than a position tolerance on each.

The general rule is worth stating in the form a designer would use. When a mechanism’s behaviour depends on a relation between features, the relation is the thing to tolerance and the setup is the thing to control. A part-by-part tolerance stack is the right instrument for a dimension and the wrong one for an alignment, and the difference is exactly the difference between a quantity that varies continuously and one that sits on a cliff.

It also says what to do when the setup cannot be controlled — a large machine, features on separate castings, an arm that is assembled rather than machined. Then the alignment is measured after assembly and the model is corrected, which is calibration rather than manufacture. The group is not recovered by that; a calibrated near-SCARA is still generic, and what the calibration buys is an accurate model of a machine that is not in a group rather than a machine that is.

What the two arms share

Ending on what is not different is worth doing, because the essay could otherwise read as an argument that a count is worthless.

Both arms have four freedoms, and the count that says so is right about both. Both have a four-dimensional reachable set, and the rank that says so is right about both. Both are positioned by multiplying four transforms, and neither has a closure equation. Every statement the site has ever made about serial chains applies to both without amendment.

The count is not wrong; it is the dimension of a group, and there are twelve groups. What this field adds is the other coordinate. And the reason that coordinate matters here rather than in the abstract is that two machines can agree on the count, disagree on the group, and be sold for different jobs at different prices to people who could not name the difference.

What it takes to build each of the twelve. The same twelve, read as a bill of materials. Six of them are one joint, because a lower pair permits the whole symmetry group of its surface and those six groups are exactly the symmetry groups surfaces have. The other five with a dimension take a chain: two slides for planar translation, three for Cartesian motion, a thread and two slides for the screw-in-a-plane group, and three parallel pins with a slide along them for Schoenflies motion — which is a SCARA arm, and is why a pick-and-place machine has four joints and not one. The group each chain produces is measured from four hundred sampled poses rather than declared, and every row agrees.
Fig. 8 The twelve, with what each costs in joints. The SCARA’s row is the four-dimensional one, and the reason it is four joints rather than one is in the third column: Schoenflies motion is not the symmetry group of any surface.

Where this goes next

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. 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. 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.

About the same objects

Not linked from either essay — found by the objects both name.

What links here

Essays that link to this one from their own argument.

The objects this essay names

Each one links to every other essay that touches it.

Displacement subgroupLie bracketMobilityRankSchoenflies motionSerial chainSubalgebraToleranceWorkspace