Machine Kinematics
solved, then drawn
What a joint is

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.

Assumes Compose two positions and see where you land and A chain multiplies.

15 min read 8 figures Two routes to the same numberExact and approximate

Everything this field measures is computed from sampled displacements, and how widely they are sampled decides the answer.

That sounds like a defect to be apologised for. It is the measurement’s most important property, and this rung is about why.

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. 1 Four revolute joints with axes at random, measured at twelve sampling ranges. Four dimensions on the left, six on the right, and a step at 10⁻⁶.

The two plateaux

The chain is four pins with unrelated axis directions. It has four joints and four freedoms, and its displacements occupy six dimensions — no proper group contains its motion.

Drive its joints over ±2.2 radians and the span comes back as six, which is the answer.

Drive them over 10−910^{-9} radians and the span comes back as four.

Both numbers are correct measurements of what was sampled. Over a range of a billionth of a radian the chain’s reachable displacements are, to every digit double precision has, a four-dimensional flat: the tangent space at the identity, spanned by the four joint screws. Over two radians they are a four-dimensional curved set that occupies all six.

Every set looks like its own tangent space near the identity. That is what a tangent space is. So a measurement made near the identity reports the tangent space whatever the mechanism is, and the difference between a chain whose motion is a group and one whose motion is not is invisible there.

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. 2 The two chains the curve is measured on, at their home positions. Sampled narrowly enough, the two panels’ captions read identically; sampled at the range every measurement in this field uses, they read four and six.

Where the step is, and why

The step sits at 10−610^{-6} radians, and its position is a fact about arithmetic rather than about mechanisms.

The departure of a set from its tangent space, over a range rr, is of order r2r^2; the set itself is of size rr. So the relative departure — which is what a rank decision on normalised vectors sees — is of order rr. When rr falls below the rank tolerance, the departure is discarded as noise and the span collapses to the tangent space’s dimension.

The tolerance here is 10−710^{-7} relative, the same one every rank decision in this field and in the screw-system machinery is made at. The step is at 10−610^{-6} for the four-pin chain and at 10−510^{-5} for the three-pin one, which is exactly the order the argument predicts.

So the curve has three regions and only two of them are about the mechanism: a plateau at the joint count where the sampling is too narrow, a transition whose position is the tolerance, and a plateau at the true answer. Every measurement in this field is made on the right-hand plateau, at ±2.2 radians, and the number is stated in the library rather than tuned.

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. 3 The right-hand plateau, for every chain in the field. Read at a narrow enough range, every bar in this chart would equal its own joint count and the chart would say nothing.

The screw system is the left-hand plateau

This is the whole reason the rung exists, and it settles the field’s relationship with its nearest neighbour.

A mechanism’s screw system is the subspace of twists available at a configuration. It is a first-order object: it is the tangent space to the motion, at a point, and it is exactly what the narrow-range measurement returns.

So the screw system cannot answer this field’s question, and no amount of care with it can. Not because it is imprecise — it is exact — but because the question is this set closed under composition is a question about finite displacements, and a tangent space is the same for every set that touches it.

The spatial field’s separation of the two overconstraints is the sophisticated version of the first-order route, and it works: it does not read the screw system at one configuration but measures how much the system turns as the mechanism moves, which is a derivative of a derivative and does carry information about the finite motion. A planar four-bar’s system stands still to 2×10−62 \times 10^{-6} degrees; Bennett’s turns 89.

That instrument and this one agree on every mechanism they both apply to. What separates them is what they can report. A system that stands still says the motion lies in some subgroup. A closure dimension of three with type GG says it lies in planar motion. And a system that turns says nothing about how far the motion is from a group, whereas the composition defect is a continuous quantity that goes to zero as the mechanism approaches one.

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.
Fig. 4 The instruments in order of what they can say. The count and the rank agree about a dimension; the closure names a group.

The failure mode, asserted rather than hidden

The site’s habit is that every assertion must also be able to reject, and a measurement with a known way of going wrong should have the way of going wrong under test rather than in a comment.

So the library asserts the failure. assertATinyRangeCannotTellThemApart takes four random pins, measures them over 10−910^{-9} radians and over 2.2, and requires the answers to be four and six — that is, it requires the instrument to be fooled by a narrow range, and requires it not to be fooled by a wide one. It also requires the crossover to be at or below 10−510^{-5}, which ties the position of the step to the rank tolerance rather than leaving it free.

An assertion that a measurement works is worth something. An assertion that it fails in exactly the stated way, at exactly the stated scale, is worth more, because it is the one that breaks if somebody quietly loosens a tolerance or narrows a default. The first version of this field’s chain measurement used a range of 1.4 radians and would still have been on the right plateau; the assertion is what makes that a checked property rather than a lucky default.

The same discipline is why the loop measurements report how many of their sampled positions actually converged. A mechanism measured from four frames and one measured from thirty-one are not equally well measured, and the count is in the figure’s own readout.

Two objects that both deserve the name

It is worth naming the two objects clearly, because the site has now got both and they are easy to conflate.

The screw system is a subspace of the six-dimensional space of twists, attached to a configuration, and it varies as the mechanism moves. Its rank is the instantaneous mobility. It is what a Jacobian’s columns span and what reciprocity acts on.

The displacement group is a subgroup of the six-dimensional group of rigid displacements, attached to the mechanism, and it does not vary at all. Its dimension is a bound on the mobility and is generally larger.

For a planar four-bar the screw system is three-dimensional at every configuration and it is the same three-dimensional subspace at every configuration — which is the drift measurement’s zero — and it is the algebra of the planar group. For a mechanism whose motion is a group, the screw system is constant and equal to the group’s algebra. That is the exact statement of how the two instruments are related, and it is why they agree.

For Bennett’s linkage the screw system is three-dimensional and turns; the displacement group is the whole of the rigid displacements; and no relation between the two is available except that the first is contained in the algebra of the second, which is vacuously true.

What a wide range costs

Sampling widely is not free, and being honest about the price is part of the measurement.

A chain has to be driven where it can go. Two radians of every joint is inside every chain in this field, and it would not be inside a machine with limited joints. A measurement of a real robot’s group would have to sample its actual range, and a machine with tight limits would give a narrow-range answer for a physical reason rather than a numerical one — its displacements really do occupy fewer dimensions, because those are the ones it reaches.

A loop has to close. Every displacement in the loop measurements comes from a converged configuration, and a loop’s travel is bounded by where the closure stops having a solution. Bennett’s linkage is measured over ±0.9 radians of its driven joint because that is what it has; the Bricard six-bar over ±0.5, and it reaches seven of the sampled positions rather than nine. A mechanism whose travel is genuinely tiny would be indistinguishable from its own tangent space, and the right response is to report the range rather than the number.

And the sampling has to be of the mechanism. A configuration that failed to converge is discarded rather than drawn, which matters more here than usual: an unsolved sample is a displacement the mechanism does not make, and the composition test would be entitled to find it outside the span. Bennett’s motion reports nine of nine sampled positions closed; a Bennett loop detuned by six per cent reports none, which is the check that the machinery notices rather than measuring a group from configurations that were never solved.

Bennett's linkage, and the group it moves inFour pins with skew axes and a condition on the lengths. Nothing about it is planar, spherical or translational. 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 **6**, so no group smaller than all the rigid displacements contains it — the displacements occupy 4 dimensions and their products leave those 4 by 0.31 of their own length. positioned by solving, not by drawing.4 joints · Bennett's linkageinside nothing smaller than SE(3)
Fig. 5 Bennett’s linkage at the end of its measured travel. Nine sampled positions, nine converged, and the range is the mechanism’s rather than a choice.

The same shape in the rolling field

The site has met this exact structure once before in another costume, and putting the two side by side is the clearest way to see what kind of measurement this is.

A rolling wheel forbids a velocity and not a position. At any instant the wheel may move in two directions out of three, and the third — sideways — is forbidden. Read that at first order and the wheel is confined to a two-dimensional set. Read it over finite motions and the wheel can be parked anywhere at any angle: the reachable set is three-dimensional, and the forbidden direction is the bracket of two permitted ones.

The two plateaux are the same two plateaux. Narrow sampling gives the tangent space, which for a wheel is two; wide sampling gives what is reachable, which is three; and the gap is the bracket.

The difference is which answer each field wants. The rolling field wants the gap to be non-zero, because that is what makes a car parkable, and it measures the exponent at which the gap is exploited — quadratic in the size of the shuffle. This field wants the gap to be zero, because that is what makes a motion a group, and it measures how far from zero the gap is.

Same operation, same two plateaux, opposite hopes. That collision is worth an essay of its own, and it is why the two brackets on this site are named for different things — one for the operation and one for the algebra it acts on.

What the curve is good for

Beyond the methodological warning, the span-against-range curve is a measurement worth having in its own right.

Its left plateau is the joint count, which is the mobility. Its right plateau is the dimension of the smallest group containing the motion. The gap between them is how much a mechanism’s finite motion exceeds its instantaneous motion, and it is a number nothing else on this site reports.

For a SCARA arm the gap is nought: four and four. For a wrist, nought: three and three. For four random pins it is two: four and six. For a Bennett linkage, driven rather than sampled freely, the gap is between one and six.

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. Three pins very nearly parallel 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. 6 The same curve for three pins that are nearly parallel. Three on the left, six on the right, and the step one decade further out — because a smaller chain departs from its tangent space more slowly.

A mechanism whose two plateaux agree is one whose behaviour is completely described by its velocities. That is a strong and useful property, it is exactly what being in a group means, and it is why the SCARA arm’s tool stays level rather than merely starting level.

Why the answer is not just “sample more widely”

A reader who has followed this far might reasonably conclude that the fix is to always sample as widely as possible, and that the narrow-range plateau is simply a mistake to be avoided. Two reasons that is not the whole story.

The narrow-range answer is the right answer to a different question, and it is the question most of kinematics asks. Instantaneous mobility, velocity ratios, singularities, transmission angles, screw systems, reciprocity — every one of them is a first-order quantity, and computing it from a wide sample would be wrong. The site’s own two routes to a Jacobian are two ways of getting the tangent space, and neither is improved by asking about finite displacements.

And a real mechanism has a range. A machine that moves through a tenth of a radian genuinely occupies a set close to its tangent space, and saying that its displacements “really” span six dimensions is a statement about a mechanism with unlimited joints rather than about the one on the bench. The honest report for such a machine is the pair: the span at its own range, and the span at full travel, with the difference being the amount of group-ness the limits are giving it for free.

That second point has a practical edge. A machine whose joints are limited to a small range is, to a good approximation, in a group even if its ideal version is not — which is one reason a badly-aligned SCARA arm with short strokes still behaves acceptably, and why the misalignment shows up as the strokes get longer.

Verify the plateau, do not trust the point

The measurement is trustworthy on a plateau and meaningless at the step, and since the step’s position is a fact about arithmetic rather than about the mechanism, a single measurement cannot say which of the three regions it landed in. That gives the field a protocol rather than a caveat.

Measure at the range, then at half of it and at double. Three numbers. If all three agree, the measurement is on a plateau and the answer is the plateau’s; if they differ, the measurement is on the step and the number means nothing about the mechanism. That is two extra measurements against the risk of quoting a transition value as a result, and the transition is the one region where a wrong answer looks exactly like a right one.

It also says what to report, which is not a number. The curve is the measurement, and a paper’s worth of it fits in two integers and a range: the left plateau, the right plateau, and the range at which the reported figure was taken. Anybody who has those three can tell whether the reported figure is a tangent-space dimension, a global dimension, or an artefact — and cannot tell any of it from the figure alone.

The gap between the plateaux is the quantity with the content. Four and four is a SCARA arm, whose velocity freedoms describe it entirely. Four and six is a random chain, whose velocities describe four dimensions and whose reachable set fills all of them. The gap counts what the derivative cannot see, and it is exactly the quantity the site’s rolling field calls a growth vector computed at the other end.

None of that makes the narrow answer worthless, which is worth repeating because the essay’s argument runs so hard the other way. The tangent space is the right object for a velocity question — what can this mechanism do now, what does its Jacobian permit, what does a controller command — and every one of those is asked at a configuration. The two plateaux answer two questions, and the measurement’s failure is not in either answer but in a reader taking one for the other.

Which is the general form. A sampled measurement of a set measures whatever the sampling reached, and a set with a tangent space of a different dimension has two answers available to the same routine. Reporting the range is what separates them, and reporting the curve is what proves the range was far enough.

A general caution

The lesson generalises past this field and is worth stating in the site’s own terms.

Any measurement that samples a set near a point measures the tangent space, and a great many measurements in kinematics do. A Jacobian is a tangent space. A sensitivity is a tangent space. A tolerance analysis that linearises about nominal is a tangent space, and the tolerance field says so explicitly and checks its linear route against a swept one for that reason.

What this field adds is a case where the tangent space and the object it touches have different dimensions, so the difference is not a matter of accuracy but of kind. Four and six are not a small error and a large one; they are two answers to two questions, and the narrow-range measurement is answering the other one perfectly.

The practical rule that comes out of it is short. Before quoting a dimension, say what was sampled and over what range. Every table in this field does, every figure’s readout carries the count of converged samples, and the library’s own defaults are written down rather than chosen at each call site. That is not fussiness: it is the difference between a number that means something and a number that means whatever the sampling happened to be.

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. 7 The smallest version of the same fact. Two pins have a two-dimensional tangent space whichever way their axes point, and the finite motion they generate is three-dimensional or six.
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. 8 And the quantity that survives the caution. The defect is measured from finite composed displacements, it is continuous, and it separates the two populations by twelve orders of magnitude without any threshold being chosen.

Where this goes next

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. Whose constraints stay put. The largest principal angle between a mechanism's screw system at the start of its motion and at each later position. The 3 mechanisms whose motion lies in a subgroup of the rigid displacements — planar, spherical, translational — never leave the same subspace, and read between 1.9e-6 and 3.0e-6 degrees, which is the precision of an arccosine near one rather than a movement. The paradoxical ones turn through 49° and 57°. Every curve runs over its own range of motion, because Bricard's linkage assembles over 120 degrees and a shared axis would hide it. Two ways to be overconstrained A planar four-bar and Bennett's four-bar report the same redundancy, the same rank and the same wrong count. One of them is overconstrained at every set of link lengths; the other at exactly one ratio and nowhere near it. The difference is not in any of the numbers so far — but it is measurable, and the measurement is an angle. 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.

About the same objects

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

What links here

The 8 of 10 essays linking to this one that name the most of the same objects.

The objects this essay names

Each one links to every other essay that touches it.

ConstraintDisplacement subgroupLie bracketRankScrew systemSubalgebraTangent spaceTwist