Machine Kinematics
solved, then drawn
Wheels, and where they may not go

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.

Assumes A constraint that takes nothing away and Almost nothing is a group.

15 min read 9 figures Two routes to the same numberThe singularities are the point

The rolling field is built on a Lie bracket. A wheel forbids a velocity and not a position, and the reason is that the forbidden sideways direction is the bracket of two permitted ones — so it is reachable at second order in the size of a shuffle, which is why a car can be parked and why the exponent is two.

The pairs field is also built on a Lie bracket. A set of twists is the algebra of a group of displacements exactly when its own brackets are already inside it, and almost no set of twists is — eighty thousand random subspaces, above one dimension, and not one closed.

Same word, same operation, and the two are not the same function. This rung is about the difference, because the collision of names is exactly the kind of thing this fleet has been caught by before.

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. 1 The pairs field’s version at its smallest: two revolute joints, and whether the bracket of their twists is already in the span. Parallel axes close at three; skew axes close at six.
What one point can and cannot see of a group. For each of the twelve: the group's dimension, the dimension of one point's orbit under it, and the difference — the stabiliser, the motions that leave that particular point exactly where it is. The orbit is the only picture a group has, and this table is the honest caption on it. Planar motion and spherical motion are both three-dimensional and both sweep a point over a two-dimensional surface, so each leaves one motion doing nothing at all: a turn about the plane's normal in one case, a turn about the radius in the other. A point does not see the whole group, and no drawing of one trajectory can be a complete picture of what a joint permits.
Fig. 2 The pairs field’s twelve outcomes, with what one point sees of each. A rolling constraint has no such table: its distribution is not the algebra of anything, so there is nothing to classify.

Two objects

The rolling field’s bracket takes two vector fields on a configuration manifold. A wheeled vehicle’s state is a position and a heading; the motions it may make are a distribution — a subspace of the tangent space at each state — and the subspace is different at each state, because which way is “forward” depends on the heading. The bracket is computed by finite differences: move along one field, then the other, then back along the first, then back along the second, and see what is left.

The pairs field’s bracket takes two twists. The twists are six-vectors, the bracket is an exact bilinear map with constant structure constants, and the answer is the same wherever it is evaluated. There is no manifold, no configuration and no differencing.

The rolling one is a bracket of fields; the pairs one is a bracket of vectors in a fixed Lie algebra. The second is a special case of the first — the algebra of a Lie group is the brackets of its left-invariant fields — and the specialisation removes everything that makes the rolling field’s version hard.

The two plateaux, in both fields

The structural similarity is real and worth drawing out, because it is the same picture twice.

In the rolling field: at any instant a wheel may move in two directions out of three. Read at first order, the reachable set is two-dimensional. Read over finite motions, the wheel can be parked anywhere at any angle and the reachable set is three-dimensional. The gap is the bracket.

In the pairs field: over a small enough sample a chain’s displacements occupy the dimension of its tangent space, which is its joint count. Over a wide sample they occupy the dimension of the smallest group containing them. The gap is the bracket.

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. 3 The pairs field’s version of the two plateaux. Four dimensions on the left, which is the tangent space and is the joint count; six on the right, which is what the finite displacements occupy.

Both fields measure a first-order object, a finite object, and the gap between them. The rolling field calls the gap a growth vector and reports the sequence (2,3)(2,3) for a wheel, (2,3,4)(2,3,4) for a car and (2,3,4,5)(2,3,4,5) for a car and trailer. The pairs field calls it the difference between a span and a closure, and reports four and six.

The two are not merely analogous, either. In both cases the first-order object is a tangent space and the finite object is what iterated brackets generate, and in both cases the reason a narrow measurement gives the wrong answer is the same: a set agrees with its tangent space to first order, by definition, so a measurement that cannot see second order cannot see the difference. The rolling field states it as the wiggle is the bracket and measures the amplitude exponent; the pairs field states it as the span is not a derivative and measures where the plateau steps.

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 finite object measured directly rather than through a dimension. Compose two displacements a mechanism reaches and see how far outside its own span the product lies: at the solver’s floor for the mechanisms whose brackets close, and at two to three tenths for the ones whose do not.

Opposite hopes

Where the two fields part company is what they want the answer to be.

The rolling field wants the bracket to be non-zero. A distribution that is closed is integrable, and an integrable velocity constraint is really a position constraint: a rail rather than a wheel. The whole subject of the field is that a wheel’s constraint is not integrable, that everything is reachable anyway, and that the cost of reaching it is quadratic in the size of the manoeuvre. Non-closure is the liberation.

The pairs field wants the bracket to be zero. A subspace that is closed is a group’s algebra, and a mechanism whose motion is a group has a motion type — planar, spherical, Schoenflies, a translation — with everything that follows: a guarantee that transports, an orbit that is a surface, an intersection rule for legs, a name from a list of twelve. Non-closure is the disqualification.

And in both fields the ordinary answer is not closed. A wheeled vehicle is generically controllable; a set of twists is generically not a subalgebra. So the two fields are looking at the same background fact from opposite sides, and each finds its exceptions interesting for opposite reasons.

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.
Fig. 5 The background both fields are read against. Above one dimension, no subspace of the twists drawn at random is closed, and every one generates the whole six.

What each bracket is measuring the failure of

A sharper way to put the difference: both brackets measure how badly something fails to be true, and the somethings are different.

The rolling field’s bracket measures the failure of a distribution to be integrable — that is, the failure of a velocity constraint to be a disguised position constraint. When it vanishes, the mechanism is confined to a surface in its configuration space and its history does not matter. When it does not, the mechanism can go anywhere and the route matters, which is the whole subject.

The pairs field’s bracket measures the failure of a subspace to be a subalgebra — that is, the failure of a set of permitted motions to be closed under doing them in succession. When it vanishes, the mechanism’s motion has a type and everything true near one configuration is true near all of them. When it does not, there is no type.

Integrability is about where a mechanism can get to; closure is about what kind of displacement it makes. A mechanism can be nonholonomic and inside a group — a rolling disc’s contact motions are constrained in the first sense and its body displacements are perfectly ordinary in the second — and it can be holonomic and inside nothing.

Where the rolling field’s is harder

Three ways, and they are why the two functions cannot be shared.

The distribution varies. A wheel’s permitted directions depend on its heading, so the bracket has to be evaluated at a configuration and can be different at another. A subspace of twists does not vary at all.

It is computed numerically. The rolling field differences the flows of two vector fields with a step of 10−510^{-5} and reads what is left over, which is a second-order quantity extracted from a difference of two nearly-equal displacements — a cancellation that field has to be careful about. The pairs field’s bracket is an algebraic identity in six numbers.

And it iterates. The rolling field’s growth vector is a sequence of dimensions — the distribution, then its brackets, then the brackets of those — and the sequence can take several rounds to fill the space. The pairs field’s closure terminates in at most five rounds by a dimension argument and in practice takes two: eighty thousand random subspaces, every one at six after the first bracket.

Span, and what it closes to. Two bars per loop: how many dimensions the reached displacements occupy, and how many they occupy after the brackets are added. For the four trivial loops the two bars are equal — the motion is already inside a group and bracketing adds nothing. For Bennett's linkage and the Bricard six-bar the first bar is four and the second is six, and the gap between the two bars is the whole of what paradoxical means: a one-degree-of-freedom motion that occupies four dimensions of displacement and generates all six.
Fig. 6 The pairs field’s version, on six mechanisms. Two bars each, the dimensions occupied and the dimensions generated, and the gap between them is what the bracket added.

A third bracket that is not here

For completeness, because a reader who knows the subject will wonder: there is a bracket in the constraint field’s neighbourhood too, and it is not either of these.

The reciprocal product pairs a twist with a wrench and vanishes when the wrench does no work on the twist. It is a bilinear form, symmetric, and it is the operation behind screw reciprocity: a mechanism’s constraints are the orthogonal complement of its freedoms under it.

It is not a bracket. It takes two screws and returns a number; a bracket takes two screws and returns a screw. Confusing the two would be a type error rather than a subtle mistake, which is why the site has never had trouble with it — but the three operations together are the reason screw-theoretic writing is hard to read, and naming which is which once is worth the paragraph.

So: a reciprocal product answers does this constraint resist this motion. A bracket of twists answers what does doing two motions in the wrong order leave over. And a bracket of vector fields answers the same question when the motions are configuration-dependent.

The naming, and why it is recorded

The two routines sit a few steps apart in the same machinery, and they are given different names on purpose: one is named for the bracket, the other for the algebra it acts on, and each says beside itself that the other exists and is not the same function.

That is not fussiness. The fleet this site belongs to keeps a record of three implementations of gauss that turned out to be three different functions sharing a word, and of a line clipper that had to be renamed because it cut against a square in fractional coordinates rather than against a rectangle in real ones. A shared name is not a shared function, and the failure mode is a later reader reaching for the wrong one and getting an answer rather than an error.

The hazard here was live. Both functions take two arguments and return something bracket-shaped; both are about kinematics; both are used to decide whether a set is closed. A reader reaching for the bracket named after the operation, and handing it two twists, would land in a routine expecting vector fields on a manifold, which would either throw or — worse — return a number.

Naming the second one for its algebra rather than for the operation is the whole of the fix, and it costs a word.

The same distinction in the two fields’ vocabulary

Both fields have a word for a constraint that fails to be what it looks like, and the words point in opposite directions.

The rolling field’s is nonholonomic: a constraint on velocities that is not the derivative of any constraint on positions. A wheel is nonholonomic; a rail is not; and one character apart is the field’s demonstration that a single sign in a constraint row decides which. What makes a constraint nonholonomic is that its permitted directions are not closed under the bracket.

The pairs field’s is paradoxical: an overconstrained mechanism whose motion is inside no proper subgroup. What makes a mechanism paradoxical is that its permitted twists are not closed under the bracket.

The same condition, and the two words are almost antonyms in their connotation. Nonholonomic is a description of something useful — every wheeled vehicle is nonholonomic and would be far less useful if it were not. Paradoxical is a description of something anomalous — Bennett’s linkage is a curiosity that took decades to be believed.

That the two subjects landed on such different words for one algebraic fact is a small piece of evidence for how independently they developed, and a reason to be careful reading either literature with the other’s expectations.

What each field takes from the other

Two transfers, and both are real.

The rolling field’s discipline about the sampling range is what the pairs field needed and did not have at first. A distribution’s integrability is a first-order question asked at a point; a group’s closure is a finite question asked over a range; and the pairs field’s first measurement was made over too narrow a range and reported a chain’s tangent space as its group. That failure is now asserted rather than avoided, which is a habit the rolling field’s careful separation of forbidden direction from unreachable direction made obvious.

And the pairs field’s constant structure is what the rolling field’s is a generalisation of. A wheeled vehicle’s configuration space is not a group, so its distribution has no constant algebra behind it — but where a system’s configuration space is a group and its constraints are left-invariant, the two collapse into one, and the growth vector becomes a filtration of subalgebras. The ball that remembers where it has been is close to that case, and the field’s treatment of it is the site’s nearest approach to the algebraic version.

One point, six groups, six shapes. A group has no picture, so here is the next best thing: fix one point of the moving body — the marked one — and draw everywhere the group can send it. A prismatic pair sends it along a line, a revolute round a circle, a helical along a helix, a cylindrical over a cylinder, a spherical over a sphere, a planar over a plane. Those six shapes are the six surfaces the previous figures drew, which is not a coincidence and is the field's first argument read backwards: a lower pair's surface is an orbit of its own group, which is exactly why the surface can slide on itself.
Fig. 7 The six shapes the pairs field’s closed cases produce. The rolling field has no equivalent, because a distribution that closes is a rail rather than a wheel and the field’s subject is the ones that do not.

What would have gone wrong

It is worth being concrete about the failure the naming prevents, because “two functions with one name” is only alarming when the consequence is spelled out.

Suppose a later essay asks whether a wheeled mechanism’s reachable displacements form a group — a perfectly sensible question, since a trolley’s configuration space really is a group and its constraints really are left-invariant. The natural thing to reach for is the bracket the rolling field already has, handed two twists.

The rolling field’s routine expects a system, a configuration and two indices into a distribution. Passing it twists would throw, which is the good case. The bad case is a caller who adapts the arguments until it returns something: a finite-difference bracket of two constant fields, evaluated at a configuration, with a step of 10−510^{-5} — which would return the right answer to about five digits and would be reported as a bracket computed exactly. A quantity that is nearly right and produced by the wrong routine is exactly the shape of error this site’s assertions exist to catch and its naming conventions exist to prevent.

The one this field uses is named for the algebra it acts on rather than for the operation, takes two six-vectors and returns one. Nothing about it invites the other use, and it names the other routine beside itself and says what that one does instead.

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. 8 The measurement the naming protects. Every row here is a bracket closure over a chain’s sampled displacements, and every one of them would have a plausible-looking wrong answer available if the wrong bracket were reachable by the same name.

One is the other, restricted

The two brackets are described as the same operation on different objects, and the relation between them is tighter than that: the pairs field’s bracket is the rolling field’s, specialised.

A rigid displacement group is a manifold, and vector fields on it can be brought back to a single tangent space by translating along the group — a left-invariant field, which is determined everywhere by its value at the identity. Take two such fields, form the rolling field’s bracket of vector fields, and the answer is another left-invariant field, whose value at the identity depends only on the two starting twists.

That dependence is the pairs field’s bracket. It has constant coefficients because left-invariance has removed every dependence on where the mechanism is, and the structure constants that result are the ones the pairs field uses. So the pairs field’s operation is the general one restricted to a group and to fields that respect it.

The restriction runs one way and only one way, and that decides which routine could substitute for which. The rolling field’s machinery would compute the pairs field’s answer correctly — differencing two flows on the displacement group, expensively, and getting the same twist. The pairs field’s would be simply wrong on a rolling system, because a wheel’s permitted directions are not left-invariant fields on a group; they turn as the wheel turns, and the constant structure constants have nowhere to record that.

That is the sharpest form of the rolling field’s is harder. It is not harder by degree — more terms, more care with steps. It is the general case, and the pairs field’s is what the general case collapses to when the manifold happens to be a group and the fields happen to respect its structure.

Which is also why the naming matters as much as the essay says. Two functions related as general and special case are exactly the pair a reader is most likely to substitute for one another, since the special one is cheaper and gives the right answer in its own setting. Names that describe the object each acts on — fields on a manifold, twists in an algebra — carry the restriction in the call site, and a name that described the operation would carry nothing.

The one-sentence version

A bracket says what is reachable that was not permitted, and two fields of this site use it to establish opposite things.

The rolling field uses it to show that a wheel’s forbidden direction is reachable, which is why a car parks. The pairs field uses it to show that a mechanism’s permitted twists are already everything reachable, which is why a planar mechanism stays in its plane. Both are the same operation; neither result transfers; and the two are named for different things here, each saying beside itself that the other exists and is not it.

The general form, for anybody working in either field: the bracket measures the difference between what is permitted and what is reachable. Whether that difference is a feature or a defect depends entirely on which of the two was being described with the other.

Four loops, and where one point of each of them goes. The orbit of one point of the moving link in four overconstrained loops, drawn from solved configurations. Sarrus's platform runs along a straight line, because its displacements are a one-dimensional group of translations. A spherical four-bar's coupler point stays on a sphere. A planar four-bar's stays in a plane. Bennett's does none of those, and the reason is not that its curve is complicated: its displacements are inside no proper subgroup at all, so there is no surface for the point to be confined to. The first three paths are orbits of groups and the fourth is not an orbit of anything.
Fig. 9 The pairs field’s outcome, in four mechanisms. Three paths that are orbits of groups — the bracket added nothing — and one that is not, because the bracket added everything.

Where this goes next

A rolling wheel, where it was driven to. The mechanism at a configuration nothing wrote down: it was reached by integrating permitted velocities from the start of the trail, and there is no equation here whose root it is. The barred line at each wheel is the direction that wheel forbids — the subject of the whole field, and the one thing a photograph of a car cannot show. The constraint residual along the drawn history is 0.0e+0. A constraint that takes nothing away A rolling wheel forbids one direction of motion and removes no coordinate from the mechanism's description. It cannot slide sideways and it can still be brought to any position at any heading — and the gap between those two sentences is the whole of this field, because in every mechanism built of pins and slides the two agree. The parking shuffle, three times over. Forward on left lock, forward on right lock, back on left lock, back on right lock — and repeat. The heading comes back to where it started and so does the position along the road, both exactly and at every leg length, because the four legs are a symmetric set. What is left is 255 mm of pure sideways translation per cycle at a leg of 0.80 m, which is exactly 4R sin(φ) tan(φ/2) with φ = s/R. Halve the leg and it quarters. Parking is an exponent Four legs — forward on left lock, forward on right lock, back on left lock, back on right lock — return a car to its own heading and to its own place along the road, exactly, and move it sideways by 4R sin φ tan(φ/2). Halve the room and the gain quarters, so the number of shuffles goes up by four and the distance driven doubles. 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

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.

ConstraintDisplacement subgroupDistributionGrowth vectorIntegrabilityLie bracketNonholonomicSubalgebraTangent spaceTwist