A name for each overconstraint

Two ways to be overconstrained left the spatial field with a working separation and no vocabulary.

Two mechanisms hold identical paperwork: a planar four-bar built as a spatial loop and Bennett’s four-bar. Four links, four revolutes, Kutzbach $-2$, rank three, three redundant constraints, mobility one. Every number matches, and the two are not alike at all.

The instrument that separated them measured how far the mechanism’s screw system turns through its own motion — a principal angle between the subspace at one configuration and at another. The planar four-bar’s stands still: $2 \times 10^{-6}$ degrees over a sweep. Bennett’s turns 89, and a Bricard six-bar’s 22.

That is a good measurement and it produces a verdict rather than a name. The verdict is subgroup or paradoxical, and the word subgroup was doing a lot of unexamined work in it: the essay that introduced the distinction said the screw system stands still for planar, spherical and translational mechanisms, and it identified those three by knowing which mechanisms they were rather than by measuring anything about their groups. This essay is what happens when the same six loops are handed a finite instrument instead of a differential one.

The finite instrument #

Take the displacements the moving link actually reaches, from converged configurations only. Take their logarithms. Find the smallest subspace containing them — the span — and then close that subspace under the Lie bracket and read the dimension.

If the mechanism’s motion lies inside a proper subgroup, every logarithm lies in that subgroup’s algebra, the span is at most its dimension, and the closure equals it. If it lies inside nothing smaller than the whole, the closure is six.

The classifier then names the answer from three integers: its dimension, how many independent rotation directions it contains, and how many pure translations. That is the whole of the classification of the twelve, and every subalgebra it can be handed is one of them.

The six loops, named #

A planar four-bar built as a spatial loop. Span three, closure three, type $G$ — planar motion. Every joint’s axis is parallel to every other, so every joint’s group lies inside one planar group, and the mechanism cannot leave it.

A spherical four-bar. Span three, closure three, type $S$ — rotations about a point. Every axis passes through one point.

A universal joint. Span three, closure three, type $S$. The same group, which is the group-theoretic form of a fact the field already had: a universal joint is a spherical four-bar with two of its links being the shaft yokes.

Sarrus’s linkage. Span one, closure one, type $T$ — a translation. Two arms of three parallel revolutes each, each confining the platform to a planar group, and the platform in the intersection of two planes: a line.

Bennett’s linkage. Span four, closure six.

A Bricard six-bar. Span four, closure six.

The Bricard row is worth a sentence of its own because it is the second paradoxical mechanism and the site has only two. It is a six-bar with a line symmetry, Kutzbach counts nought, its rank is five of six, and the spatial field’s own essay on it establishes that it needs its symmetry exactly. That its span is also four — the same as Bennett’s, from a mechanism with two more links and a completely different construction — is the kind of coincidence that is either meaningless or a clue, and this field is not in a position to say which.

Why span and closure are reported separately #

Two numbers per row rather than one, and the reason is that they differ on exactly the mechanisms that matter.

The span is how many dimensions the reached displacements occupy. The closure is how many they generate once brackets are taken. For a motion inside a group the two are equal, because the group’s algebra is already closed and the motion cannot leave it.

For Bennett they are four and six. That gap is not an artefact: a generic spatial loop’s coupler reaches a one-parameter curve whose logarithms span all six, so Bennett’s four is a genuine restriction that the exact condition on its lengths produces. It is a four-dimensional subspace that is not a subalgebra, which is an object with no standing in the subject and is nonetheless what the mechanism’s displacements occupy.

Reporting the span alone would say Bennett is more constrained than a generic loop, which is true and misleading. Reporting the closure alone would say it is inside nothing, which is true and throws away the four. Both, together, say what is actually the case: the motion is confined and the confinement is not a group.

What the naming buys #

Three things the drift could not give.

It distinguishes the trivial cases from each other. A planar four-bar and Sarrus’s linkage both report a screw system that stands still, and their groups are a three-dimensional planar group and a one-dimensional translation group. Those are different in dimension, in type, and in what the mechanism is for. The drift assigns both the same verdict.

It is checkable against the geometry. Every name it returns can be read off the drawing: parallel axes give $G$, concurrent axes give $S$, and two perpendicular planar arms give $T$ along their common perpendicular. A measurement whose answer can be independently derived is a measurement that can be wrong, which is the property this site asks for.

And it applies to a mechanism nobody has classified. Hand it any loop, and if the motion is inside a group it says which. The drift says whether, and the answer whether is only useful when somebody already knows the candidates.

The trivial ones, read off the axes #

The names are worth deriving once from the geometry, because a measurement that can be independently derived is a measurement that can be wrong.

Parallel axes give planar motion. A revolute about direction $\mathbf u$ generates a one-dimensional group inside the planar group with normal $\mathbf u$; four of them with the same $\mathbf u$ are four subgroups of one three-dimensional group; the mechanism’s motion is a product of subsets of that group and is therefore inside it. Closure three, type $G$.

Concurrent axes give spherical motion. A revolute through a point $\mathbf c$ is inside the spherical group about $\mathbf c$; four of them through one point are inside one spherical group. Closure three, type $S$.

Two perpendicular planar arms give a translation. Each arm’s joints are inside its own planar group, so each arm confines the platform to that group; the platform is in the intersection; and two planar groups with non-parallel normals meet in a one-dimensional translation group along their common perpendicular. Closure one, type $T$.

Three derivations, no mechanism solved, and the measurement agrees with all three. What the measurement adds is that the mechanism as built delivers what the axes promise — which is not free, since a loop has to assemble and its travel has to exist.

Where the two instruments agree, and why they must #

They agree on all six rows, and the agreement is not automatic. It is worth showing where it comes from.

If a mechanism’s motion lies inside a subgroup $H$, then at every configuration its available twists lie in $H$’s algebra, which is a fixed subspace. The screw system is that subspace, or part of it, and a fixed subspace does not turn. A motion inside a group forces a screw system that stands still.

The converse is the interesting direction and is nearly true. A screw system that stands still means the tangent spaces to the motion coincide everywhere, which for a connected motion means the motion is inside the subgroup generated by that fixed subspace. So a drift of zero does imply membership of a group — of the group generated by the screw system, whose dimension is the closure of the screw system under the bracket.

That last clause is the join between the two instruments, and it is exact: the drift being zero means the motion is inside the bracket closure of its own screw system. Which is what this field computes, from the other end.

The difference is that the drift measures whether the tangent space moves and this measures what the tangent space generates. When the tangent space is fixed, the two are the same statement; when it is not, the drift reports an angle and this reports a dimension of six.

The definition, restated for this field #

The spatial field can now put its own distinction in one line each.

Trivially overconstrained: all the joints lie inside one proper subgroup of the rigid displacements, and the mechanism’s redundant constraints are the ones that subgroup makes redundant. Kutzbach’s arithmetic, corrected to use the subgroup’s dimension in place of six per loop, gives the right answer — which is the number Kutzbach is missing, now available before the mechanism is assembled rather than measured from a rank afterwards.

Paradoxical: no proper subgroup contains the motion. The redundancy is a coincidence in the dimensions rather than in the group, and the mechanism exists only for exact values of its lengths and twist angles.

The second half of that is testable and is tested: a Bennett loop detuned by six per cent produces no solved configurations across the sampled travel, against nine of nine for the true one. The mechanism does not degrade; it stops existing.

The measurement’s own limits #

Three, and they are the reason the drift measurement is not being retired.

It needs solved configurations. Every displacement in it comes from a converged root of the closure equations, so a mechanism that will not assemble has nothing to compose and the test returns nothing rather than a number. That is correct behaviour and it is asserted: a Bennett loop detuned by six per cent reports zero reached configurations against nine for the true one.

It needs a range. Every set looks like its own tangent space near the identity, so a mechanism sampled over a small enough travel reports its screw system’s dimension whatever it is. The loops here are sampled over ±0.5 to ±0.9 radians of the driven joint, which is what their travel allows, and the range is part of the claim.

And its floor is the solve, not the arithmetic. A configuration is a root found to $10^{-13}$ and judged at $10^{-9}$; its logarithm inherits that. So Sarrus’s composition defect comes back at $1.6 \times 10^{-12}$ and the planar four-bar’s at $4 \times 10^{-16}$, and neither number is about the mechanism. The number that is about a mechanism is three tenths.

What is still not explained #

Naming the absence of a group is not the same as explaining what is there instead, and the field should say so.

Bennett’s linkage moves because its four lengths and four twist angles satisfy a relation exactly, and the spatial field’s own essay is about that relation. Nothing in this instrument produces the relation, predicts it, or explains why one exists. The closure test says the mechanism is not inside a group; it does not say what a one-parameter motion in a six-dimensional group with no group structure is.

The span of four is the one piece of new information about the positive side. Bennett’s reached displacements occupy four of six dimensions rather than all six, which a generic spatial loop does not — so the exact condition on the lengths leaves a trace in the shape of the motion, and this measurement can see it. What that trace means is not settled here.

An overconstraint the field has not met #

Naming the trivial cases raises a question the field can now ask and had no way to before: are there mechanisms whose motion lies in one of the other groups?

The four trivial rows use three groups — $G$, $S$ and $T$ — out of twelve. Nothing forbids a loop whose motion is inside the cylindrical group, or the two-translation group, or Schoenflies motion, and each would be an overconstrained mechanism of a kind this site has not drawn.

Some of them are familiar under other names. A mechanism whose platform translates in a plane without turning is inside $T_2$, and every X–Y stage built from links rather than ways is one. A mechanism producing pure spatial translation is inside $T_3$, and that is the specification of a Delta machine. A mechanism producing Schoenflies motion is a SCARA arm’s parallel cousin, and several exist commercially.

What the field has not done is measure any of them. They are named here as a gap rather than filled, and the instrument that would fill it is the one this essay is about: build the loop, sweep it, close the logarithms, read the type. That is a substantial piece of work on mechanisms this site has not touched, and it is worth recording as such rather than implied.

A loop of prismatics, which the field has not met #

The four trivial rows use three of the twelve groups, and the question of which of the other nine could host a trivially overconstrained loop is answerable without building anything — so it is worth answering for at least one of them, because the answer is a mechanism nobody has drawn on this site.

Take the translation group $T_3$ and ask for a closed loop all of whose joints lie inside it. A joint inside $T_3$ is a prismatic pair, so the mechanism is a closed loop of four prismatic joints, sliding along four directions that span space.

Count it. Four links, three of them moving, four prismatic joints each removing five freedoms: $6 \times 3 - 5 \times 4 = -2$. The count says the assembly is a structure with two constraints to spare.

Measure it instead. Translations commute, so the loop closes exactly when the four slide displacements sum to zero — three scalar equations in four unknowns, and the mobility is one provided the four directions span three dimensions. So the loop moves, the count says it cannot, and $\nu = 3$.

That is a trivially overconstrained mechanism inside $T_3$, arrived at by reading the lattice rather than by finding a mechanism and classifying it. Its motion is a one-parameter family of translations, its screw system is constant — every joint screw is a pure translation and none of them turns — so the drift measurement would report zero and the closure would report a translation group of dimension one.

Two things about it are worth saying plainly. It is not a machine anybody builds: four sliders in a ring is an awkward object with no obvious use, and the reason it is absent from the field is that nobody had a reason to draw it rather than that it was missed. And it is nevertheless a legitimate row of the field’s table — a fifth trivially overconstrained loop, in a fourth group, predicted from the classification and confirmed by two lines of counting.

Which is what the naming buys beyond tidiness. A verdict sorts the mechanisms somebody has; a name points at the ones nobody has, because the twelve groups are a finite list and the ones with no loop against them are a list of questions.

What this changes about the spatial field #

Two entries in the field’s own record.

The drift measurement stands, unamended. It is a first-order instrument, it works, it separates the two populations by seven orders of magnitude, and it is cheaper than a closure — one sweep and a sequence of principal angles against thirty-one solved configurations and a bracket closure. Nothing here supersedes it.

And the field’s list of overconstrained mechanisms now has a column. Planar $G$, spherical $S$, universal $S$, Sarrus $T$, Bennett none, Bricard none. Four of them are the same phenomenon under three different groups, and two of them are the phenomenon that has no explanation of this kind.

The third entry is a piece of hygiene. The field’s account of trivial overconstraint used to be a list of three special cases — planar, spherical, translational — that everybody names and nobody derives. It is now a consequence of a classification: those three are the groups that a loop’s joints can all lie inside and that a loop of revolutes can fill, and the reason there are three rather than seven is that a revolute’s group has to be inside the candidate, which the lattice settles in one reading. A list somebody compiled has become a list somebody can check.