Where the shortest loops are

A tolerance stack-up on a mechanism is a loop closure with ranges instead of numbers. The lengths go round a closed path and have to add up; each of them has a manufacturing range; and the question is what the ranges do to the output.

Which lengths are involved is decided by which loop. And which loops there are — and how short they can be made — is decided entirely by the graph, before any of the lengths exists and without reference to any of them.

How many loops, and which #

The number of independent loops is $j - n + 1$, and it is forced by the same two totals Grübler’s rule reads. A four-bar has one. A six-bar has two. An eight-link chain of one degree of freedom has three, and all sixteen of them have three.

Which loops is a different question, because the choice is not unique. Any set of three independent cycles will do as a basis, and different sets involve different links. What the graph decides is the shortest such set, and that is the quantity worth having: the fewest link dimensions any tolerance analysis on the mechanism can involve.

Finding it is a small computation with a standard answer. Horton’s construction generates a candidate cycle for every vertex and every edge — the shortest paths from that vertex to the edge’s two ends, plus the edge — and a greedy pass over the candidates by length keeps each one whose edge-vector is independent of the ones already kept. Independence is over the two-element field, so the test is Gaussian elimination on bitmasks.

Two details in that are worth naming because both are places to get it wrong. A candidate whose two shortest paths overlap is a walk rather than a cycle — it doubles back on itself, and its edge-vector has fewer edges in it than the walk has steps. Those are discarded rather than shortened, because a shortened one is a different cycle that the construction generates elsewhere anyway. And the greedy pass has to be over the candidates sorted by length: taking them in generation order produces an independent set that is a basis and is not the shortest one, which is precisely the failure this quantity exists to measure in somebody else’s stack-up.

Watt against Stephenson, once more #

The six-link case is small enough to check by hand and it separates the two chains again.

Watt’s chain has two independent loops of four links each, total eight. Its two ternary links share a pin, and each of the two binary paths between them closes a four-link loop through that shared pin.

Stephenson’s has a loop of four and a loop of five, total nine. Its ternary links are joined by three binary paths of lengths one, one and two; the two short paths close a four-loop, and any second independent loop has to use the long path and is five.

So a Stephenson six-bar’s tolerance problem involves one more link dimension than a Watt six-bar’s, at the same link count, the same pin count, the same assortment and the same number of loops. Nine numbers against eight, decided by whether two plates share a bearing — which is not a difference anybody would have looked for and is the kind a census makes visible.

Why the shortest basis is the right quantity #

Two reasons, and the second is the one that makes it a design number rather than a curiosity.

A shorter loop has fewer terms. Worst case is a sum of absolute sensitivities and the statistical estimate is a root sum of squares; either way, more dimensions in a loop means a wider band, all else equal. Eight terms against nine is about a 6 per cent difference in the root-sum-of-squares estimate for equal tolerances, and rather more in the worst case.

And the loops are what a stack-up is written on. An engineer analysing a mechanism writes down closure equations, one per loop, and the equations they write are a basis — usually the obvious one, usually not the shortest. The minimum basis is a lower bound on the work, so it says how much of a long stack-up is inherent and how much is a bad choice of loops.

That second point is the practical one. A stack-up written on a poorly chosen basis has extra terms that a different choice removes, and nothing in a spreadsheet says so. The minimum basis is the number to compare against.

Reading a basis off a chain #

The computation is standard and the answer is easy to check by eye on the cases that matter, so it is worth walking one.

Take the four-bar. One loop, and it is the whole mechanism: four links, four pins, and the closure equation says the four link vectors sum to zero. Four terms, and there is no other loop to choose.

Take Watt’s chain. Its two ternary links share a pin. Start at that pin, go out along one binary path — two links — and come back along the ternary links: four links in the loop. Do the same on the other binary path: another four. The two loops share one pin and no link, they are independent, and eight is their total.

Take Stephenson’s chain. Its ternary links are joined by three binary paths, of one link, one link and two links. Combining the two one-link paths gives a loop of four — two ternary links and two binaries. Any second independent loop must involve the two-link path, and the shortest such is the two-link path with one of the one-link paths: five links. Total nine.

At eight links the reasoning stops being doable by inspection, which is where the algorithm earns its place — and where the greedy pass matters, since a naive choice of loops on an eight-link chain routinely produces a basis two or three links longer than the minimum.

Where the same idea already appears on this site #

Two fields have met this quantity without naming it, and it is worth connecting them.

The networks field counts loops constantly, because a scissor chain or a folded sheet has hundreds of them and the loops are in the graph. What it does with them is different: it needs a basis to write the constraint Jacobian, and any basis will do because the rank does not care which one. The length of the basis matters here and not there, because a rank is a rank and a stack-up is a sum.

The practice field has been writing stack-ups on four-bars and six-bars since its own phase, and in every case the loop was obvious because there was one, or two, and no choice about them. The choice appears at eight links and it is the first size where a stack-up can be written badly.

Neither had reason to ask for a minimum. That is the pattern this whole phase keeps producing: a quantity that is trivial at the sizes a field works at, becomes a choice one size up, and becomes a computation the size after that.

The loops share links, and the stack-up has to know #

There is a subtlety in going from a basis of loops to a tolerance figure, and it is the reason the total link count is a starting point rather than an answer.

A basis of three loops on an eight-link chain involves twelve to fifteen link dimensions counted with multiplicity, and the multiplicity is the whole difficulty: eight links spread across three loops of four or five means several links appear in two loops and at least one usually appears in all three. That link has one length. Whatever error it carries enters every loop it belongs to, with the same sign and the same magnitude, at whatever sensitivity each loop gives it.

So the three loop equations are not independent as statistical objects even though they are independent as linear ones. Treating them as independent — computing a root sum of squares within each loop and then combining the loops the same way — double counts nothing and under-counts the correlation, which is the direction that flatters the answer. The shared link’s contributions can reinforce, and a statistical estimate that assumed they could not has quietly asserted that they cannot.

The correct procedure is the unglamorous one: assemble the whole thing as a single sensitivity vector over the distinct dimensions rather than as a set of per-loop sums, and take one root sum of squares over that. Then a link in three loops appears once, with a sensitivity that is the combined effect of its three appearances, and the arithmetic is right by construction.

Which gives the loop-length total its proper standing. Twelve against fifteen is a real difference and it is a difference in how many dimensions the closure equations reach, not in how many terms the final sum has — the final sum has as many terms as there are distinct dimensions involved, which is at most the link count and is often all of it. The basis decides which links are involved and how often; the sensitivity analysis decides what each is worth; and only the first of those is a fact about the graph.

The shortest basis is not always the useful one #

A second caution, and it is about what question is being asked rather than about arithmetic.

A minimum cycle basis minimises the total, and a designer usually wants something else: the shortest loop through one particular link — the one whose tolerance is being argued about, or the one that would be expensive to hold tighter. That loop may not be in the minimum basis at all, because a basis is chosen to be small overall and can perfectly well route the cheap loops around a link and leave it to a long one.

Those are different computations with different answers, and both are cheap. Asking for the whole basis answers how big is this mechanism’s tolerance problem; asking for the shortest cycle through a named link answers what does this dimension actually affect, which is the question a change request arrives as.

And a minimum basis is not unique. The number is — that is what makes it a property of the chain worth tabulating — but the loops realising it generally are not, so two engineers computing one on the same chain may write different sets of equations, both minimal, both correct, and disagree about which lengths appear where. That is harmless for the total and confusing in a review, which is a reason to record the basis alongside the number rather than only the number.

What it does not decide #

The usual caution, and it is the same one the whole topology field carries.

It says nothing about which dimension matters most. The four lengths of a four-bar do not matter equally, and which one dominates is a sensitivity computed from a Jacobian at a configuration — a dimensional quantity through and through. The graph says which lengths are in the sum and not what their coefficients are.

It says nothing about the size of the band. Twelve dimensions with tight tolerances is better than eight with loose ones. The count is a term count, not an error estimate.

And it says nothing about clearances. A clearance is a link in this site’s treatment, and adding one changes the graph — a pin with play is two links and two contacts rather than one pin — so the loop structure of the modelled mechanism is not the loop structure of the idealised one.

That last point is worth dwelling on, because it is where this rung’s quantity has to be used carefully. The minimum basis is computed on the idealised chain. A tolerance model that represents clearances explicitly has a different and larger graph, and the shortest basis of that graph is the number that bounds its stack-up.

The distribution across the census #

Across the sixteen eight-link chains the shortest basis runs from twelve links to fifteen.

Three of the sixteen achieve twelve — three loops of four each, which is the smallest possible since a four-link loop is the shortest a chain can have — and one needs fifteen. The rest are at thirteen or fourteen, with fourteen the commonest.

There is no obvious pattern in which chains land where. The three at twelve do not share an assortment, the one at fifteen is not the most symmetric, and the quantity is not correlated with the number of mechanisms a chain gives or with whether it has a closed-form driving choice. It is a fifth independent number about a chain, and like the others it has to be computed rather than inferred from the ones already known. The three at twelve have three different assortments — two quaternary links, one quaternary with two ternaries, and four ternaries — so even the coarsest description of a chain says nothing about it, and the chain needing fifteen has an automorphism group of eight against a maximum of sixteen, so symmetry says nothing either. Eight of the sixteen sit at fourteen, which is the modal answer and the one a guess would land on.

That spread is a fifth of the total, which is a real difference in a subject where a 5 per cent tolerance improvement is worth arguing about. And it is available for nothing: the computation reads the graph, it takes microseconds, and it is a filter that can be applied before any dimension exists.

Why a four-link loop is the shortest there is #

A small piece of arithmetic that explains the floor.

A loop of three links would be a triangle, and a triangle is exactly what disqualifies a graph from being a chain: its own mobility count is nought, so it is a structure rather than a mechanism. So no kinematic chain has a three-link loop, and the shortest possible is four.

That gives a clean lower bound: a chain with $L$ independent loops has a shortest basis of at least $4L$. At eight links, $L = 3$ and the bound is twelve — which three of the sixteen chains attain, so the bound is tight.

The four-bar attains it trivially: one loop, four links, and a stack-up with four terms — which is why every tolerance argument in this field’s opening rungs is about four numbers, and why moving to a six-bar doubles the work before any dimension is chosen.

What a designer does with it #

The number is small and the use of it is correspondingly modest, which is worth stating rather than inflating.

It is a tie-breaker, and the place a census is used as one. Two topologies that both satisfy a design’s requirements, both give a workable motion and both have acceptable transmission angles: the one with the shorter loop basis has fewer terms in its stack-up, and that is a reason to prefer it that costs nothing to evaluate.

It is a check on a stack-up somebody wrote. An analysis with fourteen terms on a mechanism whose minimum basis is twelve has two terms that a different choice of loops removes. Whether that is worth rewriting depends on the analysis; knowing it is possible does not.

And it bounds the problem before the problem exists. At ten links every chain has four independent loops and therefore at least sixteen link dimensions in its stack-up. That is a floor on the tolerance work a ten-link mechanism entails, available from the census row alone, and it is one of the honest reasons a designer prefers fewer links.

None of that replaces a sensitivity analysis, and none of it says which dimension to tighten. What it does is put a number on a quantity that has always been a consequence of a decision nobody wrote down — which is the same service the whole of the topology field performs, applied to the one place where this field and a graph have something to say to each other.