The crease count is a reliability budget

What a second deployment costs ends by naming a second way a deployment’s requirements eat its compaction, and it has nothing to do with fatigue. A structure that has to open once, at a distance, needs every one of its hinges to work, and a pattern folded more finely has more of them.

That is a product rather than a sum, and products of numbers below one fall fast.

What a hinge count costs #

At a thousandth of a chance of a hinge failing — which is a good hinge — eight hinges give a 99.2 per cent deployment, twenty-four give 97.6, sixty give 94.2, a hundred and twenty give 88.7 and three hundred give 74.1.

Those are not small differences and the last one is not an exotic pattern. A Miura folded eight by eight has about a hundred and thirty interior creases; folded finely enough to reach the compaction ratios this field advertises, it has several hundred. A pattern that packs superbly is a pattern with a one-in-four chance of not opening, at a hinge reliability nobody would call poor.

The arithmetic is unforgiving because it is a product. A designer improving a hinge from 0.999 to 0.9995 halves each hinge’s failure chance and buys, on three hundred hinges, a move from 74.1 per cent to 86.1 — a real gain and less than the gain from halving the crease count, which takes the same hinge to 86.1 as well. The two levers are equally strong and only one of them is usually considered, because the crease count is decided by a pattern and the hinge quality by a supplier.

The direction is what makes it a budget rather than a nuisance. Compaction rises with the fold count and the chance of getting it falls, so the crease count is a quantity being spent from two accounts at once, and there is a fold count at which the spending is best.

The fold count that is worth most #

What a mission gets is not the compaction a pattern reaches but the compaction times the probability of reaching it. Multiplying the packing curve by $p^k$ moves its peak in: at a hinge reliability of 0.99 on a sheet of ten, the packing peaks at 88 folds and the expected packing peaks at 54.

That is a large shift and a cheap one. The compaction given up between 88 folds and 54 is a fraction of a per cent, because the packing curve is nearly flat at its peak — and the deployment probability bought is tens of points, because the exponential is not flat anywhere. A pattern optimised for compaction alone is folded about sixty per cent finer than it should be, and the sixty per cent buys almost nothing.

The general shape is the familiar one for a product of a parabola and an exponential: the peak of the product sits where the parabola’s proportional slope equals the exponential’s rate. What is worth carrying is that the parabola’s peak is flat and the exponential is not, so the compromise always moves toward the exponential and the move is always cheap.

That asymmetry is the whole practical content. Near its optimum a packing curve is quadratic, so moving a third of the way down from the peak costs a few per cent; an exponential has the same proportional slope everywhere, so the same move buys the full exponential return. Any trade between a flat optimum and a steep monotone quantity resolves the same way, and a designer meeting one can take the steep side almost for nothing without computing anything.

How much the trade depends on the numbers #

Two numbers were chosen — a hinge reliability and a hinge radius — and both move the answer, so it is worth seeing the trade at a second setting.

At a hinge radius of 0.08 and a reliability of 0.995 the packing peaks at 55 folds and the expected packing at 47: a shift of fifteen per cent rather than of forty. A better hinge moves the compromise back toward the packing optimum, which is the intuitive direction and is worth confirming because the whole argument rests on the exponential being steep.

So the size of the correction is a property of the hardware and the existence of it is not. Wherever the hinge reliability is below one, the expected-value optimum is strictly below the packing optimum, and the only question is by how much.

The practical reading is that the correction is largest exactly where it is least welcome — in a finely folded pattern with many hinges, where the packing optimum is high and the exponential at that count is severe. A coarse pattern hardly notices.

The article that cannot be tested #

The third column of the first figure is the one an engineer would notice first.

A hinge reliability cannot be assumed; it has to be demonstrated. With $r$ consecutive successful tests and no failures, the ninety-five per cent lower bound on a hinge’s reliability is $0.05^{1/r}$ — so demonstrating a system reliability $R$ across $n$ hinges, each needing $R^{1/n}$, takes

$$r = \frac{\ln 0.05}{\ln R^{1/n}} = \frac{n\ln 0.05}{\ln R}$$

which grows linearly with the hinge count. For a 99 per cent system: 2,385 successes for eight hinges, 7,154 for twenty-four, 17,885 for sixty, 35,769 for a hundred and twenty, and 89,422 for three hundred.

Those numbers are worth pausing on. Seven thousand successful tests to demonstrate that a twenty-four-hinge mechanism opens ninety-nine times in a hundred is a campaign measured in years, and twenty-four hinges is a four-by-four Miura — the smallest sheet anybody would call a pattern.

And the flight article deploys once. It cannot be tested at all — testing it is deploying it — so every one of those tests is on a hinge, a coupon or an engineering model, and the argument from them to the article is an argument rather than a measurement.

That is the real cost of a fine pattern, and it is not in any compaction ratio. From a shell to a solar array finds twenty-five years between the Miura pattern being published and flying, with the gap entirely in qualifying a mechanism that has to work once at a distance of several hundred kilometres. Tens of thousands of hinge tests is what that gap is made of.

A gentler target, and it does not help much #

Relaxing the target from 99 per cent to 95 and improving the hinge by a factor of two takes the test count for a hundred and thirty hinges from about eighteen thousand to seven and a half. That is a large relaxation for a factor of two and a bit, because the test count depends on the target only through a logarithm and on the hinge count directly.

The dependence is the useful part. $r = n\ln(0.05)/\ln R$ is linear in the hinge count and logarithmic in the target, so halving the pattern’s creases halves the campaign while relaxing the target from 99 to 95 per cent takes off about a fifth. A programme trying to shorten its qualification has one effective lever and it is the pattern.

That is an unusual shape for an engineering trade. Requirements are normally the expensive thing and design the cheap one; here the requirement is logarithmic and the design is linear, so the pattern is where the campaign is decided.

Which patterns this favours #

The tube that gets built observes that every folded structure leaving a laboratory is a sheet joined to itself, which adds a seam’s own hinges to the count. Fourth of eight, and still not chosen for it measures patterns by how much compaction they buy per unit of crease and finds the Miura fourth of the eight printed patterns — nearly three times worse than the Yoshimura. Read against this, that measurement acquires a second meaning.

A pattern that converts crease into compaction efficiently is a pattern that reaches a given compaction with fewer hinges, and fewer hinges is a higher deployment probability and a shorter test campaign. So the conversion rate is not only an efficiency; it is a reliability figure of merit, and the factor of eight across the shelf is a factor of about eight in how much probability a given compaction costs.

Which adds to the reading of the Miura’s middling rate. That essay asks what selected the Miura, since its rate does not single it out. Part of the answer is what the rate does not buy it: at the same compaction a better converter would have used fewer hinges and been likelier to open, and three patterns on the shelf are better converters.

The two budgets together #

There are now two independent taxes on the crease count and it is worth adding them up, because they point the same way and nobody applies either.

Fatigue raises the hinge radius as a power of the cycle count, which lowers the best fold count. Reliability multiplies the compaction by an exponential in the fold count, which lowers the best fold count again. The two act on different terms — one on the radius and one on the count — so they compose rather than overlap.

A structure asked for twenty deployments at a hinge reliability of 0.99 is therefore folded twice over: the fatigue law takes its packing optimum from 87.6 folds to about 19.6, and the reliability weighting takes it lower still. The pattern a real requirement produces is coarse, and both of the things that coarsen it are invisible in any drawing of a crease pattern.

That is the honest summary of this whole corner of the subject. The compaction ratios the field advertises are computed for a structure that deploys once, with perfect hinges, and both of those are assumptions rather than achievements.

What the model assumes #

The hinges fail independently. They do not. A batch of hinges made together, folded together and stored together share every cause a common-mode failure could have, and the product $p^n$ is the optimistic case by exactly the amount they are correlated.

Every hinge must work. Some patterns tolerate a stuck hinge and open anyway, which is a redundancy the product ignores. Most rigid patterns with a single degree of freedom do not: one locked crease locks the sheet.

The hinge reliability does not depend on the fold count. A finer pattern has sharper hinges, which what a second deployment costs suggests are more strained and so presumably less reliable — so the two effects compound and this counts only one of them.

The pattern folds rigidly with one freedom. Panels instead of paper is where that constraint separates a paper model from a mechanism, and it is what makes a single locked crease fatal.

And the test statistics are the simplest available. Consecutive successes with no failures, a ninety-five per cent one-sided bound, and no prior; a real campaign would use a prior and would be shorter, and would have to defend the prior.

How the numbers were checked #

The system reliability must fall with the hinge count at every step, which is the claim that the crease count is a budget, and it would fail if the exponentiation had been implemented as a sum.

The test count must grow by more than tenfold across the hinge counts drawn, since the point is that it scales with the pattern’s fineness rather than being a fixed campaign.

The expected-value optimum must lie below the packing optimum — checked on the computed curves rather than argued — and the compaction given up must be small while the probability gained is not, which is the trade the figure exists to show.

And the packing curve is the same one the rest of this collection uses, so the shift is measured against a peak that other essays have already established rather than against one invented here.

What an engineer already knows, and what is new #

The product $p^n$ is not a discovery; it is the first line of any reliability textbook, and an engineer building a deployable knows it. Two things here are not standard.

The optimum moves, and cheaply. Treating compaction and reliability as separate requirements — “pack to this ratio, and be this reliable” — is the usual arrangement and it hides the fact that one is bought with the other. Multiplying them and maximising gives a fold count sixty per cent below the packing optimum at a cost of under a per cent of compaction, and that is a free improvement nobody takes because nobody writes the product down.

And the test campaign scales with the pattern. A hinge count is chosen by a geometer optimising compaction and it determines a test count that a programme manager pays for, and the two decisions are made by different people from different documents. Tens of thousands of hinge tests is what a fine pattern costs, and it does not appear in any comparison of crease patterns anywhere.

Folding that gets built sets out the requirement every deployable meets — large in use, small in transit, along a path nobody has to trust to chance — and the last clause is the one this prices. Trusting nothing to chance is expensive in proportion to the crease count, and the crease count is exactly what the first two clauses push up.

Counting the hinges of a real pattern #

A crease count sounds like a number a pattern comes with and it is not, quite, so it is worth saying what the $n$ in $p^n$ should be.

It is the number of hinges that must move for the deployment to complete — which is every interior crease of a rigid pattern with one freedom, since a locked crease locks the sheet. It is not the number of crease lines drawn, because a line crossing several panels is several hinges, and it is not the number of vertices.

For a Miura of $R$ by $C$ quads that is roughly $2RC$, so an eight-by-eight sheet is about a hundred and thirty and a sixteen-by-sixteen is about five hundred. The count grows with the area of the sheet rather than with its linear size, which is why the reliability falls so quickly as a pattern is refined: doubling the resolution quadruples the hinges and takes the deployment probability to its fourth power.

What this cannot settle #

The arithmetic is short and its limits are worth stating plainly.

It cannot supply a hinge reliability. The figures use 0.999 and 0.99 because they bracket what a good mechanical hinge might be, and the real number for a creased sheet in a folded stack after years in orbit is not something that can be computed from geometry or taken from a table.

It cannot see a hinge whose reliability depends on where it is. Which crease to push finds an error’s consequences varying by an order of magnitude across one sheet, and a failure’s consequences would vary at least as much — a locked crease at the edge is not a locked crease in the middle.

It cannot handle partial failure. A hinge that works stiffly rather than not at all is the common case in mechanisms, and the binary model has no room for a deployment that completes slowly or incompletely.

And it cannot price what a failed deployment costs. The expected-compaction optimum treats a failure as delivering nothing, which is right for a spacecraft and wrong for a laboratory demonstrator, and any other loss function moves the optimum.

Still open: where the redundancy could go #

The product $p^n$ is the arithmetic of a series system, and engineering’s usual answer to a series system is to stop it being one.

A pattern that opens with one crease stuck is a pattern whose reliability is not a product, and whether such patterns exist among rigid foldings is a question that can now be asked precisely. A rigid pattern with a single degree of freedom cannot tolerate a locked crease; one with two freedoms might, at the cost of needing two actuators — which two drivers and one freedom shows is its own difficulty when the freedoms do not match the drivers.

So there is a three-cornered trade nobody has drawn: freedoms against actuators against reliability. One freedom and one actuator is the simplest mechanism and the least tolerant; two freedoms and two actuators tolerates a stuck crease and introduces an over-determination problem of its own. Which wins is an arithmetic question with all its terms available.

Sideways from here, the test-count arithmetic suggests a design criterion nobody uses. A pattern could be chosen to minimise the tests needed for a target reliability rather than to maximise compaction, and since the tests scale with the hinge count, that criterion is the crease count itself — which is a remarkably simple thing to optimise and would produce patterns nothing in this field currently looks like.

The habit worth carrying is about quantities that multiply. When a system needs every part, count the parts before admiring the performance. A design improved by adding parts is being paid for out of an account that is not on the datasheet, and the account is exponential in exactly the number the improvement is proportional to.