Two drivers and one freedom

Only four creases decide a Miura ends on a question left standing: a sheet with one freedom driven by two actuators. It sounds like redundancy — two motors where one would do, so a failure is survivable — and it is not redundancy at all.

It is over-determination. One freedom means one number describes the sheet’s state, two actuators are two commands for that number, and two commands that disagree cannot both be met. The sheet goes somewhere; the question is where.

Where the sheet settles #

Let the two creases be geared at $g$ — an error at the first arrives at the second multiplied by $g$, which is the quantity which crease to push measures. The sheet has one freedom, so moving the first crease by $x$ moves the second by $gx$.

Two actuators of equal stiffness, commanding positions that disagree by $\delta$, store an energy proportional to the sum of the squares of the two errors. Minimising it,

$$x = \frac{g\,\delta}{1+g^2}, \qquad \text{residual at the second} = \frac{\delta}{1+g^2}$$

so the sheet moves toward the second actuator by $x$ and leaves the rest of the disagreement standing there. Both quantities depend only on the gearing, and neither has the stiffness in it once the two are equal.

The first curve peaks at $g = 1$ and falls away on both sides. That is the worst case for the sheet’s position: equally geared creases pull it furthest from the first actuator’s command. The second curve falls monotonically, so the residual left at the far crease is worst when the gearing is smallest.

The loosest coupling is the expensive one #

On a four-by-four mesh with no two vertices alike, one crease reaches the others at six distinct gearings — 0.316, 0.539, 0.784, 0.934, 1.000 and 1.761, four creases at each. The share of a disagreement left standing at the second actuator runs from 91 per cent down to 24.

So a pair chosen badly keeps nearly the whole disagreement and a pair chosen well absorbs three quarters of it. And the direction is counter-intuitive: the pairs that keep the disagreement are the weakly geared ones — the pairs where an error at one crease barely reaches the other.

The reason is that gearing works both ways. A weakly geared pair is one where moving the first crease hardly moves the second, so the sheet cannot use the first actuator’s compliance to relieve the second: it would have to move the first a great deal to fix a little at the second, and the energy says no. A strongly geared pair is one where a small move at the first fixes a large error at the second, so the sheet makes that move.

Loose coupling means the two actuators cannot help each other, which is exactly why loose coupling is comfortable in most systems and expensive here.

Reading the six values #

The six gearings on the measured mesh are worth walking, because they are not spread evenly and the spacing is the useful part.

Four creases sit at 0.316, four at 0.539, four at 0.784, four at 0.934, four at 1.000 and four at 1.761. The clustering near one — three of the six values lie between 0.78 and 1.00 — means most pairs of this mesh split a disagreement roughly evenly, keeping between half and two thirds of it. The interesting pairs are the two ends, and each end has only four creases in it.

So the design choice is not a fine gradation. It is: avoid the four creases at 0.316, prefer the four at 1.761, and treat the middle sixteen as interchangeable. That is a much simpler instruction than a table of twenty-four numbers, and it is what the distribution supports.

The clustering also explains why the effect is easy to miss. A designer who tried two arbitrary pairs would most likely draw two from the middle, see the residual split somewhere near half in both, and conclude that the pairing does not matter. The variation lives in the tails, and the tails are a third of the mesh.

And the energy runs the same way #

The stored energy at the compromise is proportional to $\delta^2/(1+g^2)$, so it falls with the gearing just as the residual does. A pair at $g = 0.316$ stores four times what a pair at $g = 1.761$ stores for the same disagreement.

That matters because stored energy is what breaks things. Two actuators fighting at a weakly geared pair are loading the sheet’s panels and hinges with a force that does no work and achieves nothing, and the loading is four times what it would be at the other end of the mesh.

The shape does not change with the size of the disagreement — both errors scale linearly with $\delta$ — so the choice of pair is a fixed multiplier on whatever the actuators’ accuracy turns out to be. Halving the actuators’ error and choosing the pair well are interchangeable improvements, and one of them is free.

What a designer would do with it #

Three rules follow and they are not the ones a redundancy argument would give.

Two actuators are not a safety measure. Folding that gets built collects structures whose whole requirement is a path nobody has to trust to chance, and a second driver on one freedom adds a way to fail rather than removing one. They cannot both be right, they load the sheet whenever they disagree, and the loading is at its worst when they are furthest apart in the mesh. A second actuator is worth having for torque or for speed, and not for redundancy.

If there are two, put them on a strongly geared pair. That is usually a pair close together in the mesh, which is the opposite of where a redundancy argument would put them — a designer spreading actuators for coverage is choosing the weakly geared pairs and the four-times loading.

And measure the gearing rather than guessing it. The spread on a single mesh is more than five to one from one crease, and there is no way to read it off the pattern. A crease’s gearing to another is a property of the chain of vertices between them, and two creases that look symmetrically placed can sit at opposite ends of the range.

The three together are a short design note and they replace a habit. The habit is to place actuators for coverage — spread across the structure, so that no region is far from one — and coverage is precisely the arrangement that maximises both the accumulated disagreement and the penalty for it. What the arithmetic wants is the opposite: drivers close together, strongly coupled, with one of them computed from the other.

Where the disagreement comes from #

A disagreement of a hundredth of a radian sounds like an actuator specification and mostly it is not.

An actuator’s own position error is one source and usually the smallest. The larger ones are in the sheet: a panel cut a hair long, a hinge with a fraction of a degree of play, a crease whose radius makes its effective angle differ from its commanded one. Which crease to push prices how far such an error travels when one crease is driven; with two drivers the same errors become a disagreement, because the two actuators are measuring the same sheet through different chains of vertices.

That reframes the gearing rule slightly. The disagreement is not a fixed number a designer controls; it is the accumulated geometric error along the path between the two driven creases, so a pair far apart in the mesh has both a larger disagreement and a worse split — the two effects compound, and both favour driving from creases close together.

It also means the disagreement grows with the sheet. A five-by-five mesh has longer chains than a three-by-three, so two actuators placed at opposite corners of a large sheet are accumulating error over a longer path and resolving it at whatever gearing the corners happen to have.

The Miura, where the question changes #

On a Miura every crease is the same crease, so the gearings take only two values — one for row creases and about 1.77 for column creases — and the pairing question nearly disappears.

What replaces it is worse. A Miura driven from most of its creases has several folded states, so two actuators are not merely disagreeing about where along one freedom the sheet should be — they may be commanding different states, which is not a disagreement a compromise can resolve. The energy argument assumes one continuous freedom parameterised by a single number, and a sheet with eight consistent configurations does not have one.

So the pairing rule applies to the mesh nobody builds and the state problem applies to the mesh everybody builds. Paper that folds itself named that split from the beginning: supplying the torque is the easy half, and deciding what the sheet becomes is the half with no force behind it.

What over-determination is not #

Three things this is often confused with are worth separating, because each has a different remedy.

It is not backlash. Backlash is a mechanism that fails to transmit a command over some range; here every command is transmitted perfectly and there are simply two of them.

It is not misalignment. A misaligned mechanism has a systematic offset that a calibration removes; two actuators on one freedom disagree afresh at every commanded position unless their commands are computed from one another, which is a control decision rather than an assembly one.

And it is not redundancy. A redundant system tolerates the loss of a component; this one is worse with both components working than with either alone, since a single actuator has nothing to fight. Only four creases decide a Miura found the other way a second actuator fails to help — on a sheet with several states two actuators may command different ones — and between them the two results say that a second driver on a one-freedom sheet needs a reason other than safety.

The remedy in every case is the same and it is a control one: compute the second command from the first through the mesh’s own kinematics, so the two agree by construction. The gearing is exactly the quantity that computation needs, which is a pleasant closing of the loop: the number that prices the disagreement is the number that prevents it.

What the model cannot show #

The energy argument is two springs and a linear relation, and a sheet is neither.

It assumes the actuators are linear springs of equal stiffness. Unequal stiffnesses tilt the split — a stiff actuator holds its command and a soft one gives way — and the whole result is then about the stiffness ratio as much as about the gearing. Nothing here computes that, and it is one term away.

It assumes the sheet is rigid between the creases. A real panel bends, which adds a third compliance in parallel with the two actuators and absorbs some of the disagreement without either of them moving. That makes the sheet more forgiving than the model and by an amount the model has no way to state.

And it assumes the gearing is constant over the disagreement. It is not: the gearing varies through the motion, and the hardest instant measures how much. Over a hundredth of a radian the variation is small; over a tenth it is not, and the linear split becomes an approximation.

What it would take to measure #

The whole argument is an energy minimisation over a linear relation, and a laboratory could check it in an afternoon.

Take a rigid mesh of the kind measured here, put position actuators on two creases, and command them to disagree by a known amount. The sheet settles somewhere; measure where, and compare against $g\delta/(1+g^2)$ with the gearing measured separately by driving one crease alone. Two numbers, one prediction, and a straightforward way to be wrong.

What would show the model up is a settled position that does not depend on the gearing, which would mean the panels’ own compliance is absorbing the disagreement rather than the actuators — the term this account leaves out and cannot bound. That is the likeliest failure and it is also the most useful, since it would say the sheet is more forgiving than the arithmetic allows and by how much.

What the model assumes #

One freedom exactly. The whole argument is that two commands exceed one freedom by one, and a sheet with two freedoms and two actuators is a different and much happier problem.

The mesh folds rigidly. One crease decides the sheet establishes the determinacy the whole argument stands on, and the hardest instant measures how the gearing varies through the motion.

The gearing is measured, not assumed. Each value comes from driving a crease a hair either side and dividing, on the mesh itself, so the six values are properties of that mesh at that angle.

The energy is quadratic in each actuator’s error. That is a linear spring and it is the simplest thing that gives a definite answer.

And the disagreement is in position rather than in force. Two position-commanded actuators fight; two torque-commanded ones do not, they simply add, and the sheet goes where the sum takes it.

How the numbers were checked #

The worst gearing for the sheet’s position is required to be one, found by scanning the curve rather than by differentiating it, so a slip in the algebra would put the peak somewhere else.

The residual is required to fall by more than a factor of ten across the gearings drawn, which is the claim that the choice of pair matters.

The gearings are required to be several and distinct on the mesh measured, since a mesh with one gearing everywhere would make the pairing question vacuous — which is exactly what the Miura is.

And the split is checked at two disagreement sizes, so the claim that both errors scale linearly with the disagreement is verified rather than read off the formula.

A third actuator, briefly #

The argument extends and the extension is not encouraging.

Three actuators on one freedom are three commands for one number, and the compromise minimises the sum of three squared errors. The sheet settles at a weighted average of the three commands, with weights proportional to the square of each crease’s gearing to the chosen parameter — so the most strongly geared actuator wins, and the others are loaded in proportion to how far they are from where it put the sheet.

That gives a rule of thumb for any number of drivers: the sheet obeys whichever actuator is most tightly coupled to the freedom, and every other actuator is a spring being stretched. Adding drivers adds torque and adds loading, and past two the loading grows faster than the torque, since each new driver disagrees with all the others.

None of that is drawn here, and it is a straightforward extension of the same energy minimisation rather than a new computation. What is not straightforward is the version with several freedoms, where the number of actuators can match the number of freedoms and the whole problem becomes well posed — which is the arrangement a deployable with more than one motion needs, and which nothing here touches.

Still open: what unequal stiffness does #

The model’s most obviously false assumption is the easiest to remove, and removing it changes the design rule.

With stiffnesses $k_1$ and $k_2$, the compromise minimises $k_1 x^2 + k_2 (gx - \delta)^2$, so the split becomes $x = k_2 g \delta / (k_1 + k_2 g^2)$ — and a designer now has two levers instead of one. A soft actuator at the weakly geared end would give way and remove most of the penalty; a stiff one there would make it worse than the equal-stiffness case.

That suggests a rule the equal-stiffness analysis cannot state: stiffness should be traded against gearing, with the actuator on the weakly coupled crease made deliberately compliant. Whether the product $k g$ is the quantity that should be matched, and what it costs in the torque each actuator can deliver, is a computation nobody has made.

That is also where the actuator’s own character would enter. Paper that folds itself sets out what a self-folding sheet’s actuator actually is — a material that changes shape rather than a motor commanding a position — and a shape-change actuator is closer to a spring with a moving rest length than to a position command, which is the compliant end of this trade rather than the stiff one.

Sideways from here, the same arithmetic prices something else entirely: a single actuator against a sheet with a manufacturing error. A panel cut a hair wrong is a permanent disagreement between the sheet’s own geometry and the actuator’s command, and it divides by the same $1/(1+g^2)$. Which creases are forgiving of a manufacturing error is therefore the same question as which pairs are forgiving of two actuators, and neither has been asked of a real pattern.

The habit worth carrying is about redundancy. Count the freedoms before calling a second driver a spare. Two drivers on two freedoms are redundant; two drivers on one freedom are a quarrel, and the quarrel’s cost is set by a coupling that has to be measured rather than assumed.