The strand that is slack

Mobility is the first thing this site computes about any mechanism, and it has always been one number. Grübler’s formula counts it from the links and joints; the rank of the constraint Jacobian measures it from the geometry; where the two disagree the formula is wrong and the mechanism is interesting. Both are properties of the mechanism, and both give the same answer wherever the mechanism happens to be standing, because a pin removes the same two freedoms whatever the links are doing.

A strand does not work like that, and the reason is the second of its two defining sentences: it holds two points at no more than a distance. So it removes a freedom where it is taut and none at all where it is slack, and whether it is taut is a fact about the current configuration rather than about the machine.

Three strands, and a region instead of a place #

Take a point in the plane, tie it to three fixed anchors, and ask where it can be.

With three rigid links the answer is nowhere. Three distance equations in two unknowns are one equation too many: unless the three lengths are chosen to satisfy a relation exactly, there is no point at the right distance from all three, and the mechanism cannot be assembled at all.

With three strands the answer is a region — the intersection of three discs — and it is a perfectly ordinary two-dimensional set.

That is the whole difference between the two kinds of member, and it is worth dwelling on how large a difference it is. The rigid version has no solutions; the strand version has a continuum. The rigid version’s assembly is a knife-edge in the space of lengths; the strand version is indifferent to the lengths over a wide range, and merely changes size.

Three mobilities, in one mechanism #

Inside the region nothing is taut. Every strand is shorter than it needs to be, none of them is doing anything, and the point has both its freedoms: mobility two.

On one of the arcs, one strand is exactly at length. The point may slide along that arc — the freedom along the arc survives — and it may move inward, into the region, but not outward. Mobility one, and one-sided.

At a corner two strands are taut together. The point is at the intersection of two circles, which is the classical two-link determination, and it has mobility nought — again one-sided, since it can still move into the region.

There is no defensible way to call this mechanism’s mobility one number. It is genuinely two in most of its own configuration space and nought on a set of three points, and that is not an artefact of the counting: it is what a one-sided constraint is.

The discrete part of the state #

There is a way to describe the mechanism honestly, and this site has met it before.

A configuration here is not a point. It is a point and an active set — which of the strands are currently taut. That label is discrete, it changes as the mechanism moves, and the equations governing the mechanism are different for each value of it. Exactly the same structure as the mechanism that waits, whose configuration is a contact label plus a solve, and whose input therefore does not determine its output.

The parallel is close enough to be useful. A ratchet’s pawl is either on a tooth flank or riding over a tip, and which of those it is doing cannot be read off the wheel angle. A strand is either taut or slack, and which of those it is doing cannot be read off the position of the point — because a point in the interior of the region has every strand slack, and a point on the boundary has one or two taut, and the boundary is where the constraint suddenly starts to exist.

The site’s own machinery still works, once the label is given #

Nothing above says the mobility cannot be computed. It says it is not a property of the mechanism, and the distinction is worth making precisely, because the machinery this site already has turns out to answer correctly as soon as it is told which strands are taut.

The constraint Jacobian for a set of distance constraints has one row per constraint: the unit vector from the anchor to the point. Its rank is what the mobility field measures. Take the rows belonging only to the active strands, and the rank of that matrix is the number of freedoms removed, everywhere:

• In the interior no strand is taut, the matrix has no rows at all, its rank is nought and the mobility is two.
• On an arc there is one row, its rank is one, and the mobility is one.
• At a corner there are two rows, they point in different directions, the rank is two and the mobility is nought.

At the degenerate length of 56.191685 mm there are three rows and the rank is still two, because three vectors in the plane cannot be independent — which is the same redundant-constraint arithmetic the constraint field is built on, arriving in a mechanism that is not over-constrained at all in any way a designer would care about.

So the answer is not that a strand mechanism is uncountable. It is that its count is a function on the configuration space rather than a number, and the argument of that function is the discrete label.

The corners, and the branches they are #

The three corners of the region are the configurations in which the point is determined, and each of them is a two-circle intersection — the oldest solve on this site.

For the 130, 130 and 120 mm case they sit at $(90.00, 93.81)$, where the two long strands are taut together; $(125.13, 35.26)$, where the first long strand and the short one are; and $(54.87, 35.26)$, the mirror of it. Each pair of circles intersects in two points, and only one of each pair is inside the third disc: the other lies outside it, which means the third strand would have to stretch to reach.

That is the same structure as the assembly modes of a rigid mechanism — two roots of a quadratic, only some of which can be reached — with one difference that matters. A rigid mechanism’s second branch is a genuinely different assembly, and getting from one to the other means taking the machine apart. Here the discarded root is not a second assembly; it is a position the mechanism cannot reach because a strand would have to lengthen, and the mechanism can be walked continuously from any corner to any other through the interior of the region without anything being disassembled.

A region is not a workspace #

One thing this region is often confused with, and it is a different object.

A workspace — the thing the parallel platform field computes — is the set of positions a mechanism can be driven to, given the ranges its actuators have. It is a statement about what the machine can achieve.

The region here is the set of positions the mechanism is permitted to occupy with the strand lengths fixed at particular values. It is a statement about what the machine cannot prevent. Change the lengths and it moves and changes shape; drive the lengths and the region sweeps out something larger, which would be the workspace.

The two coincide only for a mechanism whose constraints are all bilateral and all active, which is every rigid mechanism and no strand mechanism. It is worth keeping them apart because the intuition transfers badly in both directions: a large workspace is usually good and a large region is usually bad, since it is the amount by which the load is free to wander when nothing is holding it.

No number of strands can hold anything still #

Here is the consequence that matters most, and it is geometric rather than mechanical.

The intersection of discs is convex. So unless the region has degenerated to a single point, it has an interior, and a point sitting anywhere on its boundary — including at a corner, with two strands taut — can always move inward. Every strand it is touching goes slack when it does, and none of them objects.

Strands alone therefore never fix anything. Not three of them, not six, not a hundred: whatever the arrangement, the reachable set is convex and a body on its boundary can retreat into it. Something has to push outward — gravity on a suspended load, a spring, a body in compression, another mechanism — and the moment that something is named the argument has left this field.

That is the honest form of a fact usually stated about cable-driven machines: they need more cables than they have freedoms, and even then they need the load to be pulling in a helpful direction. The geometric half of it is above, and it is exact. The rest is a statement about wrenches and belongs where forces do.

Where the region goes #

Shortening one strand shrinks the region, and it does so smoothly right down to nothing.

The interesting length is where it reaches nought, and it has a closed form. The two 130 mm strands are taut together at the intersection of their circles, at $(90, 93.808)$; the third anchor is at $(90, 150)$; the distance between them is 56.191685 mm. At exactly that third length the three circles pass through one common point, the region is that point, and every one of the three strands is taut at once.

That is the only configuration in which three strands determine a position, and it takes a length specified to six figures to arrange. A millimetre longer and the point is free to wander over 0.67 mm² of a lens-shaped region; a millimetre shorter and the mechanism cannot be assembled at all, because the third strand cannot reach.

What the rigid mechanism would have been #

The comparison is worth completing, because it is the reason this field exists.

Three rigid links to three fixed anchors is a structure with mobility $-1$ by any count on this site: three constraints on two freedoms. It is the redundantly-constrained case the constraint field is about, and its behaviour is the one that field describes — it assembles only for lengths satisfying a relation, and a tolerance on any of the three lengths destroys the assembly.

Replace the links with strands and the redundancy stops being a problem and becomes a feature: the third strand is not fighting the other two, it is idle, and it does something only in the corner of the region where it is reached. Adding a fourth, fifth and sixth strand costs nothing and each simply trims another arc off the boundary.

This is why cable-driven machines are built with more cables than freedoms and rigid parallel machines are not. Redundant rigid constraint is a manufacturing problem; redundant unilateral constraint is free.

What a fourth strand does #

Since a strand costs nothing to add, it is worth asking what adding one buys.

Each new strand contributes one disc, and the region becomes the intersection of four rather than three. That either trims an arc off the boundary — replacing part of one strand’s arc with part of the new one’s, and adding two corners while removing none — or does nothing at all, if the new disc contains the whole existing region. The second case is the four-strand version of the peg inside the hull: a constraint that is present, correct, and never active.

Which of the two happens is decided entirely by the length. So a designer adding a cable to a rigging is not adding a constraint; they are adding a constraint conditionally, and whether it does anything depends on a length they have to get right to within the size of the region they are trying to remove.

There is no equivalent of this among rigid links, and that is the last of the differences worth listing. A rigid link added to a mechanism always does something — it either removes a freedom or over-constrains the assembly, and both are visible in every count. A strand added to a mechanism may do nothing whatever, for ever, and no count will report it.

The same inequality, in the mechanisms that have runs #

The point held by three strands is the cleanest case and not the only one. Every mechanism in this field has the same inequality somewhere.

A belt is taut because the pulleys are further apart than the belt is long; slacken the centres and the drive stops driving, without anything about the geometry becoming invalid. A tackle’s parts go slack the moment the hauling end is released. A tendon pulls a joint one way and nothing the other. A chain hangs on its slack span, which is a length inequality that is not active and a shape this field does not compute.

Two things this cannot say #

Which configuration the mechanism is in. The region is where the point may be. What decides where it actually is, out of two square millimetres or five thousand, is whatever is pulling on it, and nothing in this field knows. That is a sharper limitation than it sounds: for a rigid mechanism, giving the input determines the output, and here giving every strand’s length determines only a set.

What happens at the moment a strand goes taut. Approaching the boundary, the mechanism has two freedoms; on it, one; and the transition is instantaneous in this model because the strand is inextensible. A real strand stretches, so the transition is smooth and takes a measurable distance to complete, and that distance is a stiffness divided by a force. Not here.

A convex region has no branches #

The convexity of the region is used above to prove that strands never fix anything, and it has a second consequence that runs the other way and is worth having, because it is the compensation for the first.

A convex set is connected, and simply connected. Any two points of the region are joined by the straight segment between them, and that segment lies entirely inside. So a point held by three strands can be moved from any permitted position to any other by the most direct route available, with every intermediate position permitted, and there is nothing to plan.

Set that against the mechanism this site usually draws. A rigid linkage’s configuration set is a curve, generally with several branches, and two configurations on different branches cannot be joined at all — which is what an assembly mode is. A parallel platform has up to six assemblies for one set of motor angles and a controller that loses track of which is in trouble.

A strand mechanism has none of that. There is one region, it is convex, every configuration in it is reachable from every other, and the question which branch is the machine on does not arise. That is a genuine and unusual property, and it follows from exactly the same convexity that makes the mechanism unable to hold anything.

So the two facts are one trade stated twice. Strands give up determinacy and get connectivity. Rigid links determine a position and pay for it with branches, singularities and assembly modes; strands determine nothing and have a configuration set with no structure in it worth the name.

Which explains a practical asymmetry between the two kinds of machine. A cable-driven machine’s control problem is entirely about tension — keeping every cable taut, deciding how to share the load among more cables than freedoms — and not at all about which configuration it is in. A rigid parallel machine’s is the reverse. Two mechanisms doing similar work, with their difficulties in completely different places, and the geometry says which.

What is not modelled #

A slack strand’s shape is not computed and is not drawn. A real one hangs in a catenary decided by its weight, which this field does not have; here a slack strand is simply a length inequality that is not currently active. Nothing collides: a slack strand may pass through an anchor, through another strand, or through the point it is tied to, and none of that is checked, because the model has no strand in space to collide with. The anchors are points and the moving body is a point, so nothing rotates and no strand wraps anything in this essay — the moment the moving body has a size, the taut length stops being a distance between two points and becomes a run, and everything above has to be recomputed with the machinery from the rest of the field.