Where the input stops deciding

Every configuration on this site until this field has been the answer to a question of the form given the input, where is everything else. For a loop that is a Newton solve on closure equations; for an open chain it is a product of exponentials; for a parallel platform it is a polynomial system with several roots. The answers differ, the question does not.

The question does not work here.

Three answers to the same input #

Take the pallet angle 1.1°, which is where the corner of the left pallet sits. Over one period the mechanism is at that pallet angle three times, and the wheel is somewhere different each time:

• on the way out, with a tooth on the left pallet’s locking face, before the supplementary arc;
• on the way back, with the same tooth on the left pallet’s impulse face, just beginning the impulse;
• and much later, with the pallet swinging the other way entirely and a different tooth held on the right pallet, the left pallet touching nothing at all.

Those are not three roots of one equation. They are three different equations, because the contact is different in each: circle-against-arc on one face, circle-against-line on another, and a contact on the other pallet altogether. A solver handed the pallet angle and told to find the wheel has not been given enough information to start.

What it needs is the contact state — a discrete label naming which surfaces are touching — and with that label it has an ordinary problem again. So a configuration in this field is a pair: a label, and a solve within it.

Why this is not just a branch #

The site has met multiplicity before and it is worth being precise about the difference, because “there are several answers, pick a branch” is the standard way of dealing with it and it does not work here.

A four-bar assembled with the same four lengths and the same crank angle has two configurations, and they are the two roots of one quadratic. Which one a given linkage is in is fixed at assembly and does not change while it runs; the branch is a choice within one system of equations.

An escapement’s three answers are not roots of anything. There is no single system whose solutions they are, and they are not fixed at assembly — the mechanism moves between them, in a fixed cyclic order, twice per period. Choosing a branch does not disambiguate them because a branch is a label on a solution and these are labels on equations.

The nearest thing the site has to the escapement’s situation is the assembly modes of a mechanism that can change between them — a change-point linkage passing through a position where the branches meet. Even there, the two branches are roots of one system and the meeting is a degeneracy of that system. Here there is no system to be degenerate.

The path, and what its shape says #

The picture is the most compact statement available, and three of its features are worth reading off.

It crosses itself. That is the multiplicity above, drawn. A curve in the plane that crosses itself is not the graph of a function, and no re-parameterisation makes it one.

Some of its runs are exactly horizontal, and some are not. A horizontal run is an interval of pallet motion over which the wheel does not move — the deadbeat’s locking face doing its work. The escapement drawn here has flat faces with six degrees of draw, so its runs slope: the wheel loses ground during each supplementary arc and takes it back on the return, and the slope of those runs is the same number as the draw.

Two segments are vertical. Those are the drops, in which the wheel turns and the pallet is not touching it. They are drawn at the pallet angle of the release, and that is a deliberate incompleteness: how far the pallet has travelled by the time the wheel lands is a question about torque, and the figure says so rather than interpolating.

And it closes. After one period the pallet is back where it started and the wheel has advanced by exactly one tooth — 12.000000000° on thirty teeth, against a pitch of 12°. The path is not a closed curve in the plane, and it is a closed curve on a cylinder, which is the honest way to say that the mechanism’s state space is (pallet angle) × (wheel angle modulo a tooth).

The state machine #

Naming the labels turns the picture into something with a small, complete description.

There are six states in a period, in a fixed cycle: for each pallet in turn, locked (a tooth on the locking face), impulsing (the same tooth on the impulse face), and free (the drop, with nothing touching anything). Each transition is triggered by a geometric condition and by nothing else:

Every trigger is a length or an angle on the drawing. That is why an escapement can be specified on paper: the state machine’s transitions are dimensions, and the only thing the dimensions do not settle is when each transition happens in time.

It is also why the beat budget closes. The wheel’s travel over a period is the sum of what it does in each state, and each of those is a contact solve; three of them are non-zero and they add to half a pitch per beat exactly.

What this does to the site’s usual questions #

Several questions the site asks routinely need restating in this field, and it is worth doing so explicitly.

“What is the mechanism’s mobility?” It is not one number. A lever escapement has two degrees of freedom while its fork is empty and one while the pin is in it, and both are correct. Mobility is a property of a state, not of a mechanism.

“Where are its singularities?” The configurations where a Jacobian drops rank are still there and are still findable, and they are no longer where the interesting behaviour is. What matters here happens at the transitions, and a transition is not a singularity — nothing degenerates, a contact simply ends.

“What is its transmission function?” Piecewise, and not a function at all if the pieces are forgotten. The Geneva’s driven fraction $(n-2)/2n$ is the fraction of the input’s turn spent in one particular state; quoting it without saying which state it belongs to is what produced the $1/n$ that the first essay in this field corrects.

Hysteresis, which is what a crossing looks like from outside #

There is a word for a mechanism whose output depends on where the input has come from as well as where it is, and it is worth using because it connects this field to a great deal that is not mechanism kinematics.

Hysteresis is the crossing, seen by somebody who only has access to the input and the output. Sweep the pallet angle up and the wheel does one thing; sweep it down and it does another; the difference between the two is a loop enclosing an area, and the area is the thing that makes the mechanism useful rather than a defect in it.

For the escapement drawn above the loop is thin — the wheel’s excursion is a fraction of a degree during the lock and a couple of degrees during the impulse — and for the anchor escapement it is fatter, because the recoil is larger. For a ratchet it is as wide as the mechanism: push the lever forward and the wheel follows; pull it back and the wheel does not move at all.

The useful thing about naming it is that it says what the discrete label is, physically. It is memory. An escapement remembers which pallet is engaged; a ratchet remembers which tooth it has reached; a Geneva remembers whether the pin is in a slot. Every mechanism in this field has a small amount of state that is not a position, and every one of them was built to have it — which is why intermittent mechanisms turn up wherever something has to be counted, indexed, held or stepped, and never where something merely has to be moved.

Two ways to draw the same period #

The state path can be plotted either way round and the two versions say different things, which is worth a paragraph because the choice is not obvious.

Wheel against pallet — the figures above — is the mechanism’s own state space, and it is the version in which the drops are vertical jumps and the locks are flat treads. It has the crossing in it, which is the point.

Pallet against wheel is the same curve reflected, and in that orientation the drops become horizontal and the locks become vertical, which reads as though the pallet moved instantaneously and the wheel waited. Both are true statements about a curve and only one of them matches how the mechanism is driven: the pendulum is the input, so the pallet angle is what advances steadily, and a figure that puts it on the vertical axis is inviting the reader to read the wheel as the driver.

This site draws the input on the horizontal axis everywhere, and the reason is the same one: the crank is the argument. In this field the crank is a pendulum, and the escape wheel — which is doing all the moving, and which is what a photograph of the mechanism draws the eye to — is the output.

What is gained by admitting the discrete part #

It would be possible to insist that the mechanism really does have a single configuration space and that the labels are bookkeeping. That is true in a sense — the true configuration space is a subset of (wheel angle, pallet angle) cut out by non-penetration inequalities, and the labels name its pieces — and it is not useful, because the pieces are where every computation happens.

What admitting the labels buys is that each piece is an ordinary problem. Inside one contact state this field’s mechanisms are as tractable as anything else on the site: closed-form contacts, checked against Newton, agreeing to $10^{-15}$. It is only the boundaries between pieces that are new, and the boundaries are exactly where the subject’s own vocabulary lives — drop, lock, corner, banking, drop-in, let-off. Horology named the transitions because the transitions are where the mechanism is.

That is a general lesson about mechanisms with contacts, and it is worth stating as one. When a mechanism’s joints are all pins, its state is a configuration; when some of them are contacts, its state is a configuration and a list of which contacts are closed. The second is not a complication of the first. It is a different object, and the essays in this field have been about the list rather than about the configurations all along.

What a photograph cannot show #

One consequence of all this is practical and slightly melancholy: an intermittent mechanism cannot be understood from a still picture of it, and the whole of the historical literature is still pictures.

A drawing of a four-bar contains its entire kinematic content. The four lengths are on the page, the crank angle is on the page, and everything else follows — which is why a linkage can be copied from a plate in an 1868 catalogue and will work. A drawing of an escapement contains its dimensions and not its state, and the dimensions alone do not say which of six configurations is shown. Two plates showing the identical mechanism at the identical pallet angle may be showing two entirely different instants.

Nineteenth-century horological drawing solved this by convention rather than by geometry: escapements are drawn at the instant of drop, or at the instant of lock, and the caption says which. The convention works and it is a convention, and a reader who does not know it is looking at a picture with a third of its content missing.

This site’s figures carry the state in the caption for the same reason, and the drag carries it in the readout. It is the one place where a figure on this site needs a word to be complete, and the reason is exactly the argument above: the state has a part that is not a position, and a position is all a drawing can hold.

Six problems and a table, and three things to check #

Admitting the discrete part buys a decomposition, and the decomposition is worth stating as a verification strategy, because it makes available three checks that have no counterpart on a mechanism with one piece.

The mechanism is six continuous problems and a transition table. Each state has its own contact set, its own equations and its own solve; each transition has a trigger that is a length or an angle on the drawing. Verify the pieces and verify the table, and the whole is verified — which is exactly the decomposition a single configuration space would deny.

First check: every trigger is reached. A state whose exit condition is never satisfied is a state the mechanism enters and does not leave, which is a mechanism that stops. That is a real failure of a real escapement — a lock too deep for the pallet’s travel to clear, a drop that never arrives because the tooth cannot reach the next pallet — and it is invisible in any one state’s equations, since each of the six is perfectly well posed on its own.

Second check: no two triggers fire at once. If two exit conditions can be satisfied simultaneously the next state is ambiguous, and an ambiguous transition is a mechanism whose behaviour depends on which inequality the arithmetic tests first. That is a genuine design fault rather than an implementation detail: it means two teeth arrive at once, or a pallet releases at the instant the other engages with no interval between, and the mechanism’s behaviour is decided by whatever the parts do rather than by the geometry.

Third check: the cycle closes. After the six states the pallet must be where it started and the wheel exactly half a tooth pitch further on. That is the beat budget, and it is a check on the table rather than on any state — the six states could each be right and the cycle still fail to close if a transition is mis-ordered.

None of those three questions can be asked of a four-bar, because a four-bar has one state and no table. They are the price and the compensation of the discrete part together: the mechanism is harder to describe, and what makes it harder is also what makes it checkable in a way a continuous mechanism is not.

Which is the general form of the lesson this essay reaches. A mechanism with contacts is a hybrid object — continuous inside each contact set, discrete between them — and the right response is neither to force it into one configuration space nor to treat it as a special case. It is to compute the pieces with the ordinary machinery and to give the table the same standard of proof, since the table is where the mechanism’s character actually lives.

The question this leaves #

If the state has a discrete part, then two configurations of the mechanism can be in the same piece of the space and still not be joinable — or in different pieces and joinable through a transition. Neither of those is a question about connectivity, which is what the configuration space essay taught the site to ask.

The last essay in this field asks the other question, on the simplest mechanism in it, and the answer is that being in one piece and being able to get there are different things.

That is a stronger break with the configuration-space result than it sounds. Connectivity is a symmetric relation — a path run backwards is a path — and every claim the site has made about assembly branches, working modes and unreachable poses has leaned on that symmetry without ever naming it. A mechanism with a one-way contact breaks it while leaving the connectivity entirely intact, so the free space is one piece and half of the ordered pairs in it cannot be joined. The measurement is one line and the consequence runs back through six phases.