The mechanism drops out

A four-bar’s coupler is a bar between two pins. That is what it is made of, and it is not what it does. What it does is carry a whole plane with it — the bar, the tracing point bolted to it, the paint on its side, every point of the infinite sheet rigidly attached to those two pins — and slide that plane over the fixed plane underneath.

So does a slider-crank’s rod. So does the follower face of a cam, the coupler of a Watt linkage, a wheel rolling on a road, and a sheet of paper pushed about a table with two fingers. Near any one instant these are all the same kind of object: a planar motion, which is a point and an angle, both changing.

This field takes that object seriously and then discards the machine. Every essay before this one has been about a mechanism — what it can reach, whether it can move at all, how many ways it assembles, what it is like when built out of parts with tolerances. What follows is about the motion, and the mechanism appears in the figures as scaffolding: something had to produce the motion, and it might as well be the four-bar the site has been using since its first essay.

A motion is two functions #

Fix a frame in the moving plane. Then a point of that plane with body coordinate $\zeta$ sits, in the fixed frame, at

$$z(t) = A(t) + \zeta\,e^{i\varphi(t)},$$

where $A$ is where the moving frame’s origin has got to and $\varphi$ is how far it has turned. Complex numbers are being used here as a shorthand for the plane, so $e^{i\varphi}$ is a rotation and $\zeta$ is a fixed pair of coordinates.

Everything in this field is a derivative of that one expression. The velocity of the point is

$$z' = A' + i\varphi'\,\zeta e^{i\varphi},$$

and the whole of the instantaneous centre is the observation that this vanishes for exactly one $\zeta$ whenever $\varphi' \neq 0$. Set $z' = 0$ and solve: the point of the moving plane that is momentarily still sits at $iA'/\varphi'$ from the moving origin, and that is a formula rather than a construction.

Two things are worth noticing about it straight away. The first is that the mechanism has already disappeared. $A'$ and $\varphi'$ are properties of the motion; nothing in the expression knows how many bars there are or where the ground pivots sit. The second is that the pole is a point of the moving plane and also, at that instant, a point of the fixed plane, and it is a different point of each of them a moment later.

The prime is a crank angle, not a second #

There is no time in this field. Every derivative here is with respect to the input angle — the crank angle of the four-bar, the rod angle of a trammel — and every quantity called a velocity is a change per radian of input.

That is not a simplification, it is the correct choice, and the reason is that the geometry of a motion does not know how fast the motion is being run. Turn the crank twice as quickly and every velocity doubles, every acceleration quadruples, and the pole does not move an inch. The path a point traces is the same path. Since this field is entirely about the shapes of those paths, using the input as the parameter keeps every quantity independent of a running speed nobody stated.

Where a quantity does depend on the parameterisation, the essays say so. The pole does not. Path curvature does not. The exponents in how long a pivot can stand in for a linkage do not. What does depend on it is the acceleration of a point, which is why the other pole needs a paragraph of its own on the subject.

One matrix, four right-hand sides #

To get anywhere the derivatives of $A$ and $\varphi$ are needed, and not only the first ones. The circle of points going straight needs two derivatives; the cubic of stationary curvature needs three; the points that stay on one circle longest need four.

The four-bar’s loop is a sum of complex vectors that must vanish:

$$a\,e^{i\theta_2} + b\,e^{i\theta_3} - c\,e^{i\theta_4} - g = 0.$$

Differentiating $e^{i\theta}$ repeatedly produces a pattern worth writing out, because it is the whole trick:

$$\begin{aligned} (e^{i\varphi})' &= (i\varphi')\,e^{i\varphi} \\ (e^{i\varphi})'' &= (i\varphi'' - \varphi'^2)\,e^{i\varphi} \\ (e^{i\varphi})''' &= (i\varphi''' - 3\varphi'\varphi'' - i\varphi'^3)\,e^{i\varphi} \end{aligned}$$

In every one of these the highest derivative appears linearly and by itself, multiplied by $i$, and everything else in the bracket is built from derivatives already known. So differentiating the loop $n$ times gives two real equations — the real and imaginary parts — in the two unknowns $\theta_3^{(n)}$ and $\theta_4^{(n)}$, and the matrix of those equations is

$$\begin{bmatrix} i\,b\,e^{i\theta_3} & -i\,c\,e^{i\theta_4} \end{bmatrix},$$

which does not depend on $n$ at all. One two-by-two matrix, four right-hand sides. Factor it once and every derivative up to the fourth falls out in the same number of operations as the first.

That matrix is also the loop’s Jacobian, so a position where it is singular is a position where the mechanism is at a dead centre, and the derivatives are refused there rather than returned as very large numbers. The two failure modes coincide because they are the same failure: at a dead centre the mechanism’s motion is not differentiable in the input, and neither is anything computed from it.

Why not just difference the solve? #

There is an obvious alternative. Every position on this site comes from a Newton–Raphson solve; solve at seven nearby crank angles, take differences, and read off derivatives. It is four lines of code and it needs no algebra.

It is also the route this field uses as its check rather than as its answer, and the reason is worth stating carefully, because the obvious argument for the algebra turns out to be wrong.

The obvious argument goes: the solver stops when its residual falls below $10^{-9}$, so its answers carry an error of that size; a third derivative divides that error by $h^3$; at a step of $0.01$ radians that is $10^{-3}$, so a differenced third derivative would have about four correct digits left.

That argument is four orders of magnitude wrong, and measuring it is how the site found out. Newton’s convergence is quadratic, so the iterate that first satisfies a $10^{-9}$ test has already overshot far past it. Measured over 360 crank angles against the four-bar’s closed-form solution, the worst position error is $9.8\times10^{-14}$ and the worst residual at the moment the loop stopped asking is $9.7\times10^{-14}$ — nearly five orders inside the tolerance that was supposed to bound it. A stopping rule says where an iteration stops asking, not how good its answer is.

So differencing the solve does work: at a step of $0.01$ radians and a seven-point stencil it recovers the third derivative to $2.3\times10^{-8}$ relative. It is a genuine second opinion rather than a degraded one, which is better than the story the algebra was originally justified by.

The algebra is still the route the figures use, for two reasons that survive the correction. It needs one solve rather than seven — and seven solves at neighbouring angles are seven chances to be refused near a dead centre, where a sweep with a missing frame is not obviously a sweep with a missing frame. And it has no step size in it, so there is no number to choose and no number to get wrong.

The check that had to be rewritten #

The first version of the derivative check compared the exact derivatives against central differences of the closed-form coupler angle, using the three-point and five-point stencils that everyone writes first. The first derivative came out $2.1\times10^{-5}$ wrong.

There is nothing wrong with the arithmetic. A three-point central difference has a truncation error of order $h^2$, and at $h = 4\times10^{-3}$ that is $1.6\times10^{-5}$ — the entire discrepancy. The check was measuring its own stencil.

Two things about that are worth carrying. The first is that it would have been read the wrong way round: the quantity under test is the exact route, the quantity doing the testing is the approximate one, and a disagreement gets attributed to whichever one the reader distrusts more. The second is that the fix is free. Seven samples buy a sixth-order formula for the first two derivatives and a fourth-order one for the other two, at no extra evaluations, because the samples were already being taken. With those stencils the four derivatives agree to $9.5\times10^{-13}$, $4.1\times10^{-9}$, $3.7\times10^{-7}$ and $1.1\times10^{-5}$ — which is the accuracy the differencing can reach, and is reported as such rather than as the accuracy of the thing being checked.

That last number is why the fourth derivative is used sparingly. Everything in this field that needs it — the points with five-point circle contact — is checked a second way, by measuring a drawn path rather than by trusting a number good to five digits.

What the field is going to claim #

The rest of this field is the consequences of those derivatives, and it is worth laying out what they are, because they are more surprising than the machinery suggests.

Every point of the moving plane has its path curvature decided by one relation and two numbers. Not one relation per point: one relation, holding at every point at once, with the point’s position relative to the pole as its only input. That is Euler and Savary’s and it is 1830.

The points whose path is momentarily straight form a circle. Not approximately a circle — a circle, and the fact is derivable in one line from the structure of the second derivative rather than constructed. It is the inflection circle, it passes through the pole, and its diameter is the pole’s own speed divided by the plane’s angular rate.

The points whose path curvature has stopped changing form a cubic. The expression whose zero set that is looks like a quartic; its fourth-degree terms cancel, and the cancellation is measured rather than asserted. The cubic runs through the pole twice and through both moving pins.

Where those two curves meet is where a slide could replace the linkage. One point, at all but two of 360 positions of the site’s four-bar, and it is not where Watt or Chebyshev put their tracing points — which is measurable and was worth measuring.

And the whole thing has an exponent attached to it. Replace a coupler point by a crank pivoting at the centre of its path’s osculating circle and drive the mechanism away from the instant. The point leaves the circle like the cube of the step for an ordinary point, like the fourth power for a point of the cubic, and like the fifth for the rare points that satisfy one condition more. Those three exponents are what the two special curves are for.

The pole is not the interesting part, and it is where everything starts #

It is worth being precise about the status of the instantaneous centre in all of this, because it is the one object here a reader is likely to have met, and it is usually met in a form that makes it seem more than it is.

The pole is a fact about velocities and nothing else. At this instant the velocity field of the moving plane is exactly the velocity field of a rotation about that point. That is all. It does not follow that the plane is rotating about that point in any sense that survives to the next instant, that the point is fixed, that it has anything to do with the accelerations, or that a body pivoted there would behave as the mechanism does.

The last of those is measurable and is measured. Over 117 positions of the site’s four-bar the point with zero velocity and the point with zero acceleration are never nearer than 1.49 coupler lengths and get as far apart as 22.2. They are different points at every position and they move differently. An essay of this field is about that gap because it is a mistake with a long history and a plausible-sounding defence.

What the pole is good for is that it organises everything else. Both special curves are given most conveniently in polar form about it. Euler–Savary is a statement about distances from it. The pole’s own velocity is the diameter of the inflection circle. Its path in the fixed plane and its path in the moving plane are the two centrodes, and the entire motion is the second rolling on the first without slipping — a statement with no mechanism in it whatsoever, which is the clearest sign that the mechanism was never the subject.

Three checks that the machinery is describing the drawing #

A field whose objects are this abstract needs its assertions tied to something drawn, or it becomes a set of internally consistent numbers about nothing. Three of them do that job throughout.

The pole three ways. Kennedy’s construction intersects two lines that are already in every drawing of the mechanism — the crank extended and the rocker extended. The velocity field’s zero is a solve. The derivative route is $iA'/\varphi'$. Over 57 positions the three agree to $1.5\times10^{-13}$, and the first of them is a construction a reader can check with a ruler on the figure.

The curvature against the drawn curve. A coupler point’s path is traced by the solver at hundreds of positions and the resulting polyline’s curvature is read off by circumcircle through consecutive points. That number, which knows nothing about any derivative, matches the exact curvature; at the position where the site’s Watt linkage sits at its symmetric pose both routes give $4.644$.

The inflection count two ways. How many times a coupler curve changes the way it bends, counted from the sign of the exact curvature, and counted from the turning of the drawn polyline. On a coupler point at $(0.8, 0.2)$ of the coupler, both give two.

None of those is decoration. The worked example this site keeps returning to is a sign error in a Jacobian that produced perfectly correct pictures for months, and every quantity that was printed alongside those pictures was also correct. What caught it in the end was a measurement of something drawn. A field this far from the drawing needs that habit more than the others, not less.

What this field is not #

Three exclusions, stated here so that later essays can point at them rather than restate them.

It is not dynamics. A path’s curvature is geometry. What a body of some mass does when made to follow that path involves a force, and the boundary is the one the transmission field drew and the rolling field kept: if an argument needs to know what is pushing, it is somebody else’s. The one place it gets close is the acceleration pole, and the essay on it is careful to keep the geometry and the mechanics apart.

It is not synthesis, though it borders on it. The problem backwards is finding a linkage that produces a wanted motion. What is here is a description of a motion the linkage already produces. The two meet exactly once, in the essay where Burmester’s four-position construction is watched as its four positions coalesce, and they turn out to be the same object approached from two directions.

It is not about paths as sets. A coupler curve is a sextic and has a great deal of algebra attached to it: degrees, double points, the number of them a linkage can trace. That is the curves field, and it is about a whole curve. This field is about a neighbourhood of one point of one curve, and the two questions have almost no arithmetic in common.

The claim, stated so it can fail #

The field’s opening claim is that a mechanism is one of many ways of producing a motion and that near an instant the motion can be written down without it. That is easy to say and hard to test, so here is the version with a number in it.

Take a coupler point. Compute the centre of its path’s curvature and the radius. Throw the four-bar away and replace it with a single crank pivoting at that centre with that radius. Drive both, and measure how far apart the two points get.

The answer is that they separate like the cube of the input step, and the fitted exponent over a decade of step sizes is $3.00$. Three derivatives of the motion agreed and the fourth did not, so the disagreement shows up at third order — which is exactly what a pivot standing in for a linkage should do if the description is complete to second order and no further.

That is a claim that would fail loudly if any of the machinery were wrong. A curvature computed from a mistaken third derivative would give an exponent of two. A curvature computed from a mistaken second derivative would give one. The exponent is not a decoration on the measurement; it is the measurement, and it is the shape every later essay in this field uses to establish that a special point is special.