Machine Kinematics
solved, then drawn
The paths points trace

Where the coupler is turning

At every instant the coupler of a four-bar is rotating about a single point — not a pin, and usually not on the mechanism at all. Track that point in two different frames and you get two curves which, rolled on each other without slipping, reproduce the coupler's motion exactly. The bars are one way of producing it and not the motion itself.

Assumes What a coupler point draws and Four bars and four pins.

15 min read 7 figures Two routes to the same numberSolved, not drawnThe singularities are the point

Take a four-bar in one configuration and freeze it. The coupler — the link joining the two moving pins — is doing something: its two ends are moving in different directions at different speeds, and the link as a whole is going somewhere and turning.

There is a fact about rigid bodies in the plane that makes this simpler than it sounds. Any planar motion of a rigid body is, at any instant, a rotation about a single point. Not a rotation plus a translation; just a rotation, about a point that is generally not on the body and is different at every instant.

That point is the instantaneous centre, and finding it turns a complicated velocity field into one number and one location.

Where the coupler is pivoting, at 70°At any instant the coupler is turning about one point — not a pin, and usually not on the mechanism at all. Kennedy's theorem finds it: the crank and coupler share the pin at A, the coupler and rocker share B, so the coupler's centre relative to the frame must lie on both O₂A extended and O₄B extended, and it is where they cross. The dashed lines are that construction. The cross is a completely different route to the same point — the place where the coupler's solved velocity field is zero, computed from the Jacobian and knowing nothing about Kennedy. Across 119 positions the two agree to 2.7e-15. At this instant the centre lies outside the frame — the two construction lines are nearly parallel, the coupler is close to translating, and the pivot has run off rather than gone missing.centre outside the framepositioned by solving, not by drawing
Fig. 1 Where the coupler is pivoting at one crank angle. The dashed lines are Kennedy’s construction — the centre must lie on both O₂A extended and O₄B extended. The cross is a completely different route to the same point: the place where the coupler’s solved velocity field vanishes.

Finding it

A four-bar has four links, so it has six pairs of links and therefore six instantaneous centres — one for each pair.

Four of them are trivial. Where two links are pinned together is a point they share, so it is a point of each that is momentarily at rest relative to the other. The four pins are the centres I₁₂ (crank and frame), I₂₃ (crank and coupler), I₃₄ (coupler and rocker) and I₁₄ (rocker and frame).

The other two are not on the mechanism, and Kennedy’s theorem says where they are: for any three links, their three instantaneous centres lie on a straight line.

Apply it to the frame, the crank and the coupler. Their three centres are I₁₂ (which is O₂), I₂₃ (which is A) and I₁₃, the one being sought — so I₁₃ is on the line through O₂ and A. Apply it again to the frame, the coupler and the rocker: their centres are I₁₄ (O₄), I₃₄ (B) and I₁₃ — so I₁₃ is also on the line through O₄ and B.

Two lines, one point. The coupler’s instantaneous centre is where the crank and rocker, extended, cross.

The same argument on the other triples gives I₂₄, the centre for the crank and rocker relative to each other, at the crossing of O₂O₄ and AB.

Why Kennedy’s theorem is true

The theorem is usually stated and used without proof, and the proof is short enough to be worth having because it explains why the three in it is not arbitrary.

Take three links, 1, 2 and 3, and suppose their three pairwise centres are not collinear. Consider I₂₃, the point that is momentarily at rest as far as links 2 and 3 are concerned.

As a point of link 2, its velocity relative to link 1 is perpendicular to the line joining it to I₁₂. As a point of link 3, its velocity relative to link 1 is perpendicular to the line joining it to I₁₃. But it is the same point with the same velocity — links 2 and 3 have no relative motion there, by definition of I₂₃.

So one velocity vector is perpendicular to two different lines, which is impossible unless those lines are the same line. Hence I₁₂, I₁₃ and I₂₃ are collinear.

The argument uses three links and no more because it needs exactly two “as a point of” readings of one velocity. With four links there is no analogous statement, which is why a four-bar’s six centres are found by applying the three-link theorem four times rather than by a four-link theorem.

The second route

Kennedy’s construction is geometry. It uses no velocities and no solver.

The other route uses nothing else. The solver returns a velocity for every joint — differentiating the constraints gives a linear system with the same Jacobian the position solve already formed, so a full velocity field costs one back-substitution. From the two coupler pins’ velocities, the coupler’s angular velocity follows, and from that the point of the coupler that is momentarily at rest.

Across 180 positions of the crank the two routes agree to about 10⁻¹⁵ relative — which is arithmetic noise, and is the number the figure prints.

Positions where the two construction lines are nearly parallel are excluded from that comparison rather than allowed to dominate it. The centre is genuinely at infinity there, both routes say so, and comparing two very large numbers would measure the arithmetic rather than the geometry. How many positions were dropped is reported, because silently discarding a fraction of a sweep and quoting the agreement on the rest would be a lie of omission.

Where the coupler is pivoting, at 200°At any instant the coupler is turning about one point — not a pin, and usually not on the mechanism at all. Kennedy's theorem finds it: the crank and coupler share the pin at A, the coupler and rocker share B, so the coupler's centre relative to the frame must lie on both O₂A extended and O₄B extended, and it is where they cross. The dashed lines are that construction. The cross is a completely different route to the same point — the place where the coupler's solved velocity field is zero, computed from the Jacobian and knowing nothing about Kennedy. Across 119 positions the two agree to 2.7e-15.I₁₃two routes agree to 2.7e-15positioned by solving, not by drawing
Fig. 2 The same linkage half a turn later. The centre has moved a long way, which is what makes the locus below a curve rather than a point, and the two routes still agree to 10⁻¹⁵.

The centre at infinity

That exclusion is worth taking seriously rather than treating as a numerical nuisance, because it is a real configuration with a physical meaning.

When the crank and rocker are parallel, the two lines never meet and the instantaneous centre is at infinity. A rotation about a point infinitely far away is a translation, and that is exactly what the coupler is doing at that instant: every point of it is moving in the same direction at the same speed.

This happens twice per revolution in a typical crank-rocker, and a figure that drew a pivot ten screens away would be worse than one saying “the coupler is translating”. So the generator reports how far from parallel the two lines were, and the figure states the case rather than drawing it.

The parallelogram linkage is the extreme version: crank and rocker equal and parallel at every position, so the coupler translates throughout and the instantaneous centre never exists. That is what a parallelogram linkage is for, and it is a good example of a mechanism whose behaviour is most easily described by the fact that one of its constructions degenerates.

Accelerations do not work this way

One warning, because the instantaneous centre is so useful for velocities that it gets misapplied.

The point of the coupler at the instantaneous centre has zero velocity. It does not have zero acceleration. There is a different point — the centre of acceleration — where the acceleration vanishes, and it is generally somewhere else entirely.

The reason is that the instantaneous centre is itself moving. A point of the coupler that is momentarily at rest is about to stop being at rest, because the centre has moved on to a different point of the coupler, and the rate at which the centre travels along the centrodes contributes to every point’s acceleration.

This matters practically. A designer who computes the velocity of a coupler point by the instantaneous-centre method and then differentiates that formula with respect to time will get the wrong answer, because the formula’s centre is not a constant. The site’s velocities come from the same Jacobian the position solve used and its accelerations, where it needs them, come from differentiating the constraints again — never from differentiating a graphical construction.

The two centrodes

The instantaneous centre moves as the mechanism does. Trace it.

In the frame, it draws a curve — the fixed centrode.

In the coupler’s own frame — origin at one pin, x axis along the coupler — the same point draws a different curve, the moving centrode.

The two curves that could replace the linkage. The coupler's instantaneous centre, traced twice. In the frame it draws the fixed centrode; in the coupler's own frame — origin at A, x along A→B — the same point draws the moving centrode. The classical claim is that the coupler's motion is exactly reproduced by rolling the second curve on the first with no slipping, which makes the bars one way of producing the motion rather than the motion itself. That is testable, and the test is arc length: over each unbroken stretch the two curves cover the same distance, to 7.8e-8 of it — and the disagreement falls by a factor of 4.4 when the sampling is doubled, which is what a chord approximation to a smooth curve should do and is the reason the residue is sampling rather than slipping. Both curves run off to infinity where the coupler momentarily translates; the breaks are that, not gaps in the computation.
Fig. 3 The two curves, over a full revolution. Both run off to infinity where the coupler momentarily translates, which is why the curves have breaks rather than being closed loops.

Now the classical claim, and it is a strong one: the coupler’s motion is exactly reproduced by rolling the moving centrode on the fixed centrode without slipping. Not approximately, not near some configuration. Cut the two curves out of card, lay one on the other and roll them, and the moving curve’s plane goes through precisely the positions the coupler goes through.

That says something about what a linkage is. The bars are one way of producing a motion, and the motion itself is the rolling of one curve on another. Two completely different four-bars with the same centrodes produce the same coupler motion, and the bars are, in that sense, an implementation detail.

It is the same shift in viewpoint that Roberts’s cognate theorem forces from the other direction: three different linkages draw the same coupler curve, so the curve is the invariant and the linkage is a way of realising it.

The two curves that could replace the linkage. The coupler's instantaneous centre, traced twice. In the frame it draws the fixed centrode; in the coupler's own frame — origin at A, x along A→B — the same point draws the moving centrode. The classical claim is that the coupler's motion is exactly reproduced by rolling the second curve on the first with no slipping, which makes the bars one way of producing the motion rather than the motion itself. That is testable, and the test is arc length: over each unbroken stretch the two curves cover the same distance, to 1.4e-7 of it — and the disagreement falls by a factor of 4.4 when the sampling is doubled, which is what a chord approximation to a smooth curve should do and is the reason the residue is sampling rather than slipping. Both curves run off to infinity where the coupler momentarily translates; the breaks are that, not gaps in the computation.
Fig. 4 A third linkage’s centrodes. The arc-length agreement is measured over every unbroken stretch of each pair, and it lands at the sampling floor in all of them.

Checking the rolling

“Rolls without slipping” is a claim about arc length. If the moving curve rolled with slip, it would cover a different distance along itself than it covers along the fixed curve.

So: over every unbroken stretch where both curves are defined and bounded, measure the arc length along each and compare. They agree to about 8 × 10⁻⁸ of the length.

That number is not zero, and the interesting part is establishing what it is. Both curves are polylines through sampled points, so the comparison is limited by the sampling — and the limit is measured rather than assumed, by running the same comparison at half the resolution. The disagreement rises to 3.4 × 10⁻⁷, a factor of 4.4, which is what a chord approximation to a smooth curve does when the step is doubled. The residue is sampling, not slipping.

That discipline — quote an agreement against the resolution of the thing that measured it — is the same one the cognate comparison needs, and both were got wrong first by demanding a tolerance the measurement could not resolve.

The moving centrode is drawn in a frame that moves

There is a step in computing the moving centrode that is easy to skip and impossible to skip correctly, and it is worth spelling out because it is where the two curves differ.

The fixed centrode is straightforward: solve, find the instantaneous centre, record its coordinates. Its coordinates are already in the frame the answer is wanted in.

The moving centrode is the same point, recorded in the coupler’s frame — which means converting the centre’s position into coordinates measured along and across the coupler, with the origin at one of its pins. That conversion changes at every position of the mechanism, because the coupler has moved.

So the moving centrode is not a different point; it is the same point in a different set of axes. Read that sentence back and the rolling claim becomes almost obvious: at each instant the two curves touch at the instantaneous centre, and the instantaneous centre is by definition the point where the moving plane has zero velocity relative to the fixed one. Two curves touching at a point of zero relative velocity are rolling, not sliding. The theorem is a restatement of what “instantaneous centre” means.

That does not make the measurement pointless — a claim’s being obvious in hindsight says nothing about whether the code implements it — but it does explain why the arc-length agreement comes out at the sampling floor rather than at some interesting physical residue. There is no slipping to find.

The two curves that could replace the linkage. The coupler's instantaneous centre, traced twice. In the frame it draws the fixed centrode; in the coupler's own frame — origin at A, x along A→B — the same point draws the moving centrode. The classical claim is that the coupler's motion is exactly reproduced by rolling the second curve on the first with no slipping, which makes the bars one way of producing the motion rather than the motion itself. That is testable, and the test is arc length: over each unbroken stretch the two curves cover the same distance, to 1.0e-7 of it — and the disagreement falls by a factor of 4.0 when the sampling is doubled, which is what a chord approximation to a smooth curve should do and is the reason the residue is sampling rather than slipping. Both curves run off to infinity where the coupler momentarily translates; the breaks are that, not gaps in the computation.
Fig. 5 A different linkage’s pair of centrodes. The curves change completely with the proportions, which is the point: they are a description of the motion, and two linkages with the same centrodes produce the same motion.

What the centre is good for

Three uses, in increasing order of practicality.

Velocities, without differentiating anything. Once the instantaneous centre is known, the velocity of any point of the coupler is the angular velocity times the distance from that point to the centre, perpendicular to the line joining them. That is a graphical method, it was how velocity analysis was done before computation, and it is still the fastest way to get an approximate answer from a drawing.

Reading a coupler curve’s shape. The curve a coupler point traces has cusps exactly where the tracing point coincides with the instantaneous centre, because a point momentarily at rest is a point where the curve’s tangent is undefined. That is a genuinely useful diagnostic: a coupler curve with a cusp says that the tracing point crosses the fixed centrode, and a designer who wants a cusp — for a mechanism that has to stop and reverse — knows where to put the point.

Understanding the transmission angle. The transmission angle measures how much of a force applied along the coupler turns into useful torque on the rocker, and its relationship to the instantaneous centre is direct: the distance from the centre to the output pin is the effective lever arm. A mechanism whose instantaneous centre wanders close to the rocker pin has almost no leverage there, and that is the same fact the transmission angle reports.

Designing a motion rather than a mechanism. If the required motion is specified as a rolling of one shape on another — which is how a designer thinks about a wheel, a cam, or a rolling-contact bearing — then the centrodes are the specification, and a linkage that produces them is a way of implementing it without machining the curves.

Where the output stops, and why the return is quicker. A crank-rocker's output reaches an extreme exactly when the crank and coupler line up — stretched out, so O₂ to B is 4.50, or folded back, so it is 2.50. Nothing about the rocker enters the condition, which is why the limits can be written down rather than searched for. Those two crank angles are 40.8° and 228.5°, so the crank spends 187.7° going one way and 172.3° coming back while the rocker covers the same 40.0° both times. The ratio is 1.0894 predicted and 1.0894 measured over 7200 swept positions — a shaper cuts on the slow stroke and returns on the fast one, and this number is what the proportions are chosen to get.
Fig. 6 The configurations I₂₄ lands on a ground pivot at. The output has stopped, and the instantaneous-centre construction and the collinearity condition are describing the same two positions.

The other constructed centre

I₂₄ — the crank’s centre relative to the rocker — gets less attention than I₁₃ and is worth a paragraph, because it answers a question the transmission angle also answers and answers it differently.

It sits where the ground line O₂O₄ crosses the coupler line AB. Its use is the mechanical advantage: the ratio of the rocker’s angular velocity to the crank’s is the ratio of the distances from I₂₄ to the two ground pivots. So a single construction gives the instantaneous velocity ratio of the whole mechanism, without solving anything.

Two consequences fall straight out. When I₂₄ is far away — the coupler nearly parallel to the ground — the two distances are nearly equal and the ratio is near 1. When I₂₄ lands on one of the ground pivots, the ratio is zero or infinite, and those are the limit and toggle positions reached from a different direction: the output has stopped, or the input has no leverage at all.

This is the reason the classical texts spend so long on instantaneous centres. Before there were solvers, one straightedge construction gave the velocity ratio, the mechanical advantage and the location of every singularity, from a drawing. The solver gives better numbers and the construction gives the reasons.

Where the two curves fail

The centrodes are the coupler’s motion, and they are an awkward object in two specific ways.

They are unbounded. Every time the coupler momentarily translates, both curves run off to infinity, and a mechanism that translates twice per revolution has centrodes in several unbounded pieces. Rolling one on the other is a fine mathematical statement and not something to build.

They are not closed. A closed curve rolling on a closed curve would return to its start after a whole number of turns, which is what a pair of gears does. The centrodes generally do not close, so the “rolling” is a description rather than a construction.

Both restrictions have an important exception: when the centrodes are closed and bounded, the rolling can be built, and it is. That is what non-circular gears are — two closed centrodes cut as pitch curves and given teeth — and it is how a mechanism with a prescribed varying velocity ratio is made when a linkage will not do. The connection is worth carrying: a gear pair’s constant ratio comes from two circles rolling, and a varying ratio comes from two non-circles rolling, and the centrodes are how the two are found.

Why the centre is worth computing at all

A reasonable objection to everything above is that the solver already returns every joint’s velocity, so a construction that produces velocities more cheaply is a historical curiosity.

Two answers, and the second is the real one.

The instantaneous centre is one point instead of a field. A velocity field is a function; the centre is a location and a scalar. That compression is what makes the quantity thinkable — a designer can look at where the centre is and say immediately whether the coupler is nearly translating, whether the mechanism is near a singularity, and roughly how fast any point of the coupler is going, without evaluating anything.

And the centre is the object the motion is a statement about, in a way the joint velocities are not. Joint velocities are a property of a particular linkage; the centrodes are a property of the motion, and three different linkages producing the same coupler motion have the same centrodes. Computing the centre is therefore a step toward describing what the mechanism does rather than what it is, which is the same shift synthesis makes from the other direction.

Two poles at 66°, and only one of them is stillThe stubs here are **accelerations**, not velocities, and they vanish at a different point from the one where the velocities vanish. The cross is the velocity pole — the point the plane is turning about — and the square is the acceleration pole, the point that is momentarily not accelerating. Over a full turn of this crank the two are never nearer than 1.49 coupler lengths and get as far apart as 22.2. The instantaneous centre is about velocities and about nothing else: the point that is not moving is being accelerated, usually hard, and a body pivoting about it in the ordinary sense would not be. positioned by solving, not by drawing.closest approach 1.49 coupler lengths in a full turnpositioned by solving, not by drawing
Fig. 7 The same instant with accelerations instead of velocities. They vanish at a different point, and over a full turn of this crank the two are never nearer than one and a half coupler lengths. The centre found above is about velocities and about nothing else.

Where this goes next

Five points on one coupler. The same four-bar, with a tracing point rigidly attached to the coupler at five different places. Each curve is a sextic — degree six — and moving the attachment point a little changes it a great deal. That sensitivity is the reason coupler-curve synthesis was done with atlases of printed curves for most of the twentieth century: there is no simple inverse, so the practical method was to look one up. What a coupler point draws A point rigidly attached to the coupler of a four-bar traces a curve of degree six. Move the attachment a little and the curve changes a great deal. For most of the twentieth century the practical way to find the linkage that draws a wanted curve was to look it up in a book of printed atlases. The transmission angle through one turn. μ is the angle at B between coupler and rocker, computed from each solved position rather than from a formula. It runs from 54.3° to 100.3° for these lengths. The shaded band is the usual design rule — keep μ between 40° and 140° — and this linkage stays inside it throughout. The rule is about geometry alone: nothing here knows about friction, and a mechanism with a comfortable μ can still be a poor machine. The transmission angle The angle at which the coupler meets the rocker decides how much of an applied force becomes useful output torque and how much goes into the bearings. It is pure geometry, it is computed here from every solved position rather than from a formula, and it is the number a linkage is judged by after Grashof has said it turns. 3 four-bars, one coupler curve. Three different four-bars, with different ground pivots, different link lengths and different proportions — 0.692 and 0.799 times the size of the first. Every one of them draws this same curve. Roberts's theorem says there are always exactly three, and the construction is one complex multiplication: write the coupler point as λ = (P − A)/(B − A), put the third fixed pivot at O₂ + λ(O₄ − O₂), and the other two linkages fall out with their bars' roles permuted — what is a coupler in one is a crank in another. The curves here were traced separately, each from its own solver runs, and agree to 1.3e-5 against a sampling resolution of 1.4e-5 — which is to say, as closely as the comparison can tell. Three linkages, one curve Every coupler curve is drawn by three different four-bars, not one. The other two can be constructed from the first with a single complex multiplication, they have different proportions and different ground pivots, and the roles of their bars are permuted — what is a coupler in one is a crank in another.

What this makes readable

Essays that name this one as a prerequisite.

About the same objects

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The 8 of 20 essays linking to this one that name the most of the same objects.

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AccelerationCentrodeCoupler curveInstant centreKennedy's theoremRatioRolling without slippingTransmission angleVelocity fieldVelocity ratio