Machine Kinematics
solved, then drawn
Drawn wrongly

Two things called jamming

A four-bar's mechanical advantage peaks at 1,673 in one configuration and its transmission angle collapses in another, 222° away. Both get described as the mechanism jamming. One is enormous force output and the other is force disappearing into the bearings, and they are opposite situations.
16 min read 7 figures The singularities are the pointThe constant that is not constant

A four-bar has two configurations that get described in the same words and are not the same thing.

Two things that are not the same configurationThe mechanical advantage of a four-bar, from the velocity solution: for a lossless mechanism it is the reciprocal of the output-to-input speed ratio, so wherever the output momentarily stops the force ratio diverges. It reaches 1673 at 222° — a toggle, where crank and coupler line up, and the reason a knee-joint clamp holds with almost no effort. The transmission angle is at its worst somewhere else entirely: 46.5° at 0°, 222° away, where coupler and rocker line up instead. Both get called "the mechanism jamming" and they are opposite situations — one is enormous force output, the other is force going into the bearings.0153045600100200300crank angle (degrees)mechanical advantage (clipped at 60)toggle: advantage 1673worst μ = 46°four-bar 3.4/1.2/3/2.4, 720 solved positions222° apart
Fig. 1 Mechanical advantage through one turn, from the velocity solution. The green line marks the toggle, where the advantage reaches 1,673. The red line marks where the transmission angle is at its worst. They are 222° apart.

The toggle

A toggle, or dead centre, is where the crank and coupler become collinear. From the fixed pivot, the two of them form a single straight segment reaching to the far pin.

The output’s velocity passes through zero there, because the distance it depends on is at an extreme and its derivative vanishes. Since mechanical advantage is the reciprocal of the velocity ratio for a lossless mechanism, the advantage diverges.

This is the mechanism at its strongest. A knee-joint clamp holds at exactly this position with almost no effort at the handle; an over-centre latch stays latched because pushing on the output produces no crank torque at all.

The same configuration is why a single-cylinder engine needs a flywheel: at top dead centre the piston can exert no turning force, so something has to carry the crank past it.

The worst transmission angle

The transmission angle collapses when the coupler and rocker become collinear — a different pair of links, in a different configuration.

There the coupler’s force acts almost entirely along the rocker rather than across it, so nearly all of it goes into the output bearing and almost none produces torque. This is the mechanism at its weakest.

With friction — which this site does not model — a sufficiently poor transmission angle produces genuine locking: the mechanism will not move however hard it is pushed. That is the situation the 40° design rule exists to keep clear of.

The transmission angle through one turnμ 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.0501001500100200300crank angle (degrees)transmission angle μ (degrees)40° design limitground 4, crank 1, coupler 3.5, rocker 3μ from 54.3° to 100.3°
Fig. 2 The transmission angle through a cycle for a well-proportioned linkage, staying inside the usual band. Whether a mechanism approaches its collapse configuration is a design question separate from whether it has toggles.

Why the confusion

Both configurations stop the mechanism doing what was intended, so both get called jamming, and both are singularities in the sense that the constraint Jacobian is losing conditioning.

But the failure modes are opposite. At a toggle the mechanism cannot be driven backwards and is extremely effective driven forwards. At a poor transmission angle it is ineffective in both directions.

And the design responses are opposite. A toggle is something either exploited deliberately — clamps, latches, riveting tools — or design out by not letting the input reach it. A poor transmission angle is always a defect, and it is fixed by changing the four lengths.

What “singular” means here

Both configurations are singular, and it is worth saying precisely what that means, because the word does a lot of work in this subject and is rarely unpacked.

A mechanism’s constraint Jacobian relates small changes in the joint coordinates to small changes in the constraint residuals. At an ordinary configuration it has full rank: every direction the constraints could be violated in corresponds to some motion of the joints, and the relationship inverts cleanly.

At a singular configuration it does not. Some combination of joint motions produces no first-order change in the constraints, which is exactly the statement that the mechanism has, instantaneously, an extra freedom — or, seen the other way, that some motion the mechanism ought to have costs nothing to resist.

Three consequences follow, and they are the three phenomena on this page.

The output velocity can vanish while the input velocity does not, which is a toggle. The linear system relating them has become degenerate in that direction.

The force ratio can diverge, because for a lossless mechanism it is the reciprocal of the velocity ratio, and a zero in the denominator is exactly what the degeneracy produces.

The configuration can stop being determined at all, which is a change point. The redundant parallelogram’s flat position is this case: the constraints hold, and they no longer choose between two continuations. The solver’s response is to refuse, which is the correct answer to a question with two answers.

Measuring how close a configuration is

Because rank is a decision about which numbers count as zero, this site reports not only the rank but the gap — the ratio between the smallest pivot accepted and the largest one rejected during the elimination.

A gap in the millions means the decision was not close and the mechanism is comfortably away from a singularity. A gap near one means the matrix is nearly degenerate, and the rank the solver reports depends on where the threshold was put.

That second case is not a numerical embarrassment. It is the honest signal that the mechanism is approaching a toggle, and it is available continuously rather than only at the exact configuration — which matters, because a mechanism that passes near a singularity has large forces and poor conditioning without ever reaching the configuration a closed-form analysis would look for.

Why the vocabulary matters practically

Two situations that share a word get confused in design reviews, and the confusion has a direction: because “jamming” sounds like a failure, the toggle — which is a mechanism working extremely well — gets designed out along with the poor transmission angle.

That is a real loss. The whole family of over-centre devices depends on toggles: a toggle clamp, a latch that stays latched, a riveting tool, the locking action of a folding table leg. Every one of them is a mechanism deliberately operated at or just past the configuration where its advantage diverges, and the property being exploited is precisely that the output cannot drive the input back.

The design rules that follow are opposite and both are simple. For a poor transmission angle: change the four lengths until the worst value in the cycle is above about 40°. For a toggle: decide whether the mechanism should reach it, and either stop the input before it or place it deliberately where the holding force is wanted.

Neither rule is usable without knowing which situation the mechanism is in, which is the whole reason to separate the words.

Both are visible in the Jacobian

Neither needs special-case geometry to find. A singular configuration is where the constraint Jacobian loses rank, and the solver reports how close it came — the ratio between the smallest accepted pivot and the largest rejected one.

That is how the dead centres of a non-Grashof linkage are located on this site: the solver is asked for every input angle and declines 87 of 360, and the refusals are the mechanism’s limits rather than a numerical failure.

The angles this crank cannot reachA non-Grashof four-bar — non-Grashof (triple rocker) — asked for all 360 input angles. It assembled at 273 of them. The dial on the left marks the reachable arcs in green and the refused ones in red; the refusals are not a numerical failure but the mechanism's dead centres, where the crank and coupler line up and the linkage physically stops. Nothing about the four lengths had to be inspected to find them: the solver was asked, and declined.273of 360input anglenon-Grashof (triple rocker) · s + l exceeds p + q by 0.50green: assembles · red: refused
Fig. 3 Refusals located by asking. The red arcs are input angles at which the bars would have to change length. No closed form for the dead centres was needed — and the same method finds them for any mechanism the solver can position.

The general form

A single word covering two opposite phenomena is a sign that the underlying quantity has not been named.

Here the quantity is the mechanical advantage, and the two configurations are its two extremes: one where it diverges and one where it collapses. Saying which is meant takes one extra clause and removes the ambiguity entirely.

That pattern — a single number standing in for something that varies, and losing the interesting part — is what this whole field is about.

A four-bar at 60°, solvedGround 4, crank 1, coupler 3.5, rocker 3. Every joint position here is the output of a Newton–Raphson solve on the loop-closure equations, converged to 0.0e+0 — not a placement that looked right. Grashof's condition classifies these lengths as a crank rocker, and sweeping the crank through 360° confirms it: 120 of 120 positions assemble. The transmission angle at this instant is 66.9°.ABO₂O₄crank (input)couplerrocker (output)crank rocker · residual 0.0e+0positioned by solving, not by drawing
Fig. 4 The mechanism both configurations belong to. Turning the crank through a full revolution passes through both, and nothing about the picture at any single instant says which one is approaching.
What a sketched mechanism costsA four-bar positioned the way one is drawn by hand: the crank pin at its angle, and the rocker interpolated smoothly between its extremes because that is what the motion looks like. Plotted is how long the coupler would have to be at each position, minus how long it is. The error reaches 0.89 on a coupler of 3.5 — 25% — and a reader looking at the drawing would see nothing wrong, because every individual frame is a perfectly plausible picture of a four-bar. This is the failure the solver exists to make impossible: a configuration that does not satisfy the constraints cannot be drawn, because there is nothing to draw it from.-0.50000.50010100200300crank angle (degrees)coupler length errora rigid barcoupler 3.5, worst error 0.890every frame looks fine on its own
Fig. 5 Why both configurations are found rather than assumed. Every position here comes from a solve, so a toggle appears as a velocity passing through zero rather than as something the illustrator had to know about.
Three parallel bars, and a formula that says this cannot moveFive links and six pin joints, so Grübler's criterion gives 3(5−1) − 2(6) = 0 and calls it a structure. The Jacobian has rank 5 against 6 free coordinates, so it measures one degree of freedom — and the sweep assembles 59 of 60 positions, which settles the matter. The third bar removes no freedom because its constraint is already implied by the other two, and a formula that counts joints cannot notice that they happen to be parallel. Mechanisms of exactly this kind carry drafting machines, anglepoise lamps and locomotive coupling rods, where the redundant bar is there for load sharing and for keeping the linkage out of its change point.the redundant oneGrübler: 3(5−1) − 2(6) = 0Jacobian: 6 − rank 5 = 1
Fig. 6 A third thing that stops a mechanism, and a different one again: a redundant constraint that is not redundant once a dimension is wrong. That one jams and does not move at all.

The pattern, once more

A single word standing in for two opposite phenomena is a symptom, and the diagnosis is always the same: some quantity has not been named.

Here the quantity is the mechanical advantage, and the two configurations are its two extremes — one where it diverges and one where it collapses. Naming it takes a clause and removes the ambiguity completely.

The same diagnosis applies to a ratio quoted as a number for a mechanism that does not have one, to a complexity class quoted without an input distribution, and to a threshold quoted as an integer when the formula gives a fraction. In every case the fix is the same: say what is being measured, and the ambiguity has nowhere to live.

The transmission angle through one turnμ 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.0501001500100200300crank angle (degrees)transmission angle μ (degrees)40° design limitground 4, crank 1, coupler 3.5, rocker 3μ from 54.3° to 100.3°
Fig. 7 The quantity that decides one of the two, plotted through a full turn rather than quoted as a number. The worst value in the cycle is the figure a design is judged on, and it is not the mean.

The toggle as a design goal

Most of this essay treats the toggle as a configuration to know about. It is at least as often a configuration to aim for, and the machines that aim for it are worth listing because they make the geometry concrete.

A toggle clamp is a four-bar taken just past its toggle. Squeezing the handle drives the mechanism through the aligned position and a fraction beyond, where the clamping force acts to hold it rather than to release it. The clamp cannot be opened by pushing on the work, at any force, because the work’s line of action passes the wrong side of the pivot. Releasing requires the handle.

A knee joint — the human one and the prosthetic one — locks straight for the same reason. Standing does not require muscle to hold the knee extended, because at full extension the load line passes in front of the joint centre and the ligaments take it. The mechanism is at its toggle and the load is holding it there.

Aircraft landing gear is designed to reach an over-centre position when extended, with a small spring or actuator to hold it past the toggle. The landing load then drives the gear harder into the locked position, which is the only acceptable failure direction.

Rock crushers and press linkages run near the toggle at the moment of maximum force, so a modest hydraulic or eccentric input produces the enormous crushing force the process needs — the mechanical advantage that reaches 1,673 in the figure above is the whole design.

In every case the property being used is the same: at the toggle the output cannot back-drive the input, at any force. That is a geometric statement, not a friction one, which is why it survives lubrication, wear and load.

The two singularities, named properly

Robotics inherited these configurations and gave them precise names, and the names sort out the confusion this essay is about better than the mechanism vocabulary does.

An inverse kinematic singularity — also called a serial or type-one singularity — is where the input can move and the output cannot. That is the dead centre of a four-bar: the crank turns through, the rocker is momentarily stationary, and the mechanism has lost the ability to produce output velocity in some direction. The mechanism is at the edge of its workspace.

A direct kinematic singularity — parallel, or type two — is where the output can move and the input cannot resist it. The mechanism gains an uncontrolled freedom: the actuators are locked and the end effector still moves. That is dangerous in a way the first kind is not, because a machine that is stiff everywhere else becomes limp in one direction with no warning.

A four-bar’s toggle is the second kind viewed from the rocker and the first viewed from the crank, which is precisely why the same configuration gets two descriptions. Which one it is depends on which link is driven — the same observation the inversions make about mechanisms in general.

Both show up in the constraint Jacobian as a rank drop, and they are distinguished by which null direction appears: a direction in the output coordinates with the input fixed is the dangerous kind. That is another instance of the null space carrying information a rank number does not, and of a measured route reporting what a formula can only count.

Why the two get conflated

The vocabulary problem this essay is about is not carelessness. Both configurations really do produce a mechanism that will not move, and both really do involve a force that goes somewhere useless, so the same words fit both.

What separates them is whose motion stops and why.

At a toggle, the mechanism is momentarily unable to be driven backwards, at any force, for a purely geometric reason: the output’s line of action passes through a pivot and produces no moment about it. That is exact, it does not depend on friction, and it survives perfect lubrication. It is also usually what the designer wanted.

At a poor transmission angle, the mechanism can be driven both ways, and the force needed grows as the useful component of the coupler force shrinks. Whether it actually sticks depends on friction, joint clearance and load, so the boundary is fuzzy and application-specific. It is almost never what the designer wanted.

The measurement makes the distinction concrete: in the four-bar plotted here the two configurations are 222° apart, so a statement about “where the mechanism jams” is a statement about two different parts of the revolution depending on which sense is meant.

The practical cost of confusing them is real. A designer avoiding “jamming” by keeping μ above 40° has not avoided the toggle, which is elsewhere. A designer relying on a toggle to lock and quoting a transmission angle has quoted a number about somewhere else. Both are stated correctly only by naming the configuration, and the naming is what the vocabulary loses.

This is one of several places on this site where the technical language collapses a distinction that the measurement keeps — the others being the ratio of a linkage, which is a function rather than a number, and mobility, where one word covers a count and a measurement that can disagree.

What to check, and in what order

Reduced to a procedure, the two situations call for different checks and the checks are cheap.

For a toggle. Sweep the mechanism and look for configurations where the driving link and the coupler become collinear. Those are geometric and locatable in closed form for a four-bar; the sweep confirms them and gives the mechanical advantage nearby, which is the number that decides whether the toggle is useful or dangerous. If the mechanism is meant to lock, check the design sits past the toggle rather than at it, since at it the mechanism is on a knife edge and either side is a different behaviour.

For a poor transmission angle. Plot μ through the working range and take its minimum, not its average. Compare against a threshold chosen from the joint quality and the load — 40° for general machinery, more for precision. The design levers are the ground length and the crank length, and both trade output travel for force quality.

For both at once. Compute the constraint Jacobian’s smallest singular value through the sweep. It dips near either condition, which makes it a single screening quantity, and it does not distinguish them — so a dip is a prompt to look rather than a diagnosis. Distinguishing requires asking which null direction appeared, which is the same information the mobility measurement reports for a different purpose.

Order matters. Grashof first, because a chain that does not rotate fully has a different problem. Then the toggles, because they are geometric and their positions constrain the timing. Then μ over what is left, because it is the quality measure and quality is the last question. Each stage narrows what the next one has to consider, and each is a reading off configurations that were solved rather than drawn.

Where this goes next

0501001500100200300crank angle (degrees)transmission angle μ (degrees)40° design limitground 4, crank 1, coupler 3.5, rocker 3μ from 54.3° to 100.3° 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. -0.25000.2500.5000100200300crank angle (degrees)output ÷ input angular velocitya 20:40 gear pair, 0.500mean 0.000four-bar 4/1/3.5/3-0.40 to 0.33 through one turn The ratio that is not a number A four-bar's output-to-input speed ratio runs from −0.29 to 0.51 through one turn and changes sign on the way. Quoting a single figure for it quotes the average of that curve, which the mechanism never exhibits. A gear pair is the case where the same phrase is honest, and it is honest by construction. GrüblerJacobiantriangulated frame00agreefour-bar11agreeslider-crank11agreePeaucellier cell11agreeparallelogram + third bar01they disagree — the mechanism moves3(n−1) − 2j₁ − j₂ · free coordinates − rank(J)one row where the formula loses Counting and measuring mobility Grübler's criterion counts links and joints and never asks how long anything is. The rank of the constraint Jacobian measures the lengths and never asks what a joint is. Two calculations with no inputs in common, producing one number — which is the only arrangement under which agreement is evidence.