Sarrus, and the straight line that is exact

The straight-line problem is the oldest hard question in this subject, and the planar essays tell it as a two-hundred-year story: Watt’s approximation of 1784, Chebyshev’s better one, the theoretical result that no planar four-bar can be exact, and Peaucellier’s inversion cell of 1864 which finally was.

Something is missing from that account. In 1853, eleven years before Peaucellier, Pierre Frédéric Sarrus published a linkage that produces exact straight-line motion from six ordinary pin joints and no inversion argument at all. It is barely mentioned in most treatments, and the reason is that it does not fit: it is not planar, and the planar literature had no place to put it.

The construction, and why it works #

Take a fixed plate and a moving plate above it. Join them by a chain of three revolute joints whose axes are all parallel — call the direction x̂.

What can the moving plate do? Three parallel-axis revolutes in series form a planar chain, and a planar chain gives the plate three freedoms: it can translate in the two directions perpendicular to x̂ and it can rotate about x̂. What it cannot do is move along x̂, because none of the three joints can produce that motion.

So one chain confines the plate to a plane — the plane perpendicular to x̂.

Now add a second chain, three more revolutes, with all their axes parallel to ŷ instead. On its own it confines the plate to a plane perpendicular to ŷ.

Attach both. The plate must satisfy both constraints, so it is confined to the intersection of two planes, which is a line — and the line is the direction perpendicular to both x̂ and ŷ.

That is the entire argument. There is no approximation anywhere in it, no series expansion, no error term that gets small when the proportions are right. Two planes meet in a line, and the plate is on both planes, so it is on the line.

Compare that with what the planar mechanisms have to do. Watt’s linkage works because the coupler curve of a particular four-bar has a point of inflection, and near an inflection a curve is approximately straight. Chebyshev’s works because a different four-bar’s curve is a better approximation over a longer span. Both are arguments about how well a curve resembles a line, and both have an error that is a fixed fraction of the travel — 9% and 12% over their full strokes.

Sarrus’s argument is not about resemblance.

Measured, not asserted #

The claim “exact” is exactly the kind this site refuses to accept on the strength of a derivation. So the figure’s plate position is solved at every step, and two things are measured at each: how far the plate has turned from its starting orientation, and how far its centre has left the axis.

Over the sampled travel both come out at the 10⁻¹⁴ level, which is arithmetic noise on quantities of order one. The travel itself is about 1.5 units, so the deviation as a fraction of the stroke is around 10⁻¹⁴ — against Watt’s 9 × 10⁻² and Chebyshev’s 1.2 × 10⁻¹.

The tilt measurement took two attempts, and the first one is worth recording because it is a mistake with a recognisable signature. It compared the plate’s frame with the identity and reported a tilt of exactly π/2 at every position — a constant, which is the giveaway. A joint frame’s z axis is its own axis of rotation, so the frame in question was already a quarter turn from the identity before the mechanism moved at all. The quantity wanted is the rotation the plate has undergone, R(θ)·R(0)ᵀ, not its absolute orientation. A number that never changes as the mechanism moves is not measuring the mechanism.

What the drawing has to be careful about #

A spatial mechanism on a flat page is ambiguous in a way a planar one is not, and the Sarrus linkage is a bad case: from many viewing directions the two chains overlap and the figure reads as one folded chain rather than two perpendicular ones.

Three decisions in the figure exist to prevent that, and all three are drawing decisions rather than measurements, so they are worth declaring.

The projection is orthographic, not perspective. A perspective projection makes equal lengths unequal on the page, and a reader measuring a figure whose whole argument is about lengths would get the wrong answer. Parallel edges stay parallel here.

The joint axes are drawn, as dashed stubs, rather than left to be inferred from the bars. The argument of this essay is entirely about axis directions — three parallel here, three parallel there — and a figure that showed only the bars would be omitting the thing being claimed.

The viewing direction is stated in the figure’s description, and it is chosen so that neither chain is edge-on. There is no viewpoint from which both chains are seen undistorted; a projection is a choice, and pretending otherwise is how spatial figures mislead.

Everything else in the figure comes out of the solve. The plate’s height, the joint angles, the ghosted outlines of earlier positions: each is a converged configuration, and each carries its own residual.

Why three joints per chain and not two #

The obvious question about the construction is why each chain needs three revolutes rather than two, and the answer is a small mobility calculation that is worth doing because it shows the design has no slack in it.

A chain of n parallel-axis revolutes between the fixed plate and the moving one gives the moving plate as many planar freedoms as the chain has joints, up to the three a planar motion has. Two joints give two: the plate can reach a two-parameter set of positions in its plane, but its orientation is then determined by where it is. Three joints give the full three — position and angle independently — which is what “confines the plate to a plane and nothing more” requires.

With two joints per chain the two chains would over-determine each other: each would be dictating the plate’s orientation as well as its plane, and the two dictations would generally disagree. The mechanism would be a structure. Three per chain is the smallest number that leaves each chain saying only “the plate is in this plane” and nothing else, which is exactly what the intersection argument needs.

Four per chain would work too, and would add a second redundant constraint per chain with no new capability — more stiffness, more sensitivity to error, and an extra pin to make. Six joints total is the minimum, and it is what Sarrus used.

What a third chain would do #

Since two chains give a line, it is natural to ask what a third one gives, and the answer is a mechanism that is fully constrained — a structure.

Three chains in three mutually perpendicular planes would confine the plate to the intersection of three planes, which is a point. The plate could not move at all. That is not a useful mechanism, but it does make a point about how the argument scales: each chain removes one of the plate’s three translational freedoms after the first, and there are only so many to remove.

What is done in practice instead is to duplicate a chain in the same direction for stiffness — two x̂-chains and one ŷ-chain — which adds three more redundant constraints and no new restriction on the motion. That is a heavily overconstrained mechanism, it is stiff in a way a determinate one cannot be, and it will bind if the two x̂-chains are not accurately parallel to one another. The trade is exactly the one the field’s first essay describes, taken to its practical extreme.

What Kutzbach makes of it #

Six links, six revolute joints, one closed loop. The spatial mobility count gives

$$M = 6(6 - 1) - 5 \times 6 = 30 - 30 = 0$$

so the linkage is a structure and cannot move. It moves.

The screw system says why. The three x̂-parallel screws span a three-dimensional space; so do the three ŷ-parallel ones; but the two spans overlap, and the six screws together have rank 5 rather than 6. Mobility is 6 − 5 = 1.

One redundant constraint. That is a modest amount of overconstraint compared with the universal joint’s three or Bennett’s three, and it has a consequence worth drawing out: the Sarrus linkage is only slightly special, and being only slightly special is what makes it buildable.

The condition for it to work is that each chain’s three axes are parallel to each other, and that the two chains’ directions are not parallel to one another. Nothing else. The link lengths are free, the joints can be anywhere along their axes, and the two chains need not even be perpendicular — if they are at some other angle the plate still translates along the line where the two planes meet, just not along a line perpendicular to both.

Parallelism is a condition machining can hold, which is why this mechanism was practical in 1853 and Bennett’s, whose condition is an exact equation between four dimensions, was not practical for most of a century.

Against Peaucellier #

The two exact mechanisms deserve comparing directly, because they are exact for completely different reasons and the difference is instructive.

Peaucellier’s cell is planar. It works by realising the geometric operation of inversion in a circle mechanically: the product |OP|·|OQ| is held constant by a rhombus and two long arms, and inversion maps a circle through the centre to a straight line. The exactness is the exactness of a theorem in plane geometry, and the mechanism is an eight-bar with seven joints.

Sarrus’s is spatial. It works because two planes meet in a line. The exactness is the exactness of linear algebra, and the mechanism is a six-bar with six joints.

Both measure at the noise floor. Peaucellier’s inversion product holds to about 1.7 × 10⁻¹⁵ and its straightness to 9.8 × 10⁻¹⁶ of its span; Sarrus’s off-axis deviation is comparable. Neither is more exact than the other in any meaningful sense — both are exact, and the residues are the arithmetic’s, not the mechanism’s.

Where they differ practically is in what they cost. Peaucellier’s cell needs seven pins on eight bars and, more awkwardly, the accuracy of its output depends on two long arms being equal and a rhombus being a rhombus. Those are dimensional conditions on lengths, and a length can be wrong. Sarrus’s needs six pins whose axes are parallel in two groups, and parallelism of a bore is a different kind of manufacturing problem from equality of a length — usually an easier one, because it is set by the fixture rather than by the measurement.

The catch is the one overconstraint. A Sarrus linkage whose two chains are not quite in perpendicular planes, or whose axes within a chain are not quite parallel, has nowhere to put the error: the plate is being asked to be on two planes that do not quite intersect where the mechanism thinks they do, and the result is binding rather than a small deviation. This is the general behaviour of overconstrained mechanisms and it is the price of the stiffness they give.

The two kinds of exactness #

It is worth being precise about what “exact” means here, because the word does two jobs in this subject and confusing them is how approximate mechanisms get quoted as though they were exact.

An exact mechanism produces its output property for every configuration it can reach, as a consequence of its constraints. The Sarrus plate is on the line at every position because it is on both planes at every position. There is no stroke over which the claim is true and beyond which it degrades.

An approximate mechanism produces its output property to within some error over some range, and the error is a function of position. Watt’s linkage traces a curve with an inflection; near the inflection the curve is straight to third order, and further out it is not. The 9% figure the site quotes is the deviation over the full stroke, and quoting a smaller number for a shorter stroke is legitimate and is what designers do.

The distinction has a practical edge. An approximate linkage can be made better by using less of it. An exact one cannot be made better at all, because there is nothing to improve — and correspondingly, an exact one that is manufactured wrongly does not degrade gracefully. A Watt linkage with a bar a percent long is a slightly worse approximation. A Sarrus linkage with a chain a degree out of parallel is a mechanism that binds.

That asymmetry is the reason the approximate mechanisms did not simply disappear when the exact ones were published. Watt’s linkage was in production for a century after Peaucellier, and it was the right choice for a beam engine: four bars, three pins, and a failure mode that is a small error rather than a seized machine.

Where it is actually used #

Sarrus’s linkage is not a museum piece, which is the other reason it deserves better treatment than the planar literature gives it.

It appears wherever something must rise and fall without tilting and without a sliding guide. Scissor lifts are the family resemblance, though most are not Sarrus linkages proper. Deployable structures use it because it folds flat and extends to a definite height. Precision positioning stages use it because a pin joint has no stiction and a linear bearing does, so a mechanism that produces translation entirely from rotations avoids the failure mode that limits a slideway.

That last one is worth dwelling on because it inverts the usual reason for wanting a straight-line linkage. Watt needed one because he had no way to guide a piston rod accurately in 1784 — the technology for a straight slideway did not exist, and the linkage was a substitute for a machine tool. Modern uses of Sarrus’s linkage are not substituting for a slideway that cannot be made. They are avoiding one that can be made and behaves worse.

What the field gains from it #

The Sarrus linkage makes a general point that the planar half of the site cannot.

The straight-line problem was hard in the plane. It took two centuries, it produced a theorem saying no four-bar could do it, and the eventual answer was an eight-bar built on a piece of projective geometry. One dimension up, the same problem has a six-bar answer whose proof is one sentence, and that answer was available first.

That is not an argument that spatial mechanisms are easier. It is an argument that the plane is a constraint on the designer, not only on the mechanism — and that a problem which looks fundamental can turn out to be an artefact of where the subject agreed to work. The essays before this one are all about what can be done with bars in a plane. This one is about a problem that stops being hard when the plane is given up, and the cost of giving it up is one redundant constraint and a manufacturing condition that a fixture can hold.

The two mechanisms are worth setting against each other one last time, because the comparison is about what kind of exactness rather than about which is better. Peaucellier’s cell is exact by an algebraic identity: the product of two distances is constant, and the identity holds for the lengths it was built with and for no others. Sarrus’s is exact by membership of a group: the platform’s displacements are the intersection of two planar groups, and an intersection of groups is a group for any lengths whatever. So one mechanism is exact because its dimensions satisfy a relation and the other is exact because its axes have a relation — and the second kind survives every error in every length while the first does not. That is the same distinction the spatial field draws between the two overconstraints, arriving in the straight-line problem, and it explains why the answer that came eleven years earlier is the one with fewer bars and a more robust claim. Exactness from an identity is a coincidence held by manufacture; exactness from a group is a coincidence held by alignment, and alignment is what a workshop is organised to hold.