Sarrus linkage: rotation to straight translation without guides

You want a platform to rise and fall in a perfectly straight line, square to its base, and the first idea that comes to mind is a rail — a slider, a guide, two surfaces sliding against each other. But a printed rail rubs, wears, and needs a clearance that turns into wobble. The Sarrus linkage solves the same problem without a single sliding surface anywhere: just bars and pivots. It is an exact straight-line generator — not approximate — built entirely from rotational joints, and that is the property that makes it interesting to print. You know how to get a printed pivot right; a printed rail, almost never.

Why two perpendicular planes give you a straight line

The kinematics are more elegant than they look, and it pays to understand them before you touch a dimension. Picture a single chain of articulated links between the base and the platform: two bars joined by a pivot; one pivot at the top against the platform, another at the bottom against the base; all three joints parallel to one another. That chain on its own lets the platform move, but not in a straight line: it traces an arc within its own plane, because nothing stops it from drifting sideways in the direction perpendicular to its pivot axes. It has a surplus of freedom.

The Sarrus trick is to mount a second, identical chain rotated 90°, in a plane perpendicular to the first. What matters is what each chain forbids, not what it allows. A chain whose three pivots are parallel to the X axis blocks lateral translation along X and two of the rotations; it leaves free the translation along Y, the vertical translation, and one rotation. The second chain, rotated 90°, blocks what the first left free: the translation along Y and the leftover rotation. Between the two, the constraints combine: both lateral translations and all three rotations are blocked, and the only freedom that survives both is pure vertical translation, perpendicular to the base. What remains is a straight line. It is not a compromise or an approximation tuned so the error stays small over a stretch, the way a four-bar is; it is an exact geometric constraint. The platform cannot go anywhere else.

It pays to know what this costs: the two-chain Sarrus is an over-constrained mechanism. By the classical degree-of-freedom count, two chains of three pivots should lock the platform completely; that it moves at all, and that it moves straight, depends on a special geometric condition — the three axes of each chain strictly parallel, the two planes exactly at 90°. When that condition holds, the line is exact. When it does not, the mechanism jams or binds. That over-constraint is the underlying reason for almost everything that follows: it is what makes the Sarrus so sensitive to whether the two chains match.

This has a direct practical consequence. Because the straight motion is guaranteed by the geometry of the pivots and not by contact between flat surfaces, there is no guide to score, no track to flatten, no pair of faces to wear cycle by cycle. There is still sliding, of course — at the pin-and-hole interface of each pivot — but it is small-radius, low-speed sliding over a cylindrical surface that FDM prints far better than a pair of ground flats. The travel is not bounded by the end of a rail but by the folding of the links themselves: the platform rises until the bars approach vertical and lowers until they lie nearly flat, at which point it sits practically level with the base. Between those two extremes it travels in a straight line.

What fails is accumulated play, not a single loose pivot

Here is the part that decides whether your printed Sarrus works or wobbles. The exact straightness the kinematics promise assumes perfect pivots, with no play. Your mechanism has several pivots in series per chain, and each one contributes its small bit of play: the gap between the pin and its hole, the tenths of a millimeter the printer shaved off the fit. In the worst case those gaps add up along the chain; in practice, because they are more or less independent errors, the typical tilt grows more slowly. But there is one source of error that dominates over the summed clearances of a single chain: the mismatch between the two chains. Because the straight line is born from both being identical and at 90°, any asymmetry — one chain looser, shorter, or worse aligned than the other — breaks the cancellation, and the platform drifts toward the side of the weaker chain. This is where the over-constraint shows itself: the mechanism does not forgive a mismatch between the two halves.

That is why the design criterion is not "let each pivot turn freely" but "keep the total play and the chain-to-chain mismatch below what your application tolerates." It is the same per-side clearance arithmetic that governs any printed fit — the one you have worked through in Tolerances for moving parts — but applied to a stack of joints where each one's error is inherited by the next and, above all, where the two chains have to come out as twins. If you make the pivots print-in-place (printed already articulated, no assembly), calibrate them carefully: a clearance below the threshold where the printer welds the walls leaves you with a fused, rigid link instead of an articulated one; too much clearance does give motion, but every extra tenth accumulates into platform tilt.

That fusion threshold is not set by your machine's calibration alone: it depends a great deal on the orientation of the gap relative to the layers. A horizontal gap, between two layers resting one on top of the other, welds differently from a vertical gap traversed by the same perimeter, and a bridge or overhang at the interface changes it again. The window is narrow, and you find it through calibration and part orientation on the bed, not from a table.

When straightness truly matters, the answer is to assemble the pivots with fitted pins or bushings rather than trusting everything to print-in-place. A straight metal pin gives a much cleaner turning surface and a fit you control finely, and an embedded bushing absorbs the wear that would otherwise open up the plastic hole cycle by cycle. You do not have to do it on all of them: the most loaded pivots, or the ones that contribute most to the error chain, are enough.

Orient the links so they load along the grain

Each link of a Sarrus is, mechanically, a connecting rod: it transmits force between two pivots working in tension or compression along its axis. That is the direction in which you have to lay the bead's fibers. If you print the link so the layer line crosses its axis, you are asking the load to separate two welded layers, and the bond between layers is exactly the anisotropic weakness of FDM: the plane along which a printed part gives way first, as Layer orientation for motion spells out. Lay the links flat on the bed so the bead runs along the bar and tension follows the grain rather than crossing it.

But laying the bar flat on the bed has a cost worth confronting directly: it leaves the pivot hole's axis vertical, perpendicular to the bed, and a hole printed with its axis vertical comes out layered, stepped, with worse roundness exactly where you want a clean turn. You have two demands in tension there — the bar wants to lie down to resist, the hole wants to stand on edge to turn well — and there is no single orientation that satisfies both. The practical way out is to prioritize the link's strength (lay it down) and recover the pivot's quality another way: run a reamer through the hole, or fit a pin with a bushing that does not depend on the printed roundness. On the pivots where straightness rules, the bushing settles the conflict once and for all.

The failure mode that the flat orientation prevents is not just the clean fracture of a link. Under the compressive load that appears when the platform carries weight, a long, slender link can buckle: instead of holding when pushed in a straight line, it bows suddenly to one side and loses all its load capacity. Buckling gives no warning; it is sudden, and a link with poorly oriented layers buckles sooner because its effective section is weaker than its geometry suggests. To raise the threshold, thicken the smaller dimension of the section: buckling happens about the axis of least inertia, so widening the broad side does no good if the bar stays thin in the other direction; it buckles sideways where it is thinnest. As a rule of thumb for when buckling matters: below a length-to-depth ratio of about 10 to 1, buckling rarely governs and the link breaks first by something else; above 20 to 1 buckling rules and you should redesign the section or shorten the bar. Save the slender links for the Sarrus mechanisms that only guide a motion without holding much weight.

The third failure mode is local: an overloaded pivot that breaks. In a chain in series, not every pivot sees the same force; those in the bottom row, against the base, usually carry most of the moment of the extended platform. If one gives, the whole chain loses the constraint it provided and the mechanism stops moving straight on the spot. Identify the most loaded pivots, give them more material around the hole or run a metal pin through them, and don't leave them as the thin print-in-place link that will be the first to delaminate.

When the Sarrus is the right choice

The Sarrus shines where you want compact, foldable straight translation without paying for the friction surfaces of a slider: lift platforms that rise vertically, mechanisms that fold flat and deploy in a line, stages that need to move perpendicular to their base with no guide to maintain or lubricate. It folds almost flush with the base when the bars lie down, and it deploys to its full stroke with no rail clearance to make it tip sideways — provided you have tamed the pivot play and matched the two chains. Against a dovetail (the trapezoidal-section guide that nests one part inside another) or a sliding rail, it trades the continuous rubbing of two flat faces for the low-speed turning of a few joints — exactly what FDM does best.

It is not the choice when you need a long stroke with little folded height and a lot of cantilever load: there the links become long, buckling becomes critical, and the accumulated play of so many pivots ends up dominating. For those cases a guide with bearing surfaces distributes the moment better. But for a clean, perpendicular, rail-free lift, the Sarrus gives you a perfect straight line out of the part your printer makes better than any other: a pivot.

The next step, before printing, is to decide the orientation of each link and each pivot on the bed, because both the straightness and the buckling resistance come from there. You worked this through in Layer orientation for motion.