A Sarrus Linkage is a six-bar spatial mechanism that converts hinged folding motion into pure straight-line translation between two parallel plates. The folding camera bellows on early Kodak field cameras and the deployable solar array hinges on several CubeSats use this exact arrangement. Its purpose is to give you guided linear motion without rails, slides, or prismatic joints — only revolute hinges. The outcome is a compact stowed package that extends along one axis with no lateral wobble, ideal anywhere weight, friction, or rail contamination would kill a sliding solution.
Sarrus Linkage Interactive Calculator
Vary link length, fold angle, load, and efficiency to see vertical travel, fold span, extension, and lift torque for a symmetric Sarrus linkage.
Equation Used
The symmetric Sarrus linkage behaves like two equal folding links in each hinge chain. For link length L and fold angle theta, the plate separation is h = 2L sin(theta). The horizontal fold span is 2L cos(theta). A centered load requires quasi-static drive torque proportional to F dh/dtheta, adjusted here by efficiency.
- Two identical hinge chains are mounted 90 deg apart.
- Both chains use equal link lengths and fold symmetrically.
- theta is measured from the flat stowed position toward vertical extension.
- Load is centered on the moving top plate.
- Calculation is quasi-static and ignores hinge friction except for efficiency.
Operating Principle of the Sarrus Linkage
The Sarrus Linkage, also called the Sarrus linkage (spatial straight-line) in kinematics literature, works by constraining two parallel plates with two independent hinge chains, each containing three revolute joints whose axes lie in a single plane. One chain folds in one vertical plane, the other folds in a plane rotated 90° around the vertical axis. Because each chain on its own would allow the top plate to translate AND rotate, putting two of them in series cancels every rotational freedom and leaves a single degree of freedom: pure vertical translation. That is the entire trick. No rails, no bushings, no prismatic slider — just six revolute hinges arranged so their constraints intersect on the line you want.
Why design it this way? Revolute hinges are cheap, sealable, and tolerant of dust and vacuum. Sliding rails are not. If you need linear motion in a satellite deployment, a film-camera bellows, or a folded medical retractor that has to survive autoclaving, you do not want a prismatic joint exposed to the environment. The Sarrus linkage gives you the function of a slider using only pin joints.
Tolerances matter more than people think. The two hinge planes must be perpendicular to within roughly ±0.5° or the top plate will start to skew and bind. If the link lengths in the two chains do not match within about 0.1 mm on a 100 mm linkage, you get a parasitic tilt at full extension — the plate lifts but tips a few degrees off horizontal. Common failure modes are hinge-pin wear (introduces lateral play that compounds across three joints), out-of-plane bending of the link plates under load (which adds a rotation the second chain cannot cancel), and over-travel near full extension where the chain approaches a singular configuration and force transmission collapses to near zero.
Key Components
- Base plate: The fixed reference frame. Both hinge chains attach to it through their first revolute joints. Flatness should be held to about 0.05 mm across a 100 mm plate, otherwise the two chains start fighting each other at the limits of travel.
- Top plate (moving platform): The output member that translates in pure straight-line motion. Its mass and the load it carries set the torque demand at every hinge. Keep its centre of mass directly above the linkage centroid — offset loading multiplies the bending moment on the link plates by the moment arm.
- Hinge chain A (three coplanar revolute joints): Three hinges with parallel axes, all lying in the same vertical plane. This chain alone would let the top plate translate vertically and rotate about the hinge-axis direction. Pin diameter typically 3–6 mm for a desktop-scale build.
- Hinge chain B (three coplanar revolute joints, rotated 90°): Identical to chain A but rotated 90° around the vertical axis. Its constraints kill the rotation chain A allowed and vice versa. Hinge axis perpendicularity to chain A must be held within ±0.5°.
- Link plates: The rigid bodies between hinges. They take bending and torsion under load. Use plates stiff enough that mid-span deflection stays below 0.1% of link length under working load, or you will lose the straight-line behaviour.
- Hinge pins and bushings: Standard revolute pivots. Radial play under 0.05 mm per joint is the working target for precision builds — three joints in series mean clearances stack.
Where the Sarrus Linkage Is Used
The Sarrus linkage shows up wherever a designer needs guided straight-line motion but cannot tolerate a sliding joint. Aerospace deployables love it. So do folding optics, surgical retractors, and any compact lift where rail contamination is a deal-breaker. You will see it called a Sarrus linkage (spatial straight-line) in academic papers and a Sarrus mechanism in deployable-structure literature — same thing.
- Aerospace / spacecraft: Deployable solar array hinges on small satellites and CubeSats — for example, the MarCO mission used folding hinge mechanisms in the Sarrus family to deploy panels without rails that could gall in vacuum.
- Photography (historical): Folding plate cameras such as early Kodak and Voigtländer field cameras used Sarrus-type linkages to keep the lens board parallel to the film plane as the bellows extended.
- Medical devices: Surgical retractors and self-deploying tissue spreaders that need to open along one axis without a prismatic slide that would trap fluids or fail autoclaving.
- Robotics: Compact vertical-lift end effectors and pick-and-place Z-axes on tabletop robots where a linear slide would be too heavy or too tall in the stowed position.
- Consumer products: Pop-up phone stands, folding laptop risers, and some adjustable-height monitor arms that need rigid vertical travel from a flat-folded state.
- Architecture / deployable structures: Foldable shelters and rapidly deployable shade canopies — the Hoberman-style deployables sometimes use Sarrus chains as edge members to keep flat panels parallel during expansion.
The Formula Behind the Sarrus Linkage
The useful design equation for a Sarrus linkage relates the vertical extension of the top plate to the link length and the fold angle of each chain. At the low end of useful travel — fold angles below about 15° from the base plate — the top plate barely lifts and you get tiny vertical motion for a lot of angular sweep, which is great for fine-positioning but terrible for deployment speed. At the high end — fold angles above about 75° — you approach the singular configuration where the chain straightens, force transmission collapses, and any small load can collapse the linkage. The sweet spot for general-purpose lifting sits between 30° and 60° fold angle, where vertical travel per degree is highest and mechanical advantage stays well-behaved.
Variables
| Symbol | Meaning | Unit (SI) | Unit (Imperial) |
|---|---|---|---|
| h | Vertical height of the top plate above the base plate | m | in |
| L | Length of each link in the hinge chain (assumes equal-length links) | m | in |
| θ | Fold angle of each link relative to the base plate | rad or ° | rad or ° |
Worked Example: Sarrus Linkage in a CubeSat deployable radiator panel
You are sizing a Sarrus linkage to deploy a thermal radiator panel on a 6U CubeSat. Stowed height is 25 mm, deployed target is 140 mm, and the link length L is 75 mm. You want to know the vertical position at three operating points across deployment so the motor controller can map angle to height correctly, and you need to confirm the geometry never drives the chain into singularity under thruster vibration.
Given
- L = 75 mm
- θlow = 15 °
- θnom = 45 °
- θhigh = 75 °
Solution
Step 1 — at the nominal mid-deployment angle of 45°, plug into the height formula:
That is the cruise position where the controller can hold deployment for thermal balancing without fighting either end of the geometry. Mechanical advantage is high and the chain is well away from singularity.
Step 2 — at the low end of typical operating range, 15° fold angle:
The radiator has barely cleared its stowed pocket. You get only about 14 mm of useful clearance above the 25 mm stowed datum, and the actuator is doing a lot of angular work for very little vertical motion — a 15° change at this end produces only ~25 mm of lift. Useful for a slow, controlled break from stow, but you do not linger here.
Step 3 — at the high end of typical operating range, 75° fold angle:
Almost fully deployed. But you are now within 15° of the singular straight-link configuration at 90°, where mechanical advantage collapses and the chain becomes unable to resist even small axial loads — a launch-vibration aftershock could fold it back. Set a hard stop at 75° to 80° in the controller. Do not chase the last few millimetres by going to 88°.
Result
At nominal 45° fold angle, the top plate sits at 106. 1 mm above the base — the cruise position you want for thermal control. The range tells the story: 15° gives just 38.8 mm (barely out of stow), 45° gives 106.1 mm (sweet spot), and 75° gives 144.9 mm (close to full deployment but uncomfortably near singularity at 90°). If your measured deployment height comes in 5 mm short of predicted, the most common causes are: (1) hinge-pin radial clearance stacking across the three joints in a chain, which lets the link plates sag and reduces effective L, (2) link plates flexing under their own weight in 1g ground testing — vacuum and zero-g will recover this but you will scare yourself during qualification, or (3) the two chain planes drifting from 90° perpendicularity, introducing a parasitic tilt that the controller reads as lost height.
When to Use a Sarrus Linkage and When Not To
When you need straight-line motion between two plates, the Sarrus linkage competes against scissor lifts and prismatic linear slides. Each wins on different axes. Pick based on stowed height, contamination tolerance, load capacity, and how much travel you actually need.
| Property | Sarrus Linkage | Scissor Lift | Prismatic Linear Slide |
|---|---|---|---|
| Stowed-to-deployed ratio | Up to 6:1 with thin link plates | Up to 10:1 with multi-stage | 1:1 — slides do not collapse |
| Joint type required | Revolute only — sealable, vacuum-safe | Revolute plus one prismatic at base | Prismatic — needs rails and bushings |
| Load capacity (typical desktop scale) | 1–20 kg before link bending dominates | 5–500 kg, scales well with stage stacking | 1–1000+ kg, limited by rail rating |
| Lateral rigidity at full extension | Poor near singularity (>75°), excellent below | Moderate, degrades with stage count | Excellent at any extension |
| Cost (small-batch build) | Low — six pin hinges, two plates | Moderate — multiple pivots and a slider | Moderate-high — precision rails dominate cost |
| Contamination tolerance | Excellent — no sliding surfaces | Poor — prismatic base joint exposed | Poor — rails accumulate debris |
| Best application fit | Deployables, folding optics, vacuum/clean environments | Vehicle lifts, work platforms, stage equipment | Machine tool axes, 3D printers, high-precision positioning |
Frequently Asked Questions About Sarrus Linkage
This is almost always link-length mismatch between chain A and chain B. If chain A's three links sum to 150.0 mm centre-to-centre and chain B sums to 150.2 mm, the top plate cannot stay parallel — one side hits its limit before the other. Measure pin-to-pin distance on every link with calipers and match within 0.1 mm. The error is most visible at full extension because the parasitic tilt grows with sin(θ).
The second suspect is hinge-axis perpendicularity. If the two chain planes are off from 90° by even 1°, the constraint geometry no longer perfectly cancels rotation, and you see a steady tilt that does not go away with link matching.
Practical limit is around 80°. At 80° the mechanical advantage is already roughly 6× worse than at 45°, and any axial load on the top plate translates into very high hinge-pin forces. Past 85° the linkage is effectively a column standing on pin joints — small lateral disturbances can buckle it back through the singularity in the wrong direction.
Set your controller's deployment limit at 75°–80° and accept that you will give up the last few millimetres of theoretical travel. If you genuinely need that travel, lengthen the links instead of pushing the angle.
For 200 mm and 5 kg, a single-stage Sarrus is fine if you have ~70 mm of stowed height available (link length around 100 mm folded). It will be lighter, simpler, and have no sliding parts. A scissor lift will give you a smaller stowed height — maybe 40 mm — but adds a prismatic joint at the base and roughly twice the part count.
Pick Sarrus if the environment is dirty, vacuum, or autoclave. Pick scissor if stowed height is the dominant constraint and you have a clean operating environment.
Almost always one of two things. First, hinge-pin friction is not equal across the six joints — a tight pin in one hinge of chain A will resist motion until the input torque overcomes static friction, then break free abruptly. Disassemble and check that every pin rotates under the same finger-applied torque.
Second, link plates are touching or rubbing each other near a fold-over angle around 60°–70°. Designers often forget that link plates pass close to each other mid-stroke. Add 1–2 mm of clearance between adjacent plates or stagger them axially.
One actuator is enough — that is the whole point of the single-DOF kinematics. Drive one hinge in either chain (usually the base hinge) with a torque source, and the rest of the linkage follows by constraint. A linear actuator pushing directly on the underside of the top plate also works and tends to give better mechanical advantage near full extension.
Driving both chains independently is over-constraint and will cause binding unless your two actuators are synchronized perfectly. Do not do it.
Near stow (small θ), the formula h = 2L·sin(θ) means the height-per-degree is very small, but the force-per-degree of input torque is correspondingly very large — mechanical advantage is at its peak. So static math predicts the motor can lift huge loads at this point. Reality bites because that same geometry multiplies any friction in the hinges by the same advantage factor. Six joints with even 0.05 N·m of stiction each can swallow most of your input torque before the load sees anything.
Rule of thumb: derate predicted output force by 30–40% near stow to account for cumulative hinge friction, and use a torque margin of at least 2× when sizing the deployment motor.
References & Further Reading
- Wikipedia contributors. Sarrus linkage. Wikipedia
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