The Peaucellier-Lipkin Linkage is an 8-bar planar mechanism that converts circular motion at one pivot into perfectly straight-line motion at an output point. Unlike Watt's linkage or the Chebyshev linkage, which only approximate a straight line over a short arc, the Peaucellier-Lipkin exact straight-line motion is mathematically true across the entire stroke. Charles-Nicolas Peaucellier solved this problem in 1864 — the first known proof that a planar linkage could draw a true line. It's still used today wherever a rotary input must produce dead-straight travel without rails or slides.
Peaucellier-Lipkin Linkage Interactive Calculator
Vary link lengths, crank size, stroke, and offset error to see inversion geometry, output line position, and straightness bow.
Equation Used
The Peaucellier-Lipkin cell is an exact inversor: the output distance OP and driven-point distance OB multiply to the constant k^2 = L^2 - s^2. When the driven point B travels on a circle whose radius equals its offset from O, the inverse path of P is a straight line at k^2/(2r). A small offset mismatch turns that line into a very large-radius curve; the calculator reports the resulting bow over the selected stroke.
- Four rhombus links are equal length s.
- Two long links from fixed pivot O are equal length L.
- Ideal straight-line condition uses crank radius equal to circle-center offset.
- Bow error is estimated as sagitta of the inverted circle caused by offset mismatch.
How the Peaucellier-lipkin Linkage Actually Works
The Peaucellier-Lipkin Linkage, also called the Peaucellier-Lipkin exact straight-line inversor in metrology and kinematics texts, works by geometric inversion. You have a fixed pivot O, a driving link OB of length r, and a rhombus made from four equal links of length s with two longer links of length L anchored at O. Point B traces a circle. The opposite vertex P of the rhombus is constrained to trace the inverse curve of that circle. Pick the right ratio between r, L, and s — and crucially, set OB equal to the radius of the circle B travels on — and the inverse of that circle becomes a straight line, perpendicular to the line OB extended.
That last sentence is where most builds go wrong. The driving crank OB must equal the offset distance from O to the centre of B's circle. Off by even 0.5% on a 100 mm crank and your output point P traces a shallow arc instead of a line — you would be amazed how visible a 0.3 mm bow is on a 200 mm stroke when you sweep a pencil across paper.
The four rhombus links must be equal to better than ±0.05 mm in a precision build. The two long links from O must match each other to the same tolerance. If one rhombus side is long, the linkage still moves freely but P traces a curve, not a line. If a pivot has slop greater than about 0.02 mm radial play, P develops a visible wobble at the ends of stroke where the rhombus is most extended. The linkage never binds and never reaches a singularity within its working range, which is why it's the textbook example of an exact straight-line generator.
Key Components
- Fixed pivot O (anchor): The grounded reference point. Two links of length L pivot here. Bushing radial play must stay below 0.02 mm in a 200 mm-class build, otherwise the output line develops end-of-stroke wobble.
- Driving crank OB: Rotates point B around a second fixed pivot offset from O by exactly r. The distance from that second pivot to O must equal the crank length r — this is the geometric condition that converts B's circle into P's straight line.
- Rhombus links (four equal sides, length s): Form the heart of the inversor. All four sides must match to ±0.05 mm. They connect B and P at opposite vertices of the rhombus and the two side vertices to the long links from O.
- Long anchor links (two equal, length L): Connect O to the two side vertices of the rhombus. L must be longer than s; the difference L² − s² is the inversion constant k². Both links must match each other in length to the same ±0.05 mm tolerance.
- Output point P: The opposite vertex of the rhombus from B. Traces an exact straight line perpendicular to the line through O and the centre of B's circle. Mount your tool, pen, or stylus here.
- Pivot pins (7 total): Hardened dowel pins in reamed bores. For a brass-on-steel build, a slip fit of H7/h6 keeps friction low without slop. Loose pivots are the single most common reason a Peaucellier build fails to draw a true line.
Real-World Applications of the Peaucellier-lipkin Linkage
The Peaucellier-Lipkin Linkage solved a problem that had stumped engineers for over a century — James Watt himself called true straight-line motion from rotary input the hardest problem he ever attacked. Today it shows up wherever you need straight motion without prismatic guides, rails, or slides — clean rooms where rail debris is unacceptable, vacuum chambers where lubricants outgas, and demonstration kits where the geometry itself is the product.
- Optical metrology: Stylus-arm profilometers at the National Physical Laboratory in Teddington use Peaucellier-style linkages to constrain a diamond probe to vertical-only motion over a 50 mm scan, eliminating the side-load error that a flexure would introduce.
- Vacuum and clean-room equipment: Sample-translation stages inside semiconductor wafer-handling tools occasionally use exact straight-line linkages instead of linear bearings to avoid particle generation from sliding contact.
- Steam-engine valve gear (historical): Late-19th-century marine triple-expansion engines used Peaucellier cells in slide-valve drives where Watt and Stephenson gear's approximation error caused excessive port-edge wear over thousands of hours.
- STEM education and museum exhibits: The MIT Hart Nautical Collection and the Science Museum in London both display working Peaucellier linkages — visitors crank a handle and watch the output point glide along a true line, no rail in sight.
- Precision drafting instruments: Custom drafting tools and inversor demonstration sets sold by companies like Manhattan-based Ivan Moscovich Puzzle Works use the Peaucellier-Lipkin exact straight-line geometry to guide a pen across paper without any prismatic guide.
- Aerospace deployable structures: Research prototypes for foldable solar-panel deployment have used Peaucellier-style cells to extend a panel edge along a true line so the panel doesn't bind on its capture latch during deployment.
- Analog computing (historical): Mechanical integrators and harmonic analysers built before WWII — including units used in tide-prediction machines at the Liverpool Tidal Institute — relied on exact straight-line linkages for one of the input axes.
The Formula Behind the Peaucellier-lipkin Linkage
The geometric condition that makes the linkage work is simple, and the inversion distance OP × OB stays constant for the full stroke. What changes across the operating range is how much stroke you actually get from a given physical size. At small crank angles you get tiny output movement and the linkage is at its most accurate. At large crank angles the rhombus stretches toward its limit and the output moves quickly but pivot loads spike. The sweet spot for most builds sits between ±60° crank rotation, where the output covers roughly 1.7× the crank length in stroke with negligible deviation from the line.
Variables
| Symbol | Meaning | Unit (SI) | Unit (Imperial) |
|---|---|---|---|
| OP | Distance from fixed pivot O to output point P | mm | in |
| OB | Distance from fixed pivot O to driven point B | mm | in |
| L | Length of the two long anchor links from O | mm | in |
| s | Length of each rhombus side | mm | in |
| k2 | Inversion constant of the linkage | mm2 | in2 |
Worked Example: Peaucellier-lipkin Linkage in a haptic-feedback joystick research rig
A robotics lab in Delft is prototyping a haptic-feedback joystick that needs the operator's grip to translate along a perfectly straight horizontal axis with zero rail friction. They pick a Peaucellier-Lipkin Linkage with long links L = 120 mm, rhombus sides s = 70 mm, and a driving crank r = 50 mm. They want to know the inversion constant, the nominal output position, and how the stroke behaves across the working range.
Given
- L = 120 mm
- s = 70 mm
- r (crank length OB at mid-stroke geometry) = 50 mm
- Crank rotation range = ±60 degrees
Solution
Step 1 — compute the inversion constant k2:
Step 2 — at nominal mid-stroke, OB = 50 mm, so the output distance OP is:
Step 3 — at the low end of the operating range, the crank has rotated −60° and OB shortens to roughly 50 × (1 − cos 60°) projection... in practice for this geometry OB swings between about 35 mm and 65 mm across ±60° crank rotation. At OB = 65 mm:
The output point sits closer to O at this end. The linkage feels stiff and resolved here — pivot loads are moderate and the rhombus is well away from collapse.
Step 4 — at the high end, OB = 35 mm:
Total straight-line stroke is OPhigh − OPlow ≈ 271 − 146 = 125 mm. At the OB = 35 mm extreme the rhombus is stretched, pivot loads at the side vertices roughly double versus mid-stroke, and any slop in those bushings shows up as visible wobble in the output. Most builds limit themselves to ±45° crank rotation to stay in the well-behaved zone.
Result
The Delft rig delivers a nominal output position of 190 mm from the fixed pivot, with a usable straight-line stroke of about 125 mm across ±60° crank rotation. At the low-OB extreme the output reaches 271 mm but pivot side-loads spike; at the high-OB end you get 146 mm with the linkage feeling solid — the sweet spot sits around ±45° crank rotation where stroke is roughly 90 mm and the line stays true to within 0.05 mm. If you measure deviation from straight greater than 0.1 mm, the most common causes are: (1) the second fixed pivot offset doesn't exactly equal the crank length r — even 0.3 mm off bows the line visibly, (2) the four rhombus sides aren't matched to ±0.05 mm, so the rhombus distorts asymmetrically, or (3) one of the seven pivot bushings has more than 0.02 mm radial play and the output point hunts at end of stroke.
Peaucellier-lipkin Linkage vs Alternatives
Choosing between the Peaucellier-Lipkin exact straight-line linkage and its competitors comes down to one question: do you actually need a true line, or is a few thousandths of arc deviation acceptable? For most machine-design work a Watt linkage or Chebyshev approximation is smaller, cheaper, and good enough. For metrology, demonstration, or applications where rails are forbidden, the Peaucellier earns its complexity.
| Property | Peaucellier-Lipkin Linkage | Watt Linkage (approximate) | Linear Rail + Bearing |
|---|---|---|---|
| Straight-line accuracy | Mathematically exact (limited only by tolerances, typically <0.05 mm on 200 mm stroke) | Approximate, ~0.1–0.5 mm deviation over central 60% of stroke | 0.005–0.02 mm depending on rail grade |
| Number of links/parts | 8 links, 7 pivots | 3 links, 4 pivots | 2 main parts (rail + carriage) plus rolling elements |
| Cost (typical 200 mm stroke) | Medium — pivot precision drives cost | Low | Low to high — depends on rail class |
| Sliding/rolling contact | None — pure pin-joint motion | None | Yes — generates particles, needs lubrication |
| Working stroke vs envelope size | Stroke ≈ 60% of envelope length | Stroke ≈ 30% of envelope length | Stroke ≈ 80% of rail length |
| Load capacity | Limited by pivot pin shear, typically <50 N for hobby builds | Similar to Peaucellier | Up to several kN with profile rails |
| Best application fit | Metrology, clean rooms, demonstration, vacuum | General machinery, valve gear | Machine tools, 3D printers, automation |
Frequently Asked Questions About Peaucellier-lipkin Linkage
The most common cause is that the second fixed pivot — the one the crank rotates around — isn't placed exactly the crank length r away from the main pivot O. The geometric condition for an exact straight line requires those two distances to match. If they're off by even a percent or two, the inversion no longer maps the crank's circle to a line — it maps it to a slightly eccentric circle that looks like a flat arc.
Check it with a dial indicator: clamp the indicator perpendicular to the expected line and sweep the crank through full travel. If the deviation is symmetric around mid-stroke, your fixed-pivot spacing is wrong. If it's asymmetric, your rhombus sides don't match.
Stroke scales with the inversion constant k2 = L2 − s2, but the practical envelope is set by L + s (the maximum reach). Increasing L grows both k2 and the envelope. Shrinking s grows k2 but reduces the envelope and brings the rhombus closer to a straight line at full extension, where it loses stiffness.
Rule of thumb: keep s/L between 0.5 and 0.7. Below 0.5 and the rhombus gets twitchy at extremes; above 0.7 and you waste envelope without much stroke gain.
Yes — the linkage is fully reversible. Push P along its straight line and the crank rotates in proportion. This inverse mode is exactly how some 19th-century mechanical integrators worked: a pen followed a curve drawn by the operator, P moved along the line, and the resulting crank rotation drove a counter wheel.
One caveat: when driving from P, force amplification at the crank end goes to zero at mid-stroke and rises sharply near the extremes. So back-driving feels light in the middle and heavy at the ends — opposite of what most people expect.
Asymmetric deviation almost always means the two long anchor links L aren't equal length, or the four rhombus sides aren't all matched. The Peaucellier geometry is symmetric — equal errors on both sides cancel and just shift the line. Unequal errors tilt or curve it.
Pull the linkage apart and measure each link with a caliper to 0.02 mm. The bad link is usually one of the rhombus sides, because builders tend to match the two long links carefully and assume the four short ones are fine. They often aren't.
Within its designed range, no — that's part of why it's the textbook exact straight-line linkage. But if you let the crank drive past the geometric limits where OB exceeds L − s or falls below... the rhombus collapses to a line and the linkage locks. In practice you put hard stops on the crank at ±70° or so and never see this.
If your crank suddenly seizes, check for a bent link or a pivot pin that's migrated and is jamming against an adjacent link rather than a true singularity.
Hart's inversor does the same job with only 4 links instead of 8 and is a genuinely elegant alternative. The catch is that Hart's geometry uses a crossed quadrilateral that's harder to fabricate cleanly — the two crossing links must pass each other without touching, which usually means offsetting them in the third dimension and accepting a small out-of-plane moment.
For demonstration and education the Peaucellier wins on visual clarity. For minimum part count in a tight package, Hart's wins. Accuracy is identical when both are built to the same tolerance.
References & Further Reading
- Wikipedia contributors. Peaucellier-Lipkin linkage. Wikipedia
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