The Bricard exact straight-line linkage is an overconstrained six-bar mechanism that guides a coupler point along a mathematically perfect straight line using only revolute joints. Precision metrology and instrument-making trades rely on it where prismatic sliders would introduce stiction or wear. It works by combining specific link-length ratios so the kinematic constraints cancel sideways motion exactly, not approximately. The result is straight-line travel with no sliding contact — useful for coordinate measuring probes, dial test indicator carriers, and precision balances.
Bricard Exact Straight-line Interactive Calculator
Vary link tolerance, link length, stroke, and geometry sensitivity to see estimated straightness error and build risk for a Bricard linkage.
Equation Used
The estimate treats straightness error as proportional to relative link-length error, stroke length, and a geometry sensitivity factor. Lower tolerance, shorter stroke, or a smaller k value reduces the predicted coupler-point deviation from the ideal straight line.
- First-order tolerance estimate for a correctly proportioned Bricard linkage.
- Tolerance is the representative link-length manufacturing error.
- Sensitivity constant k depends on the exact Bricard geometry.
- Joint clearance, pivot bore error, and elastic deflection are not included.
How the Bricard Exact Straight-line Actually Works
Raoul Bricard published this mechanism in 1897 as part of his work on overconstrained linkages. The geometry is special — a generic six-bar with arbitrary link lengths will lock up or wobble, but Bricard's specific length and angle ratios make all six revolute joints rotate freely while still constraining one coupler point to an exact straight line. No sliders, no prismatic joints, no curved tracks. Just pin joints.
The trick lies in inversive geometry. The Peaucellier-Lipkin cell does the same job with eight links by inverting a circle through a point on its circumference into a straight line. Bricard's version achieves equivalent exactness with fewer links by exploiting a specific spatial-symmetry condition. If you build it, the coupler point traces a line whose straightness is limited only by your manufacturing tolerances — pivot bore concentricity, link-length accuracy, and bearing clearance. Get the link lengths wrong by 0.05 mm on a 100 mm link and you'll see a measurable bow in the traced path on a dial indicator.
What happens if tolerances drift? Two failure modes dominate. First, the linkage stops moving — overconstrained mechanisms only assemble and articulate when the geometry is satisfied within tight bounds, so a 0.1 mm error on the wrong link can bind the whole assembly. Second, if clearances are slack rather than tight, the coupler point traces a narrow ellipse instead of a line, and you lose the exactness that justified using this linkage in the first place. The whole point of choosing a Bricard over a Watt or Chebyshev approximate straight-line linkage is exactness — sloppy bushings throw that away.
Key Components
- Ground link (frame): The fixed reference link carrying two of the six revolute joints. Pivot bore concentricity must hold to within 0.01 mm on a benchtop instrument build, otherwise the overconstrained geometry refuses to articulate smoothly.
- Driving crank: The link rotated by the input — typically a hand wheel or a stepper motor in instrumented builds. Length tolerance of ±0.02 mm on a 50 mm crank keeps the coupler trace inside a 5 µm straightness band.
- Coupler links (three intermediate bars): These three bars carry the constrained coupler point. Their relative lengths must satisfy the Bricard ratio condition exactly — a 1% error here is the difference between exact straight-line motion and an arc that wanders 0.5 mm sideways across the stroke.
- Output link: The link that anchors the trace point in the geometric chain. Often extended past its pivot to mount a probe stylus or pen carrier.
- Revolute joints (six total): All six pivots use rolling-element or jewel bearings in precision builds. Radial play above 5 µm per joint compounds across the chain and rounds the corners of what should be a sharp linear trace.
Real-World Applications of the Bricard Exact Straight-line
Where do you actually use this? Anywhere a slider would compromise precision through stiction, wear, or contamination. Engineers reach for the Bricard exact straight-line linkage when they need pure pin-jointed motion in a clean-room, high-precision, or low-friction context. The mechanism is rare in heavy machinery — it's overkill — but in instrument design it earns its keep. You'll also see it as a teaching artefact in kinematics labs because it demonstrates the power of overconstrained synthesis on a tabletop.
- Precision metrology: Stylus carriers in form-measuring instruments such as the Taylor Hobson Talyrond series, where exact linear traverse without prismatic-slider stiction matters at the sub-micron level.
- Analytical balance design: Historical Mettler and Sartorius mechanical balances used Bricard-type linkages to guide the pan support in pure vertical motion without sliding contact.
- Drafting and graphic arts: High-end pantograph and ellipsograph instruments built by Stanley London used exact straight-line linkages as a sub-assembly to constrain one axis of motion.
- Kinematics education: Cornell's Reuleaux Kinematic Models collection includes overconstrained six-bar linkages used to teach the synthesis principles Bricard published in 1897.
- Aerospace test rigs: Wind-tunnel sting-balance calibration fixtures occasionally use exact straight-line linkages to apply known forces along a pure linear axis without slider friction corrupting the reference measurement.
- Watchmaking: Escapement test jigs at firms like Patek Philippe historically used pin-jointed exact straight-line guides to translate test loads onto balance staffs without contaminating sliding surfaces.
The Formula Behind the Bricard Exact Straight-line
The straightness error of a Bricard linkage is theoretically zero when link lengths are perfect. In the real world, you care about how manufacturing tolerance on the link lengths translates into deviation of the coupler point from the ideal line. At the low end of typical instrument-build tolerances (±0.005 mm on 50 mm links), straightness error stays under 1 µm across a 30 mm stroke — the sweet spot for sub-micron metrology. At the nominal hobby-machine-shop tolerance (±0.02 mm), error grows to roughly 5-10 µm. Push to the high end (±0.1 mm, typical of laser-cut sheet linkages) and you're looking at 50 µm or more — at which point you've thrown away the reason for choosing a Bricard over a cheap Watt linkage.
Variables
| Symbol | Meaning | Unit (SI) | Unit (Imperial) |
|---|---|---|---|
| δstraight | Peak deviation of the coupler point from the ideal straight line | mm or µm | in or thou |
| k | Geometric sensitivity constant for the specific Bricard configuration (typically 0.3 to 1.2 depending on link-ratio choice) | dimensionless | dimensionless |
| Δℓ | Manufacturing tolerance on the worst-case link length | mm | in |
| ℓ | Nominal length of the reference link | mm | in |
| S | Stroke length of the coupler point | mm | in |
Worked Example: Bricard Exact Straight-line in a sub-micron stylus carrier for a surface form tester
You are designing a stylus carrier for a custom surface form tester used to inspect ground gauge blocks at a calibration lab in Sheffield. The carrier must traverse a 30 mm stroke while holding a diamond stylus within 2 µm of an ideal straight line. You've chosen a Bricard exact six-bar with a nominal reference link length of 60 mm and a geometric sensitivity constant k = 0.6 derived from the Bricard ratio you selected.
Given
- ℓ = 60 mm
- S = 30 mm
- k = 0.6 dimensionless
- Δℓnominal = 0.02 mm
Solution
Step 1 — at the nominal machine-shop tolerance of ±0.02 mm on each link, compute the relative tolerance:
Step 2 — apply the sensitivity formula at nominal tolerance:
That misses the 2 µm spec. You need tighter manufacturing. Step 3 — at the low end of precision-instrument tolerance, ±0.005 mm (achievable on a Hardinge toolroom lathe with proper temperature-controlled measurement):
That clears the spec with margin. The diamond stylus tracks within the optical-flat reference and the form trace is usable. Step 4 — at the high end, laser-cut acrylic prototype tolerance of ±0.1 mm:
30 µm is a visible wobble — you'd see the coupler trace bowing off a steel rule held alongside the path. Fine for a teaching demo, useless for gauge-block inspection.
Result
Nominal straightness error at ±0. 02 mm link tolerance is 6 µm over a 30 mm stroke — three times worse than the 2 µm spec. The low-end build at ±0.005 mm achieves 1.5 µm and meets the spec; the high-end laser-cut prototype at ±0.1 mm produces 30 µm of visible bow that defeats the whole reason for picking a Bricard. If your built carrier measures, say, 12 µm of error instead of the predicted 6 µm, the most common causes are: (1) radial play above 5 µm in any single revolute pivot, which compounds geometrically across the six-bar chain, (2) thermal expansion mismatch between aluminium links and steel pivot pins shifting effective length by 5-10 µm over a 5°C lab swing, or (3) one link machined to the wrong nominal dimension entirely — a 0.1 mm mistake on the wrong bar can double the predicted error.
When to Use a Bricard Exact Straight-line and When Not To
The Bricard linkage isn't the only way to generate straight-line motion from rotary input. The two main alternatives are the Peaucellier-Lipkin cell (also exact, also pin-jointed, but eight links) and the Watt or Chebyshev approximate straight-line linkages (four links, simpler, but the trace is only straight to a tolerance, not exactly). Pick based on what your application actually demands.
| Property | Bricard exact straight-line | Peaucellier-Lipkin cell | Watt approximate straight-line |
|---|---|---|---|
| Straightness exactness | Exact (limited only by tolerance) | Exact (limited only by tolerance) | Approximate, ~0.1% of stroke deviation |
| Link count | 6 | 8 | 4 |
| Joint count | 6 revolute | 8 revolute | 4 revolute |
| Sensitivity to tolerance | High — overconstrained, requires ±0.01 mm class machining | Moderate — not overconstrained, more forgiving | Low — tolerant of ±0.1 mm sloppy builds |
| Typical stroke | 10-50 mm | 20-200 mm | 10-500 mm |
| Cost to build (precision instrument grade) | High — requires jig boring | Moderate — more parts but standard tolerances | Low — laser cutting acceptable |
| Best application fit | Sub-micron metrology, analytical balances | Larger-stroke exact motion, drafting tools | Engine linkages, walking toys, general mechanism |
Frequently Asked Questions About Bricard Exact Straight-line
This is the classic overconstrained-linkage failure. The Bricard only moves when its link lengths satisfy the geometric ratio condition exactly — within roughly 0.05% in a precision build. If even one link is 0.1 mm long on a 100 mm bar, the kinematic chain becomes statically indeterminate and binds at certain coupler positions, even though each individual pivot rotates fine on its own.
Diagnostic check: disconnect one pivot, articulate the remaining five-bar chain through the full stroke, then re-measure that pivot's hole position. If the as-built hole is more than 0.02 mm from where the geometry needs it, that's your bind point. Re-bore or shim, don't try to force it.
Pick the Bricard when you need fewer parts and a more compact envelope, and you can hold ±0.01 mm machining tolerances. Pick Peaucellier-Lipkin when stroke length exceeds about 50 mm, when you can tolerate the larger swept volume of an eight-bar mechanism, or when your shop can't reliably hit the tight tolerances the overconstrained Bricard demands.
Rule of thumb: in a clock-sized instrument, Bricard wins on compactness. In a half-metre drafting machine, Peaucellier-Lipkin wins on tolerance forgiveness because it isn't overconstrained.
That signature — a bow that reverses sign at mid-stroke — points to asymmetric pivot clearance, not link-length error. Link-length errors typically produce a monotonic deviation that grows toward one end of the stroke. A reversing bow means one or two pivots have slack that lets the coupler shift one way on the forward stroke and the opposite way on the return.
Check the input crank pivot first; it sees the highest reaction force and wears clearance fastest. A jewel bearing with 8 µm radial play will produce roughly the symptom you describe in a 50 mm-link build.
Using δ ≈ k × (Δℓ/ℓ) × S with a typical k of 0.6 and a 60 mm reference link, solving for Δℓ at δ = 5 µm and S = 40 mm gives Δℓ ≈ 0.0125 mm — call it ±0.01 mm. That's grinder territory, not mill-and-file territory. A surface grinder with a sine bar setup gets you there. A bandsaw and a file does not.
If you're prototyping, build it once with sloppy parts to verify articulation, then re-make the links to ground tolerance only when you've confirmed the geometry assembles.
You can, but watch the microstepping resolution and acceleration profile. The overconstrained chain has effectively zero compliance — any step transient that would cause a moment of mechanical lag in a four-bar shows up here as a jam or a tooth skip on the input gear. Use 1/16 microstepping or finer, and keep peak acceleration below roughly 500 mm/s² at the coupler for a typical 60 mm-link instrument build.
NEMA 17 steppers with closed-loop encoders (like the LDO or Moons' lines) work well because they can detect a missed step before the chain binds.
Steel links at 60 mm long expand roughly 0.7 µm per °C. If your linkage uses a single material throughout, expansion is uniform and the geometry stays valid — straightness barely shifts. The problem is mixed materials: aluminium links on steel pivot pins differ by about 1.4 µm per °C across a 60 mm span. A 5°C lab swing then introduces 7 µm of length mismatch, enough to push a sub-2 µm spec out of compliance.
For precision builds, match materials throughout, or specify ±0.5°C lab control if mixed-material construction is unavoidable.
References & Further Reading
- Wikipedia contributors. Straight-line mechanism. Wikipedia
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