Roberts–Chebyshev cognates are three four-bar linkages that trace the same coupler-point curve with different link arrangements. Their pivot locations and motion of individual links differ. The animation demonstrates the geometric construction, and the calculator scales the coupler triangle to your entered reference length.
Cognate Linkage Interactive Calculator
Set the coupler-triangle proportions AP:BP:AB and the reference length AB. The calculator reports the scaled dimensions and hole-tolerance ratio. Three constrained four-bars demonstrate the original and its two Roberts–Chebyshev cognates; impossible triangle proportions are flagged.
Equation Used
- Rigid links and ideal revolute joints.
- AP:BP:AB must form a non-degenerate triangle.
- Original four-bar proportions are fixed and explicitly stated.
- Hole-tolerance ratio does not predict path accuracy.
Exact ideal planar linkage construction for the illustrated proportions. Original ground/crank/coupler/rocker lengths are 1.4/0.4/1/1 times AB. Panels are separately fitted, and poses are synchronized by geometric correspondence. No claim of equal input speeds, force capacity or tolerance-induced path error.
Three constrained mechanisms, one point path
The original mechanism has fixed pivots O₁ and O₂, crank O₁A, coupler AB and rocker BO₂. P is a fixed point on the rigid triangular coupler APB. Two further mechanisms use O₁/O₃ and O₂/O₃ as their fixed pivots. Similar triangles and parallelograms determine their links.
Each animation panel solves the same corresponding configuration. Its links stay rigid and its fixed pivots remain fixed throughout rotation. The panels are separately fitted to show each mechanism clearly; matching path shape does not mean their displayed drawing scales are identical.
The illustrative original uses ground length 1.4×AB, crank 0.4×AB and rocker 1×AB. Those proportions permit a full crank rotation. Your three ratios determine AP, BP and AB, so changing them changes the tracing point and both cognates. Link crossings represent parts in separate planes.
Why use a cognate?
A different arrangement of fixed pivots can make a required point path easier to package. The same path does not guarantee equal transmission angles, forces, link orientations or input-speed relationships. Those properties require separate examination.
Use the model to compare alternative pivot arrangements for a point path. Choosing dimensions for a real machine also requires checks of transmission angles, clearances, joint forces and the required motion timing.
Triangle sizing and the construction
For entered ratios a:b:c and reference AB length L, the scale factor is k = L/c. Then AP = ak, BP = bk and AB = L. With A at (0,0) and B at (L,0), the coupler-point coordinates are x = (AP² + L² − BP²)/(2L) and y = √(AP² − x²).
Each side must be less than the sum of the other two. If that condition fails, the calculator shows the entered lengths but does not invent a closing triangle. The original four-bar still moves without a claimed P point.
For the animated construction, write q = (x + iy)/L as a complex scale and rotation, with O₁ = 0 and O₂ = d. At any original pose, P = A + q(B − A), and the third fixed pivot is O₃ = qd. The first cognate uses A₁ = q(B − A) and C₁ = qB. The second uses B₁ = d + (q − 1)(B − A) and C₂ = (1 − q)A + qd. These constructions maintain constant link lengths and the same P.
Worked 3:4:5 coupler triangle
With ratios AP:BP:AB = 3:4:5 and AB = 100 mm, the scaled sides are 60, 80 and 100 mm. P lies at (36,48) mm in the coupler frame, so q = 0.36 + 0.48i.
The demonstration ground span is 140 mm, its crank 40 mm and rocker 100 mm. The third fixed pivot O₃ is (50.4,67.2) mm relative to O₁. All three mechanisms trace the same P curve. An entered hole tolerance of 0.05 mm is 0.05% of AB; this is a simple dimension ratio, not predicted path deviation.
What the animation does and does not prove
Constant lengths, loop closure and coincident tracing points verify the ideal geometric construction. Corresponding poses are synchronized from the original crank. That does not imply that every cognate input link can be driven at the same constant speed to reproduce the same timing.
Bearing play, elastic deflection, link interference, inertia and load capacity are outside this calculation. A hole-tolerance input alone cannot determine end-point error; a sensitivity model and the individual joint deviations would be needed.
Cognate-linkage questions
Do all three panels trace exactly the same curve?
Yes, for a valid triangle in this ideal model. They use the same world-coordinate point P, reconstructed through different rigid link chains. Each panel is fitted separately for readability.
Why do some ratio combinations give a warning?
Three lengths cannot form a non-degenerate triangle unless each is smaller than the sum of the other two.
Does hole tolerance change the nominal animation?
No. It changes the reported tolerance-to-AB ratio. The nominal ideal geometry is unaffected; path error would require the individual joint deviations and a sensitivity calculation.
Are the three input links rotating at the same speed?
Not generally. Playback shows corresponding geometric configurations; input rotations are related differently in the cognates.
Can the panels use different fixed pivots?
Yes. The original uses O₁/O₂; the two cognates use O₁/O₃ and O₂/O₃. O₃ is calculated from the same coupler triangle.
Construction references
- Sherman, Hauenstein and Wampler, A General Method for Constructing Planar Cognate Mechanisms, section 5.1 and figure 2: four-bar Roberts cognates and complex-vector construction.
- Middle East Technical University, Four-Bar and Slider-Crank Cognates: geometric construction, fixed third pivot and corresponding rotations.
Building or designing a mechanism like this?
Explore the precision-engineered motion control hardware used by mechanical engineers, makers, and product designers.
