The Roberts linkage is an approximate straight-line four-bar with a triangular coupler. In the selected standard construction, the two side links, coupler base and two coupler-point legs have equal length L, while the ground pivots are 2L apart.
Roberts Linkage Interactive Calculator
Explore exact tracer coordinates for one defined standard Roberts approximate straight-line linkage.
Equation Used
- Ground length is 2L and all five moving triangle/link lengths are L.
- The upper circle-intersection assembly and outward coupler apex are selected.
- Links are rigid and joints are ideal revolutes with no clearance.
- Forces, speed, tolerances, wear and application suitability are excluded.
The secondary figure uses the same equal-length geometry and input angle.
How the selected Roberts linkage works
Two equal side links connect fixed pivots A and D to the triangular coupler at B and C. The tracer P is the third vertex of an equal-sided coupler triangle. As the input side link rocks, the closed four-bar constrains P to a coupler curve with a portion near the fixed-pivot line.
The output is approximate rather than exact straight-line motion. The calculator solves the linkage by circle intersection; it does not use the removed empirical geometry constant.
What this calculator demonstrates
Select the common moving-link length L and input-link angle from 20° to 70°. The calculator returns the exact tracer coordinates relative to the midpoint of the fixed pivots, current absolute departure from their line and a sampled maximum departure over that defined angular interval.
It does not calculate a prescribed stroke, best-fit straight line, tolerance compliance, forces, speed, joint clearance, wear, manufacturing error or application suitability.
Exact position solution
Place A = (−L,0) and D = (L,0). For input angle θ, B = A + L(cosθ,sinθ). Point C is the selected upper circle intersection satisfying |C−B| = |C−D| = L. Point P is the outward equilateral-triangle vertex satisfying |P−B| = |P−C| = L.
The displayed sample evaluates this exact construction every 0.5° from 20° through 70° and reports max |y| relative to the ground-pivot line y = 0. It is a numerical sample over that interval, not a universal maximum or tolerance.
Default selected example
For L = 40 mm and θ = 45°, the exact selected assembly gives tracer coordinates from the circle-intersection solution. Scaling L scales every coordinate and the sampled departure proportionally; it does not change the dimensionless path shape.
Scope and correction boundary
Theory of Machines and Mechanisms defines the Roberts construction using three congruent isosceles triangles, with BC = AP = PD = AD/2. The Ohio University mechanism atlas and Cornell Reuleaux collection identify the same approximate straight-line four-bar topology.
The removed calculator's Δ = kL(L/Lside)² expression and k = 0.0025 claim were not retained because no authoritative derivation or defined Roberts geometry supported them.
Questions
Is the tracer path exactly straight?
No. The selected curve only approaches the ground-pivot line over part of its motion.
What does sampled maximum mean?
It is the largest |y| found at 0.5° increments from 20° to 70° for the defined assembly and reference line.
Does changing L improve the dimensionless path?
No. It scales both travel and departure by the same factor.
Does this model include joint clearance?
No. All links are perfectly rigid and joints are ideal revolutes.
Technical references
- Theory of Machines and Mechanisms — Roberts construction from three congruent isosceles triangles.
- Ohio University Mechanism Atlas — Roberts approximate straight-line mechanism topology.
- Cornell Reuleaux Collection, Roberts Four-Bar Model — historical physical model and approximate straight-line classification.
- Kempe, How to Draw a Straight Line — historical discussion and illustration of Roberts' linkage.
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