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Coupler curve

Also known as coupler-point trajectory · path of a coupler point · sextic curve of a four-bar

The coupler curve of a four-bar linkage is the trajectory traced by a single point on the coupler link as the crank rotates through one full revolution. For a generic planar four-bar the curve is algebraic of degree six (a sextic) and can include cusps, double-points, and self-intersections. It is the source of the four-bar's expressive power and the design target of path-generation synthesis.

What it is

Pick any point P on the coupler link of a four-bar — a vertex, a midpoint, an offset along the coupler axis, anywhere. As the crank rotates from 0° to 360°, P sweeps out a closed curve in the ground frame. That curve is the coupler curve of P.

Different points on the same coupler trace different curves. The designer chooses both the four link lengths and the position of P on the coupler — five degrees of freedom in total.

Why it is interesting

The coupler curve of a four-bar is, in general, a sextic — a sixth-degree algebraic curve. That is enough geometric expressivity to:

  • Trace approximate straight lines (Roberts straight-line linkage, Watt’s linkage).
  • Trace dwell sequences where the coupler point pauses for a stretch of the crank rotation (used in textile machines, packaging mechanisms, gait-emulating prosthetics).
  • Pass through specified target points with prescribed tangents.
  • Trace closed loops with cusps used in decorative applications.

Cubic-spline interpolation of arbitrary curves is impossible; a four-bar is the smallest closed kinematic chain that can approximate a non-trivial target path with a single rotating input.

Path-generation synthesis

The synthesis problem is: given a target path (a list of n precision points the coupler should pass through, optionally with tangents), choose the four link lengths and the coupler-point offset (u, v) such that the coupler curve of P matches the target as closely as possible.

This is not closed-form. The five design variables and the non-linear nature of the coupler-curve equation mean path-generation synthesis is an optimisation problem, typically solved with differential evolution over a chamfer-distance objective.

In our pipeline:

  • Burmester theory handles rigid-body guidance — three precision positions (point + orientation), closed-form.
  • Path generation handles path matching — many precision points along a target curve, DE optimisation.
  • The two are duals — Burmester is point-and-orientation; path generation is point-only-but-many. Different chapters in the same synthesis suite, sharing notation and the dynamics-validation back-end.

Coupler-curve atlas

Workspaces of coupler curves for representative link-length ratios are shipped as a visual library — Grashof grid × branch mode × coupler-point offset. Engineers browse the atlas to pick a candidate region of design space, then refine with DE if the target curve is exotic.

Defects

A coupler curve that passes through every target point is not necessarily smooth on every target tangent. Cusps and double-points are real geometric features of generic sextics — they appear or disappear as the link lengths cross discriminant surfaces in design space. The synthesis pipeline reports cusp positions as a side-channel on every candidate so the designer can avoid them when smooth motion is required.

See also

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