The equation
For a planar four-bar linkage with link lengths a (crank), b
(coupler), c (rocker), d (ground), and crank/rocker angles θ2 and
θ4 measured from the ground link, the Freudenstein form is:
K1 cos(θ4) − K2 cos(θ2) + K3 = cos(θ2 − θ4)where the Freudenstein constants are:
K1 = d / a
K2 = d / c
K3 = (a² − b² + c² + d²) / (2ac)Three equations of this form — one per precision pair (θ2_i, θ4_i) —
solve for the three unknowns K1, K2, K3 linearly. From the
constants, the link-length ratios fall out by inversion. One ground length
is free (typically normalised to 1), so the four ratios fix the geometry
up to scale.
Why it matters
Freudenstein turned a coupled non-linear position problem into a linear
system in the constants K1, K2, K3. Function-generation synthesis —
“design a four-bar whose output angle matches a target function of the
input angle at a finite set of precision points” — collapses to a 3×3
linear solve. No iteration, no seed, no optimiser.
In our pipeline this matters because:
- It is closed-form, like Burmester is for rigid-body guidance. Microseconds of compute, deterministic, no DE population to tune.
- It exposes the Freudenstein constants as design knobs — engineers can reason about a synthesis problem in terms of three scalar invariants rather than four link lengths plus a redundancy.
- The same algebraic structure underlies our function-generation chapter and the velocity-targeted synthesis chapter — the chapters share notation and variable names with the optimiser modules. Theory and code do not drift.
Three positions, four positions, five positions
Like Burmester theory, the solution-count behaviour of Freudenstein is determined by precision-pair count:
| Precision pairs | Solution structure |
|---|---|
| 3 | One linear solve. Unique solution (up to scale). |
| 4 | Over-constrained. Generally non-zero residual; least-squares fit. |
| 5 | Strongly over-constrained. Optimiser territory. |
Three is the sweet spot for closed-form work. Four or five is where DE and Burmester (the geometric dual) take over.
Limits
Freudenstein gives the geometry that hits the angles. It does not guarantee the resulting four-bar passes:
- Grashof’s law — link-length sums may violate rotatability.
- Branch / order checks — the precision angles may require flipping assembly mode.
- Transmission angle floor — the geometric solution may have μ approaching 0° somewhere on the rotation.
So Freudenstein is necessary, not sufficient. The other classical filters apply downstream, exactly as they do for Burmester.
See also
- Four-bar linkage
- Burmester theory — geometric dual for rigid-body guidance.
- Transmission angle
- Practica case study — Freudenstein is the function-generation kernel that complements the Burmester worked example.