Glossary

Burmester theory

Also called: Burmester precision points · Burmester closed-form synthesis · rigid-body guidance synthesis

Burmester theory is the closed-form geometric method for synthesising a four-bar linkage that carries a rigid coupler body through a prescribed set of precision positions. For three positions there is a one-parameter family of solutions; for four or five positions the solutions reduce to a finite set determined by the Burmester curves of pole points. We use it to hit precision targets in microseconds — without optimisation, without seeding, without stochastic search.

What it solves

Burmester theory addresses the rigid-body guidance problem: given a sequence of precision positions (each one a position + orientation of a rigid body in the plane), find a four-bar linkage whose coupler body passes exactly through every one of them.

The geometry is closed-form. No iteration. No optimiser. No initial guess. You write down two pole points per pair of positions, intersect their perpendicular bisectors, and the moving and fixed pivots fall out algebraically. The runtime is dominated by float arithmetic — microseconds.

Why precision points matter

Three or four “the linkage must pass through exactly this position with this orientation” constraints come up everywhere:

A pick-and-place arm must clear an obstacle, deposit a part flat, and retract along a specific path.
A door-opening linkage must close flush, swing through 90°, and stop at a specific intermediate hinge angle.
A gear-shifter linkage must align three discrete positions at three detents.
An L-shaped coupler in a university practica must hit three precision poses for the homework.

For these problems an optimiser is overkill — the constraints are algebraic, the solution set is closed, and the only hard parts are the defect filters that come after the synthesis.

Solution count by position count

Precision positions	Solution structure
2	Two-parameter family — degrees of freedom in the moving and fixed pivots.
3	One-parameter family — pivot points lie on the Burmester centerpoint curve and circlepoint curve. Designer picks one.
4	Finite — solutions are intersections of two Burmester curves. Generally a small set (often 4–6).
5	Generically zero — over-constrained. Solutions exist only for special configurations.

Three positions is the sweet spot — finite design freedom, but enough constraint to make the mechanism distinctive.

Why we use it instead of DE

Differential evolution (DE), genetic algorithms, and similar stochastic search will also hit precision points — eventually. But:

They take seconds to minutes per run.
They require seeds and population tuning.
They produce slightly-off precision matches (e.g. 0.03 cm trace error) unless the population converges tightly.
They don’t expose the underlying geometric structure — the designer can’t reason about why one solution is better.

Closed-form Burmester gives:

0.0000 cm trace error at every precision point — by construction.
Microseconds of runtime — irrelevant to total pipeline cost.
A free family parameter in the 3-position case — designers can sweep it and visually pick the candidate that also satisfies Grashof, transmission angle, and the dynamics constraints.

We carry the DE-based velocity / function / path generation chapters too, because not every design problem has clean precision points. But when the problem does have them, Burmester is the right tool.

Interaction with defect filters

A Burmester solution can be Grashof or non-Grashof. It can have a 0° transmission angle at one of the precision poses (the linkage locks). It can flip the assembly branch between two precision points (the order defect). All four classical filters still apply — Burmester guarantees precision, not practicality.

In the practica case study, the Burmester synthesis generated 72,000 candidates in the family-sweep. The branch filter alone rejected 62,000. Three survived all four filters. That funnel is the value.