Burmester's theory

His approach was to compute the geometric constraints of the linkage directly from the inventor's desired movement for a floating link.

The design of a crank for the linkage now becomes finding a point in the moving floating link that when viewed in each of these specified positions has a trajectory that lies on a circle.

This formulation of the mathematical synthesis of a four-bar linkage and the solution to the resulting equations is known as Burmester Theory.

Finding these circling points requires solving five quadratic equations in five unknowns, which he did using techniques in descriptive geometry.

Thus for three task positions, a four-bar linkage is obtained for every pair of points A and B chosen as moving pivots.

Burmester's approach to the synthesis of a four-bar linkage can be formulated mathematically by introducing coordinate transformations [Ti] = [Ai, di], i = 1, ..., 5, where [A] is a 2×2 rotation matrix and d is a 2×1 translation vector, that define task positions of a moving frame M specified by the designer.

[6] The goal of the synthesis procedure is to compute the coordinates w = (wx, wy) of a moving pivot attached to the moving frame M and the coordinates of a fixed pivot G = (u, v) in the fixed frame F that have the property that w travels on a circle of radius R about G. The trajectory of w is defined by the five task positions, such that Thus, the coordinates w and G must satisfy the five equations, Eliminate the unknown radius R by subtracting the first equation from the rest to obtain the four quadratic equations in four unknowns, These synthesis equations can be solved numerically to obtain the coordinates w = (wx, wy) and G = (u, v) that locate the fixed and moving pivots of a crank that can be used as part of a four-bar linkage.

Burmester proved that there are at most four of these cranks, which can be combined to yield at most six four-bar linkages that guide the coupler through the five specified task positions.

It is useful to notice that the synthesis equations can be manipulated into the form, which is the algebraic equivalent of the condition that the fixed pivot G lies on the perpendicular bisectors of each of the four segments Wi − W1, i = 2, ..., 5.