A unified synthesis method based on solution regions for four finitely separated and mixed “Point-Order” positions

For rigid-body guidance mechanism synthesis reaching four specified task positions, Burmester curve equations are presented by a displacement matrix method. These equations are suitable for both four-finite and four-mixed position cases. With the transformation of coordinates, and based on the projective geometry theory that a series of intersections of two pencils-of-low-order curves constitute a high-order curve, Burmester curves can be expressed by the intersections of a pencil-of-circles and a pencil-of-lines. Then the expressions of Burmester points are parameterized with the angle φ by employing a solution region method. Therefore, Burmester curves can be single-valuedly and orderly expressed, and an infinite number of mechanism solutions can be expressed on a finite coordinate plane.