Design of planar four-bar linkage with n specified positions for a flapping wing robot

• Generalize Burmester's theory to arbitrary n prescribed positions
• Propose the unified expression for multiposition design
• The method is successfully used to design a flapping wing robot.

This paper focuses on the design of a flapping wing robot and proposes a unified design formula for planar four-bar linkages with arbitrary n prescribed positions. The absolute coordinates of a circle point corresponding to every prescribed position are expressed by those of the first one through matrix transformation. Because the distance between any circle point and center point is always equal to the length of the side link jointed with the fixed base, a set of quadratic equations for the distance constraints are obtained. Expanding and rearranging these equations presents a linear system for the coordinates of center point. The condition for existing solution of these equations requires that the rank of the augmented rectangle coefficient matrix, C, should always be less than 3 as there are only two unknown coordinates for the center point. The rectangle augmented coefficient matrix C is (n − 1) × 3, and therefore it can be transformed to a 3 × 3 square matrix, M, by left multiplying the transpose of C. This provides a unified expression for the design of a planar four-bar linkage with arbitrary n positions for a flapping wing.