A flapping wing driving apparatus includes at least one crank gear capstan rotatably coupled to a crank gear, the at least one crank gear capstan disposed radially offset from a center of rotation of the crank gear; a first wing capstan coupled to a first wing, the first wing capstan having a first variable-radius drive pulley portion; and a first drive linking member configured to drive the first wing capstan, the first drive linking member windably coupled between the first variable-radius drive pulley portion and one of the at least one crank gear capstan; wherein the first wing capstan is configured to non-constantly, angularly rotate responsive to a constant angular rotation of the crank gear.