A flexure hinge positioning stage with giant magnetostrictive actuator（ GMA） as the driving unit is designed,which is composed of multi- group flexure hinges and guided by elastic deformation under force. The theories of the energy equation,equation of deflection curve,axioms of statics and energy conservation equation are applied to establish the mathematic theory model of flexure hinge positioning stage. The driving force and output displacement of the flexure hinge positioning stage are analyzed on the basis of working requirements driven by GMA. The geometric parameters of radius R and minimum thickness t of flexure hinge are optimized to achieve the optimized structure parameters of the flexure hinge positioning stage. Finally,the flexure hinge positioning stage is simulated by finite element analysis and the simulation values and theoretical analysis values are compared,the accuracy of theoretical analysis is verified.