Geometrical principium of fewer-axis grinding for large complex optical mirrors

Ultraprecision grinding is an important approach to efficiently fabricate large complex optical mirrors, and five-axis grinding method is commonly used for ultraprecision grinding. However, this method can hardly meet the high stiffness requirement for grinding large mirror, especially with a diameter over 2 m. Meanwhile, the use of fewer-axis grinding solves this problem, as it reduces the number of the grinder’s axes to improve the rigidity of the system and minimize deformation for hard and brittle materials. But its characteristic of unfixed grinding point which changes with workpiece surface curvature increases geometric complexity and requires a higher geometric shape accuracy of grinding wheel. This paper parameterizes grinding wheel’s geometric shape, reveals the relationship between fewer-axis and five-axis grinding methods from the point of view of the differential geometry, and establishes virtual-axis equivalence principium of fewer-axis grinding. A quantitative method to determine grinding wheel’s geometric parameters and its shaft inclination angle is proposed based on the requirements of geometric properties of optical mirror, grinder features and grinding process. Moreover, according to the properties of Gauss curvature of curved surface, the wear law of the toric grinding wheel is found and the surface geometric error distribution due to wear is achieved for fewer-axis grinding. The correctness of the principium and method above are verified through simulations.