Geometrical Principle And Realization Of Toric Grinding For Large Optical Mirrors

In recent years,large high-resolution optical systems for earth or space observation are widely used in various fields such as military and scientific research,thus the need for large complex optical surfaces including off-axis aspheric ones are also increasing.In order to shorten the processing cycle,precision grinding will take place of lapping in the modern manufacturing process of optical mirrors and ensure that the ground mirror has the characteristics of high shape accuracy(PV: 5-8μm),low sub-surface damage(depth 8-10 m)and small surface roughness(Ra<20nm).Compared with the traditional five-axis grinding for aspheric optical mirrors,the fewer-axis toric grinding increases system stiffness and homogenizes grinding wheel’s wear by reducing the number of machine axes and changing the position of girnding point on the wheel surface,respectively.These two features of toric grinding make it easier to obtain high surface accuracy and low sub-surface damage for large optical mirrors.It thus becomes an ideal alternative to the traditional five-axis grinding.The fewer-axis grinding has been studied by many scholars both at home and abroad,but its widespread commercial use has not yet been received.One of the main obstacles is the complex geometric and kinematic relationships between toric wheel and workpiece surface.Therefore,the geometrical principle and realization method of toric grinding for large optical mirrors are deeply studied in this paper.According to the characteristics of variable contact points and curvatures in feweraxis toric grinding,this paper parameterizes grinding wheel’s geometrical shape,reveals the relationship between fewer-axis and five-axis grinding methods from the point of view of geometry,and establishes virtual-axis equivalence principium of feweraxis grinding.A quantitative method to determine grinding wheel’s geometrical parameters and its tilted-shaft angle is proposed based on the requirements of geometrical properties of optical mirror,grinder features and grinding process by methods like Gaussian mapping in differential geometry.Moreover,fewer-axis kinematic equations are established and trajectory planning problems with different machine configurations and different grinding modes are solved based on the virtualaxis equivalence principium.Error modeling and compensation is an important means to improve the machining accuracy.In this paper,a complete set of geometrical error modeling,separation,and compensation methods of tilted toric wheel is established for fewer-axis grinding of large complex optical mirrors.Based on the virtual-axis kinematic equations under error conditions,a mirror surface error model including all wheel error components is established,providing accuracy requirement basis for wheel manufacturing,installation and dressing.By linearizing the mirror surface error model and using the error information of mirror surface,solving and separating wheel error components are achieved.Then,the mirror surface with highly improved accuracy is obtained after the compensation of wheel trajectories with the error compensation function of control points.The contact area between grinding wheel and workpiece(grinding area)is one of the most important geometrical variables in toric grinding process.Compared with that of cylindrical wheel,it is three-dimensional,difficult to solve and lacks accurate and intuitive description variables(such as contact arc length for cylindrical wheel).According to the geometrical and kinematic characteristics of toric grinding,the differential representations of surfaces related with contact area including wheel surface,wheel motion enveloping surface and residual ground surface are made using differential geometry.Thus,the intuitive three-dimensional modeling of the contact area can be achieved by a few variables such as the principal curvatures of the grinding wheel surface at the grinding point,grinding depth,and so on.The contact area affects several variables such as toric grinding force and local wheel deformation during the grinding process.This paper tries to establish the models of these variables and obtains good results.The geometrical wear of the toric wheel will have a direct impact on the accuracy of the mirror surface.Two modelling methods of wheel’s geometrical wear are provided in this paper based on the characteristics of variable contact points.One is to construct wear function by the new representation of grinding ratio which uses Gaussian curvature as the core and has a simple algorithm.The other is to obtain the wear distribution of whole wheel by a method similar to the two-dimensional convolution.The latter has a more accurate modeling process and the point wear amount per unit distance is first obtained based on the 3D wheel-workpiece contact area proposed by this paper.Both of these methods try to avoid the direct operation of the surface equations by using the differential geometrical properties of the contact area to establish an intuitive,simple and uniform model.In the end,aiming at the realization of the geometrical principle in the toric grinding system,the CNC system and the CAM system for large fewer-axis optical grinder are developed,which has created favorable conditions for the large-scale application of fewer-axis toric grinding in the rapid manufacturing process of larege optics.