Research On Geometric Error Modeling And Compensation Of CNC Machine Tools Based On The Product-of-exponential Theory And Transforming Differential Changes Between Coordinate Frames

High precision has become an inevitable tendency of the development of CNC machine tools.Geometric errors are one of the main error sources of machine tools.Due to high repeatability,good systematicness and measurability,geometric error compensation becomes one important way for precision enhancement of machine tools.Therefore,it has important theoretic and practical meanings to research on how to conveniently measure the geometric errors,how to accurately model the geometric errors of machine tools and how to efficiently compensate the geometric errors.The related researches have become key topics of precision enhancement of machine tools.On the basis of existing relevant theories and achievements,this dissertation is intended to do some research studies on the key technologies of geometric errormodeling and error compensation.The dissertation uses all steps of geometric error compensation process as the main line.It starts with the studies of measurement and identification of geometric errors of rotary axes,and the six-circle method with ballbar measurement is proposed.Then,the product-of-exponential(POE)theory and transforming differential changes between coordinate frames are applied to geometric error modeling of machine tools,respectively.On this basis,different approaches of geometric error compensation are developed,including the application of the mathematical optimization and workpiece model reconstruction.The main contribution of the dissertation is summarized as follows:One geometric error identification approach of rotary axes with ballbar measurement,six-circle method,is proposed.It can obtain all ten geometric errors of each rotary axis,and is available for different rotary axes.It's concluded that the principle of six-circle method has nothing to do with error model of machine tools through analyzing the geometric error model of machine tools.The error identification matrix is established.Then,the identification formulas of all geometric errors are established based on the properties of geometric errors.Next,the impact of the set-up errors of ballbar is analyzed.The set-up errors are obtained using least square method to improve the accuracy of six-circle method.Six measurement positions of six-circle method are suitable for two rotary axes of five-axis mach,ine tools,so it's fast and convenient for five-axis machine tools.Finally,the error measurement experiments of rotary axes are carried on the five-axis machine tool.And the comparison between the ballbar readings with and without compensation verifies the feasibility of six-circle method.Geometric error modeling and compensation based on product-of-exponential theory is proposed,which is suitable for all types of machine tools.Clear physical meanings of twists in POE theory are used to model motions and geometric errors of each axis.The error twist and POE models of squareness errors are established according to the influences of squareness errors on motions of axes.Then,the integrated POE geometric error model of machine tools is established by multiplying all POE matrices in certain order.This model has the clear geometric meaning.Only one global reference coordinate system is established in the whole modeling,which can greatly simplify the modeling.What's more,the twist Jacobian of the machine tool is established for error compensation based on the property of POE theory.It avoids the singular problem,and it's convenient for calculation.Finally,the experiments on the three-axis machine tool and five-axis machine tool are conducted respectively to verify the effectiveness of the proposed error modeling and compensation.Geometric error modeling and compensation based on transforming differential changes between coordinate frames is established for precision enhancement of machine tools.Transforming differential changes between coordinate frames can represent different forms of one differential motion in different coordinate frames.Differential motion matrix of each axis relative to tool is established to calculate the error vector of geometric errors of each axis in tool coordinate frame.The sum of these error vectors is the integrated error vector of tool in its local coordinate frame.It can reflect the influences of geometric errors of each axis on the precision of the machine tool,so it has the clear physical meaning.Then,Jacobian is constructed with the differential motion matrix of each axis without extra calculation to compensate the integrated errors of tool.Finally,cutting tests are carried on the five-axis machine tool to testify the effectiveness of the proposed modeling and compensation.One NC code optimization approach based on particle swarm optimization(PSO)is proposed for geometric error compensation of machine tools.Firstly,the kinematics of the five-axis machine tool is analyzed with structural parameters of the machine tool.The postprocessing of this machine tool is established to develop the bidirectional transformation between NC codes and tool poses.Secondly,the polynomial modeling of basic geometric errors of each axis is proposed,which uses F test of regression analysis to determine the optimal polynomials.Thirdly,the nominal tool poses are introduced into the geometric error model based on forward kinematics of the machine tool.The mathematical expressions of integrated geometric error models are established with polynomials of basic geometric errors.The expressions can evaluate the compensation effect of one NC code.Moreover,PSO is used to seek the optimal NC codes.The purpose is to achieve the minimum global geometric errors.The tool poses are chose as particles,and the limited space of particles are defined.Finally,the results of workpiece cutting experiments show that the error compensation of optimizing NC codes can effectively improve the precision of machine tools.Geometric error compensation of three-axis machine tools based on numerical solution of simultaneous equations and workpiece model reconstruction is proposed.It's available for different numerical control systems of three-axis machine tools.Polynomial set of integrated geometric error models are established as functions about workpiece points based on the simple kinematics of three-axis machine tools,which introduces the nominal tool position as the error compensation goal.Numerical solution of polynomial set calculates the compensated points of workpiece model.It has high computing accuracy and efficiency.Then,one conversion approach from CAD model to STL model is proposed.It can obtain the reconstructed STL model of workpiece with compensated points.The machining with the reconstructed model realizes the geometric error compensation of whole machining process.The proposed compensation does not need special knowledge of compensation for workers.Finally,cutting experiments of the three-axis machine tool are carried out to verify the feasibility of the compensation.