Research On The Mathematical Models And Algorithms For Form And Position Errors Based On Cartesian Coordinate Measurements

CMM is great has significant meaning to the assessment of form and position errors. But it can not be reasonably applied in it for lacking of the mature assessment models. Therefore, the models and algorithms of form and position errors have been studied systematically in the Cartesian coordinates. What’s more, the visualization of the errors and assessment system are also studied.(1) The mathematical models of the straightness (spatial straightness) errors have been established by least squares and minimum zone methods. The mathematical models of the flatness error have been established by least squares and minimum zone methods. The mathematical model of the roundness (cylindrical) errors has been established by least squares circle (cylinder), minimum zone cylinder, minimum enclosing circle(cylinder) and maximum inscribed circle (cylinder). The mathematical models of the concentricity errors have been established by least squares and minimum zone methods. What’s more, parts of the models have been linearized, and their linearization errors have been analyzed.(2) The basic nature of the objective function about the mathematical models was studied systematically. It is proved by experiment that the minimum values of the objective function for the minimum zone method about the straightness errors; flatness errors and roundness errors are unique. The objective functions of the minimum zone method for cylindrical error, spatial straightness error and concentricity error are not convex functions. It is proved by experiment and theory that the linear objective functions of the minimum zone method are convex functions.(3) The algorithms for the corresponding model were established by Matlab referring to the nature of the objective functions. The model and the algorithm are proved correct and feasible on the accuracy and speed by comparing to others.(4) Three basic questions about visualization were proposed and solved. And the visualization for form and position errors was realized. Taking the visualization of cylindrical errors for example, the steps of visualization were explained.(5) The assessment system of form and position errors is developed by Matlab based on the theories above. And the assessment and visualization of errors were realized together.