Fundamental Research On High Precision Interpolation Algorithm Of Curves And Sculptured Surface For Open Architecture CNC Systems

With the aid of advanced commercial CAD/CAM systems, it is possible to design and manufacture complicated parts with sculptured surface. However, machining these parts on existing machine tools is time-consuming, complicated and prone to errors. Without general real-time curve and curve-on-surface interpolators, most CNC machine tools are ones with proprietary controllers. Moreover, they are linear and circular interpolators. It results in contour errors, huge G codes and low machining efficiency.The research of this thesis is to develop novel curve interpolation methods for high precision machining of sculptured surfaces supported by Natural Science Fund of Jiangsu Province(BK2003005). Three major achievements have been made. (1) A method of generating contour-parallel guiding curves for pen cutting of sculptured surface is proposed. (2) A real-time interpolation algorithm for the curve-on-surface——cutter contact loci directly derived from these driving curves is implemented. (3) A real-time parametric subdivision interpolation algorithm for free form curves——cutter location loci with multi-axis CNC machining is carried out.A new mode of surface machining——pen cutting of sculptured surface is proposed. This machining mode can be classified into two types. One is single path method, the other is area path method. These methods overcome the limitation of traditional machining modes. Its advantage is to improve the efficiency of surface machining and enhance the quality of the surface.The methods of generating planar cubic B spline driving curves are discussed. Firstly, the algorithms of getting single layer inner offset driving curves are introduced. Then the distance between adjacent driving curvilinear loops is studied.Two methods of generating contour-parallel driving tool paths are based on analyzing the geometrical and topologic relations of closed areas formed by control polygons. One method is based on the principle of mapping. According the similarity between the outer closed polygon and inner offset one, every vertex of the inner offset polygon can be achieved by mapping. The self-intersection of inner offsetting loops is discussed. A method of generating non self-intersection loops is given. Driving curves can be achieved by inner offset polygon. The other is the creation of cobwebby driving tool paths. Inner offset polygons can be achieved through three steps. The first step is to calculate the center of the control polygon. Then the second is to connect this center to all vertexes of the polygon and to divide all the lines into equal pieces. Finally the third is to join these equal points and to get inner offset driving curves. The second method can avoid self-intersection.An algorithm of offsetting distance between driving curves is given based on error constraints. Cutter contact loci are a set of spatial curves formed by mapping driving curves on 2D plane to 3D surface. Actual contour error between spatial curve loops is uneven due to the change of curvature. The offset distance is decided according to the curvature of surface. It is found that the contour error can be controlled within allowable range by adjusting control points of planar driving curve.In order to reduce the interpolation error and resolve the conflict of high speed machining and high precision machining, a two-stage interpolation method is proposed. The first stage is the procedure of rough interpolation. A novel direct interpolation algorithm for pen cutting of sculptured surface is proposed. Even though driving curves are expressed by NURBS, actual tool paths have no explicit mathematical expression. Due to the difficulty of direct interpolation for this type of tool paths, the given algorithm is based on the driving curves. The precondition of the algorithm is the fixed velocity of cutter contact points. The parameter of driving curves can be calculated based on Taylor series expansion according the kinematics and geometric relation between cutter contact loci and driving curves. Cutter contact points on the surface can be achieved. This algorithm can ensure that all cutter contact points lie on the surface. The simulation indicates that the interpolation time is less than 2ms. Finally analyze the non-linear error of cutter contact points. An error control method is given.In order to improve further the interpolation precision and reduce machining data size, real time parametric subdivision interpolation algorithms are proposed by applying the interpolation data of the first stage. One is implemented based on cubic parametric polynomial curve for cutter location loci. The other is carried out based on cubic B spline curve for cutter location loci. And these algorithms are analyzed and compared. Then a real time cubic spline parametric subdivision interpolation algorithm is given for angular interpolation. These algorithms are composed of three parts: section search, segment fitting and subdivision. These algorithms have the advantage for interpolation precision when high speed machining. The test experiments demonstrate that the interpolation time is less than 0.03ms. The interpolation error is less than 0.25μm when machine feedrate is 6000mm/min and subdivision amount is 20. The investigation shows that the subdivision interpolation error is smaller with better approximation and larger subdivision amount.The general and systemic achievements on pen cutting of sculptured surface have been made in this dissertation. The idea of pen cutting extends existing modes of surface machining. These efficient and valid methods for high efficiency and high precision of sculptured surface machining is applicable for aeronautic and astronautic industry.