中轴变换的几何学与最优化原理及其在数控加工中的应用
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摘要
本论文研究了2D与3D中轴变换的微分几何学、最优化原理、统一求解方法及其在数控加工中的应用。
论文从研究2D中轴变换入手,建立了中轴线与边界曲线之间法向映射的基本方程及两组微分关系式,在此基础上给出了与2D中轴变换相关的三个微分不变量之间的内在联系。考虑到分叉点所处的重要地位,论文讨论了中轴线的奇点条件与分类及其与分叉点形成机制的内在联系。依据特殊点可将复杂的中轴线划分为9类基元段,并给出了基于基本方程和微分关系式发展出的用于基元段求解的跟踪算法。
注意到2D中轴变换的特殊性质,将旋轮线理论引入到其研究中,发展出相应的运动几何学。证明了存在有一对镜像映射的动瞬心线,它们在中轴线两侧做纯滚动时,所发生出的旋轮线即为边界线,并给出了曲率应满足的结构条件。建立了动瞬心线、中轴线与半径函数及边界线之间的关系方程,讨论了动瞬心线的确定及边界线的反求。据此可揭示出2D中轴变换的内在几何性质,指出在动瞬心线一定的条件下,改变中轴线的形状,便可得一族定长、等积、保角的基本型,这是一类更为深刻的几何学性质,渴望在工程设计中得到应用。
在上述基础上,应用活动标架、发甫形式等方法研究更具难度的3D中轴变换的微分几何学。建立了中面与边界面之间映射关系的基本方程及标架变换关系；基于标架的运动,定义了梯度线、等值线、骨架线、切迹线等重要曲线,为研究提供了载体。应用标架的无穷小平移,揭示了中面与边界面之间的距离微分关系；应用标架的无穷小回转,给出了回转矢量之间的关系,进而得到七个微分不变量的表达式。在此基础上,讨论了3D中轴变换的规范型,及其在蒙日曲面上的应用。
以最优化原理为先导,以微分几何学为基础,构成了本论文研究的主体脉络。从最优化原理出发,研究了2D、3D中轴变换所共有的本质属性,在此基础上给出了鞍点维数意义下的中轴点定义及其分类。建立了2D、3D中轴变换的鞍点规划模型,并证明了其最优条件包括等值性条件与凸体包容条件。依据这一条件,构思出分步求解、逐次逼近,线性化操作、精确验收的求解思路,解决了一维、及二维最小查询中的振荡、死循环问题,攻克了在求解过程中对分叉点、骨架点、结点等特殊点的识别与确定这一难点,发展出同时适用于2D、3D各维数鞍点规划模型的统一解法。
本文最后以数控加工为落脚点,讨论了中轴变换在2.5D型腔零件及以闭式叶轮为代表的通道类零件数控加工中的应用。为适应加工的需要,依托于鞍点规划平台,文中对中轴变换进行了推广与发展。提出并研究了平面域的等距盒、通道面的通道等距盒、最大内接柱、及最佳刀轴面等概念,建立了它们的求解模型,并给出了其解法。据此发展出2.5D型腔分步加工的策略,及闭式叶轮加工的“两阶段三步骤”纵切法加工方案。
本论文课题为国家自然科学基金资助项目“中轴变换的几何学原理及其在数控加工中的应用(No.50775022)”。
In this dissertation, the differential geometry, optimal theory and algorithm of2D and3D Medial Axis Transform (MAT) are studied, and on these bases, its applications to the NC machining are explored.
Beginning from the2D MAT, the normal equidistant mapping equations and two differential relations between the boundary and the Medial Axis (MA) are first proposed. The relations of three differential invariants in the2D MAT are investigated. Considering the importance of the branch point, its generating mechanism is discussed based on the singularity condition and classifications of the MA. The MA of a closed domain, which is a complex tree-like structure, can be divided into nine types of fundamental segments according to different compositions of the irregular points. Furthermore, a tracing algorithm for the fundamental segments is derived based on the basis equation and differential relations.
Depending on the cycloid theory, the kinematics geometry of the2D MAT is generated. It is proved that there is a pair of moving centrodes under a pure rolling contact with the MA, and the two cycloids generated will be the boundary curves. Meanwhile, the conditions that the curvatures of the moving centrodes and the MA should satisfy are given. After constructing the relations between the moving centrodes, MA and radius function, boundary curves, the method how to determine the moving centrodes and the reverse method for the boundary are presented. Accordingly, the intrinsic geometric properties of2D MAT are revealed. If the moving centrodes are determined, with the change of the shape of the MA, a family of fundamental domains can be obtained with the same perimeter, area and angularity of tangent lines. This is a kind of deeper geometric properties, which are expected to be used in the field of engineering design.
Based on the above discussion, the properties of moving frame and differential form are employed to investigate the differential geometry of3D MAT, which is more difficult to be studied. Similarly, the normal equidistant mapping equations and moving frames attached to both the MA (medial axis) point and the associated boundary points are constructed firstly. In view of the movement of local frames, the gradient curve, level curve and skeletal curve etc., are defined to serve as carriers for the study. After exploring the infinitesimal translation and rotation of the moving frames, the relations of distance differentials, rotation vectors and seven differential invariants are derived. Finally, a special case, the normal form, is defined, and it shows that the3D MAT problem for the moulding surfaces can be converted into a2D one.
If the differential geometry serves as the basis, the optimal theory will serve as the precursor for this study. From the optimal theory of MAT, after exploring its saddle point properties, a redefinition and classification for the medial axis points are provided, from the perspective of the dimension of saddle point. Then, the mathematical programming method is employed to construct the saddle point programming models. Based on the optimality conditions, two solving strategies, i.e. the divide and conquer strategy and the linearization strategy, are derived. And a modified Newton method is adopted for the calculation of the minimum distance avoiding oscillation. In addition, the irregular points are identified according to the sudden changes of the solutions to the normal points. A generic algorithm for computing various medial axis points is proposed, and it is proved to be efficient and accurate by numerical examples.
Finally, as a further goal, this dissertation discusses the applications in the NC machining of2.5D pockets and closed impeller. In order to satisfy the demands for machining, it needs to extend the concept of MAT. More specifically, the offset box, maximal inscribed cylinder and the optimal cutter axial surface are presented and determined relying on the saddle point programming. Accordingly, the step machining method for the2.5D pockets, and the rip cutting idea for the closed impeller are developed.
This dissertation is supported by National Natural Science Foundation of China,"The Geometric Theories of the Medial Axis Transform and its Applications in NC Machining (No.50775022)'
论文从研究2D中轴变换入手,建立了中轴线与边界曲线之间法向映射的基本方程及两组微分关系式,在此基础上给出了与2D中轴变换相关的三个微分不变量之间的内在联系。考虑到分叉点所处的重要地位,论文讨论了中轴线的奇点条件与分类及其与分叉点形成机制的内在联系。依据特殊点可将复杂的中轴线划分为9类基元段,并给出了基于基本方程和微分关系式发展出的用于基元段求解的跟踪算法。
注意到2D中轴变换的特殊性质,将旋轮线理论引入到其研究中,发展出相应的运动几何学。证明了存在有一对镜像映射的动瞬心线,它们在中轴线两侧做纯滚动时,所发生出的旋轮线即为边界线,并给出了曲率应满足的结构条件。建立了动瞬心线、中轴线与半径函数及边界线之间的关系方程,讨论了动瞬心线的确定及边界线的反求。据此可揭示出2D中轴变换的内在几何性质,指出在动瞬心线一定的条件下,改变中轴线的形状,便可得一族定长、等积、保角的基本型,这是一类更为深刻的几何学性质,渴望在工程设计中得到应用。
在上述基础上,应用活动标架、发甫形式等方法研究更具难度的3D中轴变换的微分几何学。建立了中面与边界面之间映射关系的基本方程及标架变换关系；基于标架的运动,定义了梯度线、等值线、骨架线、切迹线等重要曲线,为研究提供了载体。应用标架的无穷小平移,揭示了中面与边界面之间的距离微分关系；应用标架的无穷小回转,给出了回转矢量之间的关系,进而得到七个微分不变量的表达式。在此基础上,讨论了3D中轴变换的规范型,及其在蒙日曲面上的应用。
以最优化原理为先导,以微分几何学为基础,构成了本论文研究的主体脉络。从最优化原理出发,研究了2D、3D中轴变换所共有的本质属性,在此基础上给出了鞍点维数意义下的中轴点定义及其分类。建立了2D、3D中轴变换的鞍点规划模型,并证明了其最优条件包括等值性条件与凸体包容条件。依据这一条件,构思出分步求解、逐次逼近,线性化操作、精确验收的求解思路,解决了一维、及二维最小查询中的振荡、死循环问题,攻克了在求解过程中对分叉点、骨架点、结点等特殊点的识别与确定这一难点,发展出同时适用于2D、3D各维数鞍点规划模型的统一解法。
本文最后以数控加工为落脚点,讨论了中轴变换在2.5D型腔零件及以闭式叶轮为代表的通道类零件数控加工中的应用。为适应加工的需要,依托于鞍点规划平台,文中对中轴变换进行了推广与发展。提出并研究了平面域的等距盒、通道面的通道等距盒、最大内接柱、及最佳刀轴面等概念,建立了它们的求解模型,并给出了其解法。据此发展出2.5D型腔分步加工的策略,及闭式叶轮加工的“两阶段三步骤”纵切法加工方案。
本论文课题为国家自然科学基金资助项目“中轴变换的几何学原理及其在数控加工中的应用(No.50775022)”。
In this dissertation, the differential geometry, optimal theory and algorithm of2D and3D Medial Axis Transform (MAT) are studied, and on these bases, its applications to the NC machining are explored.
Beginning from the2D MAT, the normal equidistant mapping equations and two differential relations between the boundary and the Medial Axis (MA) are first proposed. The relations of three differential invariants in the2D MAT are investigated. Considering the importance of the branch point, its generating mechanism is discussed based on the singularity condition and classifications of the MA. The MA of a closed domain, which is a complex tree-like structure, can be divided into nine types of fundamental segments according to different compositions of the irregular points. Furthermore, a tracing algorithm for the fundamental segments is derived based on the basis equation and differential relations.
Depending on the cycloid theory, the kinematics geometry of the2D MAT is generated. It is proved that there is a pair of moving centrodes under a pure rolling contact with the MA, and the two cycloids generated will be the boundary curves. Meanwhile, the conditions that the curvatures of the moving centrodes and the MA should satisfy are given. After constructing the relations between the moving centrodes, MA and radius function, boundary curves, the method how to determine the moving centrodes and the reverse method for the boundary are presented. Accordingly, the intrinsic geometric properties of2D MAT are revealed. If the moving centrodes are determined, with the change of the shape of the MA, a family of fundamental domains can be obtained with the same perimeter, area and angularity of tangent lines. This is a kind of deeper geometric properties, which are expected to be used in the field of engineering design.
Based on the above discussion, the properties of moving frame and differential form are employed to investigate the differential geometry of3D MAT, which is more difficult to be studied. Similarly, the normal equidistant mapping equations and moving frames attached to both the MA (medial axis) point and the associated boundary points are constructed firstly. In view of the movement of local frames, the gradient curve, level curve and skeletal curve etc., are defined to serve as carriers for the study. After exploring the infinitesimal translation and rotation of the moving frames, the relations of distance differentials, rotation vectors and seven differential invariants are derived. Finally, a special case, the normal form, is defined, and it shows that the3D MAT problem for the moulding surfaces can be converted into a2D one.
If the differential geometry serves as the basis, the optimal theory will serve as the precursor for this study. From the optimal theory of MAT, after exploring its saddle point properties, a redefinition and classification for the medial axis points are provided, from the perspective of the dimension of saddle point. Then, the mathematical programming method is employed to construct the saddle point programming models. Based on the optimality conditions, two solving strategies, i.e. the divide and conquer strategy and the linearization strategy, are derived. And a modified Newton method is adopted for the calculation of the minimum distance avoiding oscillation. In addition, the irregular points are identified according to the sudden changes of the solutions to the normal points. A generic algorithm for computing various medial axis points is proposed, and it is proved to be efficient and accurate by numerical examples.
Finally, as a further goal, this dissertation discusses the applications in the NC machining of2.5D pockets and closed impeller. In order to satisfy the demands for machining, it needs to extend the concept of MAT. More specifically, the offset box, maximal inscribed cylinder and the optimal cutter axial surface are presented and determined relying on the saddle point programming. Accordingly, the step machining method for the2.5D pockets, and the rip cutting idea for the closed impeller are developed.
This dissertation is supported by National Natural Science Foundation of China,"The Geometric Theories of the Medial Axis Transform and its Applications in NC Machining (No.50775022)'
引文
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