面向自由曲面加工轨迹规划的黎曼度量几何性质研究与应用

航空发动机叶片等自由曲面类产品广泛应用于国家战略领域。自由曲面的高效高精加工轨迹规划方法涉及数学、信息科学、机械制造等学科深度交叉。目前，如何兼顾几何精度与物理场信息、协调刀具运动状态与加工实际需求是加工轨迹规划方法面临的重要挑战。本项目改变在欧式度量空间中规划轨迹的传统，提出在黎曼度量空间中规划轨迹的新尝试。将曲面几何、刀具几何、刀具运动状态的耦合作为内蕴性质赋予黎曼流形上，由此搭建了自由曲面整体加工路径规划的统一框架。借此，黎曼几何中的数学工具（如微分型场论算子）具备了机械加工中的实际意涵，也为高效加工轨迹规划提供了便利与新思路。主要研究内容包括：加工黎曼度量的存在性条件分析与可行加工空间确定，加工黎曼度量构建与微分型场论算子计算，加工黎曼度量下的整体轨迹规划方法。本项目预期目标是：搭建自由曲面加工统一轨迹规划框架，实现高效加工并进行实例验证。

Products with freeform surfaces, such as aero-engine blades, are widely used in national strategic fields. The high-efficient and high-precision machining tool path planning method of freeform surfaces is a multidisciplinary issue involved in mathematics, information science, mechanical manufacturing and etc. At present, how to combine geometric precision with physical field information, and to coordinate the motion state of tool with the actual needs of machining is an important issue in tool path planning. Changing the tradition of trajectory planning in Euclidean metric space, this project proposes a new attempt to plan tool paths in Riemannian metric space. The coupling of surface geometry, tool geometry and tool motion state are given as intrinsic properties to the Riemannian manifold, which is called Machining-Riemannian-Metric (MRM), thus building a unified framework for the overall tool path planning of freeform surfaces. In this way, the mathematical tools in Riemannian geometry (such as differential operators of field theory) have actual meanings in machining, and also provide convenience and new approach for efficient tool path planning. The main research contents include: the existence condition analysis of the MRM and the determination of the feasible processing space, the construction of the MRM and the calculation method of the differential operators of field theory, and the overall tool path planning method using the MRM. This project aims to build a unified tool path planning framework for freeform surface machining, to achieve high-efficient machining and to carry out example verification.