Path smoothing and trajectory planning are universally applied in computer-numerical-control (CNC) machining to avoid natural discontinuity of tangency and curvature at the junctions of G01 blocks. However, most existing methods primarily focus on path-centric indicators that consider the toolpath as a continuous curve, such as contour error and manufacturing efficiency, neglecting the global machining quality and failing to avoid surface inconsistencies, such as single tool marks. This paper establishes a theoretical framework to evaluate the global continuity of toolpaths and trajectories, proposing the consistency as a surface-centric evaluation that considers toolpaths as a surface in CNC machining. In this paper, the consistency is defined as similarity between adjacent toolpaths and trajectories when facing similar input fold-paths in single-point milling. The consistency of four typical existing methods representing a broad category of typical approaches is investigated based on the developed theory. As a theoretically ideal objective, the proposed strong consistency requires a path smoothing method robust to any positional disturbance on the input fold-paths, and this paper points out that few algorithms have achieved strong consistency so far. The proposed weak consistency focusing on the tangential disturbance is practical in the industry. Filtering-based methods without contour error limitations are proved to achieve weak consistency, and smoothing methods with explicit geometric constraints fail to achieve weak consistency. To facilitate evaluation on the consistency of more complex methods, this paper proposes numerical benchmarks and quantitative indicators which can determine whether a method is consistent by numerical experiments. Conducted on a 3-axis machine tool with a ball-end milling cutter, real-world experiments show that inconsistencies in toolpaths’ position and trajectories’ feedrate causes surface inconsistencies like single tool marks. The proposed consistency theory and the carefully designed benchmarks can serve as a novel evaluation for path smoothing and trajectory planning from a global perspective, and it can help to identify areas where inconsistencies may occur in single-point milling.

中文翻译：

---

关于 CNC 加工中路径平滑和轨迹规划的一致性：以表面为中心的评估


路径平滑和轨迹规划普遍应用于计算机数控 （CNC） 加工中，以避免 G01 块连接处的切线和曲率自然不连续。然而，大多数现有方法主要关注以路径为中心的指标，这些指标将刀具路径视为连续曲线，例如轮廓误差和制造效率，而忽略了整体加工质量，无法避免曲面不一致，例如单个刀具标记。本文建立了一个理论框架来评估刀具路径和轨迹的全局连续性，提出了一致性作为一种以表面为中心的评估，将刀具路径视为 CNC 加工中的表面。在本文中，一致性定义为在单点铣削中面对相似的输入折叠路径时，相邻刀具路径和轨迹之间的相似性。基于发展的理论，研究了代表一大类典型方法的四种典型现有方法的一致性。作为一个理论上理想的目标，所提出的强一致性要求一种路径平滑方法对输入折叠路径上的任何位置干扰都具有鲁棒性，本文指出，到目前为止，很少有算法能够实现强一致性。所提出的专注于切向扰动的弱一致性在工业中是可行的。事实证明，没有等值线误差限制的基于过滤的方法可实现弱一致性，而具有显式几何约束的平滑方法无法实现弱一致性。为了便于评价复杂方法的一致性，本文提出了数值基准和定量指标，可以通过数值实验来确定方法是否一致。 在带有球头铣刀的 3 轴机床上进行的实际实验表明，刀具路径位置和轨迹进给率的不一致会导致表面不一致，如单个刀具标记。所提出的一致性理论和精心设计的基准可以从全局角度作为对路径平滑和轨迹规划的新评估，并且有助于识别单点铣削中可能出现不一致的区域。