Abstract: To make the CNC trajectory smoother and further improve the machining efficiency,a bidirectional fourth-order Taylor fitting method for smooth curves based on spatial axis was proposed.Through optimization,the curvature was gentler and the contour error of fitted curves was further reduced.The finite uniform data points were selected on the parametric curve to calculate the derivatives of parameters and the differential geometric quantities such as curvature and torsion to establish the Frenet coordinate system.Then,the forward and backward fourth-order Taylor expansion was performed on the first and last data points of the i interval respectively,and the fitting curve of coupling weight was determined according to the relative distance of the parameter space formed by the Cutter Location (CL) points at the first and last data points.Finally,the optimum trajectory in machine tool coordinate system XYZ was obtained by inverse coordinate transformation.The comparison between the original trajectory and the bidirectional fourth-order Taylor fitting showed that the contour error had decreased gradually with the increasing of the order,and the mean value of curvature,the range of curvature and the range of curvature derivatived all approach the original curve.A fitting test was carried out with a standard circle and a high-speed train wheel tread LM-32,which found that the optimized curve obtained by the proposed method was 4μm ~5um less than the contour error of the curve.

Key words: CNC machining, tool path optimization, Frenet coordinate system, Taylor expansion

摘要： 为了使数控轨迹更加光滑,以进一步提高加工效率,提出一种基于空间轴的光滑曲线双向四阶Taylor拟合方法,通过优化使曲率更平缓来进一步降低拟合曲线的轮廓误差。首先,在参数曲线上选取有限均匀的数据点计算参数各阶导数以及曲率、挠率等微分几何量,以建立Frenet坐标系;然后,在第i区间始末两个数据点分别进行前向和后向四阶Taylor展开,根据刀位点在始末两个数据点形成的参数空间的相对距离确定耦合权重的拟合曲线;最后,通过坐标反变换得到机床坐标系XYZ下的优化轨迹。通过对原始轨迹和双向四阶Taylor拟合进行比较表明,随着阶数的提高,轮廓误差逐渐降低,曲率均值、曲率极差和曲率的导数极差均趋近原始曲线。通过对标准圆和高速列车车轮踏面LM-32进行拟合测试表明,相比拟合曲线,所提方法得到的优化曲线曲率变化连续,轮廓误差小4 μm~5 μm。

关键词: 数控加工, 刀具轨迹优化, Frenet坐标系, Taylor展开