五轴加工刀具姿态球面NURBS曲线设计及优化

Sphere NURBS Curve Design and Optimization of Tool Orientations in Five-axis Machining

基于刚体空间转动的四元数表达,以有理运动描述五轴数控加工中刀具姿态变化。将刀具姿态点等同于单位球面上的点,建立单位球面上的点和四元数空间中的点之间的一一映射关系,从而将离散刀具姿态插值问题归结为在四元数空间插值曲线设计问题。设计出的单位球面上的曲线(刀具姿态曲线)具有通常三维欧式空间NURBS曲线的表达形式,可以直接输入支持双NURBS插补的高端数控系统。基于四元数空间度量,以最小二乘准则优化刀具姿态NURBS曲线,明显改善优化前复杂曲面五轴加工留下的过切线。

The motion of tool orientation is expressed in the frame of rational motion based on the parameterization of rigid body displacement with quaternion. With the identity of tool orientation points and the points on unit sphere,the 1-1 mapping from points on unit sphere to elements of quaternion space is constructed. Thus the problem of interpolation of discrete tool orientation is transformed to that of interpolatory curve design in quaternion space. The obtained NURBS curve has the same mathematical expression as that in Euclidean 3-dimension space,and as a result it can be inputted directly to some CNC system with double-NURBS interpolator. Least square method based on the quaternion space metric is applied to the tool orientations optimization. When the optimized tool orientation is used for milling,the overcut is reduced.