Abstract: 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.

Key words: Interpolation and optimization, Rational motion, Sphere NURBS curve, Tool orientation