Interpolating and Approximating Moving Frames Using B-splines

The representation of moving geometry entities is an important issue in the fields of CAD/CAM and robotic motion design. We present a method to interpolate the moving frame homogeneously using B-splines. By analysing the matrix representation of the moving frames, a simplified optimization functional is derived and a recursive iteration method is presented for approximating orthogonal frames at any position. The error caused by approximation and interpolation is given and is shown to be controllable. Since the calculation of moving frames is only related to linear equations of two or three unknowns, and the moving frames are approximated with a low-degree polynomial B-spline, the proposed approach can run very fast. The method is useful in motion design and the swept volume representation.