A Trajectory Planning Method for Polishing Optical Elements Based on a Non-Uniform Rational B-Spline Curve

Optical polishing can accurately correct the surface error through controlling the dwell time of the polishing tool on the element surface. Thus, the precision of the trajectory and the dwell time (the runtime of the trajectory) are important factors affecting the polishing quality. This study introduces a systematic interpolation method for optical polishing using a non-uniform rational B-spline (NURBS). A numerical method for solving all the control points of NURBS was proposed with the help of a successive over relaxation (SOR) iterative theory, to overcome the problem of large computation. Then, an optimisation algorithm was applied to smooth the NURBS by taking the shear jerk as the evaluation index. Finally, a trajectory interpolation scheme was investigated for guaranteeing the precision of the trajectory runtime. The experiments on a prototype showed that, compared to the linear interpolation method, there was an order of magnitude improvement in interpolation, and runtime, errors. Correspondingly, the convergence rate of the surface error of elements improved from 37.59% to 44.44%