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

Abstract

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%.

Special Feature on Advanced Mobile Robotics

“…Optical polishing also needs trajectory planning, and the accuracy of trajectory and runtime of trajectory can influence the polishing quality. Thus, Zhao and Guo [11] proposed that applying a B-spline curve method improves the performance.…”

B-Spline Curve Fitting of Hungry Predation Optimization on Ship Line Design

“…Unreasonable knot vector arrangement may lead to unacceptable shapes [10]. Researchers put higher fitting accuracy at the same knot number [11,12], fewer knot number under the same accuracy requirements [13,14] and faster operating efficiency [15,16] as the main pursuit goals of B-spline curve fitting. This paper develops a B-spline curve fitting of hunger predation optimization on ship line design (HPA).…”

Optimum time-energy-jerk trajectory planning for serial robotic manipulators by reparameterized quintic NURBS curves

“…Alternatively, non-uniform rational B-splines (NURBS) are well-known parametric curves with exact representation of various freeform curves as well as common profiles such as lines, parabolas, circles, etc., thus, being preferable candidates for trajectory planning. 17,20–23 Trajectories defined by cubic NURBS curves have been extensively adopted, 24–26 whereas, vibrations may occur to the manipulators together with high errors at the knots since cubic NURBS curves need some additional control points with prescribed second-order derivative of the endpoint, despite that they can represent elementary analytic curve accurately and the shape of the curve can be modified easily by the weights. Therefore, to achieve smooth kinematic specifications, trajectories are commonly expressed by curves with higher degrees, say quintic curves, which can make the trajectory more flexible and easier to be modified.…”