Nonuniform Rational Basis Spline Interpolation for Off-Axis Aspheric Mirror Grinding Based on Wheel Path Planning

Large-diameter aspheric mirrors made of hard and brittle materials suffer from low efficiency in the grinding stage, making them difficult to process. In this study, a nonuniform rational basis spline (NURBS) interpolation method is proposed and developed based on the grinding wheel path planning. The characteristic parameters of NURBS machining program are calculated to ensure the accuracy of curve interpolation path planning, and a series of control points are selected according to the normal vector points of the grinding wheel path. The control points were determined by the distance between the aspheric tangent and the next type value point at a fixed distance d. We then established a mathematical model for the knot point and simulated the relationship between the knot point and d. This indicated that d must be optimized according to the number of points to reduce the number of control points. According to this established model, a comparison experiment was conducted using an off-axis aspheric mirror, with a 280 mm diameter. Straight line and NURBS grinding were compared, and the NURBS interpolation method was shown to effectively improve grinding accuracy and efficiency by about 30%.

Planar tool radius compensation for CNC systems based on NURBS interpolation

This paper introduces the realization of a tool radius compensation algorithm for NURBS trajectory. First, a single-segment NURBS trajectory tool radius compensation algorithm is developed. Different from the straight line and arc trajectory, the self-intersection phenomenon is prone to happen when calculating a single NURBS tool center trajectory, and the self-intersection will cause the overcut of workpiece. To avoid this situation, the algorithm introduced in this paper can detect whether the NURBS tool center track has caused overcut, and deal with the self-processing. Second, the tool radius compensation algorithm with multi-segment NURBS trajectory is implemented. The focus of this part is the tool radius compensation of the trajectory transfer, and the trajectory transfer is divided into two types: the extension type and the shortened type. For the shortened type transfer, cross-processing is needed to avoid the overcut of workpiece at the transfer. When calculating the tool radius compensation of the shortened type, we not only need to find the intersection of the tool center trajectory of two adjacent NURBS curves, but also need to select the intersection we need when a number of intersections exist. For the extension type transfer, in order to ensure the continuity of the tool center trajectory, we need to extend the tool center trajectory or add arc-segment at the transfer. The proposed algorithm can automatically decide where to extend the tool center trajectory or add arc-segment to achieve the best efficiency. Finally, the algorithm can output the calculated NURBS tool center trajectory in the form of linear segment interpolation G code or NURBS interpolation G code according to the processing needs. Simulations on VERICUT and experiments on three-axis CNC machine tool shows the effectiveness and validation of the tool path compensation algorithm.

КІНЕМАТИЧНІ ВЛАСТИВОСТІ ТРАЄКТОРІЙ, ОПИСАНИХ NURBS СПЛАЙНАМИ

The purpose of the article is to solve a number of problems of using NURBS curves in the implementation of high-speed processing of complex products on machining equipment. NURBS-splines are becoming increasingly popular among CNC specialists, which is motivated by their wide application in the modeling of the production facility in CAD/CAM systems. Advanced manufacturers of CNC devices in the number of shaping algorithms necessarily include NURBS - interpolation. But despite the extensive capabilities of CAD/CAM systems for describing the shape of products and toolpaths of processing equipment using spline curves, many manufacturers of CNC systems use more primitive methods for describing the trajectory of the tool - piecewise linear and linear-circular. This negates the benefits of using spline curves, including smooth curvature change, which is very important when moving the tool at high speed processing. Such simplification of trajectories is associated with a number of difficulties in the implementation of spline curves in modern CNC systems. In addition, the requirements for industrial cleanliness of the surfaces of the parts is information about the processing parameters, the directions of relative movements of the tool and the workpiece on the finishing operations in order to provide technological conditions to the size of the burrs formed. It is possible that the choice of strategy of processing for obtaining the minimum size of a Burr or reduction of their number. The sizes of the turned-out agnails define methods of their elimination in technological process of production of a detail. Therefore, the provision of technological information using the analytical standard of the part is a necessary condition for ensuring the quality of products and their high reliability. NURBS - interpolation allows to obtain a compact description of the tool movement in the process of forming, which allows to correctly formulate the problem of working of complex-shaped parts.