Isophote interpolation

Abstract

The isophote is an important class of characteristic curves on a parametric surface. Accordingly, as a kind of feature based path generation method, isophote based tool paths has been proposed as an important path pattern for parametric surface machining. In this paper, an approach has been proposed for the isophote interpolation on a parametric surface. The paper has related the arc-length derivatives to time derivatives of parameters along the isophote. The results are then used to derive the parametric interpolation. The isophote curve interpolation is developed based on parametric interpolation. The proposed interpolation guarantees that interpolated points always stay on the parametric surface. An improvement interpolation has been presented to alleviate inclination errors and point deviation from the isophote. Simulations of isophote interpolation have been carried out to verify the effectiveness of the proposed algorithm. The proposed algorithm has applications in real time tool path interpolation for the machining of parametric surfaces.

Introduction

Computer numerical control (CNC) machining is an important method for machining complex three-dimensional part surfaces. Surface machining involves four main phases, namely, optimization of machining process parameters, tool path planning, path interpolation and servo control. Path interpolation is concerned of converting tool paths obtained from a tool path planning system into time-dependent commands for driving the servo control system of a CNC machine.

Traditionally, linear and circular interpolators are the most common interpolators for NC systems, because of a lower computational requirement that was important in the past when computers were slower. A well-known approach for developing these interpolators is the digital differential analyzer (DDA) approach that is based on mathematical integration [9], [11]. Qin [16] also developed a DDA algorithm for the interpolation of three-dimensional circular paths. Another interpolation approach is based on pattern recognition [4], [15]. In this approach, a set of rules is employed to determine the next step from a possible set of moves. The above-mentioned approaches were originally conceived with the assumption of hardware implementation of the interpolator. The advent of computing technologies has been driving a migration of the implementation of interpolators from hardware to software.

While linear and circular interpolators are only built-in software interpolators for a CNC system, the approximation scheme is utilized for CNC surface machining. According to the approximation scheme, once the machining tool paths are generated, line approximation scheme are used to discretize the tool paths into line segments for the generation of NC codes. Real time operation of the CNC machine uses the downloaded NC codes as input. Further, arc approximation schemes were proposed to improve the smoothness of machined surfaces. Qiu [17] proposed to use circular arc interpolation for tool path generation. Curve fitting with arc splines was suggested by Yeung [29]. Biarcs were used to approximate complex tool paths in which line and arc segments were special cases of biarcs [22].

It was then recognized that those interpolation approaches were insufficient for the machining of complex surfaces. Disadvantages of the approximation approach include significant contour and orientation errors, heavy loading of the communication and large memory requirement for the CNC controller and considerable acceleration and deceleration for the line segment executions [10], [13], [19].

Today, computing speed for CNC controllers has increased dramatically and research on free-form curve interpolators has become quite active. Consequently, a real time path approach [10], [12], [18] is proposed as an alternative to off-line path approximation schemes. This aims to equip the CNC system with built-in interpolators for complex curves. In the real time path approach, the CNC system with built-in free-form curve interpolators inputs curve geometry parameters directly and generates motion commands in real time. Accordingly machining accuracy and product quality can be greatly improved while machining time is also significantly reduced.

Parametric curve is the most important form of tool paths for surface machining. Considerable attention was paid to the interpolation of parametric curves. General geometric properties of parametric curves were related to kinematics and coordinated motion of the machine axes in [2], [3]. Zhang [30] implemented a real time non-uniform rational B-spline interpolator for a multi-axis robot. Wang [23] proposed a real time quintic spline interpolator. Koren [10] introduced the concept of surface interpolators in which the tool orientations along a path are obtained from the parametric surface model. There were also efforts to improve the feedrate accuracy [14] and to use variable feedrate for path interpolation. Farouki [5], [6] proposed and implemented real time interpolators for Pythagorean-hodograph (PH) curves that are two-dimensional curves. A constant feed and reduced angular acceleration interpolation algorithm is proposed for parametric curve in multi-axis machining [7]. Yeh [27] also proposed a parametric curve interpolator with speed control and further presented an interpolation method with confined chord error [28].

Implicit curve is another form of tool paths for surface machining. While much of the earlier work in real time curve interpolations assume the use of parametric curves, recently there has been an increase in attention to real time interpolations for implicit curves [14], [19], [20], [24], [25], [26].

Drive surface methods is an important approach for tool path generation [1]. According to drive surface methods, tool paths can be generated by intersecting a part surface with a family of drive surfaces such as planes and cylinders. But in general, the tool path obtained by intersecting a part surface with a drive surface is a hybrid curve [21]. The hybrid curve is a novel form of tool paths, which is different from parametric curves and implicit curves. Accordingly hybrid curve interpolation [21] is proposed with tool paths generated by intersecting a parametrically modeled part surface with a family of implicitly defined drive surfaces.

According to Ref. [8], the common weakness of the current tool path generation methods is feature-blindness. The usual strategy is to adopt a preset mathematical rule to generate paths for the entire surface. The geometry difference that may exist among the scattered features/geometry on a surface, however, cannot be taken into consideration. The isophote is an important class of characteristic curves on a parametric surface. Then, as a kind of feature based path topology, isophote based tool paths has been proposed as an important path pattern for parametric surface machining in Ref. [8] where a tracing method for the isophote is proposed based on grid subdivision for off-line path approximation for the machining of parametric surface.

This paper is focused on the isophote interpolation in real time for parametric surface machining. It is noted that an isophote can be represented as neither a parametric curve, nor an implicit curve, nor a hybrid curve. Therefore, existing interpolation methods are not applicable to isophote interpolation. An approach for the isophote interpolation is proposed in this paper. The strategy is to carry out the isophote interpolation based on parametric interpolation. The arc-length derivatives of the parameters are related to time derivatives of parameters with feedrate along the isophote in Section 2. Based on the results, the parametric interpolation is developed in Section 3. In Section 4, the interpolation and its improvement for isophote are developed. Details of the algorithm and examples are provided for isophote interpolation for parametric surfaces in Section 5.