An Optimal Feed Interpolator Based on G2 Continuous B\xe9zier Curves for High-Speed Machining of Linear Tool Path

Yongqiao Jin,Yuhan Wang,Sheng Zhao

doi:10.1186/s10033-019-0360-8

Abstract

A numerical control (NC) tool path of digital CAD model is widely generated as a set of short line segments in machining. However, there are three shortcomings in the linear tool path, such as discontinuities of tangency and curvature, huge number of line segments, and short lengths of line segments. These disadvantages hinder the development of high speed machining. To smooth the linear tool path and improve machining efficiency of short line segments, this paper presents an optimal feed interpolator based on G2 continuous Bézier curves for the linear tool path. First, the areas suitable for fitting are screened out based on the geometric characteristics of continuous short segments (CSSs). CSSs in every area are compressed and fitted into a G2 Continuous Bézier curve by using the least square method. Then a series of cubic Bézier curves are generated. However, the junction between adjacent Bézier curves is only G0 continuous. By adjusting the control points and inserting Bézier transition curves between adjacent Bézier curves, the G2 continuous tool path is constructed. The fitting error is estimated by the second-order Taylor formula. Without iteration, the fitting algorithm can be implemented in real-time environment. Second, the optimal feed interpolator considering the comprehensive constraints (such as the chord error constraint, the maximum normal acceleration, servo capacity of each axis, etc.) is proposed. Simulation and experiment are conducted. The results shows that the proposed method can generate smooth path, decrease the amount of segments and reduce machining time for machining of linear tool path. The proposed research provides an effective method for high-speed machining of complex 2-D/3-D profiles described by short line segments.

Highlights

1 Introduction Linear tool path is the most widespread tool path representation form to approximate the complicated surface in machining; it introduces both the tangent and curvature discontinuities at the segment junctions
With the development of machining technology in modern industry, CNC machine tools ask for high interpolation technology, but linear tool path cannot satisfy the requirements of high-speed and
NURBS fitting methods have the following deficiencies: 1) heavy computational complexity of recursive algorithm, which is difficult to meet the real-time requirements of system; 2) inevitable errors introduced by employing a truncated Taylor series

Summary

Introduction

Linear tool path is the most widespread tool path representation form to approximate the complicated surface in machining; it introduces both the tangent and curvature discontinuities at the segment junctions. Zhang et al [3], proposed a transition algorithm for two continuous linear segments based on the cubic Hermite curve fitting approach. This algorithm guaranteed the tangent continuity of the new tool path. Zhao et al [6] proposed a method for smoothing the linear tool path by using two transition cubic Bézier curves. An optimal feed interpolator based on G2 Bézier transition algorithm is proposed to fit the linear tool path, which can satisfy the following requirements: 1) G2 continuity; 2) data compression; 3) approximation control; 4) real-time performance. The curvature is continuous and there is only one curvature extremum in the Bézier curve

Definitions of Consecutive Short Segment and Breakpoint

Approximation Error Control

G2 Continuous Path Construction

The Maximum Feed Restricted by Comprehensive

Summary and Conclusions