Recently, modern manufacturing systems have been designed which can machine arbitrary parametric curves while greatly reducing data communication between CAD/CAM and CNC systems. However, a constant feedrate and chord accuracy between two interpolated points along parametric curves are generally difficult to achieve due to the non-uniform map between curves and parameters. A speed-controlled interpolation algorithm with an adaptive feedrate is proposed in this paper. Since the chord error in interpolation depends on the curve speed and the radius of curvature, the feedrate in the proposed algorithm is automatically adjusted so that a specified limit on the chord error is met. Both simulation and experimental results for non-uniform rational B-spline (NURBS) examples are provided to verify the feasibility and precision of the proposed interpolation algorithm.

https://ir.nctu.edu.tw:443/bitstream/11536/28963/1/000172662300005.pdf

Adaptive-feedrate interpolation for parametric curves with a confined chord error

In this paper, a dynamics-based interpolator with real-time look-ahead (DBLA) algorithm is proposed to generate a smooth and jerk-limited acceleration/deceleration (ACC/DEC) feedrate profile. The interpolator consists of three modules: geometric, dynamics-based, and jerk-limited modules. The geometric module can detect the local maximum/minimum (max/min) curvatures, and divide a NURBS curve into small segments according to the information of sharp corners. The feedrates at the sharp corners are determined based on confined chord errors and curvatures of the curve. The dynamics-based module utilizes a dynamics feedrate modification equation (DFME) to estimate contour errors at the sharp corners and adjusts the feedrates at the locations of the sharp corners. The jerk-limited module plans the feedrate profile of the curve according to the segments’ length and the given jerk limit. Simulations are performed to verify real-time performance of the look-ahead algorithm. Experiments using a PC-based motion controller and an X–Y table are conducted to demonstrate that high-accuracy can be achieved with the proposed dynamics-based interpolator as compared to the adaptive-feedrate and the curvature-based feedrate interpolation algorithms.

Development of a dynamics-based NURBS interpolator with real-time look-ahead algorithm

Modern CNC systems are designed with the function of machining arbitrary parametric curves to save massive data communication between CAD/CAM and CNC systems and improve their machining quality. Although available CNC interpolators for parametric curves generally achieve contouring position accuracy, the specified feedrate, which dominates the quality of the machining processes, is not guaranteed during motion. Recently, some approximation results concerning motion speed have been reported [Shpitalni M, Koren Y, Lo CC. Computer-Aided Design 1994;26(11):832–838; Bedi S, Ali I, Quan N. Trans ASME J Engng Industry 1993;115:329–336; Chou JJ, Yang DCH. Trans. ASME J Engng Industry 1991;113:305–310; Chou JJ, Yang DCH. Trans ASME J Engng Industry 1992;114:15–22]. In this paper, by deriving a compensatory parameter, the proposed interpolation algorithm has significantly improved curve speed accuracy. In real applications, the proposed algorithm results in: (1) constant speed; and (2) specified acceleration and deceleration (ACC/DEC) to meet the feedrate commands. The motion accuracy and feasibility of the present interpolator have been verified with a provided non-uniform rational B-spline (NURBS) example.

https://ir.nctu.edu.tw:443/bitstream/11536/31387/1/000081364400004.pdf

The speed-controlled interpolator for machining parametric curves

Parametric interpolation has many advantages over linear interpolation in machining curves. Real time parametric interpolation research so far has addressed achieving a uniform feed rate, confined chord errors and jerk limited trajectory planning. However, simultaneous consideration of confined chord errors that respect the acceleration and deceleration capabilities of the machine has not been attempted. In this paper, the offline detection of feed rate sensitive corners is proposed. The velocity profile in these zones is planned so that chord errors are satisfied while simultaneously accommodating the machine's acceleration and deceleration limits. Outside the zone of the feed rate sensitive corners, the feed rate is planned using the Taylor approximation. Simulation results indicate that the offline detection of feed rate sensitive corners improves parametric interpolation. For real time interpolation, the parametric curve information can be augmented with the detected feed rate sensitive corners that are stored in 2×2 matrices.

A parametric interpolator with confined chord errors, acceleration and deceleration for NC machining

Multi-axis tool paths are currently generated as a set of discrete data points consisting of a position vector, representing the tool tip, and an orientation unit vector, representing the tool axis. The CNC interpolator must convert these points into continuous machine tool axis motions. To achieve the highest quality parts, a constant feed and reduced angular acceleration must be maintained throughout the motion. This paper presents a new algorithm for off-line interpolation of the data points, followed by real-time axis command generation. The splines produced by the algorithm are C2 continuous, and independent of machine tool kinematics. Hence the motions produced will be the same on any five-axis machine tool, hexapod, or robotic arm. Three splines are computed: position, orientation, and reparameterization. The position spline is a near arc-length parameterized quintic polynomial spline. The paper introduces a near arc-length parameterized quintic spherical Bezier spline as the orientation spline. Coordinated motion is accomplished with an orientation reparameterization spline. The proposed algorithm is demonstrated using a practical example.

A constant feed and reduced angular acceleration interpolation algorithm for multi-axis machining

The paper presents a comparative study of linear and parametric interpolators used for command generation during CNC machining. For many decades, linear interpolators have been used in NC machines for command generation. Recently, a parametric interpolator has been developed that is designed for command generation for parametric curves (or surfaces). Two algorithms for the online implementation of this parametric interpolator are introduced. The comparison is mainly based on memory size, feedrate fluctuation and CPU time. Other considerations include tracking errors and jerk magnitude. Results show that the parametric interpolator is favourable in the machining of free-form geometry in terms of high speed and tight tolerancing.

Tom Kong

Papers

Parametric interpolator versus linear interpolator for precision CNC machining