Modeling and simulation of trajectory smoothing and feedrate scheduling for vibration-damping CNC machining

Highlights

• All the geometric and kinematic constraints for CNC machining are satisfied.

• G⁴ interpolative path lowers curvature extrema, bending energy, and fitting error.

• Time-optimal and jerk-smooth feedrate scheduling scheme is developed.

• The method produces smaller jerk and reduces feedrate fluctuation at the corners.

• Real-time method performs better in vibration damping & surface quality.

Abstract

The mathematical models of tool path smoothing and feedrate scheduling in Computer Numerical Control (CNC) system are critical for high-precision manufacture. Inthe existing studies, G²/G³ (curvature-continuous/-smooth) trajectory smoothing, and jerk-limited/-continuous feedrate scheduling schemes have been developed. Nonetheless, there are exist some requirements that cannot be satisfied simultaneously, including confined chord error, G01 points interpolated, analytical curvature extremum, real-time performance, kinematic time-optimality, and the smoothness of tool-path and feedrate profile. Recently the scholars found the potentials of the tool path with high-order geometric continuity and the feedrate with high-order kinematic constraints, respectively in increasing the smoothness of tool path and reducing the impact caused by axis actuators during the process. Aiming to reduce the vibration and guarantee high machining efficiency, this work proposes G⁴ (curvature-variation-smooth) interpolative trajectory model with confined chord error and analytical curvature extrema for trajectory smoothing, and employs jerk-smooth (jerk-differentiable) feedrate mode to perform time-optimal feedrate scheduling. Finally, a real-time tool path processing strategy under various geometric and kinematic constraints is developed. The simulation shows approximation error, curvature extrema, and feedrate fluctuation are reduced compared with G² transition and G³ interpolation schemes. The experimental results demonstrate advantages of the proposed method in vibration damping, surface quality, compared with the previous works.

Introduction

The mathematical models for tool path smoothing and feedrate scheduling are critical for high-precision and low-vibration machining. The tool path generated by computer-aided manufacturing (CAM) system is the piecewise linear trajectory with only G° continuity, leading to large machinery vibration at the sharp corners. To decline the impact of machine tool during the process and raise the surface quality, G^k (k ≥ 2) trajectory smoothing is indispensable [1]. Recently, the scholars [2,3] found the tool path with higher geometric continuity can raise the feedrate at the sharp corner, declining the fluctuation of feedrate and acceleration, raising the machining quality. On the other hand, the feedrate profile with high smoothness is a necessary condition to guarantee the continuity of high-order axial kinematic variables (such as acceleration, jerk, snap), increasing the smoothness of feedrate trajectory and reduces the impact caused by axis actuators [4], [5], [6]. Hence it is significant to develop a mathematical model of tool-path smoothing and feedrate scheduling for damping the machinery vibration and raising the surface quality.

The trajectory smoothing methods includes approximation schemes [7], [8], [9], [10], [11], [12], [13], [14], transition schemes [1,[15], [16], [17], [18], [19], [20]], and interpolation schemes [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33]. These works are valuable to improve the machining quality and efficiency. However, there still exist some shortcomings when these methods were employed in CNC machining. The comparison of G^k (k ≥ 2) tool-path smoothing schemes in recent studies is provided in Table 1. Besides the deficiencies in controlling the chord error, interpolating G01 points, the form of curvature extremum, and real-time performance, the existing tool-path smoothing schemes performs insufficiently in reducing the curvature extrema and raising the smoothness of tool path.

To generate G^k (k ≥ 2) tool path and compress the number of the data points, the approximation schemes using B-spline or Non-uniform Rational B-spline (NURBS) curve have been widely studied in the literatures [7], [8], [9], [10], [11], [12], [13], [14], including the Least Square Fitting (LSF) methods [7], [8], [9], [10] and the progressive and iterative approximation (PIA) methods [11], [12], [13], [14]. Park et al. [7,8] and Li et al. [9] adopted the conventional LSF methods to generate the smooth trajectory with the confined error between data points and the smoothed trajectory. Nevertheless, the chord error was unconstrained, and both computational efficiency and numerical stability were low in these works. Yeh et al. [10] developed a NURBS fitting method with LSF to generate G² trajectory in real time. The error between G01 points and the smoothed path was constrained, but the chord error was not constrained. Lin et al. [11] proposed an extended PIA method to reduce the number of control points. Deng et al. [12] further presented a progressive and iterative approximation method based on least squares fitting (LSPIA). Nonetheless, the chord error constraint was not considered in the works [11,12]. He et al. [13] incorporated the energy objective function into the LSPIA method, and proposed ELSPIA method to generate the smooth trajectory with confined chord error. Nonetheless, the algorithm was time-consuming due to the iterative least square approximation, and the chord-error conformance possibly weakens the overall smoothness of tool path. Bi et al. [14] proposed a fast B-spline fitting scheme with PIA method to generate G² tool path under confined chord error. For the approximation schemes [7], [8], [9], [10], [11], [12], [13], [14], the general deficiency is that the smoothed path has undefined analytical curvature extrema, leading to the heavy computational load in the stage of feedrate scheduling, hampering the real-time performance of tool path processing.

To conveniently control the shape of tool path, G^k (k ≥ 2) transition schemes [1,[15], [16], [17], [18], [19], [20]] are performed using the transitional curve, such as B-spline curve, Bezier curve. The authors [1,[15], [16], [17], [18]] performed G² transition scheme at the sharp corner. Nevertheless, the chord error of G² trajectory was not constrained in Ref. [15,16], and the analytical curvature extremum of G² trajectory was unavailable in Ref. [16]. Zhao [1] and Farouki [17] realized G² corner smoothing under the confined chord error in real time, and the tool path generated had the analytical curvature extrema. To generate smoother tool path and decline the jerk during the process, G³ transitional trajectory has been recently designed by the works [18], [19], [20]. Piazzi [18] and Fang [20] constructed G³ trajectory between the splines, but the trajectory had only G¹ continuity at the joints. Besides, the approximation error was unconstrained in the works [18,19]. Although the transition schemes generated the shape-preserving trajectories, the trajectory did not interpolate G01 points, leading to the loss of the featured position of original contour.

To preserve the information of featured positions of original contour, G^k (k ≥ 2) interpolation methods have been extensively studied in the works [2,[21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33]]. Compare with approximation schemes and transition schemes, interpolation methods can guarantee the smoothed trajectory passes through data points, and avoid the error between points and the smoothed trajectory. In those works [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], G² interpolative trajectories have been generated. Nevertheless, in these works [21], [22], [23], [24], [25], [26], [27], [28], the chord error was not constrained rigorously. The methods [25], [26], [27], [28], [29] were only applicable to the planar data or monotone data. Fan et al. [30,31] adopted Bezier curve to generate interpolative trajectory under confined chord error but the trajectories generated were not rigidly shape-preserving. G³ interpolation scheme has been recently studied by the works [2,32,33]. Bajaj [32] and Renka [33] generated a G³ shape-preserving interpolative trajectory for spatial point sequence but the chord error was not constrained and analytical curvature extremum was unavailable. Besides, in this work, a nonlinear optimization problem needed to be solved, and thereby the algorithm was only performed offline. Fan et al. [2] designed a G³ interpolative path for G01 trajectory. However, there still exists the space of declining curvature extrema and raising the smoothness at the sharp corners.

After generating the smooth trajectory, the geometric information of trajectory is transferred to the feedrate scheduling module in CNC system. The feedrate profile under confined jerk decreases greatly the excitation of the resonance frequencies of machine tool, declining machinery vibrations. In the works [34], [35], [36], [37], [38], jerk-limited feedrate scheduling schemes have been developed. Nonetheless the methods are only performed offline. Tajima et al. [39] generated jerk-limited trajectory under confined approximation error for corner smoothing in real time but the trajectory was not interpolative. Lai [40] and Annoni [41] performed real-time feedrate scheduling under confined jerk and fine interpolation error for NURBS trajectory. The kinematic smoothness helps to decline the flexible impact created by the axis actuator. To raise the kinematic smoothness, Fan [6], Liu [42], and Bharathi [43] realized time-optimal and jerk-continuous feedrate scheduling but these methods were performed offline. Jin et al. [44] proposed a real-time and jerk-continuous feedrate scheduling scheme but 7-phase ACC/DEC profile adopted was not time-optimal. In addition to the deficiencies in real-time performance and time-optimality, the kinematic smoothness in the existing feedrate scheduling schemes is insufficient, and thus there still exists the improving space for damping the impact caused by axis actuator.

High-smoothness tool path and feedrate profile are indispensable to generate smooth kinematic profiles, and helpful to damp the impact caused by axis actuator. Aiming to reduce the vibration and guarantee high machining efficiency, this work considers the various requirements of CNC machining to develop a real-time methodology, including G⁴-continuous tool-path interpolation scheme and time-optimal feedrate scheduling scheme with smooth jerk. The remainder of the work is arranged as follows. Section 2 proposed G⁴ trajectory model and provides the generation algorithm of interpolative trajectory. Section 3 proposes jerk-smooth feedrate mode and the look-ahead algorithm of feedrate planning. In Section 4, the advantages of G⁴ tool path and jerk-smooth feedrate mode are shown. Section 5 provides the simulation and comparison. The experiment and analysis are done in Section 6. The conclusion is presented in Section 7.