Feedrate Fluctuation Research Articles

A real-time dual NURBS interpolator with optimised control of flexible acceleration and deceleration for five-axis CNC machining

The limited computing capacity makes it difficult to plan a suitable feedrate profile in real-time for high speed and high accuracy machining of five-axis parametric toolpaths. In this paper, a real-time interpolation algorithm with optimised control of flexible acceleration and deceleration (acc-dec) for the dual NURBS toolpath is proposed. The toolpath is marked as subsegments with similar geometric properties by introducing the five-axis curvature. Machine kinematic and toolpath geometry constraints are considered in the kinematic parameter constraint model. Initial feedrate profiles are solved in a dynamic 3D window which preserves the motion performance of machine tools to a great extent. Convolution is used to smooth the initial feedrate profile to achieve a higher order continuity over the global range. Feedrate fluctuations caused by imprecise parameter interpolation are eliminated through modifying each interpolation periods. Resampling adjusts the position of interpolation points and unify the interpolation periods. All operations mentioned are in series and real-time is strictly guaranteed. Effectiveness of the developed algorithm is validated in simulations and also experimentally on a Self-developed-NC controlled 5-axis machine tool.

Jerk limited continuous tool path generation for flexible systems in non-cartesian coordinate systems

Inspired by computer numerical control (CNC) technology and drastic changes in the world of electronics and microcontrollers, the idea of using non-cartesian coordinate system for CNC machines became a topic of interest. Non-cartesian coordinate systems provide freedom of movement in systems with large working spaces where high dimensional accuracy is not a prime concern. Applications of such systems can be found in murals, where a painting is applied to a large wall of any arbitrary build material. In the present paper, a new design was introduced to automate the process of drawing murals on large surfaces. The proposed mechanism utilized chain and sprocket to overcome the size limitations of traditionally available x-y tables. In addition to that, the developed mechanism does not utilize the common robust structures used in the conventional CNC systems and the tool motion is created using flexible arms that are not heavily constrained. The effect of systems unwanted vibration has been eliminated using continuous trajectory and kinematic profiles. Parametric curve interpolation algorithms for trajectory planning were implemented to increase the flexibility of drawing complex curves. Parametric curves such as B-spline and non-uniform rational B-spline (NURBS) were employed to generate a jerk limited trajectory for motion control and extraction of kinematic profiles. Such trajectory constrains the jerk of the system and guarantees a smooth motion free of feed-rate fluctuations. Compared to other methods available for extracting the kinematic profiles, jerk limited method decreases the travel time and trajectory error. Ultimately, motion commands were produced by using kinematic profiles and the second order Taylor interpolator. STM32746ZG card was used as the main processor and two stepper motors controlled by a Bit Pattern interpolation algorithm were utilized to drive the proposed mechanism. The input pulses of stepper motors were stored as binary bits and transmitted to drivers in real-time. The feedback was also evaluated by two rotary encoders, placed at the end of the motors’ shafts. Encoder data were acquired and transmitted using a TCP/IP protocol to guarantee zero data loss during the transmission.

An Adapted NURBS Interpolator with a Switched Optimized Method of Feed-Rate Scheduling

With the increasing demand for processing precision in the manufacturing industry, feed-rate scheduling is a crucial component in achieving the processing quality of complex surfaces. A smooth feed-rate profile not only guarantees machining quality but also improves machining efficiency. Although the typical offline feed-rate scheduling method possesses good processing efficiency, it may not provide an optimal solution due to the NP-hard problem caused by the feed-rate scheduling of continuous curve segments, which easily results in excess kinetic limitations and feed-rate fluctuations in a real-time interpolation. Instead, the FIR (Finite Impulse Response) method is widely used to realize interpolation in real-time processing. However, the FIR method will filter out a large number of high-frequency signals, leading to a low-processing efficiency. Further, greater acceleration or deceleration is required to ensure the interpolation passes through the segment end at a predefined feed rate and the deceleration in the feed rate profile appears earlier, which allows the interpolation to easily exceed the kinetic limitation. At present, a simple offline or online method cannot realize the global optimization of the feed-rate profile and guarantee the machining efficiency. Moreover, the current feed-rate scheduling that considers both offline and online methods does not consider the situation that the call of offline data and online prediction data will lead to a decrease in the real-time performance of the CNC system. Further, real-time feed-rate scheduling data tend to dominate the whole interpolation process, thus reducing the effect of the offline feed-rate scheduling data. Hence, based on the tool path with C3 continuity (Cubic Continuously Differentiable), this paper first presents a basic interpolation unit relevant to the S-type interpolation feed-rate profile. Then, an offline local smooth strategy is proposed to smooth the feed-rate profile and reduce the exceeding of kinetic limitations and feed-rate fluctuations caused by frequent acceleration and deceleration. Further, a global online smoothing strategy based on the data generated by offline pre-interpolation is presented. What is more, FIR login and logout conditions are proposed to further smooth the feed-rate profile and improve the real-time performance and machining efficiency. The case study validates that the proposed method performs better in kinetic results compared with the typical offline and FIR methods in both the simulation experiment and actual machining experiments. Especially, in actual processing experiments, the proposed method obtains a 28% reduction in contour errors. Further, the proposed method compared with the FIR method obtains a 15% increase in machining efficiency but only a 4% decrease compared with the typical offline method.

Smoothing interpolation of five-axis tool path with less feedrate fluctuation and higher computation efficiency

Five-axis machine tools are widely used in aerospace, die and mold industries. Most of the time, the tool path is described by connecting a series of short linear segments. The tangency discontinuity of the linear segments leads to the discontinuity of the kinematic profiles, hence generating discontinuous interpolation points, and resulting in fluctuation of machine motion, which reduces the machining efficiency and causes excessive tracking and contouring errors. This paper aims to generate smooth interpolation points at servo sample intervals with the proposed G2 continuous five-axis corner rounding algorithm suitable for reducing feedrate fluctuation along the smoothed toolpath and improving machining efficiency, and the modified jerk-limited feedrate planning algorithm suitable for eliminating feedrate fluctuation within the feedrate planning units and improving interpolation computation efficiency. In this paper, the five-axis linear segments are first smoothed by curvature-optimized cubic Bezier splines, which reduces the number of the feedrate planning units along the smoothed toolpath and improves machining efficiency. Then, the jerk-limited feedrate planning algorithm is modified to ensure that the time interval of each different scheduled feedrate phase is an integer multiple of the axis position control loop closure time (i.e. servo sample interval), which eliminates the feedrate fluctuation within each feedrate planning unit and reduces the interpolation computation load of position command in each interpolation period (i.e. servo sample interval). At last, the effectiveness of the proposed algorithm is demonstrated with simulations and experiments following a five-axis, curved tool path on a CNC (Computer Numerical Control) machine tool.

A multi-parameter control method for maize threshing based on machine learning algorithm optimisation

A complex field environment and differences in maize density cause feed rate fluctuations in the combine. Traditional threshing methods do not adjust multiple operating parameters based on the feed rate, resulting in low threshing performance and high power consumption. A multi-parameter control method for maize threshing based on the feed rate was proposed to overcome these problems. Random forest, support vector machine, and multiple linear regression machine learning algorithms were used to estimate the threshing performance indices for different feed rates. The algorithm (random forest) was used to construct a control model for the rotor speed, threshing gap, and top cover guide vane angle. The performance of the control system was compared with a threshing system with constant parameters to verify the control model's accuracy. The results demonstrated that the threshing unit with the control system outperformed the one without. The broken grain rate and the power consumption were 2.18% and 36.22% lower, respectively, whereas the unthreshed grain rate was not significantly different. Random forest was more suitable than the support vector machine and multiple linear regression models for establishing a multi-parameter control model for maize threshing. The adjustment of multiple threshing parameters significantly improved the operating performance for different feed rates.

Five-axis toolpath interpolation method with kinematic corner smoothing and time synchronization

osMost toolpaths of high speed five-axis machining are composed of discrete tool positions and tool orientation vectors. Thus, toolpath smoothing is essential for increasing the machining efficiency and quality. Traditional smoothing methods face the difficulties of synchronization, computational load, feedrate fluctuation and overshoot due to the inefficient procedure of geometric smoothing followed by interpolation. To break these bottlenecks, a novel interpolation method for five-axis toolpaths with kinematic corner smoothing and time synchronization is proposed in this paper. Rather than smoothing the toolpaths geometrically, the translational velocity of tool position and the rotary angular velocity of tool orientation are directly scheduled in the workpiece coordinate system (WCS). Additionally, algorithms of time synchronization and interpolation are proposed to synchronize translation with rotation, and calculate interpolated tool positions and orientations efficiently. Simulation and experimental studies are conducted to validate the effectiveness and feasibility of the proposed method. Compared with a geometric corner smoothing method, the proposed method generates interpolated toolpaths perfectly respecting the geometric and kinematic constraints. Both the tangential and the axial kinematic profiles obtained by the proposed method are smooth and free of jitters at block junctions. Furthermore, the proposed method improves the machined surface quality, and reduces the computational time by over 50 % without solving optimization problems or scheduling recursively.

A novel tool path planning and feedrate scheduling algorithm for point to point linear and circular motions of CNC-milling machines

The manufacturing process on CNC machines is directed by the Numerical Control (NC) files which include the main tool paths as a combination of numerous instructions such as lines and arcs specified by the G01 and G02/G03 commands. In the literature, the lines and arcs generally are converted to a parametric curve based on the global tool path smoothing or local corner smoothing algorithms. However, the smoothing methods cause the deflection from the tool path in NC-File. In cases where the deflection from the main tool path is not acceptable, point-to-point motion is necessary instead of the smooth trajectory. This study presents a novel trajectory planning and feedrate scheduling algorithm for point-to-point control of linear and circular paths in the NC-File. The proposed method prevents the feedrate fluctuations at the junction points and generates the tool path without deflection from the NC-File. Also, the feedrate scheduling algorithm decrease the operating time of the machine significantly compared to traditional method of point-to-point motion. This study is verified on an industrial 3-Axis CNC-Milling machine by the operating of the roughing milling, drilling, slotting, and face milling process. The measurements of final product and real-time trajectory tracking results show that the proposed method is effective in industrial applications of CNC-Milling machines in which point-to-point motion is necessary.

NURBS Interpolator with Minimum Feedrate Fluctuation Based on Two-Level Parameter Compensation.

Feedrate plays a crucial role in determining the machining quality, tool life, and machining time. Thus, this research aimed to improve the accuracy of NURBS interpolator systems by minimizing feedrate fluctuations during CNC machining. Previous studies have proposed various methods to minimize these fluctuations. However, these methods often require complex calculations and are not suitable for real-time and high-precision machining applications. Given the sensitivity of the curvature-sensitive region to feedrate variations, this paper proposed a two-level parameter compensation method to eliminate the feedrate fluctuation. First, in order to address federate fluctuations in non-curvature sensitive areas with low computational costs, we employed the first-level parameter compensation (FLPC) using the Taylor series expansion method. This compensation allows us to achieve a chord trajectory for the new interpolation point that matches the original arc trajectory. Second, even in curvature-sensitive areas, feedrate fluctuations can still occur because of truncation errors in the first-level parameter compensation. To address this, we employed the Secant-based method for second-level parameter compensation (SLPC), which does not require derivative calculations and can regulate feedrate fluctuation within the fluctuation tolerance. Finally, we applied the proposed method to the simulation of butterfly-shaped NURBS curves. These simulations demonstrated that our method achieved maximum feedrate fluctuation rates below 0.01% with an average computational time of 360 us, which is sufficient for high-precision and real-time machining. Additionally, our method outperformed four other feedrate fluctuation elimination methods, highlighting its feasibility and effectiveness.

Smoothing and compressing algorithm of toolpath with complex contour in NC machining

The continuous short line segment processing method in CNC machining has the defect of frequent fluctuation of feedrate, which affects the quality and efficiency. For this problem, the local and global methods are common solutions. The local method is to insert transition curves at the corners to obtain a smooth toolpath, but this method does not essentially reduce the number of accelerations and decelerations, and it is difficult to insert transition curves between tiny straight sections. Although the global method can obtain smooth tool trajectories, the parametric interpolation method is not suitable for middle- and low-end CNC machine tools. The article proposes an adaptive tool path generation method with the original part contour as the research object and discretizes the NURBS curves into smooth trajectories consisting of large segments of straight lines and circular arcs. The article mimics the crawling characteristics of snakes and designs a double-headed snake algorithm based on the least squares method to implement this process. First, given the start and end points of the curve, two snakeheads search for the maximum linear and circular segments that satisfy the error constraint in the curve direction. The winner will then be selected as the candidate track segment through a competition mechanism. Finally, all trajectory segments are connected to obtain a smooth tool path. Experiments show that the toolpath data of the method in this paper are reduced by 75.56% compared with commercial CAM software when the contour error threshold is the same, and the contour error is reduced by more than one order of magnitude when the number of toolpath segments is equal. In addition, the method in this paper can obtain a smoother machining surface and more stable cutting force, thus achieving a win‒win situation of quality and efficiency.

Generalised NURBS interpolator with nonlinear feedrate scheduling and interpolation error compensation

Current nonuniform rational B-spline interpolating methods encounter problems pertaining to the interpolating error, feedrate profile, and machining efficiency; the optimal feedrate profile may not be obtained owing to the complex kinematics of multi-axis machine tools with hybrid structures. To address these problems and obtain an optimal feedrate profile with minimum machining time, this paper establishes a generalised representation for the nonlinear feedrate scheduling model, with static and dynamic feedrate constraints, through a numerical convex transformation. Based on the generalised model representation, a time-optimal feedrate profile can then be solved globally for both series- and parallel-structured mechanisms without problems caused by profile linear simplification or imprecise feedrate evolution. Considering the feedrate fluctuation problem caused by the difference between the theoretical arc length of the curve interpolation and the actual tool path, this paper proposes a novel co-lateral triangle deviation (CTD)-based parameter compensation interpolation algorithm. With the CTD calculation, which is related to the actual feed motion in real time, the proposed interpolation algorithm can determine the interpolating compensation distance for each interpolating segment and obtain the interpolation points using a second-order Taylor series expansion with a minimal interpolating error. Carried on a hybrid mechanism with complex kinematic constraints, the experimental and comparison results validated the effectiveness of the proposed method in feedrate scheduling and feedrate fluctuation reduction.

A C2 continuous trajectory planning method for 6-DOF rotational robot manipulators

PurposeThe purpose of this paper is to propose a smooth double-spline interpolation method for six-degree-of-freedom rotational robot manipulators, achieving the global C2 continuity of the robot trajectory.Design/methodology/approachThis paper presents a smooth double-spline interpolation method, achieving the global C2 continuity of the robot trajectory. The tool center positions and quaternion orientations are first fitted by a cubic B-spline curve and a quartic-polynomial-based quaternion spline curve, respectively. Then, a parameter synchronization model is proposed to realize the synchronous and smooth movement of the robot along the double spline curves. Finally, an extra u-s function is used to record the relationship between the B-spline parameter and its arc length parameter, which may reduce the feed rate fluctuation in interpolation. The seven segments jerk-limited feed rate profile is used to generate motion commands for algorithm validation.FindingsThe simulation and experimental results demonstrate that the proposed method is effective and can generate the global C2-continuity robot trajectory.Originality/valueThe main contributions of this paper are as follows: guarantee the C2 continuity of the position path and quaternion orientation path simultaneously; provide a parameter synchronization model to realize the synchronous and smooth movement of the robot along the double spline curves; and add an extra u-s function to realize arc length parameterization of the B-spline path, which may reduce the feed rate fluctuation in interpolation.

Development and performance evaluation of a guide vane inclination automatic control system for corn threshing unit based on feedrate monitoring

• Fluctuation in the feedrate requires proper top cover guide vane inclination control. • Proposed control system effectively decreases threshing damage. • Provides a theoretical basis to optimise threshing performance. The crop threshing process and the material movement speed are difficult to control with the fluctuation of feedrate in the longitudinal axial flow threshing unit, which results in the decline of threshing performance. We developed a corn threshing unit with an electric-hydraulic top cover guide vane inclination automatic control system based on the feedrate to overcome these problems. The optimal control model of feedrate and guide vane inclination was established for the control system: When the feedrates were 6, 8, and 10 kg/s, the optimal guide vane inclinations were 22°, 24°and 26°, respectively. Taking the constant guide vane inclination (20°) as the control group, the operational performance of the automatic control system of guide vane inclination was evaluated. The test bench results showed that the steady-state response time was 0.15 s, and the error was maintained within 0.27°, the control algorithm can meet the design requirement. Moreover, results demonstrated that the threshing unit with control system outperformed the one with the constant guide vane inclination, with the average torque range of the threshing rotor axis, average peak torque, the broken grain rate (BGR)、the unthreshed grain rate (UGR) and the grain-entrainment loss rate (EGR) decreasing by 17.39%, 19.03%, 1.63%, 0.06%, and 0.19%, respectively. Thus it can be seen that corresponding to different feedrates conditions, a proper guide vane inclination from the threshing unit with the developed automatic control system could obtain better working quality than the results with constant guide vane inclination.

Development and performance evaluation of the electric-hydraulic concave clearance control system based on maize feed rate monitoring

Complex field environments, diverse crop conditions, and varying feed rate fluctuations commonly result in a decline in the threshing performance and the clogging of the threshing cylinder for maize harvesters. In order to overcome these problems, an electric-hydraulic concave clearance automatic control system for the threshing unit was developed based on the maize feed rate monitoring, which can automatically realize the best match between the concave clearance and diverse feed rates during harvesting. The threshing performance of the electric-hydraulic control system was evaluated for varying and uneven maize feed rate fluctuations, such as the feed rate increased (6-8-10 kg/s), the feed rate decreased after an increase (6-10-8 kg/s, 8-10-6 kg/s), the feed rate increased after a decrease (8-6-10 kg/s, 10-6-8 kg/s), and the feed rate decreased (10-8-6 kg/s). In particular, the threshing rotor shaft peak torque, the range of threshing rotor shaft torque, the rate of broken grains (BGR), and the rate of unthreshed grains (UGR) with and without the electric-hydraulic control system were tested. Treatments with the electric-hydraulic control system were adjustable concave clearance with the value of 45 mm, 50 mm, and 55 mm. Treatments without the electric-hydraulic control system were constant concave clearance (50 mm). Results demonstrate that the threshing unit with the electric-hydraulic control system outperformed the one without the electric-hydraulic control system, with threshing rotor peak torque, the range of threshing rotor axis torque, the BGR, and the UGR decreasing by 18.38%, 38.27%, 2.08%, and 0.10%, respectively. Moreover, the rate of broken grains was lower than 5.00%, better than the national standard. Thus, the feed rate fluctuations and timely adjustment of the concave clearance were able to avoid blocking the rotor and improve the threshing performance compared to the constant concave clearance. Keywords: maize harvester, threshing cylinder, feed rate, concave clearance, electro-hydraulic control system DOI: 10.25165/j.ijabe.20221502.6369 Citation: Fan C L, Zhang D X, Yang L, Cui T, He X T, Zhao H H. Development and performance evaluation of the electric-hydraulic concave clearance control system based on maize feed rate monitoring. Int J Agric & Biol Eng, 2022; 15(2): 156–164.

B-Spline-Based Corner Smoothing Method to Decrease the Maximum Curvature of the Transition Curve

Abstract Toolpath represented by linear segments leads to tangency discontinuity between blocks, which results in fluctuation of feedrate and reduction of machining efficiency and quality. To eliminate these unwanted external factors, optimal corner smoothing operation is essential for computerized numerical control systems to achieve a smooth toolpath. This work proposes a corner smoothing approach by generating a B-spline transition curve with seven control points. By adjusting the position of the control points, the resulting transition curve is not limited to smooth the corner in the convex side of the corner, but shuttles back and forth between the convex and concave sides to decrease the maximum curvature, while respecting the given error tolerance. The approximation errors in convex and concave sides can be analytically calculated. Experimental results demonstrate the effectiveness of the proposed method on machining efficiency improvement.

A newly developed corner smoothing methodology based on clothoid splines for high speed machine tools

Continuous linear commands are widely executed in computer numerical control (CNC) machining. The tangential discontinuity at the junction of consecutive segments restricts the machining efficiency and deteriorates the surface quality. Corners of linear segments have been successfully blended by inserting parametric splines. There still exists challenges when the common methods are employed in the line-segment commands due to part of the following restrictions: (1) the stringent computation for iteratively calculating the arc-length; (2) the unwanted feedrate fluctuation; (3) the oversize contour deviation for separately completing curve fitting and velocity planning. A novel smoothing method based on a clothoid pair to synchronously accomplish planning of geometry blending and speed scheduling is proposed, the spline parameter of which is arc-length-parameterized. The arc-length, curvature extreme, and geometric shape of the transition curve are analytically expressed by the transition length. On these bases, the transition curve and the velocity profile are concurrently constructed based on the predefined approximation error, the reachable velocity, and normal kinematic constraints in the look-ahead stage. Then, a real-time interpolation scheduling is developed to overcome the crossing difficulties between the linear and parametric segments. Compared with existing methods, the proposed method can analytically calculate the length of transition curves for the arc-length-parameterized expression form. Furthermore, the feedrate fluctuation is eliminated in the fine interpolation. Moreover, the overlarge contour derivation produced by corner smoothing is significantly avoided. It is friendlier to the CNC system for the on-line executing smooth motion since more computing resources can be released to handle other tasks, smoother motion can be achieved and higher contour accuracy can be obtained. The experimental results also demonstrate its practicability and reliability.

A NURBS curve interpolator with small feedrate fluctuation based on arc length prediction and correction

Non-uniform rational B-spline (NURBS) curve has been widely used in manufacturing systems. A good interpolator can help the system to improve the contour accuracy and get smooth dynamics performance, but it is hard to get a balance between the interpolation performance and computational load. As the derivative and curvature of NURBS curves used in manufacturing systems are high-order continuous, it is possible to predict a desired interpolation arc length based on the relationship of historical feed chord length and its corresponding arc length in one interpolation cycle. Therefore, this paper proposes a novel interpolation method, which consists three stages. Firstly, a NURBS curve is split into several high-order continuous segments based on its degrees and control points. Secondly, a prediction model based on Newton’s divided differences interpolation equation is derived from the relationship of interpolated chord length and its corresponding arc length, so that the target arc length of the next interpolation cycle can be predicted. Finally, target parameter u of every interpolation cycle is calculated with Taylor’s expansion, whose values are corrected by iteration, and then the position of the target interpolation point can be achieved. Performance of the proposed algorithm is tested and compared with other methods, and the simulation results show the proposed method can achieve smaller velocity fluctuation with low computational load.

A Novel Local Smoothing Method for Five-Axis Machining With Time-Synchronization Feedrate Scheduling

This paper presents an analytical continuous smoothing method for the five-axis toolpath by simultaneously scheduling the tool position and tool orientation trajectories. In order to ensure the high-order continuous, the peak-controlled jerk and arclength-parameterized property, a novel curve “airthoid” is proposed for the first time. The biairthoid is involved to smooth the corners of the tool position in the workpiece coordinate system (WCS) and the corners of the tool orientation in the machine coordinate system (MCS), the geometries of which are analytically determined by the user-defined deviation errors. A time synchronization strategy is proposed to extend the duration of the predetermined cubic acceleration profile to a specified time. With the kinematic constraints of the tool position and the tool orientation, the transitional and rotational trajectories are analytically synchronized by sharing the same motion time. To comply with the constraints of the linear feed drives, an optimization strategy is conducted by adjusting the kinematics of the tool position. By doing so, the approximation errors of the tool position and tool orientation in the WCS are strictly satisfied. The analytical arclength expression of the smoothing curves is more suitable for the on-line interpolation. Due to the arclength-parametrized transition curve, the feedrate fluctuation is eliminated. With the proposed time synchronization strategy, the physical limits of the feed drives are all respected. Moreover, the high-order continuous airthoid makes the motion more smoothing-going. Simulations and experiments verify the effectiveness of the proposed algorithm.

NURBS Interpolator with Adaptive Smooth Feedrate Scheduling and Minimal Feedrate Fluctuation

For an effective NURBS interpolator, generating the smooth feedrate profile and minimizing the feedrate fluctuation are two main tasks, which have a great influence on the machining quality. In this paper, a NURBS interpolator is proposed to solve the problem with offline-online two stages, which contain curving splitting, feedrate scheduling, and real-time interpolation modules. At first, in offline module, the limited feedrate of the curve is calculated with consideration of geometric characteristics and dynamic constraints, and the curve is split into several segments according to the key regions. Thus the necessary information is stored. Then the feedrate scheduling algorithm is applied with constant feedrate scheduling method at each segment, where tangential acceleration and jerk are taken into consideration to generate an adaptive smooth S-type feedrate profile. Moreover, in real-time interpolation module, an average feedrate calculation method is proposed to determine the sampling step size. And to minimize the feedrate fluctuation, a double interpolation algorithm based on the cosine theorem is presented, which use the second-order Taylor’s expansion method two times in one interpolation period. Finally, simulations and experiments are conducted to verify the performance of the proposed interpolator.

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

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.

An Improved B-Spline Fitting Method with Arc-Length Parameterization, G2-continuous Blending, and Quality Refinement

In computer numerical control machining, tool paths are typically represented with piecewise linear segments (GO1s), which lack G1 and G2 continuity and cause fluctuation of feed rate and acceleration. To improve the continuity of tool paths, some method, e.g., B-spline tool path fitting, is preferred. However, none of the currently B-spline fitting method satisfies all tool-path fitting requirements of arc-length parameterization, G2 continuity and smoothness across the whole path. The aim of this research is to derive an industrial-strength method for B-spline tool path fitting, satisfying all of the named requirements. The improvements cover the arc-length based parameterization, G2 continuity as well as numerically measure to improve the quality of B-spline tool paths. To achieve arc-length parameterization, a preliminary fitting is added to evaluate arc-length and guide the parameterization (i.e., knot vector assignment); for joint G2-continuity, a five-control-point B-spline which satisfies chord error and shape-preserving constraints is constructed to blend adjacent B-splines; and a method based on the Hausdorff distance between the fitted B-splines and the original polylines is used to evaluate the fitting quality, and an additional refinement procedure is employed when the evaluation finds any disqualification. Simulation and experiment results prove that the proposed methods are effective.