Corner Smoothing Research Articles

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.

Kinematic Tool-Path Smoothing for 6-Axis Industrial Machining Robots

The demands for machining by industrial robots have been increasing owing to their low installation cost and high flexibility. A novel trajectory generation algorithm for high-speed and high-accuracy machining by industrial robots is proposed in this paper. Linear interpolation in the workspace and smooth trajectory generation at the corners are important in industrial machining robots. Because industrial robots are composed of rotational joints, the joint space has a nonlinear relationship with the workspace. Therefore, linear interpolation in the joint space, which has been widely used in conventional machine tools, does not guarantee linear interpolation in the actual machining workspace. This results in the degradation of the machining surface. The proposed trajectory generation algorithm based on the decoupled approach can achieve linear interpolation in the workspace by separating the position commands into Cartesian coordinates and the orientation commands into spherical coordinates. In addition, a novel corner smoothing method that generates a smooth and continuous trajectory from discrete commands is proposed in this paper. The proposed kinematic local corner smoothing generates a smooth trajectory by using a 3-segmented constant jerk profile at the corners in the joint space. The sharp corners can thereby be replaced by smooth curves. The resulting cornering error is controlled by varying the cornering duration. The simulation results demonstrate the effectiveness of the proposed kinematic smoothing algorithm in achieving linear tool motion in straight sections and in generating smooth trajectories at corner sections within the user-defined tolerance.

Special Issue on New Technologies for Robotic Manipulators and Their Industrial Applications

The industrial robot is more precisely an “automatically controlled, reprogrammable, multipurpose manipulator, programmable in three or more axes, which can be either fixed in place or mobile” (ISO 8373:2012). According to the International Federation of Robotics, by 2018, more than 400,000 new units were being installed annually, and the global average robot density in the manufacturing industry was 99 robots per 10,000 employees. More than 30% of all installed robots were in the automotive industry, the biggest customer for robots. Research on measuring and calibrating, modeling, programming and controlling, and integrating systems has been conducted to give robotic manipulators a wider variety of industrial applications. This special issue covers technical and academic efforts related to new technologies that improve the accuracy and facilitate the implementation of robotic manipulators in industrial applications. The first paper, by Ibaraki et al., outlines technical issues and future research directions for the implementation of model-based numerical compensation schemes for industrial robots. The random forest method is used by Kato et al. to construct a calibration model for positioning errors and identify industrial robots’ positioning errors. A procedure for the quasi-static compliance calibration of serial articulated industrial manipulators is proposed by Theissen et al. A review of the kinematic modeling theory and a derived algorithm to identify error sources for a six-axis industrial robot are presented by Alam et al. Nagao et al. derive a forward kinematics model and identify the kinematics parameters for the calibration of a robot-type machine tool. A novel trajectory generation algorithm, including a corner smoothing method for high-speed and high-accuracy machining by industrial robots, is proposed by Tajima et al. Sato et al. study the vibration characteristics of an industrial robot and derive a mathematical model that represents the dynamic behavior of the system. In the context of smart manufacturing, a multilayer quality inspection framework including a measurement instrument and a robot manipulator is introduced by Azamfirei et al. To support mass customization and the development of reconfigurable manufacturing systems, Inoue et al. propose an autonomous mobile robotic manipulator. Yonemoto and Suwa present an adaptive manipulation procedure to establish an automated scheduling technique that flexibly responds to unforeseen events, such as machine failures. Sasatake et al. introduce a learning system that is based on a method for calculating the similarity between tools, and they test it on a robot system for doing housework. Finally, for better knowledge of the key challenges that manufacturers experience in implementing collaborative industrial robots, an industrial survey is conducted by Andersson et al. The editors sincerely appreciate the contributions of all the authors as well as the work of the reviewers. We are confident that this special issue will further encourage research and engineering work to increase our understanding and knowledge of robotic manipulators and their industrial applications.

A corner smoothing algorithm using Trajectory Pattern Method (TPM) for high-speed and high-quality machining

In micro-line segments machining, transition curves with high harmonic components are more prone to causing vibration issues in the feed drive system, which affects machining efficiency and quality severely. To construct low harmonic trajectories, this paper proposes a corner smoothing algorithm that uses the Trajectory Pattern Method (TPM). The transition curve construction and axial motion scheduling are performed with a specified fundamental frequency in one step, which reduces the smoothing process time and avoids excitation of natural modes of vibration of the system. The synthesized trajectories and axial kinematic profiles are all smooth and only contain the selected fundamental frequency and its first two odd harmonics, which minimizes the number of high harmonic components in the required actuation forces/torques and avoids excitation of the system modes of vibration. Linear programming is used to synthesize the trajectories. The proposed algorithm is shown to achieve near time-optimal trajectories. The provided experimental analysis and comparisons demonstrate that the proposed algorithm achieves smooth axial kinematic profiles with low harmonic contents, which would improve machining efficiency and quality.

Stress minimization for lattice structures. Part I: Micro-structure design.

Lattice structures are periodic porous bodies which are becoming popular since they are a good compromise between rigidity and weight and can be built by additive manufacturing techniques. Their optimization has recently attracted some attention, based on the homogenization method, mostly for compliance minimization. The goal of our two-part work is to extend lattice optimization to stress minimization problems two-dimensionally. The present first part is devoted to the choice of a parametrized periodicity cell that will be used for structural optimization in the second part of our work. In order to avoid stress concentration, we propose a square cell microstructure with a super-ellipsoidal hole instead of the standard rectangular hole often used for compliance minimization. This type of cell is parametrized two-dimensionally by one orientation angle, two semi-axis and a corner smoothing parameter. We first analyse their influence on the stress amplification factor by performing some numerical experiments. Second, we compute the optimal corner smoothing parameter for each possible microstructure and macroscopic stress. Then, we average (with specific weights) the optimal smoothing exponent with respect to the macroscopic stress. Finally, to validate the results, we compare our optimal super-ellipsoidal hole with the Vigdergauz microstructure which is known to be optimal for stress minimization in some special cases. This article is part of the theme issue 'Topics in mathematical design of complex materials'.

Asymmetrical pythagorean-hodograph (PH) spline-based C3 continuous corner smoothing algorithm for five-axis tool paths with short segments

• A real-time C 3 continuous corner smoothing algorithm is proposed for five-axis short linear segments. • The proposed method can achieve lower curvature based on asymmetrical PH splines. • Smoothing errors around the corners are bounded under given constraints. • The arc-length can be analytically calculated to achieve real-time interpolation. • Experiments verify that much lower curvature and higher machining efficiency can be realized. Local corner smoothing is widely used for the G01 motion commands to achieve the motion continuity of transition corners. Most splines constructed by the existing local corner smoothing methods oriented for five-axis machining mainly must be connected by linear segments. However, with respect to the five axis machining of the corners with short line segments, these kinds of methods need that both ends of the constructed splines usually must be shrunk with the same proportion to avoid the overlapping of adjacent splines. This will lead to relatively large curvatures of the obtained trajectories together with relatively low machining efficiency. This article proposes a new real-time corner smoothing algorithm for five-axis machine tools by constructing the C 3 continuous asymmetrical PH splines. The typical characteristic lies in that the control points related to the two ends of the PH spline can be independently adjusted through reasonably introducing two more adjustable variables, which are realized by analytically calculating and then adding two additional control points into the corresponding control polygon. As a result, the PH spline poses the property of the asymmetry about the angular bisector. Based on this, the short linear commands of tool tip and tool orientation can be smoothed only through inserting different lengths of asymmetrical PH splines that can be connected directly. The motion variance of the smoothed tool orientation related to the tool tip displacement is analytically proved to be C 3 continuous, and thus, the synchronization between tool tip and tool orientation is guaranteed. The outstanding advantage of the algorithm for short line segments is twofold. The first is that the performance of the drives can be more fully utilized to reduce cycle time since the curvature around corners are reduced by adopting the asymmetrical profile, and the second is that the synchronization procedure is greatly simplified since the algorithm eliminates the requirement to convert the intermediate line segments into a spline between the adjacent connected PH splines. Both simulation and the experimental results demonstrate that the proposed algorithm can realize a higher machining efficiency.

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.

Feedrate blending method for five-axis linear tool path under geometric and kinematic constraints

To tackle the bottlenecks of the linear interpolation and two-step geometry-based corner smoothing methods in five-axis machining, an analytical one-step corner smoothing method based on feedrate blending is proposed in this paper. In the proposed algorithm, the maximum kinematics adjustments scheme together with the variable acceleration and jerk optimization algorithms are first adopted to determine the dynamic performances along the linear segments. Meanwhile, the smooth feedrate profiles of the tool tip position in the workpiece coordinate system (WCS) and tool orientation in the machine coordinate system (MCS) are generated by the tool tip mastered jerk-limited feedrate scheduling scheme, with which the tool tip feedrate and angular feedrate in the WCS, together with the translational and rotatory feedrate in the MCS are well bounded. Then, the blending time is calculated based on the specified tool tip error and tool orientation error in the WCS, and the feedrate profiles are blended to generate the smooth trajectories under axial kinematic constraints along the whole paths in both the WCS and MCS. Since the proposed method can generate the smooth corner trajectory by directly blending the feedrate profiles, it eliminates the construction and interpolation process of parametric splines in two-step corner smoothing algorithms and shows better tracking and real-time performances. Finally, simulation and experimental results validate the effectiveness and feasibility of the proposed method.

A Trade-off Between Energy Saving and Cycle Time Reduction by Pareto Optimal Corner Smoothing in Industrial Feed Drive Systems

Apart from improving motion accuracy, increasing productivity while reducing computer numerical control machine tools' carbon footprint is vital in the manufacturing industry. Several studies have proposed corner smoothing methods that improve cycle time for piecewise linear paths by exploiting axial limits to achieve time-optimal trajectories. Energy-saving is not considered an objective in these methods. This paper presents a method of generating Pareto optimal corner smoothing trajectories that trade-off the contradicting objectives of minimizing cycle time and energy consumption. The trajectories along linear paths and smoothed corners are respectively described using jerk limited acceleration profiles and kinematic corner smoothing methods. An energy consumption model of an industrial two-axis feed drive system is identified and used in solving the bi-objective optimization problem. From the resulting Pareto frontier, the best trade-off trajectory is selected as the one that minimizes both objectives. The proposed method's effectiveness is vindicated experimentally, where the best trade-off trajectory achieves respectively 65.87 % and 60.44 % of the time and energy-saving potentials.

Pareto Optimization of Cycle Time and Motion Accuracy in Trajectory Planning for Industrial Feed Drive Systems

Manufacturing industries aim to improve product quantity and quality. These trade-off objectives are typically manifested as cycle time reduction and motion accuracy improvement. Corner smoothing approaches that reduce the cycle time of piecewise linear trajectories are proposed in the literature. This study tackles the two objectives by Pareto-optimal corner smoothing with constraints imposed as kinematic limits, continuity conditions and user-specified cornering tolerance. Linear and cornering motions along a contour are respectively described by jerk-limited acceleration profiles and a modified kinematic corner smoothing with interrupted acceleration (KCSIA) approach. A Pareto frontier is generated by the divide and conquer algorithm, where the solution nearest to the utopia point is selected as the best trade-off solution. The effectiveness of the proposed method is validated through experiments, where the best trade-off solution reduces the maximum and average contouring errors by 47.53% and 25.40% while it increases cycle time by 2.53% compared to KCSIA.

Real-Time Local Optimal Bézier Corner Smoothing for CNC Machine Tools

A straight line or G01 is the most common path in computer numerical control (CNC) contour machining. However, non-smooth corners introduce discontinuous movement along each axis during machining. The commonly used strategy is to redesign a corner into a smooth corner path. Therefore, optimal corner smoothing should simultaneously find an optimal corner path and a corner speed that satisfied the axis motion limits. To achieve this goal, we use particle swarm optimization (PSO) to find the optimal Bezier curve and the optimal corner speed, then convert the axis constraints into polynomial inequalities and use fitness selection to deal with axis acceleration and jerk limits. For real-time operation, we establish a look-up table with the optimal corner speed and corner path models for use by a CNC controller. In the experiment, the YCM NXV560A vertical machining center was used to conduct a real machining test on a Taiwan contour, and the feasibility of the controller to generate the optimal corner strategy for various linear paths was verified.

Local corner smoothing of linear toolpath based on B-spline fitting and its piecewise interpolation

Line segment toolpaths are commonly utilized in machining of complex parts with curved surfaces. However, the first-order discontinuity of the linear toolpath results in frequent variation of the feedrate, which is bad to the machining quality. To smooth the sharp corners of linear toolpath thus avoiding above problem, this paper presents a local corner smoothing approach. Compared with existing methods, the smoothed toolpath using the proposed the is one integral B-spline, instead of alternative straight-line segments and spline segments, which make the continuity more easily to ensured. Additionally, the integral long B-spline is interpolated through a piecewise feedrate scheduling method, thus the computational burden can be released. Effectiveness of the presented method in corner smoothing and kinematic constraining is demonstrated by illustration test.

Local asymmetrical corner trajectory smoothing with bidirectional planning and adjusting algorithm for CNC machining

Abstract The linear motion command (G01) is a widely used command format in CNC machining. However, the tangential direction discontinuity at the corner junction will cause velocity fluctuations and excite the structural vibration of machine tools. Corner smoothing methods with controlled tolerance are used to obtain continuous smooth motion. Typically, most methods generate a symmetrical cornering trajectory around the angle bisector, and the trajectory generally decelerates first and then accelerates, which limits the velocity increase. In this paper, a novel local asymmetrical corner trajectory smoothing method is proposed, which can realize accelerated/decelerated cornering transition. The proposed method can obtain the analytical solution in one step, which is different from two-step geometric-based corner path smoothing, and can generate the same cornering trajectory in back-and-forth parallel toolpath. In addition, this paper proposes a bidirectional planning and adjusting algorithm for the situation where smoothed cornering paths are close to each other or even overlap. The algorithm can generate a jerk-limited trajectory by coordinating the cornering error of adjacent corners, while respecting the user-specified tolerance. Experimental results demonstrate the effectiveness of the proposed method in contour accuracy and cycle time for CNC machining along short-segmented toolpath.

An analytical G3 continuous corner smoothing method with adaptive constraints adjustments for five-axis machine tool

In this paper, an analytical local corner smoothing method is proposed with two specially designed quartic curves to generate smooth motion trajectories of five-axis linear segments. One symmetrical quartic B-spline curve is used to smooth the tool tip position in workpiece coordinate system (WCS); meanwhile, another asymmetrical quartic B-spline curve is used to smooth the tool orientation in machine coordinate system (MCS). With the proposed method, not only the transition error of tool tip position and the smoothing error of the tool orientation are mathematically constrained in WCS, but the G3 continuity of the tool tip position and tool orientation are guaranteed along the entire toolpath. Besides, the maximum possible machining error of tool tip position caused by transition error and chord error is also controlled analytically by transition error adjustment. What’s more, in feedrate scheduling, the feedrate of tool tip is moderately adjusted by smooth feedrate profile, which can limit the maximum feedrate of tool orientation and achieve continuous axial feedrate and acceleration. Finally, the effectiveness and feasibility of the proposed method have been validated via simulations.

Corner accuracy in direct ink writing with support material

3D printing methods which enable control over the position and orientation of embedded particles have promising applications in cell patterning and composite scaffolds. Extrusion-based additive manufacturing techniques such as fused deposition modeling and direct ink writing can experience particle patterning defects at corners which could hinder cell survival at corners and create unintended property gradients. Here, we propose models which predict the behavior of deposited lines at corners for moderate viscosity inks which are impacted by both capillarity and viscous dissipation. Using direct ink writing with acoustophoresis and a Carbopol-based support gel, we write polygons out of dental resin-based composite inks containing a narrow distribution of microparticles at the center of the filament. A Laplace pressure differential between the inner and outer surfaces of the corner drives corner smoothing, wherein the inner radius of the corner increases. Double deposition, or printing on the same area twice, drives corner swelling, wherein excess ink is diverted to the outer edge of the corner. Fast turns at corners produce ringing, wherein vibrations in the stage manifest in oscillations in the print path. Swelling and ringing effects are apparent in the particle distributions at corners immediately after deposition, while smoothing effects are apparent after the printed structure has had time to relax. When the nozzle returns to write a neighboring line, it imposes shear stresses which mitigate inconsistencies in microstructure at corners by erasing defects which appeared during relaxation. Using a support bath instead of layer-by-layer support suppresses microstructural corner defects.

A corner smoothing method with feedrate blending for linear segments under geometric and kinematic constraints

To eliminate the disadvantages of linear interpolation and geometry-based smoothing methods, the one-step corner smoothing method with feedrate blending algorithm for linear segments is proposed in this article. In the proposed method, the variable acceleration and jerk optimization method will be first adopted to determine the dynamic performance of each linear segment for feedrate scheduling, which takes the different axial limits into consideration. Then, the feedrate scheduling method based on the S-type acceleration and deceleration algorithm is implemented to generate smooth feedrate profiles. Moreover, the feedrate blending algorithm is adopted to calculate the corner time analytically within the specified corner error and thus to generate the smooth trajectory without inserting a smooth curve at adjacent linear segments. With the presented one-step corner smoothing method, a smoother and more accurate trajectory could be generated with faster machining time. The results of simulations and machining experiments demonstrate the effectiveness of the proposed method.

A Real-Time C3 Continuous Tool Path Smoothing and Interpolation Algorithm for Five-Axis Machine Tools

Abstract Local corner smoothing method is commonly adopted to smooth linear (G01) tool path segments in computer numerical control (CNC) machining to realize continuous motion at transition corners. However, because of the highly non-linear relation between the arc-length and the spline parameter, and the challenge to synchronize the tool tip position and tool orientation, real-time and high-order continuous five-axis tool path smoothing and interpolation algorithms have not been well studied. This paper proposes a real-time C3 continuous corner smoothing and interpolation algorithm for five-axis machine tools. The transition corners of the tool tip position and tool orientation are analytically smoothed in the workpiece coordinate system (WCS) and the machine coordinate system (MCS) by C3 continuous PH splines, respectively. The maximum deviation errors of the smoothed tool tip position and the tool orientation are both constrained in the WCS. An analytical synchronization algorithm is developed to guarantee the motion variance of the smoothed tool orientation related to the tool tip displacement is also C3 continuous. The corresponding real-time interpolation method is developed with a continuous and peak-constrained jerk. Simulation results verify that the maximum deviation errors caused by the tool path smoothing algorithm are constrained, and continuous acceleration and jerk of each axis are achieved along the entire tool path. Comparisons demonstrate that the proposed algorithms achieve lower amplitude and variance of acceleration and jerk when compared with existing methods. Experiments show that the proposed five-axis corner smoothing and interpolation algorithms are serially executed in real-time with 0.5-ms cycle.

Space corner smoothing of CNC machine tools through developing 3D general clothoid

Tool paths defined by G01/G02/G03 commands need to be smoothed to eliminate the discontinuities of velocity, acceleration and jerk at the junction points. Because of inherent problems of curve fillets, traditional corner smoothing strategies are limited to the smoothing of corners in plane. This article presents a method to smooth the space corners through blending the position, tangent, curvature and sharpness of the adjacent trajectory segments based on 3D general clothoid splines, which are analytically developed by proposing 3D clothoid with G3 continuity. The 3D general clothoid realizes extending the traditional clothoid from 2-dimension to 3-dimension, and reserves some good properties of the traditional clothoid, i.e. the curve length parameterization and the analytically expressed curvature. It can also achieve higher degree of continuity compared to the traditional 2D clothoid. Based on the proposed 3D general clothoid, a corner smoothing algorithm, which is suitable for the smoothing of any planar or space line-line, line-arc or arc-arc pairs with G3 continuity, is proposed. At the same time, a smoothing-error-constraining-algorithm is developed to constrain the smoothing error within the tolerance. Simulation and experimental results, which are obtained from the smoothing of trajectories containing space corners, trajectories containing planar corners with G02/G03 commands, and trajectories containing planar corners with only G01 commands, demonstrate that the proposed corner smoothing algorithm can effectively increase the machining quality and efficiency.

Trajectory smoothing method using reinforcement learning for computer numerical control machine tools

Abstract Tool-path codes output by computer-aided manufacturing software for high-speed machining are composed of discontinuous G01 line segments. The discontinuity of these tool movements causes computer numerical control (CNC) inefficiency. To achieve high-speed continuous motion, corner smoothing algorithms based on pre-planning methods are widely used. However, it is difficult to optimize smoothing trajectories in real-time systems. To obtain smooth trajectories efficiently, this paper proposes a neural network-based direct trajectory smoothing method. An intelligent neural network agent outputs servo commands directly based on the current tool path and running state in every cycle. To achieve direct control, motion feature and reward models were built, and reinforcement learning was used to train the neural network parameters without additional experimental data. The proposed method provides higher cutting efficiency than the local and global smoothing algorithms. Given its simple structure and low computational demands, it can easily be applied to real-time CNC systems.

Spline-Based Trajectory Generation for CNC Machines

Manufacturing of workpieces with computerized numerical control machines requires computing machine tool trajectories that fast and accurately track the desired workpiece contour. This paper presents a novel B-spline trajectory generation method for machine tools. The method solves an optimal control problem to minimize the motion time of the tool, while taking into account the velocity, acceleration, and jerk limits of the tool axes. Furthermore, it directly includes the allowed workpiece tolerance, by constraining the trajectory to lie inside a tube around the nominal geometry contour. This allows exploring the tradeoff between accuracy and productivity, while computing near-optimal trajectories. The presented method creates fluent connections between segments that build up the contour by simultaneously optimizing trajectories for multiple segments. On the other hand, limiting the amount of simultaneously optimized segments and using an efficient problem formulation keeps the computation time acceptable. The trajectory generation method is validated in simulation by comparison with industrial benchmarks, showing a reduction in machining time of more than ${\text{10}}\%$ . The comparison to a state-of-the-art corner smoothing approach shows that the presented method obtains similar or slightly faster trajectories, at a computation time that is up to 45 times lower. In addition, the method is validated experimentally on a three-axis micromilling machine. To easily generate trajectories for different workpieces and machines, the method is included in a user-friendly open-source software toolbox.