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... this paper, a methodology for generation of curvature- continuous transition curve and thereafter jerk-limited feedrate profile for the linear tool path in real-time is proposed. Table 1 presents the relationship between the proposed methodology and those available in the literatures. The remainder of this paper is organized as follows. ...
Citations
... Thus, the interpolation parameters shall be limited within the requirements of the machine tool. (20) Considering Equations (19) and (20), the maximum feed rate of each segment v max can be expressed as: ...
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.
... Various toolpath smoothing methods have been proposed to improve the continuity of consecutive linear segments, including global smoothing [6][7][8] and local smoothing [9][10][11]. Global smoothing uses an entire spline to interpolate or approximate the cutter locations (CLs). ...
G01 commands are the most commonly used toolpath format in Computer Numerical Control (CNC) machining. Due to the discontinuity between adjacent linear segments, toolpath smoothing is necessary to improve machining efficiency and machined surface quality. This paper proposes an analytical C3 continuous local toolpath smoothing and synchronization method for a five-axis hybrid robot. Position and orientation smoothing are decoupled, with quintic B-spline segments replacing the original linear segments. The curvatures of the toolpath are reduced under constrained smoothing errors by weaving transition splines along the linear segments instead of confining them within the corners. A synchronization method based on arc length parameterization tailored to the smoothing method is also proposed. Tool tip position and tool axis orientation are synchronized using another monotonic, C3 continuous spline, which achieves smoother joint motion along the toolpath. For the hybrid robot with highly nonlinear kinematics, improved results in feedrate scheduling considering joint constraints can be realized. Simulations and experiments validate the effectiveness of the proposed method.
... This is achieved by computing the parameter u i+1 , which corresponds to the next position coordinate of the interpolation cycle, based on the current position's corresponding parameter u i . Derivatives of the parameter u with respect to time t can be derived, and by expanding these first-order derivatives into a second-order Taylor series, a second-order Taylor direct interpolation formula [35] based on feed rate can be obtained: ...
NURBS curve parameter interpolation is extensively employed in precision trajectory tasks for industrial robots due to its smoother performance compared to traditional linear or circular interpolation methods. The trajectory planning systems for industrial robots necessitate four essential functional modules: first, the spline curve discretization technique ensuring chord error compliance; second, the contour scanning technique for determining the maximum feasible feed rate for multi-constraint and multi-segment paths; third, the technique for achieving a smooth feed rate profile; and fourth, the continuous curve parameter interpolation technique. Therefore, this paper proposes a high-precision planar NURBS interpolation system for industrial robots. Firstly, a segmentation method for NURBS curves based on a closed-loop chord error constraint is proposed, which segments the original global NURBS curve into a collection of Bezier curves that strictly meet the chord error constraint. Secondly, a bidirectional scanning technique is presented to meet the joint space constraint, establishing an analytical mapping between the tool tip kinematic constraint and the joint kinematic constraint. Then, based on the traditional S-shaped feed rate profile, an adaptive algorithm with a displacement constraint is introduced, considering the real-time speed adjustment requirements of robots. Finally, a compensation interpolation strategy based on arc length parameterization is adopted to solve the accumulated error problem in parameter interpolation. The effectiveness of and potential for enhancing the quality of planar machining of the proposed planar NURBS interpolation system for industrial robots are validated through simulations and experiments. The results demonstrate the system’s applicability and accuracy, and its ability to improve planar machining quality.
... The geometric path planning, indeed, is more application specific and thus less amenable to be treated in a standard way. Different strategies and ideas, a review of which can be found in [24], can be specialized for use in diverse fields such as race driver modeling [25], routing of mobile robots for agricultural applications [26], CNC milling [27,28], and fixed-based robots operating in cluttered environments [29]. ...
The ability to predict the maximal performance of an industrial robot executing non-deterministic tasks can improve process productivity through time-based planning and scheduling strategies. These strategies require the configuration and the comparison of a large number of tasks in real time for making a decision; therefore, an efficient task execution time estimation method is required. In this work, we propose the use of neural network models to approximate the task time function of a generic multi-DOF robot; the models are trained using data obtained from sophisticated motion planning algorithms that optimize the shape of the trajectory and the executed motion law, taking into account the kinematic and dynamic model of the robot. For scheduling purposes, we propose to evaluate only the neural network models, thus confining the online use of the motion planning software to the full definition of the actually scheduled task. The proposed neural network model presents a uniform interface and an implementation procedure that is easily adaptable to generic robots and tasks. The paper’s results show that the models are accurate and more efficient than the full planning pipeline, having evaluation times compatible with real-time process optimization.
... Owing to the advantages in guaranteeing the high-order continuity and controlling accurately the approximation error, the local corner smoothing methods using transition curves, such as Bézier spline Ernesto & Farouki, 2012;Sencer et al., 2015), B-spline (Bi et al., 2019;S. Sun et al., 2016;Zhao et al., 2013), non-uniform rational B-spline (NURBS) (Du & Wang, 2022;Duan & Okwudire, 2016;Lin et al., 2007), Pythagorean Hodograph (PH) curve (Q. Hu et al., 2019;X. ...
Corner rounding methods have been widely developed to pursue the smooth motions of machine tools. However, most corner rounding methods, which adopt the double inscribed transitions, still remain an inherent issue of retaining large curvatures of transition curves. Even for those double circumscribed transitions-based methods with relatively small curvatures, they also constrain excessively the transition lengths and are limited to a low-order continuity, deteriorating the feedrate and jerk of machine tools. For addressing these problems, a C3 continuous double circumscribed corner rounding (DCCR) method is proposed for five-axis linear tool path. In this method, the C3 continuous double circumscribed B-splines are specially designed to round the corners of tool position and tool orientation, whose transition lengths are analytically determined by jointly constraining the approximation errors, overlaps elimination and parameter synchronization. Moreover, the excessive constrains of transition lengths imposed by traditional methods are alleviated by fully considering the effects of overlaps and parameter synchronization, and the jerk of rotary axes is also limited with a high-order continuity. Compared to the existing double inscribed corner rounding (DICR) and DCCR methods, experiment results demonstrate that our method can improve further the feedrate while limiting the jerk of machine tools.
... On the other hand, converting the G01 short-line toolpath to a parametric curve is also an effective method to deal with the high-order discontinuity of the short-line toolpath. Zhao et al. [8] proposed a dual quaternion B-spline approximation method of dominant points to generate smooth toolpaths for five-axis CNC machine tools. Tao et al. [9] proposed an online smooth compression algorithm for complex toolpaths based on the interpolation curve fitting method. ...
To improve the application of the parametric curve tool path in surface machining, a method for generating arc-length parameterized toolpaths based on short-line toolpaths is proposed. The method first uses commercial CAM software to plan the short-line tool path. Then, the machining tool path with smooth continuous features is extracted. The smooth and continuous features are determined according to the chord length of the short-line tool path, and the extraction method is based on Chebyshev’s inequality of large numbers. After obtaining the smooth short-line trajectory, the iterative arc-length parameterized curve toolpath generation method is used to fit the discrete short-line toolpath. In the iterative process, the B-spline toolpath with parameterized chord length is obtained first. It is discretized according to the arc length difference information of the chord length B-spline. Finally, by continuously checking and inserting B-spline nodes, and fitting according to discrete points, the arc-length parameterized B-spline tool path is obtained. The feasibility of the method is verified with a part containing a sculpted surface, and two smooth arc-length parametric toolpaths are generated. In the end, the part is machined with the arc-length parameterized curve tool path.
... For local methods, the general idea is to replace the sharp corners within two adjacent segments with some smoothing blending curves. Various local smoothing algorithms have been proposed by introducing B-splines, Bézier curve, PH curves and clothoid splines [4][5][6][7][8][9][10][11][12][13]. However, if the tool path is composed of a high density of short segments, it is possible to smooth these segments by using global methods [14]. ...
... Based on the feedrate scheduling methods for linear [6] and circular toolpaths, the feedrate profile of each segment can be obtained. In the conventional situation, the motion of latter segment starts until the former one is over. ...
... The second one is tool paths containing both linear and circular segments. In order to calculate the interpolation points, interpolation method mentioned in [6] is used for B-spline method [7], and the sampling period is appointed as T s . To display the difference between resulting trajectories and original toolpaths, the contour error of each interpolation point, which is defined as the minimum distance between this point to all segments, is calculated. ...
... Due to its simplicity and convenience, the local smoothing method has been widely used to smooth the position path of three-axis machine tools [10]. Bezier [11,12] and B-spline [13] curves are commonly used to smooth sharp corners, and both of them can be directly processed into position paths composed of straight lines and arcs (G02/G03 commands). There are mainly two types of strategies to realize path smoothing for five-axis machine tools, i.e., kinematic and geometric methods [14]. ...
... Combining Eqs. (1) and (13) together and considering the properties of the B-spline curve ( ∑ 8 j=0 N j,5 (u) = 1 ), the inserted smoothed spline curve can be further expressed as the function of l a and l b : ...
In traditional multi-axis machining, the linear tool path generated by computer-aided manufacturing systems is still in a broad tool path format, resulting in discontinuities at the corners, which slow down tool motion, and also trigger tool vibration, reducing machining accuracy and surface quality. In order to ensure the C³ continuity of the tool during the multi-axis machining process, this paper proposes an analytical B-spline local smoothing method with curvature control. The tool position trajectory and tool orientation are both smoothed by a 5th spline curve with 9 control points. Optimal control points are determined for position and orientation splines to achieve C³ continuity at the junctions between the spline and linear segment while preserving the predefined geometric tolerance limits and minimizing the maximum curvature of the inserted B-spline curve. The smoothed tool path is expected to have less curvature fluctuation, which can significantly improve both the kinematic performance of the machine tool and the machining efficiency. Both computer simulations and physical printing experiments are carried out to demonstrate the improvements of our proposed method in motion smoothness and tracking accuracy in multi-axis machining tasks.
... To ensure the smoothness of the machining path and the high-order continuity of speed, CNC usually adopts "local blending/ smoothing" or "global blending/smoothing" algorithms to improve the geometric continuity of the path and then carries out speed planning and interpolation. In related research, by inserting a line segment, arc, B-spline, and Bezier curve at the adjacent tool-path [5,6], the smooth transition at the corner is realized and the speed continuity of the path corner is improved. The local smoothing has a good local property and can effectively control the contour error at the corner [6]. ...
... In related research, by inserting a line segment, arc, B-spline, and Bezier curve at the adjacent tool-path [5,6], the smooth transition at the corner is realized and the speed continuity of the path corner is improved. The local smoothing has a good local property and can effectively control the contour error at the corner [6]. Compared with the local smoothing method, the global smoothing method adopts the fitting or interpolation to fit a large number of continuous short linear segments into one or more smooth spline curves under the contour error constraint; commonly used splines include Bezier, B-spline, and NURBS curves [7][8][9]. ...
... The proposed PD-VPC algorithm complies with the set contour error requirement of 10 [μm] in all processing areas, whereas the NS-DCB algorithm has an out-of-tolerance phenomenon in local areas. The corners [1][2][3][4][5][6] in the Fig. 14 show that the contour error of the PD-VPC algorithm is smaller than that of the NS-DCB algorithm at various corners. Figure 15 also shows that the contour error of the PD-VPC algorithm is smaller than NS-DCB algorithm in most areas of the whole trajectory. ...
Three-axis machining tool-paths are generated using computer-aided manufacturing (CAM) systems and programmed by a series of G01 codes mixed with long and short segments. The computer numerical control (CNC) system needs to analyze path information according to the G-codes through path smoothing, speed planning, and interpolation process; generate smooth trajectory; and ensure the high-order continuity of velocity. This study proposes a real-time smooth trajectory generation algorithm with a simple structure and low calculation for 3-axis blending machining tool-paths. Finite impulse response filters are used to generate a smooth trajectory with bounded acceleration in real-time. Due to the delayed response of the filter, it will inevitably bring contour error. The feed rate is adjusted by considering the influence of the adjacent segments on the contour error in the case of long segments. This idea is then extended to continuous micro-segments. Finally, a direct one-step smooth trajectory generation algorithm for the blending tool-paths is realized through pre-discretization with certain rules. The algorithm can effectively control the contour error but also is suitable for blending tool-paths. Consequently, the effectiveness of the proposed algorithms is validated in simulations and also experimented on a 5-axis machine tool.
... The Bezier, B-spline, NURBS, or Pythagorean-hodograph curves are inserted into transition corners between linear toolpath segments, in order to achieve different order of motion continuity under predefined tolerance limits. Zhu et al. proposed a B-spline transition method, and the transition curve was composed of two symmetrical Bezier curves [5]. Fan et al. utilized two quartic Bezier curves to obtain an optimal continuous toolpath where the curvature extreme can be calculated [6]. ...
Linear motion commands of multi-axis computer numerical control (CNC) machine tools need to be smoothed at the transition corners, because the velocity discontinuities at corners can result in fluctuations on machine tool motions and lead to poor surface quality. However, few studies have reported on the transition between non-line contour. The paper proposes a novel two-step local corner smoothing method for non-line contour transition, including path smoothing and feedrate scheduling. In path smoothing, a non-uniform rational B-spline (NURBS) curve is adopted to smooth the adjacent non-line contour of screw rotor section while constraining the cornering error with the set tolerance. The maximum deviation between the original trajectory and smoothed toolpath can be analytically calculated. In feedrate scheduling, a jerk-limited S-curve feedrate scheduling scheme is developed based on machine tool dynamics. Finally, in order to verify the correctness of the proposed method, a series of numerical simulations and actual machining and measurement experiments are conducted, and the simulations and experimental results have showed the good performance of the proposed algorithm.
