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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
... In this article, the value of p was selected as 3 as it is the lowest degree for achieving continuity [18], [20]. Seven control points ...
... AQ4: Please provide the issue number or month for [9], [20], and [23]. ...
Achieving high-throughput uniform scanning of stick-slip actuators is challenging because the conventional sharp triangular trajectory can easily induce dynamic friction between the compliant mechanism and the end-effector, which can lead to scanning failures. To address these issues, this article proposes a smooth trajectory that integrates a linear rising edge for uniform sampling with a cubic B-spline curve for smooth transitions. By optimizing the parameters, the trajectory effectively minimizes peak acceleration and reduces high-frequency components, thereby enhancing both scanning throughput and accuracy. The experimental results demonstrated that the proposed trajec-tory yielded a scanning throughput 2.86 times greater than that achieved with the conventional triangular trajectory. Moreover, the proposed trajectory demonstrated superior accuracy in scanning across all tested frequencies. This advancement is significant for expanding the applications of stick-slip actuators in high-throughput scenarios.
... When calculating the interpolation point parameters, it is impractical to calculate each interpolation point parameter directly and accurately. In the currently studied algorithms, such as the Taylor expansion method, iterative method, and polynomial fitting method, the nonlinear relationship between the arc length displacement and the parameter values leads to interpolation speed error and feedrate fluctuation [35][36][37]. Chen et al. [38] proposed a generalized parameter interpolation method that simultaneously considered machine tool contouring errors and eliminated feedrate fluctuations. Liu et al. [39] proposed a NURBS interpolation method to minimize the feedrate fluctuation of CNC machine tools. ...
Non-uniform rational B-spline (NURBS) is a common toolpath for machining complex parts, and its feedrate scheduling directly affects the machining efficiency of computer numerical control (CNC) machine tools. To improve the machining efficiency of the NURBS toolpath, this study proposes an adaptive feedrate scheduling method. The proposed method comprises two parts: offline feedrate scheduling and online toolpath interpolation. First, the allowable feedrate is determined based on geometric and normal drivability constraints, and the NURBS toolpath is segmented into sub-curves. Second, according to the tangential drivability constraints, the allowable feedrate is updated and S-shaped feedrate rules are adopted to generate the initial feedrate profile. For any sub-curves, if the initially obtained feedrate profile exceeds the allowable feedrate, the sub-curve parameters are dynamically adjusted and transition intervals are generated. Next, a bidirectional scanning is performed to obtain the feedrate profile for the transition intervals, thereby generating a complete feedrate profile. Then, to obtain the accurate interpolation points of the NURBS toolpath during the online interpolation process, the second-order Runge–Kutta interpolation method is modified based on the feedrate profile. Finally, simulations and experiments are performed to verify the effectiveness and advantages of the proposed method. The proposed method shows a significant superiority compared to the current method that uses constant velocity transitions in large curvature intervals. In two different NURBS toolpath cases, the machine efficiency is improved by 46.73% and 22.33%, respectively.
... For optimal movement performance even in challenging scenarios, we choose a different approach traditionally adopted in machining [21,47,48]. We compute a feed rate profile for the four end-effectors of the machine adopting a feed rate limit function (FLF), which defines the maximum allowable feed rate limited by a linearization of axial acceleration and jerk constraints, denoted * and * with * = , , , respectively. ...
... Одним из наиболее значимых алгоритмов в системах ЧПУ является алгоритм опережающего просмотра (Look-ahead). Алгоритм предназначен для анализа ограничений и отклонений от расчетной траектории [Любимов, Мартинова, Обухов, 2016;Zhao, Zhu, Ding, 2013]. Результатом просмотра буфера кадров являются своевременная коррекция скорости подачи и повышение точности обработки. ...
... Considering the smoothing error of position paths, Kunz and Stilman [23] adopted circular curves to achieve G 1 continuous corner smoothing of line segments but could only guarantee the tangential continuity at the junctions. To ensure the curvature continuity of the tooltip path, Zhao et al. [24] adopted B-spline curves to perform G 2 continuous smoothing at the corner position of each line segment, but there is still the problem of discontinuous curvature derivative. To solve this problem, Fan et al. [25] made the short line paths reach G 3 continuous by inserting quartic Bézier curves at the corner points. ...
The discontinuity of tangent, curvature, and curvature derivative of the 5-axis toolpath consisting of line and arc segments defined by the G01/G02/G03 commands limits the performance of hybrid robot in high-efficiency and high-precision milling machining. To guarantee the high-order continuous and smooth machining, a C³ continuous corner smoothing 5-axis toolpath containing spatial arc segments is semi-analytically generated. Firstly, the control points of the smoothing quintic B-spline for line and arc segments are constructed analytically based on the G³ continuity conditions. Within the constraints of the established explicit smoothing error expressions, the G³ continuous smoothing for three kinds of tooltip positions, namely, spatial line-line pairs, arc-arc pairs, and line-arc pairs, is investigated. Meanwhile, the smoothing error model of tool orientation is analytically established. Then, the parameter synchronization of the tool orientation at the junction following the tooltip position is achieved by replacing the remaining segments with seven B-spline curves as synchronization curves. Finally, simulations and experiments with the typical 5-axis toolpath containing spatial arc segments are carried out to verify the validity of the proposed method in ensuring C³ continuous motion in the robot operation space and good following performance in the joint space.
... However, controlling the approximation error necessitates intricate iterative computations [6]. Additionally, the use of high-order parametric curves demands careful consideration to avoid over-fitting, which can lead to fluctuations and oscillations [7]. Yang et al. [8] tackled an optimization problem for quadratic B-spline curves, ensuring The local smoothing method involves replacing sharp transitions between linear segments with smooth curves to ensure continuous transitions, thereby eliminating abrupt corners, as illustrated in Figure 2. Local smoothing methods are advantageous because they can analytically determine the smoothing error at the tool tip position, which has led to their widespread use in three-axis machining. ...
... The corresponding trajectory in the SMM is illustrated in Figure 17, which also demonstrates a noticeable smoothing effect on the SMM trajectory. We employ the B-spline transition method proposed by Zhao et al. [7] as the control sample in this paper, which is referred to as the B-spline method, and the method in this paper is referred to as dual Bézier method. When the tool tip position smoothing error is set to 10 mm, the local trajectory of the two methods is shown in Figure 18. ...
The smoothing of linear toolpaths plays is critical in improving machining quality and efficiency in five-axis CNC machining. Existing corner-smoothing methods often overlook the impact of spline curvature fluctuations, which may lead to acceleration variations, hindering surface quality improvements. The paper presents a five-axis toolpath corner-smoothing method based on the space of master–slave movement (SMM), aiming to minimize curvature fluctuations in five-axis machining and improve surface quality. The concept of movement space in master–slave cooperative motion is introduced, where the tool tip position and tool orientation are decoupled into a main motion trajectory and two master–slave movement space trajectories. By deriving the curvature monotony conditions of a dual Bézier spline, a G2-continuous tool tip corner-smoothing curve with minimal curvature fluctuations is constructed in real-time. Subsequently, using the SMM and the asymmetric dual Bézier spline, a high-order continuous synchronization relationship between the tool tip position and tool orientation is established. Simulation tests and machining experiments show that with our smoothing algorithm, maximum acceleration values for each axis were reduced by 21.05%, while jerk was lowered by 22.31%. These results indicate that trajectory smoothing significantly reduces mechanical vibrations and improves surface quality.
... Then, non-stop continuous motion can be conducted by scheduling the velocity profiles for feed drives in the vicinity region of sharp corners. Zhao et al. [7], Han et al. [8], and Huang Jie et al. [9] involved cubic B-splines to reshape corners as G 2 continuous trajectories by analytically manipulating the control points, which explicitly controlled deviation errors and conducted the expression of curvature extremums. To achieve higher-order continuity, Fan et al. [10] and Sun et al. [11] developed Bezier curves to obtain G 3 continuity smoothed toolpath. ...
Linear commands need to be smoothed by inserting high-order continuous splines at corners to eliminate the tangential discontinuities of connection points. Although several local smoothing methods have been developed, it is still a challenge to improve the machining efficiency further, when following the transition spline with the predefined fixed proportion constraints. In fact, the curvature distribution of the transition spline has a direct impact on the transition velocity and machining efficiency. This article presents a real-time local smoothing method with curvature-optimized transition splines to smooth linear commands. The designed transition spline, composed of a biairthoid spline and a circular segment, is analytically determined by maximizing the allowable velocity and staying within the predefined geometric constraints. As a result, the curvature increment induced by only biairthoid splines is significantly prevented, and thus, faster transition motion is gracefully attained. With the analytical expression of the arc length and curvature extreme, it is friendlier to the CNC system for real-time executing smooth motion. Simulations and experiments demonstrate that the proposed method can shorten the machining time by more than 8% when compared to the traditional only biairthoid splines.
... Respecting the kinematic capabilities, S-shaped acceleration with velocity compensation is applied during both forward and backward scanning. However, unlike traditional bidirectional scanning methods [32,33], which scan the whole window in both directions, this approach begins with a one-segment forward scanning, followed by a backward scanning starting from the end of the look-ahead window. As shown in Fig. 3, the first step is one-step forward scanning, followed by a backward scanning of the whole window. ...
Round-off error is inevitable in computer numerical control (CNC) systems because it is challenging to ensure that the interpolation time is an integer multiple of the interpolation period. This error can affect both machining precision and motion smoothness. To eliminate the round-off error and improve machining speed and precision, a novel round-off error elimination method is proposed that modifies the S-shaped acceleration/deceleration (ACC/DEC) feedrate scheme. Firstly, an adapted bidirectional scanning algorithm is applied to a jerk-limited S-shaped feedrate profile to improve machining efficiency while respecting feedrate constraints. Secondly, during the interpolation stage, the velocity round-off error is first introduced. Based on the novel properties of the modified S-shaped scheme, the displacement round-off error is then analyzed and addressed using different strategies, such as modifying the S-shaped profile with one-cycle rounding-up in a specified section or segment transition, without crossing the end point. Finally, a series of simulation and real machining experiments are conducted to compare the proposed method with existing round-off error compensation algorithms. The results show that the proposed method offers more flexible and effective round-off error compensation, with significant improvements in machining efficiency and the smoothness of the motion profile, including velocity, acceleration, and jerk. The real machining experiments, conducted in a CNC system with a 250-µs real-time control period and focused on 3D complex surface engraving, verified the practicality of the proposed method.
... Planning 6R robot trajectories involves feedrate scheduling and interpolation. Though position spline [7], [8], [9] or double spline interpolation [10] can be used in 6R robot, complex nonlinear mapping between joint and Cartesian space complicates trajectory planning, resulting from the strong coupling of the six robot joints. High Cartesian feedrates may exceed joint constraints, causing vibrations, while low feedrates reduce efficiency. ...
... Section III-B presents PlanBlock, containing pre-planning data for each segmented curve within the look-ahead window. Traditionally, bidirectional scanning [7], [8] is employed for ensuring global feedrate continuity. However, this method is not suitable for look-ahead feedrate scheduling. ...
... Traditional bidirectional scanning [7], [8] is not suitable for look-ahead feedrate scheduling as feedrate unreachability occurs mainly during acceleration or deceleration, with deceleration being the time reversal of acceleration. Thus, bidirectional scanning feasibility hinges on the nature of the acceleration segment. ...
With the continuous development of robotic technology, there is an increasing demand for efficient, smooth, and precise trajectory planning and interpolation. For the robot manipulators with 6 rotational (6R) joints, smooth task paths are crucial for high-quality task and precise dynamic trajectory tracking. However, nonlinear mapping between joint and cartesian space complicates feedrate scheduling under joint constraints. Existing methods, like S-shape feedrate profiles or time-optimal approaches, are inefficient or compromise trajectory stability. This paper proposes a segment-based dynamically adaptive smooth look-ahead feedrate scheduling method based on local dynamic window and a Maximum Velocity Curve (MVC) for 6R robot manipulators. It balances the efficiency and stability of motion execution while considering spline trajectory and joint constraints. Five types of segmented velocities under the local dynamic window are identified, with adaptive smoothing strategies developed. Feedrate remains constant within segments and transitions smoothly between them, enhancing trajectory quality. The results of the smoothing preprocessing can be directly used for feedrate profile generation, ensuring smooth, non-oscillating motion while meeting performance and constraint requirements, which is better suited for real-time interpolation. Simulation and experimentation confirm the proposed method’s effectiveness.
Note to Practitioners
—The motivation of this article stems from the need to develop a feedrate scheduling method with joint jerk constraints of 6R robot manipulators for practical tasks like polishing, engraving, welding, and spraying. 6R robots, due to their nonlinear coupling kinematics, often fail to meet joint constraints. Existing methods, inefficient or requiring frequent changes in joint acceleration/jerk for time optimality, adversely affect task quality. To balance smoothness and operational efficiency for 6R robot tasks, we propose the segmented dynamic adaptive look-ahead smoothing feedrate scheduling with joint jerk constraints of 6R robot manipulators, ensuring trajectory execution within joint limits and balancing smoothness and operational efficiency, which is better suited for real-time interpolation. This method can be of great interest to readers working on precision manufacturing and robotics feedrate profile generation method.
... In contrast, local smoothing methods construct transition curves at corners to round adjacent linear tool path segments. Widely used techniques in this category include Bezier splines [14,15], B-splines [1,13,[16][17][18][19][20][21][22][23][24][25]33], Pythagorean-hodograph (PH) splines [26][27][28][29][30], and clothoid splines [31,32]. These techniques offer the advantage of achieving high-order continuity and precise control over the approximation error [16], garnering significant interest from researchers. ...
... When overlaps occurred, adjacent two splines were shortened with the same proportion. Zhao et al. [33] adopted a curvature-continuous B-spline with five control points to blend the adjacent straight lines. They addressed considerations such as curvature extrema, approximation error, and overlap elimination by formulating a linear programming problem. ...
