An adaptive parametric interpolator for trajectory planning

A real-time interpolation algorithm for trajectory planning is studied in this paper. The NURBS interpolation algorithm is proposed to confine contour errors and feedrate fluctuations. The feedrate is adjusted adaptively according to the specified acceleration/deceleration values and jerk value. A direct digital convolution method is also introduced into velocity planning for NURBS interpolator, and it is more efficient than the traditional method of polynomial functions. The feedrate at the sharp corner is smoothed by imposing limitations on the acceleration and jerk values generated in the machining process. Since the computation of the total length of NURBS curve is required for the digital convolution method, a numerical adaptive quadrature algorithm is used to approximate the integrand. Simulation results demonstrate the effectiveness of the proposed interpolator for machining curved paths.     

The feedrate scheduling of NURBS interpolator for CNC machine tools

This paper proposes an off-line feedrate scheduling method of CNC machines constrained by chord tolerance, acceleration and jerk limitations. The off-line process for curve scanning and feedrate scheduling is realized as a pre-processor, which releases the computational burden in real-time task. The proposed method first scans a non-uniform rational B-spline (NURBS) curve and finds out the crucial points with large curvature (named as critical point) or G⁰ continuity (named as breakpoint). Then, the NURBS curve is divided into several NURBS sub-curves using curve splitting method which guarantees the convergence of predictor–corrector interpolation (PCI) algorithm. The suitable feedrate at critical point is adjusted according to the limits of chord error, centripetal acceleration and jerk, and at breakpoint is adjusted based on the formulation of velocity variation. The feedrate profile corresponding to each NURBS block is constructed according to the block length and the given limits of acceleration and jerk. In addition, feedrate compensation method for short NURBS blocks is performed to make the jerk-limited feedrate profile more continuous and precise. Because the feedrate profile is established in off-line, the calculation of NURBS interpolation is extremely efficient for CNC high-speed machining. Finally, simulations and experiments with two free-form NURBS curves are conducted to verify the feasibility and applicability of the proposed method.     

NURBS曲线实时插补中的速度规划算法

大多数现有的NURBS曲线实时插补算法并未考虑速度方向的变化给各运动轴带来的影响,这会导致加工过程中单轴速度的剧烈变化.提出一种能保证各运动轴平稳运行的速度规划算法,它在满足精度要求的前提下,通过控制切向加速度和加加速度进行速度平滑,并根据各运动轴的当前速度和机床的实际性能再次调节进给速度,保证了机床的平稳运行.模拟实验证实了该算法的有效性.     

An accurate adaptive NURBS curve interpolator with real-time flexible acceleration/deceleration control

Parametric interpolation has been widely used in CNC machining because of its advantages over the traditional linear or circular interpolation. Many researchers focused on this field and have made great progress in the specific one, NURBS curve interpolation. These works greatly improved the CNC machining with constant feedrate, confined chord error and limited acceleration/deceleration. However, during CNC machining process, mechanical shocks to machine tool caused by the undesired acceleration/deceleration profile will dramatically deteriorate the surface accuracy and quality of the machined parts. This is, in most occasions, very harmful to machine tools. In this paper, an accurate adaptive NURBS curve interpolator is proposed with consideration of acceleration–deceleration control. The proposed design effectively reduces the machining shocks by constraining the machine tool jerk dynamically. Meanwhile, the constant feedrate is maintained during most time of machining process, and thus high accuracy is achieved while the feedrate profile is greatly smoothed. In order to deal with the sudden change of the acceleration/deceleration around the corner with large curvature, a real-time flexible acceleration/deceleration control scheme is introduced to adjust the feedrate correspondingly. Case study has been taken to verify the feasibility and advantages of the proposed design.     

Efficient time-optimal feedrate planning under dynamic constraints for a high-order CNC servo system

In this paper, the time-optimal feedrate planning problem under confined feedrate, axis velocity, axis acceleration, axis jerk, and axis tracking error for a high-order CNC servo system is studied. The problem is useful in that the full ability of the CNC machine is used to enhance the machining productivity while keeping the machining precision under a given level. However, the problem is computationally challenging. The main contribution of this paper is to approximate the problem nicely by a finite-state convex optimization problem which can be solved efficiently. The method consists of two key ingredients. First, a relationship between the tracking error and the input signal in a high-order CNC servo system is established. As a consequence, the tracking error constraint is reduced to a constraint on the kinematic quantities. Second, a novel method is introduced to relax the nonlinear constraints on kinematic quantities to linear ones. Experimental results are used to validate the proposed method.     

数控系统连续微段重构与插补算法研究

针对目前微段加工研究中采用的非重构微段加工方法存在的加工轨迹与设计曲线轮廓误差较大,轮廓加工精度较低,及微段节点处速度方向不连续,因此加工表面质量不高,加工过程机床振动较大的问题。在计算机数控（Computerized Numerical Control,CNC）中采用实时曲线重构与插补算法进行连续微段加工以实现对曲面的高速高精度加工。微段插补技术包括样条曲线的实时重构及递推插补算法,及建立满足加减速要求的可以直接递推的插补样条曲线的重构条件。应用微段曲线重构技术进行的样件数控加工实验中,在保证曲线轮廓加工精度达到um级精度的同时,加工速度提高了2~2.4倍。实验结果表明,实时曲线重构微段加工不仅可以实现在重构曲线的范围内进行整体加减速速度规划,提高加工效率,而且加工轨迹的进给速度的衔接平滑,轨迹光滑,表面质量好,并且利用重构的可以直接递推插补的样条曲线,有效解决了平衡了复杂算法加工过程中精度与运算速度的矛盾,提高了加工精度。     

数控加工中路径规划与速度插补综述

数控技术在现代制造工业中被广泛使用,相关研究一直为学界和业界共同关注。数控技术的传 统流程主要包含刀具路径规划和进给速度插补。为实现高速高精加工,人们通常将路径规划与速度插补中的若 干问题转换成数理优化模型,针对工程应用问题的复杂性,采用分步迭代优化的思路进行求解,但所得的结果 往往只是局部最优解。其次,路径规划与速度插补都是为了加工一个工件曲面,分两步进行处理虽然简化了计 算,但也导致不能进行整体优化。因此,为了更好地开展路径规划与速度插补一体化设计与全局最优求解的研 究,系统性地了解并学习已有的代表性工作是十分有必要的。所以将逐次介绍数控加工中刀具路径规划与速度 插补的相关方法与技术进展,包括基于端铣的加工路径规划;刀轴方向优化;G 代码加工以及拐角过渡;参数 曲线路径的进给速度规划等国内外相关研究以及最新提出的一些新型加工优化方法。     

A technique for time-jerk optimal planning of robot trajectories

A technique for optimal trajectory planning of robot manipulators is presented in this paper. In order to get the optimal trajectory, an objective function composed of two terms is minimized: a first term proportional to the total execution time and another one proportional to the integral of the squared jerk (defined as the derivative of the acceleration) along the trajectory. This latter term ensures that the resulting trajectory is smooth enough. The proposed technique enables one to take into account kinematic constraints on the robot motion, expressed as upper bounds on the absolute values of velocity, acceleration and jerk. Moreover, it does not require the total execution time of the trajectory to be set a priori. The algorithm has been tested in simulation yielding good results, also in comparison with those provided by another important trajectory planning technique.     

Optimal Velocity Planning of Wheeled Mobile Robots on Specific Paths in Static and Dynamic Environments

This paper deals with optimal temporal‐planning of wheeled mobile robots (WMRs) when navigating on predefined spatial paths. A method is proposed to generate a time‐optimal velocity profile for any spatial path in static environments or when mobile obstacles are present. The method generates a feasible trajectory to be tracked by fully exploiting velocity, acceleration and deceleration boundaries of the WMR, and by ensuring the continuity of the velocity and acceleration functions. As an additional benefit for the tracking process the jerk is also bounded. The algorithm is not time consuming, since it mostly uses closed mathematical expressions, nonetheless iteration strategies are presented to solve specific situations. However, such situations are not expected to occur when the spatial paths are planned as smooth curves. The success of the algorithm was tested by experimental and simulation results on the WMR “RAM.” © 2003 Wiley Periodicals, Inc.     

Development of an integrated look-ahead dynamics-based NURBS interpolator for high precision machinery

Methodologies for planning motion trajectory of parametric interpolation such as non-uniform rational B-spline (NURBS) curves have been proposed in the past. However, most of the algorithms were developed based on the constraints of feedrate, acceleration/deceleration (acc/dec), jerk, and chord errors. The errors caused by servo dynamics were rarely included in the design process. This paper proposes an integrated look-ahead dynamics-based (ILD) algorithm which considers geometric and servo errors simultaneously. The ILD consists of three different modules: a sharp corner detection module, a jerk-limited module, and a dynamics module. The sharp corner detection module identifies sharp corners of a curve and then divides the curve into small segments. The jerk-limited module plans the feedrate profile of each segment according to the constraints of feedrate, acc/dec, jerk, and chord errors. To ensure that the contour errors are bounded within the specified value, the dynamics module further modifies the feedrate profile based on the derived contour error equation. Simulations and experiments are performed to validate the ILD algorithm. It is shown that the ILD approach improves tracking and contour accuracies significantly compared to adaptive-feedrate and curvature-feedrate algorithms.     

Feedrate planning for machining with industrial six-axis robots

Nowadays, the adaptation of industrial robots to carry out high-speed machining operations is strongly required by the manufacturing industry. This new technology machining process demands the improvement of the overall performances of robots to achieve an accuracy level close to that realized by machine-tools. This paper presents a method of trajectory planning adapted for continuous machining by robot. The methodology used is based on a parametric interpolation of the geometry in the operational space. FIR filters properties are exploited to generate the tool feedrate with limited jerk. This planning method is validated experimentally on an industrial robot.     

Optimal trajectory planning for industrial robots

An analysis of the results of an algorithm for optimal trajectory planning of robot manipulators is described in this paper. The objective function to be minimized is a weighted sum of the integral squared jerk and the execution time. Two possible primitives for building the trajectory are considered: cubic splines or fifth-order B-splines. The proposed technique allows to set constraints on the robot motion, expressed as upper bounds on the absolute values of velocity, acceleration and jerk. The described method is then applied to a 6-d.o.f. robot (a Cartesian gantry manipulator with a spherical wrist); the results obtained using the two different primitives are presented and discussed.     

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.     

多坐标曲面加工中进给速度的优化控制

介绍了基于曲面CNC(Computer Numerical Control)直接插补方式的多坐标曲面加工 中进给速度的控制原理.综合考虑刀具相对零件表面切削进给速度的恒定,曲面形状引起的各运 动轴速度及其变化率不超过伺服驱动能力,以及饥床在启动、停止和速度变化时的平滑加减速运 动控制等因素,实现了进给速度的合理确定与控制,可有效提高曲面加工质量和加工效率.     

多坐标曲面加工中进给速度的优化控制1)

介绍了基于曲面CNC(Computer Numerical Control)直接插补方式的多坐标曲面加工中进给速度的控制原理.综合考虑刀具相对零件表面切削进给速度的恒定,曲面形状引起的各运动轴速度及其变化率不超过伺服驱动能力,以及机床在启动、停止和速度变化时的平滑加减速运动控制等因素,实现了进给速度的合理确定与控制,可有效提高曲面加工质量和加工效率.     

Experimental study of contouring accuracy for CNC machines executing curved paths with constant and curvature-dependent feedrates

Experiments to measure and improve the contouring accuracy of CNC machines executing curved paths with strong curvature and variations of curvature at high feedrates are reported, using P and PI controllers with a wide range of gains. The experiments are based on test curves exhibiting (a) a steady increase of curvature; (b) a periodic curvature variation between fixed minima and maxima; and (c) a sudden “spike” in the curvature profile. For the P controller, the curvature-dependent feedrate yields a diminution of the contour error by up to an order of magnitude, compared to constant feedrate. The curvature-dependent feedrate appears to be most advantageous in situations exemplified by case (c), since it affords a dramatic suppression of local contour error with a modest increase in traversal time. Moreover, the improvement in contour accuracy is relatively insensitive to the P gain. The results are less predictable when the curvature-dependent feedrate is used in conjunction with PI control, because of its more “active” response to the excitation arising from the varying path geometry and feedrate.     

基于混合遗传算法的工业机器人最优轨迹规划

为兼顾工业机器人工作效率与轨迹的平稳性,提出一种基于混合遗传算法的二次轨迹规划方案.通过最优时间轨迹规划得到最小执行时间,在最小执行时间内进行最优冲击轨迹规划,进而规划出一条既高效又平滑的运动轨迹.采用五次均匀B样条在关节空间进行快速插值,不仅保证了各关节速度和加速度连续性还保证了各关节冲击的连续性.连续平滑的冲击可以减少机械振动,延长机器人的工作寿命.选用PUMA560为对象进行仿真与实验,结果表明,该方案可以获得比较理想的机器人运动轨迹,所提出的混合遗传算法能有效提高全局寻优的性能和算法运行的稳定性.     

A parametric interpolator with confined chord errors, acceleration and deceleration for NC machining

Parametric interpolation has many advantages over linear interpolation in machining curves. Real time parametric interpolation research so far has addressed achieving a uniform feed rate, confined chord errors and jerk limited trajectory planning. However, simultaneous consideration of confined chord errors that respect the acceleration and deceleration capabilities of the machine has not been attempted. In this paper, the offline detection of feed rate sensitive corners is proposed. The velocity profile in these zones is planned so that chord errors are satisfied while simultaneously accommodating the machine's acceleration and deceleration limits. Outside the zone of the feed rate sensitive corners, the feed rate is planned using the Taylor approximation. Simulation results indicate that the offline detection of feed rate sensitive corners improves parametric interpolation. For real time interpolation, the parametric curve information can be augmented with the detected feed rate sensitive corners that are stored in 2×2 matrices.