空间曲线的六轴联动控制算法研究与测试

针对六坐标空间的复杂曲线的加工,提出六轴联动的控制算法。根据六轴之间的运动关系,合成其进给速度,由之推导出插补变量值,得出相应的六轴插补坐标值,实现动点的稳速控制。采用该控制算法在自主研制的六轴嵌入式数控工具磨削系统中进行锥刃磨削的多轴联动插补测试,根据锥刃铣刀的数学模型得出插补算式。仿真与实际磨削测试表明,该六轴联动控制算法切实高效,实现复杂曲线的六轴联动控制,进给的合成速度控制恒定,能确保高质量加工。     

Inverse kinematics for optimal tool orientation control in 5-axis CNC machining

The problem of determining the inputs to the rotary axes of a 5-axis CNC machine is addressed, such that relative variations of orientation between the tool axis and surface normal are minimized subject to the constraint of maintaining a constant cutting speed with a ball-end tool. In the context of an orientable-spindle machine, the results of a prior study are directly applicable to the solution of this inverse-kinematics problem. However, since they are expressed in terms of the integral of the geodesic curvature, a discrete time-step solution is proposed that yields accurate rotary-axis increments at high sampling frequencies. For an orientable-table machine, a closed-form solution that specifies the rotary-axis positions as functions of the surface normal variation along the toolpath is possible. In this context, however, the feasibility of a solution is dependent upon the surface normal along the toolpath satisfying certain orientational constraints. These inverse-kinematics solutions facilitate accurate and efficient 5-axis machining of free-form surfaces without “unnecessary” actuation of the machine rotary axes.     

Kinematic aspects of a robot-positioner system in an arc welding application

This paper focuses on the kinematic control of a redundant robotic system taking into account particularities of the arc welding technology. The considered system consists of a 6-axis industrial robot (welding tool manipulator) and a 2-axis welding positioner (workpiece manipulator) that is intended to optimise a weld joint orientation during the technological process. The particular contribution of the paper lies in the area of the positioner inverse kinematics, which is a key issue of such system off-line programming and control. It has been proposed a novel formulation and a closed-form solution of the inverse kinematic problem that deals with the explicit definition of the weld joint orientation relative to the gravity. Similar results have also been obtained for the known problem statement that is based on a unit vector transformation. For both the cases, a detailed investigation of the singularities and uniqueness-existence topics have been carried out. The presented results are implemented in a commercial software package and verified for real-life applications in the automotive industry.     

Efficient circular arc interpolation based on active tolerance control

In this paper, we present an efficient sub-optimal algorithm for fitting smooth planar parametric curves by G¹ arc splines. To fit a parametric curve by an arc spline within a prescribed tolerance, we first sample a set of points and tangents on the curve adaptively as well as with enough density, so that an interpolation biarc spline curve can be with any desired high accuracy. Then, we construct new biarc curves interpolating local triarc spirals explicitly based on the control of permitted tolerances. To reduce the segment number of fitting arc spline as much as possible, we replace the corresponding parts of the spline by the new biarc curves and compute active tolerances for new interpolation steps. By applying the local biarc curve interpolation procedure recursively and sequentially, the result circular arcs with no radius extreme are minimax-like approximation to the original curve while the arcs with radius extreme approximate the curve parts with curvature extreme well too, and we obtain a near optimal fitting arc spline in the end. Even more, the fitting arc spline has the same end points and end tangents with the original curve, and the arcs will be jointed smoothly if the original curve is composed of several smooth connected pieces. The algorithm is easy to be implemented and generally applicable to circular arc interpolation problem of all kinds of smooth parametric curves. The method can be used in wide fields such as geometric modeling, tool path generation for NC machining and robot path planning, etc. Several numerical examples are given to show the effectiveness and efficiency of the method.     

A greedy algorithm for feedrate planning of CNC machines along curved tool paths with confined jerk

In this paper, the problem of optimal feedrate planning along a curved tool path for 3-axis CNC machines with the acceleration and jerk limits for each axis and the tangential velocity bound is addressed. It is proved that the optimal feedrate planning must be “Bang–Bang” or “Bang–Bang-Singular” control, that is, at least one of the axes reaches its acceleration or jerk bound, or the tangential velocity reaches its bound throughout the motion. As a consequence, the optimal parametric velocity can be expressed as a piecewise analytic function of the curve parameter u. The explicit formula for the velocity function when a jerk reaches its bound is given by solving a second-order differential equation. Under a “greedy rule”, an algorithm for optimal jerk confined feedrate planning is presented. Experiment results show that the new algorithm can be used to reduce the machining vibration and improve the machining quality.     

Automatic generation of gouge-free and angular-velocity-compliant five-axis toolpath

Existing works in automatic generation of interference-free five-axis surface machining toolpaths bear a serious drawback — in order to avoid the obstacles, the tool is often required to make drastic change in its orientation between neighboring contact points. Such a quick change in the tool’s orientation can never be made possible in reality due to the stringent physical limit on the speed and acceleration of the rotary motions of the machine tool. The usual ad hoc solution to this problem is to smooth the toolpath in the configuration space, which, however, is prone to special situations of failure and is not able to guarantee the absolute compliance with the given angular velocity limit. In this paper we present an approach to this problem by directly involving the angular velocity limit in the search process. The presented algorithm will automatically generate a five-axis toolpath that not only is interference-free but also guarantees the angular-velocity compliance. Delicate computation and manipulation of visibility maps and their derivative data ensure that the proposed algorithm is computationally feasible with acceptable computing time and memory requirement. Test examples are given to demonstrate the promising use of the proposed solution.     

Feed-rate and trajectory optimization for CNC machine tools

The key task performed by CNCs is the generation of the time-sequence of set-points for driving each physical axis of the machine tool during program execution. This interpolation of axes movement must satisfy a number of constraints on axes dynamics (velocity, acceleration, and jerk), and on process outcome (smooth tool movement and precise tracking of the nominal tool-path at the desired feed-rate). This paper presents an algorithm for CNC kernels that aims at solving the axes interpolation problem by exploiting an Optimal Control Problem formulation. With respect to other solutions proposed in the literature, the approach presented here takes an original approach by assuming a predefined path tracking tolerance—to be added to the constraints listed above—and calculating the whole trajectory (path and feed-rate profile) that satisfies the given constraints. The effectiveness of the proposed solution is benchmarked against the trajectory generated by an industrial, state-of-the-art CNC, proving a significant advantage in efficiency and smoothness of axes velocity profiles.     

希尔伯特-黄变换中的一种新包络线算法

先介绍希尔伯特一黄变换(Hilbert Huang transform,HHT)中的包络线算法,分析了两种具代表性的算法——三次样条插值法和Akima插值法存在的问题;然后在抛物线参数样条插值法原理的启发下证明了分段光滑定理,依据该定理并结合直观的几何意义提出了一种新包络线算法——分段幂函数法。算例表明,在某些情况下新算法比三次样条插值法具有更好的“柔性”,比Akima插值法具有更好的“光滑性”,用其进行HHT分析时出现虚假频率的概率更小。     

Smooth tool path generation for 5-axis machining of triangular mesh surface with nonzero genus

NC machining of a nonzero genus triangular mesh surface is being more widely confronted than before in the manufacturing field. At present, due to the complexity of geometry computation related to tool path generation, only one path pattern of iso-planar type is adopted in real machining of such surface. To improve significantly 5-axis machining of the nonzero genus mesh surface, it is necessary to develop a more efficient and robust tool path generation method. In this paper, a new method of generating spiral or contour-parallel tool path is proposed, which is inspired by the cylindrical helix or circle which are a set of parallel lines on the rectangular region obtained by unwrapping the cylinder. According to this idea, the effective data structure and algorithm are first designed to transform a nonzero genus surface into a genus-0 surface such that the conformal map method can be used to build the bidirectional mapping between the genus-0 surface and the rectangular region. In this rectangular region, the issues of spiral or contour-parallel tool path generation fall into the category of simple straight path planning. Accordingly, the formula for calculating the parameter increment for the guide line is derived by the difference scheme on the mesh surface and an accuracy improvement method is proposed based on the edge curve interpolation for determining the cutter contact (CC) point. These guarantee that the generated tool path can meet nicely the machining requirement. To improve further the kinematic and dynamic performance of 5-axis machine tool, a method for optimizing tool orientation is also preliminarily investigated. Finally, the experiments are performed to demonstrate the proposed method and show that it can generate nicely the spiral tool path or contour-parallel tool path on the nonzero genus mesh surface and also can guarantee the smooth change of tool orientation.     

样条插值算法在耦合地球系统模式中的应用

为满足耦合地球系统模式应用的需求, 提出了一种二维样条插值算法, 并将其有效地实现成插值模块封装进地球系统建模框架（earth system modeling framework, ESMF）。该算法基于经典样条算法, 根据地球系统模式特点进行修改, 用两次一维插值扩张成二维插值, 引入极点区域外插处理, 将插值权重生成与插值结果计算两部分分离。实验结果表明, 该算法能获得高精度的插值结果, 模块化的设计使得用户可通过统一的接口来使用插值算法从而完成插值计算。     

基于局部梯度特征的自适应多结点样条图像插值

为了获得质量更好的插值图像，提出了一种新的C^2。连续的支撑区间为（-2，2）的三次多结点样条插值核函数．通过增加结点带来的自由度构造了多结点样条插值公式；分析了在适当的边界条件和约束下三次多结点样条插值的逼近阶；将一维多结点样条插值算法推广到二维，建立了用于图像数据的插值公式；如果忽视图像的局部特征，通常双三次多结点样条插值图像的边缘会有模糊的现象，为此。对多结点样条插值应用逆梯度，得到了自适应多结点样条插值算法；实验所得误差图像和实验所得图像的峰值信噪比也证实了用自适应多结点样条插值算法重建的图像具有更高的质量．     

Threading splines through 3D channels

Given a polygonal channel between obstacles in the plane or in space, we present an algorithm for generating a parametric spline curve with few pieces that traverses the channel and stays inside. While the problem without emphasis on few pieces has trivial solutions, the problem for a limited budget of pieces represents a nonlinear and continuous (‘infinite’) feasibility problem. Using tight, two-sided, piecewise linear bounds on the potential solution curves, we reformulate the problem as a finite, linear feasibility problem whose solution, by standard linear programming techniques, is a solution of the channel-fitting problem. The algorithm allows the user to specify the degree and smoothness of the solution curve and to minimize an objective function, for example, to approximately minimize the curvature of the spline. We describe in detail how to formulate and solve the problem, as well as the problem of fitting parallel curves, for a spline in Bernstein-Bézier form.     

Solving the forward kinematics of parallel manipulators with a genetic algorithm

A floating point genetic algorithm is proposed to solve the forward kinematic problem for parallel manipulators. This method, adapted from studies in the biological sciences, allows the use of inverse kinematic solutions to solve forward kinematics as an optimization problem. The method is applied to two 3-degree-of-freedom planar parallel manipulators and to a 3-degree-of-freedom spherical manipulator. The method converges to a solution within a broader search domain compared to a Newton-Raphson scheme. © 1996 John Wiley & Sons, Inc.     

Five-axis tool path generation for a flat-end tool based on iso-conic partitioning

Traditionally, for the flat-end tool, due to the intertwined dependence relationship between its axis and reference point, most 5-axis tool-path generation algorithms take a decoupled two-stage strategy: first, the so-called cutter contact (CC) curves are placed on the part surface; then, for each CC curve, tool orientations are decided that will accommodate local and/or global constraints such as minimum local gouging and global collision avoidance. For the former stage, usually simplistic “offset” methods are adopted to determine the cutter contact curves, such as the iso-parametric or iso-plane method; whereas for the latter, a common practice is to assign fixed tilt and yaw angle to the tool axis regardless the local curvature information and, in the case of considering global interference, the tool orientation is decided solely based on avoiding global collision but ignoring important local machining efficiency issues. This independence between the placement of CC curves and the determination of tool orientations, as well as the rigid way in which the tilt and yaw angle get assigned, incurs many undesired problems, such as the abrupt change of tool orientations, the reduced efficiency in machining, the reduced finishing surface quality, the unnecessary dynamic loading on the machine, etc. In this paper, we present a 5-axis tool-path generation algorithm that aims at alleviating these problems and thus improving the machining efficiency and accuracy. In our algorithm, the CC curves are contour lines on the part surface that satisfy the iso-conic property — the surface normal vectors on each CC curve fall on a right small circle on the Gaussian sphere, and the tool orientations associated to a CC curve are determined by the principle of minimum tilt (also sometimes called lead) angle that seeks fastest cutting rate without local gouging. Together with an elaborate scheme for determining the step-over distance between adjacent CC curves that seeks maximum material removal, the presented algorithm offers some plausible advantages over most existing 5-axis tool-path generation algorithms, particularly in terms of reducing the angular velocity and acceleration of the rotary axes of the machine. The simulation experiments of the proposed algorithm and their comparison with a leading commercial CAM software toolbox are also provided that demonstrate the claimed advantages.     

Medial axis computation for planar free-form shapes

We present a simple, efficient, and stable method for computing—with any desired precision—the medial axis of simply connected planar domains. The domain boundaries are assumed to be given as polynomial spline curves. Our approach combines known results from the field of geometric approximation theory with a new algorithm from the field of computational geometry. Challenging steps are (1) the approximation of the boundary spline such that the medial axis is geometrically stable, and (2) the efficient decomposition of the domain into base cases where the medial axis can be computed directly and exactly. We solve these problems via spiral biarc approximation and a randomized divide & conquer algorithm.     

A new format for 5-axis tool path computation, using Bspline curves

This article presents a new format of tool path polynomial interpolation in 5-axis machining. The linear interpolation usually used produces tangency discontinuities along the tool path, sources of decelerations of the machine tool whereas polynomial interpolation reduces the appearance of such discontinuities. The new format involves a faster tool path and a better surface quality. However, it imposes a modification of the process so as to take the interpolation format and the inverse kinematics transformation (necessary to 5-axis machining) into account. This article deals with the geometrical problem of tool path calculation. Validation tests are detailed. They show that profits concern the reduction of machining time as well as the quality of the machined surfaces. Indeed, the trajectory continuity avoids the appearance of marks and facets.     

An adaptive real-time NURBS curve interpolation for 4-axis polishing machine tool

The non-uniform rational B-spine (NURBS) curve interpolation is a key technology of the advanced computer numerical control (CNC) system. NURBS curve interpolation can realize a high-speed and high-precision machining, and it can also avoid some inevitable deficiencies of the linear and circular interpolation functions which are generally used in traditional NC system. Before the interpolation, some calculation tasks are finished, which will decrease the amount of calculation during interpolation and increase the interpolation efficiency. Further, an adaptive NURBS curve interpolation with real-time and flexible S-shaped curve acceleration/deceleration (ACC/DEC) control method is added to the interpolation algorithms. The NC machining simulation conducted with the MATLAB software and the NURBS curve interpolation experiments performed on the 4-axis polishing machine tool demonstrate the validity and correctness of the adaptive real-time NURBS curve interpolation algorithm in the CNC system.