Iso-planar interpolation for the machining of implicit surfaces

The paper presents an approach for on-line path generation and interpolation for the machining of implicit surfaces. For a given implicit surface, once the cutting plane direction and cut-in points have been selected, iso-planar tool paths and interpolated points can be calculated on-line according to the feedrate and scallop height requirements. The approach enables the tool position and orientation to be correctly calculated at each interpolated point. Validation examples are provided for the interpolation of cyclide surfaces with planar and curved boundaries.     

Angular interpolation of bi-parameter curves

This paper presents an approach to the interpolation of angular feedrate for bi-parameter curve paths in multi-axis machining. A bi-parameter curve is the intersection of a parametric surface and an implicit surface. A tool path is identified by a position curve and an orientation curve, both of which are generated based on the bi-parameter curve. The angular feedrate interpolator calculates the tool position and orientation at each sampling cycle according to the specified angular feedrates and the given tool path. The paper analytically relates the angular arc-length derivatives to the time derivatives of the parameters along the path making use of both angular feedrates and angular feed acceleration. The results are then used to interpolate the parameters of the bi-parameter curve leading to an accurate calculation of the position and orientation of the cutting tool. A general parametric surface has been used to verify the effectiveness of the algorithm. The bi-parameter curves of the surface have been computed for arbitrarily selected intersecting cylinders.     

Geometric Hermite interpolation for space curves

This paper considers the geometric Hermite interpolation for spacial curves by parametric quartic Bézier curve. In additon to position and tangent direction, the curvature vector is prescribed at each knot. We prove that under appropriate assumptions the interpolant exists locally with one degree of freedom. Moreover, we prove the interpolant is 6th order accurate.     

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。     

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.     

An algorithm for the interpolation of hybrid curves

Real time tool path generation consists of off-line design and real time interpolation of tool paths. An hybrid curve is the intersection of a parametric surface and an implicit surface. Previous work in tool path interpolation focused mainly in the interpolation of parametric curves. Tool paths designed by drive surface methods are hybrid curves which, in general, cannot be represented as parametric curves. An algorithm for the interpolation of hybrid curves is proposed in this paper. The algorithm is based on interpolation of the projection of the hybrid curve into the parametric domain. Each increment involves a second-order interpolation step augmented by iterative error reduction.Simulations of hybrid curve interpolation have been carried out. They are based on practical surfaces represented as NURB surfaces and implicit surfaces including a plane, a cylinder and a high order algebraic surface. They demonstrate that under typical machining conditions, interpolation error is well within the accuracy requirements of typical machining and that the use of one iteration error reduction can significantly reduce the path deviation. These show that the proposed algorithm is potentially useful for tool path interpolation for the machining of parametric surfaces.     

A new interpolation algorithm for tracing planar equidistant curves

The aim of this paper is to propose an interpolation algorithm for tracing the equidistant (bisector) of two planar curves. The structure of the algorithm may be adapted accordingly so as to be used either for purely computing purposes or for presentation purposes. As a computing tool, the algorithm is suitable for computation of offset intersections and construction of Voronoi diagrams. In this case the step size is adjusted appropriately in order to reach the desired position in a small number of steps but with high accuracy. As a presentation tool, it may be embedded in a CAD system, entrusted with the task of drawing equidistants or even it may be used for plotting equidistants by driving the plotting tip. In this case, a fixed step size is selected to satisfy the specific precision requirements of the presentation. The development of the algorithm is achieved by treating equidistant generation as a locus-tracing problem. Using analytic concepts and the locus-defining geometric property, we formulate two sophisticated constructive operations. The repeated application of these operations generates a succession of points on the desired path (the locus) accurately and efficiently.     

High speed interpolation for micro-line trajectory and adaptive real-time look-ahead scheme in CNC machining

CNC machining plays an important role in mechanical manufacturing.A key issue is to improve the machining feedrate while keeping the machining precision and satisfying the acceleration constraints of the CNC machine.For the consecutive micro-line segments interpolation,the velocities at the junction of two segments are the bottlenecks for the machining efficiency.This paper proposes a multi-period turning method to improve the feedrate at the junctions using the linear acceleration and deceleration mode,whi...     

Sharp error bounds for piecewise linear interpolation of planar curves

Curves are commonly drawn by piecewise linear interpolation, but to worry about the error is rather seldom. In the present paper we give a strong mathematical error analysis for curve segments with bounded curvature and length. Though the result seems very clear, the proof turned out to be unexpectedly hard, comparable to that of the famous four vertex theorem.     

GP-Stability of two-step implicit runge-kutta methods for delay differential equations

The GP-stability of two-step implicit Runge-Kutta (TIRK) methods for the numerical solution of systems of delay differential equations (DDEs) is considered. We focus on the stability behaviour of TIRK methods in the solution of the linear constant-coefficient systems of DDE with a single delay. We present a sufficient condition of GP-stability of TIRK methods.     

An optimization approach for biarc curve-fitting of B-spline curves

We present an approach to the optimal fitting of a biarc-spline to a given B-spline curve. The objective is to minimize the area between the original B-spline curve and the fitted curve. Such an objective has obvious practical implications. This approach differs from conventional biarc curve-fitting techniques in two main aspects and has some desirable features. Firstly, it exploits the inherent freedom in the choice of the biarc that can be fitted to a given pair of end-points and their tangents. The conventional approach to biarc curve-fitting introduces additional constraints, such as the minimal difference in curvature or others to uniquely determine successive biarcs. In this approach, such constraints are not imposed. Instead, the freedom is exploited in the problem formulation to achieve a better fit. Secondly, the end-points do not lie on the curve so that appropriate tolerance control can be imposed through the use of additional constraints. Almost all previous biarc-fitting methods consider end-points that are on the original curve. As a result of these two aspects, the resulting biarc curve fits closely to the original curve with relatively fewer segments. This has a desirable effect on the surface finish, verification of CNC codes and memory requirement. Numerical results of the application of this approach to several examples are presented.     

Adaptive-feedrate interpolation for parametric curves with a confined chord error

Recently, modern manufacturing systems have been designed which can machine arbitrary parametric curves while greatly reducing data communication between CAD/CAM and CNC systems. However, a constant feedrate and chord accuracy between two interpolated points along parametric curves are generally difficult to achieve due to the non-uniform map between curves and parameters. A speed-controlled interpolation algorithm with an adaptive feedrate is proposed in this paper. Since the chord error in interpolation depends on the curve speed and the radius of curvature, the feedrate in the proposed algorithm is automatically adjusted so that a specified limit on the chord error is met. Both simulation and experimental results for non-uniform rational B-spline (NURBS) examples are provided to verify the feasibility and precision of the proposed interpolation algorithm.     

有理三次三角Hermite插值样条曲线及其应用

给出一种有理三次三角Hermite插值样条曲线,具有三次Hermite插值样条相似的性质。该样条含有三角函数和形状参数,利用形状参数的不同取值可以调控插值曲线的形状,甚至不用解方程组,就能使曲线达到C2连续。此外,选择合适的控制点和形状参数,这种样条可以精确表示星形线和四叶玫瑰线等超越曲线。     

插值两个平行平面曲线的可展B样条曲面

两条位于平行平面上的分别是[n]次和[n+1]次B-样条曲线,以这两条曲线作边界生成一个直纹面,即[(n,n+1)]次B样条曲面,得到了该直纹面为可展曲面的充要条件,构造了（2,3）次可展B－样条曲面。通过对以上可展B样条曲面进行凸性分析,得到了（n,n+1）次可展B样条曲面为凸曲面的充要条件。给出了几个（2,3）次B-样条可展曲面。     

Efficient offset trimming for deformable planar curves using a dynamic hierarchy of bounding circular arcs

We present an efficient algorithm for computing a family of trimmed offsets for planar freeform curves under deformation. The algorithm is based on a dynamic bounding volume hierarchy (BVH) for the untrimmed offsets of a given planar curve, which can be generated efficiently using a hierarchy of recursive bisections of the given curve. The proposed algorithm is effective for deformable planar curves. At each time frame, we segment the input curve into monotone spiral pieces (Barton and Elber, 2011), which is the only pre-processing needed for the dynamic BVH construction. To speed up the on-line generation of dynamic BVH, we employ the bounding circular arcs (BCA) of Meek and Walton (1995) that can be computed very efficiently using the position and tangent information at the endpoints of each monotone spiral curve segment. Using several experimental results, we demonstrate the performance improvement of our algorithm over the previous biarc-based algorithm of Kim et al. (2012).     

插值平面曲线的可展B样条曲面及凸性分析

用两条位于两任意平面上的均匀B样条曲线作边界曲线,形成一个直纹面,得到了该直纹面为可展曲面的充要条件,解决了(n,n)次可展B样条曲面的设计问题,构造了(2,2)次B样条可展曲面。通过对以上可展B样条曲面进行凸性分析,得到了(n,n)次可展B样条曲面为凸曲面的充要条件。     

曲线曲面逼近与插值的统一表示

为了用一种模型实现从逼近到插值的转换,在多项式空间上构造了含一个参数的调配函数,由之定义了基于4点分段的曲线,该曲线可以理解为由相同的一组控制顶点定义的逼近曲线和插值曲线的线性组合,其中的逼近曲线为3次均匀B样条曲线,插值曲线经过除首末点以外的所有控制点。在均匀参数分割下,曲线具有C2连续性,取特殊参数时可达C3连续。在参数变化过程中,曲线各段起点、终点的位置发生改变,但这些点处的一阶、二阶导矢始终保持不变,即始终与3次B样条曲线相同。曲线形状与端点条件密切相关,而B样条曲线具有良好的保形性,这些综合因素使得曲线在形状变化的过程中始终可以较好地保持控制多边形的特征。采用张量积方法将曲线推广至曲面,曲线曲面图例显示了该方法在造型设计中的有效性。