Optimization of Corner Blending Curves

The blending or filleting of sharp corners is a common requirement in geometric design applications — motivated by aesthetic, ergonomic, kinematic, or mechanical stress considerations. Corner blending curves are usually required to exhibit a specified order of geometric continuity with the segments they connect, and to satisfy specific constraints on their curvature profiles and the extremum deviation from the original corner. The free parameters of polynomial corner curves of degree ≤6 and continuity up to G3 are exploited to solve a convex optimization problem, that minimizes a weighted sum of dimensionless measures of the mid-point curvature, maximum deviation, and the uniformity of parametric speed. It is found that large mid-point curvature weights result in undesirable bimodal curvature profiles, but emphasizing the parametric speed uniformity typically yields good corner shapes (since the curvature is strongly dependent upon parametric speed variation). A constrained optimization problem, wherein a particular value of the corner curve deviation is specified, is also addressed. Finally, the shape of Pythagorean-hodograph corner curves is compared with that of the optimized “ordinary” polynomial corner curves

Optimized NURBS Curve Based G-Code Part Program for CNC Systems

Indiana University-Purdue University Indianapolis (IUPUI)Computer Numerical Control (CNC) is widely used in many industries that needs high speed machining of the parts with high precision, accuracy and good surface finish. In order to avail this the generation of the CNC part program size will be immensely big and leads to an inefficient process, which increases the delivery time and cost of products. This work presents the automation of high-accuracy CNC tool trajectory planning from CAD to G-code generation through optimal NURBs surface approximation. The proposed optimization method finds the minimum number of NURBS control points for a given admissible theoretical cord error between the desired and manufactured surfaces. The result is a compact part program that is less sensitive to data starvation than circular and spline interpolations with potential better surface finish. The proposed approach is demonstrated with the tool path generation of an involute gear profile and a topologically optimized structure is developed using this approach and then finally it is 3D printed

Arc-Length Parameterized NURBS Tool Path Generation and Velocity Profile Planning for Accurate 3-Axis Curve Milling

In modern industrial CNC (Computer Numerical Control) machining processes, the pursuing of higher accuracy and efficiency has always been one of the most important tasks to be discussed and studied. A lot of proposed algorithms are developed in order to optimize the machining performance in either of the above focused domains. Nevertheless, there is forever a trade-off between gaining less machining error and providing higher feed rate. As for machining a free-shaped curve (e.g., Bezier curves, B-splines and NURBS) in a three-dimensional space, a better manner to balance out the aforementioned trade-offs turns out to be even more critical and essential. The conventional iterative function used for tool path generation could cause feed rate fluctuation during the actual machining, and it thus might lead to failure on constraining the error within the machining accuracy requirement. Another potential problem occurs when the machining process comes across into a relatively high curvature segment with the prescribed high feed rate, due to the machine axial acceleration limit, the machine may not be able to maintain the tool tip trajectory within the error tolerance. Therefore, a new approach to NURBS tool path generation for high feed rate machining is proposed. In this work, several criterions are set for checking the viability of the prescribed feed rate and adjusting it according to the actual shape of the objective curve and the capability of the machine. After the offline feed rate viability check and readjustment, a new iterative algorithm based on the arc-length re-parameterized NURBS function would be implemented to calculate the tool path in real-time. By using this proposed method, the feed rate fluctuation is diminished and the overall efficiency of the machining process would have been optimized under the condition of accuracy guaranteed

Modeling of one-dimensional contours with ensure of given accuracy of interpolation

UK: Запропоновано метод формування одновимірних обводів, виходячи з заданої точності інтерполяції. Максимальна абсолютна похибка інтерполяції визначається з урахуванням геометричних властивостей вихідної кривої лінії. Розглядається два різновиди похибки. По-перше, похибка, з якою сформована дискретно представлена крива, інтерполююча вихідний точковий ряд, представляє вихідну криву. По-друге, похибка, з якою інтерполююча крива представляє будь-яку криву з заданими характеристиками. RU: Предлагается метод формирования одномерных обводов исходя из заданной точности интерполяции. Максимальная абсолютная погрешность интерполяции определяется с учетом геометрических свойств исходной кривой линии. Рассматриваются две разновидности погрешности. Во-первых, погрешность, с которой сформированная дискретно представленная кривая, интерполирующая исходный точечный ряд, представляет исходную кривую. Во-вторых, погрешность, с которой интерполирующая кривая представляет любую кривую с заданными геометрическими характеристиками. EN: The purpose of research is to develop a method of forming of one-dimensional contours of with a given accuracy of interpolation. Determination of the accuracy of interpolation is based on the formation of a curve based on known geometric properties. The geometric model is formed on the assumption that if there is a curve without singular points that interpolates the points set, then there are no singular points for the original object. Such points include: inflection points, points of changes of the direction of increase of curvature, torsion, and other. The interpolating curve is formed in the form of a condensed points set consisting of an arbitrarily large number of points, determined on the basis of the possibility of interpolating their curve by a line with given characteristics. The error with which the discretely presented curve represents the original curve is estimated as the region of possible location of all curve lines interpolating the original points set whose properties are identical to those of the original curve. The error in the formation of the interpolating curve is estimated as the region of possible location of the curve which interpolate the thickened points set. The solution of the problem for a plane curve from the condition that there is no oscillation and the condition for a monotone change in the curvature is proposed. The region of the curve, which is determined by the condition of convexity of the curve, is the maximum and is the initial. The imposition of the following conditions: a monotonous change of curvature along the curve and the assignment of fixed positions of tangents and curvature values at the initial points, localizes the area of a possible solution. The developed method can be used to solve problems requiring the determination of the maximum absolute error with which the model represents the original object

Development of the method for the formation of one-dimensional contours by the assigned interpolation accuracy

EN: Geometric modeling is one of the tools for investigation of objects, phenomena and processes. The task of geometric modeling is to determine properties of an object being modeled using characteristics of a geometric model. Output data are geometric images assigned by a set of points. Their location reflects properties of the examined object. Geometric characteristics of a discretely represented geometric image (line or surface) can be given at the output points. We can obtain output by calculations or measurements at physical objects. UA: Запропоновано метод формування одновимірних обводів виходячи з заданої точності інтерполяції. Максимальна абсолютна похибка інтерполяції визначається з урахуванням геометричних властивостей вихідної кривої лінії. Розглядається два різновиди похибки. По-перше, похибка, з якою сформована дискретно представлена крива, що інтерполює вихідний точковий ряд, представляє вихідну криву. По-друге, похибка, з якою інтерполююча крива представляє будь-яку криву з заданими геометричними характеристиками. RU: Предлагается метод формирования одномерных обводов исходя из заданной точности интерполяции. Максимальная абсолютная погрешность интерполяции определяется с учетом геометрических свойств исходной кривой линии. Рассматривается две разновидности погрешности. Во-первых, погрешность, с которой сформированная дискретно представленная кривая, интерполирующая исходный точечный ряд, представляет исходную кривую. Во-вторых,погрешность, с которой интерполирующая кривая представляет любую кривую с заданными геометрическими характеристиками

A Chronology of Interpolation: From Ancient Astronomy to Modern Signal and Image Processing

This paper presents a chronological overview of the developments in interpolation theory, from the earliest times to the present date. It brings out the connections between the results obtained in different ages, thereby putting the techniques currently used in signal and image processing into historical perspective. A summary of the insights and recommendations that follow from relatively recent theoretical as well as experimental studies concludes the presentation

Shape characterisation of tool path motion

For a given application, the autonomous regulation of tool path motion by a machine’s controller can produce undesirable and unknown machining conditions. Machining parameters may therefore require a posteriori optimisation. Indeed, the methods employed are often iterative and informed by empirical evidence from machining trials. A shape characterisation of tool path motion is postulated by enforcing constraints on the kinematic equations describing velocity, acceleration and jerk. The resulting description of motion depends only upon the kinematic limits of a machine and the intrinsic shape properties of a tool path. The resulting shape schematics provide complete illustrations of the distinctive features of each of the kinematic vectors. Kinematic profiles, derived from a series of test tool path motions are compared with these shape schematics in order to provide supportive empirical evidence. The main contribution of this thesis is to demonstrate a priori shape characterisation of tool path motion. This characterisation is achieved without knowledge of the motion control algorithms implemented by a given machine’s controller. The characterisation may be employed to inform the selection of machining parameters and thereby reduce the time and the number of machining trials