Curvature Variation Energy Research Articles

Planar quartic G2 Hermite interpolation for curve modeling

We study planar quartic G2 Hermite interpolation, that is, a quartic polynomial curve interpolating two planar data points along with the associated tangent directions and curvatures. When the two specified tangent directions are non-parallel, a quartic Bézier curve interpolating such G2 data is constructed using two geometrically meaningful shape parameters which denote the magnitudes of end tangent vectors. We then determine the two parameters by minimizing a quadratic energy functional or curvature variation energy. When the two specified tangent directions are parallel, a quartic G2 interpolating curve exists only when an additional condition on G2 data is satisfied, and we propose a modified optimization approach. Finally, we demonstrate the achievable quality with a range of examples and the application to curve modeling, and it allows to locally create G2 smooth complex shapes. Compared with the existing quartic interpolation scheme, our method can generate more satisfactory results in terms of approximation accuracy and curvature profiles.

Estimating Gini Coefficient from Grouped Data Based on Shape-Preserving Cubic Hermite Interpolation of Lorenz Curve

The Lorenz curve and Gini coefficient are widely used to describe inequalities in many fields, but accurate estimation of the Gini coefficient is still difficult for grouped data with fewer groups. We proposed a shape-preserving cubic Hermite interpolation method to approximate the Lorenz curve by maximizing or minimizing the strain energy or curvature variation energy of the interpolation curve, and a method to estimate the Gini coefficient directly from the coefficients of the interpolation curve. This interpolation method can preserve the essential requirements of the Lorenz curve, i.e., non-negativity, monotonicity, and convexity, and can estimate the derivatives at intermediate points and endpoints at the same time. These methods were tested with 16 grouped quintiles or unequally spaced datasets, and the results were compared with the true Gini coefficients calculated with all census data and results estimated with other methods. Results indicate that the maximum strain energy interpolation method generally performs the best among different methods, which is applicable to both equally and unequally spaced grouped datasets with higher precision, especially for grouped data with fewer groups.

Energy-Minimizing Cubic H-Bézier Curve

Abstract In shape designing, a curve is required to pass smoothly through the given data set values. Minimum energy curves are usually known as smooth curves. The aim of this research article is to construct a G1-approximating curve with minimum energy. For this purpose, a cubic H-Bézier curve is considered for which the tangent directions are held to be unknowns. The optimal cubic H-Bézier curve is procured by minimizing the stretch energy, strain energy, and curvature variation energy by solving an optimization problem of two parameters relating to the magnitudes of endpoint tangent directions.

Smooth path planning via cubic GHT-Bézier spiral curves based on shortest distance, bending energy and curvature variation energy

<abstract> In this article, an algorithm to obtain a smooth path (free from obstacles) that can be optimized by shortest path distance, bending energy and curvature variation energy will be presented. Previously, scholars used various tools to generate smooth path such as Clothoid, Log-Aesthetic curves (LACs), and Bézier curves. The limited number of solutions from the aforementioned curves become one of the drawback to generate smooth path planning. Therefore, providing a number of solutions that can generate smooth path planning becomes the objective of this study. In this paper to generate a smooth path, five templates of spiral transition curves having three different shape parameters with monotone curvature (either increase or decrease) by cubic GHT-Bézier curves are proposed. Moreover, few examples of path planning technique via cubic GHT-Bézier spiral curve to show the flexibility of smooth path by minimization of the shortest path (minimum arc length) <bold>L</bold>, bending energy <bold>E</bold> and curvature variation energy <bold>V</bold> are presented. The superiority of cubic GHT-Bézier spiral path smoothing techniques as compared to Clothoid and LACs is also demonstrated. </abstract>

Length and curvature variation energy minimizing planar cubic G 1 Hermite interpolation curve

In this paper, we study how to construct a planar cubic G 1 Hermite interpolation curve with minimal length and curvature variation energy. This problem is solved by a bi-objective optimization model. We prove that there is a unique solution to this problem and give the expression of the solution. Some numerical examples show that the proposed method makes the curvature variation energy of the curve as small as possible when the length of the curve as short as possible.

Optimal Path Smoothing with Log-aesthetic Curves Based on Shortest Distance, Minimum Bending Energy or Curvature Variation Energy

Optimal Path Smoothing with Log-aesthetic Curves Based on Shortest Distance, Minimum Bending Energy or Curvature Variation Energy

A [formula omitted] continuous B-spline transition algorithm for CNC machining with jerk-smooth feedrate scheduling along linear segments

Toolpath is always presented as a combination of G01 linear segments in practical computer numerical control (CNC) machining. However, the inherent tangent discontinuity at junctions will lead to uncontinuous feedrate, acceleration (i.e. the change rate of feedrate), jerk (i.e. the change rate of tangential acceleration), making it difficult to realize higher machining efficiency and better machining quality. In recent studies, the improvement of practical feedrate profile continuity is beneficial to overcome the drawbacks above, especially when smooth jerk is satisfied. In this paper, it is proved that in order to obtain smooth jerk along toolpath, G4 continuous toolpath and jerk-smooth feedrate scheduling process are both essential. On this basis, a path-smoothing algorithm is proposed to generate G4 continuous toolpath via a quintic B-spline curve. In addition, the transition curve is optimized by curvature variation energy (CVE) to produce better machining quality. Meanwhile, the chord error is constrained seriously and the curvature extreme, occurring at the middle of transition curves, can be calculated analytically. In motion planning stage, based on the bidirectional scanning algorithm, a look-ahead method is utilized to determine the reachable transition feedrate extreme and a jerk-smooth feedrate scheduling scheme is also developed. Finally, simulations and experiments are conducted to demonstrate the feasibility and efficiency of the proposed method compared to point-to-point (P2P) motion, constant feedrate strategy, G2 and G3 methods respectively.

Curvature Variation Minimizing Cardinal Spline Curves

Since the free parameter and the two auxiliary points have effect on the shape of the Cardinal spline curves, a natural idea arises to find the optimal parameter and auxiliary points to obtain the smoothest curves. We use curvature variation minimization to achieve this goal in this paper. By minimizing an appropriate approximation of the curvature variation energy, the unique solution can be easily obtained. Some examples show that the Cardinal spline curves with minimum curvature variation are better than the Catmull-Rom spline curves in dealing with interpolation problems.

Planar cubic G1 and quintic G2 Hermite interpolations via curvature variation minimization

Given two data points and the associated unit tangents, cubic G1 Hermite interpolation is a simple and efficient scheme to construct fair curves by optimizing certain energy functionals. In order to obtain shape-preserving interpolation desired for applications, this paper presents cubic G1 Hermite interpolation by minimizing curvature variation energy subject to a feasible region, with the advantage of handling arbitrary G1 data. As a result, the G1 interpolating curves can always maintain specified end tangent directions by restricting the two parameters provided by G1 constraint to be positive; and the numerical solution is obtained by an iterative algorithm using the block coordinate descend method. This approach can be further extended to quintic G2 Hermite interpolation for input G2 data. A number of comparative experiments are conducted to verify the applicability and effectiveness of the proposed method.

A locally optimal transition method with analytical calculation of transition length for computer numerical control machining of short line segments

Line segments, or G01 codes, generated by computer-aided manufacturing softwares are the most widely used toolpath format for computer numerical control systems. The linear toolpath normally consists of thousands of short line segments due to the high-accuracy requirement of the machined parts. Due to the tangential and the curvature discontinuities at the junction of two segments, the feedrates at the start and the end points of line segment have to be slowed down. In order to increase the feedrates along short line segments, a locally optimal transition method is proposed, which uses a two-step strategy to generate a blended toolpath composed of cubic Bezier curves and line segments. In the first step, the optimal proportional coefficient is represented as the function of the angle between two adjacent line segments, which can be employed to minimize the curvature variation energy of the cubic Bezier curve. In the second step, the local optimization model with the aim to minimize the sum of two curvature extrema is established to determine the optimal transition length. The transition length can be analytically obtained under the constraint of the approximation error and the constraints of the line segment lengths. The simulation and experiment results demonstrate that, by comparison with the conventional transition methods, the proposed method can significantly decrease the machining time but does not increase the contouring error.

A realtime curvature-smooth interpolation scheme and motion planning for CNC machining of short line segments

G01 codes generated by CAM (Computer Aided Manufacturing) system are the most common form of tool path in CNC (Computer Numerical Control) machining. For the piecewise linear path, tangential and curvature discontinuities bring about large fluctuation of feedrate and acceleration, which produces vibration of machine tool. In recent studies, the methods for G2 (curvature-continuous) tool-path smoothing and jerk-limiting feedrate scheduling were developed. However there still exist the deficiencies when these methods are employed in CNC machining. It is difficult to simultaneously ensure that the tool path is chord-error-constrained and G01-point-interpolated in real time. In addition, heavy computational load hinders realtime processing in CNC system. Recently the scholars experimentally found the potential of G3 (curvature-smooth) trajectory and jerk-continuous motion in reducing the vibration of machinery. This work proposes a realtime tool-path smoothing algorithm, generating G3 interpolative tool path composed by mixed linear and quartic Bezier segments. The purpose of the smoothing scheme is the simultaneous considerations of G3 continuity, confined chord error, G01 points interpolated, and realtime performance. And the tool path generated is optimized in curvature variation energy (CVE) and analytical curvature extrema is available. To reduce the vibration, a high-efficient algorithm of jerk-continuous (JC) feedrate scheduling for G3 tool path is provided. Finally, a realtime tool-path processing scheme is developed, including G3 interpolation and motion planning functions. As shown in the simulation, the contour error, curvature of tool path, feedrate fluctuation and machining time are reduced compared with G2 transition scheme. The experiment on a machine tool is conducted to demonstrate the advantages of the proposed algorithm in vibration reduction and surface quality, compared with G2 transition scheme.

A note on curvature variation minimizing cubic Hermite interpolants

In the paper [1], Jaklič and Žagar studied curvature variation minimizing cubic Hermite interpolants. To match planar two-point G1 Hermite data, they obtained the optimal cubic curve by minimizing an approximate form of the curvature variation energy. In this paper, we present a simple method for this problem by minimizing the jerk energy, which is also an approximate form of the curvature variation energy. The unique solution can be easily obtained since the jerk energy is represented as a quadratic polynomial of two unknowns and is strictly convex. Finally, we prove that our method is equivalent to their method.