A parametric interpolation method with minimal feedrate fluctuation by nonuniform rational basis spline

2016 ◽  
Vol 231 (18) ◽  
pp. 3301-3317 ◽  
Author(s):  
Lei Lu ◽  
Lei Zhang ◽  
Yan Gu ◽  
Ji Zhao
Keyword(s):  
Computational Efficiency ◽  
Interpolation Method ◽  
Optimization Method ◽  
Rational Basis ◽  
Computational Load ◽  
Fitting Method ◽  
Feedrate Fluctuation ◽  
Data Points ◽  
Parametric Interpolation ◽  
Line Fitting

Because the relation between the arc length s and curve parameter u cannot be represented by explicit function for most of the curves, it is difficult to consider the accuracy, robustness, and computational efficiency for most of the parametric interpolation, especially when the curves are complex or extremes. Therefore, an off-line fitting interpolation method by using nonuniform rational basis spline is presented in this paper. As nonuniform rational basis spline has many geometry implementation tools and numerous good properties as compared to the polynomial, the required fitting accuracy can be obtained more easily than with polynomial. After the de Boor method is applied, the computational load of nonuniform rational basis spline is decreased as compared to the Taylor approximation and the higher order polynomial fitting method. In order to obtain the proper s-u fitting nonuniform rational basis spline and reduce the computational load of the fitting process, the sampled s-u data points are divided according to the properties of nonuniform rational basis spline, and in each segment, the knot vectors, control points, and weights are calculated by the iterative-optimization method. Then the s-u nonuniform rational basis spline can be applied in real-time interpolation, and the accuracy, robustness, and computational efficiency are demonstrated by simulations and experiments.

A segment feedrate profile fitting method for parametric interpolation

2016 ◽  
Vol 232 (2) ◽  
pp. 267-280 ◽  
Author(s):  
Lei Lu ◽  
Lei Zhang ◽  
Shijun Ji ◽  
Dunlan Song ◽  
Ji Zhao
Keyword(s):  
Tool Path ◽  
Least Squares Method ◽  
Machining Accuracy ◽  
The Novel ◽  
Profile Fitting ◽  
Fitting Method ◽  
Calculation Process ◽  
Data Points ◽  
Simulation And Experiment ◽  
Parametric Interpolation

There are many researches in scheduling an optimal feedrate profile under various constraints by numerical calculation. A large number of discrete feedrate data points are obtained. They are inconvenient for the parametric interpolator. Therefore, these discrete feedrate data points need to be fitted by parameter curves. Different from the regular curve fitting, the inappropriate feedrate fitting method can generate larger acceleration and jerk that seriously affect the machining accuracy and stability, although the feedrate satisfies the error requirements. In order to generate a suitable feedrate profile, a segment feedrate profile fitting method using B-spline is proposed in this article. The discrete feedrate data points are segmented in the jerk discontinuous points. In each segment, the feedrate profile is fitted by the linear least squares method. These fitted feedrate profiles are combined to generate a unified feedrate profile. The unified fitted feedrate profile and the tool path trajectory are used in the controller to command the axis. In this article, the process of parametric interpolation is separated into the arc-length calculation process and the curve parameter calculation process. Using parallel computation, the two processes are calculated simultaneously in the controller, and the computational efficiency is improved. Both simulation and experiment are carried out to verify that the fitted feedrate profile satisfies the error requirements, and the novel interpolation can be applied to the controller appropriately.

scholarly journals A parametric interpolation method based on prediction and iterative compensation

2019 ◽  
Vol 16 (1) ◽  
pp. 172988141982818 ◽  
Author(s):  
Hepeng Ni ◽  
Chengrui Zhang ◽  
Chao Chen ◽  
Tianliang Hu ◽  
Yanan Liu
Keyword(s):  
Tool Path ◽  
Interpolation Method ◽  
Second Order ◽  
Prediction Algorithm ◽  
Target Point ◽  
Original Curve ◽  
Current Period ◽  
Quintic Polynomial ◽  
Feedrate Fluctuation ◽  
Parametric Interpolation

Parametric interpolation for spline plays an increasingly important role in modern manufacturing. It is critical to develop a fast parametric interpolator with high accuracy. To improve the computational efficiency while guaranteeing low and controllable feedrate fluctuation, a novel parametric interpolation method based on prediction and iterative compensation is proposed in this article. First, the feedrate fluctuation and Taylor’s expansion are analyzed that there are two main reasons to reduce the calculation accuracy including the truncation errors caused by neglecting the high-order terms and discrepancy errors between the original curve and the actual tool path. Then, to reduce these errors, a novel parametric interpolation method is proposed with two main stages, namely, prediction and iterative compensation. In the first stage, a quintic polynomial prediction algorithm is designed based on the historical interpolation knowledge to estimate the target length used in the second-order Taylor’s expansion, which can improve the calculation accuracy and the convergence rate of iterative process. In the second stage, an iterative compensation algorithm based on the second-order Taylor’s expansion and feedrate fluctuation is designed to approach the target point. Therefore, the calculation accuracy is controllable and can satisfy the specified value through several iterations. When finishing the interpolation of current period, the historical knowledge is updated to prepare for the following interpolation. Finally, a series of simulations are conducted to evaluate the good performance in accuracy and efficiency of the proposed method.

Processing of 3D Unstructured Measurement Data for Reverse Engineering

2011 ◽  
Vol 1 (2) ◽  
pp. 42-51
Author(s):  
Yongmin Zhong
Keyword(s):  
Reverse Engineering ◽  
Interpolation Method ◽  
Measurement Data ◽  
Optimization Method ◽  
Structured Data ◽  
Unstructured Data ◽  
Data Points ◽  
Nurbs Surfaces ◽  
C1 Continuity ◽  
Segment Data

One of the most difficult problems in reverse engineering is the processing of unstructured data. NURBS (Non-uniform Rational B-splines) surfaces are a popular tool for surface modeling. However, they cannot be directly created from unstructured data, as they are defined on a four-sided domain with explicit parametric directions. Therefore, in reverse engineering, it is necessary to process unstructured data into structured data which enables the creation of NURBS surfaces. This paper presents a methodology to processing unstructured data into the structured data for creating NURBS surfaces. A projection based method is established for constructing 3D triangulation from unstructured data. An optimization method is also established to optimize the 3D triangulation to ensure that the resulted NURBS surfaces have a better form. A triangular surface interpolation method is established for constructing triangular surfaces from the triangulation. This method creates five-degree triangular surfaces with C1 continuity. A series of segment data are obtained by cutting the triangular surfaces with a series of parallel planes. Finally, the structured data is obtained by deleting repetitive data points in each segment data. Results demonstrate the efficacy of the proposed methodology.

Processing of 3D Unstructured Measurement Data for Reverse Engineering

2013 ◽  
pp. 118-127
Author(s):  
Yongmin Zhong
Keyword(s):  
Reverse Engineering ◽  
Interpolation Method ◽  
Measurement Data ◽  
Optimization Method ◽  
Structured Data ◽  
Unstructured Data ◽  
Data Points ◽  
Nurbs Surfaces ◽  
C1 Continuity ◽  
Segment Data

One of the most difficult problems in reverse engineering is the processing of unstructured data. NURBS (Non-uniform Rational B-splines) surfaces are a popular tool for surface modeling. However, they cannot be directly created from unstructured data, as they are defined on a four-sided domain with explicit parametric directions. Therefore, in reverse engineering, it is necessary to process unstructured data into structured data which enables the creation of NURBS surfaces. This paper presents a methodology to processing unstructured data into the structured data for creating NURBS surfaces. A projection based method is established for constructing 3D triangulation from unstructured data. An optimization method is also established to optimize the 3D triangulation to ensure that the resulted NURBS surfaces have a better form. A triangular surface interpolation method is established for constructing triangular surfaces from the triangulation. This method creates five-degree triangular surfaces with C1 continuity. A series of segment data are obtained by cutting the triangular surfaces with a series of parallel planes. Finally, the structured data is obtained by deleting repetitive data points in each segment data. Results demonstrate the efficacy of the proposed methodology.

Efficient Feedrate Optimization Method for Spline Toolpath With Curvature-Base Planning and Accurate Interpolating

2018 ◽  
Author(s):  
Yong Zhang ◽  
Mingyong Zhao ◽  
Peiqing Ye ◽  
Jiali Jiang ◽  
Hui Zhang
Keyword(s):  
Computational Efficiency ◽  
High Speed ◽  
Optimization Problem ◽  
Linear Optimization ◽  
Extreme Points ◽  
Optimization Method ◽  
Feedrate Optimization ◽  
Window Technique ◽  
Feedrate Fluctuation ◽  
Relationship Of

The well-designed feedrate optimization algorithm can obtain higher machining efficiency with various machining related constraints, thus, it is widely considered in the high-speed and high-precision machining. However, the low computational efficiency still limits the application of the optimization method. For the non-linear optimization problem of spline toolpath with feedrate-, actuator velocity-, acceleration- and jerk-limited, a linear approximation is adopted by a pseudo-jerk method and the efficient linear programming method can be applied to solve the optimization problem. To improve computational efficiency further, curvature-base window technique is presented and the whole spline toolpath is split at the curvature extreme points, which are also named critical points in traditional planning method. Thereafter, a novel feedback interpolation is presented based on Steffensen iterative accelerator method to eliminate the feedrate fluctuation caused by nonanalytic relationship of spline parameter and arc-length. Finally, simulations and experiments validations show that the proposed method is able to reduce computational burden and traversal time notably with multi-constraints.

MESHFREE CELL-BASED SMOOTHED POINT INTERPOLATION METHOD USING ISOPARAMETRIC PIM SHAPE FUNCTIONS AND CONDENSED RPIM SHAPE FUNCTIONS

2011 ◽  
Vol 08 (04) ◽  
pp. 705-730 ◽  
Author(s):  
G. Y. ZHANG ◽  
G. R. LIU
Keyword(s):  
Computational Efficiency ◽  
Interpolation Method ◽  
Shape Functions ◽  
Generalized Gradient ◽  
Singularity Problem ◽  
Numerical Examples ◽  
Point Interpolation Method ◽  
Point Interpolation ◽  
System Equations ◽  
Gradient Smoothing

This paper presents two novel and effective cell-based smoothed point interpolation methods (CS-PIM) using isoparametric PIM (PIM-Iso) shape functions and condensed radial PIM (RPIM-Cd) shape functions respectively. These two types of PIM shape functions can successfully overcome the singularity problem occurred in the process of creating PIM shape functions and make the constructed CS-PIM models work well with the three-node triangular meshes. Smoothed strains are obtained by performing the generalized gradient smoothing operation over each triangular background cells, because the nodal PIM shape functions can be discontinuous. The generalized smoothed Galerkin (GS-Galerkin) weakform is used to create the discretized system equations. Some numerical examples are studied to examine various properties of the present methods in terms of accuracy, convergence, and computational efficiency.

scholarly journals Target Localization Based on Bistatic T/R Pair Selection in GNSS-Based Multistatic Radar System

2021 ◽  
Vol 13 (4) ◽  
pp. 707
Author(s):  
Yu’e Shao ◽  
Hui Ma ◽  
Shenghua Zhou ◽  
Xue Wang ◽  
Michail Antoniou ◽  
...  
Keyword(s):  
Optimization Methods ◽  
Optimization Method ◽  
Satellite System ◽  
Radar System ◽  
Target Localization ◽  
Computational Load ◽  
Matrix Fusion ◽  
Global Navigation Satellite ◽  
Multistatic Radar ◽  
Combined Algorithm

To cope with the increasingly complex electromagnetic environment, multistatic radar systems, especially the passive multistatic radar, are becoming a trend of future radar development due to their advantages in anti-electronic jam, anti-destruction properties, and no electromagnetic pollution. However, one problem with this multi-source network is that it brings a huge amount of information and leads to considerable computational load. Aiming at the problem, this paper introduces the idea of selecting external illuminators in the multistatic passive radar system. Its essence is to optimize the configuration of multistatic T/R pairs. Based on this, this paper respectively proposes two multi-source optimization algorithms from the perspective of resolution unit and resolution capability, the Covariance Matrix Fusion Method and Convex Hull Optimization Method, and then uses a Global Navigation Satellite System (GNSS) as an external illuminator to verify the algorithms. The experimental results show that the two optimization methods significantly improve the accuracy of multistatic positioning, and obtain a more reasonable use of system resources. To evaluate the algorithm performance under large number of transmitting/receiving stations, further simulation was conducted, in which a combination of the two algorithms were applied and the combined algorithm has shown its effectiveness in minimize the computational load and retain the target localization precision at the same time.

Benchmarking and Field-Testing of the Distributed Quasi-Newton Derivative-Free Optimization Method for Field Development Optimization

2021 ◽  
Author(s):  
Faruk Alpak ◽  
Yixuan Wang ◽  
Guohua Gao ◽  
Vivek Jain
Keyword(s):  
Optimization Problems ◽  
Field Testing ◽  
Optimization Method ◽  
Training Data ◽  
Local Optima ◽  
Field Development ◽  
Derivative Free Optimization ◽  
Derivative Free ◽  
Data Points ◽  
Quasi Newton

Abstract Recently, a novel distributed quasi-Newton (DQN) derivative-free optimization (DFO) method was developed for generic reservoir performance optimization problems including well-location optimization (WLO) and well-control optimization (WCO). DQN is designed to effectively locate multiple local optima of highly nonlinear optimization problems. However, its performance has neither been validated by realistic applications nor compared to other DFO methods. We have integrated DQN into a versatile field-development optimization platform designed specifically for iterative workflows enabled through distributed-parallel flow simulations. DQN is benchmarked against alternative DFO techniques, namely, the Broyden–Fletcher–Goldfarb–Shanno (BFGS) method hybridized with Direct Pattern Search (BFGS-DPS), Mesh Adaptive Direct Search (MADS), Particle Swarm Optimization (PSO), and Genetic Algorithm (GA). DQN is a multi-thread optimization method that distributes an ensemble of optimization tasks among multiple high-performance-computing nodes. Thus, it can locate multiple optima of the objective function in parallel within a single run. Simulation results computed from one DQN optimization thread are shared with others by updating a unified set of training data points composed of responses (implicit variables) of all successful simulation jobs. The sensitivity matrix at the current best solution of each optimization thread is approximated by a linear-interpolation technique using all or a subset of training-data points. The gradient of the objective function is analytically computed using the estimated sensitivities of implicit variables with respect to explicit variables. The Hessian matrix is then updated using the quasi-Newton method. A new search point for each thread is solved from a trust-region subproblem for the next iteration. In contrast, other DFO methods rely on a single-thread optimization paradigm that can only locate a single optimum. To locate multiple optima, one must repeat the same optimization process multiple times starting from different initial guesses for such methods. Moreover, simulation results generated from a single-thread optimization task cannot be shared with other tasks. Benchmarking results are presented for synthetic yet challenging WLO and WCO problems. Finally, DQN method is field-tested on two realistic applications. DQN identifies the global optimum with the least number of simulations and the shortest run time on a synthetic problem with known solution. On other benchmarking problems without a known solution, DQN identified compatible local optima with reasonably smaller numbers of simulations compared to alternative techniques. Field-testing results reinforce the auspicious computational attributes of DQN. Overall, the results indicate that DQN is a novel and effective parallel algorithm for field-scale development optimization problems.

Multi-Objective Trajectory Optimization of Free-Floating Space Manipulator Using NSGA-II

2015 ◽  
Vol 713-715 ◽  
pp. 800-804 ◽  
Author(s):  
Gang Chen ◽  
Cong Wei ◽  
Qing Xuan Jia ◽  
Han Xu Sun ◽  
Bo Yang Yu
Keyword(s):  
Trajectory Optimization ◽  
Interpolation Method ◽  
Optimal Solution ◽  
Optimization Method ◽  
Joint Space ◽  
Motion Path ◽  
Objective Functions ◽  
Nsga Ii ◽  
Space Manipulator ◽  
Multi Objective

In this paper, a kind of multi-objective trajectory optimization method based on non-dominated sorting genetic algorithm II (NSGA-II) is proposed for free-floating space manipulator. The aim is to optimize the motion path of the space manipulator with joint angle constraints and joint velocity constraints. Firstly, the kinematics and dynamics model are built. Secondly, the 3-5-3 piecewise polynomial is selected as interpolation method for trajectory planning of joint space. Thirdly, three objective functions are established to simultaneously minimize execution time, energy consumption and jerk of the joints. At last, the objective functions are combined with the NSGA-II algorithm to get the Pareto optimal solution set. The effectiveness of the mentioned method is verified by simulations.

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