scholarly journals On the Involute of the Cubic Bezier Curve by Using Matrix Representation in E3

2020 ◽  
Vol 13 (2) ◽  
pp. 216-226
Author(s):  
Şeyda Kılıçoğlu ◽  
Süleyman Şenyurt
Keyword(s):  
Vector Fields ◽  
Matrix Representation ◽  
Matrix Form ◽  
Bézier Curve ◽  
Bezier Curve ◽  
Control Points ◽  
Cubic Bezier Curve

In this study we have examined, involute of the cubic Bezier curve based on the control points with matrix form in E3. Frenet vector fields and also curvatures of involute of the cubic Bezier curve are examined based on the Frenet apparatus of the first cubic Bezier curve in E3.  

Design of a Micro-Positioning Stage Using Corrugated Flexure Beam With Cubic Bézier Curve Segments

2018 ◽  
Author(s):  
Nianfeng Wang ◽  
Zhiyuan Zhang ◽  
Xianmin Zhang
Keyword(s):  
Stiffness Matrix ◽  
Matrix Method ◽  
Bézier Curve ◽  
Bezier Curve ◽  
Axial Stiffness ◽  
Control Points ◽  
Stiffness Ratio ◽  
Element Analysis ◽  
Positioning Stage ◽  
Cubic Bezier Curve

The paper introduces an analytical stiffness matrix method to model a new type of corrugated flexure (CF) beam with cubic Bézier curve segments. In order to satisfy particular design specifications, shape variation for limited geometric envelopes are often employed to alter the elastic properties of flexure hinges. In this paper, cubic Bézier curves are introduced to replace the axis of CF unit to rebuild the CF beam and the micro-positioning stage. Mohr’s integral method is applied to derive the stiffness matrix of the cubic Bézier curve segment. Modeling of the CF unit and the CF beam with cubic Bézier curve segments are further carried out through stiffness matrix method, which are confirmed by finite element analysis (FEA). Discussions about the two control points of the cubic Bézier curve segments are then conducted through search optimization, which highlights the off-axis/axial stiffness ratio and the axial compliance on the position of the two control points, to enable the micro-positioning stage both achieving high off-axis/axial stiffness ratio and large axial compliance. The derived analytical model provides a new option for the design of the CF beam.

scholarly journals Approximate Multidegree Reduction ofλ-Bézier Curves

2016 ◽  
Vol 2016 ◽  
pp. 1-12 ◽  
Author(s):  
Gang Hu ◽  
Huanxin Cao ◽  
Suxia Zhang
Keyword(s):  
Shape Control ◽  
Control Parameter ◽  
Linear Equations ◽  
Least Square ◽  
Bézier Curve ◽  
Bezier Curve ◽  
Control Points ◽  
Degree Reduction ◽  
Bezier Curves ◽  
Bézier Curves

Besides inheriting the properties of classical Bézier curves of degreen, the correspondingλ-Bézier curves have a good performance in adjusting their shapes by changing shape control parameter. In this paper, we derive an approximation algorithm for multidegree reduction ofλ-Bézier curves in theL2-norm. By analysing the properties ofλ-Bézier curves of degreen, a method which can deal with approximatingλ-Bézier curve of degreen+1byλ-Bézier curve of degreem  (m≤n)is presented. Then, in unrestricted andC0,C1constraint conditions, the new control points of approximatingλ-Bézier curve can be obtained by solving linear equations, which can minimize the least square error between the approximating curves and the original ones. Finally, several numerical examples of degree reduction are given and the errors are computed in three conditions. The results indicate that the proposed method is effective and easy to implement.

scholarly journals Path Planning Optimization for Driverless Vehicle in Parallel Parking Integrating Radial Basis Function Neural Network

2021 ◽  
Vol 11 (17) ◽  
pp. 8178
Author(s):  
Leiyan Yu ◽  
Xianyu Wang ◽  
Zeyu Hou ◽  
Zaiyou Du ◽  
Yufeng Zeng ◽  
...  
Keyword(s):  
Neural Network ◽  
Radial Basis Function ◽  
Basis Function ◽  
Bézier Curve ◽  
Bezier Curve ◽  
Control Points ◽  
Parallel Parking ◽  
Radial Basis ◽  
Initial Path ◽  
Continuous Curvature

To optimize performances such as continuous curvature, safety, and satisfying curvature constraints of the initial planning path for driverless vehicles in parallel parking, a novel method is proposed to train control points of the Bézier curve using the radial basis function neural network method. Firstly, the composition and working process of an autonomous parking system are analyzed. An experiment concerning parking space detection is conducted using an Arduino intelligent minicar with ultrasonic sensor. Based on the analysis of the parallel parking process of experienced drivers and the idea of simulating a human driver, the initial path is planned via an arc-line-arc three segment composite curve and fitted by a quintic Bézier curve to make up for the discontinuity of curvature. Then, the radial basis function neural network is established, and slopes of points of the initial path are used as input to train and obtain horizontal ordinates of four control points in the middle of the Bézier curve. Finally, simulation experiments are carried out by MATLAB, whereby parallel parking of driverless vehicle is simulated, and the effects of the proposed method are verified. Results show the trained and optimized Bézier curve as a planning path meets the requirements of continuous curvature, safety, and curvature constraints, thus improving the abilities for parallel parking in small parking spaces.

A Concurrent Search Algorithm for Multiple Phase Transition Pathways

2013 ◽  
Author(s):  
Lijuan He ◽  
Yan Wang
Keyword(s):  
Phase Transition ◽  
Search Algorithm ◽  
Minimum Energy ◽  
Saddle Points ◽  
Bézier Curve ◽  
Bezier Curve ◽  
Conjugate Directions ◽  
Control Points ◽  
True Value ◽  
Transition Paths

The challenge of accurately predicting a phase transition in computer-aided nano-design is estimating the true value of transition rate, which is determined by the saddle point with the minimum energy barrier between stable states on the potential energy surface (PES). In this paper, a new algorithm for searching the minimum energy path (MEP) is presented. Unlike existing pathway search methods, the new algorithm is able to locate both the saddle points and local minima simultaneously. Therefore no prior knowledge of the precise positions for the reactant and product on the PES is required. In addition, the algorithm is able to search multiple transition paths on the PES simultaneously. In this method, a Bézier curve is used to represent each transition path. Starting from a single Bézier curve, multiple curves with ends connected can be generated during the search process. For each Bézier curve, the reactant and product states are located by minimizing the two end control points of the curve, while the transition pathway is refined by moving the intermediate control points of the curve in the conjugate directions. A curve subdivision scheme is developed so that multiple transition paths can be located. The algorithm is demonstrated by examples.

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