Application of several computing techniques for minimum zone straightness

This research deals with the application of several algorithms to calculate the minimum zone straightness. Generally, in the evaluation of the minimum zone value of form errors such as straightness, flatness, roundness, and cylindricity, nonlinear optimization techniques are usually applied. The problem with the nonlinear technique that depends on computing algorithm is that the computing time may be prolonged. Therefore, some linear search techniques that are relatively easy to program are applied for straightness in this article. Furthermore, the problem can also be linearized by considering the characteristics of the measured profile for straightness; thus, the reduction of computing time will be achieved. Then, the problems caused by the above consideration are clarified. Consequently, the convergence criteria and comparison of results by means of several computing methods are investigated. The relationship between flatness and straightness values in some machining conditions and the comparison with the least-squares values are studied.     

Straightness and flatness evaluation using data envelopment analysis

In today's world of precision engineering, robustness and accuracy in the evaluation of the form tolerances are considered as competitive advantages for manufacturing enterprises. Amongst various methods for accurate and robust evaluation, which have been studied, nonlinear optimization methods based on operational research have proved to be successful as far as they can ensure unique and global convergence in practical applications. However, it is well known that ensuring the convergence is the most difficult thing to deal with for a nonlinear optimization technique because the performance is in general highly sensitive to parameter setting. Therefore, this paper introduces a robust linear programming formulation-based algorithm in which the performance is not dependent on the quality of parameters. Interestingly, in this algorithm, the data envelopment analysis technique is used to form a convex hull that decides the minimum enclosed zone in a robust manner. From the computational experiments, it is shown that the proposed algorithm can be a promising alternative to the traditional nonlinear optimization method for straightness and flatness evaluation.     

On-Machine Measurement of the Straightness and Tilt Errors of a Linear Slideway Using a New Four-Sensor Method

Although there are some multi-sensor methods for measuring the straightness and tilt errors of a linear slideway, they need to be further improved in some aspects, such as suppressing measurement noise and reducing precondition.In this paper, a new four-sensor method with an improved measurement system is proposed to on-machine separate the straightness and tilt errors of a linear slideway from the sensor outputs, considering the influences of the reference surface profile and the zero-adjustment values. The improved system is achieved by adjusting a single sensor to di erent positions. Based on the system, a system of linear equations is built by fusing the sensor outputs to cancel out the e ects of the straightness and tilt errors. Three constraints are then derived and supplemented into the linear system to make the coe cient matrix full rank. To restrain the sensitivity of the solution of the linear system to the measurement noise in the sensor outputs, the Tikhonov regularization method is utilized. After the surface profile is obtained from the solution, the straightness and tilt errors are identified from the sensor outputs. To analyze the e ects of the measurement noise and the positioning errors of the sensor and the linear slideway, a series of computer simulations are carried out. An experiment is conducted for validation, showing good consistency. The new four-sensor method with the improved measurement system provides a new way to measure the straightness and tilt errors of a linear slideway, which can guarantee favorable propagations of the residuals induced by the noise and the positioning errors.     

Application of neural network interval regression method for minimum zone straightness and flatness

The goal of this paper is to develop an accurate, efficient, and robust algorithm for the minimum zone (MZ) straightness and flatness. In this paper, we use an interval bias adaptive linear neural network (NN) structure together with least mean squares (LMS) learning algorithm, and an appropriate cost function to carry out the interval regression analysis. From the results, we can see that both the straightness and flatness results from the interval regression method by NN can converge closer to the definition of the MZ straightness and flatness, respectively, than that of the least-squares (LSQ) method. The interval regression method by NN developed in this paper is applicable in the linear regression analysis that has a complicated constraint, and where the LSQ method cannot be used.     

A novel approach to thick plate theory suggested by studies in foundation theory

A new approach to the statics of thick, elastic plates is presented. This approach to thick plate theory incorporates transverse normal stress and strain effects as well as those due to transverse shear deformation. The unusual boundary value problem resulting from the kinematic assumptions of this approach is obtained as a consequence of the principle of stationary potential energy. This boundary value problem is nonlinear so that the principle of superposition does not hold and it does not appear to be one which may be treated by standard methods. Therefore an iterative method of approximation is proposed. This iterative method is applied, analytically, to a problem of cylindrical plate bending for which an exact solution is known within the linear theory of elasticity. It is shown that for this particular problem the approximation method converges in two iterations to the exact elasticity solution. Since such simple results are not to be expected in general, there is some discussion of the need to develop a numerical algorithm to implement the proposed iterative method of approximation for arbitrary plate problems. Furthermore, certain theoretical difficulties with the new approach to thick plate theory are discussed.     

Optimization of a structure with contact conditions using equivalent loads

Engineering structures consist of various components, and the components interact with each other through contact. Engineers tend to consider the interaction in analysis and design. Interactions of the components have nonlinearity because of the friction force and boundary conditions. Nonlinear analysis has been developed to accommodate the contact condition. However, structural optimization using nonlinear analysis is fairly expensive, and sensitivity information is difficult to calculate. Therefore, an efficient optimization method using nonlinear analysis is needed to consider the contact condition in design. Nonlinear Response Optimization using Equivalent Loads (NROEL) has been proposed for nonlinear response structural optimization. The method was originally developed for optimization problems considering geometric/material nonlinearities. The method is modified to consider the contact nonlinearity in this research. Equivalent loads are defined as the loads for linear analysis, which generate the same response field as that of nonlinear analysis. A nonlinear response optimization problem is converted to linear response optimization with equivalent loads. The modified NROEL is verified through three examples with contact conditions. Three structural examples using the finite element method are demonstrated. They are shape optimization with stress constraints, size optimization with stress/displacement constraints and topology optimization. Reasonable results are obtained in the optimization process.     

Axially constrained beams on elastic foundation

The problem of bending in axially constrained beams on a Winkler-type elastic foundation is considered. This nonlinear problem is approached by a differential equation method as well as a finite element method. Numerical results are presented which show that for a low modulus of the foundation the linear solution for deflection (which neglects axial constraint effect) yields deflection values which may be several times higher than the exact solution. The two methods are shown to be in good agreement.     

改进蜂群算法在平面度误差评定中的应用

为了准确快速评定平面度误差,提出将改进人工蜂群( MABC)算法用于平面度误差最小区域的评定.介绍了评定平面度误差的最小包容区域法及判别准则,并给出符合最小区域条件的平面度误差评定数学模型.叙述了MABC算法,该算法在基本人工蜂群算法( ABC)模型的基础上引入两个牵引蜂和禁忌搜索策略.阐述了算法的实现步骤,通过分析选用两个经典测试函数验证了MABC算法的有效性.最后,应用MABC算法对平面度误差进行评定,其计算结果符合最小条件.对一组测量数据的评定显示,MABC算法经过0.436 s可找到最优平面,比ABC算法节省0.411 s,其计算结果比最小二乘法和遗传算法的评定结果分别小18.03μm和6.13 μm.对由三坐标机测得的5组实例同样显示,MABC算法的计算精度比遗传算法和粒子群算法更有优势,最大相差0.9 μm.实验结果表明,MABC算法在优化效率、求解质量和稳定性上优于ABC算法,计算精度优于最小二乘法、遗传算法和粒子群算法,适用于形位误差测量仪器及三坐标测量机.     

Assembly Tolerance Stack Analysis for Geometric Characteristics in Form Control – the "Catena" Method

Tolerance stack analysis involving geometric characteristics is usually difficult as it involves numerous rules and conditions. Users need to acquire an adequate knowledge of geometric dimensioning and tolerancing before making any attempt at the computation of tolerance stack analysis problem. Conventional stack form methods are very widely used in tolerance stack analysis, but the computation requires tedious work with numerous rules to remember that often complicate the problem and confuse the user. A need therefore arises to simplify the tolerance stack methodology. This paper presents a straight forward, easy-to-use graphical approach known as the "Catena" method for tolerance stack analysis, involving geometric characteristics in form control – flatness, straightness, circu-larity and cylindricity. Three examples are given to illustrate the effectiveness of the proposed method. No complicated mathematical formulae are required in deriving the solution.     

应用非线性优化算法自主标定星敏感器

考虑传统的星敏感器标定方法忽略了星敏感器的畸变与光学参数之间的相互作用而引入的额外误差,提出了一种基于非线性优化的星敏感器自主标定算法。该算法首先忽略星敏感器畸变的影响,构建目标函数,利用LevenbergMarquardt非线性优化算法优化星敏感器的光学参数;然后,将得到的光学参数估计值作为理想值,通过线性最小二乘法估计相机的镜头畸变系数;最后,将前两个步骤获得的参数作为初始值,构建目标函数,利用Levenberg-Marquardt算法同时优化光学参数和畸变系数。开展了仿真实验研究,并与最小二乘法和Samman法的标定结果做了对比,结果表明:提出的方法能够很好地实现星敏感器的自主标定。在同等测试条件下,文中算法获得的最大残差为0.015pixels,精度高于其它两种标定方法两个数量级。星敏感器外场实验还表明,提出的优化方法有效提升了星敏感器的性能。     

电梯导轨几何误差测试系统

开发了一套电梯导轨测试系统，该系统用于静态测量电梯导轨的几何参数，包括导轨三个导向面的直线度、平面度、导轨的宽度、高度、垂直度和扭曲度等。在测量系统中，5个光栅传感器通过适当配置用于测量电梯导轨三个面的直线度，两侧面的平面度和导轨的其他参数可以通过这些传感器的值经过适当的数学变换获得。用一个双频激光干涉仪标定了系统导轨的运动误差，用有限元分析方法对测量时导轨的挠度进行建模，进行了系统误差和挠度的补偿。实验表明，系统的测量误差小于0．05mm。     

Verification of form tolerances part II: Cylindricity and straightness of a median line

Most inspectors measure form tolerances as the minimum zone solution, which minimizes the maximum error between the datapoints and a reference feature. Current coordinate measuring machines verification algorithms are based on the least-squares solution, which minimizes the sum of the squared errors, resulting in a possible overestimation of the form tolerance. Therefore, although coordinate measuring machines algorithms successfully reject bad parts, they may also reject some good parts. The verification algorithms developed in this set of papers compute the minimum zone solution of a set of datapoints sampled from a part. Computing the minimum zone solution is inherently a nonlinear optimization problem. This paper develops a single verification methodology that can be applied to the cylindricity and straightness of a median line problems. The final implementable formulation solves a sequence of linear programs that converge to a local optimal solution. Given adequate initial conditions, this solution will be the minimum zone solution. This methodology is also applied to the problems of computing the minimum circumscribed cylinder and the maximum inscribed cylinder. Experimental evidence that the formulations are both robust and efficient is provided.     

时变多状态滞后不确定系统的保性能控制

针对时变多状态滞后不确定线性系统 ,研究了状态反馈保性能控制律的设计问题 ,采用线性矩阵不等式方法 ,导出了保性能控制律的存在条件和参数化表示。通过求解一个线性矩阵不等式约束的凸优化问题 ,提出了最优化保性能控制律的设计方法     

基于MATLAB的平面度评定方法

评定和计算平面度的过程 ,实质上是根据平面度的定义构造函数模型并进行函数优化求解的过程。本文利用MATLAB优化工具箱 ,实现了最小区域法、最小二乘法平面度的评定     

时域三点法测量直线度信息高精度获取方法研究

常规时域三点法在直线度误差测量中会把零位和随机误差的影响迭代放大,使得直线度重构曲线失真,无法得到正确的直线度误差。通过对常规时域三点法数据处理误差放大机理进行研究,提出了基于比较消除法的误差等效均化的处理办法,并利用仿真和实验证实了其有效性和实用性。优化了时域三点法的直线度测量,使得直线度信息高精度获取成为可能。     

A combinatorial optimization approach for evaluating minimum-zone spatial straightness errors

This paper presents a new and robust approach for the accurate evaluation of minimum-zone spatial straightness error from a set of coordinate measurement data points. The algorithm iteratively searches for the specific data points that define the minimum bound of the spatial straightness zone using combinatorial optimization. It is based on the fact that the minimum circumscribed cylinder of a point set, which is equivalent to the minimum spatial straightness zone of the measurement data, will pass through three, four, or five of the data points that constitute the convex hull vertices of the entire data set. Computed results have shown that although the presented approach may lead to increased computational time, it is robust and able to construct the exact minimum circumscribed cylinder for a given point set. The minimum-zone spatial straightness error can thus be evaluated with the best possible accuracy. The advantage of the presented algorithm is demonstrated via comparison with published computed results of existing algorithms.     

具有区间参数的不确定结构静力区间分析的一种算法

将结构系统中的不确定性参数用区间数来表示,用有限元法建立静力区间线性方程组。对该方程组的求解提出了一种区间逐步离散的方法。此方法通过令独立的不确定性参数取区间内的离散值,将区间线性方程组的求解转化为相应的确定性问题,再搜索各方程解中的最大最小值得到每个区间分量的边界。先用数学算例对该算法的正确性和有效性进行了验证,然后应用于静力区间分析的工程算例,并与其它算法进行了比较。计算结果表明该算法的计算效率和准确性较高。     

Dimensional and geometric tolerance design based on constraints

The paper presents a computer-aided approach of dimensional and geometric tolerance design. The method allows a designer to specify synthetically dimensional and geometric tolerances, including tolerance types and values. Firstly, tolerances are classified as self- and cross-referenced tolerances, and the rules for tolerance types design are presented. Secondly, the stack-up of 3D feature variation is formulated as a set of stack-up constraints (equation constraints), and the variation specified by tolerance forms tolerance constraints (inequality constraints). Tolerance value design is represented as the combinatorial optimization problem. The application of the variation and tolerance constraints to specify tolerance values is studied. Finally, a tolerance design example is used to illustrate the method.     

Evaluation of minimum zone flatness by means of nonlinear optimization techniques and its verification

This article deals with the application of some nonlinear optimization techniques for minimum zone flatness. The convergence criteria of the techniques, namely the downhill simplex method and the repetitive bracketing method, are considered. The least-squares method is also applied, and subsequently the three methods are compared from the viewpoint of computational accuracy. A surface profile measuring system and a noncontact sensor are used to obtain three-dimensional data. The measured data are expressed by means of perspective mapping. Subsequently, the relationship among the above three methods is clarified according to accuracy and efficiency of the computation. Furthermore, some examples of the relationship between the manufacturing method and the flatness value, and the technique of a skilled hand are described.     

Three-dimensional trajectory design for horizontal well based on optimal switching algorithms

This paper considers a three-dimensional trajectory design problem for horizontal well. The problem is formulated as an optimal control problem of switched systems with continuous state inequality constraints. Since the complexity of such constraints and the switching instants is unknown, it is difficult to solve the problem by standard optimization techniques. To overcome the difficulty, by a time-scaling transformation, a smoothing technique and a penalty function method, an efficient computational method is proposed for solving this problem. Convergence results show that, for a sufficiently large penalty parameter, any local optimal solution of the approximate problem is also a local optimal solution of the original problem. Two numerical examples are presented to illustrate the efficiency of the approach proposed.