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毕业（设计）论文外 文 翻 译系 别 机电工程 专 业 机械设计制造及自动化 班 级 161003 学生姓名 尚晓东 学 号 103353 指导教师 李文燕 日 期 2014.2.1 Synthetic error modeling for NC machine tools based on intelligent technologyXinhua Yao, Jianzhong Fu*, Yuetong Xu, Yong HeAbstractThe precision of machine tools is greatly constrained by errors either built into the machine tools or occurring on a periodic basis on account of temperature changes or variation in cutting forces, so it is essential to obtain these errors, and then eliminate or compensate for them. However, the interaction between many factors inducing errors, such as the heat source, thermal expansion coefficient, the machine system configuration and the running environment, creates complex behavior of a machine tool, and also makes synthetic error prediction difficult with traditional mathematics. Therefore, several modeling methods based on non-classical mathematics have been presented in recent years. The intelligent technology methods of neural network, support vector machines, Bayesian networks are the effective modeling and forecasting methods for machine errors. All these three methods were introduced in briefly in the paper, and the characteristics of them were discussed. A series of experiments were carried out to evaluate their merits and defects. Finally, some important conclusions about how to use these methods in different situations were provided. The works in this paper make a special summary of the error modeling with intelligent technology, and provide a useful guidance to further research on error compensation of NC machine tools.Keywords: synthetic error modeling; neural network; support vector machines; Bayesian networks; intelligent technology 1. IntroductionMachine errors including cutting-force induced, fixture dependent and thermal errors, which we name synthetic error here to distinguish it from each kind of individual error, significantly affect the precision of machine tools, so it is essential to obtain these errors, and then eliminate or compensate for them1. In the prediction of machine errors, most current research has focused on arriving at a direct correlation between the monitoring parameters and error data obtained through experiments. Once a correlation is established, the model is then used to forecast the error under any other condition based on the monitoring data. 2 However, some difficulties have all along remained in modeling for the influence of properties of synthetic error. Since the accuracy of a machine tool is affected by the overall effect of the various error sources mentioned earlier, the error compensation system should take into consideration the interaction between these sources rather than consider each error in isolation1. This situation leads to two problems: one is that the synthetic error becomes a time-varying nonlinear and process, and the other is that the various interactions make it hard to understand the relationships among various factors. These problems affect the model accuracy directly, and limit the use of traditional fitting methods in modeling errors. Therefore, some modeling methods based on non-classical mathematics have been presented in recent years. The intelligent technology methods of neural network, support vector machines, Bayesian networks are the effective ones of them for synthetic errors. This paper seeks to present these intelligent technology methods in synthetic error modeling and is organized as follows: in the next section, some important work has been done in this field is introduced; In Section 3, the three modeling method based on neural network, support vector machines and Bayesian networks are introduced respectively; The comparison experiment is shown in Section 4 and the result is analyzed; Finally, some conclusions are presented in Section 5.2. Previous WorkVarious intelligence techniques have been employed in the modeling of machine errors in recent 15 years. Artificial neural network (ANN) is the most important one of them. Yang et al.3 propose d a model articulation controller (CMAC) neural network to correlate machine thermal errors to temperature measurements through large amount of experiments and data analysis techniques. Chen4 used a three-layer ANN with a supervised back propagation training algorithm and the sigmoid activation function to map the calibrated thermal errors to the temperature measurements. However, the results were found to be unacceptable where input conditions were significantly different from the data set used to train the network. A neural network based on artificial resonance theory (ART-map) was used to predict and compensate the tool point errors of a 3-axis machining center by Mize and Ziegert5 using discrete temperature readings from the machines structure as input. Fu and Chen6 presented an artificial neural network method which was combine with fuzzy logic to predict thermal error. A new method of principal component analysis was presented to identify the structure and parameters of the fuzzy model. Then, the artificial neural network method was used to overcome the problem which the fuzzy model could not resolve, because the effective data number was smaller than the consequence parameter number. Yang and Ni7 used the dynamic neural network model to track nonlinear time-varying machine tool errors under various thermal conditions. Study has shown that the dynamic models have the advantage over static models in terms of model robustness to different working conditions, since the dynamics and the nonlinearity of the thermal-elastic process are captured using the temporal association networks.Support vector machine (SVM) is another useful method to model errors in machine tools for its good performance in classification and regression. Ramesh et al.8 developed a model using support vector machines to classify the error based on operating conditions. Once classified, the error was then predicted based on the temperature states. Lin et al.9 presented a modeling methodology using the least squares support vector machine (LS-SVM) model to track nonlinear time varying spindle thermal error under certain conditions. Experiments on spindle thermal deformation were conducted to evaluate the model performance in terms of model estimation accuracy and robustness. The comparison has indicated that the LS-SVM performs better than other modeling methods, such as multivariable least squares regression analysis, in terms of model accuracy and robustness. Recently, He et al.10 used SVM to predict the volumetric errors and also acquired high modeling accuracy.Bayesian networks (BN) is a probabilistic representation for uncertain relationships and is useful for modeling real-world problems such as diagnosis, forecasting, manufacturing control, etc., wherein there exist multiple cause and effect dependency relations11. It is a relatively new technique for thermal errors modeling. Ramesh et al.2 presented a hybrid SVM-BN model to account for the specific conditions inducing thermal errors. The experimental data was first classified using a BN model with a rule-based system. Once the classification had been effected, the error was predicted using a SVM model. Yao et al.12 described causal relationships of factors inducing thermal deformation by graph theory and estimated the thermal error by Bayesian statistical techniques. Due to the effective combination of domain knowledge and sampled data, the BN method could adapt to the change of running state of machine, and obtain satisfactory prediction accuracy.3. Intelligent Techniques for Synthetic Error Modeling3.1. Artificial neural networkThe artificial neural network is a kind of multiple nonlinear model in which the coefficients are called weights which are evaluated by training with an iterative technique called back-propagation. Thus, an ANN is appropriate for a system having a nonlinear relationship between multiple inputs and multiple outputs. There are three types of layer for the neural network, i.e. input layer, hidden layer, and output layer. For the synthetic error application, the input layer is designed to consist of sensor monitoring data such as temperature and force, and the output layer of the spindle errors. The hidden layers represent intermediate nodes, which are determined so that the neural model has the least errors. After the hidden layers are determined by testing the output layers, the output, y*, can be calculated as Eq.(1)6.Then, the fuzzy-neural network for spindle error y* can be constructed as Fig. 1. When the structure of the neural network is determined, some special data groups should be chosen to train this model. When the model precision is not enough, the fuzzy neural network should be trained a lot of times until the precision is satisfied.3.2. Least squares support vector machineSVM can be applied to regression by the introduction of an alternative loss function and the results appear to be very encouraging. In SVM, the basic idea is to map the data into a higher-dimensional feature space via a nonlinear mapping and then to do regression in this space. The optimization problem of the least squares support vector machine (LS-SVM) which is a modification of Vapnik SVM is formulated as Eq.(2)13: Fig. 1. Structure of the fuzzy neural network Then, optimization problem can be rewritten asFinally, nonlinear function takes the form:This LS-SVM regression leads to solving a set of linear equations, which is for many practitioners in different areas. Especially, the solution by solving a linear system is instead of quadratic programming. It can decrease the model algorithm complexity and shorten the computing time greatly.3.3. Bayesian networksA BN is a graphical model consisting of nodes which represent causes and effects in real-world situations and a set of edges each of which connects two nodes. If there exists a causal relationship between any two of the nodes, the edge would be directional leading from the cause variable to the effect variable. For such directed edges, the edge is said to be from parent to child. A BN is also called a directed acyclic graph for this reason as all the edges of the graph point in a particular direction and there is no way to start from one node, travel along a set of directed edges in the proper direction and arrive at the same node. A BN for a set of variables X=X1, ., Xn consists of a network S that encodes a set of conditional independence assertions about variables in X and a set P of local probability distributions associated with each variable11. Together, these components define the joint probability distribution for X. For a joint probability distribution, from the chain rule of probability, we have Eq.(7) Now, we assume that the number of nodes in a BN is n, the average number of parent nodes is m, and each variable owns k values on average. Then, the computation complexity of Eq.(9) is nkm, while it is kn1 to Eq.(7). Obviously, the time cost will decline by using Eq.(9) when m is far smaller than n. In general, the computation of a probability of a constructed model is known as probabilistic inference. In this section, probabilistic inference in BNs is described briefly. To a BN whose structure has been fixed, if there are N cases in sampled data, the inference can be calculated as follows14:Especially, if there is only one variable Xi to be calculated, Eq.(10) can be simplified as Fig.2. Synthetic error measurementWith the help of Bayesian statistics, prior knowledge of a domain can be combined well with sampled data. Accordingly, modeling of synthetic errors in machine tools will be implemented by two steps: (1) Construct network from prior knowledge. Here we should identify the factors that are of critical importance in inducing machine errors, analyze the interrelation among them, then describe all this information by building a directed acyclic graph, and finally assess prior probability of each variable in graph. (2) Conduct probabilities learning and inference in the network based on sampled data. The parameters of network will be refined, and the result of modeling can be calculated by probabilistic inference.4. Experiment and Discussion4.1. Experiment setupWithout loss of generality, this paper presents models for a high speed spindle of 10 KW. The models developed for this type of spindle are transferable to other spindles. The experimental setup that is used to measure synthetic error is shown in Fig. 2. Spindle error in radial and axial direction is respectively measured using two CCD laser displacement sensors (LK150H, KEYENCE), as shown in Fig. 2. To model and forecast synthetic error with three intelligent methods discussed above, there are three major factors inducing errors are taken into account:1. he temperatures of several key parts of the machine tool and the entironment, which monitored by thermal sensors (OS18B20, DALLAS);2. Volumetric error which measured by laser interferometer with vector diagonal step method;3. Machining parameters including rotation speed of spindle and the feed rate.4.2. Results and discussionIn experiments, temperatures are measured every 1 second during motion, using numerical thermal sensors that are attached to the spindle and other positions. The machine has been running for 2.5 hours and then stops to cool down for 2 hours. A set of data from displacement sensors is recorded for every 1 minute. For models comparison, three models are built using the same model training data. The modeling results for axial deformation from three methods and the actual value of machine are shown in Fig.3, and the results for radial deformation are shown in Fig.4. Furthermore, the modeling results are compared based on the mean absolute percentage error (MAPE) as defined below:Where di is the modeling value, while is the actual value.The comparison in MAPE and calculating time among the three methods are shown in Table 1. Results disclose that:1. All these three intelligent technology methods give quite a good modeling accuracy in error prediction. The MAPE of LS-SVM, for comparison, is a little better than ANN and BN methods.2. The LS-SVM model has distinct advantages over ANN and BN models in calculating time. The ANN model spends about 22 times as LS-SVM for it needs a relatively long time to train the network to achieve good prediction accuracy. The BN model calculates the probability of every error state by combining the experience and the sample data without network training, so it is faster than ANN and slower than LS-SVM method.3. In terms of models understandability, the BN method describes causal relationships of factors inducing synthetic error by graph theory, so it helps users to gain understanding about a problem domain readily. Modeling with ANN or LS-SVM may need more professional knowledge.Fig.3. Modeling comparison results in axial direction. (a) Deformation; (b) Residual error Fig.4. Modeling comparison results in radial direction. (a) Deformation; (b) Residual errorTo sum up the above arguments the LS-SVM model has comprehensive performance in forecast accuracy and calculating time, so it is extremely suitable for modeling synthetic error of NC machine tools on-line. Furthermore, taking the understandability of model into consideration, using the BN and LS-SVM methods together will be a perfect solution to modeling complex synthetic error behavior of a machine tool.Table 1. Comparison of MAPE and calculating time among the three methods5. ConclusionThis paper demonstrates three intelligent technology methods of artificial neural network, support vector machines and Bayesian networks to model and forecast synthetic errors of NC machine tools. The experiments were carried out to evaluate their merits and defects. The comparison result shows that all these three intellgent methods can work well in machine error modeling, how to use them depends on the requirements of the application. Especially, the LS-SVM model has comprehensive performance in high modeling accuracy and short calculating time, while BN model has better understandability than others. Therefore, using the BN and LS-SVM methods together may be a perfect solution to synthetic error modeling.References1 Ramesh R, Mannan MA, Poo AN. Error compensation in machine tools - a review Part I: geometric, cutting-force induced and fixture dependent errors. Int. J. Machine Tools and Manuf. 2000; 40: 1235-1256.2 Ramesh R, Mannan MA, Poo AN, Keerthi SS. Thermal error measurement and modelling in machine tools. Part II. Hybrid Bayesian Network - support vector machine model. Int. J. Adv. Manuf. Tech.2003; 43: 405-419.3 Yang S, Yuan J, Ni J. The improvement of thermal error modeling and compensation on machine tools by CMAC neural network. Int. J. Machine Tools and Manuf. 1996; 36: 527-537.4 Chen JS. Fast calibration and modelling of thermally induced machine tool errors in real machining. Int. J. Machine Tools and Manuf. 1997; 37: 159-169.5 Mize CD, Ziegert JC. Neural network thermal error compensation of a machining center. Precision Engineering 2000; 24: 338-346.6 Fu JZ, Chen ZC. Research on modeling thermal dynamic errors of precision machine based on fuzzy logic and artificial neural network. J. Zhejiang Univ. (Eng. Sci.) 2004; 38: 742-746.7 Yang H, Ni J. Dynamic neural network modeling for nonlinear, nonstationary machine tool thermally induced error. Int. J. Adv. Manuf. Tech. 2005; 45: 455-465.8 Ramesh R, Mannan MA, Poo AN. Support vector machines model for classification of thermal error in machine tools. Int. J. Adv. Manuf. Tech.2002; 20: 114-120.9 LinWQ, Fu JZ, Chen ZC, Xu YZ. Modeling of NC Machine Tool Thermal Error Based on Adaptive Best-fitting WLS-SVM. J. Mechanical Engineering 2009; 45: 178-182.10 He ZY, Yao XH, Fu JZ, Chen ZC. Volumetric Error Prediction and Compensation of NC Machine Tool Based on Least Square Support Vector Machine. Adv. Sci. Letters. 2011; 4: 2066-2070.11 Heckerman D, Breese JS. Causal independence for probability assessment and inference using Bayesian networks. IEEE Trans. on Systems, Man, and Cybernetics-Part A: Systems and Humans 1996; 26: 826-831.12 Yao XH, Fu JZ, Chen ZC. Bayesian networks modeling for thermal error of numerical control machine tools. J. of Zhejiang Univ.: Sci. A 2008; 9: 1524-1530.13 Zhang MG, Li ZM, Li WH. Study on least squares support vector machines algorithm and its application. Proceedings of the 17th IEEE Int. Conf. on Tools with Artificial Intelligence, Washington, DC: IEEE Computer Society; 2005, p.1082-1085.14 Heckerman D. Bayesian networks for data mining. Data Mining and Knowledge Discovery 1997; 1: 79-119基于智能技术的数控机床综合误差建模姚新华，付建中，许粤通，雍禾摘要机床的精度在很大程度上受到错误或内置到机床或发生在周期性基础上约束帐户的温度变化或变化的切削力，所以它是必不可少的，以取得这些错误，然后消除或补偿他们。然而，许多因素引起的误差，如热源，热膨胀之间的相互作用系数时，机器的系统配置和运行环境，创造了一种机床复杂的行为，并且还使得综合误差预测很难与传统的数学。因此，一些建模方法基于非经典数学已经呈现在最近几年。神经网络的智能化技术方法，支持向量机，贝叶斯网络是有效的建模和预测方法的机器误差。所有这三种方法进行了介绍在简短的文件，并把它们的特点进行了讨论。一系列的实验进行了评价其优点和缺陷。最后，提出了一些有关如何使用这些方法在不同情况下的重要结论。该在本文中做出的作品与智能技术的误差建模中的一个特殊的总结，并提供了有用的指导对数控机床误差补偿的进一步研究.关键词:合成误差建模;神经网络;支持向量机;贝叶斯网络;智能技术1 介绍机器错误，包括切割力引起的， 夹具依赖性和热误差，这是我们的名字 这里综合误差从各种区别 个别错误，显著影响精度 机床，所以它是必不可少的，以取得这些错误， 然后消除或补偿他们1。在 机器误差的预测，目前大多数的研究具有 专注于到达之间的直接关联 通过获得的监测参数和错误数据 实验。一旦相关性建立后，模型 然后用来预测下的任何其他误差 基于所述监测数据的条件2。 然而,一些困难一直保持在建模性能的合成的影响错误。由于机床的精度受到影响由于机床的精度受过程,另一个是不同的相互作用使其难以理解的关系各种因素之一。这些问题影响的模型直接精度,限制使用传统的拟合方法在建模错误。因此,一些建模基于非经典的数学方法近年来提出的。智能技术方法的神经网络、支持向量机贝叶斯网络是有效的合成的错误。本文试图展示这些聪明合成误差建模和技术方法组织如下:在下一节中,一些重要的介绍了在这方面的工作已经完成,在部分3三个基于神经网络的建模方法支持向量机和贝叶斯网络分别介绍,比较实验第四节所示,结果进行了分析;最后,在第五节给出了一些结论。2 以前的工作各种智能技术已经被使用机的建模错误在最近15年。人工神经网络(ANN)是最重要的其中的一个。杨et al提出了小脑模型关节控制器(小脑)神经网络关联机床热误差的温度通过大量的实验和测量数据分析技术3。陈使用了三层安监督反向传播训练算法和乙状结肠的激活函数映射温度校准热误差测量4。然而,结果被发现不可接受的输入条件有显著的地方不同的数据集用于训练网络。一个基于人工神经网络的共振理论(ART-map)是用来预测和补偿工具点错误的硬件由麦斯和加工中心Ziegert5使用离散的温度读数机器的结构作为输入。傅和陈6一个人工神经网络方法相结合用模糊逻辑来预测热误差。一种新方法提出了主成分分析确定模糊模型的结构和参数。然后,利用人工神经网络方法模糊模型不能解决这个问题解决,因为有效的数据数量小比结果参数数量。杨和倪7利用动态神经网络模型非线性时变机床误差各种热条件。研究显示,动态模型比静态模型的优势模型的鲁棒性不同的工作条件下,由于动力学和非线thermal-elastic过程使用捕获时序关联网络。支持向量机(SVM)是另一个有用的方法在机床模型错误为好在分类和回归。拉梅什等艾尔8。开发了一种利用支持向量机模型根据操作条件对错误进行分类。一次分类,然后错误预测的基础上温度状态。林等9提出了一种建模使用最小二乘支持向量方法跟踪非线性机回归模型主轴热误差在一定条件下。主轴热变形实验进行了评估模型的性能模型估计的准确性和鲁棒性。回归模型表明,执行比其他建模方法,如多变量最小二乘回归分析的模型的准确性和鲁棒性。最近,他 10使用支持向量机来预测体积错误和也获得建模精度高。 贝叶斯网络(BN)是一个概率表示不确定的关系,是有用的建模等现实问题的诊断,预测、生产控制等,在那里存在多个因果依赖关系11。这是一个相对较新的技术,热误差建模。拉梅什等。提出了一种混合SVM-BN模型考虑到特定的诱导条件热误差。实验数据分类与基于规则的系统使用十亿模型。一旦分类被影响,预测误差使用支持向量机模型。姚明12描述的因果关系因素诱导热变形的关系图论和
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