WO2022051973A1 - 基于小样本迁移学习的铣削机器人多模态频响预测方法 - Google Patents

基于小样本迁移学习的铣削机器人多模态频响预测方法 Download PDF

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WO2022051973A1
WO2022051973A1 PCT/CN2020/114433 CN2020114433W WO2022051973A1 WO 2022051973 A1 WO2022051973 A1 WO 2022051973A1 CN 2020114433 W CN2020114433 W CN 2020114433W WO 2022051973 A1 WO2022051973 A1 WO 2022051973A1
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frequency response
domain
complex
tensor
data
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杨文安
蔡旭林
郭宇
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南京航空航天大学
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    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B23—MACHINE TOOLS; METAL-WORKING NOT OTHERWISE PROVIDED FOR
    • B23C—MILLING
    • B23C3/00—Milling particular work; Special milling operations; Machines therefor
    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B23—MACHINE TOOLS; METAL-WORKING NOT OTHERWISE PROVIDED FOR
    • B23Q—DETAILS, COMPONENTS, OR ACCESSORIES FOR MACHINE TOOLS, e.g. ARRANGEMENTS FOR COPYING OR CONTROLLING; MACHINE TOOLS IN GENERAL CHARACTERISED BY THE CONSTRUCTION OF PARTICULAR DETAILS OR COMPONENTS; COMBINATIONS OR ASSOCIATIONS OF METAL-WORKING MACHINES, NOT DIRECTED TO A PARTICULAR RESULT
    • B23Q17/00—Arrangements for observing, indicating or measuring on machine tools
    • B23Q17/09—Arrangements for observing, indicating or measuring on machine tools for indicating or measuring cutting pressure or for determining cutting-tool condition, e.g. cutting ability, load on tool
    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B25—HAND TOOLS; PORTABLE POWER-DRIVEN TOOLS; MANIPULATORS
    • B25J—MANIPULATORS; CHAMBERS PROVIDED WITH MANIPULATION DEVICES
    • B25J11/00—Manipulators not otherwise provided for
    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B25—HAND TOOLS; PORTABLE POWER-DRIVEN TOOLS; MANIPULATORS
    • B25J—MANIPULATORS; CHAMBERS PROVIDED WITH MANIPULATION DEVICES
    • B25J11/00—Manipulators not otherwise provided for
    • B25J11/005—Manipulators for mechanical processing tasks
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
    • G06F17/10—Complex mathematical operations
    • G06F17/14—Fourier, Walsh or analogous domain transformations, e.g. Laplace, Hilbert, Karhunen-Loeve, transforms
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
    • G06F17/10—Complex mathematical operations
    • G06F17/14—Fourier, Walsh or analogous domain transformations, e.g. Laplace, Hilbert, Karhunen-Loeve, transforms
    • G06F17/148—Wavelet transforms
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
    • G06F17/10—Complex mathematical operations
    • G06F17/18—Complex mathematical operations for evaluating statistical data, e.g. average values, frequency distributions, probability functions, regression analysis
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06N—COMPUTING ARRANGEMENTS BASED ON SPECIFIC COMPUTATIONAL MODELS
    • G06N20/00—Machine learning
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06N—COMPUTING ARRANGEMENTS BASED ON SPECIFIC COMPUTATIONAL MODELS
    • G06N3/00—Computing arrangements based on biological models
    • G06N3/02—Neural networks
    • G06N3/04—Architecture, e.g. interconnection topology
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F2218/00—Aspects of pattern recognition specially adapted for signal processing
    • G06F2218/12—Classification; Matching
    • G06F2218/16—Classification; Matching by matching signal segments
    • G06F2218/20—Classification; Matching by matching signal segments by applying autoregressive analysis

Definitions

  • the invention belongs to the field of robot numerical control machining, and relates to a data-driven multimodal frequency response prediction method of a milling robot tool tip, in particular to a milling robot multimodal frequency response prediction method based on small sample transfer learning.
  • Milling robots are more and more widely used in flexible machining of aerospace parts due to their low cost, high efficiency, good flexibility, and large working space.
  • the poor stiffness characteristics of the robot lead to chatter vibration during the milling process, which greatly reduces the machining quality and even damages the robot.
  • the most commonly used strategy for avoiding flutter in practice is to obtain the machining stability domain by predicting the stability lobe map.
  • the multimodal frequency response of the tool nose point has an important impact on the prediction accuracy.
  • the multi-modal frequency response of the tool tip point can be obtained by hammering test, while the impact test can only obtain the static end frequency response of the processing equipment under a specific attitude, and it is impossible to carry out the tool tip during the machining process. Hammer test.
  • the invention patent "A method for predicting the frequency response of a binary tree robot milling system based on RCSA” (CN108268745B), by dividing the milling robot into substructures and flexible joints, and then solve all substructures in The modal parameters and coupling functions under the attitude to be solved are established, the finite element model of the substructure is established and the tool material parameters are optimized to obtain its frequency response function and response matrix.
  • the frequency response function of the overall structure under the attitude to be solved is experimentally measured, and
  • the response matrix of the flexible joint is obtained by inverse calculation by the IRCSA method, and the frequency response function of the milling system tool end under the attitude to be solved is calculated according to the RCSA method, but this method needs to combine the finite element analysis and the frequency response data of 32 poses;
  • invention patent "Method and system for analyzing frequency response characteristics of heavy-duty palletizing robot" (CN110549340A)
  • the positive kinematics solution and workspace are obtained by kinematic analysis of the robot system, the connection between joint space and Cartesian space is established based on Jacobian matrix
  • the static stiffness analysis of the palletizing robot uses the Lagrangian equation to establish a rigid-flexible coupled dynamic model to describe the joint flexibility of the high-speed and heavy-duty palletizing robot, and analyze the vibration mode of the robot to obtain the effect of different joint stiffness on the system.
  • Robotics and Computer-Integrated Manufacturing 61, 101852.
  • the above models all describe the linkage and joint dynamics of the robot.
  • the modal characteristics in the robot workspace were predicted based on statistical modeling methods, and the dynamic characteristics of the robot tip were sampled by hammering experiments under the discrete arm configuration, and then a Gaussian process regression model was established to obtain the results.
  • the modal properties of other points in the workspace are predicted and then used to predict tool tip vibration during milling, but this method is only used to predict first-order modal parameters and requires extensive hammering experiments in the workspace.
  • a milling robot frequency response based on small sample complex domain feature transfer was invented.
  • This method can not only accurately, quickly and effectively predict the multi-modal frequency response characteristics of the milling robot, but also does not require complex modeling, time-consuming simulation and a large number of experiments;
  • the hammering experiment is performed on several points and ends, that is, the data required for learning and training is completed, and the hammering experiment is performed on the corresponding points on the target posture body to complete the collection of target domain data, and through the regression described in this patent.
  • the prediction method can realize the high-quality, high-efficiency and high-precision prediction of the multi-modal frequency response of robotic milling.
  • the present invention provides a milling robot frequency response prediction method with strong practicability, high accuracy and convenience for small sample complex domain feature migration.
  • the difference and correlation of the dynamic characteristics between the attitudes are used to predict the terminal frequency response through the dynamic characteristics of the robot body under different attitudes.
  • the original learning data was obtained through hammering experiments, and the fourth-order complex tensors of the source and target domains of the robot frequency response characteristics were constructed. and output vector, the information expansion function based on triangular membership degree and the multi-objective gray wolf optimization algorithm are used to generate virtual samples, and the data features are extracted in the frequency domain, time domain and time-frequency domain respectively.
  • the complex tensors in the target domain are feature augmented, and the complex tensors in the source and target domains are reduced by the naive tensor quantum space learning method.
  • the multi-modal frequency response of the tool tip is predicted, and then the multi-modal frequency response characteristics of the milling robot under the target attitude are obtained.
  • the information expansion function based on triangular membership degree and the multi-objective gray wolf optimization algorithm are used to generate virtual samples to effectively increase the scale of the source samples;
  • the time domain data is obtained by performing inverse Fourier transform on the original frequency response data, and the real frequency domain data is obtained by wavelet transform, and then the data features are extracted in the frequency domain, time domain, and time-frequency domain respectively, and based on This performs feature augmentation on the complex tensors in the source and target domains, and re-normalizes the tensor feature space;
  • step 1 select n hammering points and tool tip points on the milling robot body under any two attitudes to conduct hammering test, take the positive direction of the x-axis as the excitation direction, and collect the corresponding direct frequency through the three-axis acceleration sensor.
  • a third-order complex tensor of the source domain and target domain of the robot frequency response characteristic migration is constructed.
  • the impact factor represented by the first order is the hammering point
  • the impact factor represented by the second order is the frequency response data
  • the impact factor represented by the third order is the frequency response type.
  • step 3 based on the input tensor and output vector of the migration source domain, an information expansion function based on triangular membership is used to perform the asymmetric feasible expansion domain of the input tensor feature, that is, the feature boundary generated by the virtual sample.
  • the skewness of the triangle shape is related to the relative number of samples located on both sides of the center point, where the abscissa represents different eigenvalues, and the ordinate represents the probability of occurrence of eigenvalues. Therefore, the left Skewness and right skewness can be expressed as:
  • N L and N U represent the number of samples smaller and larger than the central feature point, respectively, and sp is the skewness fine-tuning parameter.
  • sp is the skewness fine-tuning parameter.
  • min represents the minimum value of the feature
  • max represents the maximum value of the feature
  • step 3 virtual samples are generated and screened in a feasible extended domain by using a multi-objective gray wolf optimization algorithm combined with a complex kernel extreme learning machine.
  • the original samples in the source domain are randomly divided into training set D s and test set D t , and a complex kernel extreme learning machine is used to calculate the original prediction model And by adjusting the regularization parameter and cost parameter in the kernel extreme learning machine, the average absolute percentage error predicted on the test set D t is within 10%, where the average percentage error can be expressed as:
  • MAPE l is a 1 ⁇ 2m vector.
  • n vir dummy samples is randomly generated based on the asymmetric feasible expansion domain computed in the previous step and passed through the original prediction model Calculate the output of the virtual samples, and then obtain n vir virtual samples D V containing input tensors and output vectors; combine the virtual samples D V and the training set D s into a comprehensive sample set, and use a complex kernel extreme learning machine to calculate new predictions
  • the model H KELM is tested on the test set D t . If the average absolute percentage error is still within 10%, it is determined as a valid virtual sample, otherwise the sample is rejected.
  • the gray wolf algorithm is mainly used to control the average absolute percentage error of prediction to the minimum by means of iterative optimization.
  • the gray wolf optimization algorithm is optimized by imitating the hierarchy and hunting strategy in the wolf group.
  • Each wolf in the wolf group is defined as a solution, and the wolf corresponding to the current optimal solution, optimal solution and sub-optimal solution is defined as ⁇ , ⁇ and ⁇ wolves, and the rest of the individuals are defined as ⁇ wolves.
  • the wolves approach the global optimal solution under the guidance of ⁇ , ⁇ and ⁇ wolves.
  • the guidance equation is as follows:
  • X represents the position of the gray wolf
  • X p represents the guiding position of the prey
  • t is the number of iterations
  • C and A are the guiding coefficients:
  • r 1 and r 2 are random numbers in the range of [0, 1], and a is a control parameter whose value is in the range of [0, 2] and decreases linearly with the algorithm iteration.
  • the multi-objective gray wolf optimization algorithm needs to be used to solve the problem.
  • the importance of modal parameters increases with the order of increases and decreases, and the natural frequency is more important to ensure the machining stability than the damping ratio. Therefore, it is proposed to use the weight addition method to convert the multi-objective into a single objective, then the multi-objective optimization problem can be expressed as :
  • the set value of the weight ⁇ l can be adjusted according to the importance of the modal parameters, and x r represents the rth eigenvalue of the input tensor.
  • the time-domain data is obtained by performing inverse Fourier transform on the original frequency response data, and the real-frequency domain data is obtained by wavelet transform, and then the data features are analyzed in the frequency domain, time domain, and time-frequency domain respectively. Extraction, mainly including: variance, skewness, kurtosis, spectral skewness, spectral kurtosis and average energy, etc. Based on this, the complex tensors in the source domain and the target domain are feature augmented, and the tensor feature space is re-scaled. Normalize.
  • step 5 the shared invariant tensor quantum space of the complex tensors in the source domain and the target domain is obtained by the naive tensor quantum space learning method, so as to effectively reduce the tensor dimension.
  • each sample is a third-order tensor.
  • it can be merged into a fourth-order tensor ⁇ s , in which the influence of extra dimensions
  • the factor is the robot pose, and similarly, the third-order tensor in the target domain is expanded to a fourth-order tensor ⁇ t .
  • G s and G t represent the tensor quantum space of ⁇ s and ⁇ t , respectively, I represents the identity matrix, and U represents the invariant tensor quantum space.
  • I represents the identity matrix
  • U represents the invariant tensor quantum space.
  • step 6 a complex kernel extreme learning machine based on conjugate augmentation input is constructed, conjugate augmentation processing is performed on the input complex tensor, and the computational capability of the extreme learning machine for complex problems is further improved through the complex Gaussian kernel function .
  • the basic kernel extreme learning machine can be expressed as:
  • C is the cost parameter, and its value range is ⁇ 2 -24 ,2 -23 ,2 -22 ,...,2 24 ,2 25 ⁇
  • ⁇ is the input matrix
  • ⁇ is the output vector
  • K(x i , x j ) represents the complex Gaussian kernel function
  • the complex Gaussian kernel function used is expressed as:
  • ⁇ is a complex number, which can be selected according to the characteristics of the training samples
  • ⁇ is a regularization parameter
  • its value range is ⁇ 2 -24 ,2 - 23 ,2 -22 ,...,2 24 ,2 25 ⁇
  • a complex kernel extreme learning machine based on conjugate augmentation input is constructed.
  • the network input after conjugate augmentation processing is:
  • the extreme learning machine is mainly used to assist the generation of virtual samples and the regression prediction of multimodal frequency response in the present invention.
  • the method of the present invention is aimed at the problem of unsatisfactory prediction accuracy caused by too small sample size in the multi-modal frequency response prediction problem of milling robots with small sample complex domain feature migration, and uses an information expansion function based on triangular membership to determine the input
  • the asymmetric feasible extension domain of tensor features is the feature boundary generated by virtual samples, and virtual samples are generated based on this, and the virtual samples are screened in combination with the multi-objective gray wolf optimization algorithm, and the sample size is improved on the premise of ensuring the quality of virtual samples. , and then solve the problem of inaccurate prediction of tool frequency response caused by small samples;
  • the method of the present invention is aimed at the problem of unsatisfactory prediction caused by too single original sample features in the multi-modal frequency response prediction of milling robots with small sample complex domain feature migration, and is obtained by performing inverse Fourier transform on the original frequency response data. time domain data, and perform wavelet transform on the time domain signal to obtain real frequency domain data, and then extract data features in the frequency domain, time domain, and time-frequency domain respectively, and based on this, complex data in the source domain and target domain are extracted.
  • the feature augmentation is carried out to increase the diversity of sample features, thereby improving the accuracy of tool frequency response prediction;
  • the method of the present invention makes full use of the second-order statistics of complex tensors by performing conjugate augmentation processing on the input complex tensors , and invented a new complex Gaussian kernel function, which can well solve the problem of mutual covariance cancellation of complex Gaussian processes, and can effectively improve the complex applicability and regression prediction accuracy of extreme learning.
  • FIG. 1 is a flowchart of a method for predicting the frequency response of a milling robot involved in the method of the present invention.
  • the method for predicting the frequency response of a milling robot with a small sample complex domain feature migration involved in the method of the present invention includes the following steps:
  • Step 1 Select n hammering points and tool tip points on the body under any two attitudes of the milling robot for hammering test to obtain the frequency response data as the migration source, and also test the n hammering points on the body in the target attitude. Conduct hammer test to obtain frequency response data as migration target;
  • Step 2 Use the frequency response data of the hammering point on the milling robot body to construct a third-order complex tensor of the source domain and target domain of the robot's frequency response characteristic migration, based on the least squares complex exponential method for the multimodal frequency response of the tool tip point Carry out multi-order modal parameter identification, and then construct the labels of the data in the migration source domain;
  • a third-order complex tensor of the source domain and target domain of the robot frequency response characteristic migration is constructed.
  • the impact factor represented by the first order is the hammering point
  • the impact factor represented by the second order is the frequency response data
  • the impact factor represented by the third order is the frequency response type.
  • Step 3 Based on the input tensor and output vector of the migration source domain, use the triangular membership-based information expansion function and the multi-objective gray wolf optimization algorithm to generate virtual samples to effectively increase the scale of the source samples;
  • step 3 based on the input tensor and output vector of the migration source domain, an information expansion function based on triangular membership is used to perform an asymmetric feasible expansion domain of the input tensor feature, that is, the feature boundary generated by the virtual sample.
  • the skewness of the triangle shape is related to the relative number of samples located on both sides of the center point, where the abscissa represents different eigenvalues, and the ordinate represents the probability of occurrence of eigenvalues. Therefore, the left Skewness and right skewness can be expressed as:
  • N L and N U represent the number of samples smaller and larger than the central feature point, respectively, and sp is the skewness fine-tuning parameter.
  • sp is the skewness fine-tuning parameter.
  • min represents the minimum value of the feature
  • max represents the maximum value of the feature
  • step 3 the original samples in the source domain are randomly divided into training set D s and test set D t , and a complex kernel extreme learning machine is used to calculate the original prediction model And by adjusting the regularization parameter and cost parameter in the kernel extreme learning machine, the average absolute percentage error predicted on the test set D t is within 10%, where the average percentage error can be expressed as:
  • MAPE l is a 1 ⁇ 2m vector.
  • n vir dummy samples is randomly generated based on the asymmetric feasible expansion domain computed in the previous step and passed through the original prediction model Calculate the output of the virtual samples, and then obtain n vir virtual samples D V containing input tensors and output vectors; combine the virtual samples D V and the training set D s into a comprehensive sample set, and use a complex kernel extreme learning machine to calculate new predictions
  • the model H KELM is tested on the test set D t . If the average absolute percentage error is still within 10%, it is determined as a valid virtual sample, otherwise the sample is rejected.
  • the gray wolf algorithm is mainly used to control the average absolute percentage error of prediction to the minimum by means of iterative optimization.
  • the gray wolf optimization algorithm is optimized by imitating the hierarchy and hunting strategy in the wolf group.
  • Each wolf in the wolf group is defined as a solution, and the wolf corresponding to the current optimal solution, optimal solution and sub-optimal solution is defined as ⁇ , ⁇ and ⁇ wolves, and the rest of the individuals are defined as ⁇ wolves.
  • the wolves approach the global optimal solution under the guidance of ⁇ , ⁇ and ⁇ wolves.
  • the guidance equation is as follows:
  • X represents the position of the gray wolf
  • X p represents the guiding position of the prey
  • t is the number of iterations
  • C and A are the guiding coefficients:
  • r 1 and r 2 are random numbers in the range of [0, 1], and a is a control parameter whose value is in the range of [0, 2] and decreases linearly with the algorithm iteration.
  • the multi-objective gray wolf optimization algorithm needs to be used to solve the problem.
  • the importance of modal parameters increases with the order of increases and decreases, and the natural frequency is more important to ensure the machining stability than the damping ratio. Therefore, it is proposed to use the weight addition method to convert the multi-objective into a single objective, then the multi-objective optimization problem can be expressed as :
  • the set value of the weight ⁇ l can be adjusted according to the importance of the modal parameters, and x r represents the rth eigenvalue of the input tensor.
  • Step 4 The time domain data is obtained by performing inverse Fourier transform on the original frequency response data, and the real frequency domain data is obtained by wavelet transform, and then the data features are extracted in the frequency domain, time domain, and time-frequency domain respectively, and Based on this, feature augmentation is performed on the complex tensors in the source and target domains;
  • step 4 data features are extracted in the frequency domain, time domain, and time-frequency domain, mainly including: variance, skewness, kurtosis, spectral skewness, spectral kurtosis, and average energy, etc. Based on this Perform feature augmentation on the complex tensors in the source and target domains, and re-normalize the tensor feature space.
  • Step 5 Obtain the shared invariant tensor quantum space of complex tensors in the source domain and the target domain through the naive tensor quantum space learning method to effectively reduce the tensor dimension;
  • G s and G t represent the tensor quantum space of ⁇ s and ⁇ t , respectively, I represents the identity matrix, and U represents the invariant tensor quantum space.
  • I represents the identity matrix
  • U represents the invariant tensor quantum space.
  • Step 6 Construct a complex kernel extreme learning machine based on conjugate augmentation input, and perform conjugate augmentation processing on the input complex tensor to make full use of the second-order statistics of the complex tensor, and then use the learning machine to determine the target pose.
  • the multimodal frequency response of the tool tip is predicted.
  • step six the basic kernel extreme learning machine can be expressed as:
  • C is the cost parameter, and its value range is ⁇ 2 -24 ,2 -23 ,2 -22 ,...,2 24 ,2 25 ⁇
  • ⁇ is the input matrix
  • ⁇ is the output vector
  • K(x i , x j ) represents the complex Gaussian kernel function
  • the complex Gaussian kernel function used is expressed as:
  • ⁇ is a complex number, which can be selected according to the characteristics of the training samples
  • ⁇ is a regularization parameter
  • its value range is ⁇ 2 -24 ,2 - 23 ,2 -22 ,...,2 24 ,2 25 ⁇
  • a complex kernel extreme learning machine based on conjugate augmentation input is constructed.
  • the network input after conjugate augmentation processing is:
  • the extreme learning machine is mainly used to assist the generation of virtual samples and the regression prediction of multimodal frequency response in the present invention.

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Abstract

本发明公开了一种基于小样本迁移学习的铣削机器人多模态频响预测方法，包括以下步骤：选取铣削机器人任意两个姿态下本体上的若干点和刀尖点进行锤击测试，同时对目标姿态下本体上的若干点进行锤击测试，以得到迁移源和迁移目标数据；构建机器人频响特性迁移源域和目标域的三阶复数张量，基于最小二乘复指数法对刀尖点的多模态频响进行多阶模态参数识别，进而构建迁移源域中数据的标签；基于迁移源域的输入张量和输出向量，采用基于三角隶属度的信息扩展函数和多目标灰狼优化算法生成虚拟样本；分别在频域、时域和时频域中进行数据特征的提取，并基于此对源域与目标域中的复数张量进行特征增广；通过朴素张量子空间学习法进行源域与目标域中的复数张量的降维；构建基于共轭增广输入的复数核极限学习机对目标姿态的刀尖多模态频响进行预测。

Description

基于小样本迁移学习的铣削机器人多模态频响预测方法 技术领域
本发明属于机器人数控加工领域，涉及到一种数据驱动的铣削机器人刀尖点的多模态频响预测方法，特别涉及一种基于小样本迁移学习的铣削机器人多模态频响预测方法。
背景技术
铣削机器人因其成本低、效率高、灵活性好、工作空间大等优点，在航空航天零件柔性加工中得到越来越广泛的应用。然而机器人较差的刚度特性，导致铣削过程中极易发生颤振，进而极大地降低加工质量，甚至损坏机器人。在实际中最常用的用于避免颤振的策略是通过预测稳定叶瓣图来获得加工稳定域。刀尖点的多模态频响作为这些预测方法的重要输入，对预测精度有重要影响。一般来讲，刀尖点的多模态频响可通过锤击试验来获得的，而冲击试验只能得到加工设备在特定姿态下的静态末端频响，更无法在加工过程中对刀尖进行锤击测试。
为解决上述问题，进而实现机器人铣削加工的稳定高效。近年来，大量的加工机器人末端频响预测方法及技术被提出，大致可分为模型驱动和数据驱动两种类型。然而，构建完备的机器人动力学模型几乎是不可能的，因此很难保障刀尖点的多模态频响的预测精度，此外，模型中需要关节刚度、关节阻尼比等参数作为输入，数据准备需要花费大量的人力物力。随着机器学习算法的发展，数据驱动的加工机器人刀尖点的多模态频响预测方法成为了一种很好的替代方案，被越来越广泛地研究。
针对现有的技术进行技术总结发现：发明专利“一种基于RCSA的二叉树机器人铣削系统频响预测方法”(CN108268745B)，通过将铣削机器人划分为子结构和柔性结合部，进而求解所有子结构在待求解姿态下的模态参数和耦合函数，建立子结构的有限元模型并优化其刀具材料参数，获取其频响函数和响应矩阵，实验测得待求解姿态下整体结构的频响函数，并通过IRCSA方法反算得到柔性结合部的响应矩阵，根据RCSA方法计算待求解姿态下的铣削系统刀具端频响函数，但该方法需要结合有限元分析和32个位姿的频响数据；发明专利“重载码垛机器人频响特性分析方法及系统”(CN110549340A)，通过对机器人系统进行运动学分析得到运动学正解和工作空间，基于雅可比矩阵建立关节空间和笛卡尔空间的联系，并对码垛机器人静刚度分析，利用拉格朗日方程建立刚柔耦合动力学模型，对高速重载码垛机器人的关节柔性进行描述，并对机器人的振动模态进行分析，获得不同关节刚度对系统频响的影响规律；发明专利“一种机器人模态参数辨识与动态特性分析的方法”(CN111002313A)，通过往复加减速 使机器人产生振动响应，并采用随机减量法获取关节轴的自由响应，根据误差函数最小条件以及系统极点和模态参数之间的关系，获取关节轴的模态参数和频响函数，进而基于机器人运动学方程建立机器人末端频响函数与各个关节轴的频响函数之间的非线性关系，建立机器人末端位姿与各个关节轴的关节角之间的转化关系，进而获得机器人末端频响函数与各个关节轴的频响函数之间的非线性关系；期刊论文“Huynh H N,Assadi H,Rivière L E,Verlinden O&Ahmadi K(2020)Modelling the dynamics of industrial robots for milling operations.Robotics and Computer-Integrated Manufacturing,61,101852.”，提出了一种相对比较完备的机加工机器人多体动力学模型，通过多输入多输出识别方法、机器人的计算机辅助设计模型和实验模态分析识别模型中的连杆和转子的惯性以及关节刚度和阻尼参数，该模型虽然可以有效地预测机器人末端的四阶模态参数，但是模型的构建和求解过程复杂且耗时，十分的不方便。以上这些模型均描述了机器人的连杆和关节动力学，然而，但是由于齿轮齿隙、连杆质量分布的不均匀性、机器人的结构刚度、阻尼和惯性特性存在不确定性，难以精确模拟机器人在执行复杂任务时刀尖的动态特性，此外，这些模型往往实验耗时和过程复杂的问题，并不能很好的适应加工环境的可变性。而基于数据驱动的加工机器人刀尖点的多模态频响预测方法成为了一种很好的替代方案。期刊论文“Nguyen V,Cvitanic T&Melkote S(2019)Data-Driven Modeling of the Modal Properties of a Six-Degrees-of-Freedom Industrial Robot and Its Application to Robotic Milling.Journal of Manufacturing Science and Engineering,141(12),121006.”,基于统计建模方法对机器人工作空间内的模态特性进行了预测，通过离散臂配置下进行的锤击实验来采样机器人刀尖动力学特性，进而建立一个高斯过程回归模型，以预测工作空间中其他点的模态特性，然后用于预测铣削过程中的刀尖振动，但是该方法仅用于预测一阶模态参数，而且需要在工作空间内进行大量的锤击实验。
针对现有机加工机器人频响预测方法中问题，为使预测方法快速有效地预测不同加工姿态和加工环境下系统的多阶模态特性，发明了一种基于小样本复数域特征迁移的铣削机器人频响预测方法，该方法不仅能准确快速有效地预测铣削机器人的多模态频响特性，而且不需要复杂的建模、耗时的仿真和大量的实验；仅仅需要对任意两个姿态下机器人本体上若干点和末端进行锤击实验，即完成学习训练需要的数据，并对目标姿态本体上对应的点进行锤击实验，即可完成目标域数据的采集，并通过本专利中所述的回归预测方法，即可实现机器人铣削加工多多模态频响的高质高效高精度预测。
发明内容
本发明针对现有机加工机器人频响预测方法的不足，提供了一种实用性强、 准确性高且方便易用的小样本复数域特征迁移的铣削机器人频响预测方法，该方法利用铣削机器人各个姿态之间动力学特性的差异性与相关性，通过机器人本体不同姿态下动力学特性来预测末端频响。通过锤击实验获取原始学习数据，构建机器人频响特性迁移源域和目标域的四阶复数张量，基于最小二乘复指数法构建迁移源域中数据的标签，基于迁移源域的输入张量和输出向量，采用基于三角隶属度的信息扩展函数和多目标灰狼优化算法生成虚拟样本，并分别在频域、时域和时频域中进行数据特征的提取，基于此对源域与目标域中的复数张量进行特征增广，通过朴素张量子空间学习法进行源域与目标域中的复数张量的降维，采用构建的共轭增广输入复数核极限学习机对目标姿态的刀尖多模态频响进行预测，进而获得目标姿态下铣削机器人的多模态频响特性。
为本发明采用以下技术方案实现：
小样本复数域特征迁移的铣削机器人频响预测方法，其特征在于包括以下步骤：
(1)选取铣削机器人任意两个姿态下本体上的n个锤击点和刀尖点进行锤击测试，以得到频响数据作为迁移源，同样对目标姿态下本体上的n个锤击点进行锤击测试，以得到频响数据作为迁移目标；
(2)利用铣削机器人本体上锤击点的频响数据构建机器人频响特性迁移源域和目标域的三阶复数张量，基于最小二乘复指数法对刀尖点的多模态频响进行多阶模态参数识别，进而构建迁移源域中数据的标签；
(3)基于迁移源域的输入张量和输出向量，采用基于三角隶属度的信息扩展函数和多目标灰狼优化算法生成虚拟样本，以有效提升源样本的规模；
(4)通过对原始频响数据进行傅里叶逆变换得到时域数据，通过小波变换得到实频域数据，进而分别在频域、时域、时频域中进行数据特征的提取，并基于此对源域与目标域中的复数张量进行特征增广，并重新对张量特征空间进行归一化处理；
(5)通过朴素张量子空间学习法获取源域与目标域中的复数张量的共享不变张量子空间，以有效降低张量维度；
(6)构建基于共轭增广输入的复数核极限学习机，对输入的复数张量进行共轭增广处理，以充分利用复数张量的二阶统计量，进而采用该学习机对目标姿态的刀尖多模态频响进行预测。
优选地，步骤一中选取任意两个姿态下铣削机器人本体上的n个锤击点和刀尖点进行锤击测试，以x轴正方向为激励方向，通过三轴加速度传感器采集相应的直接频响
Figure PCTCN2020114433-appb-000001
与交叉频响
Figure PCTCN2020114433-appb-000002
作为原始迁移源域的数据，其中i＝1,2,L,n+1，同样对目标姿态下铣削机器人本体上相同的n个锤击点以x轴正 方向为激励方向进行锤击测试，并通过加速度传感器采集频响
Figure PCTCN2020114433-appb-000003
作为迁移目标域的数据，其中i＝1,2,L,n。
优选地，步骤二中构建机器人频响特性迁移源域和目标域的三阶复数张量
Figure PCTCN2020114433-appb-000004
第一阶表示的影响因子为锤击点，第二阶表示的影响因子为频响数据，第三阶表示的影响因子为频响类型，基于最小二乘复指数法对刀尖点的多模态频响进行多阶模态参数识别，获得刀尖点的多模态频响数据中m阶模态的固有频率f j与阻尼比ζ j，其中j＝1,2,L,m，进而构建迁移源域的输出向量SO∈R 1×2m。
优选地，步骤三中基于迁移源域的输入张量和输出向量，采用基于三角隶属度的信息扩展函数进行输入张量特征的非对称可行扩展域，即虚拟样本生成的特征边界。首先需计算源域中样本的中心点，由于本发明中源样本为两个姿态下的直接频响和交叉频响数据中的特征SI k，k＝1,2,L,6因此，中心点可以表示为：
Figure PCTCN2020114433-appb-000005
三角隶属度的信息扩展函数中，三角形形状的偏斜度与位于中心点两侧的样本的相对数量有关，其中横坐标表示不同的特征值，纵坐标表示特征值的发生可能性，因此，左偏度和右偏度可表示为：
Figure PCTCN2020114433-appb-000006
Figure PCTCN2020114433-appb-000007
其中，N L和N U分别表示比中心特征点小和大的样本数目，s p为偏斜度微调参数。则非对称可行扩展域的上下边界可以表示为：
Figure PCTCN2020114433-appb-000008
Figure PCTCN2020114433-appb-000009
其中，min表示特征的最小值，max表示特征的最大值。
优选的，步骤三中通过多目标灰狼优化算法结合复数核极限学习机的方法在可行扩展域中进行虚拟样本的生成与筛选。
首先将源域中的原始样本随机划分为训练集D s与测试集D t，并采用复数核极限学习机计算原始预测模型
Figure PCTCN2020114433-appb-000010
并通过调整核极限学习机中的正则化参数与成本参数将在测试集D t上预测的平均绝对百分比误差在10％以内，其中平均百分比误差可表示为：
Figure PCTCN2020114433-appb-000011
其中，y l为真实输出向量中的第l个参数，
Figure PCTCN2020114433-appb-000012
为预测输出向量中的第l个参数，在本发明专利中，MAPE l为1×2m的向量。
基于上一步中计算的非对称可行扩展域随机生成n vir个虚拟样本的输入，并通过原始预测模型
Figure PCTCN2020114433-appb-000013
计算虚拟样本的输出，进而得到n vir个含有输入张量与输出向量的虚拟样本D V；将虚拟样本D V与训练集D s合并为综合样本集，采用复数核极限学习机计算新的预测模型H KELM,内通过在测试集D t上进行测试，若平均绝对百分比误差仍然在10％以内，则判定为有效虚拟样本，否则剔除该样本。
灰狼算法主要用于通过迭代寻优的方式，将预测平均绝对百分比误差控制到最小。灰狼优化算法通过模仿狼群中的等级制度与猎食策略进行优化的，狼群中的每匹狼都定义为一个解，定义当前的最优解、优解、次优解对应的狼为α、β和δ狼，其余个体定义为ω狼，狼群在α、β和δ狼的引导下向全局最优解逼近，引导方程如下：
D p＝C·X p(t)-X(t)
Figure PCTCN2020114433-appb-000014
其中,X表示灰狼的位置，X p表示猎物的引导位置，t为迭代次数，C和A为引导系数：
A＝2gagr 1-a
C＝2gr 2
其中，r 1和r 2为[0,1]范围内的随机数，a为取值在[0,2]范围内且随着算法迭代线性递减的控制参数。
由于本发明中共涉及到2m个刀尖点的多模态频响参数，因此，需采用多目标灰狼优化算法进行求解，在本发明中，考虑到模态参数的重要性随着阶次的递增而递减，同时固有频率相对于阻尼比而言，对于保障加工稳定性更为重要，因此，拟采用权重相加的方法，将多目标转化为单个目标，则该多目标优化问题可表示为：
Figure PCTCN2020114433-appb-000015
subject to UB r≤x r≤LB r
-0.1≤((y l-y l)-y l)≤0.1
其中，权重ω l的设定值可依据模态参数的重要性进行调整，x r表示输入张量的第r个特征值。
优选地，步骤四中通过对原始频响数据进行傅里叶逆变换得到时域数据，并通过小波变换得到实频域数据，进而分别在频域、时域、时频域中进行数据特征的提取，主要包括：方差、偏度、峰度、光谱偏度、光谱峰度和平均能量等，基于此对源域与目标域中的复数张量进行特征增广，并重新对张量特征空间进行归一化处理。
优选地，步骤五中通过朴素张量子空间学习法获取源域与目标域中的复数张量的共享不变张量子空间，以有效降低张量维度。
在本发明中，假定源域中加入虚拟样本之后的综合样本集
Figure PCTCN2020114433-appb-000016
含有N s个样本，其中z＝1,2,L,N s，且每个样本均为三阶张量，为便于降维，可以合并成为四阶张量Ω s，其中多出维度的影响因子为机器人姿态，同样的，将目标域中的三阶张量扩展为四阶张量Ω t。假定源域张量和目标域张量共享一个张量子空间U＝{U (k)}，其中k＝1,2,3，在Tucker分解的基础上，求U等价于求解如下优化问题：
Figure PCTCN2020114433-appb-000017
Figure PCTCN2020114433-appb-000018
其中，G s和G t分别表示Ω s和Ω t的张量子空间，I表示单位矩阵，U表示不变张量子空间。利用Tucker分解算法可以有效求解上述方程，只要确定了最优U值，则可通过下式得到G s：
G s＝Ω s× 1U (1)T× 1U (2)T× 1U (3)T
类似的程序也可用于推导G t,至此，即可实现张量的降维，有效地降低回归预测的计算量，进而大幅提升预测效率。
优选地，步骤六中构建基于共轭增广输入的复数核极限学习机，对输入的复数张量进行共轭增广处理，并通过复数高斯核函数进一步提升极限学习机对复数问题的计算能力。
首先，基本的核极限学习机可表示为：
Figure PCTCN2020114433-appb-000019
其中，C为成本参数，其取值范围为{2 -24,2 -23,2 -22,...,2 24,2 25}，
Figure PCTCN2020114433-appb-000020
为输入矩阵，Τ为输出向量，K(x i,x j)表示复数高斯核函数，Ω ELM为核矩阵，其中Ω ij＝K(x i,x j)，在本发明中，为更好地适应复数回归问题，所采用地复数高斯核函数表示为：
Figure PCTCN2020114433-appb-000021
其中d x＝x i-x j，v r和v rj为实数，μ为复数，可根据训练样本特征进行具体的选择，γ为正则化参数，其取值范围为{2 -24,2 -23,2 -22,...,2 24,2 25}
此外，为充分利用样本中复数张量的二阶统计量，构建了基于共轭增广输入的复数核极限学习机，该方法机制简单，但是能较好地提升复数域学习回归性能。以第k个样本的输入为例，经过共轭增广处理之后地网络输入为：
Figure PCTCN2020114433-appb-000022
至此，完成了基于共轭增广输入的复数核极限学习机，该极限学习机在本发明中主要用于辅助虚拟样本的生成以及多模态频响的回归预测。
相较于现有技术，本发明提供的技术方案具有如下有益效果：
(1)本发明的方法针对小样本复数域特征迁移的铣削机器人多模态频响预测问题中样本规模过小所导致的预测精度不理想的问题，采用基于三角隶属度的信息扩展函数确定输入张量特征的非对称可行扩展域，即虚拟样本生成的特征边界，并基于此生成虚拟样本，并结合多目标灰狼优化算法对虚拟样本进行筛选，在保障虚拟样本质量的前提下提升样本规模，进而解决小样本所带来的刀具频响预测不精确问题；
(2)本发明的方法针对小样本复数域特征迁移的铣削机器人多模态频响预测中原始样本特征过于单一所导致的预测不理想问题，通过对原始频响数据进行傅里叶逆变换得到时域数据，并对时域信号进行小波变换得到实频域数据，进而分别在频域、时域、时频域中进行数据特征的提取，并基于此对源域与目标域中的复数张量进行特征增广，以提升样本特征的多样性，进而提升刀具频响预测精度；
(3)本发明的方法针对现有的核极限学习机在复数回归问题中的不适用性，为充分利用复数张量的二阶统计量，通过对输入的复数张量进行共轭增广处理，并发明了全新的复数高斯核函数，该函数可以很好地解决复数高斯过程互协方差抵消问题，可有效提升极限学习的复数适用性和回归预测精度。
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此处所说明的附图用来提供对本发明的进一步理解，构成本发明的一部分，本发明的示意性实施例及其说明用于解释本发明，并不构成对本发明的不当限定。在附图中：
图1为本发明所述方法中涉及到的铣削机器人频响预测方法流程图。
具体实施方式
为了使本发明所要解决的技术问题、技术方案及有益效果更加清楚、明白，以下结合附图，对本发明进行进一步详细说明。
如图1所示，本发明所述方法中涉及到的小样本复数域特征迁移的铣削机器人频响预测方法包含以下步骤：
步骤一：选取铣削机器人任意两个姿态下本体上的n个锤击点和刀尖点进行锤击测试，以得到频响数据作为迁移源，同样对目标姿态下本体上的n个锤击点进行锤击测试，以得到频响数据作为迁移目标；
具体地，在步骤一中，选取任意两个姿态下铣削机器人本体上的n个锤击点和刀尖点进行锤击测试，以x轴正方向为激励方向，通过三轴加速度传感器采集相应的直接频响
Figure PCTCN2020114433-appb-000023
与交叉频响
Figure PCTCN2020114433-appb-000024
作为原始迁移源域的数据，其中i＝1,2,L,n+1，同样对目标姿态下铣削机器人本体上相同的n个锤击点以x轴正方向为激励方向进行锤击测试，并通过加速度传感器采集频响
Figure PCTCN2020114433-appb-000025
作为迁移目标域的数据，其中i＝1,2,L,n。
步骤二：利用铣削机器人本体上锤击点的频响数据构建机器人频响特性迁移源域和目标域的三阶复数张量，基于最小二乘复指数法对刀尖点的多模态频响进行多阶模态参数识别，进而构建迁移源域中数据的标签；
具体地，在步骤二中，构建机器人频响特性迁移源域和目标域的三阶复数张量
Figure PCTCN2020114433-appb-000026
第一阶表示的影响因子为锤击点，第二阶表示的影响因子为频响数据，第三阶表示的影响因子为频响类型，基于最小二乘复指数法对刀尖点的多模态频响进行多阶模态参数识别，获得刀尖点的多模态频响数据中m阶模态的固有频率f j与阻尼比ζ j，其中j＝1,2,L,m，进而构建迁移源域的输出向量SO∈R 1×2m。
步骤三：基于迁移源域的输入张量和输出向量，采用基于三角隶属度的信息扩展函数和多目标灰狼优化算法生成虚拟样本，以有效提升源样本的规模；
具体地，在步骤三中，基于迁移源域的输入张量和输出向量，采用基于三角隶属度的信息扩展函数进行输入张量特征的非对称可行扩展域，即虚拟样本生成的特征边界。首先需计算源域中样本的中心点，由于本发明中源样本为两个姿态下的直接频响和交叉频响数据中的特征SI k，k＝1,2,L,6因此，中心点可以表示为：
Figure PCTCN2020114433-appb-000027
三角隶属度的信息扩展函数中，三角形形状的偏斜度与位于中心点两侧的样 本的相对数量有关，其中横坐标表示不同的特征值，纵坐标表示特征值的发生可能性，因此，左偏度和右偏度可表示为：
Figure PCTCN2020114433-appb-000028
Figure PCTCN2020114433-appb-000029
其中，N L和N U分别表示比中心特征点小和大的样本数目，s p为偏斜度微调参数。则非对称可行扩展域的上下边界可以表示为：
Figure PCTCN2020114433-appb-000030
Figure PCTCN2020114433-appb-000031
其中，min表示特征的最小值，max表示特征的最大值。
具体地，在步骤三中，首先将源域中的原始样本随机划分为训练集D s与测试集D t，并采用复数核极限学习机计算原始预测模型
Figure PCTCN2020114433-appb-000032
并通过调整核极限学习机中的正则化参数与成本参数将在测试集D t上预测的平均绝对百分比误差在10％以内，其中平均百分比误差可表示为：
Figure PCTCN2020114433-appb-000033
其中，y l为真实输出向量中的第l个参数，
Figure PCTCN2020114433-appb-000034
为预测输出向量中的第l个参数，在本发明专利中，MAPE l为1×2m的向量。
基于上一步中计算的非对称可行扩展域随机生成n vir个虚拟样本的输入，并通过原始预测模型
Figure PCTCN2020114433-appb-000035
计算虚拟样本的输出，进而得到n vir个含有输入张量与输出向量的虚拟样本D V；将虚拟样本D V与训练集D s合并为综合样本集，采用复数核极限学习机计算新的预测模型H KELM,内通过在测试集D t上进行测试，若平均绝对百分比误差仍然在10％以内，则判定为有效虚拟样本，否则剔除该样本。
灰狼算法主要用于通过迭代寻优的方式，将预测平均绝对百分比误差控制到最小。灰狼优化算法通过模仿狼群中的等级制度与猎食策略进行优化的，狼群中的每匹狼都定义为一个解，定义当前的最优解、优解、次优解对应的狼为α、β和δ狼，其余个体定义为ω狼，狼群在α、β和δ狼的引导下向全局最优解逼近，引导方程如下：
D p＝C·X p(t)-X(t)
Figure PCTCN2020114433-appb-000036
其中,X表示灰狼的位置，X p表示猎物的引导位置，t为迭代次数，C和A为引导系数：
A＝2gagr 1-a
C＝2gr 2
其中，r 1和r 2为[0,1]范围内的随机数，a为取值在[0,2]范围内且随着算法迭代线性递减的控制参数。
由于本发明中共涉及到2m个刀尖点的多模态频响参数，因此，需采用多目标灰狼优化算法进行求解，在本发明中，考虑到模态参数的重要性随着阶次的递增而递减，同时固有频率相对于阻尼比而言，对于保障加工稳定性更为重要，因此，拟采用权重相加的方法，将多目标转化为单个目标，则该多目标优化问题可表示为：
Figure PCTCN2020114433-appb-000037
subject to UB r≤x r≤LB r
-0.1≤((y l-y l)-y l)≤0.1
其中，权重ω l的设定值可依据模态参数的重要性进行调整，x r表示输入张量的第r个特征值。
步骤四：通过对原始频响数据进行傅里叶逆变换得到时域数据，并通过小波变换得到实频域数据，进而分别在频域、时域、时频域中进行数据特征的提取，并基于此对源域与目标域中的复数张量进行特征增广；
具体地，在步骤四中，在频域、时域、时频域中进行数据特征的提取，主要包括：方差、偏度、峰度、光谱偏度、光谱峰度和平均能量等，基于此对源域与目标域中的复数张量进行特征增广，并重新对张量特征空间进行归一化处理。
步骤五：通过朴素张量子空间学习法获取源域与目标域中的复数张量的共享不变张量子空间，以有效降低张量维度；
具体地，在步骤五中，假定源域中加入虚拟样本之后的综合样本集
Figure PCTCN2020114433-appb-000038
含有N s个样本，其中z＝1,2,L,N s，且每个样本均为三阶张量，为便于降维，可以合并成为四阶张量Ω s，其中多出维度的影响因子为机器人姿态，同样的，将目标域中的三阶张量扩展为四阶张量Ω t。假定源域张量和目标域张量共享一个张量子空间U＝{U (k)}，其中k＝1,2,3，在Tucker分解的基础上，求U等价于求解如下优化问题：
Figure PCTCN2020114433-appb-000039
Figure PCTCN2020114433-appb-000040
其中，G s和G t分别表示Ω s和Ω t的张量子空间，I表示单位矩阵，U表示不变 张量子空间。利用Tucker分解算法可以有效求解上述方程，只要确定了最优U值，则可通过下式得到G s：
G s＝Ω s× 1U (1)T× 1U (2)T× 1U (3)T
类似的程序也可用于推导G t,至此，即可实现张量的降维，有效地降低回归预测的计算量，进而大幅提升预测效率。
步骤六：构建基于共轭增广输入的复数核极限学习机，对输入的复数张量进行共轭增广处理，以充分利用复数张量的二阶统计量，进而采用该学习机对目标姿态的刀尖多模态频响进行预测。
具体地，在步骤六中，基本的核极限学习机可表示为：
Figure PCTCN2020114433-appb-000041
其中，C为成本参数，其取值范围为{2 -24,2 -23,2 -22,...,2 24,2 25}，
Figure PCTCN2020114433-appb-000042
为输入矩阵，Τ为输出向量，K(x i,x j)表示复数高斯核函数，Ω ELM为核矩阵，其中Ω ij＝K(x i,x j)，在本发明中，为更好地适应复数回归问题，所采用地复数高斯核函数表示为：
Figure PCTCN2020114433-appb-000043
其中d x＝x i-x j，v r和v rj为实数，μ为复数，可根据训练样本特征进行具体的选择，γ为正则化参数，其取值范围为{2 -24,2 -23,2 -22,...,2 24,2 25}
此外，为充分利用样本中复数张量的二阶统计量，构建了基于共轭增广输入的复数核极限学习机，该方法机制简单，但是能较好地提升复数域学习回归性能。以第k个样本的输入为例，经过共轭增广处理之后地网络输入为：
Figure PCTCN2020114433-appb-000044
至此，完成了基于共轭增广输入的复数核极限学习机，该极限学习机在本发明中主要用于辅助虚拟样本的生成以及多模态频响的回归预测。
上述说明示出并描述了本发明的实施方法，如前所述，应当理解本发明并非局限于文本所披露的形式，不应看作是对其他实施例的排除，而可用于各种其他组合、修改和环境，并能够在本文所述发明构想范围内，通过上述教导或相关领域的技术或知识进行改动。而本领域人员所进行的改动和变化不脱离本发明的精 神和范围，则都应在本发明所附权利要求的保护范围内。

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  1. 基于小样本迁移学习的铣削机器人多模态频响预测方法其特征在于，包括以下步骤：
    步骤一：选取铣削机器人任意两个姿态下本体上的n个锤击点和刀尖点进行锤击测试，以得到频响数据作为迁移源，同样对目标姿态下本体上的n个锤击点进行锤击测试，以得到频响数据作为迁移目标；
    步骤二：利用铣削机器人本体上锤击点的频响数据构建机器人频响特性迁移源域和目标域的三阶复数张量，基于最小二乘复指数法对刀尖点的多模态频响进行多阶模态参数识别，进而构建迁移源域中数据的标签；
    步骤三：基于迁移源域的输入张量和输出向量，采用基于三角隶属度的信息扩展函数和多目标灰狼优化算法生成虚拟样本，以有效提升源样本的规模；
    步骤四：通过对原始频响数据进行傅里叶逆变换得到时域数据，通过小波变换得到实频域数据，进而分别在频域、时域、时频域中进行数据特征的提取，并基于此对源域与目标域中的复数张量进行特征增广，并重新对张量特征空间进行归一化处理；
    步骤五：通过朴素张量子空间学习法获取源域与目标域中的复数张量的共享不变张量子空间，以有效降低张量维度；
    步骤六：构建基于共轭增广输入的复数核极限学习机，对输入的复数张量进行共轭增广处理，以充分利用复数张量的二阶统计量，进而采用该学习机对目标姿态下刀尖点的多模态频响进行预测。
  2. 根据权利要求1所述的基于小样本迁移学习的铣削机器人多模态频响预测方法，其特征在于：所述步骤一中选取任意两个姿态下铣削机器人本体上的n个锤击点和刀尖点进行锤击测试，以x轴正方向为激励方向，通过三轴加速度传感器采集相应的直接频响与交叉频响作为原始迁移源域的数据，同样对目标姿态下铣削机器人本体上的n个锤击点进行锤击测试，以得到频响数据作为迁移目标域的数据。
  3. 根据权利要求1所述的基于小样本迁移学习的铣削机器人多模态频响预测方法，其特征在于：所述步骤二中构建机器人频响特性源域和目标域的三阶复数张量
    Figure PCTCN2020114433-appb-100001
    第一阶表示的影响因子为锤击点，第二阶表示的影响因子为频响数据，第三阶表示的影响因子为频响类型，基于最小二乘复指数法对刀尖点的多模态频响进行多阶模态参数识别，进而构建迁移源域的输出向量。
  4. 根据权利要求1所述的基于小样本迁移学习的铣削机器人多模态频响预测方法，其特征在于：所述步骤三中基于迁移源域的输入张量和输出向量，采用基于三角隶属度的信息扩展函数进行输入张量特征的非对称可行扩展域，进而通过多目标灰狼优化算法结合复数核极限学习机的方法在可行扩展域中进行虚拟样本 的生成与筛选，以有效提升样本规模。
  5. 根据权利要求1所述的基于小样本迁移学习的铣削机器人多模态频响预测方法，其特征在于：所述步骤四中通过对原始频响数据进行傅里叶逆变换得到时域数据，通过小波变换得到实频域数据，进而分别在频域、时域、时频域中进行数据特征的提取，主要包括：方差、偏度、峰度、光谱偏度、光谱峰度和平均能量等，基于此对源域与目标域中的复数张量进行特征增广，并重新对张量特征空间进行归一化处理。
  6. 根据权利要求1所述的基于小样本迁移学习的铣削机器人多模态频响预测方法，其特征在于：所述步骤五中通过朴素张量子空间学习法获取源域与目标域中的复数张量的共享不变张量子空间，以有效降低张量维度。
  7. 根据权利要求1所述的基于小样本迁移学习的铣削机器人多模态频响预测方法，其特征在于：所述步骤六中构建基于共轭增广输入的复数核极限学习机，对输入的复数张量进行共轭增广处理，以充分利用复数张量的二阶统计量，并通过复数高斯核函数进一步提升极限学习机对复数问题的计算能力，进而采用该学习机对铣削机器人在目标姿态下刀尖点的多模态频响进行回归预测。
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