# Accurate Adaptive Compensation Method for Mechanical Structure Error of the Blade Measuring System

Authors: Wei Shao; Peng Peng; Awei Zhou; Quanquan Zhu; Di Zhao
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

In view of the high precision requirement for mechanical structure of aeronautical blade measuring system, this paper proposes a laser interferometer to measure the error of the spatial nodes of the measuring system based on a comprehensive analysis of domestic and foreign error compensation methods for the measuring system. The optimized algorithm backpropagation (BP) neural network (OA-BPNN) compensation method is utilized to adaptively compensate for the systematic error of the mechanical system. Compared with the traditional polynomial fitting and genetic algorithm BP neural network (GA-BPNN) algorithm, the results show that the OA-BPNN algorithm is characterized by the best adaptability, precision, and efficiency for the adaptive error compensation. The spatial errors in the XYZ directions are reduced from 10.9, 60.1, and 84.2μm to 1.3, 4.0, and 2.4 μm, respectively. The method is of great theoretical significance and practical value.


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## Body

## 1. Introduction

As the blade is one of the essential components of the engine, the measurement of the blade with high efficiency, precision, and consistency in the manufacturing process is fundamental to enhance the manufacturing quality [1, 2]. At present, the blade measuring system with the coordinate measuring machine as the main part finds wide use. However, in the measurement process, it is difficult to achieve the high precision required for blade measurement [3–5], which is mainly due to the fact that the mechanical structure error of the measuring machine increases the measurement error [6]. The error compensation is an effective method to improve the overall accuracy of the measuring machine[7]. Therefore, in order to improve the accuracy of the blade measuring system, it is important to compensate for the mechanical structural error of the measuring system.The error of the mechanical system mainly includes geometric error, force deformation error, thermal error, and dynamic error [8, 9]. These deformations change the spatial position of the measuring head, causing error in the data and affecting the accuracy of the measuring system. Thermal error is caused by the deformation of the mechanism due to temperature changes or temperature differences at different locations [10, 11]. The dynamic error results from the dynamic deformation and vibration of the mechanical structural components due to the inertial force caused by the different speeds of the mechanism during the measurement process [12]. The blade measuring system in this paper realizes blade measurement by using the cone-optical polarization holography technology and the cylindrical coordinate mechanical structure [13]. The heat source is small and the environment temperature is constant, so the thermal error is small. The turntable rotates and the measuring head follows in the measurement, so the dynamic error is small. As a result, mainly geometric error and force deformation error influence the system accuracy.Many experts and scholars have studied the compensation for the error of the mechanical structure of the measurement system. R. Hocken [14] expressed the single error of the mechanical structure with a fitting polynomial, which can obviously improve the measurement accuracy of the measuring system. However, the error models of different machines are different, so this method is not adaptive. Guo Junjie et al. [15] analyzed the spatial error of the mechanical structure of the measuring system and proposed an algorithm for separating 21 geometric errors. Lin Shuwen et al. [16]. used the coordinate kinematic chain method for system analysis to compensate for the error of the entire system based on the kinematic chain of the space coordinate system. Literature [17–21] deals with geometric error and force deformation error of the measuring machine based on meshing [17–21]. The above methods compensate for error by least square or cubic interpolation, so the accuracy of compensation is easily affected by the different mechanical structures and the mesh spacing.In this paper, based on previous research, the laser interferometer is used to measure the error of the spatial nodes of the mechanical structure, and the optimized backpropagation neural network (OA-BPNN) algorithm is applied to adaptively compensate for the error. Compared with the polynomial fitting and genetic backpropagation neural network (GA-BPNN) algorithm, the OA-BPNN algorithm boasts best adaptability, precision, and efficiency for the adaptive error compensation. The method is of great theoretical significance and practical value.


## 2. Error Analysis of the Blade Measuring System

The blade measuring system designed in this paper is shown in Figure1. The system, with the cylindrical coordinate measuring machine as the main part, is mainly composed of three linear axes, a rotation axis, a grating scale, and a drive device. To enhance the measurement accuracy of the system, it is necessary to analyze the mechanical structure and establish the corresponding mathematical model of the structure [22]. As shown in Figure 2, assuming that the system workbench and each axis are ideal rigid bodies with constant state during the measurement process, the basic coordinate system XYZ is established on the workbench. The coordinate system X1Y1Z1 translated along the Y-axis is established at the Y-axis origin; the coordinate system X2Y2Z2 translated along the X-axis is established at the X-axis origin; the coordinate system X3Y3Z3 translated along the Z-axis is established at the Z-axis origin.Figure 1
Photo of the blade measuring system.Figure 2
Simplified coordinate system of the measuring machine.(1) Workbench. (2) Rotary table. (3) X-axis slider. (4) Z-axis column. (5) Z-axis slider. (6) Measuring head. (7) Support of the measuring head.In the measurement process, the measuring head in the position M (xm, ym, zm) in the basic coordinate system moves by x, y, z along the X-, Y-, Z-axes to the new point N (xn, yn, zn) when the measuring machine moves. At this time, the coordinates of point N are (x+xm, y+ym, z+zm), but in reality, its position is affected by the motion error and the deformation of the measuring machine. As a result, the theoretical coordinates of the N point of the measuring head are (x+xm, y+ym, z+zm) and the actual coordinates are (xn, yn, zn), so the difference between the actual value and the theoretical value is the measurement error.Therefore, during the measurement process, after a point in the local coordinate system X1Y1Z1 has moved by y in the Y direction, the coordinates of origin O1 in the coordinate system XYZ can be represented by the following vector:(1)OO1=Y=δxyy+δyyδzyIn the above formula,δx(y), δy(y), δz(y) are the positioning errors in the X, Y, Z directions, respectively.At the same time, theoretically, the coordinate system X1Y1Z1 has a rotation angle error with respect to XYZ. Assuming that the coordinate system XYZ rotates by the angles θ, φ, and ω around Z, Y, and X axes, respectively, the relationship between the two local coordinate systems can be presented by the rotation matrix R:(2)x1y1z1=RxyzThe rotation matrix R is a trigonometric function matrix involving the three angles, and the relationship matrix is as follows:(3)R=cos⁡θsin⁡θ0-sin⁡θcos⁡θ0001cos⁡φ0-sin⁡φ010sin⁡φ0cos⁡φ1000cos⁡ωsin⁡ω0-sin⁡ωcos⁡ωIn the above relationship, since the angular error is very small during the actual movement, the matric involving the three angles can be simplified as follows:(4)R=θθ0-θθ0001φ0-φ010φ0φ1000ωω0-ωω=1θ-φ-θ1ωφ-ω1The three anglesθ, φ, and ω are angular motion errors εZ(y), εy(y), and εx(y), respectively, and the rotation matrix caused by the movement in the Y direction can be represented by R (y) as follows:(5)Ry=1εzy-εyy-εzy1εxyεyy-εxy1Similarly, the relationship of the coordinate change of the coordinate system in the Z direction can be obtained as follows:(6)O1O2=Z=εxzεyz-zαyzz+εzzThe rotation matrix can be obtained:(7)Rz=1εzz-εyz-εzz1εxzεyz-εxz1Similarly, the relationship of the coordinate change of the coordinate system in the X direction can be obtained as follows:(8)O2O3=X=x+εxxεyx-yαxyεzx-zαxzThe rotation matrix can be obtained:(9)Rx=1εzx-εyx-εzx1εxxεyx-εxx1In the coordinate system XYZ, the measuring head moves from M to N, and the coordinates of point N on the head are (Xn, Yn, Zn). The position of M in the X3Y3Z3 coordinate system can be represented by O3M (xm, ym, zm). The point is transformed into a point in the coordinate system X2Y2Z2, and so the coordinates of point M in the coordinate system X2Y2Z2 can be obtained as follows:(10)O2M=O2O3=R-1xO3MThe point M is transformed from the coordinate system X2Y2Z2 into the coordinate system X1Y1Z1, and then the coordinates of point M in the coordinate system X1Y1Z1 are obtained:(11)O1M=O1O2=R-1zO2MThe OM is the position of point M in the XYZ system; the coordinates of OM are(12)OM=OO1=R-1yO1MAt this time, point M is point N that moves to this point, so the following equation can be derived:(13)ON=OM+ΔδThe equation can be organized to find the coordinates of point N:(14)ON=OO1+R-1xO1M+Δδ=OO1+R-1xO1O2+R-1zO2O3+R-1xO3M+ΔδΔ δ is an additional term due to complex reasons. The additional term, which is a complex deformation caused by uncertain factors in the measurement process, exists theoretically but cannot be described by certain mathematical formulas and numerical values.(15)Δδ=δxx,z,εxx,zδyy,z,εyy,zδzx,z,δzy,zSubstitution of the above equations into the calculation equation ON and the expansion of the equation yields a three-dimensional matrix expression about ON:(16)ON=δxyy+δyyδzy+1εzy-εyy-εzy1εxyεyy-εxy1+εxzεyz-zαyzz+εzz+1εzz-εyz-εzz1εxzεyz-εxz1x+εxxεyx-yαxyεzx-zαxz+1εzx-εyx-εzx1εxxεyx-εxx1xmymzm+δxx,z,εxx,zδyy,z,εyy,zδzx,z,δzy,zIn the above equation, the three actual coordinates of ON areXn, Yn, and Zn, and the theoretical coordinates of point N of the measuring head are (x+xm, y+ym, z+zm). Then the systematic error can be obtained as follows:(17)Δx=xn-x+xmΔy=yn-y+ymΔz=zn-z+zmFrom (16) and (17), it can be derived that the positioning errors of the measuring head in X, Y, and Z directions are obtained by the nonlinear influence of the motion errors, deformation errors, and other uncertain errors of the measuring machine in the three directions. In the actual process, different deformation by different actions and influences by random factors arise at different positions due to the fact that the system is not an ideal rigid body and that the component structure is complex. Therefore, there are problems such as limited model theory, poor versatility, complex modeling, long time, low robustness, and unsuitability for real-time compensation for errors [23]. Consequently, this paper proposes the optimized algorithm backpropagation neural network (OA-BPNN) adaptive compensation method for mechanical structure error of the blade measuring system. The error value predicted by the OA-BPNN algorithm is utilized as the error compensation amount of the blade measuring system, which can effectively improve the measurement accuracy and error compensation rate.


## 3. Design of the Backpropagation (BP) Neural Network Compensation Algorithm

### 3.1. BP Neural Network

The BP neural network is a multilayer feedforward neural network, which features forward propagation of signals and backpropagation of errors. In the forward propagation, the input signals are passed layer by layer from the input layer via the hidden layer to the output layer. The neuronal state of each layer only affects the state of neurons of the next layer. If the output layer does not obtain the expected outputs, then it goes into the back propagation and adjusts the network weights and thresholds based on the predicted error so that the predicted values by the BP neural network continuously approaches the expected outputs. The topology of the BP neural network is shown in Figure3 and it can be regarded as a nonlinear function. The network input values and predicted values are the independent and dependent variables of the function. When the number of input nodes is n and the number of output nodes is m, the BP neural network expresses the function mapping relationship from n independent variables to m dependent variables[24].Figure 3
Topology of the BP neural network.


### 3.2. Optimized Algorithm Backpropagation Neural Network (OA-BPNN) Compensation Algorithm

The structure of the error prediction model for the blade measuring system constructed in this paper is shown in Figure4, according to the characteristics of the errors in the previous section. It has one input layer node, two hidden layers that both has 5 nodes, and one output layer node.X is the input value, that is, the node of the error measurement. W0i is the weight from the input layer to the hidden layer. Wij is the weight from one hidden layer to the other hidden layer. Wj3 is the weight from the hidden layer to the output layer.Y is the output value of the BP neural network, that is, the predicted value of the measurement error.Figure 4
Structure of the OA-BPNN error prediction model.Based on the above structural relationship in Figure4, (18) and (19) are used to express the propagation and weights in the BP neural network.(18)Hi=fw0ix+aii=1,2,…,5;(19)Hj=f∑i=15wijHi+ajj=1,2,…,5,wheref is the activation function of the hidden layer, which has a variety of expressions. The function chosen in this paper is the unipolar Sigmoid function. (20)fx=11+e-xBased on the outputH of the hidden layer, the weightwj3 and thresholda3 are used to calculated the predicted outputO.(21)O=∑j=15Hjwj3+a3The function chosen for the activation function of the output layer is the Purelin function g(x)=x, so the input of the output layer is equal to the output. According to the network predicted outputO and the expected outputY, the network predicted error e is calculated and the predicted error is calculated using the square sum error function.(22)e=Y-O(23)E=12e2Then the weights and thresholds of the network are continuously adjusted by backpropagation, and finally the global error coefficient is minimized. The network connection weightsw0i,wij,wj3 are updated according to the network predicted error ei, and the learning rate is γ, as shown in the following.(24)w0i=w0i+γ∂E∂w0ii=1,2,…,5;(25)wij=wij+γ∂E∂wiji=1,2,…,5;j=1,2,…,5;(26)wj3=wj3+γ∂E∂wj3j=1,2,…,5;In the process of error backpropagation, the goal is to obtain the minimum value of the error function, namely,min⁡E, so the gradient descent method is used to calculate it. In (25), ∂E/∂wij is calculated as follows, and it is calculated in the same way in (24) and (26).(27)∂E∂wij=∂E∂Hj×∂Hj∂wij∂E∂Hj=Y-O∂O∂Hj=-wj3e∂Hj∂wij=∂f∑i=15wijHi+aj∂wij=Hj1-HjHiw i j is updated as equation (28): (28)wij=wij+γHj1-HjHiwj3ei,j=1,2,…,5;Thresholds and weights are updated in a similar way. The network node thresholdsai,aj,a3 and learning rate γ are updated according to the network predicted errore.(29)ai=ai+γ∂E∂aii=1,2,…,5;(30)aj=aj+γ∂E∂ajj=1,2,…,5;(31)a3=a3+γ∂E∂a3In (30), ∂E/∂aj is calculated as follows, and (29) and (31) are calculated in the same way.(32)∂E∂aj=∂E∂Hj×∂Hj∂aj∂E∂Hj=Y-O∂O∂Hj=-wj3e∂Hj∂aj=∂f∑i=15wijHi+aj∂aj=Hj1-HjThe final updated equation foraj is (33)aj=aj+γHj1-Hjwj3ej=1,2,…,5;Finally, its convergence is based on whether the adjacent two errors are smaller than the specified value. If it converges, the output of the first hidden layer is recalculated until the requirement is satisfied.


## 3.1. BP Neural Network

The BP neural network is a multilayer feedforward neural network, which features forward propagation of signals and backpropagation of errors. In the forward propagation, the input signals are passed layer by layer from the input layer via the hidden layer to the output layer. The neuronal state of each layer only affects the state of neurons of the next layer. If the output layer does not obtain the expected outputs, then it goes into the back propagation and adjusts the network weights and thresholds based on the predicted error so that the predicted values by the BP neural network continuously approaches the expected outputs. The topology of the BP neural network is shown in Figure3 and it can be regarded as a nonlinear function. The network input values and predicted values are the independent and dependent variables of the function. When the number of input nodes is n and the number of output nodes is m, the BP neural network expresses the function mapping relationship from n independent variables to m dependent variables[24].Figure 3
Topology of the BP neural network.


## 3.2. Optimized Algorithm Backpropagation Neural Network (OA-BPNN) Compensation Algorithm

The structure of the error prediction model for the blade measuring system constructed in this paper is shown in Figure4, according to the characteristics of the errors in the previous section. It has one input layer node, two hidden layers that both has 5 nodes, and one output layer node.X is the input value, that is, the node of the error measurement. W0i is the weight from the input layer to the hidden layer. Wij is the weight from one hidden layer to the other hidden layer. Wj3 is the weight from the hidden layer to the output layer.Y is the output value of the BP neural network, that is, the predicted value of the measurement error.Figure 4
Structure of the OA-BPNN error prediction model.Based on the above structural relationship in Figure4, (18) and (19) are used to express the propagation and weights in the BP neural network.(18)Hi=fw0ix+aii=1,2,…,5;(19)Hj=f∑i=15wijHi+ajj=1,2,…,5,wheref is the activation function of the hidden layer, which has a variety of expressions. The function chosen in this paper is the unipolar Sigmoid function. (20)fx=11+e-xBased on the outputH of the hidden layer, the weightwj3 and thresholda3 are used to calculated the predicted outputO.(21)O=∑j=15Hjwj3+a3The function chosen for the activation function of the output layer is the Purelin function g(x)=x, so the input of the output layer is equal to the output. According to the network predicted outputO and the expected outputY, the network predicted error e is calculated and the predicted error is calculated using the square sum error function.(22)e=Y-O(23)E=12e2Then the weights and thresholds of the network are continuously adjusted by backpropagation, and finally the global error coefficient is minimized. The network connection weightsw0i,wij,wj3 are updated according to the network predicted error ei, and the learning rate is γ, as shown in the following.(24)w0i=w0i+γ∂E∂w0ii=1,2,…,5;(25)wij=wij+γ∂E∂wiji=1,2,…,5;j=1,2,…,5;(26)wj3=wj3+γ∂E∂wj3j=1,2,…,5;In the process of error backpropagation, the goal is to obtain the minimum value of the error function, namely,min⁡E, so the gradient descent method is used to calculate it. In (25), ∂E/∂wij is calculated as follows, and it is calculated in the same way in (24) and (26).(27)∂E∂wij=∂E∂Hj×∂Hj∂wij∂E∂Hj=Y-O∂O∂Hj=-wj3e∂Hj∂wij=∂f∑i=15wijHi+aj∂wij=Hj1-HjHiw i j is updated as equation (28): (28)wij=wij+γHj1-HjHiwj3ei,j=1,2,…,5;Thresholds and weights are updated in a similar way. The network node thresholdsai,aj,a3 and learning rate γ are updated according to the network predicted errore.(29)ai=ai+γ∂E∂aii=1,2,…,5;(30)aj=aj+γ∂E∂ajj=1,2,…,5;(31)a3=a3+γ∂E∂a3In (30), ∂E/∂aj is calculated as follows, and (29) and (31) are calculated in the same way.(32)∂E∂aj=∂E∂Hj×∂Hj∂aj∂E∂Hj=Y-O∂O∂Hj=-wj3e∂Hj∂aj=∂f∑i=15wijHi+aj∂aj=Hj1-HjThe final updated equation foraj is (33)aj=aj+γHj1-Hjwj3ej=1,2,…,5;Finally, its convergence is based on whether the adjacent two errors are smaller than the specified value. If it converges, the output of the first hidden layer is recalculated until the requirement is satisfied.


## 4. Error Compensation Experiment

### 4.1. Design of the Experiment

In this experiment, shown in Figure5, a laser interferometer is used to measure and verify the spatial nodes error of the mechanical structure of the measurement system. The laser interferometer is characterized by high precision, the linear resolution of up to 0.02μm, and large measuring range and can perform continuous measurement, so it is widely used in error measurement of machine tools or measuring machines [25]. The measurement accuracy in the experiment is related to the parameters of the surrounding environment, including air temperature, air pressure, and relative humidity [26]. Therefore, the matching devices of the laser interferometer, such as air temperature sensor, air pressure sensor, and relative humidity sensor, should be input to the socket of the host of the laser transmitter, and the relevant compensation coefficients in the measurement software should be set. The three-direction movement range of the measurement mechanism constitutes a measurement space, where the error differs in different directions and at different positions [27]. Therefore, the entire space is divided in this paper. Each direction is evenly divided by 30 selected points spaced 4mm apart. The measurement spatial nodes are shown in Figure 6.Figure 5
Experiment system.Figure 6
Cloud of the measurement spatial nodes.In the experiment, the three-dimensional coordinate error of each node in the space can be measured. Figure7 shows the error of each nodes on the 16 lines in the Y direction, with the maximum being 60.1 μm.Figure 7
Error distribution in the Y direction.


### 4.2. Data Processing and Comparison Analysis

After the measurement, linear error and nonlinear error are included in the data. The linear error can be corrected. However, it is hard to attain a standard model to compensate for the nonlinear error. The least square polynomial fitting, genetic algorithm BP neural network (GA-BPNN) algorithm, and the OA-BPNN algorithm are adopted to fit the systematic error data of the mechanical structure and the fitting results are compared.For the X direction, three algorithms are used for error data fitting and comparison. As shown in Figure8, graphs (a), (b), and (c) are the error data fittings and residual plots of the three algorithms, respectively. Graph (d) is the residual comparison of the three fitting methods. It can be seen that the proposed OA-BPNN fitting error is completely consistent with the curve trends of other methods.Figure 8
Data fitting comparison in the X direction.
(a)
Sixth-order polynomial fitting results (b)
OA-BPNN fitting results (c)
GA-BPNN fitting results (d)
Residual graph of the three fitting methodsFor the Y direction, three algorithms are used for error data fitting and comparison. As shown in Figure9, graphs (a), (b), and (c) are the error data fittings and residual plots of the three algorithms, respectively. Graph (d) is the residual comparison of the three fitting methods. It can be seen that the proposed OA-BPNN fitting error is completely consistent with the curve trends of other methods.Figure 9
Data fitting comparison in the Y direction.
(a)
Fifth-order polynomial fitting results (b)
OA-BPNN fitting results (c)
GA-BPNN fitting results (d)
Residual graph of the three fitting methodsFor the Z direction, three algorithms are used for error data fitting and comparison. As shown in Figure10, graphs (a), (b), and (c) are the error data fittings and residual plots of the three algorithms, respectively. Graph (d) is the residual comparison of the three fitting methods. It can be seen that the proposed OA-BPNN fitting error is completely consistent with the curve trends of other methods.Figure 10
Data fitting comparison in the Z direction.
(a)
Fourth-order polynomial fitting results (b)
OA-BPNN fitting results (c)
GA-BPNN fitting results (d)
Residual graph of the three fitting methodsIt can be seen from Figures8, 9, and 10 that when the error term in each direction is the minimum in the least square polynomial fitting, the X direction fits to a 6th-order polynomial fitting, the Y direction fits to a 5th-order polynomial fitting, and the Z direction fits to a 4th-order polynomial fitting. The GA-BPNN algorithm and the proposed OA-BPNN algorithm are adaptive to different error data fittings. The trends of the three fitting methods are basically the same, and there is no significant difference in accuracy.


### 4.3. Analysis of the Error Compensation Results

The polynomial fitting, proposed OA-BPNN, and GA-BPNN compensation algorithms are used to compensate for errors in the XYZ directions. Ten experiments are performed and the results are shown in Tables1, 2, and 3. Although the maximum absolute errors and mean square errors by the three algorithms are almost the same, the GA-BPNN fitting takes too long and the polynomial fitting is not an adaptive error compensation method because the best fitting high-order terms should be determined in the polynomial fitting, which depends on personnel experience.Table 1
Compensation results in the X direction.
X Sixth-orderPolynomial Fitting OA-BPNN GA-BPNN Time (s) 0.2 1.1 60.8 MAE (μm) 1.4 1.3 1.4 MSE (μm) 0.4 0.3 0.4Table 2
Compensation results in the Y direction.
Y fifth-orderPolynomial Fitting OA-BPNN GA-BPNN Time (s) 0.2 1.3 61.6 MAE (μm) 4.1 4.0 4.1 MSE (μm) 2.8 2.5 2.8Table 3
Compensation results in the Z direction.
Z fourth-orderPolynomial Fitting OA-BPNN GA-BPNN Time (s) 0.2 1.1 61.3 MAE (μm) 2.4 2.4 2.5 MSE (μm) 1.6 1.5 1.8In summary, the OA-BPNN algorithm used in this paper is characterized by the best adaptability, precision, and efficiency for the adaptive error compensation. The maximum errors (MAE) in the XYZ directions are reduced by 88.1%, 93.7%, and 97.2%, from 10.9, 60.1, and 84.2μm to 1.3, 4.0, and 2.4μm, respectively. Mean square error (MSE) is similar.


## 4.1. Design of the Experiment

In this experiment, shown in Figure5, a laser interferometer is used to measure and verify the spatial nodes error of the mechanical structure of the measurement system. The laser interferometer is characterized by high precision, the linear resolution of up to 0.02μm, and large measuring range and can perform continuous measurement, so it is widely used in error measurement of machine tools or measuring machines [25]. The measurement accuracy in the experiment is related to the parameters of the surrounding environment, including air temperature, air pressure, and relative humidity [26]. Therefore, the matching devices of the laser interferometer, such as air temperature sensor, air pressure sensor, and relative humidity sensor, should be input to the socket of the host of the laser transmitter, and the relevant compensation coefficients in the measurement software should be set. The three-direction movement range of the measurement mechanism constitutes a measurement space, where the error differs in different directions and at different positions [27]. Therefore, the entire space is divided in this paper. Each direction is evenly divided by 30 selected points spaced 4mm apart. The measurement spatial nodes are shown in Figure 6.Figure 5
Experiment system.Figure 6
Cloud of the measurement spatial nodes.In the experiment, the three-dimensional coordinate error of each node in the space can be measured. Figure7 shows the error of each nodes on the 16 lines in the Y direction, with the maximum being 60.1 μm.Figure 7
Error distribution in the Y direction.


## 4.2. Data Processing and Comparison Analysis

After the measurement, linear error and nonlinear error are included in the data. The linear error can be corrected. However, it is hard to attain a standard model to compensate for the nonlinear error. The least square polynomial fitting, genetic algorithm BP neural network (GA-BPNN) algorithm, and the OA-BPNN algorithm are adopted to fit the systematic error data of the mechanical structure and the fitting results are compared.For the X direction, three algorithms are used for error data fitting and comparison. As shown in Figure8, graphs (a), (b), and (c) are the error data fittings and residual plots of the three algorithms, respectively. Graph (d) is the residual comparison of the three fitting methods. It can be seen that the proposed OA-BPNN fitting error is completely consistent with the curve trends of other methods.Figure 8
Data fitting comparison in the X direction.
(a)
Sixth-order polynomial fitting results (b)
OA-BPNN fitting results (c)
GA-BPNN fitting results (d)
Residual graph of the three fitting methodsFor the Y direction, three algorithms are used for error data fitting and comparison. As shown in Figure9, graphs (a), (b), and (c) are the error data fittings and residual plots of the three algorithms, respectively. Graph (d) is the residual comparison of the three fitting methods. It can be seen that the proposed OA-BPNN fitting error is completely consistent with the curve trends of other methods.Figure 9
Data fitting comparison in the Y direction.
(a)
Fifth-order polynomial fitting results (b)
OA-BPNN fitting results (c)
GA-BPNN fitting results (d)
Residual graph of the three fitting methodsFor the Z direction, three algorithms are used for error data fitting and comparison. As shown in Figure10, graphs (a), (b), and (c) are the error data fittings and residual plots of the three algorithms, respectively. Graph (d) is the residual comparison of the three fitting methods. It can be seen that the proposed OA-BPNN fitting error is completely consistent with the curve trends of other methods.Figure 10
Data fitting comparison in the Z direction.
(a)
Fourth-order polynomial fitting results (b)
OA-BPNN fitting results (c)
GA-BPNN fitting results (d)
Residual graph of the three fitting methodsIt can be seen from Figures8, 9, and 10 that when the error term in each direction is the minimum in the least square polynomial fitting, the X direction fits to a 6th-order polynomial fitting, the Y direction fits to a 5th-order polynomial fitting, and the Z direction fits to a 4th-order polynomial fitting. The GA-BPNN algorithm and the proposed OA-BPNN algorithm are adaptive to different error data fittings. The trends of the three fitting methods are basically the same, and there is no significant difference in accuracy.


## 4.3. Analysis of the Error Compensation Results

The polynomial fitting, proposed OA-BPNN, and GA-BPNN compensation algorithms are used to compensate for errors in the XYZ directions. Ten experiments are performed and the results are shown in Tables1, 2, and 3. Although the maximum absolute errors and mean square errors by the three algorithms are almost the same, the GA-BPNN fitting takes too long and the polynomial fitting is not an adaptive error compensation method because the best fitting high-order terms should be determined in the polynomial fitting, which depends on personnel experience.Table 1
Compensation results in the X direction.
X Sixth-orderPolynomial Fitting OA-BPNN GA-BPNN Time (s) 0.2 1.1 60.8 MAE (μm) 1.4 1.3 1.4 MSE (μm) 0.4 0.3 0.4Table 2
Compensation results in the Y direction.
Y fifth-orderPolynomial Fitting OA-BPNN GA-BPNN Time (s) 0.2 1.3 61.6 MAE (μm) 4.1 4.0 4.1 MSE (μm) 2.8 2.5 2.8Table 3
Compensation results in the Z direction.
Z fourth-orderPolynomial Fitting OA-BPNN GA-BPNN Time (s) 0.2 1.1 61.3 MAE (μm) 2.4 2.4 2.5 MSE (μm) 1.6 1.5 1.8In summary, the OA-BPNN algorithm used in this paper is characterized by the best adaptability, precision, and efficiency for the adaptive error compensation. The maximum errors (MAE) in the XYZ directions are reduced by 88.1%, 93.7%, and 97.2%, from 10.9, 60.1, and 84.2μm to 1.3, 4.0, and 2.4μm, respectively. Mean square error (MSE) is similar.


## 5. Conclusions

In order to compensate for the error in the blade measuring system, this paper proposes a laser interferometer to measure the error of the spatial nodes of the measurement system based on the findings of previous studies. The OA-BPNN compensation method is utilized to adaptively compensates for the systematic error of the mechanical system. Compared with the traditional polynomial fitting and genetic BP neural network (GA-BPNN) algorithm, the results show that the optimized backpropagation neural network (OA-BPNN) algorithm is characterized by the best adaptability, precision, and efficiency for the self-adaptive error compensation. The maximum errors in the XYZ directions are reduced by 88.1%, 93.7%, and 97.2%, from 10.9, 60.1, and 84.2μm to 1.3, 4.0, and 2.4μm, respectively. The method is of great theoretical significance and practical value for error compensation of other measuring systems.



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*Source: 7527392-2018-10-17.xml*