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

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 84.2 μm to 1.3, 4.0, and 2.4 μm, respectively. The method is of great theoretical significance and practical value.


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][4][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][18][19][20][21] deals with geometric error and force deformation error of the measuring machine based on meshing [17][18][19][20][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.

Error Analysis of the Blade Measuring System
The blade measuring system designed in this paper is shown in Figure 1.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 X 1 Y 1 Z 1 translated along the Y-axis is established at the Y-axis origin; the coordinate system X 2 Y 2 Z 2  translated along the X-axis is established at the X-axis origin; the coordinate system X 3 Y 3 Z 3 translated along the Z-axis is established at the Z-axis origin.
In the measurement process, the measuring head in the position M (x m , y m , z m ) in the basic coordinate system moves by x, y, z along the X-, Y-, Z-axes to the new point N (x n , y n , z n ) when the measuring machine moves.At this time, the coordinates of point N are (x+x m , y+y m , z+z m ), 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+x m , y+y m , z+z m ) and the actual coordinates are (x n , y n , z n ), 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 X 1 Y 1 Z 1 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: In the above formula,   (),   (),   () are the positioning errors in the X, Y, Z directions, respectively.
At the same time, theoretically, the coordinate system X 1 Y 1 Z 1 has a rotation angle error with respect to XYZ.Assuming that the coordinate system XYZ rotates by the angles , , and  around Z, Y, and X axes, respectively, the relationship between the two local coordinate systems can be presented by the rotation matrix R: The rotation matrix R is a trigonometric function matrix involving the three angles, and the relationship matrix is as follows: In 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: The three angles , , and  are angular motion errors  Z (y),  y (y), and  x (y), respectively, and the rotation matrix caused by the movement in the Y direction can be represented by R (y) as follows: Similarly, the relationship of the coordinate change of the coordinate system in the Z direction can be obtained as follows: The rotation matrix can be obtained: Similarly, the relationship of the coordinate change of the coordinate system in the X direction can be obtained as follows: The rotation matrix can be obtained: In the coordinate system XYZ, the measuring head moves from M to N, and the coordinates of point N on the head are (X n , Y n , Z n ).The position of M in the X 3 Y 3 Z 3 coordinate system can be represented by O 3 M (x m , y m , z m ).The point is transformed into a point in the coordinate system X 2 Y 2 Z 2 , and so the coordinates of point M in the coordinate system X 2 Y 2 Z 2 can be obtained as follows: The point M is transformed from the coordinate system X 2 Y 2 Z 2 into the coordinate system X 1 Y 1 Z 1 , and then the coordinates of point M in the coordinate system X 1 Y 1 Z 1 are obtained: The OM is the position of point M in the XYZ system; the coordinates of OM are At this time, point M is point N that moves to this point, so the following equation can be derived: The equation can be organized to find the coordinates of point N: 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.
Substitution of the above equations into the calculation equation ON and the expansion of the equation yields a threedimensional matrix expression about ON: In the above equation, the three actual coordinates of ON are X n , Y n , and Z n , and the theoretical coordinates of point N of the measuring head are (x+x m , y+y m , z+z m ).Then the systematic error can be obtained as follows: From ( 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.

Input Layer Hidden Layer Output Layer
Figure 3: Topology of the BP neural network.3 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].

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 Figure 4, 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. 0i is the weight from the input layer to the hidden layer. ij is the weight from one hidden layer to the other hidden layer. j3 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.Based on the above structural relationship in Figure 4, ( 18) and ( 19) are used to express the propagation and weights in the BP neural network.
=  ( 0  +   )  = 1, 2, . . ., 5; (18) Input Layer Hidden Layer Hidden Layer Output Layer  where f 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.
Based on the output H of the hidden layer, the weight w 3 and threshold a 3 are used to calculated the predicted output O.
The 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 output O and the expected output Y, the network predicted error e is calculated and the predicted error is calculated using the square sum error function.
Then the weights and thresholds of the network are continuously adjusted by backpropagation, and finally the global error coefficient is minimized.The network connection weights w 0 , w  , w 3 are updated according to the network predicted error   , and the learning rate is , as shown in the following.
=   +      = 1, 2, . . ., 5;  = 1, 2, . . ., 5; (25) In the process of error backpropagation, the goal is to obtain the minimum value of the error function, namely, min , so the gradient descent method is used to calculate it.In (25), /  is calculated as follows, and it is calculated in the same way in ( 24) and (26).
is updated as equation ( 28 In (30), /  is calculated as follows, and (29) and (31) are calculated in the same way.
The final updated equation for   is 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.

Error Compensation Experiment
4.1.Design of the Experiment.In this experiment, shown in Figure 5, 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 0.02m, 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.
In the experiment, the three-dimensional coordinate error of each node in the space can be measured.Figure 7 shows the error of each nodes on the 16 lines in the Y direction, with the maximum being 60.1 m.

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 Figure 8, 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.
For the Y direction, three algorithms are used for error data fitting and comparison.As shown in Figure 9, 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.
For the Z direction, three algorithms are used for error data fitting and comparison.As shown in Figure 10, 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.
It can be seen from Figures 8, 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.shown in Tables 1, 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.In 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%,

Figure 1 :
Figure 1: Photo of the blade measuring system.

Figure 4 :
Figure 4: Structure of the OA-BPNN error prediction model.

Figure 6 :Figure 7 :
Figure 6: Cloud of the measurement spatial nodes.

Figure 8 :
Figure 8: Data fitting comparison in the X direction.

Table 1 :
Compensation results in the X direction.

Table 2 :
Compensation results in the Y direction.