In the field of aerospace engineering, wind tunnel testing has been a critical method for analyzing and evaluating the aeroelastic characteristics of aircraft. The development and refinement of this technology have significantly advanced aerospace capabilities. Wind tunnel tests allow engineers to simulate the force conditions experienced by aircraft during actual flight in a controlled ground environment. During the design phase, aeroelastic wind tunnel testing verifies whether the structural strength and performance of the aircraft meet requirements and provides guidance for optimization. As a fundamental tool in aerospace progress, wind tunnel testing has become an essential technical assurance in modern aircraft design and validation.
Within wind tunnel research, a key focus is understanding the force conditions on aircraft models under various airflow conditions. The six-axis force/torque sensor (hereafter referred to as six-axis force sensor) can simultaneously detect three orthogonal forces \( F_x \), \( F_y \), \( F_z \) and three orthogonal moments \( M_x \), \( M_y \), \( M_z \) in the Oxy Cartesian coordinate system. This sensor is characterized by its rich perceptual information and high measurement accuracy. The application of six-axis force sensors enables researchers to comprehensively understand the effects of wind loads, including lift, drag, lateral forces, and torque, on the model. These data are crucial for accurately assessing aerodynamic performance and structural response, thereby enhancing design reliability and efficiency. However, during operation, factors such as structural design, manufacturing precision, and installation accuracy can significantly impact measurement accuracy. Signal interference between different measurement directions, known as inter-dimensional coupling, often occurs, limiting the sensor’s application in high-precision and advanced fields. Therefore, reducing or eliminating inter-dimensional coupling is a key challenge in the development of sensor technology.
In recent years, various decoupling algorithms have been proposed by researchers worldwide to improve sensor accuracy. Depending on the sensor type, different decoupling algorithms must be selected to achieve optimal performance. Currently, decoupling algorithms for six-axis force sensors can be classified into static linear decoupling and static nonlinear decoupling. Linear decoupling algorithms employ traditional mathematical methods to correct output signals but require the sensor system to be linear, leading to certain limitations and instability. In contrast, nonlinear decoupling algorithms perform decoupling through multi-function mapping, do not rely on the linear characteristics of the sensor, can be applied to more complex systems, and offer higher accuracy. Consequently, researchers have proposed various nonlinear decoupling algorithms for different configurations of six-axis force sensors.
This paper addresses the issue of decreased force measurement performance in piezoelectric six-axis force/torque sensors caused by inter-dimensional coupling by proposing a decoupling algorithm based on the Long Short-Term Memory (LSTM) neural network. The research utilizes a four-point supported piezoelectric six-axis force sensor developed by our research group. The algorithm aims to optimize the multi-dimensional nonlinear characteristics of the sensor output, establishing an accurate mapping relationship between input and output after decoupling.
Measurement Principle of the Six-Axis Force Sensor and Calibration Experiment
The four-point supported piezoelectric six-axis force sensor used in this study employs quartz crystal sheets as the piezoelectric material. Due to the low piezoelectric coefficient of a single crystal, force-measuring crystal groups are assembled by combining two crystal sheets of the same cut type to measure a single-axis force, ensuring high measurement sensitivity for the sensor. The structure of a single-axis force-measuring crystal group is illustrated in the figure. Three different directional force-measuring crystal groups are combined using conductive adhesive to form the unit crystal group of the six-axis force sensor.
The measurement principle of the piezoelectric six-axis force sensor is based on the piezoelectric effect of quartz crystals. When a quartz crystal sheet is subjected to an external force and deforms, internal polarization occurs, generating equal and opposite charges on the relative surfaces of the crystal sheet. The charge amount is proportional to the external force, an effect generally referred to as the direct piezoelectric effect. The physical diagram of the four-point supported piezoelectric six-axis force sensor is shown in the figure.
By simultaneously measuring the co-directional forces in the four symmetrically arranged unit crystal groups and performing algebraic summation, the three orthogonal forces \( F_x \), \( F_y \), \( F_z \) are obtained. The three orthogonal moments \( M_x \), \( M_y \), \( M_z \) are derived from the directional component forces and the positional relationships of the force-measuring crystal groups. The charge signals generated by the crystal groups are converted into voltage signals through charge amplifiers and then into digital signals via signal conditioners.
According to the spatial force system balance principle, let \( F_{ij} \) be the measured force of a single unit crystal group, where \( i \) denotes the force measurement direction of the single-axis crystal group, \( i = x, y, z \); and \( j \) denotes the unit force-measuring crystal group number, \( j = 1 \sim 4 \). Define \( a \) as the distance between the spatial external force and the sensor center. The three-dimensional forces \( F_x \), \( F_y \), \( F_z \) and three-dimensional moments \( M_x \), \( M_y \), \( M_z \) can be expressed as:
$$ F_x = F_{x1} + F_{x2} + F_{x3} + F_{x4} $$
$$ F_y = F_{y1} + F_{y2} + F_{y3} + F_{y4} $$
$$ F_z = F_{z1} + F_{z2} + F_{z3} + F_{z4} $$
$$ M_x = a (F_{z2} – F_{z4}) $$
$$ M_y = a (F_{z3} – F_{z1}) $$
$$ M_z = a (F_{y1} – F_{y3} + F_{x4} – F_{x2}) $$
Sensor calibration is a method to determine the accuracy and precision of a sensor under test by using a standard sensor. The calibration experiment establishes an accurate relationship between the input and output of the sensor under test. Simultaneously, the input forces and output forces of the sensor can serve as the dataset for the decoupling experiment.
During the static calibration experiment, a self-developed six-axis force sensor static calibration platform was used as the load application tool. The sensor under test was fixed to an adapter plate via bolts, and the adapter plate was fixed to the loading platform via bolts. During the experiment, stable force/torque loads were applied to the sensor through the loading head, as shown in the figure, where \( H \) represents the height difference between the loading point position and the lower plane of the sensor. The hardware for the static calibration experiment consisted of the six-axis force sensor static calibration platform, a standard sensor (range 50 kN, sensitivity (2.0 ± 0.005) mV/V), the sensor prototype, a JY-61902B data acquisition card from JianYi Technology, a YE5850 charge amplifier from Jiangsu Lianneng, and a computer terminal. The standard sensor was connected to the loading head via threads and had a numerical display to obtain the applied load.
The stepwise loading method with equal intervals was employed for the sensor: the loading force \( F_z \) along the z-direction ranged from 0 to 20 kN with a step size of 4 kN; the loading force \( F_x \) along the x-direction ranged from 0 to 3 kN with a step size of 0.6 kN; the loading force \( F_y \) along the y-direction ranged from 0 to 3 kN with a step size of 0.6 kN; the applied torque \( M_z \) ranged from 0 to 150 N·m with a step size of 30 N·m; the applied bending moment \( M_x \) ranged from 0 to 480 N·m with a step size of 96 N·m; the applied bending moment \( M_y \) ranged from 0 to 150 N·m with a step size of 30 N·m.
Loads were applied sequentially along the six force directions up to the full scale, and the experiment was conducted three times. The repeatability error \( R_N \) was used to evaluate the deviation of the three experimental results, expressed as:
$$ R_N = \Delta r_{\text{max}} / F_{\text{max}} $$
where \( \Delta r_{\text{max}} \) is the maximum deviation of the three experimental results under the same load, and \( F_{\text{max}} \) is the full-scale load. The calculation from the three experimental results showed that the maximum repeatability error was 1.55%. According to the international standard ISO376, the calibration repeatability error for force sensors must be less than 2%, so this sensor meets the experimental requirements. To minimize experimental error, the average of the three experiments was taken as the final experimental result, and the data were recorded by computer. The calibration curves between the sensor input load and output load are shown in the figures.
Analysis of the calibration curves reveals significant inter-dimensional coupling in the six-axis force sensor. For instance, when force \( F_x \) is applied, a substantial interfering moment \( M_y \) is generated due to the height difference \( H \) between the loading position and the sensor’s lower plane coordinate system. Similarly, applying force \( F_y \) produces a significant interfering moment \( M_x \). When bending moment \( M_x \) is applied, a portion of the lateral load component results in a large interfering lateral force \( F_y \). Applying bending moment \( M_y \) generates a large interfering lateral force \( F_x \). These observations confirm the presence of inter-dimensional coupling, necessitating effective decoupling algorithms to improve measurement accuracy.
Nonlinear Decoupling Principle and Establishment of Coupling Indicators
In the study of sensor measurement indicators, evaluating measurement accuracy is crucial. Unlike the coupling error in single-axis force sensors, the coupling error in six-axis force/torque sensors includes two forms: structural coupling and error coupling. Structural coupling is primarily caused by calibration errors and manufacturing errors and cannot be reduced through algorithms. Error coupling is mainly caused by interference errors between different measurement directions when the sensor outputs signals. The measurement accuracy of error coupling in the sensor can be evaluated using Type I error and Type II error. Type I error represents the ratio of the difference between the force/moment applied on the sensor and the force/moment actually measured by the sensor in that direction to the full scale of that direction, also known as static nonlinear error. It reflects the deviation between the measured value and the actual applied value in the loading direction of the six-axis force sensor and the linearity of the six dimensions, i.e., the within-group error. If \( i \) denotes the force measurement direction, the Type I error of the sensor can be expressed as:
$$ e_{F,i} = (F_{K,i} – F_{M,i}) / F_{Q,i} $$
$$ e_{M,i} = (M_{K,i} – M_{M,i}) / M_{Q,i}, \quad i = x, y, z $$
where \( e_{F,i} \), \( e_{M,i} \) are the Type I errors for force and moment, respectively; \( F_{K,i} \), \( M_{K,i} \) are the measured force and moment in direction \( i \), respectively; \( F_{M,i} \), \( M_{M,i} \) are the actually applied force and moment in direction \( i \), respectively; \( F_{Q,i} \), \( M_{Q,i} \) are the full-scale values of force and moment in direction \( i \), respectively.
Type II error represents the ratio of the additional interfering force/moment generated in other directions when a force or moment is applied in one direction to the full scale of that direction. Type II error is also known as static cross-coupling error, reflecting the degree of interference between the six dimensions of the six-axis force sensor, i.e., the between-group error. If \( i \), \( j \) denote the force measurement directions, the Type II error produced by direction \( i \) on direction \( j \) can be expressed as:
$$ e_{F,ij} = F_{ij} / F_{Q,i} $$
$$ e_{M,ij} = M_{ij} / M_{Q,i}, \quad i = x, y, z, \quad j = x, y, z, \quad i \neq j $$
where \( e_{F,ij} \), \( e_{M,ij} \) are the Type II errors for force and moment, respectively; \( F_{ij} \), \( M_{ij} \) are the force and moment measured in direction \( j \) when no load is applied in direction \( j \) but a load is applied in direction \( i \), respectively.
The percentage coupling matrix is used to represent the Type I and Type II errors of the six-axis force sensor, expressed as:
$$ E_{\alpha\beta} = \begin{bmatrix}
E_{11} & E_{12} & E_{13} & E_{14} & E_{15} & E_{16} \\
E_{21} & E_{22} & E_{23} & E_{24} & E_{25} & E_{26} \\
E_{31} & E_{32} & E_{33} & E_{34} & E_{35} & E_{36} \\
E_{41} & E_{42} & E_{43} & E_{44} & E_{45} & E_{46} \\
E_{51} & E_{52} & E_{53} & E_{54} & E_{55} & E_{56} \\
E_{61} & E_{62} & E_{63} & E_{64} & E_{65} & E_{66}
\end{bmatrix} $$
where \( \alpha \), \( \beta \) are the row and column numbers of the matrix, \( \alpha = 1 \sim 6 \), \( \beta = 1 \sim 6 \). \( \alpha = 1 \sim 3 \) corresponds to applying force in the x, y, z directions, \( \alpha = 4 \sim 6 \) corresponds to applying moment in the x, y, z directions, \( \beta = 1 \sim 6 \) are the errors of the obtained force in the x, y, z directions and the errors of the obtained moment in the x, y, z directions, respectively. Thus, when \( \alpha = \beta \), the diagonal of the matrix represents the Type I error of the sensor; when \( \alpha \neq \beta \), the elements in the matrix represent the Type II error of the sensor, i.e., the coupling error.
Artificial Neural Network (ANN) is an intelligent learning model that simulates the working method of the human brain nervous system, reflecting the brain’s ability in parallel and distributed information processing, thereby achieving computer intelligence. ANN can reveal internal connections between samples by calculating and analyzing logical relationships and probability distributions between data samples.
Neural networks possess powerful nonlinear processing capabilities and can reflect complex mapping relationships through simple logical correspondences between samples. By providing the neural network with a certain amount of experimental data for training, a multi-dimensional coupling relationship based on the training data can be established, thereby achieving nonlinear decoupling. This paper adopts a decoupling algorithm based on the LSTM neural network and, combined with sensor calibration experiments, conducts nonlinear decoupling research on the piezoelectric six-axis force sensor. The flowchart of the sensor nonlinear decoupling algorithm principle is shown in the figure.
Decoupling Algorithm Based on LSTM Neural Network
The Recurrent Neural Network (RNN) can capture and remember contextual information, using information from the previous time step and the current time step to jointly calculate the output result at the current time step. This enables the model to generate more accurate and coherent outputs. The LSTM neural network generally consists of an input layer, a hidden layer, and an output layer. The implementation steps are: samples are input from the input layer, processed in the hidden layer, and output to the output layer; if the error between the decoupled data and the actual applied force is too large, forward propagation is performed to enter the next cycle until the error between the decoupled data and the actual applied data is less than the preset value before stopping. The structure diagram of the RNN cyclic neural network is shown in the figure, where \( x \) is the input layer, \( h \) is the hidden layer, \( o \) is the output layer, \( W \), \( U \), \( V \) are the weight matrices of the input layer, hidden layer, and output layer, respectively, and \( t \) is time.
In the RNN cyclic neural network, at \( t = 1 \) second, the hidden layer at time 0 is initialized, and the weight matrices of the input layer, hidden layer, and output layer are initialized. Then the following calculations are performed:
$$ h_1 = f(U x_1 + W h_0 + b’_i) $$
$$ o_1 = g(V h_1 + b’_o) $$
where \( h_1 \), \( o_1 \) are the hidden layer and output layer at \( t = 1 \) second, respectively; \( f \) is the activation function of the hidden layer, usually the tanh function; \( g \) is the activation function of the output layer, usually the softmax function; \( W = [W_1, W_2, \ldots, W_n]^T \), \( U = [U_1, U_2, \ldots, U_n]^T \), \( V = [V_1, V_2, \ldots, V_n]^T \) are the weight matrices of the input layer, hidden layer, and output layer, respectively; \( n \) is the number of rows of the weight matrix, its size determined by the amount of sample data and the depth of the neural network; \( b’_i \), \( b’_o \) are the threshold matrices of the input layer and output layer neurons, respectively.
After time \( t \), the final prediction result of the model is as follows:
$$ y_t = g(V h_t + b’_o) $$
where \( y_t \) is the output layer at time \( t \), i.e., the final prediction result; \( h_t \) is the hidden layer at time \( t \).
In RNN neural network computation, the thresholds \( b’_i \), \( b’_o \) and weights \( U \), \( W \), \( V \) used at each time step are the same, which is an important prerequisite for cyclic computation in RNN neural networks.
The Long Short-Term Memory neural network is a special type of RNN network. Traditional RNN networks encounter gradient vanishing and gradient explosion problems when processing long-term dependencies, causing weights in deeper layers to fail to update and preventing continued learning, making it difficult for the neural network to remember and utilize information over long time spans.
The LSTM neural network addresses the issues of long-term dependencies, gradient vanishing, and gradient explosion in traditional RNNs by designing the structure within the RNN hidden layer, adding a memory cell state mechanism and a gating mechanism. It can effectively handle noise and other influencing factors in sequences. The figure shows the LSTM neural network structure model. Here, \( C \) is the cell state, \( f_t \) is the forget gate output vector, \( f_t \in [0,1] \), \( \sigma \) is the sigmoid function, \( i_t \) is the input gate output vector, \( i_t \in [0,1] \), \( \tilde{C}_t \) is the new cell state, \( o_t \) is the output gate output vector, and “pointwise” denotes the point multiplication operation.
The key to the LSTM neural network lies in the cell state mechanism, which regulates information transmission through the gating mechanism in the hidden layer. The gating mechanism includes a sigmoid function layer and a pointwise multiplication operation. The sigmoid function compresses the input sample data to the interval [0,1], used to determine the importance of information, where 1 means completely remember and 0 means completely forget.
The gating mechanism of the LSTM neural network consists of a forget gate, an input gate, and an output gate. Because the hidden layer contains four interactive layers and interacts in a unique way, each gate has independent weights and thresholds. When a signal is input into the LSTM network, the first step is to decide the information to be discarded by the system. This process is carried out by the forget gate, as shown below:
$$ f_t = \sigma(W_f \cdot [h_{t-1}, x_t] + b_f) $$
where \( W_f \) is the weight matrix of the forget gate; \( h_{t-1} \) is the output of the previous time step; \( x_t \) is the input of the current time step; \( b_f \) is the threshold matrix of the forget gate.
The input gate determines which part of the new information will be stored in the cell. It consists of two parts: the first part is the sigmoid layer, which determines the value to be updated, with the same effect as the forget gate; the second part is the tanh layer, which is the unit update layer, used to create a new cell state \( \tilde{C}_t \). The two parts can be expressed as:
$$ i_t = \sigma(W_i \cdot [h_{t-1}, x_t] + b_i) $$
$$ \tilde{C}_t = \tanh(W_C \cdot [h_{t-1}, x_t] + b_C) $$
where \( W_i \) is the weight matrix of the input gate; \( b_i \) is the threshold matrix of the input gate; \( W_C \) is the weight matrix of the update unit; \( b_C \) is the threshold matrix of the update unit.
After completing the calculations of the forget gate and the input gate, the cell state needs to be updated. At this point, the output information of the forget gate and the input gate is added to the cell state, i.e., updating \( C_{t-1} \) to \( C_t \), as shown below:
$$ C_t = f_t * C_{t-1} + i_t * \tilde{C}_t $$
The output gate is responsible for determining the output at the current moment. Its composition and principle are the same as those of the input gate. The final output value is \( o_t \), expressed as:
$$ o_t = \sigma(W_o \cdot [h_{t-1}, x_t] + b_o) $$
$$ h_t = o_t * \tanh(C_t) $$
where \( W_o \) is the weight matrix of the output gate; \( b_o \) is the threshold matrix of the output gate.
Using the LSTM neural network toolbox in MATLAB software, the basic model of the LSTM neural network structure can be conveniently constructed. This paper employs a 5-layer LSTM neural network to build the sensor nonlinear decoupling algorithm model.
The sample dataset required for the decoupling experiment is obtained from the calibration experiment. The experiment yielded 5 sets of calibration data, each set consisting of a 6×6 matrix, including 36 data points (6 in each loading direction). The sample data are input into the decoupling algorithm in groups. 80% of the calibration data is used to train the LSTM neural network decoupling algorithm, and 20% is used to test the accuracy of the decoupling algorithm. To enhance the training speed of the neural network and improve the convergence of the model, the Mapminmax function in MATLAB is used to normalize the required experimental data, adjusting the sample numbers to the range [-1, 1]. This step not only improves computational efficiency but also helps enhance the overall accuracy and generalization ability of the model, thereby ensuring the reliability of the decoupling algorithm. The sample normalization formula is as follows:
$$ y = \frac{(y_{\text{max}} – y_{\text{min}})(x – x_{\text{min}})}{x_{\text{max}} – x_{\text{min}}} + y_{\text{min}} $$
where \( y \) is the normalized data; \( y_{\text{max}} \), \( y_{\text{min}} \) are the maximum and minimum values of the target interval, respectively; \( x \) is the data to be normalized; \( x_{\text{max}} \), \( x_{\text{min}} \) are the maximum and minimum values of the input data, respectively.
Since the LSTM neural network is suitable for processing time series data, the calibration data needs to be converted in format. When using this network, the Num2cell function in MATLAB can be used to convert the six-dimensional calibration data into a cell array to meet the input requirements of the LSTM network. The six force values of the converted cell array calibration data are used as the input layer data of the neural network, and the six force values of the calibration loading are used as the output layer data of the neural network. In the gating mechanism of the hidden layer, the sigmoid function is used as the activation function. When generating update information, the hyperbolic tangent function tanh is used as the activation function. Then, a Relu function layer is added to nonlinearly activate the output data of the fully connected layer. Finally, the data are output through the fully connected layer and the regression layer. The flowchart of the nonlinear decoupling algorithm based on the LSTM neural network is shown in the figure.
When building the decoupling algorithm model, due to the uncertainty of the model, the optimal number of neurons for training cannot be accurately obtained. Therefore, the initial number of hidden layer neurons is preset to 10 based on experience, and the network is trained separately after every increase of 10 neurons. The optimal solution for the number of hidden layer neurons is determined by the results of multiple trainings. The maximum number of iterations is set to 3000, the gradient threshold is set to 1, and the initial learning rate is set to 0.05. After training 2000 times, the learning rate is reduced, and the learning rate drop factor is set to 0.01. In each iteration, 256 samples are randomly selected for training, and other parameters are set to default values. The root mean square error \( E_{\text{RMSE}} \) is used to evaluate the impact of the number of hidden layer neurons on the decoupling accuracy. The calculation formula is as follows:
$$ E_{\text{RMSE}} = \sqrt{\frac{1}{n} \sum_{i=1}^{n} (U_i – U’_i)^2} $$
where \( n \) is the number of samples; \( U_i \), \( U’_i \) are the true value and predicted value of the i-th sample, respectively.
After multiple trainings of the neural network and changing the number of hidden layer neurons, the root mean square error of the decoupling algorithm is obtained, as shown in the figure. It can be seen from the figure that changes in the number of neurons have a significant impact on the decoupling algorithm. When the number of neurons is 50, the root mean square error of the decoupling algorithm is minimized at 7.7.
Using the above settings, the training of the LSTM neural network decoupling algorithm is completed. The number of elements in the weight matrices of the input layer, hidden layer, and output layer are 3600, 10000, and 600, respectively. The number of elements in the threshold matrices of the input layer and output layer are 200 and 12, respectively. The training error performance curve of the LSTM neural network is shown in the figure.
Analysis of Decoupling Effect and Precision
The LSTM neural network decoupling algorithm is used to perform multi-dimensional nonlinear optimization on the sensor, and the decoupled results are shown in the figures. From the figures, it can be seen that after decoupling the output forces of the six-axis force sensor using the LSTM neural network decoupling algorithm, when loads are applied in each direction separately, the output force/moment curves of the sensor show good force measurement standards and linearity. In addition, when a load is applied in a single direction, the output forces in other directions are relatively small, indicating that the inter-dimensional coupling effect of the sensor has been effectively suppressed. The experiment shows that the LSTM neural network decoupling algorithm significantly improves the decoupling performance of the six-axis force sensor, making the load measurement in each direction more accurate. This algorithm not only enhances measurement accuracy and reliability but also further verifies the powerful capability of the LSTM neural network in handling nonlinear coupling problems.
To evaluate the error coupling value and measurement accuracy of the LSTM neural network decoupling algorithm, first, 20% of the calibration samples \( U’ \) that did not participate in training are selected for the decoupling algorithm accuracy test. Then, the sample \( U’ \) is used as the input layer data, and the output force matrix \( F’_{\text{LSTM}} \) is obtained after decoupling by the LSTM neural network. Finally, based on the input force \( F’ \) of the neural network and the decoupled output force \( F’_{\text{LSTM}} \), the percentage coupling matrix \( E_{\text{LSTM}} \) of the LSTM neural network decoupling algorithm is calculated using equations (2) and (3). The specific expressions of the test sample data \( U’ \), the output force matrix \( F’_{\text{LSTM}} \), and the percentage coupling matrix \( E_{\text{LSTM}} \) are as follows:
$$ U’ = \begin{bmatrix}
6825.30 & 8.10 & -22.00 & 303.33 & 4150.33 & -443.33 \\
-222.90 & 5936.40 & 67.00 & -2923.00 & -403.67 & 747.33 \\
-704.10 & -715.20 & 7790.00 & 410.00 & -819.33 & -29.00 \\
54.60 & -616.50 & 12.00 & 654.31 & 54.20 & -0.78 \\
599.40 & -24.90 & -8.00 & 42.00 & 642.00 & -0.02 \\
-12.90 & -13.50 & -14.00 & -5.10 & -51.84 & -453.21
\end{bmatrix} $$
$$ F’_{\text{LSTM}} = \begin{bmatrix}
2991.6418 & 0.0687 & 0.8195 & 2.6373 & 3.8999 & -0.1093 \\
1.3868 & 2987.4141 & 5.2376 & 0.8660 & 0.9754 & 5.7765 \\
-19.9974 & 4.2284 & 19889.8420 & 8.8203 & 12.5945 & 19.8698 \\
-0.2578 & 0.2530 & 0.3180 & 478.2531 & 0.3946 & 0.2547 \\
-0.1701 & 0.6145 & 1.0888 & -0.0505 & 478.2741 & 0.8822 \\
-0.1677 & 0.0449 & 0.1298 & 0.0670 & 0.0946 & 149.4749
\end{bmatrix} $$
$$ E_{\text{LSTM}} = \begin{bmatrix}
0.2786 & 0.0023 & 0.0273 & 0.0879 & 0.1300 & 0.0036 \\
0.0462 & 0.4195 & 0.0289 & 0.0300 & 0.0325 & 0.1925 \\
0.1000 & 0.0211 & 0.5508 & 0.0441 & 0.0630 & 0.0993 \\
0.0537 & 0.0527 & 0.0662 & 0.3639 & 0.0221 & 0.0221 \\
0.0354 & 0.1280 & 0.2268 & 0.0105 & 0.3596 & 0.1838 \\
0.1118 & 0.0300 & 0.0865 & 0.0447 & 0.0630 & 0.3500
\end{bmatrix} $$
From the equation, it can be seen that after decoupling by the LSTM neural network, the maximum static nonlinear error of the four-point support sensor is 0.55%, occurring when \( F_z \) is loaded; the maximum static cross-coupling error is 0.23%, occurring when \( M_y \) is loaded, at which time the interference of \( F_x \) is the greatest.
To verify the decoupling performance of the Long Short-Term Memory neural network decoupling algorithm, based on the calibration experiment data, the RBF nonlinear decoupling algorithm and the Least Squares (LS) linear decoupling algorithm are selected for comparative analysis. Table 1 shows the comparison of Type I and Type II errors in the form of a percentage coupling matrix for the six-axis force sensor obtained using the three decoupling algorithms. The comparison of the maximum Type I and Type II errors of the three decoupling algorithms is shown in the figure.
| Algorithm | Error Matrix (%) |
|---|---|
| LSTM |
$$ \begin{bmatrix} 0.279 & 0.002 & 0.027 & 0.088 & 0.130 & 0.004 \\ 0.046 & 0.420 & 0.029 & 0.029 & 0.033 & 0.193 \\ 0.100 & 0.021 & 0.551 & 0.044 & 0.063 & 0.099 \\ 0.054 & 0.053 & 0.066 & 0.364 & 0.022 & 0.022 \\ 0.035 & 0.128 & 0.227 & 0.011 & 0.360 & 0.184 \\ 0.112 & 0.030 & 0.087 & 0.045 & 0.063 & 0.350 \end{bmatrix} $$ |
| RBF |
$$ \begin{bmatrix} 0.933 & 0.113 & 0.019 & 0.201 & 1.533 & 0.029 \\ 0.453 & 0.103 & 0.147 & 0.166 & 0.685 & 0.601 \\ 0.411 & 0.010 & 0.109 & 0.011 & 0.225 & 0.556 \\ 0.010 & 0.067 & 0.181 & 0.482 & 0.046 & 1.163 \\ 1.001 & 0.044 & 0.007 & 0.012 & 1.142 & 0.040 \\ 0.054 & 0.426 & 0.227 & 0.206 & 0.160 & 0.088 \end{bmatrix} $$ |
| LS |
$$ \begin{bmatrix} 1.671 & 2.651 & 1.356 & 1.256 & 0.688 & 0.189 \\ 0.556 & 1.865 & 0.287 & 2.220 & 0.627 & 0.184 \\ 0.370 & 0.004 & 2.995 & 0.052 & 0.947 & 0.056 \\ 2.218 & 0.120 & 0.255 & 2.655 & 1.006 & 0.362 \\ 0.514 & 1.346 & 1.115 & 0.565 & 0.652 & 0.085 \\ 3.255 & 2.222 & 0.377 & 0.478 & 1.246 & 4.263 \end{bmatrix} $$ |
From the figure, it can be seen that when using the LS linear decoupling method, the decoupling effect is poor, with the maximum Type I and Type II errors after decoupling being 4.26% and 3.25%, respectively. After using the RBF neural network for nonlinear decoupling, the maximum Type I and Type II errors are 1.29% and 1.53%, respectively. After using the decoupling algorithm based on the LSTM neural network to decouple the six-axis force sensor, the maximum Type I and Type II errors of the sensor are 0.55% and 0.23%, respectively. The results show that the decoupling algorithm based on the LSTM neural network for the six-axis force sensor achieves ideal results, meeting the requirement that both types of errors in sensor use are below 2%, and significantly improving the measurement accuracy and performance of the six-axis force sensor.
Conclusion
To effectively improve the measurement accuracy of piezoelectric six-axis force/torque sensors and reduce inter-dimensional coupling errors, this paper conducted research on sensor nonlinear decoupling algorithms based on the LSTM neural network. The main conclusions are as follows.
Based on the piezoelectric effect, the measurement principle of the four-point supported piezoelectric six-axis force sensor was analyzed, and the causes of inter-dimensional coupling were explained in conjunction with its calibration experiments, establishing sensor accuracy measurement indicators.
The principle of sensor nonlinear decoupling was analyzed, and a decoupling algorithm for the six-axis force sensor based on the LSTM neural network was proposed. An LSTM decoupling algorithm model was established, and the overall accuracy and generalization ability of the algorithm were improved using the Mapminmax and Num2cell functions in MATLAB.
Using the root mean square error as an indicator, the impact of different numbers of hidden layer neurons on the accuracy of the neural network was analyzed and compared. The results show that this indicator effectively demonstrates the overall decoupling effect of the algorithm. When the number of neurons is 50, the root mean square error is minimized at 7.7.
Based on the obtained calibration data, the LSTM, RBF, and LS algorithms were trained and validated separately, and the decoupling results of the three models were compared. The results show that after decoupling by the LSTM decoupling algorithm, the maximum Type I and Type II errors of the sensor are reduced to 0.55% and 0.28%, respectively, significantly reducing the coupling error.
The research results effectively reduce the inter-dimensional coupling problem in the nonlinear measurement system of the sensor, providing a new solution and theoretical basis for the development of high-precision force sensors and more accurate wind tunnel tests. The application of the LSTM neural network decoupling algorithm in six-axis force sensors demonstrates significant potential for enhancing measurement precision in demanding fields such as aerospace engineering.