A robot joint torque prediction method based on BP neural network and SG filtering

Abstract

The application provides a robot joint torque prediction method based on a BP neural network and an SG filter. It relates to the field of robot dynamics. The method comprises the following steps: performing experiments by using a single-degree-of-freedom robot experimental platform, collecting data in a starting stage and a constant-speed running stage; constructing a data sample library, pre-processing the collected data, and then dividing the data into a training set, a verification set and a test set; constructing an initial BP neural network model, setting structure parameters and training parameters; training the initial BP neural network model by using the training set, verifying the model performance by using the verification set, so as to obtain an effective model after training; inputting the test set data into the trained BP neural network model to obtain a prediction result; and further processing the output result by using an SG filter to improve the prediction accuracy.

Description

Technical Field

[0001] This invention relates to the field of robot dynamics, and in particular proposes a method for predicting robot joint torques based on BP neural networks and SG filtering. Background Technology

[0002] Robots possess excellent collaboration and flexibility, enabling them to assist or even replace human labor. However, when interacting with their environment, robots are susceptible to external forces, which can lead to task failure or even robot damage. To ensure that robots can safely, stably, and accurately complete interactive tasks, obtaining information about the robot's joint torques for effective force control is crucial.

[0003] The most direct way to obtain robot joint torque is to install force / torque sensors at the end effector or base, or at each joint. However, sensor installation and maintenance are difficult, which not only increases system complexity but also significantly raises costs. Furthermore, sensor integration requires consideration of communication, mechanisms, electrical technologies, and algorithm research, which is also a time-consuming process.

[0004] Methods based on dynamic model prediction can indirectly obtain robot joint torques. This method does not require additional sensors and is relatively low-cost. However, this method introduces joint dynamics, which brings difficulties in modeling joint friction and identifying nonlinear models, thus limiting its application scope.

[0005] With advancements in artificial intelligence algorithms, numerous machine learning and statistical algorithms have emerged to construct data-driven mathematical models, replacing traditional physical models. Among these, neural networks have a wide range of applications. They can learn the behavior of a system through data and possess excellent nonlinear mapping capabilities. Therefore, neural networks are well-suited for predicting joint torques in robots. Summary of the Invention

[0006] To address the aforementioned problems, this invention provides a robot joint torque prediction method based on BP neural network and SG filtering. The aim is to propose a solution that is simple to model, has good versatility, and excellent adaptability, in order to solve the problems of expensive sensors and difficult friction modeling in existing robot joint torque monitoring methods.

[0007] To achieve the above objectives, this invention provides a robot joint torque prediction method based on BP neural network and SG filtering, comprising:

[0008] Step S1: Conduct experiments using a single-degree-of-freedom robot experimental platform to collect data during the startup and constant-speed operation phases;

[0009] Step S2: Construct a data sample library, preprocess the collected data, and then divide the data into training set, validation set, and test set;

[0010] Step S3: Construct the initial BP neural network model and set the structural parameters and training parameters;

[0011] Step S4: Train the initial BP neural network model using the training set and verify the model performance using the validation set to obtain an effective trained model.

[0012] Step S5: Input the test set data into the trained BP neural network model to obtain the prediction results. To improve the prediction accuracy, the output results are further processed by SG filtering.

[0013] Preferably, the experiment is conducted using a single-degree-of-freedom robot experimental platform to collect data during the startup phase and the constant-speed operation phase. Specifically, corresponding experimental schemes are designed for the startup phase and the constant-speed operation phase, with each experimental scheme containing multiple sub-experiments and data acquired with a sampling period of 2ms.

[0014] Preferably, a data sample library is constructed, the collected data is preprocessed, and then the data is divided into training set, validation set, and test set. Specifically, a sample database is constructed for different experimental schemes, including joint angular velocity. angular acceleration The current I of the motor driver and the joint torque τ. In the preprocessing stage, a normalization method is used to process the data, transforming it into a uniform scale. Subsequently, 85% of the data is used as the training set to train the BP neural network model, while the remaining 15% is used as the validation set to evaluate the performance of the BP neural network model. Experimental data collected under different experimental conditions than the training and validation sets is used as the test set. The normalization formula is as follows:

[0015]

[0016] Among them, y * Here, y represents the normalized data, and y represents the original data. max y represents the maximum value in the original data. min This is the minimum value in the original data.

[0017] Preferably, an initial BP neural network model is constructed, and structural parameters and training parameters are set. Specifically, the initial BPNN model is constructed, and structural parameters and training parameters are set. The structural parameters include the number of input layer nodes N. in Number of hidden layers M and number of nodes N m Number of output layer nodes N out And the activation function f. Training parameters include the maximum number of iterations k.target Error threshold L target Gradient threshold g target Stop adding conditions ξ target The initial values ​​of the weight update coefficients μ0 and the threshold μ target Damping coefficient β and initial weight Ω 0 .

[0018] Where, k target The maximum number of iterations the algorithm can execute, to avoid infinite loops or excessively long training times.

[0019] L target The target error value for training. When the error of the neural network reaches or falls below this threshold, the algorithm will stop training.

[0020] g target When the norm of the gradient is less than g target If the loss function of the neural network has converged to a local minimum, then training should be stopped.

[0021] ξ target When the validation set error increases consecutively by ξ times target It can be assumed that the performance of the neural network has reached a bottleneck, and further training may lead to overfitting.

[0022] μ0, μ target The initial value of the weight update coefficient is set to μ0, and the value of μ is updated in each iteration. The larger the value of μ, the better the convergence effect of the algorithm. When the value of μ reaches or exceeds the preset threshold, it indicates that the algorithm has reached a satisfactory level in terms of performance, so training stops.

[0023] β: The coefficient used to update μ.

[0024] Preferably, the initial BP neural network model is trained using the training set, and the model performance is verified using the validation set to obtain a trained and effective model. The specific steps are as follows:

[0025] Step S41: Input the training set data into the neural network model for forward propagation. The neurons in the input layer are responsible for receiving the input vector from the normalized training set. The input vector includes joint angular velocities. angular acceleration and the current I of the motor driver * The input signal is then passed to the hidden layer, which is responsible for performing nonlinear transformations on the input signal. This transformation is then passed to the output layer, which is responsible for generating the network's predictions.

[0026] Step S42: Backpropagation of the error is performed using the Levenberg-Marquardt algorithm to adjust the weights.

[0027] Step S43: Determine whether the training of the neural network meets the stopping condition. The neural network is trained through continuous forward and backward propagation to adjust the network parameters to minimize the loss function. In order to effectively manage computing resources and prevent overfitting, training must be terminated at an appropriate time.

[0028] Preferably, the test set data is input into the trained BP neural network model to obtain the prediction results. To improve the prediction accuracy, the output results are further processed by SG filtering. The specific steps are as follows:

[0029] Step S51, test the angular velocity of the concentrated joints. angular acceleration The current I of the motor driver is normalized as an input vector and then input into a trained BP neural network to obtain the output result.

[0030] Step S52: Perform inverse normalization on the output results to restore the output results to the scale of the original observations, and obtain the predicted torque τ. p .

[0031] Step S53: The prediction results are smoothed using SG filtering to obtain the final prediction result.

[0032] Compared with the prior art, the beneficial effects of the present invention are:

[0033] (1) Normalizing the sample data improves the stability and convergence speed of model training.

[0034] (2) BP neural network modeling is simple and practical, avoiding the complexity of modeling physical models.

[0035] (3) The generalization ability of the model was verified by using test set data that was not used in training.

[0036] (4) The Savitzky Golay filtering algorithm based on the least squares method increases the model’s resistance to noise interference while preserving the trend information of the data, thereby ultimately improving the prediction accuracy.

[0037] Other advantages, objectives, and features of the present invention will be set forth in part in the description which follows. Attached Figure Description

[0038] To facilitate understanding by those skilled in the art, the present invention will be further described below with reference to the accompanying drawings.

[0039] Figure 1This is a flowchart of a robot joint torque prediction method based on BP neural network and SG filtering according to the present invention;

[0040] Figure 2 Diagram of the experimental setup;

[0041] Figure 3 This is a diagram of a BP neural network structure;

[0042] Figure 4 Neuron structure diagram;

[0043] Figure 5 Here is a flowchart of the BP neural network training process;

[0044] Figure 6 This is a schematic diagram illustrating the performance of the BP neural network and SG algorithm on the test set during the constant-rate phase.

[0045] Figure 7 This is a schematic diagram illustrating the performance of the BP neural network and SG algorithm on the test set during the constant-rate phase.

[0046] Figure 8 The error in the test set performance of the BP neural network and SG algorithm during the constant-rate phase. Figure 1 ;

[0047] Figure 9 The error in the test set performance of the BP neural network and SG algorithm during the constant-rate phase. Figure 2 ;

[0048] Figure 10 This is a graph showing the performance of the BP neural network and SG algorithm on the test set during the startup phase;

[0049] Figure 11 This is an error graph showing the performance of the BP neural network and SG algorithm on the test set during the startup phase. Detailed Implementation

[0050] The technical solution of the present invention will be clearly and completely described below with reference to the embodiments. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0051] like Figures 1-11 As shown, the present invention provides a method for predicting robot joint torques based on BP neural network and SG filtering, the implementation steps of which include:

[0052] Step S1: Conduct experiments using a single-degree-of-freedom robot experimental platform to collect data during the startup and constant-speed operation phases;

[0053] Step S2: Construct a data sample library, preprocess the collected data, and then divide the data into training set, validation set, and test set;

[0054] Step S3: Construct the initial BP neural network model and set the structural parameters and training parameters;

[0055] Step S4: Train the initial BP neural network model using the training set and verify the model performance using the validation set to obtain an effective trained model.

[0056] Step S5: Input the test set data into the trained BP neural network model to obtain the prediction results. To improve the prediction accuracy, the output results are further processed by SG filtering.

[0057] In this example, the experiment is conducted using a single-degree-of-freedom robot experimental platform to collect data during the start-up and constant-speed operation phases. The specific details are as follows: The platform is constructed as shown in the attached diagram. Figure 2 The experimental platform shown mainly includes: a power supply, an EL3064 microcontroller, a Beckhoff controller, a joint module, a torque sensor, and a magnetic particle brake. The joint module consists of a motor, a harmonic reducer, and hardware for monitoring state variables. Furthermore, the magnetic particle brake provides a stable load to simulate actual working conditions, and the torque sensor records the load value in real time during the experiment. When considering joint motion, the motion is divided into two main phases: the initiation phase and the constant-speed phase. The constant-speed phase is when the module performs its main task. In this phase, the module's acceleration fluctuates around 0 and lasts for a long time. In contrast, in the initiation phase, the module needs to overcome the inertia of the motor and the additional load, and the acceleration has a relatively significant value. Therefore, to simplify the training process and enhance the model's flexibility, two independent BPNNs were used, corresponding to the initiation and constant-speed phases respectively, and corresponding experimental schemes were designed, as follows:

[0058] The constant-speed phase experiment involved eight sets of experiments designed based on different steady-state speeds. The joint was moved at a constant rotational speed of 30° / s, 50° / s, 70° / s, 90° / s, 110° / s, 130° / s, 150° / s, and 170° / s, respectively. Simultaneously, the load was adjusted in each experiment, gradually increasing from no load to maximum load and then gradually decreasing back to no load. Throughout the experiment, experimental data was acquired with a sampling period of 2ms as the training and validation sets. The constant-speed phase experiment was then conducted again at a steady-state speed of 80° / s, and 6 seconds of experimental data were collected as the test set.

[0059] The specific content of the start-up phase experiment is as follows: Six sets of experiments were designed based on different start-up loads to record the acceleration phase of the joint carrying different loads from rest. Before the joint starts, a fixed load is applied to the joint by setting the input voltage of the magnetic powder brake to 1, 2, 3, 4, 5, and 6V respectively. In each set of experiments, as the joint starts, it gradually accelerates, eventually reaching speeds of 30° / s, 50° / s, 70° / s, 90° / s, 110° / s, 130° / s, 150° / s, and 170° / s respectively. Throughout the experiment, experimental data was acquired with a sampling period of 2ms as the training and validation sets. The start-up phase experiment was then conducted again under the conditions of a target speed of 80° / s after start-up and an input voltage of 3.5V for the magnetic powder brake, and the experimental data was collected as the test set.

[0060] In this example, a data sample library is constructed, the collected data is preprocessed, and then the data is divided into training set, validation set, and test set. Specifically, data from the start-up phase and the constant velocity phase are collected, and sample databases for each phase are constructed, including joint angular velocities. angular acceleration The current I of the motor driver and the joint torque τ are then used for data preprocessing. A normalization method is employed to transform the data into a uniform scale, as shown in the following normalization formula:

[0061]

[0062] Among them, y * Here, y represents the normalized data, and y represents the original data. max y represents the maximum value in the original data. min This is the minimum value in the original data.

[0063] Subsequently, 85% of the data was used as the training set to train the parameters of the BP neural network model. The model adjusted its weights by learning the relationship between input and output on the training set to minimize prediction error. The remaining 15% of the data was used as the validation set to test the performance of the BP neural network model after training and to prevent overfitting. To verify the generalization ability of the model after training, experimental data collected under different conditions than the training and validation sets was used as the test set. The model's performance on the test set will help confirm whether the model can accurately predict the dynamic behavior of joint movements on unseen samples.

[0064] In this example, an initial BP neural network model is constructed, and its structural and training parameters are set. Specifically, for ease of computation, both initial network models use the same structural and training parameters. The BP neural network model structural parameters include the number of input layer nodes N. in Number of hidden layers M and number of nodes N mNumber of output layer nodes N out And the activation function f. (See attached image) Figure 3 As shown, the input layer is the first layer of the neural network, responsible for receiving the raw input data. Each node in the input layer corresponds to a feature of the input data, namely the joint angular velocity. angular acceleration and driver current I * Following this are several hidden layers arranged sequentially, located between the input and output layers. These hidden layers are responsible for learning the features of the data. The nodes in the hidden layers use weights and non-linear activation functions to perform non-linear transformations of the input signal. The output layer is the last layer of the neural network, responsible for generating the model's predictions. The design of the multi-layer structure aims to reflect the nonlinear mapping capability of the model, enabling it to capture the complex relationship between input features and output, thereby improving the accuracy of the neural network in torque prediction.

[0065] More complex hidden layers generally result in better network fitting, but overly complex layers can lead to overfitting and reduce the network's generalization ability. The design of hidden layers relies heavily on empirical knowledge and can be referenced using the following formula:

[0066]

[0067] Where N m N in and N ou t represents the number of neurons in the hidden layer, input layer, and output layer, respectively, and b is a constant between 0 and 10.

[0068] Training parameters include the maximum number of iterations k. target Error threshold L target Gradient threshold g target Stop adding conditions ξ target The initial values ​​of the weight update coefficients μ0 and the threshold μ target Damping coefficient β and initial weight Ω 0 .

[0069] Where, k target The maximum number of iterations the algorithm can execute, to avoid infinite loops or excessively long training times.

[0070] L target The target error value for training. When the error of the neural network reaches or falls below this threshold, the algorithm will stop training.

[0071] g target When the norm of the gradient is less than g target If the loss function of the neural network has converged to a local minimum, then training should be stopped.

[0072] ξ target When the validation set error increases consecutively by ξ times target It can be assumed that the performance of the neural network has reached a bottleneck, and further training may lead to overfitting.

[0073] μ0, μ target The initial value of the weight update coefficient is set to μ0, and the value of μ is updated in each iteration. The larger the value of μ, the better the convergence effect of the algorithm. When the value of μ reaches or exceeds the preset threshold, it indicates that the algorithm has reached a satisfactory level in terms of performance, so training stops.

[0074] β: The coefficient used to update μ.

[0075] In this example, an initial BP neural network model is trained using a training set, and its performance is validated using a validation set to obtain a trained and effective model. The training process of the BP neural network is as follows: Figure 5 As shown, it includes two processes: forward propagation and backward propagation. The specific steps are as follows:

[0076] Step S41, forward propagation is the process in a neural network that receives input data and generates torque prediction values. In this process, each set of data in the training and validation sets passes through the various layers of the network in sequence, including the input layer, hidden layers, and finally reaches the output layer. The entire forward propagation process effectively utilizes the weights and activation functions of each layer to map the input to the output, thus completing the torque prediction.

[0077] Figure 4 This demonstrates a single neuron model, whose function can be represented mathematically:

[0078]

[0079] in, It is the output of the j-th neuron in the m-th layer, f m It is the activation function of the m-th layer. These are the weights connecting the j-th neuron in layer m to the i-th neuron in layer (m-1). All weights in the neural network are shown below:

[0080]

[0081] The output layer's predicted torque value is:

[0082]

[0083] Where, τ p * This is the neural network's prediction of the torque.

[0084] To facilitate subsequent calculations, the mathematical operation Φ is defined to reshape the weight ω into a column vector Ω of length n.

[0085] Ω=Φ(ω)

[0086] Step S42, backpropagation is the process used in a neural network to adjust and update the network weights. Initial weights Ω 0 The values ​​are randomly generated. After one forward propagation, the BP neural network produces a predicted value for the torque; however, this predicted value has a large error compared to the observed value. To optimize the model and enable it to predict the output more accurately, the Levenberg-Marquardt algorithm is used to minimize the loss function, thereby adjusting and updating the weights in the network.

[0087] The error between the predicted and observed torque values ​​is shown below:

[0088]

[0089] Among them, e r (Ω) represents the error between the predicted and observed torque values ​​obtained by the BP neural network, τ * Let r be the torque observation value recorded by the sensor, and r be the r-th data set in the training set.

[0090] The loss function of a neural network is shown below:

[0091]

[0092] Where L(Ω) is the loss function of the neural network, and N tr Number of training set samples

[0093] The Levenberg-Marquardt algorithm updates the weights using the following formula:

[0094] Ω k+1 =Ω k -[J T (Ω k )J(Ω k )+μU] -1 g(Ω k )

[0095] Where k is the number of iterations, U is the identity matrix, and g(Ω) k J(Ω) represents the gradient. μ is the weight update factor; when the performance of the neural network improves, the update operation of μ is to multiply it by β. Conversely, when the performance of the neural network deteriorates, the update operation of μ is to divide it by β. k The specific mathematical form of the Jacobian matrix of the loss function is shown below:

[0096]

[0097] Step S43: Determine if the training of the neural network meets the stopping condition. The neural network is trained through continuous forward and backward propagation to adjust network parameters to minimize the loss function. To effectively manage computational resources and prevent overfitting, training is stopped at an appropriate time. After each training round, the network performance is evaluated. If any of the following conditions are met, the BP neural network model is considered to have reached its optimal state, and training can be stopped:

[0098]

[0099] Where k is the iteration number, L(Ω) k Let g|(Ω) be the loss function. k )| is the magnitude of the gradient, and ξ is the number of times the validation set error increases consecutively.

[0100] In this example, the test set data is input into the trained BP neural network model to obtain the prediction results. To improve the prediction accuracy, the output results are further processed using SG filtering. The specific steps are as follows:

[0101] Step S51: The test set data was not used to train the model, therefore it can be used to evaluate the model's generalization ability. The joint angular velocities in the test set... angular acceleration The current I of the motor driver is normalized as an input vector and then input into a trained BP neural network to obtain the output result.

[0102] Step S52: Perform inverse normalization on the output results to restore the output results to the scale of the original observations, and obtain the predicted torque τ. p .

[0103] Step S53: The prediction results are smoothed using SG filtering to obtain the final prediction result. The specific steps of SG filtering are as follows:

[0104] Step S531: Define a window that includes the data point to be filtered, and h sets of data before and after this data point.

[0105]

[0106] in, The sampling interval is denoted as .

[0107] Step S532: Fit the data in the above window using the least squares method.

[0108] Define a q-th order polynomial as the fitting curve, as shown in the following equation:

[0109]

[0110] Establish an error model:

[0111]

[0112] E is about γ = [γ0, γ1, ..., γ q ] T Since E is a quadratic nonnegative function, its minimum value is obtained where the partial derivative of E with respect to γ ​​is equal to 0, as shown in the following equation:

[0113]

[0114] In matrix form, it can be expressed as follows:

[0115]

[0116] The above formula can be simplified to A T Aγ=A T τ p Further simplification yields γ = (A T A) -1 A T τ p

[0117] Step S533, the result after filtering at the point to be filtered can be expressed mathematically as follows:

[0118]

[0119] By translating the window and repeating the above operations, the filtering process of the neural network prediction results can be completed.

[0120] The following is a specific embodiment. According to the problem that this invention aims to solve, the detailed parameter settings of the BP neural network and the SG filtering method are shown in Table 1:

[0121] Table 1 Algorithm Parameters

[0122]

[0123] The BP neural network model is trained using the training and validation sets according to the steps described above. Then, the test set is processed according to step S5 above to obtain the torque τ predicted by the neural network. p And the final predicted torque after SG filtering The two predicted values ​​are compared with the observed torque values, and the residuals are calculated. Figure 6 and Figure 7 The prediction results for increased and decreased joint torques on the constant speed test set are presented separately. Figure 8 and Figure 9 This presents the corresponding error analysis. Additionally, Figure 10 The prediction results on the test set during the startup phase are shown. Figure 11 This demonstrates the corresponding error analysis.

[0124] Based on the above comparison, Table 2 lists the evaluation comparison of the prediction results of the BP neural network and the filtered network on the test set.

[0125] Table 2 Error Analysis of Test Set Prediction Results

[0126]

[0127] Clearly, the performance of the BP neural network on the test set demonstrates that it can predict joint torques. Furthermore, the SG filtering significantly improves the prediction accuracy, yielding excellent results. Therefore, a method based on BPNN and SG filtering can be used to effectively predict joint torques during robot operation, providing a foundation for in-depth research on robot torque control.

[0128] The preferred embodiments of the present invention disclosed above are merely illustrative of the invention. These preferred embodiments do not exhaustively describe all details, nor do they limit the invention to any specific implementation. Clearly, many modifications and variations can be made based on the content of this specification. This specification selects and specifically describes these embodiments to better explain the principles and practical applications of the invention, thereby enabling those skilled in the art to better understand and utilize the invention. The invention is limited only by the claims and their full scope and equivalents.

Claims

1. A method for predicting robot joint torques based on BP neural network and SG filtering, characterized in that, The method includes: S1: Experiments were conducted using a single-degree-of-freedom robot experimental platform to collect data during the startup and constant-speed operation phases. Specifically, corresponding experimental schemes were designed for the startup and constant-speed operation phases, with each scheme containing multiple sub-experiments. Experimental data were acquired with a sampling period of 2ms. S2: Construct a data sample library. Preprocess the collected data, then divide it into training, validation, and test sets. The specific process for constructing the data sample library involves building separate sample databases using different experimental schemes, specifically including joint angular velocities. angular acceleration The current of the motor driver and joint torque In the preprocessing stage, the data is processed using a normalization formula to transform it into a uniform scale; then 85% of the data is used as the training set to train the BP neural network model, while the remaining 15% is used as the validation set to evaluate the performance of the BP neural network model; experimental data collected under different experimental conditions than the training and validation sets is used as the test set. S3: Construct the initial BP neural network model and set the structural parameters and training parameters; S4: Train the initial BP neural network model using the training set and validate the model performance using the validation set to obtain an effective trained model. S5: Input the test set data into the trained BP neural network model to obtain the prediction results. To improve the prediction accuracy, the output results are further processed by SG filtering, specifically including: S51: Test the angular velocity of the concentrated joints angular acceleration The current of the motor driver The input vector is normalized and then fed into a trained BP neural network to obtain the output result. ; S52: Perform inverse normalization on the output results to restore the output results to the scale of the original observations, and obtain the predicted torque. ; S53: The prediction results are smoothed using SG filtering to obtain the final prediction result. .

2. The robot joint torque prediction method based on BP neural network and SG filtering according to claim 1, characterized in that, The normalization formula is: ， in, For the normalized data, This is the original data. The maximum value in the original data. This is the minimum value in the original data.

3. The robot joint torque prediction method based on BP neural network and SG filtering according to claim 1, characterized in that, The structural parameters mentioned in step S3 include the number of input layer nodes. Number of hidden layers and number of nodes Number of output layer nodes and activation function The training parameters include the maximum number of iterations. Error threshold Gradient threshold Stop adding conditions Initial values ​​of weight update coefficients and threshold Damping coefficient and initial weights .

4. The robot joint torque prediction method based on BP neural network and SG filtering according to claim 1, characterized in that, S4 specifically includes: S41: Input the training data into the neural network model for forward propagation. The neurons in the input layer are responsible for receiving the input vector from the normalized training set. The input vector includes joint angular velocities. angular acceleration and the current of the motor driver The input signal is then passed to the hidden layer, which is responsible for performing nonlinear transformations of the input signal. The signal is then passed to the output layer, which is responsible for generating the network's prediction results. S42: Use the Levenberg-Marquardt algorithm to backpropagate the error and adjust the weights; S43: Determine whether the training of the neural network meets the stopping condition; the neural network is trained through continuous forward and backward propagation to adjust the network parameters to minimize the loss function; in order to effectively manage computing resources and prevent overfitting, training must be terminated at an appropriate time.

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