3-prs parallel robot driving force estimation method

Abstract

The application provides a 3-PRS parallel robot driving force estimation method, a 3-PRS parallel robot system is regarded as a black box model, and kinematics parameters including slider displacement, speed, acceleration, jerk and end posture are taken as inputs, and the slider driving force is taken as output. The estimation method comprises two steps: obtaining a data set and a label value, training a regression convolutional neural network model and estimating the driving force. The driving force estimation model comprises three regression convolutional neural network, each network corresponds to the driving force estimation of one slider, avoids lengthy dynamic calculation formula, and has high precision level; the estimation method regards the whole system of the 3-PRS parallel robot as a black box, and does not consider the specific structure of the robot, and can be suitable for the estimation of other control quantities of other types of robots, and has strong universality.

Description

TECHNICAL FIELD

[0001] The present application relates to the field of robot control, in particular to a 3-PRS parallel robot driving force estimation method. BACKGROUND

[0002] Parallel robot, also known as parallel machine tool, virtual axis machine tool, hereinafter collectively referred to as parallel robot, is one of the two branches of robots. The composition of parallel robot includes fixed platform, moving platform and two or more independent kinematic chains. The fixed platform and the moving platform are connected to each other through the kinematic chain to form at least one closed loop structure and have two or more degrees of freedom. Since the parallel robot is composed of multiple branch chains, it not only has no accumulated error, but also has the advantages of large carrying capacity, stable structure, high precision, rich functions, compact structure, etc. 3-PRS (P for moving pair, R for rotating pair, S for spherical pair) type parallel robot is a typical representative of low degree of freedom parallel mechanism, which has the advantages of simple structure, large stiffness, strong carrying capacity, small cumulative error, high motion precision and good flexibility, and is suitable for mechanical machining, medical devices, micro-nano operation and other occasions. The moving platform and the fixed platform of the 3-PRS parallel robot are connected by three identical and uniformly distributed branch chains, and the moving pair is the power input end, whose characteristics vary greatly with different arrangement modes. By changing the arrangement mode of P, R and S kinematic pairs, or using flexible hinges, or taking 3-PRS parallel mechanism as a basic platform, a large number of parallel or series-parallel hybrid robots with different characteristics can be derived, so it has broad application prospect.

[0003] The performance of parallel robot depends largely on control. In the dynamic control of 3-PRS parallel robot, the driving force of the three sliders is controlled to realize the dynamic adjustment of the pose of the robot end effector, and the real-time estimation of the driving force will become the core to improve the control performance of the robot. By establishing the kinematics and dynamics theoretical model of the robot, the model is calibrated through experiment to improve the accuracy of the model, so as to design the controller to improve the performance of the robot. The dynamics model of parallel robot involves coupling between parallel branch chains, derivative operation, etc., and the accurate solution in digital control system faces many technical problems, which becomes one of the key technical problems to improve the performance of the robot. In order to greatly improve the response speed and accuracy of 3-PRS parallel robot, an effective 3-PRS parallel robot driving force estimation method is needed to improve the dynamic performance of the robot and further improve the quality of the robot. SUMMARY

[0004] The present application is to solve the above technical problems, and provides a 3-PRS parallel robot driving force estimation method. The technical scheme of the present application is as follows:

[0005] The application discloses a 3-PRS parallel robot driving force estimation method, which regards a 3-PRS parallel robot system as a black box model, takes kinematic parameters including slider displacement, velocity, acceleration, jerk and end posture as input, and takes slider driving force as output.

[0006] Further, the estimation method comprises two steps: step A: obtaining a data set and a label value, and step B: training a regression convolutional neural network model and estimating the driving force.

[0007] Further, the method of step A is to set a driving function of the 3-PRS parallel robot, add the driving function to the end of the robot, take the moving platform as a driving part, generate motion according to the driving function, obtain kinematic parameters when the motion ends, convert the obtained parameters into a gray image, the number of the gray images corresponds to the label value one by one, take the slider as a driving part to obtain the driving force for moving the robot, and take the driving force as the label value of the converted gray image.

[0008] Further, the method of step B is to establish a regression convolutional neural network algorithm framework, establish a data set according to the gray image converted from the kinematic parameters and perform training, obtain a driving force estimation model through the training, input the gray image converted from the numerical value into the trained model for testing, and directly estimate the slider driving force through the model.

[0009] Further, in step B, there are three regression convolutional neural networks, and each network corresponds to the driving force of one slider.

[0010] Further, the displacement, velocity, acceleration and jerk of the three sliders are 12 data, and the end posture is 2 data, so that a total of 14 data are obtained; the driving force of the slider is 3 data.

[0011] Further, the slider displacement, velocity, acceleration and jerk data are standardized to the interval [0, 1]. The standardized data are converted into a gray image, and the conversion process is as follows:

[0012] Step S1: the 14 standardized data of the slider displacement, velocity, acceleration, jerk and end posture are combined into a row vector, and the row vector is represented as n 1×14 ;

[0013] Step S2: the row vector n 1×14 is expanded by 100 times, and the expanded vector becomes a matrix, represented as n 100×14 , according to the expanded matrix n 100×14 , each column value is further expanded by 7 times, and the matrix represented by the expanded value is represented as n 100×98 ;

[0014] Step S3: set the gray scale image pixel to 100*100, and add matrix 0 on the matrix of dimension 100*100 100×98 n 100×2 , and convert the matrix n 100×100 into a gray scale image with pixel of 100*100.

[0015] The present application has the following advantages:

[0016] (1) The driving force estimation method provided by the present application adopts a convolutional neural network framework for calculation, avoiding lengthy dynamic calculation formulas and having a high level of precision, thereby providing a brand-new method for driving force estimation of 3-PRS parallel robots.

[0017] (2) The driving force estimation method provided by the present application regards the entire system of a 3-PRS parallel robot as a black box, without considering the specific structure of the robot, and can be applied to the estimation of other control quantities of other types of robots, thereby having strong universality.

[0018] (3) The driving force estimation method provided by the present application takes the kinematic parameters of the robot as input and the driving force as output, and can construct a driving force estimation model through simulation or experiment, thereby facilitating engineering and technical application.

[0019] (4) The calculation precision of the driving force estimation method provided by the present application can be improved by increasing the number of training samples and changing the structure of the convolutional neural network. BRIEF DESCRIPTION OF DRAWINGS

[0020] In order to more clearly illustrate the technical solutions in the embodiments of the present application or the prior art, the following will briefly introduce the drawings needed to be used in the embodiment or prior art description. Obviously, the drawings in the following description only constitute some embodiments of the present application, and those skilled in the art can also obtain other drawings according to these drawings without creative labor.

[0021] Figure 1 It is a schematic diagram of a 3-PRS parallel mechanism;

[0022] Figure 2 It is a black box model of a 3-PRS parallel robot;

[0023] Figure 3 It is a step of driving force estimation based on a regression convolutional neural network;

[0024] Figure 4 It is a total framework of driving force estimation;

[0025] Figure 5 It is the displacement of the three sliders;

[0026] Figure 6 Velocity of three sliders;

[0027] Figure 7 Acceleration of three sliders;

[0028] Figure 8 Jerk of three sliders;

[0029] Figure 9 End pose;

[0030] Figure 10 Driving force of three sliders;

[0031] Figure 11 Normalized value of three sliders displacement;

[0032] Figure 12 Normalized value of three sliders velocity;

[0033] Figure 13 Normalized value of three sliders acceleration;

[0034] Figure 14 Normalized value of three sliders jerk;

[0035] Figure 15 Normalized value of end pose;

[0036] Figure 16 Value expansion and conversion process;

[0037] Figure 17 Randomly selected 25 grayscale images of value conversion;

[0038] Figure 18 Regression convolutional neural network configuration;

[0039] Figure 19 Overall comparison of actual and estimated values of slider 1 driving force;

[0040] Figure 20 Error of actual and estimated values of slider 1 driving force;

[0041] Figure 21 Overall comparison of actual and estimated values of slider 2 driving force;

[0042] Figure 22 Error of actual and estimated values of slider 2 driving force;

[0043] Figure 23 Overall comparison of actual and estimated values of slider 3 driving force;

[0044] Figure 24 Error of actual and estimated values of slider 3 driving force.

[0045] Figure:

[0046] 1 - fixed platform 2 - moving platform 3 - column 4 - slider

[0047] 5 - revolute pair 6 - connecting rod 7 - spherical pair 8 - tool. DETAILED DESCRIPTION

[0048] The technical solutions in the embodiments of the present application will be clearly and completely described below with reference to the drawings in the embodiments of the present application. Obviously, the described embodiments are only part of the embodiments of the present application, rather than all the embodiments. Based on the embodiments in the present application, all other embodiments obtained by those skilled in the art without creative labor fall within the scope of protection of the present application.

[0049] 3 - structural diagram of PRS parallel mechanism Figure 1 The system is composed of a fixed platform 1, a moving platform 2, three vertical columns 3, three sliders 4, three revolute pairs 5, three connecting rods 6, and three spherical pairs 7. Among them, the slider 4 slides on the vertical guide rail of the column 3, the slider 4 is connected with the connecting rod 6 through the revolute pair 5, the connecting rod 6 is connected with the moving platform 2 through the spherical pair 7, and the tool 8 is fixed at the geometric center of the moving platform 2.

[0050] Figure 1 In the figure, the fixed platform 1 is simplified as a triangle B1B2B3, the moving platform 2 is simplified as a triangle P1P2P3, and the three vertical columns are simplified as B1C1, B2C2, and B3C3. i C i (i = 1, 2, 3), the three connecting rods 6 are simplified as C1P1, C2P2, and C3P3. i P i (i = 1, 2, 3), and the three sliders 4 and the revolute pairs 5 are all simplified as C1, C2, and C3.

[0051] The equivalent action shapes of the fixed platform 1 and the moving platform 2 are both equilateral triangles, that is, B1B2 = B2B3 = B1B3, P1P2 = P2P3 = P1P3, and the tool 8 is fixed at the geometric center o T of the moving platform P1P2P3 and is perpendicular to the moving platform P1P2P3.

[0052] Figure 1 In the figure, l1, l2, l3, θ1, θ2, and θ3 are the lengths of the connecting rods C1P1, C2P2, and C3P3 and the angles between the columns B1C1, B2C2, and B3C3, respectively; the circumradius of the triangle B1B2B3 and the triangle P1P2P3 is R and r, respectively; the distance from the tool tip P to the moving platform P1P2P3 is o T P is h; is h 1 , h 2 , h 3The height of the three sliders 4 is also called the displacement of slider 4.

[0053] Establish a static coordinate system OXYZ and a moving coordinate system o on the fixed platform 1 and the moving platform 2 respectively. T xyz, where O is at the midpoint of B2B3, the X-axis coincides with OB1, and the Z-axis is vertically upward; the x-axis is perpendicular to O. T P1 coincides, and the z-axis is perpendicular to the moving platform 2 and moves upward.

[0054] The pose motion law of the moving platform 2 is determined by the slider displacement h of the three sliders 4. i The motion law of (i = 1, 2, 3) is determined. To overcome the technical problem of difficulty in estimating the driving force of the slider, this invention provides a method for estimating the driving force of a 3-PRS parallel robot, the process of which is described below.

[0055] Treating the 3-PRS parallel robot system as a black box model, such as Figure 2 As shown. The displacement h, velocity v, acceleration a, jerk a', and end pose (α, β, z) of slider 4 are included. t The kinematic parameters of the slider are used as input, and the slider driving force τ is used as output. The resulting driving force is applied to the robot to control the robot.

[0056] This invention combines a 3-PRS parallel robot with deep learning to achieve dynamic control, employing a regressive convolutional neural network. Figure 2 This part of the 3-PRS parallel robot system's functionality involves directly estimating the driving force of the sliders based on kinematic parameters. During control, this driving force is then reapplied to the three sliders to achieve parallel robot control. The specific steps for driving force estimation are as follows: Figure 3 As shown.

[0057] The estimation method provided by this invention includes two steps: Step A: Obtain the dataset and label values; Step B: Train the regression convolutional neural network model and estimate the driving force.

[0058] The method in step A is to set the driving function of the 3-PRS parallel robot, add the driving function to the end effector of the robot, such as the center of the moving platform; use the moving platform as the driving component to generate motion according to the driving function, and obtain the kinematic parameters after the motion is completed; convert the obtained parameters into grayscale images using an appropriate method, with the number of grayscale images corresponding one-to-one with the label values; use the slider as the driving component to obtain the driving force that makes the robot move, and use the driving force as the label value to train together with the converted grayscale images.

[0059] The method of step B is to establish a regression convolution neural network algorithm framework according to the characteristics of the 3-PRS parallel robot, establish a data set according to the gray image converted according to the kinematic parameters and train, and obtain a driving force estimation model after training; the gray image converted according to the numerical conversion is input into the trained model for testing, and the size of the slider driving force is directly estimated through the model. Although the jerk is not obviously expressed in the process of numerical solving of dynamics, the jerk represents the change amount of acceleration, which can not only increase the information amount contained in the converted image, but also improve the accuracy in the driving force estimation process, so the 3-PRS parallel robot driving force estimation method provided by the application also takes the jerk as an input item.

[0060] In step B, there are three regression convolution neural networks, each network corresponding to a driving force.

[0061] The total framework for estimating the driving force of the three sliders 4 in the 3-PRS parallel robot is as shown in Figure 4 According to the set end driving function, the kinematic parameters of the three sliders are obtained, and the obtained data is expanded to form a numerical matrix, and the matrix is converted into a gray image. The gray image is input into the corresponding regression convolution neural network after being calibrated by different driving force values, and finally the driving force values τ1, τ2 and τ3 of the three sliders 4 at different times can be estimated.

[0062] Through the regression convolution neural network for slider driving force estimation, the corresponding data set is first obtained. The number and accuracy of the data set have a decisive influence on the driving force estimation accuracy of the parallel robot. In this embodiment, the data set is obtained by simulation, and this method has the following characteristics:

[0063] (1) The simulation method for obtaining the data set has strong stability, and can collect data at any time or in any state without being affected by external factors.

[0064] (2) When simulating, it is relatively convenient to modify the parameters of each part, and the motion state of the end can be changed as long as the robot constraints allow.

[0065] The kinematic parameters can be obtained by setting the motion trajectory of the end of the 3-PRS parallel robot, and the data set for driving force estimation of the regression convolution neural network can be made by using these data. Since the three degrees of freedom of the robot include two rotation degrees of freedom α, β and one movement degree of freedom z t , the driving function of the end of the robot can be set as

[0066]

[0067] In the formula, t is time.

[0068] Taking a period of 2 seconds for the final motion, the kinematic parameters of the slider are as follows: Figures 5-9 As shown in the diagram, slider 1, slider 2, and slider 3 represent sliders on three vertical pillars.

[0069] In ADAMS, since directly defining the end effector function of the 3-PRS parallel robot cannot obtain the magnitude of the driving force on the slider, it is necessary to obtain the slider driving force through conversion. The motion trajectory of the three sliders 4 obtained by defining the robot end effector function is used as a spline curve, and this spline curve is added as the slider driving function. The end effector function defined according to equation (1) is canceled, and the slider driving function is used as the main driving function to make the robot perform simulated motion. After the motion is completed, the driving force on the slider can be directly measured. Taking the slider motion of 2s as a cycle, the driving force curves of the three sliders 4 are as follows. Figure 10 As shown.

[0070] To each Figure 10 Data is collected from the curves, and the number of images required for training is used as the basis for data extraction. Several points are extracted from the displacement, velocity, acceleration, jerk, and end-effector attitude curves, and the same number of points are extracted from the three driving force curves accordingly. These points are saved as data. At the same time interval, the 14 data points of displacement, velocity, acceleration, jerk, and end-effector attitude for the three sliders 4 correspond to the driving forces of the three sliders.

[0071] Since the conversion is from numerical values ​​to images and does not contain other information, the converted images are saved as grayscale images. Grayscale images reduce brightness features, which can speed up training. To ensure the accuracy of the convolutional neural network during model training, the pixel dimensions of the images in the dataset are uniformly defined as 100×100. This image can be converted using a 100×100 dimension numerical matrix. Of course, other pixel dimensions can also be used, such as 100×98.

[0072] This embodiment uses MATLAB as the tool for matrix-to-image conversion. To ensure proper display and accuracy during the conversion, the data needs to be converted to floating-point numbers beforehand. The `imshow` function is used for image conversion. This function displays floating-point data in the range of 0 to 1; values ​​greater than 1 are displayed incorrectly as white, and values ​​less than 0 are displayed incorrectly as black. Therefore, during the matrix-to-image conversion process, each data point is first converted to a standard format.

[0073] The data is converted into a standard format. Since displacement, velocity, acceleration, and jerk all include negative values, the 0 axis in the middle is moved to the bottom end of the entire curve, so that all the data becomes positive. In order to make all the data uniformly distributed on [0, 1] to facilitate image conversion, all the data is scaled down or scaled up, as follows:

[0074] (1) The 0 axis of the displacement in the middle is moved to the bottom end of the curve, and all the data is scaled down by 100 times, as shown in Figure 5 Figure 11

[0075] (2) The 0 axis of the velocity in the middle is moved to the bottom end of the curve, and all the data is scaled down by 1000 times, as shown in Figure 6 Figure 12

[0076] (3) The 0 axis of the acceleration in the middle is moved to the bottom end of the curve, and all the data is scaled down by 1000 times, as shown in Figure 7 Figure 13

[0077] (4) The 0 axis of the jerk in the middle is moved to the bottom end of the curve, and all the data is scaled down by 2500 times, as shown in Figure 8 Figure 14

[0078] (5) The 0 axis of the robot end pose parameter in the middle is moved to the bottom end of the curve, although the data of the curve is between 0 and 1, in order to make the curve more uniformly distributed between 0 and 1, the data is scaled up by 2 times, as shown in Figure 9 Figure 15

[0079] The standardized data is converted into a gray scale image, and the conversion process is as follows:

[0080] (1) The 14 standardized data of displacement, velocity, acceleration, jerk, and end pose are combined into a row vector, which is represented as n 1×14 .

[0081] (2) The row vector n 1×14 is expanded by 100 times, and the expanded vector becomes a matrix, represented as n 100×14 , that is, each row of n 100×14 is equal to the row vector n 1×14 . According to the expanded matrix n 100×14 , each column value is expanded by 7 times to become a matrix, represented as n 100×98 . The first to seventh columns of n 100×98 are equal to n​​​​​​​​​​100×14 1st column of n 100×98 8th to 14th column of n, each column equals n 100×14 2nd column of n 100×98 15th to 21st column of n, each column equals n 100×14 3rd column of n, and so on 100×98 92nd to 98th column of n, each column equals n 100×14 14th column of n.

[0082] (3) Set the pixel of gray image as 100x100, add matrix 0 100×98 to the matrix of n 100×2 dimension, add two columns of 0 to the rightmost of n 100×98 dimension; convert the matrix of n 100×100 dimension to the gray image of 100x100 pixel. The value expansion and conversion process is shown in Figure 16 .

[0083] The value of the multiple of the reduction in the above conversion process is only illustrative. In the application process, a suitable multiple needs to be selected to ensure that the 14 kinematic parameters are converted to the interval [0, 1] in the required workspace of the 3-PRS parallel robot.

[0084] The above algorithm is the process of converting numerical values to a gray image. First, assign the row vector including 14 conversion values to a, and expand the vector twice to expand it into a 100x98 matrix assigned to A. Assign the value 0 to a 100x2 matrix assigned to B. Merge A and B into a 100x100 matrix C, and finally convert the matrix to a gray image through scaling. Because the image converted by MATLAB will appear a white border, which will affect the pixel value of the finally generated image, thereby destroying the effect of the regression convolution neural network training, so the generated image cannot be saved directly. Add the imwrite(C, 'path\filename.jpg') instruction at the end of the program to get the ideal gray image, where C represents the saved gray image, "path" represents the location where the image is directly stored, and "filename.jpg" refers to the naming of the image and the format of the image is "jpg" which can also be other formats (jpeg, bmp, tif...).

[0085] Randomly select 25 gray images converted from numerical values, as Figure 17The displacement, velocity, acceleration, jerk and attitude parameters of the 3-PRS parallel robot are different in different pose states, so the depth of the converted gray image stripes is different. The smaller the value before conversion, the darker the stripe color, and the larger the value, the lighter the stripe color. The number of data sets is 300, of which the number of training sets is set to 250, and the number of validation sets is set to 50, and all the gray images are used as the test set.

[0086] The image training set is preprocessed. The regression convolutional neural network obtains the final convergence effect through multiple layers of superposition. During this period, the change of each layer parameter will cause the change of adjacent layer data, which is affected by the effect of multiple layers of accumulation, and the output of the final high layer will produce huge changes. In order to reduce the influence of output distribution change, a normalization strategy is adopted to map all data to an interval to reduce the oscillation of the model and speed up the convergence speed. The driving force value of the slider is normalized, that is, the value size is distributed between [0, 1]. The driving force value in the middle is reduced by 100 times, that is, all data meet the distribution requirement between [0, 1]. Figure 10 The data set picture is named "number_label.jpg" before image training, where label represents the label value after the driving force is reduced by 100 times.

[0087] The regression convolutional neural network is used to estimate the driving force of the 3-PRS parallel robot. The network includes convolutional layers, pooling layers and fully connected layers. According to the characteristics of the activation function, PReLU is selected as the activation function of the convolutional layer, Sigmoid is selected as the activation function of the output layer, Figure 18 The overall network configuration is shown in the figure, in which the resolution of the input image is 100x100, the number of convolutional layers is 3, the number of pooling layers is 3, and the number of fully connected layers is 2. The first layer of convolution kernel size is 5x5, and the number is 96; The second layer of convolution kernel size is 3x3, and the number is 96; The third layer of convolution kernel size is 3x3, and the number is 128. In the regression convolutional neural network, the pooling layer is periodically inserted, and the maximum pooling effect is used.

[0088] The driving force of the slider 4 is estimated, as shown in Figures 19-24 The experimental results are shown in the figure.

[0089] The regression convolutional neural network model estimation method is tested using all converted images, including the comparison of actual value and estimated value and error percentage. The comparison of actual value and estimated value is shown in Figures 19-24 The regression convolutional neural network has high precision level. The error percentage is shown in Figures 19-24The absolute values of the maximum errors of the three driving forces are 0.74%, 0.82% and 0.64% respectively, and thus it can be proved that the driving force estimation method of the slider 4 based on the regression convolutional neural network has good effects.

Claims

1. A 3-PRS parallel robot driving force estimation method, the 3-PRS parallel robot consisting of a fixed platform, a moving platform, three vertical columns, three sliders, three revolute pairs, three links, and three spherical pairs, wherein, The slider slides on the vertical guide rail of the column, the slider is connected with the connecting rod through a rotary pair, and the connecting rod is connected with the moving platform through a spherical pair, characterized in that a 3-PRS parallel robot is combined with deep learning to realize dynamic control, a regression convolutional neural network is adopted to complete the same, slider displacement, velocity, acceleration, jerk and end pose are taken as inputs, and slider driving force is taken as output.

2. The 3-PRS parallel robot driving force estimation method according to claim 1, wherein The estimation method comprises two steps: step A: obtaining slider displacement, velocity, acceleration, jerk and end pose to make a data set, and taking the driving force of the slider as a label; and step B: training a regression convolutional neural network model and estimating the driving force.

3. The 3-PRS parallel robot driving force estimation method according to claim 2, characterized by, The method of step A is to set a driving function of the 3-PRS parallel robot, add the driving function to the end of the robot, take the moving platform as a driving part, generate motion according to the driving function, obtain kinematic parameters when the motion ends, convert the obtained parameters into a gray image, and the number of gray images corresponds to the label value one by one; The slider is taken as a driving part to obtain the driving force for moving the robot, and the driving force is taken as the label value of the converted gray image; The method of step B is to establish a regression convolutional neural network algorithm framework, establish a data set according to the gray image converted from the kinematic parameters and train the same, obtain a driving force estimation model through training, input the gray image converted from the numerical value into the trained model for testing, and directly estimate the slider driving force through the model; In step B, there are three regression convolutional neural networks, each network corresponding to the driving force of one slider; The displacement, velocity, acceleration and jerk of the three sliders are 12 data, plus the end pose 2 data, a total of 14 data; the driving force of the slider is 3 data; The slider displacement, velocity, acceleration and jerk data are all standardized to the interval [0, 1]; The standardized data is converted into a gray image, and the conversion process is as follows: Step S1: The 14 standardized data of displacement, velocity, acceleration, jerk, and end pose are combined into a row vector, which is denoted as ; Step S2: Convert the row vectors Each number is multiplied by 100, and the resulting vector becomes a matrix, represented as follows: In other words, Each row is equal to a row vector According to the expanded matrix Further expansion, by increasing the number of values ​​in each column by a factor of 7, results in a matrix represented as follows: ; Columns 1 to 7, each column equals Column 1; Columns 8 to 14, each column equals Column 2; Columns 15 to 21, each column equals The third column; and so on for the following columns. Columns 92 to 98, each column equals Column 14; Step S3: Set the gray scale image pixel to , in , , , the rightmost of the matrix is added with two columns of 0; the matrix is converted into a gray scale image with pixels .​

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