RBF (Radial Basis Function)-based three-dimensional surface numerical fitting and series mechanical arm control method

Abstract

The invention discloses a three-dimensional surface numerical fitting and series mechanical arm control method based on an RBF (Radial Basis Function), which achieves the purpose of controlling a multi-joint series mechanical arm to realize high-precision polishing on a complex free-form surface through physical modeling and numerical simulation, accurate prediction of a machining surface, planning of a polishing path in a three-dimensional space and establishment of a D-H matrix.

Description

Technical Field

[0001] The invention relates to the field of measurement and control technology and instrument technology, and in particular to a RBF-based three-dimensional surface numerical fitting and serial robot arm control method. Background Art

[0002] Optical free-form surfaces have high surface freedom and strong phase difference balancing ability and are widely used. However, the difficulty of polishing increases with the gradual increase in processing requirements, and the difficulty of accurate fitting also increases continuously. Using radial basis functions represented by Gaussian functions for three-dimensional surface fitting can solve the related shortcomings of the currently commonly used optical free-form surface characterization models, such as insufficient accuracy, lack of locality, and unsuitability for fitting surfaces with local characterization changes.

[0003] In addition, CNC polishing is an automated polishing method that uses tools to remove material from the workpiece surface in a certain path under the control of a computer; it is widely used in the processing of components of various precision instruments, and has the advantages of high automation and high precision. In the processing of optical-grade ultra-smooth surfaces, high-precision polishing is required on complex free-form surfaces, but large errors will occur when processing some complex free-form surfaces. The existing polishing methods have polishing paths that can achieve geometric uniform coverage on the processed surface, but cannot achieve physical uniform coverage, resulting in uncontrollable overlap of adjacent polishing belts, and the inability to remove material evenly, which in turn leads to problems such as under-polishing or over-polishing.

[0004] The present invention is a CNC polishing method based on a serial robot arm as a platform. Through physical modeling and numerical simulation, the machining surface is accurately predicted, the polishing path is planned in three-dimensional space, and a DH matrix is ​​established to achieve the purpose of controlling a multi-joint serial robot arm to realize high-precision polishing on a complex free-form surface and solve the above-mentioned problems. Summary of the invention

[0005] In view of the above-mentioned defects or shortcomings in the prior art, it is expected to provide a RBF-based three-dimensional surface numerical fitting and serial robot arm control method, which accurately predicts the machining surface through physical modeling and numerical simulation, plans the polishing path in three-dimensional space, and establishes a DH matrix, so as to achieve the purpose of controlling a multi-joint serial robot arm to achieve high-precision polishing on complex free-form surfaces.

[0006] The present invention provides a 3D surface numerical fitting and serial robot arm control method based on RBF, comprising the following steps:

[0007] 1) Uniformly sample the planes to be polished and fitted to obtain a sampling data set;

[0008] 2) Using machine learning Gaussian radial basis function to perform a first fitting on the sampled data set and the surface function of the target optical free-form surface to be obtained, and obtain the first actual surface function represented by the matrix, that is, the first actual surface to be polished;

[0009] 3) Superimposing the first actual surface to be polished and the target optical free-form surface, and extracting the space between them;

[0010] 4) Convert the sandwiched space into a grayscale image;

[0011] 5) Processing the grayscale image to obtain a quantized square matrix;

[0012] 6) extracting high-frequency data from the quantized square matrix and performing reverse restoration on the data to obtain the coordinates of the projection points of the points requiring incremental sampling on the xy reference plane, that is, obtaining incremental sampling data;

[0013] 7) adding the incremental sampling data to the sampling data set in step 1) to obtain an incremental sampling data set;

[0014] 8) performing a second fitting on the incremental sampling data set and the surface function of the target optical free-form surface using a machine learning Gaussian radial basis function to obtain a second actual surface function formula represented by a matrix, that is, obtaining a second actual surface to be polished;

[0015] 9) Plan the polishing path of the serial robot arm.

[0016] Furthermore, it is characterized in that the sampling method of the sampling data set in step 1) is to place the plane required for polishing and fitting in a spatial rectangular coordinate system, and use the xy plane as a reference plane for uniform sampling to obtain the sampling data set.

[0017] Furthermore, it is characterized in that the superposition and overlap method in step 3) is selected according to the actual working conditions to minimize the total volume of the removed part, or to minimize the maximum polishing thickness of the removed part, that is, the shortest length in the normal direction of the target surface.

[0018] Furthermore, it is characterized in that the grayscale of the grayscale image at a certain point is obtained by normalizing the thickness at the same position in step 3).

[0019] Furthermore, it is characterized in that the grayscale image in step 5) is processed by performing a two-dimensional discrete cosine transform on the grayscale image to obtain the quantization matrix.

[0020] Furthermore, the specific planning method of step 9) is to use the standard DH inverse kinematics formula of the multi-axis robot arm to compensate for gravity, elastic deformation of the robot arm and repositioning accuracy to derive multiple sets of solutions, and use the particle swarm algorithm to iterate, and obtain an optimal polishing path in which the number of joints used at each moment is the minimum and the sum of the integrals of all joint rotation angles over time is the minimum, and output the joints of the robot arm, the spatial position of the polishing head and the normal vector direction of the polishing head corresponding to the path and the moment.

[0021] Compared with the prior art, the beneficial effects of the present invention are:

[0022] (1) The RBF-based three-dimensional surface numerical fitting and serial manipulator control method of the present invention utilizes machine learning Gaussian radial basis functions. The mapping from input to output of the network is nonlinear, while the network output is linear with respect to adjustable parameters. The weight of the network can be directly solved by a set of linear equations. It combines the advantages of nonlinear mapping and linear models, greatly speeds up the learning speed and avoids the local minimum problem.

[0023] In addition, for a fixed center point and variance, its value is only related to the distance from the sample point to the center point. The farther the distance, the smaller the function value. Therefore, when a sample point is input, only the neurons belonging to the basis functions near the center point will be activated, and the activation degree of the neurons belonging to the basis functions far away from the center point is very low, and their parameters do not need to be adjusted. When facing new sample inputs, this local approximation network only needs to update the parameters and weights of the basis functions near the center point, and the amount of calculation is small. Therefore, it maintains a high computational efficiency while maintaining good fitting accuracy, has strong locality, and is more suitable for characterizing surfaces with local changes.

[0024] (2) The two-dimensional discrete cosine transform has a certain robustness to noise, which reduces the impact of noise on high-frequency information to a certain extent, and provides a method for analyzing and interpreting signals or images in the frequency domain. By visualizing the coefficient matrix obtained by the two-dimensional discrete cosine transform, the method provides better scalability. At the same time, because the two-dimensional discrete cosine transform requires normalization, the probability of over-throwing is reduced.

[0025] (3) This method is easy to use and has good scalability. It can be used for all robotic arms with 3 joints or more. When sampling smooth optical free-form surfaces, it does not require a high number of data points and has a high tolerance rate.

[0026] (4) The RBF-based three-dimensional surface numerical fitting and serial robot arm control method of the present invention can plan the serial robot arm polishing path only by collecting the required calibration data through experiments, such as the rotation angle range of each joint of the robot arm, the length of each connecting rod, the repositioning accuracy, etc.;

[0027] For the purpose of wide application, the accuracy of path planning can be further increased by adding the gravitational acceleration of the work site, the Young's modulus of the robotic arm material, the joint damping coefficient or the friction coefficient, and compensating for the errors caused by the mechanical structure.

[0028] It should be understood that the contents described in the summary of the invention are not intended to limit the key or important features of the embodiments of the present invention, nor are they intended to limit the scope of the present invention. Other features of the present invention will become easily understood through the following description. BRIEF DESCRIPTION OF THE DRAWINGS

[0029] Other features, objects and advantages of the present invention will become more apparent from the detailed description of non-limiting embodiments made with reference to the following drawings:

[0030] Figure 1 A flow chart of a three-dimensional surface numerical fitting and serial manipulator control method based on radial basis functions provided by the present invention;

[0031] Figure 2 It is a representation diagram of the target free-form surface, the actual free-form surface and the part to be discarded;

[0032] Figure 3 is the grayscale image obtained after preprocessing and normalization of the discarded part;

[0033] Figure 4 The schematic diagram of the structure of the six-joint robotic arm used is shown in FIG.

[0034] Figure 5 Flowchart for planning the polishing trajectory of a multi-joint robotic arm using particle swarm optimization;

[0035] Figure 6 Schematic diagram of the particle swarm algorithm used to plan the polishing path for the robot arm;

[0036] Figure 7 This is a curve diagram of the corresponding relationship between the rotation angle of each joint of the robot arm provided by the present invention and time. DETAILED DESCRIPTION

[0037] The present invention will be further described in detail below in conjunction with the accompanying drawings and embodiments. It is to be understood that the specific embodiments described herein are only used to explain the relevant invention, rather than to limit the invention. It should also be noted that, for ease of description, only the parts related to the invention are shown in the accompanying drawings.

[0038] It should be noted that, in the absence of conflict, the embodiments of the present invention and the features in the embodiments can be combined with each other. The present invention will be described in detail below with reference to the accompanying drawings and in combination with the embodiments.

[0039] Please refer to Figures 1 to 7The embodiment of the present invention provides a three-dimensional surface numerical fitting and serial robot arm control method based on RBF, comprising the following steps:

[0040] 1) Uniformly sample the planes to be polished and fitted to obtain a sampling data set;

[0041] The sampling method of the sampling data set is to place the plane required for polishing and fitting in a spatial rectangular coordinate system, and perform uniform sampling with the xy plane as a reference plane to obtain the sampling data set.

[0042] 2) Using machine learning Gaussian radial basis function to perform a first fitting on the sampled data set and the surface function of the target optical free-form surface to be obtained, and obtain the first actual surface function represented by the matrix, that is, the first actual surface to be polished;

[0043] 3) Superimposing the first actual surface to be polished and the target optical free-form surface, and extracting the space between them;

[0044] The method of superposition and overlap is selected according to the actual working conditions to minimize the total volume of the removed part, or to minimize the maximum polishing thickness of the removed part, that is, to minimize the length in the normal direction of the target surface.

[0045] 4) Convert the sandwiched space into a grayscale image;

[0046] The grayscale of the grayscale image at a certain point is obtained by normalizing the thickness at the same position in step 3).

[0047] 5) Processing the grayscale image to obtain a quantized square matrix;

[0048] Specifically, the grayscale image is processed by performing a two-dimensional discrete cosine transform on the grayscale image to obtain the quantization matrix.

[0049] 6) extracting high-frequency data from the quantized square matrix and performing reverse restoration on the data to obtain the coordinates of the projection points of the points requiring incremental sampling on the xy reference plane, that is, obtaining incremental sampling data;

[0050] 7) adding the incremental sampling data to the sampling data set in step 1) to obtain an incremental sampling data set;

[0051] 8) performing a second fitting on the incremental sampling data set and the surface function of the target optical free-form surface using a machine learning Gaussian radial basis function to obtain a second actual surface function formula represented by a matrix, that is, obtaining a second actual surface to be polished;

[0052] 9) Plan the polishing path of the serial robot arm.

[0053] The specific planning method is to use the standard DH inverse kinematics formula of the multi-axis robot arm to compensate for gravity, elastic deformation of the robot arm and repositioning accuracy to derive multiple sets of solutions, and use the particle swarm algorithm to iterate to obtain an optimal polishing path with the minimum number of joints used at each moment and the minimum sum of the integrals of all joint rotation angles over time. The path and the corresponding moments of the robot arm joints, the spatial position of the polishing head and the normal direction of the polishing head are output.

[0054] The present invention is further described below with reference to the accompanying drawings and examples.

[0055] The radial basis function is a real-valued function whose value depends only on the distance to the origin. It can also be defined by the distance to a certain center point c; ||xc|| is the Euclidean distance between any point x and a certain center point c, and the relevant relationship is as follows:

[0056] φ(x,c)=φ(||xc||)

[0057] The present invention uses a Gaussian radial basis function; wherein σ is a hyperparameter of the Gaussian kernel function, which controls the width of the kernel function, and the related relationship is as follows:

[0058]

[0059] In this neural network, multiple input values ​​are processed in one or more neurons to calculate the output value. The "calculation" can be the linear weighted sum of several inputs. The input vector is first mapped from low dimension to high dimension using radial basis function, and then the sum of these radial basis functions is used to approximate a given function. In this approximation process, each hidden unit is regarded as a radial basis function, and then linear weighted approximation is performed.

[0060] From a geometric point of view, it is equivalent to restoring a continuous hypersurface based on sparse given sample data points; given n different data points on the surface in three-dimensional space, we can find the weighted sum of a series of radial basis functions of the surface equation that approximates the original surface; the value of the surface at a given point must satisfy the sample value, so as to achieve the purpose of fitting the free surface with radial basis functions.

[0061] The machine learning Gaussian radial basis function network usually has three layers: input layer, hidden layer and a nonlinear activation function and linear radial basis neural network output layer (here is a two-layer neural network); its basic idea is to use several machine learning Gaussian radial basis functions to form a hidden layer space, so that the input vector can be directly mapped to the hidden space without the need for weight connection; assuming that the hidden layer has N neurons, the output vector h of the hidden layer can be expressed as:

[0062] h=[h1,h2,……,h N ] T

[0063] where h i is the output of the i-th hidden neuron, and the calculation relationship is as follows:

[0064]

[0065] Among them, c i and σ i are the center point and variance of the ith radial basis function respectively; c i is the center of the function; as the distance between the two vectors increases, the function decreases monotonically, and has the characteristic of attenuating from the center to the periphery; σ controls the range of action of the Gaussian radial basis function. The larger its value, the larger the local influence range of the function, and it is easier to capture the global structure in the data.

[0066] The determination of the center point of the Gaussian radial basis function in machine learning also determines the mapping relationship; here, the Gaussian radial basis function neural network mapping is nonlinear, and the activation value of the hidden layer is calculated by the distance between the input vector and the center point of the hidden layer neuron. The mapping from the hidden layer space to the output space is linear, that is, the output of the network is the linear weighted sum of the hidden unit outputs, and the weight here is the network adjustable parameter, which is similar to other neural networks, that is, the function value y satisfies the following relationship:

[0067] y=W T h+b

[0068] Where W T is the weight matrix, h is the output vector, and b is the bias vector.

[0069] The hidden layer maps the vector from low dimension to high dimension, so that the data can be linearly separable, and the weights of the network can be directly solved by the linear equations. It combines the advantages of nonlinear mapping and linear models, greatly speeding up learning and avoiding local minima problems.

[0070] First simulate the network and find the input vector c with the largest error i , use this vector as the center point p of the first neuron in the hidden layer i , which means that the output of the network is now very sensitive to changes near this point; at the same time, the weight ω corresponding to the output layer i will also change to minimize the overall error. Repeat this process to optimize p i and ω i , until the mean square error MSE of the network satisfies the following relationship:

[0071]

[0072] (where o i The output value predicted by the network) is lower than the preset target value.

[0073] The number of basis functions, that is, the number of samples, is known;

[0074] The first layer weights are set to P', S is the spread of the radial basis function, and the first layer biases are all set to 0.8326 / S, resulting in a radial basis function span of 0.5 at the weighted input of + / -S; the second layer weights ω{2,1} and biases b{2} are obtained by simulating the first layer output A{1} and then solving the following linear expression:

[0075] [ω{2,1};b{2}]*[A{1};ones]=T

[0076] In the invention, according to the objective function, a Gaussian radial basis function neural network is used to perform data training to obtain an output; the difference between the objective function and the output of the RBF neural network is calculated, and its minimum value is subtracted from the difference to obtain the interlayer (the minimum value of the interlayer obtained in this way is 0);

[0077] Then draw the objective function, the neural network fitting results and the three interlayer results (such as Figure 2 ) for subsequent use;

[0078] Find the maximum and minimum values ​​in the fitted surface parameter matrix; then normalize the matrix and get some integers between 0 and 255, which correspond to the pixel value range of the 8-bit grayscale image. Next, use the batch normalization method to display the matrix in the form of an image;

[0079] The relevant formula for normalization using batch standardization is as follows:

[0080] Enter some coordinate data:

[0081] ξ={x 1...m};

[0082] Data representing the corresponding position of gray value: γ, β;

[0083] The result is output as the function value:

[0084] {y i =BN γ,β (x i )}

[0085] Calculate the mean μ ξ :

[0086]

[0087] Calculate the variance

[0088]

[0089] Normalize each element:

[0090]

[0091] Perform a scaling and shift operation; where β is the shift corresponding to the mean and γ is the scale parameter corresponding to the variance:

[0092]

[0093] In summary, the function of converting a two-dimensional matrix into a grayscale image and displaying it is realized; the grayscale image is as follows Figure 3 .

[0094] Two-dimensional discrete cosine transform is a commonly used signal processing technology. Its principle is based on block processing of signals, dividing the continuous signal into several small blocks and transforming each small block using the cosine function. In addition, digital images are all real number matrices, so the original signal can be represented in the frequency domain to obtain the coefficients in the frequency domain.

[0095] The specific formula is as follows, where N is the number of rows and columns, f(x,y) is the N*N image grayscale matrix, u and v are the grid numbers in the x and y directions respectively; the result F is also an N*N matrix:

[0096]

[0097]

[0098] These coefficients represent the energy distribution of the signal at different frequencies. Therefore, we can use fewer coefficients to represent the original signal, thereby achieving signal compression. The two-dimensional discrete cosine transform has a certain robustness to noise. As a transformation method, it can reduce the impact of noise on high-frequency information.

[0099] The value in the upper left corner of the quantization table is larger, and the value in the lower right corner is smaller. It is necessary to keep the high-frequency component and suppress the low-frequency component. The reason for taking the high-frequency part is that the part of the machine learning Gaussian radial basis function with large gradient changes will be affected by the insufficient sampling rate, and the fitting result of the RBF network will be affected, because the grayscale gradient in these areas is large, and more data points are needed to accurately represent it. In the two-dimensional discrete cosine transform, the high-frequency coefficient is the part with higher frequency in the two-dimensional discrete cosine transform result. These high-frequency coefficients usually correspond to the details and edge information in the image, that is, the part with large grayscale gradient. After the two-dimensional discrete cosine transform, the low-frequency information is concentrated in the upper left corner of the matrix, and the high-frequency information is concentrated in the lower right corner. Therefore, the details can be retained by discarding the low-frequency part (corresponding to the smooth area in the image) and retaining the high-frequency part. The extracted high-frequency part is resampled to increase the number of data points in these areas so that the RBF network can be more accurate when fitting.

[0100] The present invention is actually applicable to a tandem robot with more than 3 joints. You need to input the number of connecting rods and joints (in this example, 7 connecting rods and 6 joints, such as Figure 4 ), each link is connected by a precise joint to ensure the flexibility and stability of the overall structure; for the purpose of computational efficiency or model simplification, you can choose to ignore the collision properties, friction coefficient, gravity effect, Young's modulus, damping coefficient, etc., or set them to corresponding values ​​for compensation for the accuracy of the calculation.

[0101] Here, DH parameters are used to describe the geometric relationship between the joints and links of the robot arm to simplify the kinematic description. Each joint and link is assigned four parameters: each link is associated with the previous link through a coordinate transformation matrix, which is based on these four parameters; for a robot with n joints, a total transformation matrix can be obtained by continuously multiplying the transformation matrix of each link, which describes the position and orientation of the end effector relative to the base; inverse kinematics, that is, the corresponding joint variables are found when the end posture matrix of the robot in three-dimensional space is known, and its equation can be expressed as:

[0102]

[0103] The geometric structure of the application object of the present invention should satisfy the existence of an inverse kinematics closed solution;

[0104] The specific steps for solving the closed-form solution of inverse kinematics are as follows:

[0105] The first step is to solve for θ1.

[0106] m T nis the transformation matrix between the joints, describing the coordinate system change from the mth to the nth joint, which can help us deduce the coordinate system of the base from the reference coordinate system to the coordinate system of the end effector step by step, so as to obtain the position and posture of the end effector, which is also the core of kinematic calculation; because the coordinate transformation from the initial coordinate system to the sixth joint can be regarded as a recursion from the base to the first joint, the first joint to the second joint, and so on, so it is obvious that 0 T6= 0 T1 1 T2 2 T3 3 T4 4 T5 5 T6, and the following relationship can be obtained:

[0107] L= 0 T1 -10 T6= 1 T2 2 T3 3 T4 4 T5 5 T6

[0108] Wherein, the left and right parts of the equation are represented by L and R respectively, let L(i,j) and R(i,j) represent the i-th row and j-th column element of matrices L and R respectively, i,j=1,2,…,4;

[0109] By searching for equations that can be used to solve inverse kinematics through symbolic operations, we know that:

[0110]

[0111] The joint angle θ1 can be calculated:

[0112] θ1=atan2(p y ,p x )

[0113] The second step is to solve θ3:

[0114]

[0115] Since θ i It is known that, let k1 = L(1,4), k2 = L(2,4), from L 2 (1,4)+L 2 (2,4)=R 2 (1,4)+R 2 (2,4) gives:

[0116] 2a2a3cos(θ3)-2a2d4sin(θ3)+a2 2 +a3 2 +d42 =k 2 1+k 2 2

[0117] By trigonometric substitution, we can find the two values ​​of θ3:

[0118] θ 31,32 =atan2(a3,d4)-atan2(k,±sprt(a3 2 +d4 2 -k 2 ))

[0119] The third step is to solve θ2 to get:

[0120]

[0121] Solving the system of equations yields:

[0122] θ2=atan2(k1,k2)-θ3

[0123] In the program, the particle swarm optimization algorithm is used to find a set of joint angles of the robot arm that can make the end effector of the robot arm move along a predetermined path while satisfying a series of constraints, such as joint limits and dynamic performance requirements; in the application, each particle represents a set of possible solutions (or "postures") of the robot arm's joint angles; the "position" of the particle here refers to the vector of the joint angles, and the "speed" of the particle is the vector of the change of the joint angles; the particle swarm algorithm finds the optimal solution by simulating the behavior of a flock of birds looking for food, that is, the optimal joint angle configuration here, to achieve the goal of path planning; the implementation method is as follows Figure 5 .

[0124] In the motion planning of the robot arm, each particle represents a possible joint angle configuration. The starting point and the end point define the motion task of the robot arm, which is to move from one position to another. In the particle swarm algorithm, the "position" of each particle represents a set of joint angles, while the "velocity" represents the rate of change of these angles over time. At the beginning of the algorithm, the particle swarm is randomly initialized, and the position and velocity of each particle are assigned an initial value, which will be adjusted according to the iterative process of the algorithm. In each iteration, the velocity of the particle is updated based on its own historical best position (individual optimal) and the historical best position of the entire group (global optimal). This update process encourages particles to explore new positions and also takes into account the excellent positions found previously, such as Figure 6 .

[0125] In the application of robot arms, it is sufficient to find a set of joint angles that allows the robot's end effector to reach the target position with the minimum total joint rotation angle or the least number of joint rotations. The optimal solution can be selected from multiple possible solutions, which can improve the accuracy of movement while meeting the task requirements.

[0126] Through path planning and inverse kinematics optimization, we will obtain information such as joint angle, speed, acceleration, and the relationship between the position of each joint of the robot and the motion angle and time, which characterizes the detailed information of the robot during the motion process:

[0127] The X, Y, and Z coordinates of the robot end effector in the global or base coordinate system.

[0128] cosθ: Direction cosines of the end effector’s attitude, which is related to the end effector’s attitude. cosU, cosV, and cosW represent the cosine of the angle between the end effector and the X-axis of the global coordinate system.

[0129] This data format allows the robot's control system to directly read and execute the corresponding movement, guiding the actual movement of the robot, such as Figure 7 .

[0130] In the description of this specification, the terms "connection", "installation", "fixation" and the like should be understood in a broad sense. For example, "connection" can be a fixed connection, a detachable connection, or an integral connection; it can be a direct connection or an indirect connection through an intermediate medium. For ordinary technicians in this field, the specific meanings of the above terms in this application can be understood according to specific circumstances.

[0131] In the description of this specification, the description of the terms "one embodiment", "some embodiments", etc. means that the specific features, structures, materials or characteristics described in conjunction with the embodiment or example are included in at least one embodiment or example of the present application. In this specification, the schematic representation of the above terms does not necessarily refer to the same embodiment or example. Moreover, the specific features, structures, materials or characteristics described can be combined in any one or more embodiments or examples in a suitable manner.

[0132] The above are only preferred embodiments of the present application and are not intended to limit the present application. For those skilled in the art, the present application may have various modifications and variations. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present application shall be included in the protection scope of the present application.

Claims

1. A three-dimensional surface numerical fitting and serial robot arm control method based on RBF, characterized in that: The steps include: 1) uniformly sampling the planes to be polished and fitted to obtain a sampling data set; 2) Using machine learning Gaussian radial basis function to perform a first fitting on the sampled data set and the surface function of the target optical free-form surface to be obtained, and obtain the first actual surface function represented by the matrix, that is, the first actual surface to be polished; 3) Superimposing the first actual surface to be polished and the target optical free-form surface, and extracting the space between them; 4) Convert the sandwiched space into a grayscale image; 5) Processing the grayscale image to obtain a quantized square matrix; 6) extracting high-frequency data from the quantized square matrix and performing reverse restoration on the data to obtain the coordinates of the projection points of the points that need to be incrementally sampled on the xy reference plane, that is, obtaining incremental sampling data; 7) adding the incremental sampling data to the sampling data set in step 1) to obtain an incremental sampling data set; 8) performing a second fitting on the incremental sampling data set and the surface function of the target optical free-form surface using a machine learning Gaussian radial basis function to obtain a second actual surface function formula represented by a matrix, that is, obtaining a second actual surface to be polished; 9) Plan the polishing path of the serial robot arm.

2. The RBF-based three-dimensional surface numerical fitting and serial robot arm control method according to claim 1, characterized in that: The sampling method of the sampling data set in step 1) is to place the plane required for polishing and fitting in a spatial rectangular coordinate system, and perform uniform sampling using the xy plane as a reference plane to obtain the sampling data set.

3. The RBF-based three-dimensional surface numerical fitting and serial robot arm control method according to claim 1, characterized in that: The overlapping method in step 3) is selected according to the actual working conditions to minimize the total volume of the removed part, or to minimize the maximum polishing thickness of the removed part, that is, the shortest length in the normal direction of the target surface.

4. The RBF-based three-dimensional surface numerical fitting and serial robot arm control method according to claim 1, characterized in that: The grayscale of the grayscale image at a certain point is obtained by normalizing the thickness at the same position in step 3).

5. The RBF-based three-dimensional surface numerical fitting and serial robot arm control method according to claim 1, characterized in that: The grayscale image in step 5) is processed by performing a two-dimensional discrete cosine transform on the grayscale image to obtain the quantization matrix.

6. The RBF-based three-dimensional surface numerical fitting and serial robot arm control method according to claim 1, characterized in that: The specific planning method of step 9) is to use the standard DH inverse kinematics formula of the multi-axis robot arm to compensate for gravity, elastic deformation of the robot arm and repositioning accuracy to derive multiple sets of solutions, and use the particle swarm algorithm to iterate, and obtain an optimal polishing path in which the number of joints used at each moment is the minimum and the sum of the integrals of all joint rotation angles over time is the minimum, and output the joints of the robot arm, the spatial position of the polishing head and the normal vector direction of the polishing head corresponding to the path and the moment.

Images