A forward kinematics analysis method based on RBF neural network to improve Newton's iterative algorithm

Abstract

The present invention provides a forward kinematics analysis method based on an improved Newton iteration algorithm based on an RBF neural network, comprising the following steps: S1, establishing a rod length formula of a hybrid polishing mechanism in an absolute coordinate system through a closed-loop vector method, and obtaining an inverse kinematics equation and a nonlinear equation group; S2, selecting the center of the RBF neural network, and randomly selecting sampling points as training data; S3, using the estimated value obtained by the neural network training as an initial value for iteration to perform iterative calculations through the forward kinematics equation until the accuracy meets the output result. The present invention uses the predicted value of the RBF neural network training as the initial value for iteration of the Newton iteration for iterative solution, which can avoid the insufficient accuracy of the RBF neural network due to the insufficient number of training samples, and can also avoid the dependence of the Newton iteration method on the initial value for iteration.

Description

Technical Field

[0001] The invention belongs to the technical field of computing models, and in particular relates to a forward kinematics analysis method based on an improved Newton iterative algorithm based on a RBF neural network. Background Art

[0002] As a machine tool, the hybrid robot needs to control the position of the tool head in real time during the machining process to ensure the surface quality of the free-form surface, and this motion control is based on the forward kinematics analysis of the mechanism. Therefore, it is particularly important to carry out forward kinematics analysis of the hybrid polishing robot. However, the forward kinematic solution of the hybrid mechanism is difficult to solve and has low solution efficiency. Therefore, it is necessary to propose an accurate and reliable kinematic solution calculation method for the hybrid mechanism.

[0003] Newton iteration is an important method for solving nonlinear equations. By giving specific initial values ​​for iteration, the nonlinear equations are gradually transformed into linear equations for solution. The principle of kinematics positive solution is Newton iteration to solve nonlinear equations. However, this method is more dependent on the selection of initial values. If the initial values ​​are not selected appropriately, the iteration results will not converge.

[0004] Artificial neural network is an information computing model that imitates the basic characteristics and structural theory of biological neural networks. Its basic unit is neurons, which are connected in parallel to form a neural network. By imitating the way biological brains process information and the interaction of neurons, the system has a self-learning ability similar to that of biological brains. In a neural network, a neuron can receive multiple input signals, obtain output signals in a specific processing method, and then transmit the signals to other neurons in a nonlinear way. Therefore, neural networks have high accuracy and strong robustness for controlling uncertain systems due to their ability to approximate functionals with arbitrary precision. Feedforward neural networks are widely used in neural networks due to their simple structure. For example, BP neural networks, RBF neural networks, multilayer perceptron neural networks, etc. are all feedforward networks. In recent years, with the development of intelligent algorithms, more and more intelligent algorithms are applied to the kinematics of mechanisms. RBF networks can approximate arbitrary nonlinear functions, can handle the difficult-to-analyze regularities in the system, have good generalization ability, and have a fast learning convergence speed. They have been successfully applied to nonlinear function approximation, time series analysis, data classification, pattern recognition, information processing, image processing, system modeling, control and fault diagnosis. Summary of the invention

[0005] The technical problem to be solved by the present invention is to provide a forward kinematics analysis method based on an improved Newton iteration algorithm based on an RBF neural network in view of the deficiencies in the above-mentioned prior art. The calculation method uses the predicted value of the RBF neural network training as the initial value of the Newton iteration for iterative solution, which can avoid the insufficient accuracy of the RBF neural network due to an insufficient number of training samples, and can also avoid the dependence of the Newton iteration method on the initial value of the iteration. The design is scientific and reasonable, the practicability is strong, the calculation accuracy is high, and the calculation efficiency is high.

[0006] In order to solve the above technical problems, the technical solution adopted by the present invention is: a forward kinematics analysis method based on RBF neural network improved Newton iterative algorithm, characterized in that it includes the following operation steps:

[0007] S1. Using the closed-loop vector method, a rod length formula of the hybrid polishing mechanism is established in an absolute coordinate system to obtain an inverse kinematics equation. Based on the inverse equation, a nonlinear equation group is established as a forward kinematics equation;

[0008] S2. Select the center of the RBF neural network and use the Gaussian function as the basis function. According to the inverse solution of kinematics, randomly select j sampling points within the specified range of motion as the training data of the neural network. Most of the sampling points are used as training samples, and the remaining few sampling points are used as test samples.

[0009] S3. Use the estimated value obtained by neural network training as the initial value for iteration and perform iterative calculation through the forward kinematics equation, and record the difference value of each iteration |h i -h i-1 |, and determine the iterative difference |h i -h i-1 | is less than the precision requirement ε, if |h i -h i-1 | is less than the precision requirement ε, the iteration value is output and the process ends; otherwise, the iteration continues until the precision meets the output result.

[0010] Preferably, the S1 specifically includes the following steps:

[0011] S101. Draw a schematic diagram of the hybrid mechanism for the XY-3-RPS hybrid polishing platform mechanism model. i and P i (i=1,2,3) are the centers of the revolute pair and the spherical pair of the parallel platform, respectively. The triangle ΔA formed by them 1 A 2 A 3 and ΔP 1 P 2 P 3They are represented as fixed platform and moving platform respectively. The circumscribed circle radius of the fixed platform is represented by R and r respectively. Each branch chain is represented by vector A i P i To indicate, the X-axis tandem platform is indicated by the mobile pair M, and the Y-axis tandem platform is indicated by the mobile pair N;

[0012] S102, for the structure of the XY-3-RPS hybrid polishing robot, establish the moving platform coordinate system {C 1}、Set the platform coordinate system {C 2}、Absolute coordinate system{C 0}, the coordinate origins are the geometric center C of the moving platform 1 , determine the geometric center C of the platform 2 , move the center C of the pair M 0 ;

[0013] S103, use the closed loop vector method to perform kinematic analysis. The starting point of the motion loop is the origin C of the fixed platform coordinate system, which first passes through the hinge point A between the drive rod and the fixed platform. i , and then through the hinge point P between the driving rod and the moving platform i , the end point is the coordinate origin C of the moving platform coordinate system 1 ;

[0014] S104, 3-RPS parallel mechanism has three branches in total, each of which is an independent closed motion loop. Each motion loop is represented by a vector form, CC 1 +C 1 P i =CA i +A i P i , rewritten as

[0015] S105. Since each branch chain is constrained by the rotation pair, and the unit vector j of the rotation pair axis i It is always perpendicular to the branch chain, so the constraint equation of the branch chain can be obtained

[0016] S106: Get x c ,y c ,γ about z c , α, β,

[0017]

[0018] y C =-rcosβsinγ

[0019]

[0020] S107. Since the third-order matrix can only represent pure rotational motion, in order to represent the translational motion of the hybrid mechanism, it needs to be generalized to a fourth-order homogeneous coordinate, that is,

[0021]

[0022] S108, the fixed platform coordinate system only has translation along the X and Y axes relative to the absolute coordinate system. Since the 3-RPS parallel platform moves along the Y platform of the series cross slide as a whole, the moving platform has no relative movement along the X and Y axes relative to the fixed platform. Therefore, the transformation matrix of the moving platform coordinate system relative to the absolute coordinate system is:

[0023]

[0024]

[0025] S109, by substituting the formula into the solution, we can get the driving rod length l of the XY-3-RPS hybrid polishing mechanism. i , expressed as:

[0026]

[0027] In the formula, Point A i Position in absolute coordinate system;

[0028] S110, the driving displacement x, y of the series cross slide and the displacement l of the parallel driving rod i , solve the position z of the center of the machine tool end operating platform C , α, β, and then the rod length formula of the hybrid polishing mechanism is obtained

[0029]

[0030] Preferably, S2 specifically includes the following steps:

[0031] S201, use the self-organizing center selection method to select the RBF neural network function center, select the Gaussian function as the basis function, and the function expression is:

[0032]

[0033] Where x = [x i ] T (i=1,2,…,n) is the input of the neural network. The input parameter is the length of the three driving rods of the XY-3-RPS hybrid polishing machine. n represents the number of training samples. The connection weight between the input layer and the hidden layer is 1. h=[h j ] T is the output of the hidden layer of the neural network, j is the number of nodes in the hidden layer; u=[uj ] T is the center vector of the jth neuron basis function in the hidden layer, v = [v j ] T is the width of the j-th basis function of the hidden layer;

[0034] S202, determine the output of the RBF neural network is where w = [w m ] T is the weight from the hidden layer to the output layer, y m is the actual output of the neural network;

[0035] S203, first initialize the network, randomly select j training as cluster centers u in the kinematic inverse solution of the hybrid mechanism j ;

[0036] S204, grouping according to the nearest neighbor rule, i Each cluster set φ assigned to the sample according to the Euclidean distance from the cluster center i (i=1,2,…,n);

[0037] S205, then adjust the cluster center and calculate the cluster set φ p The average value of the training samples in is used as the new cluster center. If the new cluster center does not change, the obtained cluster center is the final basis function center of the RBF neural network. If not, return to the previous step and solve the cluster center again.

[0038] S206, basis function width solution, since the selected basis function of the RBF neural network is a Gaussian function, the solution formula for the Gaussian basis function width is u max is the maximum distance between the selected centers;

[0039] S207, calculate weights: The weights from the hidden layer to the output layer are calculated using the least squares method, and the calculation formula is:

[0040] S208. According to the inverse solution results of kinematics, 5000 sampling points are randomly selected within the specified motion range as training data for the neural network, of which 4800 sampling points are used as training samples and the remaining 200 sampling points are used as test samples.

[0041] Preferably, S3 specifically includes the following steps:

[0042] S301, solve the forward kinematics equations of the mechanism, that is, solve the β position z of the center of the terminal operation platform C The problem of solving the roots of the equations F i (hi )-F i (h i-1 )=F i `(h i-1 )(h i -h i-1 );

[0043] S302, solve equation F i (h i )-F i (h i-1 )=F i `(h i-1 )(h i -h i-1 ) is converted into matrix form

[0044]

[0045] in

[0046] is the Jacobian matrix of the XY-3-RPS hybrid polishing mechanism;

[0047] S303, take the estimated value of the test sample training as the initial value of the Newton iteration, perform iterative calculation through the forward kinematics equation, and record the difference value of each iteration |h i -h i-1 |, and determine the iterative difference |h i -h i-1 Is it less than the accuracy requirement ε? i -h i-1 | is less than the precision requirement ε, the iterative value is output, and at this time h i It is the forward kinematics equations of the hybrid mechanism F i (α,β,z C )=0, otherwise, the iteration continues until the accuracy satisfies the output result.

[0048] Compared with the prior art, the present invention has the following advantages:

[0049] 1. The present invention utilizes the characteristics and advantages of Newton iteration method and RBF neural network algorithm, and uses the predicted value of RBF neural network training as the initial value of Newton iteration for iterative solution, which can avoid the insufficient accuracy of RBF neural network due to insufficient number of training samples, and avoid the dependence of Newton iteration method on the initial value of iteration.

[0050] 2. The present invention is scientifically and reasonably designed, highly practical, and can be widely used in aspects such as workspace, singular configuration, error compensation, and motion control of hybrid robots.

[0051] The present invention is further described in detail below with reference to the accompanying drawings and embodiments. BRIEF DESCRIPTION OF THE DRAWINGS

[0052] Figure 1 It is a flow chart of the present invention.

[0053] Figure 2 It is a structural diagram of the XY-3-RPS hybrid polishing mechanism in the present invention.

[0054] Figure 3 This is the geometric principle diagram of Newton's iteration algorithm.

[0055] Figure 4 This is the flow chart of Newton iteration method.

[0056] Figure 5 It is the structure diagram of RBF neural network.

[0057] Figure 6 is the error curve before improvement.

[0058] Figure 7 It is the error curve of this embodiment after improvement.

[0059] Figure 8 It is the improved iteration number curve of this implementation.

[0060] Fig. 9 It is an improved iteration time curve of this implementation. DETAILED DESCRIPTION

[0061] like Figure 1 As shown, the present invention comprises the following steps:

[0062] S1. Using the closed-loop vector method, a rod length formula of the hybrid polishing mechanism is established in an absolute coordinate system to obtain an inverse kinematics equation. Based on the inverse equation, a nonlinear equation group is established as a forward kinematics equation;

[0063] S2. Select the center of the RBF neural network and use the Gaussian function as the basis function. According to the inverse solution of kinematics, randomly select j sampling points within the specified range of motion as the training data of the neural network. Most of the sampling points are used as training samples, and the remaining few sampling points are used as test samples.

[0064] S3. Use the estimated value obtained by neural network training as the initial value for iteration and perform iterative calculation through the forward kinematics equation, and record the difference value of each iteration |h i -h i-1 |, and determine the iterative difference |h i -h i-1 | is less than the precision requirement ε, if |h i -h i-1| is less than the precision requirement ε, the iteration value is output and the process ends; otherwise, the iteration continues until the precision meets the output result.

[0065] In this embodiment, S1 specifically includes the following operation steps:

[0066] S101, such as Figure 2 As shown in the figure, for the XY-3-RPS hybrid polishing platform mechanism model, a structural diagram of the hybrid mechanism is drawn. i and P i (i=1,2,3) are the centers of the revolute pair and the spherical pair of the parallel platform, respectively. The triangle ΔA formed by them 1 A 2 A 3 and ΔP 1 P 2 P 3 They are represented as fixed platform and moving platform respectively. The circumscribed circle radius of the fixed platform is represented by R and r respectively. Each branch chain is represented by vector A i P i To indicate, the X-axis tandem platform is indicated by the mobile pair M, and the Y-axis tandem platform is indicated by the mobile pair N;

[0067] S102, for the structure of the XY-3-RPS hybrid polishing robot, establish the moving platform coordinate system {C 1}、Set the platform coordinate system {C 2}、Absolute coordinate system{C 0}, the coordinate origins are the geometric center C of the moving platform 1 , determine the geometric center C of the platform 2 , move the center C of the pair M 0 ;

[0068] S103, use the closed loop vector method to perform kinematic analysis. The starting point of the motion loop is the origin C of the fixed platform coordinate system, which first passes through the hinge point A between the drive rod and the fixed platform. i , and then through the hinge point P between the driving rod and the moving platform i , the end point is the coordinate origin C of the moving platform coordinate system 1 ;

[0069] S104, 3-RPS parallel mechanism has three branches in total, each of which is an independent closed motion loop. Each motion loop is represented by a vector form, CC 1 +C 1 P i =CA i +A i P i , rewritten as

[0070] S105. Since each branch chain is constrained by the rotation pair, and the unit vector j of the rotation pair axis i It is always perpendicular to the branch chain, so the constraint equation of the branch chain can be obtained

[0071] S106: Get x c ,y c ,γ about z c , α, β,

[0072]

[0073] y C =-rcosβsinγ

[0074]

[0075] S107. Since the third-order matrix can only represent pure rotational motion, in order to represent the translational motion of the hybrid mechanism, it needs to be generalized to a fourth-order homogeneous coordinate, that is,

[0076]

[0077] S108, the fixed platform coordinate system only has translation along the X and Y axes relative to the absolute coordinate system. Since the 3-RPS parallel platform moves along the Y platform of the series cross slide as a whole, the moving platform has no relative movement along the X and Y axes relative to the fixed platform. Therefore, the transformation matrix of the moving platform coordinate system relative to the absolute coordinate system is:

[0078]

[0079]

[0080] S109, by substituting the formula into the solution, we can get the driving rod length l of the XY-3-RPS hybrid polishing mechanism. i , expressed as:

[0081]

[0082] In the formula, Point A i Position in absolute coordinate system;

[0083] S110, the driving displacement x, y of the series cross slide and the displacement l of the parallel driving rod i , solve the position z of the center of the machine tool end operating platform C , α, β, and then the rod length formula of the hybrid polishing mechanism is obtained

[0084]

[0085] In this embodiment, S2 specifically includes the following operation steps:

[0086] S201, use the self-organizing center selection method to select the RBF neural network function center, select the Gaussian function as the basis function, and the function expression is:

[0087]

[0088] Where x = [x i ] T (i=1,2,…,n) is the input of the neural network. The input parameter is the length of the three driving rods of the XY-3-RPS hybrid polishing machine. n represents the number of training samples. The connection weight between the input layer and the hidden layer is 1. h=[h j ] T is the output of the hidden layer of the neural network, j is the number of nodes in the hidden layer; u=[u j ] T is the center vector of the jth neuron basis function in the hidden layer, v = [v j ] T is the width of the j-th basis function of the hidden layer;

[0089] S202, determine the output of the RBF neural network is where w = [w m ] T is the weight from the hidden layer to the output layer, y m is the actual output of the neural network;

[0090] S203, first initialize the network, randomly select j training as cluster centers u in the kinematic inverse solution of the hybrid mechanism j ;

[0091] S204, grouping according to the nearest neighbor rule, i Each cluster set φ assigned to the sample according to the Euclidean distance from the cluster center i (i=1,2,…,n);

[0092] S205, then adjust the cluster center and calculate the cluster set φ p The average value of the training samples in is used as the new cluster center. If the new cluster center does not change, the obtained cluster center is the final basis function center of the RBF neural network. If not, return to the previous step and solve the cluster center again.

[0093] S206, basis function width solution, since the selected basis function of the RBF neural network is a Gaussian function, the solution formula for the Gaussian basis function width is u max is the maximum distance between the selected centers;

[0094] S207, calculate weights: The weights from the hidden layer to the output layer are calculated using the least squares method, and the calculation formula is:

[0095] S208. According to the inverse solution results of kinematics, 5000 sampling points are randomly selected within the specified motion range as training data for the neural network, of which 4800 sampling points are used as training samples and the remaining 200 sampling points are used as test samples.

[0096] In this embodiment, S3 specifically includes the following operation steps:

[0097] S301, solve the forward kinematics equations of the mechanism, that is, solve the β position z of the center of the terminal operation platform C The problem of solving the roots of the equations F i (h i )-F i (h i-1 )=F i `(h i-1 )(h i -h i-1 );

[0098] S302, solve equation F i (h i )-F i (h i-1 )=F i `(h i-1 )(h i -h i-1 ) is converted into matrix form

[0099]

[0100] in

[0101] is the Jacobian matrix of the XY-3-RPS hybrid polishing mechanism;

[0102] S303, take the estimated value of the test sample training as the initial value of the Newton iteration, perform iterative calculation through the forward kinematics equation, and record the difference value of each iteration |h i -h i-1 |, and determine the iterative difference |h i -h i-1 Is it less than the accuracy requirement ε? i -h i-1 | is less than the precision requirement ε, the iterative value is output, and at this time h i It is the forward kinematics equations of the hybrid mechanism F i (α,β,z C)=0, otherwise, the iteration continues until the accuracy satisfies the output result.

[0103] Experimental tests were performed to verify the algorithm:

[0104] The theoretical methods used in this program are:

[0105] (1) Closed-loop vector method. Linkage kinematics analysis based on the closed-loop vector method. When using the vector method to establish the position equation of the mechanism, the rods must be represented by vectors and a closed vector polygon of the mechanism must be made. The sum of the vectors must be equal to zero. The number of branches of the parallel mechanism is the number of independent closed motion loops. The motion loop has been indicated by a red line. The starting and ending points of the motion loop are both the origins of the fixed and static platform coordinate systems, passing through the hinge between the platform and the fixed and static platform. The structural diagram embodied by this method is as follows: Figure 2 shown.

[0106] (2) Figure 3 and Figure 4 As shown in the figure, Newton iteration method is one of the most widely used algorithms in numerical methods. The basic idea of ​​solving it is to gradually transform the nonlinear equations into linear equations, and find the approximate solution by giving a specific initial value of the iteration and using a finite number of iterations. The essence of the forward kinematic equation of the hybrid polishing mechanism is to solve a set of implicit nonlinear equations. The solution to such problems can often only be obtained by numerical methods to get an approximate solution.

[0107] (3) Self-organizing center selection method. According to the different methods of selecting the center of the radial basis function, there are many learning methods for the RBF network, such as random center selection method, self-organizing center selection method, supervised center selection method and orthogonal least squares method. This method consists of two stages: the first is the self-organizing learning stage, which is an unsupervised learning process to solve the center and variance of the hidden layer basis function; the second is the supervised learning stage, which solves the weights between the hidden layer and the output layer. Here, the Gaussian function is used. As a basis function, where x = [x i ] T (i=1,2,…,n) is the input of the neural network. The input parameter is the length of the three driving rods of the XY-3-RPS hybrid polishing machine. n represents the number of training samples. The connection weight between the input layer and the hidden layer is 1. h=[h j ] T is the output of the hidden layer of the neural network, j is the number of nodes in the hidden layer; u=[u j ] T is the center vector of the jth neuron basis function in the hidden layer, v = [v j ] T The width of the jth neuron basis function in the hidden layer is also called the variance. The corresponding RBF neural network structure from the input layer to the hidden layer and then to the output layer is as follows: Figure 5 shown.

[0108] Performance comparison of several algorithms:

[0109] The setting of various algorithm parameters in the theoretical method:

[0110] like Figure 5 As shown in the figure, since the training of RBF neural network requires a large amount of sample data, multiple sets of inverse solutions are randomly selected within the motion range of the XY-3-RPS hybrid polishing machine for neural network training. The training accuracy of the neural network is related to the number of samples. The more samples there are, the higher the accuracy of the training results, and the corresponding running speed is slower; conversely, if the number of samples is too small, the accuracy of the training results will also be low. Therefore, the accuracy of the RBF neural network is often limited by the number of samples, resulting in the accuracy of the results not being as high as that of the Newton iteration method. However, the Newton iteration method is also limited by the selection of initial values. Selecting appropriate initial values ​​will greatly reduce the number and time of algorithm iterations, and can obtain high-precision results more quickly. The various parameters of the algorithm are set as follows:

[0111] Within the specified range of motion, 5000 sampling points are randomly selected as training data for the neural network, of which 4800 sampling points are used as training samples and the remaining 200 sampling points are used as test samples. The estimated value of the test sample training is used as the initial value of the Newton iteration to perform iterative calculation through the forward kinematics equation, and the difference of each iteration is recorded. The iterative algebra of the Newton iteration method of the hybrid algorithm in the present invention varies according to the specific function and the dimension of the function variable.

[0112] The error curve before improvement is as follows: Figure 6 The improved error curve is shown in Figure 7 The mean square error is shown in Table 1, and the iteration number curve is shown in Figure 8 As shown, the iteration time curve is Fig. 9 The number of iterations and time before and after the improvement are shown in Table 2.

[0113]

[0114] Table 1 Mean square error

[0115]

[0116] Table 2

[0117] The above is only a preferred embodiment of the present invention and does not limit the present invention in any way. Any simple modification, change and equivalent change made to the above embodiment according to the technical essence of the invention still falls within the protection scope of the technical solution of the present invention.

Claims

1. A forward kinematics analysis method based on RBF neural network improved Newton iterative algorithm, characterized in that: The steps include: S1. Using the closed-loop vector method, a rod length formula of the hybrid polishing mechanism is established in an absolute coordinate system to obtain an inverse kinematics equation. Based on the inverse equation, a nonlinear equation group is established as a forward kinematics equation; S2. Select the center of the RBF neural network and use the Gaussian function as the basis function. According to the inverse solution of kinematics, randomly select j sampling points within the specified range of motion as the training data of the neural network, and divide the j sampling points into training samples and test samples; S3. Use the estimated value obtained by neural network training as the initial value for iteration and perform iterative calculation through the forward kinematics equation, and record the difference value of each iteration |h i -h i-1 |, and determine the iterative difference |h i -h i-1 | is less than the precision requirement ε, if |h i -h i-1 | is less than the precision requirement ε, then the iteration value is output and the process ends; otherwise, the iteration continues until the precision meets the output result; The S1 specifically includes the following steps: S101. Draw a structural diagram of the hybrid polishing platform mechanism model. i and P i (i=1,2,3) are the centers of the revolute pair and the spherical pair of the parallel platform respectively. The triangles ΔA1A2A3 and ΔP1P2P3 formed by them are represented as the fixed platform and the moving platform respectively. The circumscribed circle radius of the fixed platform is represented by R and r respectively. Each branch chain is represented by the vector A i P i To indicate, the X-axis tandem platform is indicated by the mobile pair M, and the Y-axis tandem platform is indicated by the mobile pair N; S102, for the structure of the hybrid polishing robot, a moving platform coordinate system {C1}, a fixed platform coordinate system {C2}, and an absolute coordinate system {C0} are established respectively, and the coordinate origins are the geometric center C1 of the moving platform, the geometric center C2 of the fixed platform, and the center C0 of the moving pair M; S103, use the closed loop vector method to perform kinematic analysis. The starting point of the motion loop is the origin C of the fixed platform coordinate system, which first passes through the hinge point A between the drive rod and the fixed platform. i , and then through the hinge point P between the driving rod and the moving platform i , the end point is the coordinate origin C1 of the moving platform coordinate system; S104, the parallel mechanism has three branches in total, each of which is an independent closed motion loop. Each motion loop is represented by a vector form as CC1+C1P i =CA i +A i P i , and rewrite it as Among them, L i Represents vector A i P i , a i Representation vector CA i , p i Represents the vector C1p i , C R C1 represents the rotation matrix of the moving coordinate system in the fixed platform coordinate system, and C1 represents the geometric center origin of the moving platform; S105. Each branch chain is subject to the constraint of the rotation pair, and the unit vector j of the rotation pair axis i It is always perpendicular to the branch chain, so the constraint equation of the branch chain can be obtained S106: Get x c ,y c ,γ about z c , α, β, and C =-rcosβsinγ Among them, x c Indicates the distance that the origin of the fixed platform coordinate system C moves along the x-axis, y c Indicates the distance that the origin of the fixed platform coordinate system C moves along the y axis, z c represents the distance that the origin of the fixed platform coordinate system C moves along the z-axis, α represents the rotation angle of the moving platform coordinate system C1 relative to the X-axis of the fixed platform coordinate system C, β represents the rotation angle of the moving platform coordinate system C1 relative to the Y-axis of the fixed platform coordinate system C, and γ represents the rotation angle of the moving platform coordinate system C1 relative to the Z-axis of the fixed platform coordinate system C; S107. Since the third-order matrix can only represent pure rotational motion, in order to represent the translational motion of the hybrid mechanism, x c ,y c ,γ about z c The expressions of , α, β are generalized to 4th-order homogeneous coordinates, that is, Where cβ represents cosβ, cα represents cosα, cγ represents cosγ; sβ represents sinβ, sα represents sinα, sγ represents sinγ; S108, the fixed platform coordinate system only has translation along the X and Y axes relative to the absolute coordinate system, and the moving platform has no relative movement along the X and Y axes relative to the fixed platform. Therefore, the transformation matrix of the moving platform coordinate system relative to the absolute coordinate system is: Among them, Z M Indicates the height of the X-axis tandem platform in the Z-axis direction. N Indicates the height of the Y-axis serial platform in the Z-axis direction, cβ indicates cosβ, cα indicates cosα, cγ indicates cosγ; sβ indicates sinβ, sα indicates sinα, sγ indicates sinγ; S109, by substituting the formula into the solution, we can get the driving rod length l of the XY-3-RPS hybrid polishing mechanism. i , expressed as: In the formula, Point A i The position in the absolute coordinate system, Represents the homogeneous transformation matrix of the moving platform coordinate system C1 relative to the absolute coordinate system C0; S110, the driving displacement of the series cross slide is x, y, and the displacement of the parallel driving rod is l i , solve the position z of the center of the machine tool end operating platform C , α, β, and then the rod length formula of the hybrid polishing mechanism is obtained The S3 specifically includes the following steps: S301, solve the forward kinematics equations of the mechanism, that is, solve the β position z of the center of the terminal operation platform C The problem of solving the roots of the equations F i (h i )-F i (h i-1 )=F i `(h i-1 )(h i -h i-1 ); S302, solve equation F i (h i )-F i (h i-1 )=F i `(h i-1 )(h i -h i-1 ) is converted into matrix form in is the Jacobian matrix of the XY-3-RPS hybrid polishing mechanism; S303, take the estimated value of the test sample training as the initial value of the Newton iteration, perform iterative calculation through the forward kinematics equation, and record the difference value of each iteration |h i -h i-1 |, and determine the iterative difference |h i -h i-1 | is less than the accuracy requirement ε, when |h i -h i-1 | is less than the precision requirement ε, the iterative value is output, and at this time h i It is the forward kinematics equations of the hybrid mechanism F i (α,β,z C )=0, otherwise, the iteration continues until the accuracy satisfies the output result.

2. The forward kinematics analysis method based on RBF neural network improved Newton iterative algorithm according to claim 1, characterized in that: The S2 specifically includes the following steps: S201, use the self-organizing center selection method to select the RBF neural network function center, select the Gaussian function as the basis function, and the function expression is: Where x = [x i ] T (i=1,2,…,n) is the input of the neural network. The input parameter is the length of the three driving rods of the hybrid polishing machine. n represents the number of training samples. The connection weight between the input layer and the hidden layer is 1. h=[h j ] T is the output of the hidden layer of the neural network, j is the number of nodes in the hidden layer; u=[u j ] T is the center vector of the jth neuron basis function in the hidden layer, v = [v j ] T is the width of the j-th basis function of the hidden layer; S202, determine the output of the RBF neural network is where w = [w m ] T is the weight from the hidden layer to the output layer, y m is the actual output of the neural network; S203, first initialize the network, randomly select j training as cluster centers u in the kinematic inverse solution of the hybrid mechanism j ; S204, grouping according to the nearest neighbor rule, i Each cluster set φ assigned to the sample according to the Euclidean distance from the cluster center i (i=1,2,…,n); S205, adjust the cluster center and calculate the cluster set φ p The average value of the training samples in is used as the new cluster center. If the new cluster center does not change, the obtained cluster center is the final basis function center of the RBF neural network. If the new cluster center changes, return to the previous step and solve the cluster center again. S206. Solution of basis function width: The solution formula of Gaussian basis function width is: u max is the maximum distance between the selected centers; S207, calculate weights: The weights from the hidden layer to the output layer are calculated using the least squares method, and the calculation formula is: S208. According to the inverse solution results of kinematics, 5000 sampling points are randomly selected within the specified motion range as training data for the neural network, of which 4800 sampling points are used as training samples and the remaining 200 sampling points are used as test samples.

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