CN115098978A - RBF neural network-based forward kinematics analysis method for improving Newton iterative algorithm - Google Patents

Abstract

The invention provides a forward kinematics analysis method based on an RBF neural network improved Newton iteration algorithm, which comprises the following steps: s1, establishing a rod length formula of the series-parallel polishing mechanism under an absolute coordinate system through a closed-loop vector method, and solving a kinematic inverse solution equation and a nonlinear equation set; s2, selecting an RBF neural network center, and randomly selecting sampling points as training data; and S3, performing iterative computation by using the estimated value obtained by neural network training as an iterative initial value through a positive kinematics equation until the precision meets the output result. The method utilizes the predicted value of RBF neural network training as the initial iteration value of Newton iteration to carry out iterative solution, thereby not only avoiding the insufficient precision of the RBF neural network caused by insufficient training sample number, but also avoiding the dependency of Newton iteration method on the initial iteration value.

Description

A Forward Kinematics Analysis of Improved Newton Iterative Algorithm Based on RBF Neural Network method

technical field

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

Background technique

As a machine tool, the hybrid robot needs to control the pose of the machining tool head in real time in 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 positive solution of the kinematics of the hybrid mechanism has problems such as difficulty in solving and low efficiency. Therefore, it is necessary to propose an accurate and reliable kinematics calculation method for the hybrid mechanism.

The Newton iteration method is an important method for solving nonlinear equations. By giving a specific initial value of iteration, the nonlinear equations are gradually transformed into linear equations for solving. The principle of positive kinematics solution is that Newton iteratively solves nonlinear equations. However, this method is more dependent on the selection of the initial value. If the initial value is not selected properly, it will cause the non-convergence of the iterative results.

Artificial neural network is an information computing model by imitating the basic characteristics and structure theory of biological neural network. The basic unit is the neuron, and the neurons are connected in parallel to form a neural network. By imitating the way the biological brain processes information and the interaction of neurons, the system has the self-learning ability similar to the biological brain. In a neural network, a neuron can accept multiple input signals, obtain the output signal in a specific processing method, and then transmit the signal to other neurons in a nonlinear way. Therefore, by virtue of its functional approximation capability of arbitrary precision, neural network has high precision and strong robustness for controlling uncertain systems. Feedforward neural network is widely used in neural network due to its simple structure, such as BP neural network, RBF neural network, multilayer perceptron neural network, etc., all belong to feedforward network. In recent years, with the development of intelligent algorithms, more and more intelligent algorithms are applied to the kinematics of the mechanism. The RBF network can approximate any nonlinear function, can handle the unanalyzable regularities in the system, has good generalization ability, and has a fast learning convergence speed. It has been successfully applied to nonlinear function approximation, time series analysis, Data classification, pattern recognition, information processing, image processing, system modeling, control and fault diagnosis, etc.

SUMMARY OF THE INVENTION

The technical problem to be solved by the present invention is to provide a forward kinematics analysis method based on the RBF neural network improved Newton iterative algorithm for the deficiencies of the above-mentioned prior art. The calculation method uses the predicted value trained by the RBF neural network as the The iterative solution of the iterative initial value can avoid the insufficient accuracy of the RBF neural network due to the insufficient number of training samples, and can avoid the dependence of the Newton iteration method on the initial value of the iteration. High computational efficiency.

In order to solve the above-mentioned technical problems, the technical scheme adopted in the present invention is: a forward kinematics analysis method based on RBF neural network improvement Newton iterative algorithm, is characterized in that, comprises the following operation steps:

S1. Using the closed-loop vector method, the rod length formula of the hybrid polishing mechanism is established in the absolute coordinate system, and the kinematic inverse solution equation is obtained. On the basis of the inverse solution equation, the nonlinear equation system is established as the forward kinematic equation;

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

S3. Use the estimated value obtained by neural network training as the initial iterative value to perform iterative calculation through the forward kinematics equation, record each iteration difference |h _i -h _i-1 |, and judge the iterative difference |h _i - Whether h _i-1 | is less than the accuracy requirement ε, if |h _i -h _i-1 | is less than the accuracy requirement ε, output the iteration value and end the process, otherwise, continue to iterate until the accuracy meets the output result.

Preferably, the S1 specifically includes the following operation steps:

S101. Draw a schematic structural diagram of the hybrid mechanism for the XY-3-RPS hybrid polishing platform mechanism model. A _i and P _i (i=1, 2, 3) are the centers of the rotating pair and the ball pair of the parallel platform, respectively. , the triangles ΔA ₁ A ₂ A ₃ and ΔP ₁ P ₂ P ₃ formed by this are represented as the fixed platform and the moving platform, respectively, the circumradius of the fixed platform is represented by R and r respectively, and each branch is represented by a vector A _i It is represented by _Pi , the X-direction series platform is represented by the moving pair M, and the Y-direction series platform is represented by the moving pair N;

S102. For the structure of the XY-3-RPS hybrid polishing robot, establish a moving platform coordinate system {C ₁ }, a fixed platform coordinate system {C ₂ }, and an absolute coordinate system {C ₀ } respectively, and the coordinate origins are respectively the moving platform. The geometric center C ₁ , the geometric center C ₂ of the fixed platform, and the center C ₀ of the moving pair M;

S103, using the closed-loop vector method for kinematics analysis, the starting point of the motion loop is the origin C of the coordinate system of the fixed platform, first passes through the hinge point A _i of the driving rod and the fixed platform, and then passes through the hinge point P _i of the driving rod and the moving platform, The end point is the coordinate origin C ₁ of the moving platform coordinate system;

S104, 3-RPS parallel mechanism has a total of three branch chains, each of which is an independent closed motion loop, and each motion loop is represented by a vector form, CC ₁ +C ₁ P _i =CA _i +A _i P _i , after rewrite as

S105. Since each branch chain will be constrained by the rotation pair, and the unit vector _ji of the rotation pair axis is always perpendicular to the branch chain, the constraint equation of the branch chain can be obtained

S106: Obtain the expressions of x _c , y _c , γ about z _c , α, β,

y _C = -rcosβsinγ

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 the fourth-order homogeneous coordinates, that is,

S108. The fixed platform coordinate system only translates along the X and Y axes relative to the absolute coordinate system. Since the 3-RPS parallel platform moves as a whole with the Y platform of the series cross slide, the moving platform does not move along the X, Y axes relative to the fixed platform. The relative movement of the Y axis, therefore, the transformation matrix of the moving platform coordinate system relative to the absolute coordinate system is:

S109. The driving rod length l _i of the XY-3-RPS hybrid polishing mechanism can be obtained by substituting the formula into the solution and transformation, which is expressed as:

为点A_i在绝对坐标系下的位置；In the formula,

is the position of point A _i in the absolute coordinate system;

S110. Calculate the driving displacement x, y of the series cross slide and the parallel driving rod displacement _li to solve the pose z _C , α, β of the center of the operation platform at the end of the machine tool, and then obtain the rod length formula of the hybrid polishing mechanism

Preferably, the S2 specifically includes the following operation steps:

S201. Use the self-organizing center selection method to select the function center of the RBF neural network, select the Gaussian function as the basis function, and the function expression is

Among them, x=[x _i ] ^T (i=1,2,...,n) is the input of the neural network, the input parameter is the rod length value of the three driving rods of the XY-3-RPS hybrid polishing machine, and n represents the training sample The number of , 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 hidden layer nodes; u=[u _j ] ^T is the hidden layer The center vector of the jth neuron basis function of the layer, v=[v _j ] ^T is the width of the jth element basis function of the hidden layer;

其中w＝[w_m]^T是隐含层到输出层的权值，y_m为神经网络的实际输出；S202. Determine the output of the RBF neural network as

Where w=[w _m ] ^T is the weight from the hidden layer to the output layer, and y _m is the actual output of the neural network;

S203, initialize the network first, randomly select j trainings as the cluster center u _j in the kinematic inverse solution of the hybrid mechanism;

S204, grouping according to the nearest neighbor rule, and assigning x _i to each cluster set φ _i (i=1,2,...,n) of the sample according to the Euclidean distance from the cluster center;

S205, then adjust the cluster center, and calculate the average value of the training samples in the cluster set φ _p as the new cluster center. If the new cluster center no longer changes, the obtained cluster center is the RBF neural network The final basis function center of , if not, return to the previous step and re-solve the cluster center;

u_max为所选取中心之间的最大距离；S206, solve the width of the basis function, since the basis function of the selected RBF neural network is a Gaussian function, the solution formula for the width of the Gaussian basis function

u _max is the maximum distance between the selected centers;

S207. Calculate the weight: the weight from the hidden layer to the output layer is calculated by the least square method, and the calculation formula is

S208. According to the inverse kinematics solution result, within a specified motion range, randomly select 5000 sampling points as training data of the neural network, of which 4800 sampling points are used as training samples, and the remaining 200 sampling points are used as test samples.

Preferably, the S3 specifically includes the following operation steps:

S301. Convert the problem of solving the forward kinematic equations of the mechanism, that is, the problem of solving the β pose z _C , α and β equation roots of the center of the terminal operating platform, into solving the equation F _i (hi )-F _i ( _{hi -1} )=F _i `(h _i-1 )(h _i -h _i-1 );

S302, solve the equation F _i (h _i )-F _i (h _i-1 )=F _i `(h _i-1 )(h _i -h _i-1 ) and convert it into matrix form to obtain

in

为XY-3-RPS混联抛光机构的雅克比矩阵；

It is the Jacobian matrix of the XY-3-RPS mixed 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 kinematic equation, record the difference value of each iteration |h _i -h _i-1 |, and judge the iterative difference value |h _i - Whether h _i-1 is less than the accuracy requirement ε, when |h _i -h _i-1 | is smaller than the accuracy requirement ε, the iterative value is output, and h _i at this time is the forward kinematic equation system F _i ( α, β, z _C ) = 0 approximate solution, otherwise, continue to iterate until the accuracy meets the output result.

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

1. The present invention utilizes the characteristics and advantages of the Newton iteration method and the RBF neural network algorithm, and uses the predicted value of the RBF neural network training as the iterative initial value of the Newton iteration to iteratively solve, so as to avoid the RBF neural network caused by the insufficient number of training samples. The accuracy is insufficient, and it can avoid the dependence of the Newton iteration method on the initial value of the iteration.

2. The invention has scientific and reasonable design and strong practicability, and can be widely used in the aspects of working space, singular configuration, error compensation and motion control of the hybrid robot.

The present invention will be described in further detail below with reference to the accompanying drawings and embodiments.

Description of drawings

Figure 1 is a flow chart of the present invention.

FIG. 2 is a schematic diagram of the structure of the XY-3-RPS hybrid polishing mechanism in the present invention.

Figure 3 is a geometric schematic diagram of Newton's iterative algorithm.

Figure 4 is a flow chart of the Newton iteration method.

Figure 5 is a structural diagram of the RBF neural network.

Figure 6 is the error curve before improvement.

FIG. 7 is an error curve of the improved embodiment.

FIG. 8 is a graph of the number of iterations of the improved embodiment.

FIG. 9 is an iterative time curve of the improved embodiment.

Detailed ways

As shown in Figure 1, the present invention includes the following operation steps:

S1. Using the closed-loop vector method, the rod length formula of the hybrid polishing mechanism is established in the absolute coordinate system, and the kinematic inverse solution equation is obtained. On the basis of the inverse solution equation, the nonlinear equation system is established as the forward kinematic equation;

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

S3. Use the estimated value obtained by neural network training as the initial iterative value to perform iterative calculation through the forward kinematics equation, record each iteration difference |h _i -h _i-1 |, and judge the iterative difference |h _i - Whether h _i-1 | is less than the accuracy requirement ε, if |h _i -h _i-1 | is less than the accuracy requirement ε, output the iteration value and end the process, otherwise, continue to iterate until the accuracy meets the output result.

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

S101. As shown in Figure 2, for the mechanism model of the XY-3-RPS hybrid polishing platform, draw a schematic structural diagram of the hybrid mechanism. A _i and P _i (i=1, 2, 3) are the rotation of the parallel platform respectively. The center of the pair and the ball pair, and the triangles ΔA ₁ A ₂ A ₃ and ΔP ₁ P ₂ P ₃ formed by this are represented as the fixed platform and the moving platform, respectively, and the radius of the circumscribed circle of the fixed platform is represented by R and r respectively. The branch chain is represented by the vector A _i P _i , the X-direction series platform is represented by the moving pair M, and the Y-direction series platform is represented by the moving pair N;

S102. For the structure of the XY-3-RPS hybrid polishing robot, establish a moving platform coordinate system {C ₁ }, a fixed platform coordinate system {C ₂ }, and an absolute coordinate system {C ₀ } respectively, and the coordinate origins are respectively the moving platform. The geometric center C ₁ , the geometric center C ₂ of the fixed platform, and the center C ₀ of the moving pair M;

S103, using the closed-loop vector method for kinematics analysis, the starting point of the motion loop is the origin C of the coordinate system of the fixed platform, first passes through the hinge point A _i of the driving rod and the fixed platform, and then passes through the hinge point P _i of the driving rod and the moving platform, The end point is the coordinate origin C ₁ of the moving platform coordinate system;

S104, 3-RPS parallel mechanism has a total of three branch chains, each of which is an independent closed motion loop, and each motion loop is represented by a vector form, CC ₁ +C ₁ P _i =CA _i +A _i P _i , after rewrite as

S105. Since each branch chain will be constrained by the rotation pair, and the unit vector _ji of the rotation pair axis is always perpendicular to the branch chain, the constraint equation of the branch chain can be obtained

S106: Obtain the expressions of x _c , y _c , γ about z _c , α, β,

y _C = -rcosβsinγ

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 the fourth-order homogeneous coordinates, that is,

S108. The fixed platform coordinate system only translates along the X and Y axes relative to the absolute coordinate system. Since the 3-RPS parallel platform moves as a whole with the Y platform of the series cross slide, the moving platform does not move along the X, Y axes relative to the fixed platform. The relative movement of the Y axis, therefore, the transformation matrix of the moving platform coordinate system relative to the absolute coordinate system is:

S109. The driving rod length l _i of the XY-3-RPS hybrid polishing mechanism can be obtained by substituting the formula into the solution and transformation, which is expressed as:

为点A_i在绝对坐标系下的位置；In the formula,

is the position of point A _i in the absolute coordinate system;

S110. Calculate the driving displacement x, y of the series cross slide and the parallel driving rod displacement _li to solve the pose z _C , α, β of the center of the operation platform at the end of the machine tool, and then obtain the rod length formula of the hybrid polishing mechanism

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

S201. Use the self-organizing center selection method to select the function center of the RBF neural network, select the Gaussian function as the basis function, and the function expression is

Among them, x=[x _i ] ^T (i=1,2,...,n) is the input of the neural network, the input parameter is the rod length value of the three driving rods of the XY-3-RPS hybrid polishing machine, and n represents the training sample The number of , 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 hidden layer nodes; u=[u _j ] ^T is the hidden layer The center vector of the jth neuron basis function of the layer, v=[v _j ] ^T is the width of the jth element basis function of the hidden layer;

其中w＝[w_m]^T是隐含层到输出层的权值，y_m为神经网络的实际输出；S202. Determine the output of the RBF neural network as

Where w=[w _m ] ^T is the weight from the hidden layer to the output layer, and y _m is the actual output of the neural network;

S203, initialize the network first, randomly select j trainings as the cluster center u _j in the kinematic inverse solution of the hybrid mechanism;

S204, grouping according to the nearest neighbor rule, and assigning x _i to each cluster set φ _i (i=1,2,...,n) of the sample according to the Euclidean distance from the cluster center;

S205, then adjust the cluster center, and calculate the average value of the training samples in the cluster set φ _p as the new cluster center. If the new cluster center no longer changes, the obtained cluster center is the RBF neural network The final basis function center of , if not, return to the previous step and re-solve the cluster center;

u_max为所选取中心之间的最大距离；S206, solve the width of the basis function, since the basis function of the selected RBF neural network is a Gaussian function, the solution formula for the width of the Gaussian basis function

u _max is the maximum distance between the selected centers;

S207. Calculate the weight: the weight from the hidden layer to the output layer is calculated by the least square method, and the calculation formula is

S208. According to the inverse kinematics solution result, within a specified motion range, randomly select 5000 sampling points as training data of the neural network, of which 4800 sampling points are used as training samples, and the remaining 200 sampling points are used as test samples.

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

S301. Convert the problem of solving the forward kinematic equations of the mechanism, that is, the problem of solving the β pose z _C , α and β equation roots of the center of the terminal operating platform, into solving the equation F _i (hi )-F _i ( _{hi -1} )=F _i `(h _i-1 )(h _i -h _i-1 );

S302, solve the equation F _i (h _i )-F _i (h _i-1 )=F _i `(h _i-1 )(h _i -h _i-1 ) and convert it into matrix form to obtain

in

为XY-3-RPS混联抛光机构的雅克比矩阵；

It is the Jacobian matrix of the XY-3-RPS mixed 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 kinematic equation, record the difference value of each iteration |h _i -h _i-1 |, and judge the iterative difference value |h _i - Whether h _i-1 is less than the accuracy requirement ε, when |h _i -h _i-1 | is smaller than the accuracy requirement ε, the iterative value is output, and h _i at this time is the forward kinematic equation system F _i ( α, β, z _C ) = 0 approximate solution, otherwise, continue to iterate until the accuracy meets the output result.

Carry out experimental tests to verify the algorithm:

The theoretical methods used in this scheme are:

(1) Closed-loop vector method, link kinematics analysis based on closed-loop vector method When the vector method is used to establish the position equation of the mechanism, the rod must be represented by a vector, and the closed vector polygon of the mechanism is made. The sum of the vectors must be equal to zero. The number of branches of the parallel mechanism, that is, the number of independent closed motion rings. The motion ring has been shown with a red line. The starting point and end point of the motion loop are the origin of the coordinate system of the fixed and static platforms, passing through the hinge point between the platform and the fixed and static platform. The structure diagram of this method is shown in Figure 2.

(2) As shown in Figure 3 and Figure 4, the Newton iteration method is one of the most widely used algorithms in numerical methods. The basic idea of its solution is to gradually convert nonlinear equations into linear equations. The initial value, using a finite number of iterations, to obtain an approximate solution. The essence of the forward kinematic equations of the hybrid polishing mechanism is to solve a set of implicit nonlinear equations. Solving such problems can often only be approximated by numerical methods.

作为基函数，其中，x＝[x_i]^T(i＝1,2,…,n)为神经网络的输入，输入参数为XY-3-RPS混联抛光机床三根驱动杆的杆长值，n表示训练样本的数量。输入层与隐含层的连接权值为1。h＝[h_j]^T为神经网络隐含层的输出，j为隐含层节点数；u＝[u_j]^T是隐含层第j个神经元基函数的中心向量，v＝[v_j]^T为隐含层第j个神经元基函数的宽度也称作为方差，此方法对应的从输入层到隐含层再到输出层对应的RBF神经网络结构图如图5所示。(3) Self-organizing center selection method. According to different radial basis function center selection methods, RBF network has a variety of learning methods, such as random selection center method, self-organizing center selection method, supervised center selection method and orthogonal least squares method. Multiplication, etc. The method consists of two stages: one is the self-organized learning stage, this stage is the unsupervised learning process, and the center and variance of the underlying function of the hidden layer are solved; the second is the tutored learning stage, which solves the hidden layer to the output layer. The weight between , choose Gaussian function here

As the basis function, where x=[x _i ] ^T (i=1,2,...,n) is the input of the neural network, and the input parameter is the rod length value 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 basis function of the jth neuron in the hidden layer, v=[v _j ] ^T is the width of the basis function of the jth neuron in the hidden layer, also known as the variance. The corresponding RBF neural network structure diagram from the input layer to the hidden layer to the output layer corresponding to this method is shown in Figure 5.

Performance comparison of several algorithms:

The settings of the parameters of the algorithm in the theoretical method:

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

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

The error curve before improvement is shown in Figure 6, and the error curve after improvement is shown in Figure 7. The mean square error is shown in Table 1, the iteration number curve is shown in Figure 8, the iteration time curve is shown in Figure 9, and the iteration number and time before and after improvement are shown in Table 2.

Table 1 Mean Square Error

Table 2

The above descriptions are only preferred embodiments of the present invention, and do not limit the present invention in any way. Any simple modifications, changes and equivalent changes made to the above embodiments according to the technical essence of the invention still fall within the protection scope of the technical solutions of the present invention.

Claims (4)

1. a forward kinematics analysis method based on RBF neural network improvement Newton iterative algorithm, is characterized in that, comprises the following operation steps: S1. Using the closed-loop vector method, the rod length formula of the hybrid polishing mechanism is established in the absolute coordinate system, and the kinematic inverse solution equation is obtained. On the basis of the inverse solution equation, a nonlinear equation system is established as the forward kinematic equation; S2. Select the center of the RBF neural network, and select the Gaussian function as the basis function. According to the inverse solution result of kinematics, within the specified motion range, randomly select j sampling points 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 iterative value to perform iterative calculation through the forward kinematics equation, record each iteration difference |h _i -h _i-1 |, and judge the iterative difference |h _i - Whether h _i-1 | is less than the accuracy requirement ε, if |h _i -h _i-1 | is less than the accuracy requirement ε, output the iteration value and end the process, otherwise, continue to iterate until the accuracy meets the output result. 2. a kind of forward kinematics analysis method based on RBF neural network improvement Newton iterative algorithm according to claim 1, is characterized in that, described S1 specifically comprises the following operation steps: S101. Draw a schematic structural diagram of the hybrid mechanism according to the mechanism model of the hybrid polishing platform. A _i and P _i (i=1, 2, 3) are the centers of the rotating pair and the ball pair of the parallel platform, respectively. The triangles ΔA ₁ A ₂ A ₃ and ΔP ₁ P ₂ P ₃ are respectively represented as a fixed platform and a moving platform, the circumradius of the fixed platform is represented by R and r respectively, and each branch is represented by a vector A _i P _i , The X-direction series platform is represented by the moving pair M, and the Y-direction series platform is represented by the moving pair N; S102. For the structure of the hybrid polishing robot, establish a moving platform coordinate system {C ₁ }, a fixed platform coordinate system {C ₂ }, and an absolute coordinate system {C ₀ } respectively, and the coordinate origins are the geometric center C ₁ , The geometric center C ₂ of the fixed platform and the center C ₀ of the moving pair M; S103, using the closed-loop vector method for kinematics analysis, the starting point of the motion loop is the origin C of the coordinate system of the fixed platform, first passes through the hinge point A _i of the driving rod and the fixed platform, and then passes through the hinge point P _i of the driving rod and the moving platform, The end point is the coordinate origin C ₁ of the moving platform coordinate system; S104. The parallel mechanism has a total of three branch chains, each of which is an independent closed motion loop, and each motion loop is expressed in vector form as CC ₁ +C ₁ P _i =CA _i +A _i P _i , and rewritten as

S105. Each branch chain will be constrained by the rotation pair, and the unit vector _ji of the rotation pair axis is always perpendicular to the branch chain, so the constraint equation of the branch chain can be obtained

S106: Obtain the expressions of x _c , y _c , γ about z _c , α, β,

y _C = -r cosβsinγ

S107. Limited to the third-order matrix, it can only represent pure rotational motion. In order to represent the translational motion of the hybrid mechanism, the expressions of x _c , y _c , and γ about z _c , α, and β are generalized to the fourth-order homogeneous coordinates, which is

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. The transformation matrix is:

S109. The driving rod length l _i of the XY-3-RPS hybrid polishing mechanism can be obtained by substituting the formula into the solution and transformation, which is expressed as:

In the formula,

is the position of point A _i in the absolute coordinate system; S110, the drive displacement of the series cross slide is x, y, the parallel drive rod displacement is _li , solve the pose z _C , α, β of the center of the operation platform at the end of the machine tool, and then obtain the rod length formula of the hybrid polishing mechanism

3. a kind of forward kinematics analysis method based on RBF neural network improvement Newton iterative algorithm according to claim 1 is characterized in that, described S2 specifically comprises the following operation steps: S201. Use the self-organizing center selection method to select the function center of the RBF neural network, select the Gaussian function as the basis function, and the function expression is

Among them, x=[x _i ] ^T (i=1,2,...,n) is the input of the neural network, the input parameter is the rod length value 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 jth hidden layer The center vector of the neuron basis functions, v=[v _j ] ^T is the width of the jth element basis function of the hidden layer; S202. Determine the output of the RBF neural network as

Where w=[w _m ] ^T is the weight from the hidden layer to the output layer, and y _m is the actual output of the neural network; S203, initialize the network first, randomly select j trainings as the cluster center u _j in the kinematic inverse solution of the hybrid mechanism; S204, grouping according to the nearest neighbor rule, and assigning x _i to each cluster set φ _i (i=1,2,...,n) of the sample according to the Euclidean distance from the cluster center; S205. Adjust the cluster center, and calculate the average value of the training samples in the cluster set φ _p as the new cluster center. If the new cluster center no longer changes, the obtained cluster center is the value of the RBF neural network. The final basis function center, if the new cluster center changes, return to the previous step and re-solve the cluster center; S206, basis function width solution: the solution formula for the Gaussian basis function width is:

u _max is the maximum distance between the selected centers; S207. Calculate the weight: the weight from the hidden layer to the output layer is calculated by the least square method, and the calculation formula is

S208. According to the inverse kinematics solution result, within a specified motion range, randomly select 5000 sampling points as training data of the neural network, of which 4800 sampling points are used as training samples, and the remaining 200 sampling points are used as test samples. 4. a kind of forward kinematics analysis method based on RBF neural network improvement Newton iterative algorithm according to claim 1, is characterized in that, described S3 specifically comprises following operation steps: S301. Convert the problem of solving the forward kinematic equations of the mechanism, that is, the problem of solving the β pose z _C , α and β equation roots of the center of the terminal operating platform, into solving the equation F _i (hi )-F _i ( _{hi -1} )=F _i `(h <sub>i-1</sub> )(h <sub>i</sub> -h <sub>i-1</sub> ); S302, solve the equation F <sub>i</sub> (h <sub>i</sub> )-F <sub>i</sub> (h <sub>i-1</sub> )=F <sub>i</sub> `(h _i-1 )(h _i -h _i-1 ) and convert it into matrix form to obtain

in

It is the Jacobian matrix of the XY-3-RPS mixed 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 kinematic equation, record the difference value of each iteration |h _i -h _i-1 |, and judge the iterative difference value |h _i - Whether h _i-1 | is less than the accuracy requirement ε, when |h _i -h _i-1 | is smaller than the accuracy requirement ε, the iterative value is output, and h _i at this time is the forward kinematic equation system F _i of the hybrid mechanism (α,β,z _C )=0 approximate solution, otherwise, continue to iterate until the accuracy meets the output result.

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