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

RBF neural network-based forward kinematics analysis method for improving Newton iterative algorithm Download PDF

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CN115098978A
CN115098978A CN202210876632.9A CN202210876632A CN115098978A CN 115098978 A CN115098978 A CN 115098978A CN 202210876632 A CN202210876632 A CN 202210876632A CN 115098978 A CN115098978 A CN 115098978A
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宋孝宗
付海涛
王笑荣
包振良
贺亚洲
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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

RBF neural network-based forward kinematics analysis method for improving Newton iterative algorithm
Technical Field
The invention belongs to the technical field of calculation models, and particularly relates to a forward kinematics analysis method based on an RBF neural network improved Newton iteration algorithm.
Background
The hybrid robot is used as a machine tool and needs to control the pose of a machining tool head in real time in the machining process so as to ensure the surface quality of a free-form surface, and the motion control is established on the basis of forward kinematics analysis of a mechanism. Therefore, it becomes important to perform forward kinematic analysis on the parallel-serial polishing robot. However, the problems of difficult solution, low solution efficiency and the like exist in the kinematic positive solution of the series-parallel mechanism, and an accurate and reliable kinematic solution calculation method of the series-parallel mechanism needs to be provided for the problems.
The Newton iteration method is an important method for solving a nonlinear equation set, the nonlinear equation set is gradually converted into a linear equation to be solved by giving a specific iteration initial value, and the positive solution principle of the kinematics is that the Newton iteration solves the nonlinear equation set. However, this method relies on the selection of the initial value, and if the initial value is not properly selected, the iterative result is not converged.
An artificial neural network is an information computation model by mimicking the basic features and structural theory of a biological neural network. The basic unit is neuron, the neurons are connected in parallel to form a neural network, and the system has self-learning capability similar to biological brain by simulating the information processing mode of the biological brain and the interaction of the neurons. In a neural network, a neuron can receive a plurality of input signals, obtain an output signal by a specific processing mode, and transmit the signal to other neurons by a nonlinear mode. Therefore, the neural network has high precision and strong robustness for controlling an uncertain system by virtue of the functional approximation capability of the neural network with any precision. Due to the simple structure of the feedforward neural network, the feedforward neural network is widely used in the neural network, and for example, a BP neural network, an RBF neural network, a multilayer perceptron neural network and the like belong to feedforward type networks. 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 process the regularity which is difficult to analyze in the system, has good generalization capability and fast learning convergence speed, and has been successfully applied to nonlinear function approximation, time sequence analysis, data classification, mode identification, information processing, image processing, system modeling, control, fault diagnosis and the like.
Disclosure of Invention
The invention aims to solve the technical problem of providing a forward kinematics analysis method based on an RBF neural network improved Newton iteration algorithm aiming at the defects of the prior art, the calculation method utilizes the predicted value of RBF neural network training as the iteration initial value of Newton iteration to carry out iteration solution, not only can the insufficient precision of the RBF neural network caused by the insufficient number of training samples be avoided, but also the dependence of the Newton iteration method on the iteration initial value can be avoided, and the method has the advantages of scientific and reasonable design, strong practicability, high calculation precision and high calculation efficiency.
In order to solve the technical problems, the technical scheme adopted by the invention is as follows: a forward kinematics analysis method based on an RBF neural network improved Newton iterative algorithm is characterized by comprising the following operation steps:
s1, establishing a rod length formula of the series-parallel polishing mechanism under an absolute coordinate system by using a closed-loop vector method to obtain a kinematic inverse solution equation, and establishing a nonlinear equation set as a forward kinematic equation on the basis of the inverse solution equation;
s2, selecting an RBF neural network center, selecting a Gaussian function as a basis function, and randomly selecting j sampling points as training data of the neural network in a specified motion range according to the inverse solution result of kinematics, wherein most sampling points are used as training samples, and the rest few sampling points are used as test samples;
s3, taking the estimated value obtained by the neural network training as an iteration initial value, performing iterative calculation through a forward kinematics equation, and recording each iteration difference | h i -h i-1 And judging an iterative difference value | h i -h i-1 If | is less than the precision requirement ε, if | h i -h i-1 If | is smaller than the precision requirement epsilon, outputting an iteration value and ending the process, otherwise, continuing the iteration until the precision meets the output result.
Preferably, the S1 specifically includes the following steps:
s101, drawing a structural sketch of a series-parallel mechanism aiming at the XY-3-RPS series-parallel polishing platform mechanism model, A i And P i (i is 1,2,3) is the center of the rotating pair and the ball pair of the parallel platform respectively, and the triangle delta A is formed by the centers 1 A 2 A 3 And Δ P 1 P 2 P 3 Respectively expressed as a fixed platform and a movable platform, the radiuses of circumscribed circles of the fixed platform are respectively expressed by R and R, and each branched chain is expressed by a vector A i P i The X-direction tandem platform is represented by a moving pair M, and the Y-direction tandem platform is represented by a moving pair N;
s102, respectively establishing a movable platform coordinate system { C) for the structure of the XY-3-RPS series-parallel polishing robot 1 The coordinate system of the fixed platform { C } 2 Absolute coordinate system { C } 0 The origin of coordinates is the geometric center C of the movable platform respectively 1 Geometric center C of fixed platform 2 Center C of sliding pair M 0 ；
S103, performing kinematic analysis by adopting a closed loop vector method, wherein the starting point of a motion loop is the original point C of a fixed platform coordinate system, and the motion loop firstly passes through a hinge point A of a driving rod and a fixed platform i Then passes through the hinge point P of the driving rod and the movable platform i The end point is the coordinate origin C of the coordinate system of the movable platform 1 ；
The S104 and 3-RPS parallel mechanism has three branches in totalChains, each being an independent closed motion loop, each motion loop being represented in vector form, CC 1 +C 1 P i ＝CA i +A i P i Is rewritten as
Figure BDA0003762421300000031
S105, each branched chain is restrained by a rotating pair, and the unit vector j of the axis of the rotating pair i Is always vertical to the branched chain, so the constraint equation of the branched chain can be obtained
Figure BDA0003762421300000032
S106: to obtain x c 、y c Gamma with respect to z c The expressions of alpha and beta are shown in the specification,
Figure BDA0003762421300000033
y C ＝-rcosβsinγ
Figure BDA0003762421300000034
s107, since the 3 rd order matrix can only represent pure rotation motion, in order to represent the translation motion of the hybrid mechanism, the 3 rd order matrix needs to be generalized into 4 th order homogeneous coordinates, namely
Figure BDA0003762421300000035
S108, the fixed platform coordinate system only translates along the X, Y axial direction relative to the absolute coordinate system, and the 3-RPS parallel platform moves integrally along the Y platform of the serial cross sliding platform, so that the movable platform does not move relatively to the fixed platform along the X, Y axial direction, and therefore, the transformation matrix of the movable platform coordinate system relative to the absolute coordinate system is as follows:
Figure BDA0003762421300000041
Figure BDA0003762421300000042
s109, solving and converting by substituting a formula to obtain the length l of the driving rod of the XY-3-RPS series-parallel polishing mechanism i Expressed as:
Figure BDA0003762421300000043
in the formula (I), the compound is shown in the specification,
Figure BDA0003762421300000044
is point A i A position in an absolute coordinate system;
s110, driving displacements x and y of the series-connection cross sliding table and parallel-connection driving rod displacement l i Solving the pose z of the center of the machine tool tail end operating platform C Alpha and beta to obtain the rod length formula of the series-parallel polishing mechanism
Figure BDA0003762421300000045
Preferably, the S2 specifically includes the following steps:
s201, selecting a RBF neural network function center by adopting a self-organizing center selection method, selecting a Gaussian function as a basis function, and obtaining a function expression of
Figure BDA0003762421300000046
Wherein x is [ x ] i ] T (i is 1,2, …, n) is input into the neural network, the input parameter is the length of the three driving rods of the XY-3-RPS series-parallel polishing machine, n represents the number of training samples, the connection weight of the input layer and the hidden layer is 1, h is [ h ═ h [, n ] is j ] T Is the output of the hidden layer of the neural network, j is the hidden layerThe number of nodes; u ═ u j ] T Is the central vector of the base function of the jth neuron in the hidden layer, v ═ v j ] T Width of the jth primitive function of the hidden layer;
s202, determining the output of the RBF neural network as
Figure BDA0003762421300000051
Wherein w ═ w m ] T Is the weight, y, from hidden layer to output layer m Is the actual output of the neural network;
s203, network initialization is firstly carried out, and j training numbers are randomly selected as clustering centers u in the inverse kinematics solution of the series-parallel mechanism j ；
S204, grouping the x according to the nearest neighbor rule i Respective cluster sets phi assigned to the samples according to Euclidean distances from the cluster center i (i-1, 2, …, n);
s205, then adjusting the clustering center and calculating a clustering set phi p Taking the average value of the middle training sample as a new clustering center, if the new clustering center does not change any more, obtaining the clustering center which is the final basis function center of the RBF neural network, if not, returning to the previous step, and solving the clustering center again;
s206, solving the width of the basis function, wherein the selected basis function of the RBF neural network is a Gaussian function, and the solving formula of the width of the Gaussian basis function
Figure BDA0003762421300000052
u max The maximum distance between the selected centers;
s207, calculating a weight: the weight from the hidden layer to the output layer is calculated by a least square method, and the calculation formula is
Figure BDA0003762421300000053
And S208, according to the inverse solution result of the kinematics, randomly selecting 5000 sampling points in a specified movement range as training data of the neural network, wherein 4800 sampling points are used as training samples, and the rest 200 sampling points are used as test samples.
Preferably, the S3 specifically includes the following steps:
s301, solving a forward kinematics equation set of a solving mechanism, namely solving the beta pose z of the center of the terminal operating platform C The problem of the root of the alpha and beta equations is converted into a solution equation F i (h i )-F i (h i-1 )＝F i `(h i-1 )(h i -h i-1 )；
S302, solving equation F i (h i )-F i (h i-1 )＝F i `(h i-1 )(h i -h i-1 ) Conversion into matrix form
Figure BDA0003762421300000061
Wherein
Figure BDA0003762421300000062
Is a Jacobian matrix of the XY-3-RPS series-parallel polishing mechanism;
s303, taking an estimated value of test sample training as an initial value of Newton iteration to carry out iterative computation through a positive kinematic equation, and recording an iterative difference | h of each iteration i -h i-1 And judging an iterative difference value | h i -h i-1 Whether less than precision requirement epsilon, when | h i -h i-1 When | is less than the precision requirement epsilon, outputting an iteration value, and h at the moment i Namely a forward kinematics equation set F of the series-parallel mechanism i (α,β,z C ) And if the result is not the same as the output result, the iteration is continued until the precision meets the output result.
Compared with the prior art, the invention has the following advantages:
1. the method utilizes the characteristics and advantages of the Newton iteration method and the RBF neural network algorithm, and utilizes the predicted value of the RBF neural network training as the iteration initial value of the Newton iteration to carry out iteration solution, thereby not only avoiding the insufficient precision of the RBF neural network caused by the insufficient number of training samples, but also avoiding the dependency of the Newton iteration method on the iteration initial value.
2. The invention has scientific and reasonable design and strong practicability, and can be widely applied to the aspects of working space, singular configuration, error compensation, motion control and the like of the series-parallel robot.
The present invention will be described in further detail with reference to the accompanying drawings and examples.
Drawings
FIG. 1 is a flow chart of the present invention.
FIG. 2 is a schematic diagram of the XY-3-RPS series-parallel polishing mechanism of the present invention.
Figure 3 is a geometric schematic of a newton iterative algorithm.
FIG. 4 is a block diagram of a Newton's iterative method flow.
Fig. 5 is a diagram of the RBF neural network architecture.
Fig. 6 is an error curve before improvement.
Fig. 7 is an error curve of the present embodiment after modification.
Fig. 8 is a modified iteration number curve of the present embodiment.
Fig. 9 is an improved iteration time curve of the present embodiment.
Detailed Description
As shown in fig. 1, the present invention comprises the following operation steps:
s1, establishing a rod length formula of the series-parallel polishing mechanism under an absolute coordinate system by using a closed-loop vector method to obtain a kinematic inverse solution equation, and establishing a nonlinear equation set as a forward kinematic equation on the basis of the inverse solution equation;
s2, selecting an RBF neural network center, selecting a Gaussian function as a basis function, and randomly selecting j sampling points as training data of the neural network in a specified motion range according to the inverse solution result of kinematics, wherein most sampling points are used as training samples, and the rest few sampling points are used as test samples;
s3, taking the estimated value obtained by the neural network training as an iteration initial value to carry out iterative computation through a forward kinematics equation, and recording eachSub-iteration difference | h i -h i-1 And judging an iterative difference value | h i -h i-1 Whether | is less than the precision requirement epsilon, if | h i -h i-1 If | is smaller than the precision requirement epsilon, outputting an iteration value and ending the process, otherwise, continuing the iteration until the precision meets the output result.
In this embodiment, the S1 specifically includes the following operation steps:
s101, as shown in figure 2, aiming at the XY-3-RPS parallel-serial polishing platform mechanism model, drawing a structural sketch of a parallel-serial mechanism, A i And P i (i is 1,2,3) is the center of the rotating pair and the ball pair of the parallel platform respectively, and the triangle delta A is formed by the centers 1 A 2 A 3 And Δ P 1 P 2 P 3 Respectively expressed as a fixed platform and a movable platform, the radiuses of circumscribed circles of the fixed platform are respectively expressed by R and R, and each branched chain is expressed by a vector A i P i The X-direction tandem platform is represented by a moving pair M, and the Y-direction tandem platform is represented by a moving pair N;
s102, respectively establishing a movable platform coordinate system { C) for the structure of the XY-3-RPS series-parallel polishing robot 1 The coordinate system of the fixed platform { C } 2 Absolute coordinate system { C } 0 The origin of coordinates is the geometric center C of the movable platform respectively 1 Geometric center C of fixed platform 2 Center C of sliding pair M 0 ；
S103, performing kinematic analysis by adopting a closed loop vector method, wherein the starting point of a motion loop is the original point C of a fixed platform coordinate system, and the motion loop firstly passes through a hinge point A of a driving rod and a fixed platform i Then passes through the hinge point P of the driving rod and the movable platform i The end point is the coordinate origin C of the coordinate system of the movable platform 1 ；
The S104 and 3-RPS parallel mechanism has three branched chains in total, each branched chain is an independent closed motion ring, each motion ring is represented in a vector form, and CC 1 +C 1 P i ＝CA i +A i P i Is rewritten as
Figure BDA0003762421300000081
S105、Because each branched chain is restrained by a revolute pair, and the unit vector j of the axis of the revolute pair i Always perpendicular to the branched chain, so that the constraint equation of the branched chain can be obtained
Figure BDA0003762421300000082
S106: to obtain x c 、y c Gamma with respect to z c The expressions of alpha and beta are shown in the specification,
Figure BDA0003762421300000083
y C ＝-rcosβsinγ
Figure BDA0003762421300000084
s107, since the 3 rd order matrix can only represent pure rotation motion, in order to represent translation motion of the hybrid mechanism, the 3 rd order matrix needs to be generalized into 4 th order homogeneous coordinates, namely
Figure BDA0003762421300000085
S108, the fixed platform coordinate system only translates along the X, Y axial direction relative to the absolute coordinate system, and because the 3-RPS parallel platform moves along the Y platform of the serial cross sliding platform integrally, the movable platform does not move along the X, Y axial direction relative to the fixed platform, therefore, the transformation matrix of the movable platform coordinate system relative to the absolute coordinate system is:
Figure BDA0003762421300000091
Figure BDA0003762421300000092
s109, solving and converting by substituting a formula to obtain XY-3-RPS series-parallel throwingDrive rod length l of optical mechanism i Expressed as:
Figure BDA0003762421300000093
in the formula (I), the compound is shown in the specification,
Figure BDA0003762421300000094
is point A i A position in an absolute coordinate system;
s110, driving displacement x and y of the series cross sliding table and parallel driving rod displacement l i Solving the pose z of the machine tool end operating platform center C Alpha and beta to obtain the rod length formula of the series-parallel polishing mechanism
Figure BDA0003762421300000095
In this embodiment, the S2 specifically includes the following operation steps:
s201, selecting a RBF neural network function center by adopting a self-organizing center selection method, selecting a Gaussian function as a basis function, and obtaining a function expression of
Figure BDA0003762421300000096
Wherein x is [ x ] i ] T (i ═ 1,2, …, n) is input into the neural network, the input parameter is the length of the driving rod of XY-3-RPS series-parallel polishing machine, n represents the number of training samples, the connection weight of the input layer and the hidden layer is 1, h ═ h j ] T J is the output of the hidden layer of the neural network, and is the node number of the hidden layer; u ═ u j ] T Is the central vector of the base function of the jth neuron in the hidden layer, v ═ v j ] T Width of the jth primitive function of the hidden layer;
s202, determining the output of the RBF neural network as
Figure BDA0003762421300000097
Wherein w ═ w m ] T Is the weight, y, from hidden layer to output layer m Is the actual output of the neural network;
s203, network initialization is firstly carried out, and j training numbers are randomly selected as clustering centers u in the inverse kinematics solution of the series-parallel mechanism j ；
S204, grouping the x according to the nearest neighbor rule i Respective cluster sets phi assigned to the samples according to Euclidean distances from the cluster center i (i-1, 2, …, n);
s205, then adjusting the clustering center and calculating a clustering set phi p Taking the average value of the middle training sample as a new clustering center, if the new clustering center does not change any more, obtaining the clustering center which is the final basis function center of the RBF neural network, if not, returning to the previous step, and solving the clustering center again;
s206, solving the width of the basis function, wherein the selected basis function of the RBF neural network is a Gaussian function, and the solving formula of the width of the Gaussian basis function
Figure BDA0003762421300000101
u max The maximum distance between the selected centers;
s207, calculating a weight: the weight from the hidden layer to the output layer is calculated by a least square method, and the calculation formula is
Figure BDA0003762421300000102
And S208, according to the inverse solution result of the kinematics, randomly selecting 5000 sampling points in a specified movement range as training data of the neural network, wherein 4800 sampling points are used as training samples, and the rest 200 sampling points are used as test samples.
In this embodiment, the S3 specifically includes the following operation steps:
s301, solving a forward kinematics equation set of a solving mechanism, namely solving the beta pose z of the center of the terminal operating platform C Problem transformation of alpha and beta equation rootsTo solve equation F i (h i )-F i (h i-1 )＝F i `(h i-1 )(h i -h i-1 )；
S302, solving equation F i (h i )-F i (h i-1 )＝F i `(h i-1 )(h i -h i-1 ) Conversion into matrix form
Figure BDA0003762421300000103
Wherein
Figure BDA0003762421300000111
Is a Jacobian matrix of the XY-3-RPS series-parallel polishing mechanism;
s303, taking an estimated value of test sample training as an initial value of Newton iteration to carry out iterative computation through a positive kinematic equation, and recording an iterative difference | h of each iteration i -h i-1 And judging an iterative difference value | h i -h i-1 Whether less than precision requirement epsilon, when | h i -h i-1 When | is less than the precision requirement epsilon, outputting an iteration value, and h at the moment i Namely a forward kinematics equation set F of the series-parallel mechanism i (α,β,z C ) And if the result is equal to the approximate solution of 0, otherwise, the iteration is continued until the precision meets the output result.
Experimental tests were performed to verify the algorithm:
the theoretical method used in the scheme comprises the following steps:
(1) a closed-loop vector method is characterized in that when a position equation of a mechanism is established by using the vector method based on the link kinematic analysis of the closed-loop vector method, a rod piece needs to be represented by a vector, a closed vector polygon of the mechanism is made, and the sum of the vectors is necessarily equal to zero. The number of the branched chains of the parallel mechanism is the number of independent closed moving rings. The kinematic rings have been indicated with red lines. The starting point and the end point of the moving ring are the original points of the fixed platform coordinate system and the static platform coordinate system, and the structural sketch embodied by the method is shown in figure 2 after passing through the hinge point of the platform and the fixed platform.
(2) As shown in fig. 3 and 4, the newton iteration method is one of the most widely used algorithms in the numerical method, and the basic idea of solution is to gradually convert a nonlinear equation set into a linear equation, and obtain an approximate solution by giving a specific initial value of iteration and using a finite number of iterations. The nature of the positive kinematic equation of the series-parallel polishing mechanism is to solve a set of implicit nonlinear equations. Solving such problems often only allows an approximate solution to be obtained by numerical methods.
(3) The RBF network has a plurality of learning methods, such as a random selection center method, a self-organization selection center method, a supervised selection center method, an orthogonal least square method and the like, according to different radial basis function center selection methods. The method consists of two stages: the method comprises the following steps that firstly, a self-organizing learning stage is a guiding-free learning process, and the center and the variance of a hidden layer basis function are solved; the second is a teacher learning stage, which solves the weight from the hidden layer to the output layer, and selects the Gaussian function
Figure BDA0003762421300000121
As a basis function, where x ═ x i ] T And (i is 1,2, …, n) is input into the neural network, the input parameter is the rod length value of the three driving rods of the XY-3-RPS series-parallel polishing machine tool, and n represents the number of training samples. The connection weight of the input layer and the hidden layer is 1. h ═ h j ] T J is the output of the hidden layer of the neural network, and j is the number of nodes of the hidden layer; u ═ u j ] T Is the central vector of the base function of the jth neuron in the hidden layer, v ═ v j ] T For the width of the jth neuron basis function of the hidden layer also called variance, the structure diagram of the corresponding RBF neural network from the input layer to the hidden layer to the output layer is shown in FIG. 5.
Performance comparisons for several algorithms:
the theoretical method comprises the following steps of setting parameters of an algorithm:
as shown in FIG. 5, because a large amount of sample data is needed for training the RBF neural network, multiple sets of inverse solutions are randomly selected within the motion range of the XY-3-RPS hybrid polishing machine tool for training the neural network. The training precision of the neural network is related to the number of samples, the more the number of samples is, the higher the precision of the training result is, and the slower the corresponding running speed is; conversely, if the number of samples is small, the accuracy of the training result is also low. Therefore, the RBF neural network is often limited by the number of samples, and the accuracy of the result 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, and selecting an appropriate initial value can reduce the iteration times and time of the algorithm to a great extent, and can obtain a high-precision result more quickly, wherein each parameter of the algorithm is set as follows:
in a specified motion range, 5000 sampling points are randomly selected to serve as training data of the neural network, 4800 sampling points serve as training samples, the rest 200 sampling points serve as test samples, an estimated value of test sample training serves as an initial value of Newton iteration, iterative calculation is carried out through a positive kinematic equation, and each iteration difference value is recorded.
The error curve before the improvement is obtained is shown in fig. 6, and the error curve after the improvement is shown in fig. 7. The mean square error is shown in table 1, the iteration number curve is shown in fig. 8, the iteration time curve is shown in fig. 9, and the iteration number and time before and after improvement are shown in table 2.
Figure BDA0003762421300000131
TABLE 1 mean square error
Figure BDA0003762421300000132
TABLE 2
The above description is only for the preferred embodiment of the present invention, and is not intended to limit the present invention in any way. Any simple modification, change and equivalent changes of the above embodiments according to the technical essence of the invention are still within the protection scope of the technical solution of the invention.

Claims (4)

1. A forward kinematics analysis method based on an RBF neural network improved Newton iterative algorithm is characterized by comprising the following operation steps:
s1, establishing a rod length formula of the series-parallel polishing mechanism under an absolute coordinate system by using a closed-loop vector method to obtain a kinematic inverse solution equation, and establishing a nonlinear equation set as a forward kinematic equation on the basis of the inverse solution equation;
s2, selecting an RBF neural network center, selecting a Gaussian function as a basis function, randomly selecting j sampling points as training data of the neural network within a specified motion range according to an inverse solution result of kinematics, and dividing the j sampling points into training samples and testing samples;
s3, taking the estimated value obtained by the neural network training as an iteration initial value, performing iterative calculation through a forward kinematics equation, and recording each iteration difference | h i -h i-1 And judging an iteration difference value | h i -h i-1 Whether | is less than the precision requirement epsilon, if | h i -h i-1 If | is smaller than the precision requirement epsilon, outputting an iteration value and ending the flow, otherwise, continuing the iteration until the precision meets the output result.
2. The forward kinematics analysis method based on the RBF neural network modified Newton' S iterative algorithm of claim 1, wherein the S1 specifically comprises the following operation steps:
s101, aiming at the model of the parallel-serial polishing platform mechanism, drawing a structural sketch of the parallel-serial mechanism, A i And P i (i is 1,2,3) is the center of the rotating pair and the ball pair of the parallel platform respectively, and the triangle delta A is formed by the centers 1 A 2 A 3 And Δ P 1 P 2 P 3 Respectively expressed as a fixed platform and a movable platform, the radiuses of circumscribed circles of the fixed platform are respectively expressed by R and R, and each branched chain is expressed by a vector A i P i The X-direction tandem platform is represented by a moving pair M, and the Y-direction tandem platform is represented by a moving pair N;
s102, pair-parallel connectionThe structure of the polishing robot is that a movable platform coordinate system { C is respectively established 1 The coordinate system of the fixed platform { C } 2 } absolute coordinate system { C 0 The origin of coordinates is the geometric center C of the movable platform respectively 1 Geometric center C of fixed platform 2 Center C of sliding pair M 0 ；
S103, performing kinematic analysis by adopting a closed loop vector method, wherein the starting point of a motion loop is the original point C of a fixed platform coordinate system, and the motion loop firstly passes through a hinge point A of a driving rod and a fixed platform i Then passes through the hinge point P of the driving rod and the movable platform i The end point is the coordinate origin C of the coordinate system of the movable platform 1 ；
S104, the parallel mechanism has three branched chains in total, each branched chain is an independent closed motion ring, and each motion ring is represented as CC in a vector form 1 +C 1 P i ＝CA i +A i P i And is rewritten as
Figure FDA0003762421290000028
S105, each branched chain is constrained by a revolute pair, and the unit vector j of the axis of the revolute pair i Always perpendicular to the branched chain, so that the constraint equation of the branched chain can be obtained
Figure FDA0003762421290000029
S106,: to obtain x c 、y c Gamma with respect to z c And the expressions of alpha and beta,
Figure FDA0003762421290000021
y C ＝-r cosβsinγ
Figure FDA0003762421290000022
s107, the movement is limited to 3-order matrix and only can represent pure rotation, and the translation of the series-parallel mechanism is representedMove, x c 、y c Gamma with respect to z c The expressions α, β are generalized to 4 th order homogeneous coordinates, i.e.
Figure FDA0003762421290000023
S108, the fixed platform coordinate system only translates along the X, Y axial direction relative to the absolute coordinate system, and the movable platform does not move relative to the fixed platform along the X, Y axial direction, so that a transformation matrix of the movable platform coordinate system relative to the absolute coordinate system is as follows:
Figure FDA0003762421290000024
Figure FDA0003762421290000025
s109, solving and converting by substituting a formula to obtain the length l of the driving rod of the XY-3-RPS series-parallel polishing mechanism i Expressed as:
Figure FDA0003762421290000026
in the formula (I), the compound is shown in the specification,
Figure FDA0003762421290000027
is point A i A position in an absolute coordinate system;
s110, the driving displacement of the serial cross sliding table is x and y, and the displacement of the parallel driving rod is l i Solving the pose z of the center of the machine tool tail end operating platform C Alpha and beta to obtain the rod length formula of the series-parallel polishing mechanism
Figure FDA0003762421290000031
3. The forward kinematics analysis method based on the RBF neural network modified Newton' S iterative algorithm of claim 1, wherein the S2 specifically comprises the following operation steps:
s201, selecting a RBF neural network function center by adopting a self-organizing center selection method, selecting a Gaussian function as a basis function, and obtaining a function expression of
Figure FDA0003762421290000032
Wherein x is [ x ] i ] T (i ═ 1, 2., n) is input of the neural network, the input parameter is the length value of three driving rods of the parallel-series polishing machine tool, n represents the number of training samples, the connection weight value of the input layer and the hidden layer is 1, h ═ h j ] T J is the output of the hidden layer of the neural network, and j is the number of nodes of the hidden layer; u ═ u j ] T Is the central vector of the base function of the jth neuron in the hidden layer, v ═ v j ] T Width of the jth primitive function of the hidden layer;
s202, determining the output of the RBF neural network as
Figure FDA0003762421290000033
Wherein w ═ w m ] T Is the weight, y, from hidden layer to output layer m Is the actual output of the neural network;
s203, network initialization is firstly carried out, and j training numbers are randomly selected as clustering centers u in the inverse kinematics solution of the series-parallel mechanism j ；
S204, grouping the x according to the nearest neighbor rule i Respective cluster sets phi assigned to the samples according to Euclidean distances from the cluster center i (i-1, 2, …, n);
s205, adjusting a clustering center, and calculating a clustering set phi p Taking the average value of the middle training sample as a new clustering center, and if the new clustering center is not changed any more, obtaining the clustering center which is the new clustering centerIf the new clustering center changes, returning to the previous step and solving the clustering center again;
s206, solving the width of the basis function: the solution formula of the Gaussian function width is
Figure FDA0003762421290000034
u max The maximum distance between the selected centers is taken;
s207, calculating a weight: the weight from the hidden layer to the output layer is calculated by a least square method, and the calculation formula is
Figure FDA0003762421290000035
And S208, according to the inverse solution result of the kinematics, randomly selecting 5000 sampling points as training data of the neural network in a specified motion range, wherein 4800 sampling points are used as training samples, and the rest 200 sampling points are used as test samples.
4. The forward kinematics analysis method based on the RBF neural network modified Newton' S iterative algorithm of claim 1, wherein the S3 specifically comprises the following operation steps:
s301, solving a forward kinematics equation set of a solving mechanism, namely solving the beta pose z of the center of the terminal operating platform C The problem of the root of the alpha and beta equations is converted into a solution equation F i (h i )-F i (h i-1 )＝F i `(h i-1 )(h i -h i-1 )；
S302, solving equation F i (h i )-F i (h i-1 )＝F i `(h i-1 )(h i -h i-1 ) Conversion into matrix form
Figure FDA0003762421290000041
Wherein
Figure FDA0003762421290000042
Is a Jacobian matrix of the XY-3-RPS series-parallel polishing mechanism;
s303, taking an estimated value of test sample training as an initial value of Newton iteration to carry out iterative computation through a positive kinematic equation, and recording an iterative difference | h of each iteration i -h i-1 And judging an iterative difference value | h i -h i-1 Whether | is less than the precision requirement epsilon, when | h i -h i-1 When | is less than the precision requirement epsilon, outputting an iteration value, and h at the moment i Namely a forward kinematics equation set F of the series-parallel mechanism i (α,β,z C ) And if the result is not the same as the output result, the iteration is continued until the precision meets the output result.
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