CN109783931B - Permanent magnet spherical motor electromagnetic torque modeling method based on Gaussian process regression - Google Patents
Permanent magnet spherical motor electromagnetic torque modeling method based on Gaussian process regression Download PDFInfo
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Abstract
The invention discloses a permanent magnet spherical motor electromagnetic torque modeling method based on Gaussian process regression, which comprises the following steps of: step one, establishing a spherical motor model in electromagnetic analysis software according to a mechanical structure of a permanent magnet spherical motor; step two, defining a Gaussian process in the function space; analyzing the experimental data according to the distribution rule of the permanent magnets of the motor rotor, and determining a kernel function as a self-correlation determined two-dimensional Gaussian kernel function; step four, obtaining the optimal hyperparameter of Gaussian process regression by adopting a maximum likelihood estimation method, and determining an electromagnetic torque model f of the permanent magnet spherical motor coil 1 1* (ii) a And step five, repeating the step one to the step four, determining the electromagnetic torque models of other coils, and finally determining the electromagnetic torque model of the permanent magnet spherical motor. The invention reduces the difficulty of modeling, reduces the time cost for acquiring data, and has universality for spherical motors with different structures.
Description
Technical Field
The invention belongs to the technical field of modeling of electromagnetic torque of a permanent magnet spherical motor, and particularly relates to a modeling method of the electromagnetic torque of the permanent magnet spherical motor based on Gaussian process regression.
Background
With the rapid development of the industrial manufacturing industry and the rapid rise of the labor cost, robots, mechanical arms, panoramic cameras and the like can realize wide needs and applications of high-precision servo motion devices in multidimensional space. The conventional device capable of realizing multi-degree-of-freedom motion is generally formed by combining a plurality of single-degree-of-freedom motors with a mechanical transmission mechanism. This construction results in, on the one hand, an increase in the volume and weight of the system and a reduction in the rigidity and mechanical reliability. On the other hand, the complex structure can cause slow response and poor dynamic performance of the motion control system. Therefore, researchers at home and abroad propose a spherical motor capable of realizing multi-degree-of-freedom motion. According to different working principles, the motor can be divided into an induction spherical motor, a variable reluctance spherical motor, a permanent magnet spherical motor and the like. Among them, the permanent magnet spherical motor receives much attention because of its simple structure, small, light in weight and so on the advantages.
The establishment of the electromagnetic torque model is an indispensable part in a permanent magnet spherical motor control system. The electromagnetic torque model is generally established by adopting a finite element method to carry out electromagnetic analysis, and for a conventional motor, a three-dimensional structure can be converted into a two-dimensional structure, and units are divided according to a periodic rule, so that the calculation amount is reduced, and the calculation process is simplified. However, the conventional method cannot be adopted due to the unique spherical structure of the permanent magnet spherical motor and the corresponding distribution of the coils and the permanent magnets. At present, there are two methods for establishing an electromagnetic torque model for a spherical motor. Firstly, three-dimensional modeling is carried out on a single coil and a permanent magnet, electromagnetic analysis is carried out by adopting a finite element method or an equivalent magnetic circuit method, and an electromagnetic torque model is further established; and secondly, establishing a complete spherical motor three-dimensional model, and performing electromagnetic analysis by using electromagnetic analysis software based on finite elements so as to establish an electromagnetic torque model. However, the above modeling method has the following drawbacks:
1. the modeling method aiming at the single coil and the permanent magnet has strong dependence on the shape, the magnetizing direction and the arrangement structure of the permanent magnet, and lacks universality on spherical motors with different structures.
2. Although the modeling method adopting the complete spherical motor three-dimensional model can meet the requirements of spherical motors with various structures, the calculation amount is very large, the calculation process is slow, and more time cost is consumed.
Disclosure of Invention
Aiming at the defects of the technology, the invention provides a permanent magnet spherical motor electromagnetic torque modeling method based on Gaussian process regression.
The invention adopts the following technical scheme for solving the technical problems:
a permanent magnet spherical motor electromagnetic torque modeling method based on Gaussian process regression comprises the following steps:
step one, a spherical motor model is established in electromagnetic analysis software according to a mechanical structure of a permanent magnet spherical motor, and electromagnetic torque generated by electrifying a coil 1 when a rotor rotates at different angles is obtained through random experiments and serves as experiment data. Defining an angle vector of rotor rotation as a model input, an electromagnetic torque vector generated by electrifying the coil 1 as a model output, and dividing the experimental data into a training set D { (x) i ，u i ) 1, n and test set
Wherein x is i And
angle vectors, u, of rotation of the rotors in the training set and in the test set, respectively i And
respectively electrifying the training set and the test concentrated coil to generate electromagnetic torque;
step two, defining a Gaussian process in the function space, and obtaining the posterior distribution f of the electromagnetic torque model for describing the coil 1 by utilizing the conditional distribution property of the Gaussian distribution 1* The description is as follows:
wherein X is all X in the training set i Constructed vector of u is all u in the training set i Constructed vector, X * Is to test all
The vector of the composition is then calculated,
and
are respectively f 1* Mean and variance of;
analyzing experimental data according to the distribution rule of the permanent magnet of the motor rotor, and determining a kernel function applicable to the Gaussian process regression-based modeling method as a two-dimensional Gaussian kernel function determined by autocorrelation;
step four, adopting a maximum likelihood estimation method, obtaining the optimal hyperparameter of the Gaussian process regression by utilizing the maximum value self-adaptive learning of the training sample negative log-edge likelihood function, and determining the electromagnetic torque model f of the permanent magnet spherical motor coil 1 1* Said hyper-parameter comprises a noise variance
Variance of signal
And a length gauge matrix Θ.
And step five, repeating the step one to the step four, determining the electromagnetic torque models of other coils, and finally determining the electromagnetic torque model of the permanent magnet spherical motor.
The spherical motor control system utilizes an electromagnetic torque model established by the permanent magnet spherical motor electromagnetic torque modeling method based on Gaussian process regression to control.
Compared with the prior art, the invention has the advantages that:
1. the invention adopts Gaussian process regression to carry out the electromagnetic torque modeling of the permanent magnet spherical motor without depending on a mechanism model. The method has universality for spherical motors with different structures, and reduces the difficulty of electromagnetic torque modeling of the spherical motors.
2. The invention adopts a random experiment method to obtain the training data, ensures the completeness of the training data on the basis of limited training data, and reduces the time cost for obtaining the data.
Drawings
FIG. 1 is a mechanical structure diagram of a spherical motor;
FIG. 2 is an electromagnetic analysis model of a spherical motor;
FIG. 3 is a diagram of the electromagnetic torque profile of coil 1;
fig. 4 is a comparison graph of the electromagnetic torque of the coil 1 and the actual electromagnetic torque test point.
Detailed Description
The invention is further illustrated with reference to the following figures and examples.
In the embodiment, the permanent magnet spherical motor is derived from a three-degree-of-freedom permanent magnet spherical motor prototype, as shown in fig. 1. The permanent magnet synchronous motor consists of a spherical rotor, a spherical shell-shaped stator and a rotor output shaft, wherein the cylindrical permanent magnet is divided into four layers which are uniformly embedded on the spherical surface of the rotor, 10 permanent magnets are distributed on each layer, and N-level and S-level permanent magnets are arranged in a staggered mode. The hollow cylindrical coil is divided into two layers which are uniformly embedded in the stator spherical shell, and each layer has 12 coils. When the spherical rotor rotates at different angles, the electromagnetic torque generated by the coil through unit current changes. The invention provides a permanent magnet spherical motor electromagnetic torque modeling method based on Gaussian process regression, which does not need to rely on a mechanism model to carry out electromagnetic torque modeling on a permanent magnet spherical motor and comprises the following steps:
step one, a spherical motor model is established in electromagnetic analysis software ANSYS electromagnetic properties Maxwell3D according to the mechanical structure of the permanent magnet spherical motor, as shown in FIG. 2. A spherical coordinate system is adopted to describe the space positions of the permanent magnet and the coil, and the angle vector x of the rotor rotation is defined to be (theta, phi) to describe the relative motion of the permanent magnet and the coil. Electromagnetic torque generated by electrifying the coil 1 under 1000 groups of different angle vectors is obtained as experimental data through random experiments. Defining an angle vector x of rotor rotation as a model input, and electromagnetic torque u generated by electrifying a coil 1 as a model output, and dividing the experimental data into a training set D { (x) i ，u i ) 1., 800} and test set
Wherein x is i And
are respectively the angle vector of the rotation of the training set and the test set i And
respectively electrifying the training set and the test concentrated coil to generate electromagnetic torque;
step two, for this embodiment, consider the following general regression model with noise:
u＝f(x)+ε
where ε is the noise signal satisfying the Gaussian distribution,
f (x) is a function map with respect to x. Under the function space, a Gaussian process is defined to describe the function distribution of f (x), namely
Where m (x) is a mean function and k (x, x') is a kernel function. m (x) and k (x, x') satisfy the following relationship:
definition f * ＝f(X * ) Predicting a point distribution function for the electromagnetic torque, wherein X * Is all in the test set
The constructed vector. According to the nature of Gaussian distribution, an electromagnetic torque training point u can be obtained i And electromagnetic torque prediction point
The joint prior distribution of (a) is:
wherein u ═ u 1 ，...，u 800 ] T Is a vector of 800 × 1, X ═ X 1 ，...，x 800 ] T Is a vector of 800X 1, X * ＝[u 1 ，...，u 200 ] T Is a 200 × 1 vector, m (x) ═ m (x) 1 )，...，m(x 800 )] T Is a mean vector of 800 x 1,
is a 200 × 1 mean vector, K (X, X) is a 800 × 800 matrix, K (X, X) * ) Is a 800X 200 matrix, K (X) * X) is a 200X 800 matrix, K (X) * ，X * ) A 200 x 200 matrix. Consider the mean vector m (X), m (X) * ) Are all 0, and the conditional distribution property of Gaussian distribution is utilized to obtain the posterior distribution f of an electromagnetic torque model for describing the coil 1 1* The description is as follows:
wherein the content of the first and second substances,
and
the description is as follows:
and step three, analyzing the existing experimental data according to the distribution rule of the uniform arrangement of the permanent magnets of the motor rotor, and determining the kernel function applicable to the regression modeling method based on the Gaussian process according to the periodic rule of the change of the experimental data. In this embodiment, an autocorrelation determination two-dimensional gaussian kernel (SEard) is described as follows:
wherein the content of the first and second substances,
for the signal variance, Θ is a length gauge matrix, and Θ ═ diag (l) 1 ,l 2 )。
Step four, defining the hyper-parameters
Is a hyperparametric vector where { Θ } represents an element in the length scale matrix θ. And obtaining the optimal hyperparameter of the Gaussian process regression by using the maximum value self-adaptive learning of the negative log-edge likelihood function of the training sample by adopting a maximum likelihood estimation method. The method comprises the following specific steps:
first, a negative log-edge likelihood function is established as follows:
where p (y | X, θ) is the conditional probability of the training data,
then, a partial derivative is obtained for the hyper-parameter theta under the function, and the partial derivative of L (theta) with respect to the hyper-parameter theta is obtained in the form:
wherein the content of the first and second substances,
the minimum value of L (theta) is calculated by using the partial derivative to obtain the optimal hyper-parameter, as shown in Table 1.
TABLE 1
Finally, an electromagnetic torque model f of the permanent magnet spherical motor coil 1 is determined 1* 。
Step five, repeating the step one to the step four, and respectively determining the electromagnetic torque models f of the coil 2, the coil 3 and the coil 24 2* ，f 3* ，…，f 24* And finally determining the electromagnetic torque model of the permanent magnet spherical motor, which is described as follows:
wherein U is the total electromagnetic torque of the permanent magnet spherical motor, I j The energizing current of the jth coil.
Figure 3 is a coil 1 electromagnetic torque distribution diagram obtained by using the permanent magnet spherical motor electromagnetic torque model,
fig. 4 is a comparison diagram of a coil 1 electromagnetic torque and actual electromagnetic torque test point obtained by using the permanent magnet spherical motor electromagnetic torque model.
The accuracy of the prediction can be evaluated by calculating the decision coefficient, interpreting the coefficient of variation, normalizing the mean square error and normalizing the root mean square error. Table 2 shows the coefficients of determination, the explanatory variation coefficients, the normalized mean square error and the normalized root mean square error of the electromagnetic torque model.
TABLE 2
| Determining coefficients | Interpretation of coefficient of variation | Normalized mean square error | Normalized root mean square error |
| 0.99794 | 0.97997 | 0.00203 | 0.01628 |
It can be seen that the coefficient of determination and the coefficient of interpretation variation of the electromagnetic torque model both exceed 0.99, and the normalized mean square error are both smaller than 0.02, which indicates that the electromagnetic torque model can correctly describe the relationship between the electromagnetic torque and the rotor position change of the permanent magnet spherical motor in the embodiment. The spherical motor control system utilizes an electromagnetic torque model established by the permanent magnet spherical motor electromagnetic torque modeling method based on Gaussian process regression to control.
Claims (2)
1. A permanent magnet spherical motor electromagnetic torque modeling method based on Gaussian process regression is characterized by comprising the following steps:
step one, a spherical motor model is established in electromagnetic analysis software according to a mechanical structure of a permanent magnet spherical motor, when different angles of rotation of a rotor are obtained through random experiments, the size of electromagnetic torque generated by electrifying a coil 1 is used as experimental data, an angle vector of rotation of the rotor is defined as model input, an electromagnetic torque vector generated by electrifying the coil 1 is used as model output, and the experimental data are divided into a training set D { (x) i ，u i ) 1, …, n and test set
Wherein x is i And
angle vectors, u, of rotation of the rotors in the training set and in the test set, respectively i And
electromagnetic torques generated by electrifying the training set and the test concentration coil 1 respectively;
step two, defining a Gaussian process in the function space, and obtaining f of the electromagnetic torque model describing the coil 1 by using the conditional distribution property of Gaussian distribution 1* The description is as follows:
wherein X is all X in the training set i Constructed vector, u being all u in the training set i Constructed vector, X * Is all in the test set
The vector of the composition is then calculated,
and
are respectively f 1* Mean and variance of;
thirdly, according to the distribution rule of the permanent magnets of the motor rotor, analyzing experimental data, and determining a kernel function applicable to the Gaussian process regression-based modeling method as a two-dimensional Gaussian kernel function for autocorrelation determination;
step four, adopting a maximum likelihood estimation method, obtaining the optimal hyperparameter of the Gaussian process regression by utilizing the maximum value self-adaptive learning of the negative log-edge likelihood function of the training sample, and determining the electromagnetic torque of the permanent magnet spherical motor coil 1Model f 1* Said hyper-parameter comprising a noise variance
Variance of signal
And a length gauge matrix Θ;
step five, repeating the step one to the step four, and respectively determining the electromagnetic torque models f of the coil 2, the coil 3 and the coil k 2 *，f 3 *，…，f k Finally, determining the electromagnetic torque model of the permanent magnet spherical motor as
2. A spherical motor control system, characterized in that the spherical motor control system utilizes the electromagnetic torque model established by the permanent magnet spherical motor electromagnetic torque modeling method based on gaussian process regression of claim 1 for control.
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| CN101520857A (en) * | 2009-03-31 | 2009-09-02 | 天津大学 | Inverse kinematics resolution method of permanent magnetism spherical electric motor on the basis of neural network |
| CN108595744A (en) * | 2018-03-02 | 2018-09-28 | 中国科学院空间应用工程与技术中心 | The electromagnetic actuator Equivalent Magnetic Field intensity modeling method returned based on Gaussian process |
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| CN101520857A (en) * | 2009-03-31 | 2009-09-02 | 天津大学 | Inverse kinematics resolution method of permanent magnetism spherical electric motor on the basis of neural network |
| CN108595744A (en) * | 2018-03-02 | 2018-09-28 | 中国科学院空间应用工程与技术中心 | The electromagnetic actuator Equivalent Magnetic Field intensity modeling method returned based on Gaussian process |
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