CN101520857A - Inverse kinematics resolution method of permanent magnetism spherical electric motor on the basis of neural network - Google Patents

Abstract

The invention belongs to the technical field of inverse kinematics resolution of a permanent magnetism spherical electric motor, in particular relates to an inverse kinematics resolution method of a permanent magnetism spherical electric motor on the basis of a neural network. A positive direction kinematics model of the permanent magnetism spherical electric motor is determined through deduction, a feedforward neural network is adopted to build a model for the inverse kinematics of the permanent magnetism spherical electric motor, an input layer is provided with three nerve centers, three coordinate figures of positions of a rotor of the permanent magnetism spherical electric motor in Cartesian space are respectively inputted, an output layer is also provided with three nerve centers, and numerical numbers of three eulerian angles of the permanent magnetism spherical electric motor are respectively outputted. The number of nerve centers of a hidden layer is undetermined. The method obtains a training sample used for training the feedforward neural network according to the positive kinematics model of the permanent magnetism spherical electric motor, adopts a Levenberg-Marquardt optimizing algorithm and uses the training sample to train the feedforward neural network so as to determine the structure of the neural network, and can avoid the complicated positive kinematics inverse operation.

Description

A kind of permanent magnet spherical motor inverse kinematics method for solving based on neural network

Technical field:

The invention belongs to the technical field that the permanent magnet spherical motor inverse kinematics is found the solution, relate to a kind of neural network of utilizing and carry out the method that its inverse kinematics is found the solution.

Background technology:

For a long time, people have carried out extensive and deep research to the servo motor control system of motion in one dimension, are that the electric power servo-drive system of core has obtained widespread use with the servomotor.But research and application at modern space flight, military affairs, chemical industry, industrial automation and intelligent robot etc. all need to realize multifreedom motion more and more.Make the complicated precision apparatus that moves for robot, mechanical arm etc., the servo-drive system that is made of conventional one dimension motor then seems too complicated.Traditional single-degree-of-freedom drive motor will be realized the motion that two degree of freedom are above, needs two or more servomotors to realize by the complicated mechanical gearing.This can make that not only the total system volume is big, Heavy Weight, and the dynamic and static properties of system is also relatively poor.So, can provide the spherical motor of multifreedom motion to obtain people's extensive concern.Permanent magnet spherical motor has simple in structure, and volume is little, and is in light weight, and loss is little, and the energy index height is convenient to characteristics such as control, can be applied to the precision apparatus that joint of robot etc. is done the space multi-dimensional movement.

The orientation of permanent magnet spherical motor rotor and motion conditions can define with Eulerian angle.The kinematics of permanent magnet spherical motor is divided into positive motion and learns and inverse kinematics.Wherein, the former finds the solution on the rotor certain any motion conditions according to the situation of change of Eulerian angle in each degree of freedom, is the problem of finding the solution of the mapping from the Eulerian angle space to cartesian space; The latter then is opposite process, is the problem of finding the solution of the mapping from cartesian space to the Eulerian angle space, just sets up the problem of kinematics inversion model.It is that permanent magnet spherical motor is carried out motion control that the inverse kinematics of permanent magnet spherical motor is found the solution problem, motion analysis, the basis of off-line programing and trajectory planning.

For the inverse kinematics problem of permanent magnet spherical motor, do not see document as yet and deliver.Inverse kinematics for robot is found the solution problem, and the most direct thinking is the positive motion of permanent magnet spherical motor to be learned carry out inversion operation.Yet this method calculation of complex is found the solution difficulty.

Summary of the invention

The object of the present invention is to provide a kind of simple permanent magnet spherical motor inverse kinematics method for solving, can avoid complicated positive motion and learn inversion operation.

For this reason, the present invention adopts following technical scheme:

3. the permanent magnet spherical motor inverse kinematics method for solving based on neural network carries out according to following steps: the first step: utilize Eulerian angle θ=(α, beta, gamma) ^TDefine spherical spinner rotation front-back direction and change, determine the initial value θ of Eulerian angle ₀With related rotor-position point as initial condition after, one group of corresponding coordinate figure (x of Eulerian angle θ _e, y _e, z _e) ^T, set up the direct kinematics equation (x of permanent magnet spherical motor _e, y _e, z _e) ^T=A (x _i, y _i, z _i) ^T, $A=\left(\begin{array}{ccc}\mathrm{c\βc\γ}& \mathrm{s\αs\βc\γ}+\mathrm{s\γc\α}& -\mathrm{s\βc\αc\γ}+\mathrm{s\αs\γ}\\ -\mathrm{c\βs\γ}& -\mathrm{s\βs\α}+\mathrm{c\αc\γ}& \mathrm{s\γc\αc\β}+\mathrm{s\αs\γ}\\ \mathrm{s\β}& -\mathrm{s\αc\β}& \mathrm{c\αc\β}\end{array}\right),$ Wherein, angle cosine cos brief note is c, and the sinusoidal sin brief note of angle is s;

Second step: adopt feedforward neural network that the permanent magnet spherical motor inverse kinematics is carried out modeling, input layer has three neurons, imports three coordinate figures of permanent magnet spherical motor rotor-position in cartesian space respectively; Output layer also has three neurons, exports the numerical value of three Eulerian angle of permanent magnet spherical motor respectively, and the neuron number of hidden layer is undetermined;

The 3rd step: learn model according to the positive motion of permanent magnet spherical motor and obtain being used for training sample that this feedforward neural network is trained, this training sample is made up of the cartesian space coordinate figure of different Eulerian angle correspondences, has comprised the situation when the permanent magnet spherical motor rotor is done spinning motion, nutating;

The 4th step: feedforward neural network is trained with training sample, adopt the Levenberg-Marquardt optimized Algorithm, and convergence precision when relatively this neural network is got different hidden layer number in the training and used training step number, get and make this neural network, determine the structure of this neural network thus the hidden layer neuron number that convergence precision is the highest and used training step number is minimum of training sample.

The above-mentioned permanent magnet spherical motor inverse kinematics method for solving based on neural network in the 4th step, follows these steps to carry out:

(1) data of training sample is sent into neural network, by the output formula of hidden layer node j $a_{1,j}=f_1(\underset{i=1}{\overset{3}{\mathrm{\Σ}}}{w1}_{i,j}{X}_i-{b1}_j)$ Output formula with output layer node k ${\mathrm{\θ}}_k=f_2(\underset{j=1}{\overset{N}{\mathrm{\Σ}}}{w2}_{j,k}{a}_{1,j}-{b2}_k)$ Calculate the output of computational grid, application error function E obtains the error sum of squares of all samples in the training set then, in the formula, and f ₁Be hidden layer transport function, X _iBe the output of i input layer, w1 _{I, j}Be the connection weights between i input layer and j the hidden layer node, bl _jBe the threshold value of j hidden layer neuron, j=1 ..., N, N are the hidden layer node number;

(2) calculate the Jacobi matrix J of error to the weights differential;

(3) adjust formula Δ w=(J according to network weight ^TJ+ μ J) ^-1J ^TE obtains the weights adjustment amount, and in the formula, e is an error vector; J is the Jacobi matrix of error to the weights differential; μ is a scalar;

(4) recomputate error sum of squares with adjusted weights,, keep the weights that this step adjusts constant if new quadratic sum, is then used μ-μ-as new learning rate less than the quadratic sum of calculating in (1) step; Otherwise, with μ+μ ⁺As new learning rate, adjust formula according to described network weight and recomputate the weights adjustment amount, wherein, μ ⁺And μ ^-Be respectively the amount of adjusting upward and the downward adjustment amount of learning rate.

Beneficial effect of the present invention is:

1. feedforward neural network method that the inverse kinematics of permanent magnet spherical motor is carried out modeling avoided complicated inversion operation and derivation, and training sample obtains easily.

2. neural network has very strong nonlinear function approximation capability and generalization ability.With neural network the permanent magnet spherical motor inverse kinematics is carried out modeling, can reach precision arbitrarily.For the generalization ability of this neural network when dissatisfied, only need to change training sample, know satisfied till, comparatively simple.

3. for the modeling of diverse location point on its spherical spinner, only need to change the coordinate figure (x in the propulsion equation _i, y _i, z _i) ^T, obtain corresponding propulsion mathematic(al) function F, draw training sample then and this neural network is trained get final product, implement comparatively convenient.

4. the feedforward neural network is simple in structure, and the neuronic corresponding relation of input layer and output layer is obvious, and the Levenberg-Marquardt optimized Algorithm fast convergence rate that is adopted, and the convergence precision height has improved the real-time and the reliability of the method.

5. by the hidden layer neuron number of relatively having determined this neural network, guaranteed the optimization of network structure to training effect.

Description of drawings:

Fig. 1 feedforward neural network structure;

Fig. 2 network training result;

The desired output of Fig. 3 Eulerian angle and actual contrast of exporting;

The space tracking figure of Fig. 4 desired output;

Desired output in Fig. 5 nutation movement and actual contrast of exporting;

The space tracking figure of Fig. 6 nutation movement desired output.

Embodiment

Verified, 2 layers of feedforward neural network (promptly having only a hidden layer) can approach nonlinear function arbitrarily.The method has been avoided complicated calculating derivation, adopts feedforward neural network that the inverse kinematics of permanent magnet spherical motor is found the solution.According to the training effect of neural network, determined the structure of this neural network to learning sample.Simulating, verifying the validity of this method.

Below in conjunction with two embodiment and accompanying drawing the present invention is done further detailed description.

Embodiment 1

Permanent magnet spherical motor has simple in structure, and volume is little, and is in light weight, and loss is little, and the energy index height is convenient to characteristics such as control, can be applied to the precision apparatus that joint of robot etc. is done the space multi-dimensional movement.Its mechanical construction drawing and rotor thereof are synoptic diagram such as Fig. 1 of three-degree-of-freedom motion, and be shown in Figure 2.

The orientation of permanent magnet spherical motor rotor and motion conditions can define with Eulerian angle.The kinematics of permanent magnet spherical motor is divided into positive motion and learns and inverse kinematics.Wherein, the former finds the solution on the rotor certain any motion conditions according to the situation of change of Eulerian angle in each degree of freedom, is the problem of finding the solution of the mapping from the Eulerian angle space to cartesian space; The latter then is opposite process, is the problem of finding the solution of the mapping from cartesian space to the Eulerian angle space, just sets up the problem of kinematics inversion model.It is that permanent magnet spherical motor is carried out motion control that the inverse kinematics of permanent magnet spherical motor is found the solution problem, motion analysis, the basis of off-line programing and trajectory planning.

The position of stator of permanent magnet spherical motor and rotor-position define with rest frame xyz and moving coordinate system dqp respectively, and the p coordinate axis in rotor of output shaft axle and the dqp system overlaps.Wherein, the x axle is that the rotation alpha angle is to x from xyz ₁y ₁z ₁System, the y axle is from x ₁y ₁z ₁Rotation β angle is to x ₂y ₂z ₂System, the z axle is from x ₂y ₂z ₂System rotation γ angle is to dqp system, institute produce angle θ=(α, beta, gamma) ^TBe called the generalized Euler angle.Institute produces to such an extent that rotation matrix is A, and is as follows:

$A=\left(\begin{array}{ccc}\mathrm{c\βc\γ}& \mathrm{s\αs\βc\γ}+\mathrm{s\γc\α}& -\mathrm{s\βc\αc\γ}+\mathrm{s\αs\γ}\\ -\mathrm{c\βs\γ}& -\mathrm{s\βs\α}+\mathrm{c\αc\γ}& \mathrm{s\γc\αc\β}+\mathrm{s\αs\γ}\\ \mathrm{s\β}& -\mathrm{s\αc\β}& \mathrm{c\αc\β}\end{array}\right)---\left(1\right)$

Wherein, angle cosine cos brief note is c, and the sinusoidal sin brief note of angle is s.This rotation matrix satisfies following relational expression:

(x _e，y _e，z _e) ^T＝A(x _i，y _i，z _i) ^T (2)

Wherein, (x _i, y _i, z _i) ^TCertain some initial position coordinate in the xyz coordinate system on the expression rotor, (x _e, y _e, z _e) ^TRepresent this postrotational position coordinates in the xyz coordinate system.Eulerian angle can be expressed as form:

θ＝θ ₀+Δθ (3)

Wherein, θ ₀The initial value of expression Eulerian angle, its value has determined to find the solution the starting condition of inverse kinematics; Δ θ represents the variable quantity of Eulerian angle.By formula (3) as can be known, at θ ₀Under=0 the starting condition, when Δ θ=0, θ=0.At this moment, matrix A is a unit matrix, i.e. coordinate figure (x in the formula (1) _i, y _i, z _i) ^T(x _e, y _e, z _e) ^TEquate.According to the definition of aforementioned Eulerian angle, under this starting condition, establishing the rotor sphere diameter is r as can be known, (0,0, r) ^TThe reference position point of promptly representing rotor of output shaft axle.As can be seen, a given starting condition and coordinate figure (x _i, y _i, z _i) ^TAfter, one group of corresponding coordinate figure (x of Eulerian angle θ _e, y _e, z _e) ^TTherefore, in the rotary course of permanent magnet spherical motor, (x _e, y _e, z _e) ^TNumerical value change along with the variation of Eulerian angle.

So, spherical motor rotatablely move rotor surface arbitrarily any motion conditions just can obtain by through type (2).Formula (2) is the direct kinematics equation of permanent magnet spherical motor.For determine arbitrarily on the permanent magnet spherical motor a bit, t at a time, the position vector X of this point (t)=(x _e(t), y _e(t), z _e(t)) ^T∈ R ³With Eulerian angle variable θ (t)=(α (t), β (t), γ (t)) ^TBetween relation can be expressed as follows:

X＝F(θ) (4)

Wherein, F represents the propulsion mathematic(al) function of this point on the permanent magnet spherical motor rotor surface.F is one, and (the concrete value of m depends on the initial coordinate values (x that permanent magnet spherical motor is carried out kinematics analysis and finds the solution for 3 * m) matrix, 1≤m≤3 wherein _i, y _i, z _i) ^T

A kind of is the differential kinematic relation of setting up permanent magnet spherical motor against separating strategy, thereby finds the solution the Eulerian angle velocity vector according to its space velocity vector in the xyz coordinate system.Differentiated simultaneously in formula (4) both sides, can get:

$\stackrel{\·}{X}\left(t\right)=J\left(\mathrm{\θ}\left(t\right)\right)\stackrel{\·}{\mathrm{\θ}}\left(t\right)---\left(5\right)$

In the formula $\stackrel{\&CenterDot;}{X}\left(t\right)=\mathrm{dX}\left(t\right)/\mathrm{dt},$ $\stackrel{\&CenterDot;}{\mathrm{\&theta;}}\left(t\right)=\mathrm{d\&theta;}\left(t\right)/\mathrm{dt},$ $J\left(\mathrm{\&theta;}\left(t\right)\right)=\&PartialD;F/\&PartialD;\mathrm{\&theta;}\&Element;{\mathrm{<figure-callout\; id="3"\; label="R"\; filenames="A200910068314E00131.png,A200910068314E00141.png"\; state="\{\{state\}\}">R</figure-callout>}}^{3\&times;3}$ Being the matrix of (3 * 3), is the Jacobi matrix of permanent magnet spherical motor.

So can obtain the Eulerian angle velocity vector:

$\stackrel{\&CenterDot;}{\mathrm{\&theta;}}\left(t\right)=J^{-1}\left(\mathrm{\&theta;}\left(t\right)\right)\stackrel{\&CenterDot;}{X}\left(t\right)---\left(6\right)$

J in the formula ^-1(θ (t)) is the inverse matrix of Jacobi matrix.

According to starting condition quadratured simultaneously in formula (6) both sides then, can obtain the inverse kinematics equation of permanent magnet spherical motor.

Ask the Jacobi matrix and the inverse matrix thereof of permanent magnet spherical motor with said method, calculate more complicated, find the solution the comparison difficulty.

Verified, 2 layers of feedforward neural network can approach nonlinear function arbitrarily.Therefore consider to set up permanent magnet spherical motor inverse kinematics model with 2 layers of feedforward neural network.

By the analysis of front as can be known, the inversion model of permanent magnet spherical motor is input as its postrotational coordinate figure (x _e, y _e, z _e) ^T, export the pairing Eulerian angle of this rotary course.Therefore, input layer and the output layer that the inverse kinematics of permanent magnet spherical motor is carried out the neural network of modeling all has 3 nodes.Its hidden layer node number will be determined the training effect of sample data according to neural network.Neural network structure as shown in Figure 1.

The BP learning algorithm is the backpropagation by the network output error, adjusts and the weights that are connected of revising network, and makes error reach minimum.Its adjustment process comprises forward calculation and error back propagation.

1) forward calculation

Hidden layer node j is output as:

$a_{1,j}=f_1(\underset{i=1}{\overset{3}{\mathrm{\&Sigma;}}}{w1}_{i,j}{X}_i-{b1}_j)---\left(7\right)$

Wherein, f ₁Be hidden layer transport function, X _iBe the output of i input layer, w1 _{I, j}Be the connection weights between i input layer and j the hidden layer node, b1 _jBe the threshold value of j hidden layer neuron, j=1 ..., N, N are the hidden layer node number.

Output layer node k is output as:

${\mathrm{\&theta;}}_k=f_2(\underset{j=1}{\overset{N}{\mathrm{\&Sigma;}}}{w2}_{j,k}{a}_{1,j}-{b2}_k)---\left(8\right)$

Wherein, f ₂Be hidden layer transport function, w2 _{J, k}Be the connection weights between j hidden layer node and k the output layer node, b2 _kBe the threshold value of k output layer node, k=1,2,3.

2) error back propagation

Definition error back propagation function is:

$E=\frac{1}{2}\underset{k=1}{\overset{3}{\mathrm{\&Sigma;}}}{(T_k-{\mathrm{\&theta;}}_k)}^2---\left(9\right)$

Wherein, T _kIt is the training objective value of k output node.

In order to make the error minimum, adopt steepest gradient descent method to adjust weights, revise the weight w 1 between output layer and the hidden layer earlier, and then revise the weight w 2 between hidden layer and the input layer.

The adjustment amount of weights is:

$\mathrm{\&Delta;w}(n+1)=-\mathrm{\&eta;}\frac{\&PartialD;E}{\&PartialD;w\left(n\right)}---\left(10\right)$

Wherein, n represents iterations, and η represents learning rate.

The correction weights are:

w(n+1)＝w(n)+Δw(n+1) (11)

What traditional BP algorithm adopted is steepest gradient descent method correction weights, and training process i.e. inclined-plane from a certain starting point along error function reaches smallest point gradually to make it error be zero.If the training sample information capacity is bigger, then iterations is more in error inverse process, certainly will influence computing velocity and precision.So, here introduce Levenberg-Marquardt optimized Algorithm (brief note is the L-M optimized Algorithm), its basic thought is to make its each iteration no longer along single negative gradient direction, but permissible error is searched for along the direction that worsens, optimize network weight by self-adaptation adjustment between steepest gradient descent method and Gauss-Newton method simultaneously, network can effectively be restrained, improve the speed of convergence and the generalization ability of network greatly.The detailed introduction of relevant this method, can be referring to following two pieces of papers:

[1] Chang Liang separates and opens a position, Wang Shaobo, Xiao Zhijuan. optimize the groundwater dynamic simulation and prediction [J] of neural network algorithm based on LM. and underground water, 2005,27 (5): 380-383.

[2] Liu Guohai sets up. based on the agricultural induction motor Direct Torque Control of neural network [J]. and agricultural engineering journal, 2001,17 (4): 131-134.

The L-M optimized Algorithm is called damped least square method again, and its network weight is adjusted formula and is:

Δw＝(J ^TJ+μJ) ^-1J ^Te (12)

In the formula (12), e is an error vector; J is the Jacobi matrix of error to the weights differential; μ is a scalar, and when μ increased, algorithm approached to have the method for steepest descent of less learning rate, and when μ dropped to 0, algorithm had just become Gauss-Newton method.

The iterative step of L-M algorithm is:

1. network is sent in all inputs, by the output of formula (7) and formula (8) computational grid, application error function E obtains the error sum of squares of all samples in the training set then.

2. calculate the Jacobi matrix J of error to the weights differential.

3. obtain the weights adjustment amount according to formula (12).

4. recomputate error sum of squares with adjusted weights, if new quadratic sum is then used μ-μ less than the quadratic sum of calculating in the 1st step ^-As new learning rate, keep the weights that this step adjusts constant; Otherwise, with μ+μ ⁺As new learning rate, recomputate the weights adjustment amount according to formula (12).Wherein, μ ⁺And μ ^-Be respectively the amount of adjusting upward and the downward adjustment amount of learning rate.

The front has been discussed the method that neural network is found the solution the permanent magnet spherical motor inverse dynamics of using in more detail, below determines the concrete structure of neural network by emulation, and verifies the validity of this method.The root diameter of establishing permanent magnet spherical motor in the emulation is a unit 1.If the location point that permanent magnet spherical motor inverse kinematics problem is found the solution is epitrochanterian output shaft location point, then when the initial value of Eulerian angle was zero, the initial coordinate of this location point was (0,0,1) point, i.e. (x _i, y _i, z _i) ^T=(0,0,1) ^T

For training network, provide some groups of Eulerian angle numerical value, can obtain corresponding spatial value by the direct kinematics (2) of permanent magnet spherical motor, thereby obtain training sample.Training sample has 500 groups of data, the state when having comprised motions such as the rotation of permanent magnet spherical motor rotor do, nutating.The part training sample is as shown in table 1.

Table 1 train samples data

Table 1 The training data of the neural network

Hidden layer number difference, the speed of convergence of neural network will be different.The number that the feedforward neural network hidden layer node is established in the front is N, determines the hidden layer node number according to the actual effect of network training below.Determining of hidden layer number can be with reference to following experimental formula:

$N=\sqrt{{\mathrm{<figure-callout\; id="1"\; label="N"\; filenames="A200910068314E00131.png,A200910068314E00141.png"\; state="\{\{state\}\}">N</figure-callout>}}_1+{\mathrm{<figure-callout\; id="2"\; label="N"\; filenames="A200910068314E00131.png,A200910068314E00141.png"\; state="\{\{state\}\}">N</figure-callout>}}_2}+a---\left(13\right)$

Wherein, N is the hidden layer node number, N ₁And N ₂Be respectively input layer and output layer interstitial content, α is the constant between 1 to 10.By the analysis of front as can be known, the input layer and the output layer node number average that are used for permanent magnet spherical motor is carried out the feedforward neural network that inverse kinematics finds the solution are 3, i.e. N ₁=3, N ₂=3.So N gets 4,5, certain number in 6,7,8,9,10,11,12,13.Adopt the L-M optimized Algorithm that neural network is trained.The numerical value of the training step number of network and the training error function E that determines by formula (9) when table 2 contrast N gets different numerical value.

Table 2 neural metwork training error and number of times

Table 2 The training errors and epochs of neural

network with different values of N

Interstitial content	4	5	6	7
					Training error	0.00998749	0.00941588	0.00857883	0.00928299
Frequency of training	301	70	61	59

8	9	10	11	12	13
						0.00579235	0.00979151	0.00953425	0.00996886	0.00837573	0.00892396
10	29	29	28	31	36

As can be seen from Table 2, when the hidden layer node number got 8, neural network was the fastest to the speed of convergence of sample data, and precision is the highest.Therefore, be used for the feedforward neural network hidden layer number that the permanent magnet spherical motor inverse kinematics finds the solution and be set at 8.The hidden layer number be 8 o'clock training result as shown in Figure 2.

Below the validity of this neural network is verified.Givenly satisfy starting condition θ ₀=0 Eulerian angle θ=(α, beta, gamma) ^TVariation track, the positive motion by permanent magnet spherical motor is learned can obtain its corresponding space tracking; With this space tracking oppositely being found the solution, can obtain the change curve of Eulerian angle through the feedforward neural network of aforementioned learning sample training.Compare with given Eulerian angle variation track with this change curve then.Comparative result as shown in Figure 3, wherein, Fig. 3 (a), Fig. 3 (b), Fig. 3 (c) represents three Eulerian angle α respectively, the set-point of beta, gamma and the output valve of inverse kinematics being found the solution with neural network.

As seen from Figure 3, be 0 o'clock at the Eulerian angle initial value, this neural network can solve the Eulerian angle of space tracking correspondence, and higher solving precision is arranged.Fig. 4 is the pairing space tracking of Eulerian angle variation track given among Fig. 3.

As seen from Figure 4, the starting point of the given pairing space tracking of Eulerian angle variation track is (0,0, the 1) point among the rest frame xyz.This is explanation just, is that the reference position of rotor of output shaft axle is (x under zero the condition at the initial value of Eulerian angle _i, y _i, z _i) ^T=(0,0,1) ^T, verified the correctness of preceding surface analysis.

Embodiment 2

Below investigate the situation that neural network is found the solution the permanent magnet spherical motor inverse kinematics under the non-vanishing situation of Eulerian angle initial value.Nutation movement is one of the operating mode that can investigate spherical motor torque controllability.Eulerian angle variation track when given permanent magnet spherical motor is done nutation movement can be obtained the space tracking of nutation movement according to propulsion.The situation of change of Eulerian angle is found the solution according to space tracking with trained neural network then.Contrast given Eulerian angle variation track and neural network and find the solution the Eulerian angle situation of change that draws, the result of contrast as shown in Figure 5, wherein, Fig. 5 (a), (b), (c) respectively corresponding three Eulerian angle α, the contrast situation of beta, gamma.

As seen from Figure 5, be not 0 o'clock at the Eulerian angle initial value, this neural network still can reappear the Eulerian angle variation track of space tracking correspondence preferably.Fig. 6 has provided the space motion path of nutation movement rotor shaft position.

As seen from Figure 6, the origin coordinates of the rotor of output shaft axle location point of permanent magnet spherical motor is not at (0,0,1) point.In fact, its origin coordinates is put corresponding Eulerian angle initial value θ ₀

Aforementioned neural network is that the inverse kinematics of rotor of output shaft axle location point is carried out modeling, has realized the real-time identification to Eulerian angle.By the analysis of front as can be known, for the point of the diverse location on the spherical electric motor rotor, the propulsion mathematic(al) function F shown in the formula (4) is different.Therefore, carry out modeling to its inverse kinematics with neural network, used training sample also is different.

Claims (2)

1. the permanent magnet spherical motor inverse kinematics method for solving based on neural network carries out according to following steps: the first step: utilize Eulerian angle θ=(α, beta, gamma) ^TDefine spherical spinner rotation front-back direction and change, determine the initial value θ of Eulerian angle ₀With related rotor-position point as initial condition after, one group of corresponding coordinate figure (x of Eulerian angle θ _e, y _e, z _e) ^T, set up the direct kinematics equation (x of permanent magnet spherical motor _e, y _e, z _e) ^T=A (x _i, y _i, z _i) ^T,

Wherein, angle cosine cos brief note is c, and the sinusoidal sin brief note of angle is s;

Second step: adopt feedforward neural network that the permanent magnet spherical motor inverse kinematics is carried out modeling, input layer has three neurons, imports three coordinate figures of permanent magnet spherical motor rotor-position in cartesian space respectively; Output layer also has three neurons, exports the numerical value of three Eulerian angle of permanent magnet spherical motor respectively, and the neuron number of hidden layer is undetermined;

The 3rd step: learn model according to the positive motion of permanent magnet spherical motor and obtain being used for training sample that this feedforward neural network is trained, this training sample is made up of the cartesian space coordinate figure of different Eulerian angle correspondences, has comprised the situation when the permanent magnet spherical motor rotor is done spinning motion, nutating;

The 4th step: feedforward neural network is trained with training sample, adopt the Levenberg-Marquardt optimized Algorithm, and convergence precision when relatively this neural network is got different hidden layer number in the training and used training step number, get and make this neural network, determine the structure of this neural network thus the hidden layer neuron number that convergence precision is the highest and used training step number is minimum of training sample.

2. the permanent magnet spherical motor inverse kinematics method for solving based on neural network according to claim 1 is characterized in that, in the 4th step, follows these steps to carry out:

(1) data of training sample is sent into neural network, by the output formula of hidden layer node j

Output formula with output layer node k

Calculate the output of computational grid, application error function E obtains the error sum of squares of all samples in the training set then, in the formula, and f ₁Be hidden layer transport function, X _iBe the output of i input layer, w1 _{I, j}Be the connection weights between i input layer and j the hidden layer node, b1 _jBe the threshold value of j hidden layer neuron, j=1 ..., N, N are the hidden layer node number;

(2) calculate the Jacobi matrix J of error to the weights differential;

(3) adjust formula Δ w=(J according to network weight ^TJ+ μ J) ^-1J ^TE obtains the weights adjustment amount, and in the formula, e is an error vector; J is the Jacobi matrix of error to the weights differential; μ is a scalar;

(4) recomputate error sum of squares with adjusted weights, if new quadratic sum is then used μ-μ less than the quadratic sum of calculating in (1) step ^-As new learning rate, keep the weights that this step adjusts constant; Otherwise, with μ+μ ⁺As new learning rate, adjust formula according to described network weight and recomputate the weights adjustment amount, wherein, μ ⁺And μ ^-Be respectively the amount of adjusting upward and the downward adjustment amount of learning rate.

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