CN111843626B - Gauss model-based hysteresis modeling method, system and medium for air-driven actuator - Google Patents

Gauss model-based hysteresis modeling method, system and medium for air-driven actuator Download PDF

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CN111843626B
CN111843626B CN202010686629.1A CN202010686629A CN111843626B CN 111843626 B CN111843626 B CN 111843626B CN 202010686629 A CN202010686629 A CN 202010686629A CN 111843626 B CN111843626 B CN 111843626B
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丁烨
罗霄
丁汉
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Shanghai Moxiang Machinery Technology Co ltd
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    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B24—GRINDING; POLISHING
    • B24B—MACHINES, DEVICES, OR PROCESSES FOR GRINDING OR POLISHING; DRESSING OR CONDITIONING OF ABRADING SURFACES; FEEDING OF GRINDING, POLISHING, OR LAPPING AGENTS
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    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B24—GRINDING; POLISHING
    • B24B—MACHINES, DEVICES, OR PROCESSES FOR GRINDING OR POLISHING; DRESSING OR CONDITIONING OF ABRADING SURFACES; FEEDING OF GRINDING, POLISHING, OR LAPPING AGENTS
    • B24B47/00—Drives or gearings; Equipment therefor
    • B24B47/10—Drives or gearings; Equipment therefor for rotating or reciprocating working-spindles carrying grinding wheels or workpieces
    • B24B47/14—Drives or gearings; Equipment therefor for rotating or reciprocating working-spindles carrying grinding wheels or workpieces by liquid or gas pressure
    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B24—GRINDING; POLISHING
    • B24B—MACHINES, DEVICES, OR PROCESSES FOR GRINDING OR POLISHING; DRESSING OR CONDITIONING OF ABRADING SURFACES; FEEDING OF GRINDING, POLISHING, OR LAPPING AGENTS
    • B24B49/00—Measuring or gauging equipment for controlling the feed movement of the grinding tool or work; Arrangements of indicating or measuring equipment, e.g. for indicating the start of the grinding operation
    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B24—GRINDING; POLISHING
    • B24B—MACHINES, DEVICES, OR PROCESSES FOR GRINDING OR POLISHING; DRESSING OR CONDITIONING OF ABRADING SURFACES; FEEDING OF GRINDING, POLISHING, OR LAPPING AGENTS
    • B24B51/00—Arrangements for automatic control of a series of individual steps in grinding a workpiece
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Abstract

The invention provides a gas-driven actuator hysteresis modeling method, system and medium based on a Gaussian model, comprising the following steps: step 1: constructing a pneumatic actuator; step 2: establishing a training model of the gas-driven actuator based on a Gaussian process regression model; and step 3: training the training model to obtain training model parameters; and 4, step 4: constructing a hysteresis model differential equation based on a Gaussian process regression model; and 5: solving a hysteresis model differential equation through a numerical method according to the parameters of the training model, and predicting the output force; step 6: and solving an inverse model of the hysteresis model according to the trained training model, compensating the hysteresis nonlinearity, and controlling the output force of the pneumatic actuator by controlling the air pressure. The method accurately establishes the mathematical model for describing hysteresis without considering the specific shape of the hysteresis loop, and has the advantages of few parameters, good identification precision and high efficiency.

Description

Gauss model-based hysteresis modeling method, system and medium for air-driven actuator
Technical Field
The invention relates to the technical field of hysteresis modeling, in particular to a gas-driven actuator hysteresis modeling method, system and medium based on a Gaussian model. In particular to a gas drive active force control polishing actuator hysteresis modeling method based on Gaussian process regression.
Background
Polishing is one of the most common machining methods, aims to reduce the surface roughness of a workpiece and improve the flatness of the surface of the workpiece, and is commonly used for polishing and machining complex parts such as automobile hubs. The common grinding mode is to install the grinding disc and the main shaft thereof at the tail end of the industrial robot so as to meet the requirement that the workpiece grinding usually needs a larger working space. In the case of selected process parameters, the precise maintenance of the normal contact force of the sanding head with the workpiece at a predetermined value is one of the most critical factors in determining the quality of the sanding. However, a general industrial robot has insufficient rigidity of the body, low motion precision and no high-precision force control function, so that in order to ensure the polishing quality, an active force control polishing actuator needs to be installed at the tail end of the industrial robot to output the expected polishing force, and at the moment, the robot body only takes charge of the spatial motion with low precision requirement.
Considering that the problem that the output signal of the force sensor has larger noise and the like under the polishing working condition and the advantages of high power-mass ratio, light weight and the like of the air driving mode, the main power control polishing actuator body does not carry a force sensor and provides output force by controlling the air pressure of the air cylinder. In practice, there is a rate-independent nonlinear hysteresis between the input pressure and the output force of the pneumatic actuator. Therefore, in order to achieve a sanding actuator capable of accurately outputting a desired force, it is necessary to establish a correlation model that accurately describes the dynamic hysteresis relationship between the input and output, while also facilitating use in control. At present, a phenomenological model is mostly adopted for an actuator hysteresis modeling method, and the defects that the selection of the model depends on the shape of a hysteresis loop, the determination of model parameters needs strong subjectivity and the like exist, so that the identification precision and the identification efficiency are influenced. Therefore, the high-precision and high-efficiency hysteresis modeling method based on the Gaussian process regression is provided, and has very important significance and prospect for improving the force control precision of the actuator and the polishing quality.
Patent document CN104991997A (application number: 201510319039.4) discloses a generalized rate-dependent P-I lag model optimized by an adaptive differential evolution algorithm, which is used for describing lag characteristics in a micro-nano control platform, and the model is a discretization model constructed by weighted superposition of a finite number of generalized Play operators and a finite number of weight coefficients and by introducing an input function of odd power and a derivative of the input function; the voltage signal given by the computer and the displacement signal measured by the grating sensor are used as input and output data for identifying the model; by respectively introducing two different smooth functions to describe the variation factor and the cross probability factor, a fast-convergence self-adaptive differential evolution algorithm is provided and used for identifying the parameters of the model.
Disclosure of Invention
Aiming at the defects in the prior art, the invention aims to provide a gas-driven actuator hysteresis modeling method, system and medium based on a Gaussian model.
The gas-driven actuator hysteresis modeling method based on the Gaussian model comprises the following steps:
step 1: constructing an air-driven actuator facing the grinding, polishing and deburring processing field;
step 2: establishing a training model of the gas-driven actuator based on a Gaussian process regression model;
and step 3: measuring the input and the output of the pneumatic actuator through a force sensor and a pressure sensor, sampling to obtain an observation data set as the input of a training model, and training the training model to obtain parameters of the training model;
and 4, step 4: constructing a hysteresis model differential equation based on a Gaussian process regression model;
and 5: solving a hysteresis model differential equation through a numerical method according to the parameters of the training model, and predicting the output force;
step 6: solving an inverse model of the hysteresis model according to the trained training model, compensating the hysteresis nonlinearity, and controlling the output force of the pneumatic actuator by controlling the air pressure;
the pneumatic actuator takes a cylinder with a friction degree meeting the preset requirement as a driving element of the pneumatic actuator, takes an electric pressure proportional valve as a pneumatic control unit, takes a pressure sensor as a signal feedback element, and constrains a single-degree-of-freedom linear motion track according to a preset linear guide rail;
the pneumatic actuator does not carry a force sensor.
Preferably, the expression of the training model is:
Figure BDA0002587803370000021
p represents the input pressure and F represents the output force;
Figure BDA0002587803370000022
representing a non-linear continuous equation to be identified, the function f returning from the Gaussian processIn the step of establishing the network, the user can establish the network,
Figure BDA0002587803370000023
a domain representing a function f; dF represents the rate of change of the output force, dP represents the rate of change of the input pressure;
using y as dF/dP as random variable of output space of training model, using pressure P and output force F as two-dimensional random variable of input space of training model, using x as [ F P ]]TExpressing that a function f (x) in the differential equation is a set of mappings from an input space two-dimensional random variable x to an output space random variable y, and the expression is as follows:
f(x)～GP(m(x),k(x,x'))
wherein m (x) is 0,
Figure BDA0002587803370000031
GP represents a Gaussian process; m (x) represents a mean function; x' represents a two-dimensional random variable different from x; sigmafRepresenting one of the gaussian process parameters to be identified; exp represents an exponential function; d represents the total dimension of the two-dimensional random variable x; d represents the sign of the d-th component of the random variable; x is the number ofdA d-th component representing a random variable; x is the number ofd' means other than xdThe d-th component of the random variable of (a); w is adAnd representing the weight value corresponding to the d component of the random variable.
Preferably, the step 3 comprises:
inputting air pressures with different amplitudes into an air driving actuator, collecting output force, and estimating a random variable y of an output space through output pressure, wherein the expression is as follows:
Figure BDA0002587803370000032
an observation dataset is obtained, which is recorded as:
D＝{X,y}＝{(xi,yi)|i＝1,...,n}＝{([Fi,Pi]T,(dF/dP)i)|i＝1,...,n}；
wherein, yiAn observed value representing an ith output space random variable; x is the number ofiAn observed value representing an ith input space random variable; fiAn observed value representing an ith output force; i represents the sequence tags of the observation training set; n represents the total number of samples of the observation training set.
Preferably, the output spatial random variable y is equal to the sum of the value of the function f and the gaussian noise epsilon:
y＝f(x)+ε,
Figure BDA0002587803370000033
wherein N represents a Gaussian distribution;
Figure BDA0002587803370000034
representing the variance of the gaussian noise;
the variance of the output spatial random variable y is:
Figure BDA0002587803370000035
wherein K (X, X) represents a covariance matrix between observation points; i represents an identity matrix;
in the observation data set X and in the test data set X*Then, the observed value y and the predicted value f of the spatial random variable are output*The joint gaussian distribution of (a) is:
Figure BDA0002587803370000036
wherein, K (X, X)*) And K (X)*,X*) Respectively representing random input variable X of the training observation set and random input variable X of the training observation set, random input variable X of the training observation set and random input variable X of the test set*Random input variable X of test set*And test set random input variable X*A covariance matrix between;
obtaining test data set input according to conditional probability and marginalized attribute of Gaussian distributionRandom variable X*Predicted value f of*The formula is:
Figure BDA0002587803370000041
wherein:
Figure BDA0002587803370000042
Figure BDA0002587803370000043
e represents the expectation of a predicted value;
the method is simplified as follows: k ═ K (X, X), K*＝K(X,X*)。
Preferably, when the random input variable of the test set has only one test point x*When k (x)*)＝k*Denoted as test point x*And the covariance matrix of the random input variables of the n training sets is obtained:
Figure BDA0002587803370000044
k denotes the test point x*A covariance matrix of random input variables with n training sets; t represents the transpose of the matrix;
the above formula is a linear combination of n kernel functions, each kernel function takes a training point as a core, and is recorded as:
Figure BDA0002587803370000045
wherein,
Figure BDA0002587803370000046
prediction value f representing random variable of output space*The expected value of (d); alpha is alphaiRepresenting the weight of the ith kernel function;
Figure BDA0002587803370000047
obtaining a display expression form of a hysteresis input and output differential equation:
Figure BDA0002587803370000048
F*representing the output force of the prediction set; p*An input pressure representing a prediction set;
Figure BDA0002587803370000049
expressing the predicted value of the random variable of the output space;
Figure BDA00025878033700000410
a prediction set representing input spatial random variables;
rising curve f of hysteresis loopa(F, P) and the descending Curve Fd(F, P) respectively identifying to obtain an inverse model of the hysteresis model, wherein the inverse model is expressed as:
Figure BDA0002587803370000051
preferably, the step 5 comprises:
all training model parameters are represented by the vector θ:
Figure BDA0002587803370000052
Figure BDA0002587803370000053
representing one of the pre-identified parameters of the Gaussian process; w represents a weight matrix corresponding to an independent variable in the Gaussian kernel function;
the likelihood function p (y | X) obeys gaussian distribution, and the Log likelihood function is obtained as:
Figure BDA0002587803370000054
determining the value of each component of the parameter vector { theta } according to the partial derivative of the Log likelihood function, and obtaining:
Figure BDA0002587803370000055
wherein α ═ K-1y；
Figure BDA0002587803370000056
Is a partial derivative symbol; thetajRepresenting the jth pre-recognition model parameter; tr denotes the traces of the matrix.
Preferably, the pressure signal P is given*Solving the inverse model of the hysteresis model by numerical methods
Figure BDA0002587803370000057
Deriving the output force F of the prediction set*；
F(k+1)*＝P(k)*+f(F(k)*,P(k)*)ΔP(k)*
Figure BDA0002587803370000058
Where k is a sequence of sampling points, h1Represents the step size, h1Is set to 1;
Figure BDA0002587803370000059
preferably, the step 6 includes:
solving the inverse model of hysteresis, expressed as:
Figure BDA00025878033700000510
output force F from a prediction set*The input pressure P of the required prediction set is calculated by a numerical method*The formula is as follows:
Figure BDA0002587803370000061
Figure BDA0002587803370000062
where k denotes the sampling sequence, h2Represents a step size; f. ofaA differential equation corresponding to a rising curve representing a hysteresis loop; Δ F (k)*Representing a value of change for a given expected output force.
The invention provides a gas-driven actuator hysteresis modeling system based on a Gaussian model, which comprises:
module M1: constructing an air-driven actuator facing the grinding, polishing and deburring processing field;
module M2: establishing a training model of the gas-driven actuator based on a Gaussian process regression model;
module M3: measuring the input and the output of the pneumatic actuator through a force sensor and a pressure sensor, sampling to obtain an observation data set as the input of a training model, and training the training model to obtain parameters of the training model;
module M4: constructing a hysteresis model differential equation based on a Gaussian process regression model;
module M5: solving a hysteresis model differential equation through a numerical method according to the parameters of the training model, and predicting the output force;
module M6: solving an inverse model of the hysteresis model according to the trained training model, compensating the hysteresis nonlinearity, and controlling the output force of the pneumatic actuator by controlling the air pressure;
the pneumatic actuator takes a cylinder with a friction degree meeting the preset requirement as a driving element of the pneumatic actuator, takes an electric pressure proportional valve as a pneumatic control unit, takes a pressure sensor as a signal feedback element, and constrains a single-degree-of-freedom linear motion track according to a preset linear guide rail;
the pneumatic actuator does not carry a force sensor.
According to the present invention, a computer-readable storage medium is provided, in which a computer program is stored, which, when being executed by a processor, carries out the steps of the method as described above.
Compared with the prior art, the invention has the following beneficial effects:
1. the invention provides a gas drive active force control polishing actuator hysteresis modeling method based on Gaussian process regression, which can conveniently, efficiently and accurately establish the relation between hysteresis nonlinear input and output;
2. in the aspect of modeling, compared with a common phenomenological model modeling method, the method does not need to consider the specific shape of a hysteresis loop and correct the model due to the change of the shape of the hysteresis loop, does not need to draw up some model parameters by means of subjective experience, and obviously improves the identification precision and efficiency, and has few model parameters and no hyper-parameters;
3. in the aspect of control, the hysteresis model is easy to invert, can be conveniently used in a control system, and can be used for finishing accurate control on the output force of the actuator without a force sensor for the pneumatic driving active force control polishing actuator, so that the polishing processing quality and the economic benefit are greatly improved.
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Other features, objects and advantages of the invention will become more apparent upon reading of the detailed description of non-limiting embodiments with reference to the following drawings:
FIG. 1 is a schematic diagram of the implementation and control principle of a polishing process integrated system with a pneumatically-actuated active-control polishing actuator;
FIG. 2 is a graph of hysteresis between the output forces of an air actuated actuator given different amplitude, rate of change air pressure signals;
FIG. 3 is a schematic diagram of the acquisition of a training observation set during the identification of a hysteresis nonlinear model of an air-driven actuator;
FIG. 4 is a schematic diagram of a hysteresis inverse model used in a controller to compensate for hysteresis non-linearities;
FIG. 5 is a graph of the recognition fitting effect of the hysteresis modeling method under actual experimental data;
wherein, Ascend represents a curve when the air pressure change rate is positive, i.e., an ascending curve; descan represents a curve in which the rate of change in air pressure is negative, i.e., a decreasing curve.
Detailed Description
The present invention will be described in detail with reference to specific examples. The following examples will assist those skilled in the art in further understanding the invention, but are not intended to limit the invention in any way. It should be noted that it would be obvious to those skilled in the art that various changes and modifications can be made without departing from the spirit of the invention. All falling within the scope of the present invention.
Example (b):
aiming at the defects in the prior art, the invention provides a hysteresis modeling method based on data driving. The method combines the inherent dynamic nonlinear characteristics of the hysteresis phenomenon, utilizes Gaussian process regression to establish a first order differential equation expressing the relation between the hysteresis input and output, then utilizes the experiment to obtain a corresponding data set to train a Gaussian process regression model, deduces an input and output differential equation explicit expression through conditional probability, and utilizes a numerical method to solve the differential equation to complete the prediction of the corresponding output under the condition of given input. The gas-driven polishing actuator is directly compensated by utilizing the solving hysteresis inverse model, so that the accurate semi-closed-loop control of the output force is realized by closed-loop control of the air pressure. Therefore, a hysteresis describing mathematical model is accurately established without considering the specific shape of the hysteresis ring, and the accurate force control of the force-controlled polishing actuator is realized, so that a rough spatial position is provided by an industrial robot, a polishing mode that the air-driven polishing actuator outputs expected polishing force is feasible, the polishing processing quality is greatly improved, and good economic benefit is obtained.
Please refer to fig. 1, fig. 2, fig. 3 and fig. 4.
Aiming at providing a rough space pose of a workpiece in a grinding process by an industrial robot, an active force control actuator arranged at the tail end of the robot provides a grinding mode with expected constant grinding force. A single-degree-of-freedom force control polishing actuator which is based on an air cylinder and used as a driving element and carried by a body powerless sensor is designed.
Establishing a first-order differential equation of input and output of the pneumatic actuator by utilizing a Gaussian process regression model based on the rate-independent characteristic of the hysteresis system;
testing the force output by the air driving actuator under the action of low-frequency and air pressure signals with different amplitudes to obtain a data set of a training model;
obtaining a hysteresis model differential equation display expression by utilizing Gaussian process regression;
training a Gaussian process regression model by using a data set to obtain model parameters;
under the given input, a numerical method is utilized to solve a hysteresis differential equation to complete the prediction of corresponding output;
and solving an inverse model of the hysteresis model, and finishing accurate control on the output force of the air-driven actuator by controlling air pressure through corresponding direct compensation.
Specifically, the embodiment provides a semi-closed loop gas drive active force control polishing actuator hysteresis modeling method based on gaussian process regression, which includes the following steps:
step 1: a single-degree-of-freedom force control polishing actuator which is based on a cylinder as a driving element and carried by a body powerless sensor is designed;
step 2: in order to describe and compensate the nonlinear hysteresis between the input air pressure and the output force of the air driver, a first-order differential equation of the input and the output of the air driver actuator is established by utilizing a Gaussian process regression model based on the rate-independent characteristic of a hysteresis system;
and step 3: measuring the force output by the air driving actuator under the action of low-frequency and air pressure signals with different amplitudes to obtain a data set of a training model;
and 4, step 4: obtaining a hysteresis model differential equation display expression by utilizing Gaussian process regression;
and 5: training a Gaussian process regression model by using a data set to obtain model parameters;
step 6: under the given input, the prediction of the corresponding output is completed by using a numerical method;
and 7: solving an inverse model of the hysteresis model, and finishing accurate control on the output force of the air-driven actuator by controlling air pressure through corresponding direct compensation;
preferably, the step 1 specifically comprises:
a low-friction cylinder is used as a driving element of a force control polishing actuator, an electric pressure proportional valve is used as a pneumatic control unit, a pressure sensor is used as a signal feedback element, and a corresponding linear guide rail is designed to provide single-degree-of-freedom linear motion track constraint. The actuator body does not carry a force sensor. And, the above-mentioned original paper is integrated, so that it can be installed at the end of the industrial robot. In the process of polishing the workpiece, the industrial robot receives a position instruction and provides a space pose required by the polishing process for the air-driven actuator.
Preferably, the step 2 specifically comprises:
considering the inherent dynamic property of the hysteresis phenomenon and combining the hysteresis rate-independent characteristic of the system, it is assumed that a first order differential equation exists between the input pressure F and the output force P, as follows:
Figure BDA0002587803370000091
in combination with the hysteresis-related feature, the above equation can be written in a more concise form:
Figure BDA0002587803370000092
wherein,
Figure BDA0002587803370000093
representing a non-linear equation of continuity to be identified. Regardless of the actual physical mechanism of the pneumatic actuator, the function f is built by regression through the Gaussian process and passes through the corresponding training setLearning results in a particular display expression.
According to the statistical learning method, let y be dF/dP as the random variable of the model output space, let pressure P and output force F as the two-dimensional random variable of the model input space, and use x as [ F P ]]TAnd (4) showing. Is recorded as:
Figure BDA0002587803370000094
f in the differential equation is the set of input space x to output space y mappings. The distribution of the function f is described by a gaussian process as:
f(x)～GP(m(x),k(x,x'))
here, assuming that the mean of this gaussian process is 0, the kernel function k (x, x') takes the most common square exponential function of the form:
Figure BDA0002587803370000095
in the above equation, the input spatial argument is two-dimensional, i.e., D ═ 2.
Preferably, the step 3 specifically comprises:
and inputting a series of air pressures with different amplitudes into the air supply driving actuator, and acquiring signals of the pressure sensor at the moment. Typically, the barometric pressure signal is in the form of a series of triangular waves. And meanwhile, acquiring a signal value of an external force sensor in contact with the force output end of the actuator. The model output spatial random variable y ═ dF/dP is estimated by:
Figure BDA0002587803370000096
at this time, an observation data set, i.e., a training set, of the model is obtained. Recording as follows:
D＝{X,y}＝{(xi,yi)|i＝1,...,n}＝{([Fi,Pi]T,(dF/dP)i)|i＝1,...,n}
the above set can be regarded as being independently and identically distributed by the gaussian process of the function f. The observation points follow a potentially gaussian process.
Preferably, the step 4 specifically includes:
assuming that the random variable y of the model output space, dF/dP, is equal to the sum of the value of the function f and the gaussian noise epsilon:
y＝f(x)+ε,
Figure BDA0002587803370000101
wherein the mean value of the Gaussian noise epsilon is 0 and the variance is used
Figure BDA0002587803370000102
And (4) showing. Namely:
Figure BDA0002587803370000103
the correlation assumption according to function f, therefore, the variance of the random variable y is:
Figure BDA0002587803370000104
where K (X, X) represents the covariance matrix between the observation points. Each element of this matrix is computed from a kernel function.
In the observation data set X and in the test data set X*Then, the observed value y and the predicted value f of the random variable are output*The joint gaussian distribution of (a) is:
Figure BDA0002587803370000105
wherein, K (X, X)*) And K (X)*,X*) Respectively representing random input variable X and random input variable X of training set, random input variable X of training set and random input variable X of test set*Random input variable X of test set*And test set random input variable X*The covariance matrix in between.
Obtaining an input random variable X under a test data set according to the conditional probability and the marginalized attribute of Gaussian distribution*Predicted value f of*Distribution of (a):
Figure BDA0002587803370000106
wherein:
Figure BDA0002587803370000107
Figure BDA0002587803370000108
using a more compact expression line:
K＝K(X,X),K*＝K(X,X*)
assume that the test set has only one test point x for its immediate input variable*，k(x*)＝k*Denoted as test point x*And a covariance matrix of the n training set random input variables. Thus, the above equation can be written as:
Figure BDA0002587803370000111
thus, the above equation can be viewed as a linear combination of n kernel functions, each kernel function having a training point as a core. Is recorded as:
Figure BDA0002587803370000112
wherein,
Figure BDA0002587803370000113
thus, the above formula is equivalent to:
Figure BDA0002587803370000114
therefore, the display expression of the hysteresis input and output differential equation is equivalently obtained. Wherein F, P represents the force and pressure values of the training set. Since the gaussian process regression is a non-parametric model, F, P here is equivalent to a known constant in the model.
Considering that the hysteresis loop is generally divided into a rising curve and a falling curve, in order to increase the prediction accuracy and the recognition efficiency, the rising curve f of the hysteresis loop is seta(F, P) and the descending Curve Fd(F, P) are respectively identified as:
Figure BDA0002587803370000115
preferably, the step 5 specifically includes:
in step 4, model parameters in the expression are displayed, mainly in the kernel function
Figure BDA0002587803370000116
And wd(d 1,2), and variance of gaussian noise
Figure BDA0002587803370000117
W is represented by { W }d＝1、wd＝2. All parameters are represented by a vector θ, i.e.:
Figure BDA0002587803370000118
because the likelihood function p (y | X) obeys gaussian distribution, its Log likelihood function is obtained as:
Figure BDA0002587803370000119
and obtaining the specific value of the model parameter through a maximization Log likelihood function. Here, the values of the components of the parameter vector θ are determined by calculating the partial derivatives of the Log likelihood function, that is:
Figure BDA0002587803370000121
wherein α ═ K-1y。
Preferably, the step 6 specifically includes:
given a pressure signal P*The corresponding output force F can be obtained by solving the sectional differential equation by a numerical method*。
F(k+1)*＝P(k)*+f(F(k)*,P(k)*)ΔP(k)*
Figure BDA0002587803370000122
Where k is the sequence of sample points. h is1Indicating the step size, can be set to 1. f (F (k)*,P(k)*) Specifically, the formula is as follows:
Figure BDA0002587803370000123
in the above equation, F, P represents the values of the training set pressure and output force as known constants.
Preferably, the step 7 specifically includes:
in order to apply the model describing the hysteresis nonlinearity directly to this control system, the hysteresis inverse model needs to be solved. The inverse model can be expressed as:
Figure BDA0002587803370000124
that is, in conjunction with the foregoing derivation and utilizing the inverse model described above, at a given desired output force F*In the case of (2), by the numerical valueThe method can calculate the required input pressure P of the hysteresis nonlinear system*. Namely:
Figure BDA0002587803370000125
Figure BDA0002587803370000126
where k denotes the sampling sequence, h2The step size is indicated.
The inverse model is directly used in a controller, so that the hysteresis nonlinearity between input and output can be compensated, and the accurate control of the output force of the pneumatic driving power control polishing actuator is completed. Since the actuator body does not mount a force sensor, the force output is controlled in an open loop manner. The control system controls the output force by controlling the input pressure, so the whole control system belongs to semi-open loop control. Assuming that the pressure can be controlled accurately, theoretically, the more accurate the hysteresis model describes the hysteresis non-linearity capability, the more accurate the control of the output force.
The following description of the embodiments of the present invention will be made with reference to examples of specific embodiments. According to the steps 1-4 in the invention content, the air pressure signals with different amplitudes of the air drive driving force control sander sample model are supplied, the force output by the air drive actuator is measured, and a training set is obtained. Wherein, the training set is divided into two parts according to the positive and negative of the pressure change rate dP. According to the invention, as shown in step 5, the model parameter value theta is obtained by using the training set when the hysteresis loop differential equation rises and fallsa、θdThe result is:
Figure BDA0002587803370000131
Figure BDA0002587803370000132
the hysteresis differential equation is solved through step 6 to obtain the fitting effect of the gaussian process regression model on hysteresis nonlinearity, as shown in fig. 5. The result shows that the gas drive actuator hysteresis modeling method based on the Gaussian process regression has high fitting precision.
The invention provides a gas-driven actuator hysteresis modeling system based on a Gaussian model, which comprises:
module M1: constructing an air-driven actuator facing the grinding, polishing and deburring processing field;
module M2: establishing a training model of the gas-driven actuator based on a Gaussian process regression model;
module M3: measuring the input and the output of the pneumatic actuator through a force sensor and a pressure sensor, sampling to obtain an observation data set as the input of a training model, and training the training model to obtain parameters of the training model;
module M4: constructing a hysteresis model differential equation based on a Gaussian process regression model;
module M5: solving a hysteresis model differential equation through a numerical method according to the parameters of the training model, and predicting the output force;
module M6: solving an inverse model of the hysteresis model according to the trained training model, compensating the hysteresis nonlinearity, and controlling the output force of the pneumatic actuator by controlling the air pressure;
the pneumatic actuator takes a cylinder with a friction degree meeting the preset requirement as a driving element of the pneumatic actuator, takes an electric pressure proportional valve as a pneumatic control unit, takes a pressure sensor as a signal feedback element, and constrains a single-degree-of-freedom linear motion track according to a preset linear guide rail;
the pneumatic actuator does not carry a force sensor.
According to the present invention, a computer-readable storage medium is provided, in which a computer program is stored, which, when being executed by a processor, carries out the steps of the method as described above.
Those skilled in the art will appreciate that, in addition to implementing the systems, apparatus, and various modules thereof provided by the present invention in purely computer readable program code, the same procedures can be implemented entirely by logically programming method steps such that the systems, apparatus, and various modules thereof are provided in the form of logic gates, switches, application specific integrated circuits, programmable logic controllers, embedded microcontrollers and the like. Therefore, the system, the device and the modules thereof provided by the present invention can be considered as a hardware component, and the modules included in the system, the device and the modules thereof for implementing various programs can also be considered as structures in the hardware component; modules for performing various functions may also be considered to be both software programs for performing the methods and structures within hardware components.
The foregoing description of specific embodiments of the present invention has been presented. It is to be understood that the present invention is not limited to the specific embodiments described above, and that various changes or modifications may be made by one skilled in the art within the scope of the appended claims without departing from the spirit of the invention. The embodiments and features of the embodiments of the present application may be combined with each other arbitrarily without conflict.

Claims (10)

1. A gas-driven actuator hysteresis modeling method based on a Gaussian model is characterized by comprising the following steps:
step 1: constructing an air-driven actuator facing the grinding, polishing and deburring processing field;
step 2: establishing a training model of the gas-driven actuator based on a Gaussian process regression model;
and step 3: measuring the input and the output of the pneumatic actuator through a force sensor and a pressure sensor, sampling to obtain an observation data set as the input of a training model, and training the training model to obtain parameters of the training model;
and 4, step 4: constructing a hysteresis model differential equation based on a Gaussian process regression model;
and 5: solving a hysteresis model differential equation through a numerical method according to the parameters of the training model, and predicting the output force;
step 6: solving an inverse model of the hysteresis model according to the trained training model, compensating the hysteresis nonlinearity, and controlling the output force of the pneumatic actuator by controlling the air pressure;
the pneumatic actuator takes a cylinder with a friction degree meeting the preset requirement as a driving element of the pneumatic actuator, takes an electric pressure proportional valve as a pneumatic control unit, takes a pressure sensor as a signal feedback element, and constrains a single-degree-of-freedom linear motion track according to a preset linear guide rail;
the pneumatic actuator does not carry a force sensor.
2. The gas-driven actuator hysteresis modeling method based on the gaussian model as recited in claim 1, wherein the expression of the training model is as follows:
Figure FDA0002587803360000011
p represents the input pressure and F represents the output force; f:
Figure FDA0002587803360000012
representing a non-linear continuous equation to be identified, the function f is established by a Gaussian process regression,
Figure FDA0002587803360000013
a domain representing a function f; dF represents the rate of change of the output force, dP represents the rate of change of the input pressure;
using y as dF/dP as random variable of output space of training model, using pressure P and output force F as two-dimensional random variable of input space of training model, using x as [ F P ]]TExpressing that a function f (x) in the differential equation is a set of mappings from an input space two-dimensional random variable x to an output space random variable y, and the expression is as follows:
f(x)～GP(m(x),k(x,x'))
wherein m (x) is 0,
Figure FDA0002587803360000014
GP represents a Gaussian process; m (x) represents a mean function; x' represents a radical different fromA two-dimensional random variable of x; sigmafRepresenting one of the gaussian process parameters to be identified; exp represents an exponential function; d represents the total dimension of the two-dimensional random variable x; d represents the sign of the d-th component of the random variable; x is the number ofdA d-th component representing a random variable; x is the number ofd' means other than xdThe d-th component of the random variable of (a); w is adAnd representing the weight value corresponding to the d component of the random variable.
3. The gas-driven actuator hysteresis modeling method based on the gaussian model as recited in claim 2, wherein the step 3 comprises:
inputting air pressures with different amplitudes into an air driving actuator, collecting output force, and estimating a random variable y of an output space through output pressure, wherein the expression is as follows:
Figure FDA0002587803360000021
an observation dataset is obtained, which is recorded as:
D＝{X,y}＝{(xi,yi)|i＝1,...,n}＝{([Fi,Pi]T,(dF/dP)i)|i＝1,...,n}；
wherein, yiAn observed value representing an ith output space random variable; x is the number ofiAn observed value representing an ith input space random variable; fiAn observed value representing an ith output force; i represents the sequence tags of the observation training set; n represents the total number of samples of the observation training set.
4. The Gaussian model-based hysteresis modeling method for air-actuated actuators according to claim 3, characterized in that the output spatial random variable y is equal to the sum of the value of the function f and the Gaussian noise ε:
y＝f(x)+ε,
Figure FDA0002587803360000022
wherein N represents GaussDistributing;
Figure FDA0002587803360000023
representing the variance of the gaussian noise;
the variance of the output spatial random variable y is:
Figure FDA0002587803360000024
wherein K (X, X) represents a covariance matrix between observation points; i represents an identity matrix;
in the observation data set X and in the test data set X*Then, the observed value y and the predicted value f of the spatial random variable are output*The joint gaussian distribution of (a) is:
Figure FDA0002587803360000025
wherein, K (X, X)*) And K (X)*,X*) Respectively representing random input variable X of the training observation set and random input variable X of the training observation set, random input variable X of the training observation set and random input variable X of the test set*Random input variable X of test set*And test set random input variable X*A covariance matrix between;
obtaining a test data set input random variable X according to the conditional probability and the marginalized attribute of Gaussian distribution*Predicted value f of*The formula is:
Figure FDA0002587803360000031
wherein:
Figure FDA0002587803360000032
Figure FDA0002587803360000033
e represents the expectation of a predicted value;
the method is simplified as follows: k ═ K (X, X), K*＝K(X,X*)。
5. The Gaussian model-based hysteresis modeling method for air-actuated actuator of claim 4, wherein when the random input variable of the test set has only one test point x*When k (x)*)＝k*Denoted as test point x*And the covariance matrix of the random input variables of the n training sets is obtained:
Figure FDA0002587803360000034
k denotes the test point x*A covariance matrix of random input variables with n training sets; t represents the transpose of the matrix;
the above formula is a linear combination of n kernel functions, each kernel function takes a training point as a core, and is recorded as:
Figure FDA0002587803360000035
wherein,
Figure FDA0002587803360000036
prediction value f representing random variable of output space*The expected value of (d); alpha is alphaiRepresenting the weight of the ith kernel function;
Figure FDA0002587803360000037
obtaining a display expression form of a hysteresis input and output differential equation:
Figure FDA0002587803360000038
F*representing the output force of the prediction set; p*An input pressure representing a prediction set;
Figure FDA0002587803360000039
expressing the predicted value of the random variable of the output space;
Figure FDA00025878033600000310
a prediction set representing input spatial random variables;
rising curve f of hysteresis loopa(F, P) and the descending Curve Fd(F, P) respectively identifying to obtain an inverse model of the hysteresis model, wherein the inverse model is expressed as:
Figure FDA00025878033600000311
6. the Gaussian model-based gas driven actuator hysteresis modeling method of claim 5, wherein the step 5 comprises:
all training model parameters are represented by the vector θ:
Figure FDA0002587803360000041
Figure FDA0002587803360000042
representing one of the pre-identified parameters of the Gaussian process; w represents a weight matrix corresponding to an independent variable in the Gaussian kernel function;
the likelihood function p (y | X) obeys gaussian distribution, and the Log likelihood function is obtained as:
Figure FDA0002587803360000043
determining the value of each component of the parameter vector { theta } according to the partial derivative of the Log likelihood function, and obtaining:
Figure FDA0002587803360000044
wherein α ═ K-1y；
Figure FDA0002587803360000045
Is a partial derivative symbol; thetajRepresenting the jth pre-recognition model parameter; tr denotes the traces of the matrix.
7. The Gaussian model-based gas actuated actuator hysteresis modeling method as claimed in claim 6 wherein a pressure signal P is given*Solving the inverse model of the hysteresis model by numerical methods
Figure FDA0002587803360000046
Deriving the output force F of the prediction set*；
F(k+1)*＝P(k)*+f(F(k)*,P(k)*)ΔP(k)*
Figure FDA0002587803360000047
Where k is a sequence of sampling points, h1Represents the step size, h1Is set to 1;
Figure FDA0002587803360000048
8. the gaussian model-based gas driven actuator hysteresis modeling method according to claim 7, wherein said step 6 comprises:
solving the inverse model of hysteresis, expressed as:
Figure FDA0002587803360000049
output force F from a prediction set*The input pressure P of the required prediction set is calculated by a numerical method*The formula is as follows:
Figure FDA0002587803360000051
Figure FDA0002587803360000052
where k denotes the sampling sequence, h2Represents a step size; f. ofaA differential equation corresponding to a rising curve representing a hysteresis loop; Δ F (k)*Representing a value of change for a given expected output force.
9. A gas-driven actuator hysteresis modeling system based on a Gaussian model, comprising:
module M1: constructing an air-driven actuator facing the grinding, polishing and deburring processing field;
module M2: establishing a training model of the gas-driven actuator based on a Gaussian process regression model;
module M3: measuring the input and the output of the pneumatic actuator through a force sensor and a pressure sensor, sampling to obtain an observation data set as the input of a training model, and training the training model to obtain parameters of the training model;
module M4: constructing a hysteresis model differential equation based on a Gaussian process regression model;
module M5: solving a hysteresis model differential equation through a numerical method according to the parameters of the training model, and predicting the output force;
module M6: solving an inverse model of the hysteresis model according to the trained training model, compensating the hysteresis nonlinearity, and controlling the output force of the pneumatic actuator by controlling the air pressure;
the pneumatic actuator takes a cylinder with a friction degree meeting the preset requirement as a driving element of the pneumatic actuator, takes an electric pressure proportional valve as a pneumatic control unit, takes a pressure sensor as a signal feedback element, and constrains a single-degree-of-freedom linear motion track according to a preset linear guide rail;
the pneumatic actuator does not carry a force sensor.
10. A computer-readable storage medium, in which a computer program is stored which, when being executed by a processor, carries out the steps of the method according to any one of claims 1 to 8.
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