CN112589798B - Soft robot state feedback control method based on dielectric elastomer actuator - Google Patents

Soft robot state feedback control method based on dielectric elastomer actuator Download PDF

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CN112589798B
CN112589798B CN202011448335.1A CN202011448335A CN112589798B CN 112589798 B CN112589798 B CN 112589798B CN 202011448335 A CN202011448335 A CN 202011448335A CN 112589798 B CN112589798 B CN 112589798B
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dielectric elastomer
elastomer actuator
neural network
gaussian
radial basis
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CN112589798A (en
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彭滔
鲁云花
周鹏
肖汉光
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Chongqing University of Technology
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    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B25—HAND TOOLS; PORTABLE POWER-DRIVEN TOOLS; MANIPULATORS
    • B25J—MANIPULATORS; CHAMBERS PROVIDED WITH MANIPULATION DEVICES
    • B25J9/00—Programme-controlled manipulators
    • B25J9/16—Programme controls
    • B25J9/1628—Programme controls characterised by the control loop
    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B25—HAND TOOLS; PORTABLE POWER-DRIVEN TOOLS; MANIPULATORS
    • B25J—MANIPULATORS; CHAMBERS PROVIDED WITH MANIPULATION DEVICES
    • B25J9/00—Programme-controlled manipulators
    • B25J9/16—Programme controls

Abstract

The invention discloses a state feedback control method of a soft robot based on a dielectric elastomer actuator, which establishes a dynamic control model of the dielectric elastomer actuator by utilizing a virtual work simulation mode, then establishes a Gaussian radial basis function neural network approximator for an unknown system function in the dynamic control model, and dynamically adjusts the parameters of the Gaussian radial basis function neural network approximator on line by correspondingly designing the parameter update rate thereof to realize the online approximation of the unknown system function, and finally establishes a state feedback intelligent controller embedded with the Gaussian radial basis function neural network approximator in a state feedback control mode to dynamically adjust the control input of the dielectric elastomer actuator on line so that a state tracking error is converged to zero, thereby realizing the control target of the soft robot and enhancing the control precision and response speed of the control system, the adaptability and intelligence of the dielectric elastomer actuator-based soft robot are improved.

Description

Soft robot state feedback control method based on dielectric elastomer actuator
Technical Field
The invention relates to the technical field of soft robots, in particular to a state feedback control method of a soft robot based on a dielectric elastomer actuator.
Background
Different from the traditional robot taking a rigid structure as a main part, the soft robot processed by adopting soft materials can be bent, twisted and stretched continuously, and can make up the defects that the rigid robot has a complex structure, limited flexibility, poor man-machine interaction safety and adaptability and can finish the tasks which cannot be realized by a plurality of traditional rigid robots in special applications such as limited space, extreme environment, grabbing complexity, frangibility and the like. Therefore, the soft robot has a great application prospect and becomes a hotspot research direction in the robot field. Meanwhile, the software robot is determined to have infinite freedom degree by the intrinsic property of the soft material, so that the software robot is closer to natural creatures in the aspects of bionic structure and bionic motion, and the software robot is greatly different from a rigid robot in the aspects of design, manufacture, sensing, driving, control and the like.
Dielectric Elastomer (DE) is an intelligent material, which has the advantages of high strain response, high energy density, and fast response, and is widely used in various fields, such as artificial muscle, bionic robot, energy generator, etc. A Dielectric Elastomer (DE) is embedded in a soft robot, and a voltage is applied to control the deformation of the DE under the action of an electric field, so that the DE becomes a Dielectric Elastomer Actuator (DEA) to enable the soft robot to complete various actions and motions. Thus, the motion and motion control problems of a soft robot can essentially be transformed into controlling the deformation of the DEA.
The establishment of a kinematic and dynamic model of the DEA is the basis for the system to recognize and understand the characteristics of the DEA, and is also more favorable for ensuring that a controller designed for the DEA has better control quality. Mathematical models of physical systems are generally classified into two types, namely, a phenomenological model and a phenomenological model. The phenomenological model is established based on physical mechanisms, emphasizes on disclosing the essence of a physical system and has the advantages of accuracy and definition, and the phenomenological model is established based on data of experimental phenomena and emphasizes on describing the physical phenomena, thereby having the advantages of simplicity and effectiveness. For example, Sou proposes a unique model of DEA by analyzing the energy conversion mechanism during DEA deformation using continuous mechanics theory and thermodynamics theory. Sarban et al describes the electrical part of the DEA with capacitors and resistors, describes the mechanical part of the DEA with springs and dampers, and then experimentally analyzes the behavior of the physical components of the DEA (e.g. resistors, capacitors, springs and dampers) to create a phenomenological model.
Currently, the results of the dielectric elastomer actuators are focused mainly on the material and physical properties, while the control results are few. Some researchers have proposed control strategies using the above described phenomenological and phenomenological models of dielectric elastomer actuators, but most of them are implemented using open-loop control using pressure control of pressure sensors or using volume control of strain sensors to drive a section of the soft body of a robot. However, in general, there are some unmeasured parameters in the dielectric elastomer actuator, and some estimates of them need to be considered in the control process. In this respect, it is necessary to use some achievable control of the dielectric elastomer actuator, using a control strategy that can use the dynamics of the dielectric elastomer actuator, tolerating parameter uncertainties and limited measurement conditions.
State feedback control is a feature of modern control theory in the control field. The state feedback control means that each state variable of the system is multiplied by a corresponding feedback coefficient, fed back to an input end and added with a reference input, and the sum is used as a control signal of a controlled system. A state variable of a system can exhibit its internal characteristics of the entire system without knowing the internal structure of the system. Therefore, compared with the traditional output feedback control, the state feedback control can be more excellent and more effective to control the system, so that the system can work stably and normally.
Generally, some unknown parameters and immeasurable information exist in the dielectric elastomer actuator, and the unknown dynamic information and the estimated value or observed value of the state quantity of the system caused by the unknown parameters or the immeasurable information need to be considered in the control process. Therefore, there is a pressing need to take into account the unknown dynamics and parametric uncertainties of the dielectric elastomer actuators in the control process, while at the same time there is a need to design a high quality controller and control method thereof that is useful in such situations.
Disclosure of Invention
Aiming at the defects in the prior art, the technical problems to be solved by the invention are as follows: how to provide a state feedback control method of a soft robot based on a dielectric elastomer actuator to enhance the adaptivity and intelligence of a control system of the dielectric elastomer actuator so as to improve the action control precision and response speed of the soft robot based on the dielectric elastomer actuator.
In order to solve the technical problems, the invention adopts the following technical scheme:
a state feedback control method of a soft robot based on a dielectric elastomer actuator comprises the following steps:
establishing a dynamic control model of the dielectric elastomer actuator by using a virtual work simulation mode;
constructing a Gaussian radial basis function neural network approximator for an unknown system function in a dynamic control model, and dynamically adjusting parameters of the Gaussian radial basis function neural network approximator on line by designing a parameter update rate of the Gaussian radial basis function neural network approximator to realize on-line approximation of the unknown system function;
and a state feedback intelligent controller embedded with the Gaussian radial basis function neural network approximator is constructed in a state feedback control mode and used for dynamically adjusting the control input of the dielectric elastomer actuator on line so that the state tracking error is converged to zero, thereby realizing the control target of the soft robot.
In the method for controlling the state feedback of the soft robot based on the dielectric elastomer actuator, as an optimal scheme, the designed parameter update rate of the Gaussian radial basis function neural network approximator dynamically adjusts three parameters of a weight vector of the neural network of the Gaussian radial basis function neural network approximator, a central point vector of the Gaussian function and a variance vector of the Gaussian function on line.
In the above method for controlling feedback of state of soft robot based on dielectric elastomer actuator, as a preferable solution, the dynamic control model of the dielectric elastomer actuator is:
Figure GDA0003345727000000031
wherein λ is a state quantity output by the dielectric elastomer actuator, i.e. a ratio of a transverse length value after deformation of the dielectric elastic film to a transverse length value before deformation;
Figure GDA0003345727000000032
and
Figure GDA0003345727000000033
first order differential and second order differential of the state quantity λ are respectively expressed; u is the control input of the dielectric elastomer actuator, namely a state feedback intelligent controller to be established; f. of1(λ)、f2(lambda) and f3(λ) is an unknown system function, whose respective expression is as follows:
Figure GDA0003345727000000034
wherein, the operation abbreviations
Figure GDA0003345727000000035
L is a value of a transverse length before deformation of the dielectric elastic film, L3Thickness of the dielectric elastic film before deformation; p is the transverse tension to which the dielectric elastic film is subjected when deformed; κ is the dielectric constant of the dielectric elastomer actuator; j. the design is a squaremIs the limit of the transverse deformation ratio of the dielectric elastomer actuator after deformation relative to the actuator before deformation; ρ is the density of the dielectric elastomer actuator; c is the damping coefficient of the dielectric elastomer actuator;
Figure GDA0003345727000000036
is the amount of change in the deformation process of the dielectric constant κ over the state quantity λ.
In the state feedback control method of the soft robot based on the dielectric elastomer actuator, as a preferable scheme, the state feedback intelligent controller is:
Figure GDA0003345727000000037
wherein k is2And k3The second adjustable design parameter and the third adjustable design parameter of the state feedback intelligent controller u are respectively;
Figure GDA0003345727000000038
are respectively aimed at f1(λ)、f2(lambda) and f3(lambda) constructed Gaussian radial basis function neural network approximators, i.e.
Figure GDA0003345727000000039
Is directed to fi(λ) constructing a gaussian radial basis function neural network approximator, i being 1,2, 3; wherein, WiIs the weight vector expectation, G, of the radial basis function neural networkiIs a gaussian function vector expectation value Gi(λ,Ξi,Δi)，ΞiIs the central point vector expectation, Δ, of the Gaussian functioniIs a variance vector of a Gaussian functionAn expected value;
Figure GDA00033457270000000310
Figure GDA00033457270000000311
and
Figure GDA00033457270000000312
respectively, the expected value W of the weight vectoriGaussian function vector expected value GiCentral point vector expectation xi of Gaussian functioniSum gaussian function variance vector expected value Δi(ii) an approximate estimate of (d); the upper right corner mark T is a transposed symbol;
Figure GDA0003345727000000041
is a Gaussian radial basis function neural network approximator
Figure GDA0003345727000000042
Estimate fiThe upper bound value of the remainder of (lambda),
Figure GDA0003345727000000043
is the optimum value of the coefficient of the upper bound of the remainder, phiiIs that
Figure GDA0003345727000000044
1,2, 3;
abbreviation operator
Figure GDA0003345727000000045
Wherein e and
Figure GDA0003345727000000046
error is tracked for state, and e ═ λd-λ，
Figure GDA0003345727000000047
λd、
Figure GDA0003345727000000048
And
Figure GDA0003345727000000049
are respectively lambda,
Figure GDA00033457270000000410
And
Figure GDA00033457270000000411
to track the expected state value, k1Is a first adjustable design parameter of the state feedback intelligent controller; abbreviation operator
Figure GDA00033457270000000412
In the state feedback control method of the soft robot based on the dielectric elastomer actuator, as a preferred scheme, when the estimated parameters of the gaussian radial basis function neural network approximator are dynamically updated on line, the parameter update rate of the estimated parameters is as follows:
Figure GDA00033457270000000428
Figure GDA00033457270000000413
Figure GDA00033457270000000414
Figure GDA00033457270000000415
Figure GDA00033457270000000416
Figure GDA00033457270000000417
Figure GDA00033457270000000418
Figure GDA00033457270000000419
Figure GDA00033457270000000420
Figure GDA00033457270000000421
Figure GDA00033457270000000422
Figure GDA00033457270000000423
wherein the content of the first and second substances,
Figure GDA00033457270000000424
are respectively as
Figure GDA00033457270000000425
And phiiI is 1,2, 3;
Figure GDA00033457270000000426
approximate estimation values respectively representing gaussian function vectors
Figure GDA00033457270000000427
Differentiation values of xi and delta, respectively; gamma-shapedimAn adjustable update rate parameter of a Gaussian radial basis function neural network approximator, i is 1,2,3, and m is 1,2,3, 4; wherein, gamma is1m,Γ2m,Γ3m∈Rn×n，Γi4∈R4 ×4(ii) a n is the number of nodes of the radial basis function neural network, and the value range is a non-zero natural number.
In summary, the state feedback control method for the soft robot based on the dielectric elastomer actuator provided by the invention derives the trajectory tracking closed-loop control system of the soft robot by defining the state tracking error, so as to establish a dynamic control model of the dielectric elastomer actuator by using a virtual work simulation mode; secondly, constructing a Gaussian radial basis function neural network approximator for an unknown system function in the dynamic control model, and dynamically adjusting parameters of the Gaussian radial basis function neural network approximator on line by designing the parameter update rate of the Gaussian radial basis function neural network approximator to realize on-line approximation of the unknown system function; finally, a state feedback intelligent controller embedded with the Gaussian radial basis function neural network approximator is constructed in a state feedback control mode and used for dynamically adjusting the control input of the dielectric elastomer actuator on line so that the state tracking error converges to zero, and therefore the control target of the soft robot is achieved; the state feedback control method can effectively enhance the control precision and response speed of the control system, and effectively improve the adaptivity and intelligence of the soft robot based on the dielectric elastomer actuator.
Drawings
FIG. 1 is a diagram of a dielectric elastomer actuator before and after deformation; fig. 1(a) shows a state before deformation of the dielectric elastomer actuator, and fig. 1(b) shows a state after deformation of the dielectric elastomer actuator.
Fig. 2 is a schematic block diagram of a control logic structure of the dielectric elastomer actuator-based soft robot state feedback control method of the present invention, that is, a schematic block diagram of a dielectric elastomer actuator-based soft robot state feedback control system of the present invention.
Fig. 3 is a graph showing a trace of a state quantity λ output from a dielectric elastomer actuator in a simulation experiment according to the present invention.
FIG. 4 shows the state of the output of the dielectric elastomer actuator in the simulation experiment of the present inventionFirst order differential of quantity
Figure GDA0003345727000000051
Trace graph of (a).
FIG. 5 shows the state tracking errors e and e in the simulation experiment of the present invention
Figure GDA0003345727000000052
A graph of (a).
FIG. 6 shows the state quantities λ and λ of the simulation experiment of the present invention
Figure GDA0003345727000000053
The two-dimensional plane of (a) tracks the graph.
FIG. 7 is a graph of control input u for a dielectric elastomer actuator in a simulation of the present invention.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention will be described in further detail with reference to the accompanying drawings.
1. Summary of the invention
The invention provides a state feedback control method of a soft robot based on a dielectric elastomer actuator, which mainly aims at a locus tracking control task of the soft robot taking the dielectric elastomer as the actuator by utilizing a radial basis function neural network (RBF NN) and researches the design problem of a state feedback intelligent controller. Firstly, in order to realize the controller design of a dielectric elastomer actuator system, a dynamic control model of the dielectric elastomer actuator needs to be established by utilizing a virtual work simulation mode, wherein the model describes the elastic energy of the dielectric elastomer by utilizing a Gent model; then, constructing a Gaussian radial basis function neural network approximator (RBF NNs approximator for short) for the unknown system function in the dynamic control model to carry out online estimation; finally, on the basis of the RBF NNs approximator, in order to realize the track tracking control of the soft robot based on the dielectric elastomer actuator, a state feedback intelligent controller embedded with the Gaussian radial basis function neural network approximator is designed and constructed in a state feedback control mode to adjust and control the control input of the dielectric elastomer actuator, the parameter updating rate of the RBF NNs approximator is designed to dynamically update and adjust the estimated parameters of the RBF NNs approximator to approach an unknown system function on line, so that the control input of the dielectric elastomer actuator is dynamically adjusted on line, and the state tracking error is converged to zero to realize the control target of the soft robot. According to the Lyapunov theorem, the closed-loop state tracking error control of the method can be proved to be converged to zero, which shows that the method effectively realizes the track tracking control target of the soft robot, and can intuitively see that the adaptability and intelligence of the dielectric elastomer actuator control system are enhanced through convergence simulation.
2. Dynamic control virtual model of dielectric elastomer actuator
And establishing a dynamic control model of the dielectric elastomer actuator based on a virtual work simulation mode. Fig. 1 is a state diagram before and after deformation of the dielectric elastomer actuator, in which fig. 1(a) is a state before deformation of the dielectric elastomer actuator, and fig. 1(b) is a state after deformation of the dielectric elastomer actuator. In FIG. 1, L1，L2And L3Is the length, width, height, l, of the dielectric elastomer actuator before deformation1，l2And l3Is the length, width, height, P, of the dielectric elastomer actuator after deformation1And P2The tensile force is applied to the dielectric elastomer actuator in the transverse length and width directions when the dielectric elastomer actuator is deformed, phi is the control voltage of the dielectric elastomer actuator,
Figure GDA0003345727000000061
is the corresponding electrical quantity.
Order to
Figure GDA0003345727000000062
And
Figure GDA0003345727000000063
λ1、λ2、λ3the ratio of the length, width and height of the dielectric elastomer actuator after deformation relative to the length, width and height of the actuator before deformation, respectivelyIncompressibility of dielectric elastomer actuators, having λ1λ2λ3＝1；
Electric quantity
Figure GDA0003345727000000064
The relationship to the voltage Φ is:
Figure GDA0003345727000000065
where T is the ambient temperature, κ (λ)1,λ2And T) is the dielectric constant of the dielectric elastomer actuator, which is λ1，λ2And a non-linear function of T.
When the length and width of the dielectric elastomer actuator occur separately
Figure GDA0003345727000000066
And
Figure GDA0003345727000000067
will change the tension respectively when deformed
Figure GDA0003345727000000068
And
Figure GDA0003345727000000069
the deformation process of the voltage is changed by
Figure GDA00033457270000000610
Therefore, the deformation process variation of the electric quantity
Figure GDA00033457270000000611
Comprises the following steps:
Figure GDA00033457270000000612
(1)
the inertial forces along the x and y directions of the coordinate system in the dielectric elastomer actuator are respectively ρ L2L3x2(d2λ1/dt2) And ρ L1L3y2(d2λ2/dt2) Damping forces are cx (d λ)1Dt) and cy (d lambda)2Dt). Thus, the inertial and damping forces act as:
Figure GDA0003345727000000072
Figure GDA0003345727000000073
where ρ is the density of the dielectric elastomer actuator and c is the damping coefficient of the dielectric elastomer actuator.
Change in deformation process of free energy W of dielectric elastomer actuator
Figure GDA0003345727000000077
Under the influence of voltage, pulling force, inertial force and damping force, then there are:
Figure GDA0003345727000000074
and W is the elastic energy WelaAnd electric energy WeleConsists of the following components:
Figure GDA0003345727000000075
where μ (T) is the shear modulus of the dielectric elastomer actuator as a function of ambient temperature T, JmIs the dielectric elastomer actuator deformation limit.
Note 1: the elastic energy in the above process is described using the Gent model. In addition to the Gent model, there are other models that describe elastic energy, such as the Neo-Hookean model, the Mooney-Rivlin model, the Ogden model, the Arruda-Boyce model, and the like.
Substituting formula (1) into formula (2) includes:
Figure GDA0003345727000000076
according to the formula (4), there are:
Figure GDA0003345727000000081
Figure GDA0003345727000000082
from equations (3), (5) and (6), the virtual model for dynamic control of the dielectric elastomer actuator can be derived as:
Figure GDA0003345727000000083
Figure GDA0003345727000000084
without loss of generality, the dielectric elastomer actuator can be generally considered isotropic, and for simplicity of representation, L can be generalized1＝L2＝L，P1＝P2P and λ1＝λ2λ. Thus, κ (λ)1,λ2T) can be reduced to κ (λ, T),
Figure GDA0003345727000000085
is the amount of change in the deformation process of the dielectric constant κ over the state quantity λ. According to (7) and (8), a kinetic model can be established for the DEA as follows:
Figure GDA0003345727000000086
for the sake of simplicity of representation, let κ (λ, T) and μ (T) be abbreviated as κ and μ hereinafter, the shape of the dielectric constant κ over the state quantity λThe variation of the process is abbreviated as
Figure GDA0003345727000000087
The system (9) is rewritten into an input/output form, and includes:
Figure GDA0003345727000000088
wherein λ is a state quantity output by the dielectric elastomer actuator (i.e. a ratio of a transverse length value after deformation of the dielectric elastic film to a transverse length value before deformation);
Figure GDA0003345727000000091
and
Figure GDA0003345727000000092
first order differential and second order differential of the state quantity λ are respectively expressed; u is phi2The control input of the dielectric elastomer actuator is the state feedback intelligent controller to be established; l is a value of a transverse length before deformation of the dielectric elastic film, L3Thickness of the dielectric elastic film before deformation; p is the transverse tension to which the dielectric elastic film is subjected when deformed; κ is the dielectric constant of the dielectric elastomer actuator; j. the design is a squaremIs the limit of the transverse deformation ratio of the dielectric elastomer actuator after deformation relative to the actuator before deformation; ρ is the density of the dielectric elastomer actuator; c is the damping coefficient of the dielectric elastomer actuator;
Figure GDA0003345727000000093
is the amount of change in the deformation process of the dielectric constant κ over the state quantity λ.
Order to
Figure GDA0003345727000000094
Wherein the abbreviation operator
Figure GDA0003345727000000095
To simplify the representation, let fi(λ) is abbreviated as fiAnd i is 1,2 and 3. Due to P, kappa and JmAnd c are difficult to measure, so that fiAnd (lambda) is an unknown system function, and i is 1,2 and 3.
Then according to (11), the system control module (10) of the DEA can be written as:
Figure GDA0003345727000000096
3. control target
The track tracking of the soft robot is to track the target track
Figure GDA0003345727000000097
The soft robot is driven to enable the real track to reach the target track by controlling the voltage loaded on the flexible electrode. Lambda [ alpha ]d、
Figure GDA0003345727000000098
Are respectively lambda,
Figure GDA0003345727000000099
To track the expected state values. Namely, for the system (12), the controller u is designed so that
Figure GDA00033457270000000910
According to the principle of feedback control, trajectory tracking may be defined as
Figure GDA00033457270000000911
e and
Figure GDA00033457270000000912
error is tracked for state, and order
Figure GDA0003345727000000101
Wherein k is1Is a first adjustable design parameter of the state feedback intelligent controller;
according to (15) and (12), there are
Figure GDA0003345727000000102
If the controller u can be designed for the system (16) so that t → ∞ has
Figure GDA0003345727000000103
That is, equation (5) is satisfied, and the control target of the trajectory tracking is completed.
4. Controller and approximator parameter update rate design
The present invention designs a state feedback controller based on an RBF NN approximator to enable a control target (17) of a system (16) to be established. Due to f in the system (16)iIs an unknown function, i is 1,2 and 3, and can not be directly used in the control process, so that the unknown function f is approximated on line by using a Gaussian RBF NN in the inventioni。
Before designing the controller, two important properties of the RBF NN are first introduced.
Property 1, linear parametric form: RBF NN belongs to a class of linear parametric neural networks and can be expressed in the form
Figure GDA0003345727000000104
Wherein W ═ W1,w2,...,wn]T∈RnIs a weight vector; n is the number of neural network nodes; g (λ, xi, Δ) [ [ G (λ, xi)1,δ1),...,g(x,ξn,δn)]T∈RnIs a Gaussian function vector, xi ═ xi1,ξ2,...,ξn]T∈RnIs the central point vector of the Gaussian function，Δ＝[δ1,δ2,...,δn]T∈RnIs a vector of variance of the Gaussian function, an
Figure GDA0003345727000000105
Property 1, best-fit property: according to the literature "J.park, I.W.Sandberg," Universal adaptation using radial-basis-function networks [ J.]As known from Neural computation, vol.3, No.2, pp.246-257,1991', RBF NN can approximate any non-linear function in a tight set to any small error range. In the literature "F.Girosi, T.Poggio," Networks and the best adaptation property [ J ]]The best approximation property of RBF NN, namely, the best approximation property, is set forth in Biological cybernetics, vol.63, No.2, pp.169-176,1990 ″
Figure GDA0003345727000000106
And
Figure GDA0003345727000000107
so that
Figure GDA0003345727000000108
Wherein eiIs an approximation error, and a sufficiently large n can make the error arbitrarily small, i.e.
Figure GDA0003345727000000111
For the sake of simplicity of presentation, the following description will be given
Figure GDA0003345727000000112
It is briefly described as
Figure GDA0003345727000000113
Note 2: in addition to gaussian functions, there are also some radial basis functions that are often used as activation functions for RBF NNs. Such as an inverted S-function, a multiple quadratic function, an inverted quadratic function, etc.
But RBF NN ideal parameters
Figure GDA0003345727000000114
And
Figure GDA0003345727000000115
is unknown and cannot be used in the control process. Thus, an RBF neural network is constructed
Figure GDA0003345727000000116
On-line approximation of unknown function fiWherein
Figure GDA0003345727000000117
And
Figure GDA0003345727000000118
are respectively the expected weight
Figure GDA0003345727000000119
Figure GDA00033457270000001110
And
Figure GDA00033457270000001111
is estimated. For the sake of simplicity of presentation, the following description will be given
Figure GDA00033457270000001112
It is briefly described as
Figure GDA00033457270000001113
That is to say, the parameter update rate of the designed gaussian radial basis function neural network approximator mainly performs online dynamic adjustment on three parameters, namely, a neural network weight vector, a gaussian central point vector and a gaussian variance vector of the gaussian radial basis function neural network approximator.
According to the best approximation property, there are
Figure GDA00033457270000001114
Wherein the content of the first and second substances,
Figure GDA00033457270000001115
Figure GDA00033457270000001116
Figure GDA00033457270000001117
is a Gaussian radial basis function neural network approximator
Figure GDA00033457270000001118
Estimate fiThe upper bound value of the remainder of (lambda),
Figure GDA00033457270000001119
is the optimum value of the coefficient of the upper bound of the remainder, phiiIs that
Figure GDA00033457270000001120
1,2, 3;
wherein the content of the first and second substances,
Figure GDA00033457270000001121
are respectively aimed at f1(λ)、f2(lambda) and f3(lambda) constructed Gaussian radial basis function neural network approximators, i.e.
Figure GDA00033457270000001122
Is directed to fi(λ) constructing a gaussian radial basis function neural network approximator, i being 1,2, 3; wherein, WiIs the weight vector expectation, G, of the radial basis function neural networkiIs a gaussian function vector expectation value Gi(λ,Ξi,Δi)，ΞiIs the central point vector expectation, Δ, of the Gaussian functioniIs a gaussian function variance vector expected value;
Figure GDA00033457270000001123
Figure GDA00033457270000001124
and
Figure GDA00033457270000001125
respectively, the expected value W of the weight vectoriGaussian function vector expected value GiCentral point vector expectation xi of Gaussian functioniSum gaussian function variance vector expected value Δi(ii) an approximate estimate of (d); the upper right hand symbol T is the transposed symbol.
For the system (8), the following intelligent controllers are designed using RBF NNs.
Figure GDA00033457270000001126
Wherein k is2And k3Respectively, a second adjustable design parameter and a third adjustable design parameter of the state feedback intelligent controller; phi is aiIs that
Figure GDA00033457270000001127
Is approximated by an estimate of (a), defining
Figure GDA00033457270000001128
For the estimated parameters in the intelligent controller (21), the following update rate is designed
Figure GDA0003345727000000121
Figure GDA0003345727000000122
Figure GDA0003345727000000123
Figure GDA0003345727000000124
Figure GDA0003345727000000125
Figure GDA0003345727000000126
Figure GDA0003345727000000127
Figure GDA0003345727000000128
Figure GDA0003345727000000129
Figure GDA00033457270000001210
Figure GDA00033457270000001211
Figure GDA00033457270000001212
Wherein the content of the first and second substances,
Figure GDA00033457270000001213
are respectively as
Figure GDA00033457270000001214
And phiiI is 1,2, 3;
Figure GDA00033457270000001215
approximate estimation values respectively representing gaussian function vectors
Figure GDA00033457270000001216
Differentiation values of xi and delta, respectively; gamma-shapedimAn adjustable update rate parameter of a Gaussian radial basis function neural network approximator, i is 1,2,3, and m is 1,2,3, 4; wherein, gamma is1m,Γ2m,Γ3m∈Rn×n，Γi4∈R4 ×4(ii) a n is the number of nodes of the radial basis function neural network, and the value range is a non-zero natural number.
From the DEA system, the controller designed and the estimated parameter update rate, it can be seen that the system control structure is shown in fig. 2.
Theorem 1: for the system (16), a controller (21) and parameter update rates (22) - (33) are applied, with appropriate selection of the design parameter kiNumber of neural network nodes n and update rate parameter
Figure GDA00033457270000001217
Can make the system closed-loop state error e and
Figure GDA00033457270000001218
and converging to zero, thereby realizing the track tracking control target of the soft robot.
And (3) convergence derivation:
substituting the formula (21) into the formula (16), and arranging the formula (20) according to
Figure GDA0003345727000000131
Consider the following candidate lyapunov function:
Figure GDA0003345727000000132
the two sides of formula (27) are derived to obtain
Figure GDA0003345727000000133
Substituting expressions (14) - (26) into expression (28) to simplify
Figure GDA0003345727000000134
According to
Figure GDA0003345727000000135
The (37) is simplified to obtain
Figure GDA0003345727000000136
Selecting a suitable k2So that
Figure GDA0003345727000000141
Then there is
Figure GDA0003345727000000142
Wherein if and only if r is 0 has
Figure GDA0003345727000000143
By combining the above analysis and according to the Lyapunov stability theory, it can be known that r belongs to L2And t → ∞ has r → 0, thus obtaining e and
Figure GDA00033457270000001410
converges to 0 with t → ∞. Therefore, the method effectively realizes the track tracking control target of the software robot with DEA as the driver.
5. Simulation verification
The simulation research is carried out on the controller designed by DEA-driven soft robot trajectory tracking to verify the correctness and the validity of the controller.
According to the literature "GUGGI K.Dielctric elastomer activators [ J]Chemistry,2001 ", the following DEA model parameters were selected: l is 0.02m, L3＝0.01m，ρ＝960kg/m3，Jm＝70，μ(T)＝0.097MPa，c＝1.2。
According to the experimental results in the documents "SHENG J J, CHEN H L, LIU L, et al, dynamic electrochemical performance of visco-electronic direct reactors [ J ], Journal of Applied Physics,2013,114(13): 134101-1-134101-8", there are
Figure GDA0003345727000000144
Figure GDA0003345727000000145
Wherein κ0＝8.85×10-12(F/m) is the vacuum dielectric constant; kappa∞＝2.1，T＝300，aw＝-0.1658，bw＝-0.04086，cw＝-0.003027。
Selecting a tracking target track as lambdad＝-0.5cost+2，
Figure GDA0003345727000000146
And
Figure GDA0003345727000000147
initial quantity of system state is lambda01.5 and
Figure GDA0003345727000000148
thus, a tensile force is
Figure GDA0003345727000000149
Selecting system design parameter n as 10, k1＝650，k2＝2，k3＝2， Γ im1,2,3,4, {10} (i ═ 1,2, 3; m ═ 1,2,3, 4.); setting the weight, center point, variance, and φ of RBF NNs by Matlab's rand functioniThe initial values of (a) are:
W1:1.66,1.04,1.85,1.93,1.68,1.76,1.74,1.39,1.66,1.17；
Ξ1:1.71,1.03,1.28,1.05,1.10,1.82,1.69,1.32,1.95；
Δ1:1.44,1.38,1.77,1.80,1.19,1.49,1.45,1.65,1.71,1.75；
φ1:1.28,1.68,1.66,1.16；
W2:1.12,1.50,1.96,1.34,1.59,1.22,1.75,1.26,1.51,1.70；
Ξ2:1.89,1.96,1.55,1.14,1.15,1.26,1.84,1.25,1.81,1.24；
Δ2:1.93,1.35,1.20,1.25,1.62,1.47,1.35,1.83,1.59,1.55；
φ2:1.92,1.29,1.76,1.75；
W3:1.38,1.57,1.08,1.05,1.53,1.78,1.93,1.13,1.57,1.47；
Ξ3:1.01,1.34,1.16,1.79,1.31,1.53,1.17,1.60,1.26,1.65；
Δ3:1.69,1.75,1.45,1.08,1.23,1.91,1.15,1.83,1.54,2.00；
φ3:1.08,1.44,1.11,1.96。
during the control, these parameters are adjusted to the final values according to the update rates (14) - (25):
W1:238.19,238.45,238.16,237.64,237.83,238.02,237.69,238.03,238.00,237.69；
Ξ1:-1.55,-2.32,-2.61,-2.72,-2.62,-2.34,-2.17,-1.93,-2.16,-2.21；
Δ1:0.38,0.38,0.38,0.38,0.38,0.38,0.38,0.38,0.38,0.38；
φ1:562.03,279769.50,2692.40,2046.81；
W2:659.66,659.77,659.66,659.55,660.07,659.27,660.21,660.08,659.87,659.85；
Ξ2:1.45,1.92,2.24,1.97,1.88,1.66,1.45,1.69,1.84,1.65；
Δ2:1.48,1.62,1.31,1.59,1.61,1.56,1.43,1.47,1.47,1.54；
φ2:664.37,698916.20,2452.46,3599.24；
W3:124.89,127.29,125.28,127.66,125.29,124.81,125.24,124.64,126.30,125.10；
Ξ3:2.76,-1.04,2.77,-0.89,2.86,3.06,2.68,2.64,-0.59,2.25；
Δ3:0.23,0.23,0.23,0.23,0.23,0.23,0.23,0.23,0.23,0.23；
φ3:154.37,201223.21,698.31,761.36。
the results of the simulation, shown in fig. 3-7, where fig. 3 is a state quantity λ tracking curve; FIG. 4 is the first order differential of the state quantity λ
Figure GDA0003345727000000151
The tracking curve of (2); FIG. 5 is a state tracking error e and
Figure GDA0003345727000000152
a curve; FIG. 6 is the state quantities λ and
Figure GDA0003345727000000153
the two-dimensional plane tracking curve of (1); FIG. 7 is a control input u-curve for a dielectric elastomer actuator.
As can be seen from fig. 3 and 4, the system states λ and
Figure GDA0003345727000000154
very close to the target value lambdadAnd
Figure GDA0003345727000000155
as can be seen in FIG. 5, the state error falls within 0.015 within about 2 seconds, converging to 0 within about 5 seconds, which is consistent with the trajectory tracking of FIG. 6; it is shown from fig. 7 that the system control input u is smooth, smooth and stable during the control process, which is advantageous for practical system applications.
Therefore, the control strategy provided by the invention is correct and effective, has high response speed, and also shows that the track tracking control target of the software robot is realized.
6. Summary of the invention
The invention provides a state feedback control method of a soft robot based on a dielectric elastomer actuator, which provides an intelligent control strategy for the problem of the track tracking control of the soft robot based on the drive of the dielectric elastomer actuator. The control strategy is based on a dynamic model of a dielectric elastomer actuator, a soft robot track tracking closed-loop control system is derived by defining state tracking errors, so that the dynamic control model of the dielectric elastomer actuator is established in a virtual work simulation mode, and the model describes the elastic energy of the dielectric elastomer by using a Gent model; secondly, constructing a Gaussian radial basis function neural network approximator for an unknown system function in the dynamic control model, and dynamically adjusting parameters of the Gaussian radial basis function neural network approximator on line by designing the parameter update rate of the Gaussian radial basis function neural network approximator to realize on-line approximation of the unknown system function; and finally, constructing a state feedback intelligent controller embedded with the Gaussian radial basis function neural network approximator in a state feedback control mode, and dynamically adjusting the control input of the dielectric elastomer actuator on line to make the state tracking error converge to zero, thereby realizing the control target of the soft robot. According to the Lyapunov stability theory, the track tracking error of the closed-loop control error of the method is proved to be capable of effectively converging to zero when the control parameter, the RBF NNs node number and the update rate parameter are properly selected, and simulation research shows that the control strategy provided by the invention has good control precision and response speed. Therefore, the state feedback control method of the soft robot based on the dielectric elastomer actuator can effectively enhance the control precision and response speed of the control system and effectively improve the adaptivity and intelligence of the soft robot based on the dielectric elastomer actuator.
Finally, it is noted that the above-mentioned embodiments illustrate rather than limit the invention, and that, while the invention has been described with reference to preferred embodiments thereof, it will be understood by those skilled in the art that various changes in form and details may be made therein without departing from the spirit and scope of the invention as defined by the appended claims.

Claims (4)

1. A state feedback control method of a soft robot based on a dielectric elastomer actuator is characterized by comprising the following steps:
establishing a dynamic control model of the dielectric elastomer actuator by using a virtual work simulation mode; the dynamic control model of the dielectric elastomer actuator is as follows:
Figure FDA0003360021370000011
wherein λ is a state quantity output by the dielectric elastomer actuator, i.e. a ratio of a transverse length value after deformation of the dielectric elastic film to a transverse length value before deformation;
Figure FDA0003360021370000012
and
Figure FDA0003360021370000013
first order differential and second order differential of the state quantity λ are respectively expressed; u is the control input of the dielectric elastomer actuator, namely a state feedback intelligent controller to be established; f. of1(λ)、f2(lambda) and f3(λ) is an unknown system function, whose respective expression is as follows:
Figure FDA0003360021370000014
wherein, the operation abbreviations
Figure FDA0003360021370000015
L is a value of a transverse length before deformation of the dielectric elastic film，L3Thickness of the dielectric elastic film before deformation; p is the transverse tension to which the dielectric elastic film is subjected when deformed; κ is the dielectric constant of the dielectric elastomer actuator; j. the design is a squaremIs the limit of the transverse deformation ratio of the dielectric elastomer actuator after deformation relative to the actuator before deformation; ρ is the density of the dielectric elastomer actuator; c is the damping coefficient of the dielectric elastomer actuator;
Figure FDA0003360021370000016
is the deformation process variation of the dielectric constant k on the state quantity lambda;
constructing a Gaussian radial basis function neural network approximator for an unknown system function in a dynamic control model, and dynamically adjusting parameters of the Gaussian radial basis function neural network approximator on line by designing a parameter update rate of the Gaussian radial basis function neural network approximator to realize on-line approximation of the unknown system function;
and a state feedback intelligent controller embedded with the Gaussian radial basis function neural network approximator is constructed in a state feedback control mode and used for dynamically adjusting the control input of the dielectric elastomer actuator on line so that the state tracking error is converged to zero, thereby realizing the control target of the soft robot.
2. The dielectric elastomer actuator-based soft robot state feedback control method of claim 1, wherein the designed parameter update rate of the gaussian radial basis function neural network approximator is an on-line dynamic adjustment of three parameters of a weight vector of the gaussian radial basis function neural network approximator, a central point vector of the gaussian, and a variance vector of the gaussian.
3. The dielectric elastomer actuator based soft robot state feedback control method of claim 1, wherein the state feedback intelligent controller is:
Figure FDA0003360021370000021
wherein k is2And k3The second adjustable design parameter and the third adjustable design parameter of the state feedback intelligent controller u are respectively;
Figure FDA0003360021370000022
are respectively aimed at f1(λ)、f2(lambda) and f3(lambda) constructed Gaussian radial basis function neural network approximators, i.e.
Figure FDA0003360021370000023
Is directed to fi(λ) constructing a gaussian radial basis function neural network approximator, i being 1,2, 3; wherein, WiIs the weight vector expectation, G, of the radial basis function neural networkiIs a gaussian function vector expectation value Gi(λ,Ξi,Δi)，ΞiIs the central point vector expectation, Δ, of the Gaussian functioniIs a gaussian function variance vector expected value;
Figure FDA0003360021370000024
Figure FDA0003360021370000025
and
Figure FDA0003360021370000026
respectively, the expected value W of the weight vectoriGaussian function vector expected value GiCentral point vector expectation xi of Gaussian functioniSum gaussian function variance vector expected value Δi(ii) an approximate estimate of (d); the upper right corner mark T is a transposed symbol;
Figure FDA0003360021370000027
is a Gaussian radial basis function neural network approximator
Figure FDA0003360021370000028
Estimate fiThe upper bound value of the remainder of (lambda),
Figure FDA0003360021370000029
is the optimum value of the coefficient of the upper bound of the remainder, phiiIs that
Figure FDA00033600213700000210
1,2, 3;
abbreviation operator
Figure FDA00033600213700000211
Wherein e and
Figure FDA00033600213700000212
error is tracked for state, and e ═ λd-λ，
Figure FDA00033600213700000213
λd、
Figure FDA00033600213700000214
And
Figure FDA00033600213700000215
are respectively lambda,
Figure FDA00033600213700000216
And
Figure FDA00033600213700000217
to track the expected state value, k1Is a first adjustable design parameter of the state feedback intelligent controller; abbreviation operator
Figure FDA00033600213700000218
4. The dielectric elastomer actuator-based soft robot state feedback control method according to claim 3, wherein when dynamically updating the estimated parameters of the Gaussian radial basis function neural network approximator on-line, the parameter update rate of the estimated parameters is:
Figure FDA00033600213700000219
Figure FDA00033600213700000220
Figure FDA00033600213700000221
Figure FDA00033600213700000222
Figure FDA00033600213700000223
Figure FDA00033600213700000224
Figure FDA00033600213700000225
Figure FDA0003360021370000031
Figure FDA0003360021370000032
Figure FDA0003360021370000033
Figure FDA0003360021370000034
Figure FDA0003360021370000035
wherein the content of the first and second substances,
Figure FDA0003360021370000036
are respectively as
Figure FDA0003360021370000037
And phiiI is 1,2, 3;
Figure FDA0003360021370000038
approximate estimation values respectively representing gaussian function vectors
Figure FDA0003360021370000039
Differentiation values of xi and delta, respectively; gamma-shapedimAn adjustable update rate parameter of a Gaussian radial basis function neural network approximator, i is 1,2,3, and m is 1,2,3, 4; wherein, gamma is1m,Γ2m,Γ3m∈Rn×n，Γi4∈R4×4(ii) a n is the number of nodes of the radial basis function neural network, and the value range is a non-zero natural number.
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