CN109465825B - RBF neural network self-adaptive dynamic surface control method for flexible joint of mechanical arm - Google Patents

RBF neural network self-adaptive dynamic surface control method for flexible joint of mechanical arm Download PDF

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CN109465825B
CN109465825B CN201811330519.0A CN201811330519A CN109465825B CN 109465825 B CN109465825 B CN 109465825B CN 201811330519 A CN201811330519 A CN 201811330519A CN 109465825 B CN109465825 B CN 109465825B
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neural network
rbf neural
mechanical arm
flexible joint
tracking error
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CN109465825A (en
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李鸿一
肖文彬
周琪
鲁仁全
曹亮
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Guangdong 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
    • B25J9/163—Programme controls characterised by the control loop learning, adaptive, model based, rule based expert control

Abstract

The invention relates to a RBF neural network self-adaptive dynamic surface control method based on a flexible joint of a mechanical arm, and belongs to the technical field of artificial intelligence and intelligent control. The invention carries out modeling aiming at the flexible joint of the mechanical arm and designs the controller by combining the RBF neural network and the dynamic surface technology by using a self-adaptive control method. The RBF neural network is used for compensating the uncertainty of system parameters, the weight of the neural network is adjusted by using the self-adaptive law, the approximation capability of the RBF neural network to a nonlinear function is improved, the requirement on a precise dynamic model of the mechanical arm is eliminated, and a position tracking control algorithm suitable for a light mechanical arm with a flexible joint is researched. Finally, the design controller is verified through a simulation example, which shows that the invention can ensure that the joint can effectively track a given signal under the condition of limited output torque of the robot arm, the tracking error is not restricted within a certain range, and all signals are semi-globally bounded.

Description

RBF neural network self-adaptive dynamic surface control method for flexible joint of mechanical arm
Technical Field
The invention belongs to the field of artificial intelligence and control, and particularly relates to a RBF neural network self-adaptive dynamic surface control method for a flexible joint of a mechanical arm.
Background
Since the 60's of the 20 th century, robots have found wide industrial applications such as machining, arc welding, spot welding, painting, assembly, inspection, aerospace, space exploration, and the like. Industrial robots have long occupied an important position in industrial automation lines. However, as the application range of the robot is expanded, people find that the fixed and repetitive operation of the industrial robot cannot meet the requirements of more and more flexible tasks, such as the operation that the work task changes along with the position and the task requirements are different. Therefore, engineering designs a light mechanical arm that can meet such special requirements. The light mechanical arm is mostly applied to the unknown working environment in advance, the accurate position of the peripheral object relative to the light mechanical arm can not be obtained, the high positioning precision is needed, and the flexibility of the equipment is considered in the design process based on the consideration of the human-computer exchange safety. The light mechanical arm is generally formed by connecting a plurality of connecting rods through rotating or moving joints, the flexibility of the robot mainly comes from the flexibility of the connecting rods and the flexibility of the joints, the flexibility of the joints refers to the twisting deformation of a transmission mechanism and a joint rotating shaft, and generally, the flexible joints can be used as connecting objects such as springs and the like by using elastic objects. Although the flexibility of the joint can be reduced by mechanical design, the effect is not very good from the present results.
In industrial production, it is not sufficient to consider only the movement of the rigid bars, and not the elasticity into the movement, and even in some designs, the flexibility of the mechanical arms is not considered, resulting in instability of the closed loop system. In response to this problem, in recent years, the problem of motion control of robots having flexible joints has been studied by many experts and has been developed. First, in 1987, Spong proposed a simplified model of a flexible joint, i.e., treating it as a linear spring, which greatly motivated control studies of flexible joint robots. The PD or PID and the flexible joint compensation controller are applied to the flexible robot joint control according to the control idea of the rigid robot. However, this method is relatively complicated in the aspect of stability certification, and although the controller is relatively simple, the control accuracy is not high; with the progress of research, feedback linearization and robust adaptive control for controlling a flexible joint robot are noticed by many scholars. Although the feedback linearization method is a feasible method for controlling the flexible joint robot, the tracking performance of the method depends heavily on the accuracy of the system model, however, the accurate modeling of the flexible joint robot is difficult. Therefore, feedback linearization and methods such as a self-adaptive technology, fuzzy control, a neural network and the like are combined to eliminate the accurate requirement on the model, the robot with the flexible joint can be well controlled, and a relatively ideal effect is achieved. For a flexible joint robot, besides uncertainty and output feedback control research existing in a system, due to the limitation of output power of a joint motor, saturation limitation exists on output torque of a robot joint driver, and although research is carried out in many documents, no better solution exists at present.
Disclosure of Invention
Aiming at the problems in the prior art, the invention comprehensively considers the problems of joint flexibility, limited output torque, external interference and the like, and provides the self-adaptive dynamic surface control method based on the RBF neural network for the flexible joint of the mechanical arm.
In order to realize the task, the invention adopts the following technical scheme:
a RBF neural network self-adaptive dynamic surface control method for a flexible joint of a mechanical arm comprises the following steps:
step 1, modeling a flexible joint of a mechanical arm;
step 2, converting an equation model obtained by modeling into a state equation according to the physical characteristics of the flexible joint of the mechanical arm;
step 3, defining a first tracking error, designing a first virtual controller, inputting a signal of the first virtual controller into a first-order low-pass filter to obtain a new state variable x2dThe first virtual controller is replaced to carry out the next calculation, and the calculation amount is reduced;
step 4, defining a second tracking error, approximating a nonlinear function containing uncertain parameters by using the first RBF neural network, and designing a self-adaptive law of the first RBF neural network weight; designing a second virtual controller, and passing the signal of the second virtual controller through a second first-order low-pass filter to obtain a new variable x3d；
Step 5, defining a third tracking error, designing a third virtual controller, inputting a signal of the third virtual controller into a third first-order low-pass filter to obtain a new state variable x4d；
Step 6, defining a fourth tracking error, and performing compensation approximation on the fourth tracking error by using a second RBF neural network according to the state equation to design a self-adaptive law of the weight of the second RBF neural network;
and 7, designing an actual controller.
Further, the equation model of the flexible joint of the mechanical arm is as follows:
Figure GDA0003103984710000031
in the above formula, q and theta represent the rotation angle positions of the connecting rod and the rotor respectively,
Figure GDA0003103984710000032
rotational speed expressed as the rotational angle of the connecting rod, I, J representing the moment of inertia of the connecting rod and the rotor, respectively, CHRepresenting the gravity vector of the robot arm, BmExpressing the viscous friction coefficient of a motor shaft, K expressing the joint stiffness coefficient of the mechanical arm, M, g and l respectively representing the mass of a connecting rod, the gravity acceleration and the distance from the joint to the mass center of the connecting rod, and taumIs the motor torque input;
Figure GDA0003103984710000033
the first and second derivatives of q are provided,
Figure GDA0003103984710000034
the second derivative of theta.
Further, the state equation in step 2 is:
Figure GDA0003103984710000035
in the above formula, x ═ x1,x2,x3,x4]TIn order to be a state variable, the state variable,
Figure GDA0003103984710000036
are respectively x1,x2,x3,x4The first derivative of (a) is,
Figure GDA0003103984710000037
Figure GDA0003103984710000038
delta (t) is external bounded interference and satisfies that delta (t) < rho, and rho is a constant; g (x)1,x2)，f(x1,x3) A non-linear function of unknown but bounded specific form; suppose that the flexible joint of the mechanical arm needs to track an input signal x1dFirst derivative, second derivative thereof
Figure GDA0003103984710000039
All exist and satisfy
Figure GDA00031039847100000310
ξ is a constant.
Further, the step 3 specifically includes:
step 3.1, defining a first tracking error: s1＝x1-x1dDerived by derivation
Figure GDA00031039847100000311
The virtual controller is taken as:
Figure GDA00031039847100000312
wherein c is1To design parameters, and satisfy c1＞0；
Step 3.2, mixing
Figure GDA00031039847100000313
Input into the first order low-pass filter to obtain new state variable x2d；
Step 3.3, defining the filtering error e of the first virtual controller2The expression is as follows:
Figure GDA00031039847100000314
then its derivative can be found as:
Figure GDA0003103984710000041
further, the step 4 specifically includes:
step 4.1, define a second tracking error: s2＝x2-x2dAnd obtaining a derivative:
Figure GDA0003103984710000042
step 4.2, establishing a first RBF neural network;
step 4.3, approximating the unknown function by using the first RBF neural network described in step 4.2
Figure GDA0003103984710000043
Figure GDA0003103984710000044
Wherein
Figure GDA0003103984710000045
Represents the optimal value h when the first RBF neural network weight approaches the unknown function1(x1,x2) Is the neuron activation function of the neural network, epsilon1 *Representing the error of the approximation;
step 4.4, design the second virtual controller
Figure GDA0003103984710000046
Wherein
Figure GDA0003103984710000047
Is that
Figure GDA0003103984710000048
Estimated value of k2，λ2、c2Is a design parameter and satisfies k2＞0，λ2＞0、c2＞0；
Step 4.5, design adaptive law
Figure GDA0003103984710000049
Wherein gamma is1Is a positive definite symmetric matrix, σ1To design parameters and satisfy sigma1＞0，K2To the tracking error S2A constraint on size;
step 4.6, mixing
Figure GDA00031039847100000410
Input into a second first-order low-pass filter to obtain a new variable x3d；
Step 4.7, defining the filtering error e of the second virtual controller3The expression is as follows:
Figure GDA00031039847100000411
then its derivative can be found as:
Figure GDA00031039847100000412
further, the step 5 specifically includes:
step 5.1, define a third tracking error: s3＝x3-x3dDerived by derivation
Figure GDA00031039847100000413
Designing virtual controllers
Figure GDA00031039847100000414
λ3、c3Is a design parameter and satisfies lambda3＞0，c3＞0；
Step 5.2, mixing
Figure GDA00031039847100000415
Inputting the third first-order low-pass filter to obtain a new state variable x4d；
Step 5.3, defining a third virtual controller filtering error e4The expression is as follows:
Figure GDA00031039847100000416
then its derivative can be found as:
Figure GDA0003103984710000051
further, the step 6 specifically includes:
step 6.1, define the fourth tracking error: s4＝x4-x4dDerived by derivation
Figure GDA0003103984710000052
Step 6.2, constructing a second RBF neural network pair nonlinear function
Figure GDA0003103984710000053
Carrying out approximation:
Figure GDA0003103984710000054
wherein
Figure GDA0003103984710000055
Representing the approximation of the optimal value of the nonlinear function, ε, by a second RBF neural network weight2 *Error of the approximation, h2(x1,x3) An activation function for neurons of a second RBF neural network;
step 6.3, define the vector
Figure GDA0003103984710000056
Where ρ is an upper bound of Δ (t) external interference, ω is a design constant and satisfies ω > 0, c4To design a constant, and satisfy c4And (3) designing an adaptive law as follows:
Figure GDA0003103984710000057
wherein gamma is2Is a positive definite symmetric matrix, σ2To design the parameters, sigma is satisfied2＞0,K4To the error S4A constraint on size;
further, the step 7 specifically includes:
the actual controller is designed as follows:
Figure GDA0003103984710000058
wherein
Figure GDA0003103984710000059
Is theta2Estimate of (a) ("lambda4Is a design parameter and satisfies lambda4＞0。
Compared with the prior art, the invention has the following technical characteristics:
1. according to the invention, by combining a 'recursive' design method of the backstepping control, a first-order filter is introduced in each step to calculate the derivative of the virtual control item to obtain a new state variable, so that the problem of item expansion caused by repeated derivation of the virtual control by the backstepping control method is avoided; the designed controller and self-adaptive law can further ensure that the tracking error of the joint rotation angle is restricted within a certain range.
2. Since an accurate mathematical model of the flexible joint robot arm cannot be obtained, when a stable flexible joint robot controller is designed and constructed, the influence caused by external interference and compensation parameter uncertainty needs to be considered, and a position control algorithm suitable for the light robot arm with the flexible joint is researched. The method makes full use of the approximation capability of the RBF neural network to the nonlinear function, designs the controller by using the approximation error, and adjusts the weight of the neural network by using the self-adaptive law, thereby eliminating the need of a precise dynamic model of the mechanical arm. Finally, the design controller is verified through a simulation example, the effectiveness of the method is proved, the joint can be ensured to effectively track a given signal under the condition that the output torque of the mechanical arm is limited, and all signals are semi-globally bounded.
Drawings
FIG. 1 is a schematic structural view of a flexible joint of a robotic arm;
FIG. 2 is a schematic diagram of a neural network;
FIG. 3 is a diagram illustrating a neural network identifying a non-linear structure;
FIG. 4 is a schematic diagram of adaptive control of an RBF neural network;
FIG. 5 is a schematic illustration of a tracking effect;
FIG. 6 is a schematic of a tracking error;
FIG. 7 is a neural network approximating a non-linear function
Figure GDA0003103984710000061
Schematic diagram of；
FIG. 8 is a neural network approximating a non-linear function
Figure GDA0003103984710000062
A schematic diagram;
FIG. 9 is an output control torque indicator τmIntention is.
Detailed Description
The invention provides a dynamic surface control method of a self-adaptive RBF neural network based on Lyapunov stability analysis aiming at a mechanical arm with a flexible joint, the method fully utilizes the approximation capability of the RBF neural network to compensate the problem caused by inaccurate modeling, eliminates the requirement on an accurate dynamic model, solves the problem of calculation expansion caused by backstepping adopted by the traditional self-adaptive control by utilizing a dynamic surface technology, and provides a proper controller by considering the limited torque output of the mechanical arm and the uncertain external interference; the virtual controller virtually disassembles a complex mechanical arm system into a plurality of subsystems, decomposes modeling, controller design and stability analysis of the complex system into the subsystems, and finally dynamically connects the subsystems through virtual power flow. The specific method of the invention is as follows:
a RBF neural network self-adaptive dynamic surface control method for a flexible joint of a mechanical arm comprises the following steps:
step 1, modeling is carried out on flexible joints of mechanical arms
This scheme adopts the flexible mechanical arm joint structure of current this field generally accepted: the "spring-link" model, as shown in fig. 1, establishes an equation model of the flexible joint of the mechanical arm as follows:
Figure GDA0003103984710000071
in the above formula, q and theta represent the rotation angle positions of the connecting rod and the rotor respectively,
Figure GDA0003103984710000072
rotational speed, expressed as the angle of rotation of the connecting rod, I, J respectivelyMoment of inertia of connecting rod and rotor, CHRepresenting the gravity vector of the robot arm, BmExpressing the viscous friction coefficient of a motor shaft, K expressing the joint stiffness coefficient of the mechanical arm, M, g and l respectively representing the mass of a connecting rod, the gravity acceleration and the distance from the joint to the mass center of the connecting rod, and taumIs the motor torque input;
Figure GDA0003103984710000073
the first and second derivatives of q are provided,
Figure GDA0003103984710000074
is the second derivative of θ; in the following formula, the parameter superscript dots represent the derivative of the parameter, and the number of dots is the order of the derivative, which is not described again.
The joint stiffness is larger, the joint flexibility is small, and q and theta are closer; the smaller K indicates that the joint stiffness is smaller, the joint flexibility is larger, and the difference between q and theta is larger.
The control target of the scheme is that a given signal can be tracked based on the rotation angle of the connecting rod, the tracking error is small, and all signals of the system are guaranteed to be semi-globally bounded.
Step 2, converting the equation model established in the step 1 into a state equation according to the physical characteristics of the flexible joint of the mechanical arm, wherein the state equation is specifically expressed as follows:
Figure GDA0003103984710000075
in the above formula, x ═ x1,x2,x3,x4]TIn order to be a state variable, the state variable,
Figure GDA0003103984710000076
are respectively x1,x2,x3,x4The first derivative of (a) is,
Figure GDA0003103984710000077
Figure GDA0003103984710000078
delta (t) is external bounded interference and satisfies that delta (t) < rho, and rho is a constant; g (x)1,x2)，f(x1,x3) Is a non-linear function of unknown specific form but bounded, where the unknown specific form is due to g (x)1,x2)，f(x1,x3) C in the expressionH、BmThe specific form of K, J, I is unknown, and in the present method a non-linear function g (x)1,x2)，f(x1,x3) Respectively utilizing RBF neural networks to carry out approximation; suppose that the flexible joint of the mechanical arm needs to track an input signal x1dFirst derivative, second derivative thereof
Figure GDA0003103984710000081
All exist and satisfy
Figure GDA0003103984710000082
ξ is a constant.
And aiming at the state equation established in the step, combining the recursion of a backstepping method and designing the controller by utilizing the dynamic surface technology.
Step 3, defining a first tracking error, designing a first virtual controller, inputting a signal of the first virtual controller into a first-order low-pass filter to obtain a new state variable x2dThe first virtual controller is replaced to carry out the next calculation, and the calculation amount is reduced; the method comprises the following specific steps:
step 3.1, defining a first tracking error: s1＝x1-x1dDerived by derivation
Figure GDA0003103984710000083
The virtual controller is taken as:
Figure GDA0003103984710000084
wherein c is1To design parameters, and satisfy c1＞0。
Step 3.2, mixing
Figure GDA0003103984710000085
Input into the first order low-pass filter to obtain new state variable x2d(ii) a Wherein the first order filter is represented as:
Figure GDA0003103984710000086
in the above formula, x2dFor new state variables obtained by a first order filter, τ2Is a filter time constant of a filter, wherein
Figure GDA0003103984710000087
Representing through virtual controllers
Figure GDA0003103984710000088
X of the filtered signal2dAre equal; by x2dInstead of the former
Figure GDA0003103984710000089
In deriving the actual controller, the next calculation is performed.
Step 3.3, defining the filtering error e of the first virtual controller2The expression is as follows:
Figure GDA00031039847100000810
then its derivative can be found as:
Figure GDA00031039847100000811
step 4, defining a second tracking error, wherein the system state equation contains a nonlinear function, so that the nonlinear function containing uncertain parameters is approximated by using a first RBF neural network to construct a Lyapunov function, and the adaptive law of the weight of the neural network is designed through the stability analysis of the Lyapunov so as to reduce the approximation error as much as possible; designing a second virtual controller, and passing the signal of the second virtual controller through a second first-order low-pass filter to obtain a new variable x3d(ii) a Utensil for cleaning buttockThe body is as follows:
step 4.1, define a second tracking error: s2＝x2-x2dAnd obtaining a derivative:
Figure GDA0003103984710000091
step 4.2, establishing a first RBF neural network
Due to the fact that
Figure GDA0003103984710000092
Contains all non-linear terms g (x)1,x2) Therefore, the first RBF neural network is adopted to approximate the first RBF neural network, the first RBF neural network is a multi-input single-output neural network, the structure of the first RBF neural network is shown in fig. 2, and the approximation principle of the first RBF neural network is shown in fig. 3 and 4. In an RBF network architecture, X ═ X1,x2,…,xi]TSetting radial basis vector H of hidden layer of RBF network as [ H ] for input layer vector of network1,h2,…,hn]TWherein h isjIs a Gaussian function, is a neuron activation function of a neural network, and has the expression:
Figure GDA0003103984710000093
j is 1,2, …, n; wherein the center vector of the j-th node of the network is cj＝[cj1,cj2,…,cjn]TThe base width vector of the network is: bj＝[bj1,bj2,…,bjn]T，bjIs the sum of the width parameters of node j and has bjIs greater than 0. The network weight vector is: thetaj＝[θj1,θj2,…,θjn]T. The output of the RBF network is: y isn(t)＝θ1h1+θ2h2+…+θnhn。
Step 4.3, because
Figure GDA0003103984710000094
Unknown, therefore using steps4.2 approximating the unknown function by the first RBF neural network
Figure GDA0003103984710000095
Wherein
Figure GDA0003103984710000096
Represents the optimal value h when the weight of the neural network approaches the unknown function1(x1,x2) Is a neuron activation function of the first RBF neural network1 *The effect graph of the nonlinear function approximated by the neural network is shown in fig. 8.
Step 4.4, design the second virtual controller
Figure GDA0003103984710000097
Wherein
Figure GDA0003103984710000098
Is that
Figure GDA0003103984710000099
Estimated value of k2，λ2、c2Is a design parameter and satisfies lambda2＞0、c2＞0。
Step 4.5, design adaptive law
Figure GDA00031039847100000910
Wherein gamma is1Is a positive definite symmetric matrix, σ1To design parameters and satisfy sigma1＞0，K2To the tracking error S2Size constraints.
Step 4.6, mixing
Figure GDA00031039847100000911
Input into a second first-order low-pass filter to obtain a new variable x3d(ii) a Wherein the filter can be expressed as:
Figure GDA0003103984710000101
wherein, tau3For the filter time constant of the filter, a new state variable x is obtained3d；
Step 4.7, defining the filtering error e of the second virtual controller3The expression is as follows:
Figure GDA0003103984710000102
then its derivative can be found as:
Figure GDA0003103984710000103
step 5, defining a third tracking error, designing a third virtual controller, inputting a signal of the third virtual controller into a third first-order low-pass filter to obtain a new state variable x4dThe method comprises the following specific steps:
step 5.1, define a third tracking error: s3＝x3-x3dDerived by derivation
Figure GDA0003103984710000104
Designing virtual controllers
Figure GDA0003103984710000105
λ3、c3Is a design parameter and satisfies lambda3＞0，c3＞0；
Step 5.2, mixing
Figure GDA00031039847100001010
Inputting the third first-order low-pass filter to obtain a new state variable x4d(ii) a Wherein the filter can be expressed as:
Figure GDA0003103984710000106
wherein, tau4For the filter time constant of the filter, a new state variable x is obtained4d。
Step 5.3, defining a third virtual controller filtering error e4Expression ofComprises the following steps:
Figure GDA0003103984710000107
then its derivative can be found as:
Figure GDA0003103984710000108
and 6, defining a fourth tracking error, generating a nonlinear function in the state equation according to the state equation, performing compensation approximation on the nonlinear function by using a second RBF neural network, wherein the approximation effect is as shown in FIG. 8, and designing an interference compensation term by considering external interference delta (t). Constructing a Lyapunov function, and designing an adaptive law of a neural network weight through the stability analysis of the Lyapunov function to make an approximation error of a nonlinear function as small as possible; the method comprises the following specific steps:
step 6.1, define the fourth tracking error: s4＝x4-x4dDerived by derivation
Figure GDA0003103984710000109
Step 6.2, since m2And f (x)1,x3) Unknown, therefore constructing a second RBF neural network versus a nonlinear function
Figure GDA0003103984710000111
Carrying out approximation:
Figure GDA0003103984710000112
wherein
Figure GDA0003103984710000113
Representing the approximation of the optimal value of the nonlinear function, ε, by a second RBF neural network weight2 *Error of the approximation, h2(x1,x3) For the activation function of the neurons of the second RBF neural network, the neural network approximates a non-linear function effect graph, as shown in FIG. 8; the first RBF neural network and the second RBF neural network have the same network structure, and the difference between the step 6.2 and the step 4.3 is that the input and the output of the two neural networks are different, and the first neural network is differentIs x1，x2Output is
Figure GDA0003103984710000114
The input to the second neural network is x1，x3The output is
Figure GDA0003103984710000115
In the scheme, the parameter upper right corner plus x represents the approximate optimal value.
Step 6.3, define the vector
Figure GDA0003103984710000116
Where ρ is an upper bound of Δ (t) external interference, ω is a design constant and satisfies ω > 0, c4To design a constant, and satisfy c4And (3) designing an adaptive law as follows:
Figure GDA0003103984710000117
wherein gamma is2Is a positive definite symmetric matrix, σ2To design the parameters, sigma is satisfied2＞0,K4To the error S4Size constraints.
And 7, designing an actual controller as follows:
Figure GDA0003103984710000118
wherein
Figure GDA0003103984710000119
Is theta2Estimate of (a) ("lambda4Is a design parameter and satisfies lambda4＞0。
In practical application, the torque controller input of the mechanical arm flexible joint motor is as follows: tau ismAs shown in fig. 9, the motor torque of the flexible joint of the mechanical arm is controlled to further affect the rotation of the link and the rotor of the joint, so that the link of the mechanical arm can track the given signal x1dThe tracking effect is shown in fig. 5, and the tracking error is shown in fig. 6.
Simulation experiment:
in this simulation experiment, the control target is to make the lever linkage angle q followTrace an ideal trajectory x1dLet sint assume that the external interference is Δ 0.2cos (2 t). According to the actual system, the physical parameters of the system in the model adopted in this example can be selected as: 1.0kg m when J is equal to I2，Mgl＝5N·m，K＝40N·m/rad，CH＝1N，Bm＝0.001，ω＝0.01，K2＝0.2，K4＝0.5，c1＝6，c2＝80，c3＝34，c4Q is 5, 0.35. The initial state of the system is selected as x ═ 0,0,0,0]TThe time constant of the filter is chosen to be tau2＝τ3＝τ40.005, take Γ1＝[2,2,2,2,2,2,2]Since external disturbances are taken into account at the rotor angle, Γ in the adaptive law when using the RBF to remove uncertainty errors2＝[10,10,10,10,10,10,10,0.5]. For two nonlinear functions to be approximated, the first RBF approximation network is selected as: 2-7-1. Due to the function to be approximated
Figure GDA0003103984710000121
In relation to two state variables, two inputs are selected, 7 neurons are selected, and finally a network output is obtained. All initial values are selected to be 0 according to x1，x3The value range of the Gaussian function is selected as follows: [ -2.4,2.4]I.e. by
Figure GDA0003103984710000122
Wherein the first RBF approximation network is selected as: 2-7-1. Due to the function to be approximated
Figure GDA0003103984710000123
In relation to two state variables, two inputs are selected, 7 neurons are selected, and finally a network output is obtained. All initial values are selected to be 0 according to x1，x3The value range of the Gaussian function is selected as follows: [ -2.4,2.4]I.e. by
Figure GDA0003103984710000124
And (4) analyzing results:
from the equation of state in step 2, consider the following tight set:
Figure GDA0003103984710000125
Figure GDA0003103984710000126
wherein q is an arbitrary positive number, and,
Figure GDA0003103984710000127
respectively represent
Figure GDA0003103984710000128
To theta1、
Figure GDA0003103984710000129
To theta2Estimation error in estimation is satisfied
Figure GDA00031039847100001210
Figure GDA00031039847100001211
Selecting Lyapunov function
Figure GDA00031039847100001212
According to the Lyapunov stability theorem, if the initial condition V (0) is less than or equal to q, adjusting the parameter ci，τi，ε，σ1，σ2，Γ1，Γ2All signals of the system can be made to be semi-globally consistent and bounded, and the tracking error of the system can be converged to be close to the origin. Note:
Figure GDA0003103984710000131
respectively represent x1d，x2d，x2d，x3d，x4dThe corresponding first derivative, and the second derivative. x is the number of2d(0)，x3d(0)，x4d(0) Denotes x2d，x3d，x4dThe initial value of (c).
In the method, a non-linear function g (x)1,x2)，f(x1,x3) The model is unknown in a specific form because all unknown parameters are contained and the actual flexible joint model of the mechanical arm is more complex. However, when the tracking control algorithm is simulated and verified, unknown parameters need to be given values, but the parameters are still nonlinear and need to be approximated by a neural network respectively.
The method adopts two different neural networks to respectively approximate two nonlinear functions, and the self-adaptive law is used for adjusting the weight theta when the neural networks approximate the nonlinear functions1Improving the approaching performance; the control law is divided into a virtual controller and an actual controller, which is specific to the backstepping method, the virtual controller is used for deriving actual control to ensure the stability of the system, and the virtual controller can be understood as ensuring the stability of each subsystem, namely the virtual controller is used for ensuring the stability of each subsystem
Figure GDA0003103984710000132
And finally, the deduced actual control can ensure the stability of the whole system, and the stability of the system can be proved according to the Lyapunov stability theorem.
When the scheme is applied to reality and a motor controller of the mechanical arm is designed, a virtual controller needs to be designed
Figure GDA0003103984710000133
Figure GDA0003103984710000134
Adaptive law:
Figure GDA0003103984710000135
and the actual controller: tau ismFor controlling the motor torque of the system. Finally, the controller can ensure that the system output, namely the connecting rod of the mechanical arm can track the given reference signal x1dThe tracking effect is shown in fig. 5, and the tracking error is shown in fig. 6.

Claims (7)

1. A RBF neural network self-adaptive dynamic surface control method of a flexible joint of a mechanical arm is characterized by comprising the following steps:
step 1, modeling a flexible joint of a mechanical arm; the equation model of the flexible joint of the mechanical arm is as follows:
Figure FDA0003103984700000011
in the above formula, q and theta represent the rotation angle positions of the connecting rod and the rotor respectively,
Figure FDA0003103984700000012
rotational speed expressed as the rotational angle of the connecting rod, I, J representing the moment of inertia of the connecting rod and the rotor, respectively, CHRepresenting the gravity vector of the robot arm, BmExpressing the viscous friction coefficient of a motor shaft, K expressing the joint stiffness coefficient of the mechanical arm, M, g and l respectively representing the mass of a connecting rod, the gravity acceleration and the distance from the joint to the mass center of the connecting rod, and taumIs the motor torque input;
Figure FDA0003103984700000013
the first and second derivatives of q are provided,
Figure FDA0003103984700000014
is the second derivative of θ;
step 2, converting an equation model obtained by modeling into a state equation according to the physical characteristics of the flexible joint of the mechanical arm;
step 3, defining a first tracking error, designing a first virtual controller, inputting a signal of the first virtual controller into a first-order low-pass filter to obtain a new state variable x2dThe first virtual controller is replaced to carry out the next calculation, and the calculation amount is reduced;
step 4, defining a second tracking error, approximating a nonlinear function containing uncertain parameters by using the first RBF neural network, and designing the weight of the first RBF neural networkAn adaptation law; designing a second virtual controller, and passing the signal of the second virtual controller through a second first-order low-pass filter to obtain a new variable x3d；
Step 5, defining a third tracking error, designing a third virtual controller, inputting a signal of the third virtual controller into a third first-order low-pass filter to obtain a new state variable x4d；
Step 6, defining a fourth tracking error, and performing compensation approximation on the fourth tracking error by using a second RBF neural network according to the state equation to design a self-adaptive law of the weight of the second RBF neural network;
and 7, designing an actual controller.
2. The RBF neural network adaptive dynamic surface control method for the flexible joint of the mechanical arm as claimed in claim 1, wherein the state equation in step 2 is:
Figure FDA0003103984700000021
in the above formula, x ═ x1,x2,x3,x4]TIn order to be a state variable, the state variable,
Figure FDA0003103984700000022
are respectively x1,x2,x3,x4The first derivative of (a) is,
Figure FDA0003103984700000023
Figure FDA0003103984700000024
delta (t) is external bounded interference and satisfies that delta (t) < rho, and rho is a constant; g (x)1,x2)，f(x1,x3) A non-linear function of unknown but bounded specific form; suppose that the flexible joint of the mechanical arm needs to track an input signal x1dFirst derivative, second derivative thereof
Figure FDA0003103984700000025
All exist and satisfy
Figure FDA0003103984700000026
ξ is a constant.
3. The RBF neural network adaptive dynamic surface control method for the flexible joint of the mechanical arm as claimed in claim 2, wherein said step 3 specifically comprises:
step 3.1, defining a first tracking error: s1＝x1-x1dDerived by derivation
Figure FDA0003103984700000027
The virtual controller is taken as:
Figure FDA0003103984700000028
wherein c is1To design parameters, and satisfy c1＞0；
Step 3.2, mixing
Figure FDA0003103984700000029
Input into the first order low-pass filter to obtain new state variable x2d；
Step 3.3, defining the filtering error e of the first virtual controller2The expression is as follows:
Figure FDA00031039847000000210
then its derivative can be found as:
Figure FDA00031039847000000211
τ2is the filter time constant of the filter.
4. The RBF neural network adaptive dynamic surface control method for the flexible joint of the mechanical arm as claimed in claim 3, wherein said step 4 specifically comprises:
step 4.1, define a second tracking error: s2＝x2-x2dAnd obtaining a derivative:
Figure FDA00031039847000000212
step 4.2, establishing a first RBF neural network;
step 4.3, approximating the unknown function by using the first RBF neural network described in step 4.2
Figure FDA00031039847000000213
Figure FDA00031039847000000214
Wherein
Figure FDA00031039847000000215
Represents the optimal value h when the first RBF neural network weight approaches the unknown function1(x1,x2) Is the neuron activation function of the neural network, epsilon1 *Representing the error of the approximation;
step 4.4, design the second virtual controller
Figure FDA0003103984700000031
Wherein
Figure FDA0003103984700000032
Is that
Figure FDA0003103984700000033
Estimated value of k2，λ2、c2Is a design parameter and satisfies lambda2＞0、c2＞0；
Step 4.5, design adaptive law
Figure FDA0003103984700000034
Wherein gamma is1Is a positive definite symmetric matrix, σ1To design parameters and satisfy sigma1＞0，K2To the tracking error S2A constraint on size;
step 4.6, mixing
Figure FDA0003103984700000035
Input into a second first-order low-pass filter to obtain a new variable x3d；
Step 4.7, defining the filtering error e of the second virtual controller3The expression is as follows:
Figure FDA0003103984700000036
then its derivative can be found as:
Figure FDA0003103984700000037
τ3is the filter time constant of the filter.
5. The RBF neural network adaptive dynamic surface control method for the flexible joint of the mechanical arm as claimed in claim 4, wherein said step 5 specifically comprises:
step 5.1, define a third tracking error: s3＝x3-x3dDerived by derivation
Figure FDA0003103984700000038
Designing virtual controllers
Figure FDA0003103984700000039
λ3、c3Is a design parameter and satisfies lambda3＞0，c3＞0；
Step 5.2, mixing
Figure FDA00031039847000000310
Inputting the third first-order low-pass filter to obtain a new state variable x4d；
Step 5.3, define the third virtualController filter error e4The expression is as follows:
Figure FDA00031039847000000311
then its derivative can be found as:
Figure FDA00031039847000000312
τ4is the filter time constant of the filter.
6. The RBF neural network adaptive dynamic surface control method for the flexible joint of the mechanical arm as claimed in claim 5, wherein said step 6 specifically comprises:
step 6.1, define the fourth tracking error: s4＝x4-x4dDerived by derivation
Figure FDA00031039847000000313
Step 6.2, constructing a second RBF neural network pair nonlinear function
Figure FDA00031039847000000314
Carrying out approximation:
Figure FDA00031039847000000315
wherein
Figure FDA00031039847000000316
Representing the approximation of the optimal value of the nonlinear function, ε, by a second RBF neural network weight2 *Error of the approximation, h2(x1,x3) An activation function for neurons of a second RBF neural network;
step 6.3, define the vector
Figure FDA0003103984700000041
Where ρ is an upper bound of Δ (t) external interference, ω is a design constant and satisfies ω > 0, c4To design a constant, and satisfy c4＞0，The self-adaptation law is designed as follows:
Figure FDA0003103984700000042
wherein gamma is2Is a positive definite symmetric matrix, σ2To design the parameters, sigma is satisfied2＞0,K4To the error S4Size constraints.
7. The RBF neural network adaptive dynamic surface control method for the flexible joint of the mechanical arm as claimed in claim 6, wherein said step 7 specifically comprises:
the actual controller is designed as follows:
Figure FDA0003103984700000043
wherein
Figure FDA0003103984700000044
Is theta2Estimate of (a) ("lambda4Is a design parameter and satisfies lambda4＞0。
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