CN110879589A - A fault-tolerant control method and system for a manipulator based on backstepping strategy and synovial strategy - Google Patents

Abstract

The invention discloses a mechanical arm fault-tolerant control method and system based on a backstepping strategy and a sliding mode strategy, which comprises the following steps: establishing a fault dynamic model of the mechanical arm system; constructing a virtual controller by utilizing a backstepping design method based on the model; then, a first-order synovial differentiator is used for effectively estimating the derivative of the virtual controller; processing unknown parameters and fault parameters existing in the system model by using self-adaptive estimation; and designing the final controller by a backstepping design method by means of the results of self-adaptive estimation and sliding film estimation and a hyperbolic function. According to the method, the first-order synovial differentiator is applied to the backstepping design, the problem of 'explosion complexity' of derivation of a virtual controller is effectively avoided, and the aim of controlling the rotation angle of the mechanical arm to be bounded to track a target signal is fulfilled under the condition that the mechanical arm controller breaks down.

Description

A fault-tolerant control method and system for a manipulator based on backstepping strategy and synovial strategy

technical field

The invention belongs to the technical field of control, and in particular relates to a fault-tolerant control method of a robotic arm based on a backstepping strategy and a synovial strategy.

Background technique

The statements in this section merely provide background information related to the present disclosure and do not necessarily constitute prior art.

The inventor found in the research that with the development of robots, the research on the control system of the manipulator has also received extensive attention. Controller failure is a common problem in engineering applications. If the controller of the robotic arm fails to be dealt with in a timely and effective manner, the consequences will be disastrous.

In recent years, in order to ensure the reliability and safety of engineering, the research on fault-tolerant control of manipulators has become a subject that has attracted much attention and is challenging. On the one hand, the structural design technology represented by backstepping design was proposed in the 1980s, and since backstepping design was introduced to the manipulator system by researchers, it has gradually become an inexorable method for studying problems related to manipulator systems. lack of basic methods.

On the other hand, in recent years, the sliding mode protocol has gradually attracted the attention of scholars due to its advantages of fast response speed and simple physical implementation, and has produced a series of research results; the first-order synovial differential developed from sliding mode control It is an important way to solve the difficulty of "complexity explosion" in the derivation of virtual controller in typical backstepping design method. However, this method has not been applied to the research of solving the failure of the controller of the manipulator.

SUMMARY OF THE INVENTION

In order to overcome the above-mentioned shortcomings of the prior art, the present invention provides a fault-tolerant control method of a manipulator based on a backstepping strategy and a synovial strategy, and a fault-tolerant control strategy of the manipulator designed by means of an adaptive method and hyperbolic function characteristics.

To achieve the above object, one or more embodiments of the present invention provide the following technical solutions:

A fault-tolerant control method of a manipulator based on a backstepping strategy and a synovial strategy, including:

Establish a dynamic model of the failure of the robotic arm system;

Based on the above model, using the backstepping design method to construct a virtual controller;

Use first-order synovial differentiator to effectively estimate the derivative of virtual controller;

Use adaptive estimation to deal with unknown parameters and fault parameters existing in the system model;

Using the backstepping design method again, the final controller is designed by means of the results of the adaptive estimation, the synovial estimation, and the hyperbolic function.

A further technical solution, the failure dynamic model of the robotic arm system:

Where: J is the moment of inertia of the engine, M is the mass of the rigid connecting rod, q is the angle of the rigid connecting rod, B is the damping coefficient, l is the length of the axial center, and g is the acceleration of gravity; where:

为故障参数，v(t)为设计控制器.

和

分别代表故障开始时间和结束时间，并且

和

m为故障发生的总个数，s是start，e是end。u(t) is the actual control input, 0＜k-≤k _j ≤1,

is the fault parameter, and v(t) is the design controller.

and

represent the failure start time and end time, respectively, and

and

m is the total number of failures, s is start, and e is end.

The system is further represented as the following mathematical model:

表示未知参数。where: x ₁ =q,

Indicates an unknown parameter.

In practical engineering, the design goal of the manipulator is to design the controller (u) so that the system rotation angle (q) can reach the desired rotation angle (r) under the action of the control. In practical applications, the force generated by the controller often causes unpredictable failures. Therefore, it is necessary to consider how the design controller uses its own system-related information to achieve the design goal in the event of a failure, and then the fault-tolerant control in this scheme is proposed. The relevant information of the manipulator system has been quantified during modeling, so in this design scheme, the design of the compensator (v) can directly borrow the system quantification information (the moment of inertia of the engine, the mass of the rigid link, the damping coefficient) , the length of the axial center) to design the effective control of the manipulator output (ie the rotation angle q), and finally control the difference between the actual effect and the desired control target (r) within a certain range. effect within.

A further technical solution is to construct a virtual controller by applying a backstepping design method;

Introduce coordinate transformation:

y ₁ =x ₁ -r(t), y ₂ =x ₂ -l(t)

Choose a Lyapunov preparatory function:

Construct the virtual controller as:

Among them: r(t) is the tracking signal, l(t) is the virtual controller, and c ₁ is the design parameter.

A further technical solution is to use the first-order synovial differentiator to effectively estimate the derivative of the virtual controller: the first-order synovial differentiator has the form:

h ₀ and h ₁ are the two states of the differentiator system, and C ₀ and C ₁ are the two design parameters of the system.

A further technical solution uses adaptive control to process parameters:

The adaptive update rate of the unknown parameters inherent in the system is designed as:

为θ＝(θ₁，θ₂)^T的估计，Г、Λ为二维的设计矩阵。in:

is the estimation of θ=(θ ₁ , θ ₂ ) ^T , and Г and Λ are two-dimensional design matrices.

A further technical solution, the adaptive update rate of the relevant quantities of the system controller fault parameters:

First, the relevant symbol definitions are given:

The adaptive update rate of both is designed as:

分别为

p的估计，γ₁、γ₂、c₂、σ₁、σ₂为调节参数。in:

respectively

The estimation of p, γ ₁ , γ ₂ , c ₂ , σ ₁ , and σ ₂ are adjustment parameters.

A further technical solution, constructing the Lyapunov function:

分别为对应的估计误差，利用自适应估计、滑膜估计的结果，并设计带有双曲函数的控制器：in:

are the corresponding estimation errors, using the results of adaptive estimation and synovial estimation, and design a controller with a hyperbolic function:

According to the setting of the adaptive update rate of the unknown parameters of the system and the estimation setting of the virtual controller through the first-order synovial differentiator, the derivative of the Lyapunov preparatory function can be obtained to satisfy:

Among them: k and Δ are the substitute symbols of relevant parameters.

The invention discloses a design strategy of a manipulator fault-tolerant controller based on a backstepping strategy and a synovial strategy.

One or more of the above technical solutions have the following beneficial effects:

(1) The present disclosure reduces the requirement for the tracking target of the rotation angle of the manipulator, only the first derivative of the tracking target function needs to exist and is bounded, and it is not necessary to obtain its display form.

(2) The Λ matrix in the design of the present disclosure replaces the parameters in the traditional scheme, so that the adaptive effect of the unknown parameters is enhanced, and the phenomenon that the previous parameter estimation effect is not good is improved.

(3) The present disclosure applies the first-order synovial differentiator to the backstepping design, which effectively avoids the "explosive complexity" problem of virtual controller derivation.

(4) This disclosure applies the method of combining the first-order synovial differentiator and the backstepping design to the fault-tolerant control of the robotic arm system for the first time, effectively compensating for the controller fault, and ensuring the accuracy of the rotation angle control in the robotic arm system. The stability improves the safety of the robotic arm system control.

(5) The present disclosure applies the hyperbolic function tanh(x) to the fault-tolerant control of the robotic arm system for the first time, which effectively improves the accuracy of the tracking error.

设为未知的情况，有效的增大了研究对象的广泛性。(6) When the present disclosure constructs the model, the system parameters are

Setting it as an unknown situation effectively increases the breadth of the research objects.

Description of drawings

The accompanying drawings forming a part of the present invention are used to provide further understanding of the present invention, and the exemplary embodiments of the present invention and their descriptions are used to explain the present invention, and do not constitute an improper limitation of the present invention.

FIG. 1 is a schematic structural diagram of a robotic arm system according to an embodiment of the present disclosure;

2 is an error diagram of a manipulator rotation angle q tracking target signal r according to an embodiment of the disclosure;

3 is an estimation error diagram of a first-order synovial differentiator according to an embodiment of the present disclosure;

FIG. 4(a)-FIG. 4(d) are adaptive estimation diagrams of unknown parameters and related quantities of fault parameters according to an embodiment of the present disclosure;

FIG. 5 is a flowchart of an embodiment of the present disclosure.

Detailed ways

It should be noted that the following detailed description is exemplary and intended to provide further explanation of the invention. Unless otherwise defined, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this invention belongs.

It should be noted that the terminology used herein is for the purpose of describing specific embodiments only, and is not intended to limit the exemplary embodiments according to the present invention. As used herein, unless the context clearly dictates otherwise, the singular is intended to include the plural as well, furthermore, it is to be understood that when the terms "comprising" and/or "including" are used in this specification, it indicates that There are features, steps, operations, devices, components and/or combinations thereof.

Embodiments of the invention and features of the embodiments may be combined with each other without conflict.

The general idea proposed by the present invention:

Referring to FIG. 5 , a fault-tolerant control strategy of the manipulator is designed based on the combination of the backstepping design method and the first-order synovial differentiator method, and with the help of the adaptive method and hyperbolic function characteristics. Firstly, according to the internal operation mechanism of the robot arm system, the mathematical model of the robot arm system is established; secondly, the virtual controller is constructed according to the backstepping design method; the derivative of the virtual controller is effectively estimated with the help of the first-order synovial differentiator; Estimate the unknown parameters existing in the system, including the inherent unknown parameters and fault parameters of the system, and perform appropriate estimation processing; go back to the backstepping design method again, use the results of adaptive estimation and synovial estimation, and use the hyperbolic function to design the final design. The controller can effectively achieve the goal of fault-tolerant control of the manipulator system. This method not only effectively avoids the "complex explosion" problem and weakens the requirement of tracking signals, but also enhances the range of research targets and improves the efficiency of adaptation.

Example 1

Referring to FIG. 1, the present embodiment discloses a fault-tolerant control method of a robotic arm based on a backstepping strategy and a synovial strategy. First, a fault dynamic model of the robotic arm system is established according to the internal operating mechanism of the robotic arm system:

Where: J is the moment of inertia of the engine, M is the mass of the rigid link, q is the angle of the rigid link, B is the damping coefficient, l is the length of the axial center, and g is the acceleration of gravity . in:

为故障参数，v(t)为设计控制器，

和

分别代表故障开始时间和结束时间，并且

和

Here: 0<k_≤k _j ≤1,

is the fault parameter, v(t) is the design controller,

and

represent the failure start time and end time, respectively, and

and

In practical engineering, the design input force v(t) of the robotic arm system often has some failures, which makes the design input force inconsistent with the actual input force u(t), and it is fully considered that there is a certain relationship between v(t) and u(t). , so it is modeled as

k _j is the constant coefficient of the designed controller, and there is a size limit: greater than zero, less than or equal to 1, because it is unknown, it is called an unknown fault parameter, each j can represent k _j as a constant of different sizes, m and The role of n in other designs is the same;

也是一个未知的常量故障参数，但是其大小没有限制，表示与v(t)无关的故障量，但一般情况下

的值并不会太大；

is also an unknown constant fault parameter, but its size is not limited, representing the amount of fault independent of v(t), but in the general case

The value of is not too large;

则为控制器并没有发生故障。When k _j =1,

The controller is not faulty.

j=1, 2, ..., m: indicates that the magnitudes of the two fault parameters in each time period can be different, and there are m fault time periods in total.

和

表示故障参数为多个连续的常数组成随时可变得故障。

and

Indicates that the fault parameter is composed of multiple continuous constants that can become faulty at any time.

The system is further represented as a mathematical model that can use the backstepping design method:

表示未知参数；并且根据实际情况x₁为系统的输出信号。where: x ₁ =q,

Represents unknown parameters; and x ₁ is the output signal of the system according to the actual situation.

In practical engineering, the signal that can be directly measured or obtained is called the output signal. In the system of the present invention, the signal that can be directly obtained is the angle q of the manipulator, and since x ₁ =q in modeling, then x ₁ is the output signal of the manipulator system.

来表示未知参数，并在以下的设计步骤中通过自适应的方式将其进行估计。In practice, some information in the manipulator system (mass of rigid link, damping coefficient, length of axial center) may not be available, so these possibly unknown quantities should be fully considered in engineering design. for mold

to represent the unknown parameters, which are estimated adaptively in the following design steps.

Apply the backstepping design method to construct a virtual controller:

It can be seen from the backstepping design method that the coordinate transformation should be introduced first:

y ₁ =x ₁ -r(t), y ₂ =x ₂ -l(t)

Construct a suitable Lyapunov preparatory function, here choose:

In order to make the final Lyapunov preparation function meet certain conditions, the virtual controller is constructed here as:

Among them: r(t) is the target signal, l(t) is the virtual controller, and c ₁ is a design parameter greater than 1. The tracking signal can also be designed according to the actual tracking target. For example, if the tracking signal is set to 0, the output effect of the system is near 0. The tracking error of the simulation example of this scheme is shown in FIG. 2 .

Referring to Fig. 3, the derivative of the virtual controller constructed by the backstepping design method is effectively estimated by means of a first-order synovial differentiator

The first-order synovial differentiator has the form:

Here: sign is a sign function; C ₀ , C ₁ are two design parameters, the settings of which are related to the virtual controller, and the specific size needs to be debugged; h ₀ , h ₁ are the two states of the first-order synovial differentiator system. According to the relevant knowledge of the first-order synovial differentiator, it can be known that as long as the parameters C ₀ and C ₁ are adjusted, as well as the initial values h ₀ (0), h of the two system states h ₀ and h ₁ in the first-order synovial differentiator ₁ (0), the derivative of the virtual controller l(t) can be effectively estimated with η ₀ .

Referring to Fig. 4(a)-Fig. 4(d), the adaptive control is used to properly process the fault parameters or the unknown parameters existing in the system. By adjusting the adaptive rate and the parameters in a section of the synovial differentiator, Estimate the set unknowns that need to be estimated:

The adaptive update rate of the unknown parameters inherent in the system is designed as:

为θ＝(θ₁，θ₂)^T的估计，Г、Λ为二维的设计矩阵，为了便于操作方便，此处两个设计矩阵以对角矩阵为优，并且Λ矩阵元素的大小与系统参数有关，需要具体调试。in:

is the estimation of θ=(θ ₁ , θ ₂ ) ^T , Г and Λ are two-dimensional design matrices. For the convenience of operation, the two design matrices here are preferably diagonal matrices, and the size of the Λ matrix elements is related to the system The parameters are related and need to be debugged.

The design idea of the adaptive update rate of the relevant quantities of the fault parameters of the system controller: It is convenient to effectively adapt the fault parameters. Here, the relevant symbol definitions are given first:

β ₌ inf _1≤j≤m {kj},

出现导致控制力量v(t)偏差最大(即

带来的作用)情况下能达到控制目标。故障最严重时能达到控制目标则一般故障发生情况下也能达到控制目标。The purpose of defining the relevant symbols here is to take into account the existence of the fault parameter k _j to minimize the control force v(t) (that is, the effect of inf _1≤j≤m {k _j }), and

appears to result in the largest deviation of the control force v(t) (i.e.

effect) can achieve the control objective. The control target can be achieved when the fault is the most serious, and the control target can also be achieved when the general fault occurs.

Since the actual values of the two quantities are unknown, adaptive estimation is required here, and the adaptive update rates of the two are designed as:

分别为

p的估计，γ₁、γ₂、c₂、σ₁、σ₂为调节参数，这里c₂只需大于1即可，但σ₁、σ₂的值需要根据系统固有参数的设置而具体调试。in:

respectively

For the estimation of p, γ ₁ , γ ₂ , c ₂ , σ ₁ , and σ ₂ are adjustment parameters, where c ₂ only needs to be greater than 1, but the values of σ ₁ and σ ₂ need to be specifically adjusted according to the settings of the inherent parameters of the system .

Returning to the backstepping design method again, according to the corresponding control theory research basis, the Lyapunov function is constructed here:

分别为对应的估计误差。借助上述实施方案中自适应估计、滑膜估计的结果，并借助双曲函数

(∈为任意正数)的性质设计最终的控制器：in:

are the corresponding estimation errors, respectively. With the help of the results of the adaptive estimation, the synovial estimation in the above embodiment, and with the help of the hyperbolic function

(∈ is any positive number) to design the final controller:

When modeling the system, the first step is to quantify the internal related information of the system (mass of the rigid link, damping coefficient, length of the axial center), so in actual engineering, when designing the control force for the robotic arm system, it is also This information should be quantified first, and then the design of v(t) in this scheme is used to achieve the goal of fault-tolerant control.

According to the setting of the adaptive update rate of the unknown parameters of the system in the above scheme and the estimation setting of the virtual controller through the first-order synovial differentiator, the derivative of the Lyapunov preparatory function can be obtained to satisfy:

Among them: k and Δ are parameters, which can be obtained according to the relevant theoretical research knowledge in the control field, thus achieving the goal of fault-tolerant control of the robotic arm system.

From this, it can be known that by applying the designed control force v to the system, it is possible to realize the bounded tracking of the output signal x ₁ to the target signal r in the event of a failure of the robotic arm system.

According to the specific form of the designed controller v, in practical engineering applications, the engineering parameters that need to be adjusted are mainly: Λ, σ ₁ , σ ₂ , C ₀ , C ₁ . Although there is no range of parameter values, if these debugging parameters are set incorrectly, the system output signal _{x 1} (ie the rotation angle q of the manipulator) cannot be effectively controlled. The actual tracking effect knows whether these debug parameters are close to their corresponding actual values.

h₀(0)＝-20、h₁(0)＝-2。Here, you may wish to set it during simulation: the tracking target is r(t)=sin(t), and the initial values are x ₁ (0)=0.5, x ₂ (0)=0,

h ₀ (0)=-20, h ₁ (0)=-2.

(即β＝2，p＝10、θ₁＝-2、θ₂＝-10)，c₁＝20、c₂＝20、E＝5、Г＝diag{1，1}、Λ＝diag{0.4，0.013}、γ₁＝1、γ₂＝1、σ₁＝0.012、σ₂＝0.07、C₀＝8、C₁＝10。The parameters are: B=0.5, J=M=0.25, l=1, g=10, k _j =0.5,

(ie β=2, p=10, θ ₁ =-2, θ ₂ =-10), c ₁ =20, c ₂ =20, E=5, Г=diag{1,1}, Λ=diag{ 0.4, 0.013}, γ ₁ =1, γ ₂ =1, σ ₁ =0.012, σ ₂ =0.07, C ₀ =8, C ₁ =10.

The simulation result is the tracking error between the output signal x ₁ of the manipulator system (that is, the rotation angle q of the manipulator) and the target signal r.

In another embodiment, the present invention discloses a manipulator fault-tolerant system based on the backstepping method and the synovial method, including a controller, and the controller utilizes the above-mentioned manipulator based on the backstepping strategy and the synovial strategy. The fault-tolerant control method involves acquisition.

Embodiment 2

The purpose of this embodiment is to provide a computing device, including a memory, a processor and a computer program stored in the memory and running on the processor, the processor implements the following steps when executing the program, including:

Establish a dynamic model of the failure of the robotic arm system;

Based on the above model, using the backstepping design method to construct a virtual controller;

Use first-order synovial differentiator to effectively estimate the derivative of virtual controller;

Use adaptive estimation to deal with unknown parameters and fault parameters existing in the system model;

Using the backstepping design method again, the final controller is designed by means of the results of the adaptive estimation, the synovial estimation, and the hyperbolic function.

Embodiment 3

The purpose of this embodiment is to provide a computer-readable storage medium.

A computer-readable storage medium on which a computer program is stored, the program executes the following steps when executed by a processor:

Establish a dynamic model of the failure of the robotic arm system;

Based on the above model, using the backstepping design method to construct a virtual controller;

Use first-order synovial differentiator to effectively estimate the derivative of virtual controller;

Use adaptive estimation to deal with unknown parameters and fault parameters existing in the system model;

Using the backstepping design method again, the final controller is designed by means of the results of the adaptive estimation, the synovial estimation, and the hyperbolic function.

The steps involved in the apparatus of the above embodiment correspond to the method embodiment 1, and the specific implementation can refer to the relevant description part of the embodiment 1. The term "computer-readable storage medium" should be understood to include a single medium or multiple media including one or more sets of instructions; it should also be understood to include any medium capable of storing, encoding or carrying for use by a processor The executed instruction set causes the processor to perform any of the methods of the present invention.

Although the specific embodiments of the present invention have been described above in conjunction with the accompanying drawings, they do not limit the scope of protection of the present invention. Those skilled in the art should understand that on the basis of the technical solutions of the present invention, those skilled in the art do not need to pay creative work. Various modifications or deformations that can be made are still within the protection scope of the present invention.

Claims (10)

1. A mechanical arm fault-tolerant control method based on a backstepping strategy and a sliding mode strategy is characterized by comprising the following steps: establishing a fault dynamic model of the mechanical arm system;

constructing a virtual controller by utilizing a backstepping design method based on the model;

effectively estimating the derivative of the virtual controller by using a first-order synovial differentiator;

processing unknown parameters and fault parameters existing in the system model by using self-adaptive estimation;

and designing the final controller by a backstepping design method by means of the results of self-adaptive estimation and sliding film estimation and a hyperbolic function.

2. The method for controlling the fault tolerance of the mechanical arm based on the backstepping strategy and the slip film strategy as claimed in claim 1, wherein the fault dynamic model of the mechanical arm system is as follows:

wherein: j represents the moment of inertia of the engine, M is the mass of the rigid link, q is the angle of the rigid link, B is the damping coefficient, l is the length of the axial center, g is the gravitational acceleration;

wherein:

u (t) is the actual control input, 0 < k_-≤k_j≤1、

V (t) for the fault parameters, a design controller,

and

respectively represent a fault start time and an end time, and

and

the system is further represented as the following mathematical model:

wherein: x is the number of₁＝q、

Representing an unknown parameter.

3. The fault-tolerant control method for the mechanical arm based on the backstepping strategy and the slip film strategy as claimed in claim 2, wherein a virtual controller is constructed by applying a backstepping design method:

introducing coordinate transformation:

y₁＝x₁-r(t)，y₂＝x₂-l(t)

selecting a Lyapunov preparation function:

the virtual controller is constructed as follows:

wherein: r (t) is a tracking signal, l (t) is a virtual controller, c₁Are design parameters.

4. A method for fault-tolerant control of mechanical arms based on a backstepping strategy and a slip film strategy according to claim 1 or 3, characterized in that a first-order slip film differentiator is used to effectively estimate the derivative of the virtual controller: the first-order synovial differentiator is formed as:

5. the method for controlling the fault tolerance of the mechanical arm based on the backstepping strategy and the slip film strategy as claimed in claim 3, wherein the parameters are processed by using adaptive control:

the adaptive update rate of the unknown parameters inherent to the system is designed as follows:

wherein:

is theta ═ theta₁，θ₂)^TГ, Λ is a two-dimensional design matrix.

6. The method for controlling the fault tolerance of the mechanical arm based on the backstepping strategy and the slip film strategy as claimed in claim 3, wherein the adaptive update rate of the fault parameter related quantity of the system controller is as follows:

first, the associated symbol definition is given:

β＝inf_1≤j≤m{k_j}，

the adaptive update rate of the two is designed as follows:

wherein:

are respectively as

Estimation of p, γ₁、γ₂、c₂、σ₁、σ₂To adjust the parameters.

7. The method for controlling the fault tolerance of the mechanical arm based on the backstepping strategy and the slip film strategy as claimed in claim 3, wherein a Lyapunov function is constructed:

wherein:

respectively utilizing the results of adaptive estimation and synovial estimation for corresponding estimation errors, and designing a controller with a hyperbolic function:

according to the setting of the self-adaptive update rate of the unknown parameters of the system and the estimation setting of the virtual controller through a first-order synovial differentiator, the derivative of the Lyapunov preparation function obtained by derivation can meet the following conditions:

wherein: k. Δ is a substitute sign for the relevant parameter.

8. A fault-tolerant control system for a mechanical arm based on a backstepping strategy and a slip film strategy, which comprises a controller, and is characterized in that the controller is obtained by using the fault-tolerant control method for the mechanical arm based on the backstepping strategy and the slip film strategy as claimed in any one of claims 1 to 7.

9. A computing device comprising a memory, a processor, and a computer program stored on the memory and executable on the processor, wherein the processor executes the program to perform steps comprising:

establishing a fault dynamic model of the mechanical arm system;

constructing a virtual controller by utilizing a backstepping design method based on the model;

effectively estimating the derivative of the virtual controller by using a first-order synovial differentiator;

processing unknown parameters and fault parameters existing in the system model by using self-adaptive estimation;

and designing the final controller by a backstepping design method by means of the results of self-adaptive estimation and sliding film estimation and a hyperbolic function.

10. A computer-readable storage medium, having a computer program stored thereon, the program, when executed by a processor, performing the steps of:

establishing a fault dynamic model of the mechanical arm system;

constructing a virtual controller by utilizing a backstepping design method based on the model;

effectively estimating the derivative of the virtual controller by using a first-order synovial differentiator;

processing unknown parameters and fault parameters existing in the system model by using self-adaptive estimation;

and designing the final controller by a backstepping design method by means of the results of self-adaptive estimation and sliding film estimation and a hyperbolic function.

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