CN115202189A - Variable-structure finite-time vibration suppression method for fractional-order manipulator model - Google Patents

Abstract

The invention discloses a variable structure finite time vibration suppression method of a fractional order mechanical arm model, which comprises the following steps: establishing a fractional order mechanical arm model by a dynamic equation of the mechanical arm model; design of non-linear fractional order P (ID) by combining sliding mode control technology ^α A terminal sliding mode surface; considering uncertainty of lumped parameters of a fractional order mechanical arm system, and determining a self-adaptive estimation law of unknown parameters of a fractional order mechanical arm model; considering the influence of fan-shaped nonlinear input and combining with the self-adaptive estimation law of unknown parameters of the fractional order mechanical arm model, designing a finite time controller; and applying the established self-adaptive estimation law of the unknown parameters of the fractional order mechanical arm model and the designed finite time controller to the fractional order mechanical arm model to realize the finite time vibration suppression. The variable structure limited time vibration suppression method can realizeAnd the finite time vibration suppression of the fractional order mechanical arm model realizes that the state track of the fractional order mechanical arm model converges to an equilibrium state in finite time.

Description

A Variable Structure Finite Time Vibration Suppression Method for Fractional Manipulator Model

technical field

The invention relates to the technical field of manipulator model control, in particular to a variable-structure finite-time vibration suppression method of a fractional-order manipulator model.

Background technique

Robotic arm systems are widely used because they can perform repetitive tasks and have high precision in hazardous areas. For example, in the medical field, robotic arms are used for minimally invasive interventions, neurosurgery, orthopedics, and urology patients. It has a very good therapeutic effect; at present, an enhanced robotic arm device has been used for limb rehabilitation, which can replace physical therapists; in the field of aerospace, the robotic arm has played an important role in the construction of the space station and space science experiments ; In the field of industrial production, the robotic arm can be competent for tasks that cannot be accomplished by humans. For personnel in special operations, the use of the robotic arm greatly ensures the safety of production and people's lives. Therefore, it is of great practical significance to design an appropriate control law to realize the trajectory planning of the robotic arm system.

SUMMARY OF THE INVENTION

In view of this, the present invention provides a variable-structure finite-time vibration suppression method for a fractional-order manipulator model. The fractional-order calculus theory is introduced into the construction of the sliding mode surface, and the established P(ID) ^α terminal sliding mode surface can be The finite-time vibration suppression problem of fractional manipulator system is well solved, which fills the gap in the research results of fractional manipulator system.

In order to realize the above-mentioned technical purpose, the present invention is realized through the following technical solutions: a variable structure finite-time vibration suppression method of a fractional-order mechanical arm model, comprising the following steps:

Step 1. Select the state variables in the two-degree-of-freedom manipulator structure, transform the dynamic equation of the manipulator structure model into an integer-order state space model of the manipulator structure model, and combine the fractional calculus theory to determine the fractional-order manipulator model. ;

Step 2. Using PID control technology and sliding mode control theory, combined with the state variables in step 1, establish a terminal sliding mode surface of fractional order P(ID) ^α . When the state trajectory of the fractional manipulator model in step 1 reaches fractional order P (ID) ^α terminal sliding mode surface, the fractional derivative is obtained for the fractional P(ID) ^α terminal sliding mode surface, and the desired sliding mode equation is obtained;

Step 3. Considering the influence of the uncertainty of the lumped parameters of the fractional manipulator model, establish an adaptive estimation law for the unknown parameters of the fractional manipulator model according to the fractional P(ID) ^α terminal sliding surface;

Step 4. Design a finite-time controller by considering the influence of the fan-shaped nonlinear input and combining with the adaptive estimation law of the unknown parameters of the fractional-order manipulator model;

Step 5: Apply the adaptive estimation law of the unknown parameters of the established fractional-order manipulator model and the designed finite-time controller to the fractional-order manipulator model to realize finite-time vibration suppression.

Further, step 1 includes the following sub-steps:

Step 11. Obtain the actual joint position θ measured by the encoder, the rotational inertia J of the actuator, the viscous friction coefficient B of the actuator, the efficiency η of the gearbox, the gear ratio υ, and the torque from the structure of the two-degree-of-freedom manipulator. Constant K _t , back electromotive force constant _Ke , resistance R, torque τ on the control manipulator joint, armature voltage input u, establish the dynamic equation of the manipulator structure model:

为θ的一阶导数，

为θ的二阶导数；in,

is the first derivative of θ,

is the second derivative of θ;

作为第二状态变量x₂，将机械臂结构模型的动力学方程转变为机械臂结构模型的整数阶状态空间模型：Step 12, take θ as the first state variable x ₁ ,

As the second state variable x ₂ , the dynamic equation of the manipulator structure model is transformed into an integer-order state space model of the manipulator structure model:

Step 13. Use the Caputo fractional calculus operator D ^α to describe the integer-order state space model of the manipulator structure model in step 12, and consider the influence of the fan-shaped nonlinear input φ(u) and the fraction related to time t. The influence of the lumped parameter uncertainty ΔL(X,t) of the first-order manipulator model to determine the fractional-order manipulator model:

Among them, α∈(0,1) is the fractional order, and X=[x ₁ ,x ₂ ] ^T is the state vector of the fractional manipulator model.

Further, the fractional-order manipulator model lumped parameter uncertainty ΔL(X, t) related to time t includes: model error, parameter fluctuation, unmodeled dynamics and external disturbance, the ΔL(X, t )Satisfy:

|ΔL(X,t)|≤γ||X||+λ (4)

Among them, γ is the first unknown parameter, and λ is the second unknown parameter.

Further, the sector-shaped nonlinear input φ(u) is continuous in the interval [δ ₁ ,δ ₂ ] and satisfies:

δ ₁ u ² ≤uφ(u)≤δ ₂ u ² (5)

Among them, δ ₁ is the first slope, δ ₂ is the second slope, and u is the finite-time controller.

Further, step 2 includes the following substeps:

Step 21. Using PID control technology and sliding mode control theory, combined with the state variables in step 12, establish a fractional order P(ID) ^α terminal sliding mode surface:

q ₁ s+q ₂ D ^α s+q ₃ Iα[(|s|+|D ^α s| ^σ )sgn(D ^α s)]=k _p x ₁ +k _d D ^α x ₁ +k _i I ^α [(|x ₁ |+|D ^α x ₁ | ^ρ )sgn(D ^α x ₁ )] (6)

Among them, s is the sliding mode surface variable, I ^α is the fractional integral operator, sgn() is the sign function, q ₁ is the third positive real number, q ₂ is the fourth positive real number, q ₃ is the fifth positive real number, σ ∈(0,1) is the first fixed value, ρ∈(0,1) is the second fixed value, k _p > 0 is the fractional order P(ID) ^α terminal sliding mode scale coefficient, k _d is the fractional order Differential coefficient of P(ID) ^α terminal sliding mode surface, k _i > 0 is the integral coefficient of fractional P(ID) ^α terminal sliding mode surface;

Step 22. When the state trajectory of the fractional manipulator model reaches the sliding mode surface, there is s=D ^α s=D ^2α s=0, and calculate the α-order fractional derivative for the fractional-order P(ID) ^α terminal sliding mode surface:

k _p D ^α x ₁ +k _d D ^2α x ₁ +k _i (|x ₁ |+|D ^α x ₁ | ^ρ )sgn(D ^α x ₁ )=0 (7)

Step 23. Combining the fractional differential property D ^2α x ₁ =D ^α (D ^α x ₁ )=D ^α x ₂ , the expected sliding mode equation is obtained:

Further, the establishment process of the adaptive estimation law of the unknown parameters of the fractional-order manipulator model is as follows:

为第一未知参数γ的估计值，

为第二未知参数λ的估计值，X＝[x₁,x₂]^T为分数阶机械臂模型的状态矢量，m＞0为第一估计律增益，n＞0为第二估计律增益，k_d为分数阶P(ID)^α终端滑模面的微分系数，s为滑模面变量。where D ^α is the Caputo-type fractional calculus operator,

is the estimated value of the first unknown parameter γ,

is the estimated value of the second unknown parameter λ, X=[x ₁ , x ₂ ] ^T is the state vector of the fractional-order manipulator model, m>0 is the gain of the first estimation law, n>0 is the gain of the second estimation law, k _d is the differential coefficient of the fractional order P(ID) ^α terminal sliding mode surface, and s is the sliding mode surface variable.

Further, the limited time controller is:

为有限时间控制器系数，ε(t)有限时间控制器结构变量，s为滑模面变量，sgn(D^αs)为符号函数，k_d为分数阶P(ID)^α终端滑模面的微分系数，q₁为第三正实数，q₂为第四正实数，D^α为Caputo型分数阶微积分算子，q₃为第五正实数，B为执行机构的粘性摩擦系数，K_e为反电动势常数，x₂为第二状态变量，J为执行器的转动惯量，τ为控制机械手关节上的力矩，

为第一未知参数γ的估计值，X＝[x₁,x₂]^T为分数阶机械臂模型的状态矢量，

为第二未知参数λ的估计值，k_i为分数阶P(ID)^α终端滑模面的积分系数，x₁为第一状态变量，ρ∈(0,1)为第二固定值。Among them, J is the moment of inertia of the actuator, R is the resistance, η is the efficiency of the gearbox, υ is the gear ratio, K _t is the torque constant,

is the finite-time controller coefficient, ε(t) is the finite-time controller structural variable, s is the sliding mode surface variable, sgn(D ^α s) is the sign function, and k _d is the fractional order P(ID) ^α terminal sliding mode surface Differential coefficient, q ₁ is the third positive real number, q ₂ is the fourth positive real number, D ^α is the Caputo fractional calculus operator, q ₃ is the fifth positive real number, B is the viscous friction coefficient of the actuator, K _e is the back electromotive force constant, x ₂ is the second state variable, J is the moment of inertia of the actuator, τ is the torque on the control manipulator joint,

is the estimated value of the first unknown parameter γ, X=[x ₁ ,x ₂ ] ^T is the state vector of the fractional manipulator model,

is the estimated value of the second unknown parameter λ, _ki is the integral coefficient of the fractional P(ID) ^α terminal sliding mode surface, x ₁ is the first state variable, and ρ∈(0,1) is the second fixed value.

Compared with the prior art, the present invention has the following beneficial effects:

(1) The variable-structure finite-time vibration suppression method of the fractional-order manipulator model of the present invention combines the terminal sliding mode surface designed by traditional PID technology and sliding mode control technology, which includes the advantages of strong robustness and adaptability of traditional PID control. , and also has the advantages of fast convergence speed of sliding mode control. At the same time, the constructed fractional P(ID) ^α terminal sliding mode surface contains fractional calculus operators. By adjusting the calculus order, the convergence time can be effectively adjusted, and the control good effect;

(2) The variable-structure finite-time vibration suppression method of the fractional-order manipulator model of the present invention realizes finite-time control through the fractional-order manipulator model. The fractional-order model of the manipulator structure is studied for the first time, and the fractional manipulator model is fully considered in the research process. The uncertainty of lumped parameters is affected, and an appropriate adaptive estimation law is designed for the unknown upper bound of the uncertainty of lumped parameters, which greatly improves the identification effect of unknown parameters;

(3) The variable-structure finite-time vibration suppression method of the fractional-order manipulator model of the present invention verifies the finite-time stability of the approaching stage and the sliding-mode stage by selecting a suitable form of Lyapunov function, that is, the approaching stage and the sliding-mode stage are both to converge in finite time;

(4) The variable structure finite-time vibration suppression method of the fractional-order manipulator model of the present invention fully considers the influence of sector nonlinear input on the fractional-order manipulator model, and the finite-time controller is designed, which can well overcome the nonlinear input, set The adverse effects brought by the uncertainty of the total parameters improve the robustness of the controlled system;

(5) The variable-structure finite-time vibration suppression method of the fractional-order manipulator model of the present invention can effectively suppress the vibration phenomenon that occurs inside the fractional-order manipulator model within a given time, obtain the desired output and performance indicators, and save control costs ,Improve economic efficiency.

Description of drawings

Fig. 1 is the flow chart of the variable-structure finite-time vibration suppression method of the fractional-order manipulator model of the present invention;

Figure 2 is a structural diagram of a sector nonlinear input function.

Detailed ways

In order to illustrate the purpose and advantages of the present invention more clearly, the following describes the application principle of the present invention in detail with reference to the accompanying drawings.

Figure 1 is a flowchart of the variable structure finite time vibration suppression method of the fractional-order manipulator model of the present invention, and the variable structure finite time vibration suppression method specifically includes the following steps:

Step 1. Select the state variables in the two-degree-of-freedom manipulator structure, transform the dynamic equation of the manipulator structure model into an integer-order state space model of the manipulator structure model, and combine the fractional calculus theory to determine the fractional-order manipulator model. , the fractional operator is especially suitable for describing the system with memory characteristics. The movement process of the manipulator involves a large number of memory characteristics. Therefore, the mathematical modeling of the manipulator model by fractional calculus can obtain more accurate results; the specific steps include the following sub-steps :

Step 11. Obtain the actual joint position θ measured by the encoder, the rotational inertia J of the actuator, the viscous friction coefficient B of the actuator, the efficiency η of the gearbox, the gear ratio υ, and the torque from the structure of the two-degree-of-freedom manipulator. Constant K _t , back electromotive force constant _Ke , resistance R, torque τ on the control manipulator joint, armature voltage input u, establish the dynamic equation of the manipulator structure model:

为θ的一阶导数，

为θ的二阶导数；in,

is the first derivative of θ,

is the second derivative of θ;

作为第二状态变量x₂，将机械臂结构模型的动力学方程转变为机械臂结构模型的整数阶状态空间模型：Step 12, take θ as the first state variable x ₁ ,

As the second state variable x ₂ , the dynamic equation of the manipulator structure model is transformed into an integer-order state space model of the manipulator structure model:

Step 13. Use the Caputo fractional calculus operator D ^α to describe the integer-order state space model of the manipulator structure model in step 12, and consider the influence of the fan-shaped nonlinear input φ(u) and the fraction related to time t. The influence of the lumped parameter uncertainty ΔL(X,t) of the first-order manipulator model to determine the fractional-order manipulator model:

Among them, α∈(0,1) is the fractional order, and X=[x ₁ ,x ₂ ] ^T is the state vector of the fractional manipulator model.

In the present invention, the fractional order manipulator model lumped parameter uncertainty ΔL(X, t) related to time t includes: model error, parameter fluctuation, unmodeled dynamics and external disturbance. For the convenience of controller design and derivation, it is assumed that ΔL (X,t) is bounded and satisfies:

|ΔL(X,t)|≤γ||X||+λ (4)

Among them, γ is the first unknown parameter, and λ is the second unknown parameter.

As shown in Fig. 2, the fan-shaped nonlinear input φ(u) in the present invention is nonlinear and continuous in the fan-shaped region enclosed by the first slope δ ₁ and the second slope δ ₂ , which is often encountered in the current controller execution process. One of the typical nonlinear characteristics, such nonlinear characteristics will seriously affect the performance of the controlled system, and even lead to the instability of the mechanical arm structure model. Therefore, it must be considered in the controller design. In the present invention, the fan-shaped nonlinear input φ(u) satisfies:

δ ₁ u ² ≤uφ(u)＜δ ₂ u ² (5)

Among them, u is a finite-time controller.

Step 2. Using PID control technology and sliding mode control theory, combined with the state variables in step 1, establish a terminal sliding mode surface of fractional order P(ID) ^α . When the state trajectory of the fractional manipulator model in step 1 reaches fractional order P (ID) ^α terminal sliding mode surface, the fractional derivative of the fractional P(ID) ^α terminal sliding mode surface is obtained, and the desired sliding mode equation is obtained. The designed fractional P(ID) ^α terminal sliding mode surface combines the Traditional PID technology and sliding mode control technology have the advantages of fast convergence speed and strong robustness; it includes the following sub-steps:

Step 21. Using PID control technology and sliding mode control theory, combined with the state variables in step 12, establish a fractional order P(ID) ^α terminal sliding mode surface:

q ₁ s+q ₂ D ^α s+q ₃ I ^α [(|s|+|D ^α s| ^σ )sgn(D ^α s)]=k _p x ₁ +k _d D ^α x ₁ +k _i I ^α [(|x ₁ |+|D ^α x ₁ | ^ρ )sgn(D ^α x ₁ )] (6)

Among them, s is the sliding mode surface variable, I ^α is the fractional integral operator, sgn() is the sign function, q ₁ is the third positive real number, q ₂ is the fourth positive real number, q ₃ is the fifth positive real number, σ ∈(0,1) is the first fixed value, ρ∈(0,1) is the second fixed value, k _p > 0 is the fractional order P(ID) ^α terminal sliding mode scale coefficient, k _d is the fractional order Differential coefficient of P(ID) ^α terminal sliding mode surface, k _i > 0 is the integral coefficient of fractional P(ID) ^α terminal sliding mode surface;

Step 22. When the state trajectory of the fractional manipulator model reaches the fractional P(ID) ^α terminal sliding mode surface, the sliding mode motion will be performed along the fractional P(ID) ^α terminal sliding mode surface until the movement reaches the origin. After reaching the fractional P(ID) ^α terminal sliding mode surface, s=D ^α s=D ^2α s=0 is satisfied, and the α-order fractional derivative is obtained for the fractional P(ID) ^α terminal sliding mode surface:

k _p D ^α x ₁ +k _d D ^2α x ₁ +k _i (|x ₁ |+|D ^α x ₁ | ^ρ )sgn(D ^α x ₁ )=0 (7)

Step 23. Combining the fractional differential property D ^2α x ₁ =D ^α (D ^α x ₁ )=D ^α x ₂ , the expected sliding mode equation is obtained:

来验证期望滑模态方程的有限时间稳定性。For the sliding mode stage in the control process of the manipulator structure, the Lyapunov function is selected

to verify the finite-time stability of the expected sliding mode equation.

Calculate the α-order fractional derivative for V ₁ (t), and according to the characteristics of the fractional operator, we get:

Let k _i >k _d , formula (11) simplifies to

Choose a helper function

Transform formula (12) into:

得：According to the research results of Stabilization of a class of cascade nonlinear switched systems with application to chaotic systems, there is a constant η ₁ > 0 such that

have to:

According to the research of Graph theory-based finite-time synchronization of fractional-order complex dynamical networks, the system whose fractional derivative of energy function satisfies Eq. (15), its sliding mode will converge to the origin in finite time _t2 :

Among them, t ₁ is the time to reach the sliding surface, and Γ() is the gamma function.

Step 3. Considering the influence of the uncertainty of the lumped parameters of the fractional manipulator model, according to the fractional P(ID) ^α terminal sliding surface, establish an adaptive estimation law for the unknown parameters of the fractional manipulator model, which can well predict unknown parameters. The parameter upper bound improves the robustness of the system; the establishment process of the adaptive estimation law of the unknown parameters of the fractional-order manipulator model in the present invention is as follows:

为第一未知参数γ的估计值，

为第二未知参数λ的估计值，m＞0为第一估计率增益，n＞0为第二估计律增益，m、n的取值大小直接决定未知参数辨识速率。in,

is the estimated value of the first unknown parameter γ,

is the estimated value of the second unknown parameter λ, m>0 is the first estimation rate gain, n>0 is the second estimation law gain, and the values of m and n directly determine the unknown parameter identification rate.

Step 4. Considering the influence of the fan-shaped nonlinear input and combining the adaptive estimation law of the unknown parameters of the fractional manipulator model, a finite-time controller is designed, which can suppress the vibration phenomenon generated by the system within a given time and improve the reliability of the system operation; The finite time controller in the present invention is:

为有限时间控制器系数，ε(t)为有限时间控制器结构变量，sgn(D^αs)为符号函数，其值与D^αs有关，当D^αs＞0，sgn(D^αs)＝1，当D^αs＝0，sgn(D^αs)＝0，当D^αs＜0，sgn(D^αs)＝-1。in,

is the finite-time controller coefficient, ε(t) is the finite-time controller structural variable, sgn(D ^α s) is the sign function, and its value is related to D ^α s, when D ^α s>0, sgn(D ^α s) =1, when D ^α s=0, sgn(D ^α s)=0, and when D ^α s<0, sgn(D ^α s)=−1.

Combined with formula (5), we get:

且机械臂结构参数都为正，进一步可得：because

And the structural parameters of the manipulator are all positive, we can further obtain:

-sgn(D ^α s)φ(u)≥ε(t) (18)

Multiply both sides of the inequality of formula (18) by |D ^α s| at the same time, and according to |D ^α s|sgn(D ^α s)=D ^α s, we can get:

Next, verify the finite-time nature of the approach phase:

Choose the Lyapunov function

为未知上界参数γ的估计误差，即估计值

和实际值γ的差，

为未知上界参数λ的估计误差，即估计值

和实际值λ的差，

in,

is the estimated error of the unknown upper bound parameter γ, that is, the estimated value

difference from the actual value γ,

is the estimated error of the unknown upper bound parameter λ, that is, the estimated value

difference from the actual value λ,

Calculate the α-order fractional derivative of formula (20), and combine formulas (3), (6), (9), we can get:

According to the relation (19) satisfied by the designed finite-time controller (10) and the fan-shaped nonlinear input, formula (21) is further simplified as:

Similar to formula (12), it can be obtained that the trajectory of the fractional manipulator model will reach the sliding mode surface within a finite time t ₁ , and continue to move to the origin along the sliding mode surface, at this time, the system vibration is effectively suppressed:

Among them, η ₂ >0 is an auxiliary parameter.

To sum up, the approach stage and the sliding mode stage are verified to be stable in a limited time, that is, the entire control stage is stable in a limited time.

Step 5: Apply the adaptive estimation law of the unknown parameters of the established fractional-order manipulator model and the designed finite-time controller to the fractional-order manipulator model to realize finite-time vibration suppression.

The variable-structure finite-time vibration suppression method of the fractional-order manipulator model of the present invention can realize the finite-time vibration suppression of the fractional-order manipulator model, and realize that the state trajectory of the fractional manipulator model converges to an equilibrium state within a limited time. Especially when the system has input nonlinear characteristics, the designed finite-time controller can improve the robustness of the controlled system and improve the performance of the fractional-order manipulator model. Compared with the traditional sliding mode control technology, the design of the fractional P(ID) ^α terminal sliding mode surface can flexibly adjust the transition time by adjusting the fractional calculus order.

The above are only preferred embodiments of the present invention, and the protection scope of the present invention is not limited to the above-mentioned embodiments, and all technical solutions that belong to the idea of the present invention belong to the protection scope of the present invention. It should be pointed out that for those skilled in the art, some improvements and modifications without departing from the principle of the present invention should be regarded as the protection scope of the present invention.

Claims (7)

1. a variable structure finite time vibration suppression method of a fractional-order mechanical arm model, is characterized in that, comprises the following steps: Step 1. Select the state variables in the two-degree-of-freedom manipulator structure, transform the dynamic equation of the manipulator structure model into an integer-order state space model of the manipulator structure model, and combine the fractional calculus theory to determine the fractional-order manipulator model. ; Step 2. Using PID control technology and sliding mode control theory, combined with the state variables in step 1, establish a terminal sliding mode surface of fractional order P(ID) ^α . When the state trajectory of the fractional manipulator model in step 1 reaches fractional order P (ID) ^α terminal sliding mode surface, the fractional derivative is obtained for the fractional P(ID) ^α terminal sliding mode surface, and the desired sliding mode equation is obtained; Step 3. Considering the influence of the uncertainty of the lumped parameters of the fractional manipulator model, establish an adaptive estimation law for the unknown parameters of the fractional manipulator model according to the fractional P(ID) ^α terminal sliding surface; Step 4. Design a finite-time controller by considering the influence of the fan-shaped nonlinear input and combining with the adaptive estimation law of the unknown parameters of the fractional-order manipulator model; Step 5: Apply the adaptive estimation law of the unknown parameters of the established fractional-order manipulator model and the designed finite-time controller to the fractional-order manipulator model to realize finite-time vibration suppression. 2. according to the variable structure finite time vibration suppression method of the described fractional-order manipulator model of claim 1, it is characterized in that, step 1 comprises the following substeps: Step 11. Obtain the actual joint position θ measured by the encoder, the rotational inertia J of the actuator, the viscous friction coefficient B of the actuator, the efficiency η of the gearbox, the gear ratio υ, and the torque from the structure of the two-degree-of-freedom manipulator. Constant K _t , back electromotive force constant _Ke , resistance R, torque τ on the control manipulator joint, armature voltage input u, establish the dynamic equation of the manipulator structure model:

in,

is the first derivative of θ,

is the second derivative of θ; Step 12, take θ as the first state variable x ₁ ,

As the second state variable x ₂ , the dynamic equation of the manipulator structure model is transformed into an integer-order state space model of the manipulator structure model:

Step 13. Use the Caputo fractional calculus operator D ^α to describe the integer-order state space model of the manipulator structure model in step 12, and consider the influence of the fan-shaped nonlinear input φ(u) and the fraction related to time t. The influence of the lumped parameter uncertainty ΔL(X, t) of the first-order manipulator model to determine the fractional-order manipulator model:

Among them, α∈(0, 1) is the fractional order, and X=[x ₁ , x ₂ ] ^T is the state vector of the fractional manipulator model. 3. The variable-structure finite-time vibration suppression method of the fractional-order manipulator model according to claim 2, wherein the fractional-order manipulator model lumped parameter uncertainty ΔL(X, t) related to time t Including: model errors, parameter fluctuations, unmodeled dynamics, and external disturbances, the ΔL(X, t) satisfies: |ΔL(X, t)|≤γ||X||+λ (4) Among them, γ is the first unknown parameter, and λ is the second unknown parameter. 4. The variable-structure finite-time vibration suppression method of the fractional-order manipulator model according to claim 2, wherein the sector nonlinear input φ(u) is continuous in the interval [δ ₁ , δ ₂ ], and satisfies: δ ₁ u ² ≤uφ(u)≤δ ₂ u ² (5) Among them, δ ₁ is the first slope, δ ₂ is the second slope, and u is the finite-time controller. 5. according to the variable structure finite time vibration suppression method of the described fractional-order manipulator model of claim 2, it is characterized in that, step 2 comprises the following substeps: Step 21. Using PID control technology and sliding mode control theory, combined with the state variables in step 12, establish a fractional order P(ID) ^α terminal sliding mode surface: q ₁ s+q ₂ D ^α s+q ₃ I ^α [(|s|+|D ^α s| ^σ )sgn(D ^α s)]=k _p x ₁ +k _d D ^α x ₁ +k _i I ^α [(|x ₁ |+|D ^α x ₁ | ^ρ )sgn(D ^α x ₁ )] (6) Among them, s is the sliding mode surface variable, I ^α is the fractional integral operator, sgn() is the sign function, q ₁ is the third positive real number, q ₂ is the fourth positive real number, q ₃ is the fifth positive real number, σ ∈(0,1) is the first fixed value, ρ∈(0,1) is the second fixed value, k _p > 0 is the fractional order P(ID) ^α terminal sliding mode scale coefficient, k _d is the fractional order Differential coefficient of P(ID) ^α terminal sliding mode surface, k _i > 0 is the integral coefficient of fractional P(ID) ^α terminal sliding mode surface; Step 22. When the state trajectory of the fractional manipulator model reaches the sliding mode surface, there is s=D ^α s=D ^2α s=0, and calculate the α-order fractional derivative for the fractional-order P(ID) ^α terminal sliding mode surface: k _p D ^α x ₁ +k _d D ^2α x ₁ +k _i (|x ₁ |+|D ^α x ₁ | ^ρ )sgn(D ^α x ₁ )=0 (7) Step 23. Combining the fractional differential property D ^2α x ₁ =D ^α (D ^α x ₁ )=D ^α x ₂ , the expected sliding mode equation is obtained:

6. according to the variable structure finite time vibration suppression method of the described fractional-order manipulator model of claim 1, it is characterized in that, the establishment process of the self-adaptive estimation law of described fractional-order manipulator model unknown parameter is:

where D ^α is the Caputo-type fractional calculus operator,

is the estimated value of the first unknown parameter γ,

is the estimated value of the second unknown parameter λ, X=[x ₁ , x ₂ ] ^T is the state vector of the fractional-order manipulator model, m>0 is the gain of the first estimation law, n>0 is the gain of the second estimation law, k _d is the differential coefficient of the fractional order P(ID) ^α terminal sliding mode surface, and s is the sliding mode surface variable. 7. The variable-structure finite-time vibration suppression method of the fractional-order manipulator model according to claim 1, wherein the finite-time controller is:

Among them, J is the moment of inertia of the actuator, R is the resistance, η is the efficiency of the gearbox, υ is the gear ratio, K _t is the torque constant,

is the finite-time controller coefficient, ε(t) is the finite-time controller structural variable, s is the sliding mode surface variable, sgn(D ^α s) is the sign function, and k _d is the fractional order P(ID) ^α terminal sliding mode surface Differential coefficient, q ₁ is the third positive real number, q ₂ is the fourth positive real number, D ^α is the Caputo fractional calculus operator, q ₃ is the fifth positive real number, B is the viscous friction coefficient of the actuator, K _e is the back electromotive force constant, x ₂ is the second state variable, J is the moment of inertia of the actuator, τ is the torque on the control manipulator joint,

is the estimated value of the first unknown parameter γ, X=[x ₁ , x ₂ ] ^T is the state vector of the fractional manipulator model,

is the estimated value of the second unknown parameter λ, _ki is the integral coefficient of the fractional P(ID) ^α terminal sliding mode surface, x ₁ is the first state variable, and ρ∈(0,1) is the second fixed value.

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