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

Abstract

The present invention discloses a variable structure finite time vibration suppression method for a fractional order manipulator model, comprising: establishing a fractional order manipulator model by using a dynamic equation of the manipulator model; designing a nonlinear fractional order P(ID) by combining a sliding mode control technology; α Terminal sliding surface; considering the uncertainty of the lumped parameters of the fractional-order manipulator system, determine the adaptive estimation law of the unknown parameters of the fractional-order manipulator model; considering the influence of the fan-shaped nonlinear input and combining the adaptive estimation law of the unknown parameters of the fractional-order manipulator model, design a finite-time controller; apply the established adaptive estimation law of the unknown parameters of the fractional-order manipulator model and the designed finite-time controller to the fractional-order manipulator model to achieve finite-time vibration suppression. The variable structure finite-time vibration suppression method of the present invention can achieve finite-time vibration suppression of the fractional-order manipulator model, and realize that the state trajectory of the fractional-order manipulator model converges to the equilibrium state within a finite time.

Description

Technical Field

[0001] The present invention relates to the technical field of robot model control, and in particular to a variable structure finite time vibration suppression method for a fractional order robot model. Background Art

[0002] Robotic arm systems are widely used because they can perform repetitive tasks and have high precision in dangerous areas. For example, in the medical field, robotic arms are used for minimally invasive interventions and have very good therapeutic effects on patients in neurosurgery, orthopedics, and urology. At present, an enhanced robotic arm device has been used for limb rehabilitation, which can replace physical therapists. In the aerospace field, robotic arms have played an important role in the construction of space stations and space science experiments. In the field of industrial production, robotic arms can perform tasks that cannot be completed by humans. For personnel in special operations, the use of robotic arms greatly guarantees safe production and people's lives. Therefore, it is of great practical significance to design appropriate control laws to realize trajectory planning of robotic arm systems. Summary of the invention

[0003] In view of this, the present invention provides a variable structure finite time vibration suppression method for a fractional order manipulator model, which applies fractional order calculus theory to the construction of sliding surface and establishes a P(ID) α The terminal sliding surface can effectively solve the finite-time vibration suppression problem of fractional-order manipulator systems, filling the gap in research results in this aspect of fractional-order manipulator systems.

[0004] To achieve the above technical objectives, the present invention is implemented through the following technical solutions: a variable structure finite time vibration suppression method for a fractional order manipulator model, comprising the following steps:

[0005] 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 determine the fractional-order manipulator model by combining the fractional-order calculus theory;

[0006] Step 2: Use PID control technology and sliding mode control theory, combined with the state variables in step 1, to establish a fractional order P(ID) α Terminal sliding surface, when the state trajectory of the fractional-order robot model in step 1 reaches the fractional-order P(ID) α When the terminal sliding surface is α The fractional derivative of the terminal sliding surface is calculated to obtain the expected sliding mode equation;

[0007] Step 3: Consider the uncertainty of the lumped parameters of the fractional-order manipulator model, according to the fractional-order P(ID) αTerminal sliding surface, establish the adaptive estimation law of unknown parameters of fractional-order manipulator model;

[0008] Step 4: Considering the influence of fan-shaped nonlinear input and combining the adaptive estimation law of unknown parameters of the fractional-order manipulator model, a finite-time controller is designed;

[0009] Step 5: Apply the established adaptive estimation law of the unknown parameters of the fractional-order manipulator model and the designed finite-time controller to the fractional-order manipulator model to achieve finite-time vibration suppression.

[0010] Furthermore, step 1 includes the following sub-steps:

[0011] Step 11: Obtain the actual joint position θ measured by the encoder, the moment of 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 constant K from the two-degree-of-freedom robotic arm structure. t , Back EMF constant K e , resistance R, control torque τ on the manipulator joint, armature voltage input u, and establish the dynamic equation of the manipulator structure model:

[0012]

[0013] in, is the first-order derivative of θ, is the second-order derivative of θ;

[0014] Step 12: Take θ as the first state variable x1, As the second state variable x2, the dynamic equation of the manipulator structure model is transformed into an integer-order state space model of the manipulator structure model:

[0015]

[0016] Step 13: The integer-order state space model of the manipulator structure model in step 12 is transformed into the integer-order state space model using the Caputo-type fractional-order calculus operator D α Describe it, consider the influence of the fan-shaped nonlinear input φ(u) and the influence of the lumped parameter uncertainty ΔL(X,t) of the fractional-order manipulator model related to time t, and determine the fractional-order manipulator model:

[0017]

[0018] Among them, α∈(0,1) is the fractional order, X=[x1,x2] T is the state vector of the fractional-order robot model.

[0019] Furthermore, the lumped parameter uncertainty ΔL(X, t) of the fractional-order manipulator model related to time t includes: model error, parameter fluctuation, unmodeled dynamics and external disturbance, and the ΔL(X, t) satisfies:

[0020] |ΔL(X,t)|≤γ||X||+λ (4)

[0021] Among them, γ is the first unknown parameter and λ is the second unknown parameter.

[0022] Furthermore, the fan-shaped nonlinear input φ(u) is continuous in the interval [δ1,δ2] and satisfies:

[0023] δ1u 2 ≤uφ(u)≤δ2u 2 (5)

[0024] Among them, δ1 is the first slope, and δ2 is the second slope.

[0025] Furthermore, step 2 includes the following sub-steps:

[0026] Step 21: Use PID control technology and sliding mode control theory, combined with the state variables in step 12, to establish the fractional order P(ID) α Terminal sliding surface:

[0027] q1s+q2D α s+q3I α [(|s|+|D α s| σ )sgn(D α s)]=k p x1+k d D α x1+k i I α [(|x1|+|D α x1| ρ )sgn(D α x1)] (6)

[0028] Among them, s is the sliding surface variable, I α is the fractional integral operator, sgn() is the sign function, q1 is the third positive real number, q2 is the fourth positive real number, q3 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 fractional order P(ID) α Proportional coefficient of terminal sliding surface, k d is the fractional order P(ID) α Differential coefficient of terminal sliding surface, k i >0 is fractional order P(ID) αIntegration coefficient of terminal sliding surface;

[0029] Step 22: When the state trajectory of the fractional-order robot model reaches the sliding surface, s = D α s=D 2α s=0,logarithmic fractional order P(ID) α The terminal sliding surface obtains the α-order fractional derivative:

[0030] k p D α x1+k d D 2α x1+k i (|x1|+|D α x1| ρ )sgn(D α x1)=0 (7)

[0031] Step 23: Combine the fractional differential properties D 2α x1=D α (D α x1)=D α x2, we get the expected sliding mode equation:

[0032]

[0033] Furthermore, the process of establishing the adaptive estimation law of the unknown parameters of the fractional-order manipulator model is as follows:

[0034]

[0035] Among them, D α is a 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=[x1,x2] T is the state vector of the fractional-order manipulator model, m>0 is the first estimation law gain, n>0 is the second estimation law gain, k d is the fractional order P(ID) α The differential coefficient of the terminal sliding surface, s is the sliding surface variable.

[0036] Furthermore, the finite time controller is:

[0037]

[0038] Where 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 structure variable, s is the sliding surface variable, sgn(Dα s) is the symbol function, k d is the fractional order P(ID) α The differential coefficient of the terminal sliding surface, q1 is the third positive real number, q2 is the fourth positive real number, D α is a Caputo-type fractional calculus operator, q3 is the fifth positive real number, B is the viscous friction coefficient of the actuator, K e is the back electromotive force constant, x2 is the second state variable, J is the moment of inertia of the actuator, τ is the torque on the joint of the control manipulator, is the estimated value of the first unknown parameter γ, X=[x1,x2] T is the state vector of the fractional-order robot model, is the estimated value of the second unknown parameter λ, k i is the fractional order P(ID) α The integral coefficient of the terminal sliding surface, x1 is the first state variable, and ρ∈(0,1) is the second fixed value.

[0039] Compared with the prior art, the present invention has the following beneficial effects:

[0040] (1) The terminal sliding surface designed by the variable structure finite time vibration suppression method of the fractional-order manipulator model of the present invention combines the traditional PID technology and the sliding mode control technology, which has the advantages of strong robustness and adaptability of the traditional PID control, and the advantages of fast convergence speed of the sliding mode control. At the same time, the constructed fractional-order P(ID) α The terminal sliding surface contains a fractional-order calculus operator. By adjusting the calculus order, the convergence time can be effectively adjusted, and the control effect is good.

[0041] (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. The research process fully considers the influence of the lumped parameter uncertainty in the fractional-order manipulator model, and designs a suitable adaptive estimation law for the unknown upper bound of the lumped parameter uncertainty, which greatly improves the identification effect of the unknown parameters.

[0042] (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, both the approaching stage and the sliding mode stage converge in finite time;

[0043] (4) The variable structure finite time vibration suppression method of the fractional order manipulator model of the present invention fully considers the influence of the fan-shaped nonlinear input on the fractional order manipulator model, and designs a finite time controller, which can well overcome the adverse effects of nonlinear input and lumped parameter uncertainty, and improve the robustness of the controlled system;

[0044] (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 occurring inside the fractional-order manipulator model within a given time, obtain the desired output and performance indicators, save control costs, and improve economic benefits. BRIEF DESCRIPTION OF THE DRAWINGS

[0045] Figure 1 It is a flow chart of the variable structure finite time vibration suppression method of the fractional order manipulator model of the present invention;

[0046] Figure 2 This is the structural diagram of the fan-shaped nonlinear input function. DETAILED DESCRIPTION

[0047] To more clearly illustrate the purpose and advantages of the present invention, the application principle of the present invention is described in detail below in conjunction with the accompanying drawings.

[0048] like Figure 1 The flowchart of the variable structure finite time vibration suppression method of the fractional order manipulator model of the present invention is as follows:

[0049] 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 determine the fractional-order manipulator model in combination with the fractional-order calculus theory. The fractional-order operator is particularly suitable for describing systems with memory characteristics. The manipulator motion process involves a large number of memory characteristic links. Therefore, using fractional-order calculus to mathematically model the manipulator model can obtain more accurate results. The specific steps include the following:

[0050] Step 11: Obtain the actual joint position θ measured by the encoder, the moment of 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 constant K from the two-degree-of-freedom robotic arm structure. t , Back EMF constant K e , resistance R, control torque τ on the manipulator joint, armature voltage input u, and establish the dynamic equation of the manipulator structure model:

[0051]

[0052] in, is the first-order derivative of θ, is the second-order derivative of θ;

[0053] Step 12: Take θ as the first state variable x1, As the second state variable x2, the dynamic equation of the manipulator structure model is transformed into an integer-order state space model of the manipulator structure model:

[0054]

[0055] Step 13: The integer-order state space model of the manipulator structure model in step 12 is transformed into the integer-order state space model using the Caputo-type fractional-order calculus operator D α Describe it, consider the influence of the fan-shaped nonlinear input φ(u) and the influence of the lumped parameter uncertainty ΔL(X,t) of the fractional-order manipulator model related to time t, and determine the fractional-order manipulator model:

[0056]

[0057] Among them, α∈(0,1) is the fractional order, X=[x1,x2] T is the state vector of the fractional-order robot model.

[0058] The lumped parameter uncertainty ΔL(X, t) of the fractional-order manipulator model related to time t in the present invention includes: model error, parameter fluctuation, unmodeled dynamics and external disturbance. To facilitate the derivation of controller design, it is assumed that ΔL(X, t) is bounded and satisfies:

[0059] |ΔL(X,t)|≤γ||X||+λ (4)

[0060] Among them, γ is the first unknown parameter and λ is the second unknown parameter.

[0061] like Figure 2 In the present invention, the fan-shaped nonlinear input φ(u) is nonlinear and continuous in the fan-shaped area surrounded by the first slope δ1 and the second slope δ2. It is one of the typical nonlinear characteristics often encountered in the current controller execution process. This type of nonlinear characteristic will seriously affect the performance of the controlled system and even cause the instability of the mechanical arm structure model. Therefore, it must be considered when designing the controller. In the present invention, the fan-shaped nonlinear input φ(u) satisfies:

[0062] δ1u 2 ≤uφ(u)≤δ2u 2 (5)

[0063] Step 2: Use PID control technology and sliding mode control theory, combined with the state variables in step 1, to establish a fractional order P(ID) α Terminal sliding surface, when the state trajectory of the fractional-order robot model in step 1 reaches the fractional-order P(ID) α Terminal sliding surface, fractional order P(ID) αThe fractional derivative of the terminal sliding surface is obtained to obtain the expected sliding mode equation. The designed fractional order P(ID) α The terminal sliding surface combines traditional PID technology and sliding mode control technology, and has the advantages of fast convergence speed and strong robustness. It specifically includes the following sub-steps:

[0064] Step 21: Use PID control technology and sliding mode control theory, combined with the state variables in step 12, to establish the fractional order P(ID) α Terminal sliding surface:

[0065] q1s+q2D α s+q3I α [(|s|+|D α s| σ )sgn(D α s)]=k p x1+k d D α x1+k i I α [(|x1|+|D α x1| ρ )sgn(D α x1)] (6)

[0066] Among them, s is the sliding surface variable, I α is the fractional integral operator, sgn() is the sign function, q1 is the third positive real number, q2 is the fourth positive real number, q3 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 fractional order P(ID) α Proportional coefficient of terminal sliding surface, k d is the fractional order P(ID) α Differential coefficient of terminal sliding surface, k i >0 is fractional order P(ID) α Integration coefficient of terminal sliding surface;

[0067] Step 22: When the state trajectory of the fractional-order robot model reaches the fractional-order P(ID) α The terminal sliding surface will be along the fractional order P(ID) α The terminal sliding surface performs sliding motion until it moves to the origin. It reaches the fractional order P(ID) α After the terminal sliding surface satisfies s=D α s=D 2α s=0,logarithmic fractional order P(ID) α The terminal sliding surface obtains the α-order fractional derivative:

[0068] k p D α x1+kd D 2α x1+k i (|x1|+|D α x1| ρ )sgn(D α x1)=0 (7)

[0069] Step 23: Combine the fractional differential properties D 2α x1=D α (D α x1)=D α x2, we get the expected sliding mode equation:

[0070]

[0071] For the sliding mode stage in the process of manipulator structure control, the Lyapunov function is selected To verify the finite-time stability of the desired sliding mode equation.

[0072] Calculate the α-order fractional derivative of V1(t), and according to the characteristics of fractional operators, we get:

[0073]

[0074] Let k i >k d , formula (11) is simplified to

[0075]

[0076] Select helper function

[0077] Transform formula (12) into:

[0078]

[0079] According to the research results of Stabilization of a class of cascade nonlinear switched systems with application to chaotic systems, there exists a constant η1>0 such that have to:

[0080]

[0081] According to the study of Graph theory-based finite-time synchronization of fractional-order complex dynamical networks, for a system whose fractional-order derivative of the energy function satisfies equation (15), its sliding mode will converge to the origin within a finite time t2:

[0082]

[0083] Among them, t1 is the time to reach the sliding surface, and Γ() is the gamma function.

[0084] Step 3: Consider the influence of lumped parameter uncertainty of fractional-order manipulator model, according to fractional-order P(ID) α The terminal sliding surface is used to establish the adaptive estimation law of the unknown parameters of the fractional-order manipulator model, which can well predict the upper bound of the unknown parameters and improve 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:

[0085]

[0086] 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.

[0087] Step 4: Considering the influence of the fan-shaped nonlinear input and combining the adaptive estimation law of the unknown parameters of the fractional-order manipulator model, a finite-time controller is designed to 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:

[0088]

[0089] in, is the finite time controller coefficient, ε(t) is the finite time controller structure variable, sgn(D α s) is a sign function, whose value is related to D α s related, when D α s>0,sgn(D α s)=1, when D α s=0,sgn(D α s)=0, when D α s<0,sgn(D α s)=-1.

[0090] Combining formula (5), we get:

[0091]

[0092] because And the structural parameters of the robot arm are all positive, and we can further obtain:

[0093] -sgn(D α s)φ(u)≥ε(t) (18)

[0094] Multiply both sides of the inequality in formula (18) by |D α s|, and according to |D α s|sgn(D α s)=D α s, we can get:

[0095]

[0096] Next, verify the finite time characteristics of the approach phase:

[0097] Select the Lyapunov function

[0098]

[0099] in, is the estimated error of the unknown upper bound parameter γ, that is, the estimated value The difference between the actual value γ and is the estimated error of the unknown upper bound parameter λ, that is, the estimated value The difference between the actual value λ and

[0100] By taking the α-order fractional derivative of equation (20) and combining it with equations (3), (6) and (9), we can obtain:

[0101]

[0102] According to the designed finite time controller (10) and the relationship (19) satisfied by the sector nonlinear input, formula (21) is further simplified as:

[0103]

[0104] Similar to equation (12), the trajectory of the fractional-order manipulator model will reach the sliding surface within a finite time t1 and continue to move along the sliding surface to the origin. At this time, the system vibration is effectively suppressed:

[0105]

[0106] Among them, η2>0 is an auxiliary parameter.

[0107] To sum up, it is proved that both the approaching stage and the sliding mode stage are finite-time stable, that is, the entire control stage is finite-time stable.

[0108] Step 5: Apply the established adaptive estimation law of the unknown parameters of the fractional-order manipulator model and the designed finite-time controller to the fractional-order manipulator model to achieve finite-time vibration suppression.

[0109] 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 order manipulator model converges to the equilibrium state within a finite 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 fractional order P(ID) α The design of the terminal sliding surface can flexibly adjust the transition time by adjusting the fractional calculus order.

[0110] The above are only preferred embodiments of the present invention, and the protection scope of the present invention is not limited to the above embodiments. All technical solutions under the concept of the present invention belong to the protection scope of the present invention. It should be pointed out that for ordinary technicians in this technical field, 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

1. A variable structure finite time vibration suppression method for a fractional order manipulator model, characterized in that: The following steps are involved: 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 determine the fractional-order manipulator model by combining the fractional-order calculus theory; including the following sub-steps: Step 11: Obtain the actual joint position θ measured by the encoder, the moment of 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 constant K from the two-degree-of-freedom robotic arm structure. t , Back EMF constant K e , resistance R, control torque τ on the manipulator joint, armature voltage input u, and establish the dynamic equation of the manipulator structure model: in, is the first-order derivative of θ, is the second-order derivative of θ; Step 12: Take θ as the first state variable x1, As the second state variable x2, the dynamic equation of the manipulator structure model is transformed into an integer-order state space model of the manipulator structure model: Step 13: The integer-order state space model of the manipulator structure model in step 12 is transformed into the integer-order state space model using the Caputo-type fractional-order calculus operator D α Describe it, consider the influence of the fan-shaped nonlinear input φ(u) and the influence of the lumped parameter uncertainty ΔL(X,t) of the fractional-order manipulator model related to time t, and determine the fractional-order manipulator model: Among them, α∈(0,1) is the fractional order, X=[x1,x2] T is the state vector of the fractional-order robot model; Step 2: Use PID control technology and sliding mode control theory, combined with the state variables in step 1, to establish a fractional order P(ID) α Terminal sliding surface, when the state trajectory of the fractional-order robot model in step 1 reaches the fractional-order P(ID) α When the terminal sliding surface is α The fractional derivative of the terminal sliding surface is calculated to obtain the expected sliding mode equation; Step 3: Consider the uncertainty of the lumped parameters of the fractional-order manipulator model, according to the fractional-order P(ID) α Terminal sliding surface, establish the adaptive estimation law of unknown parameters of fractional-order manipulator model; Step 4: Considering the influence of fan-shaped nonlinear input and combining the adaptive estimation law of unknown parameters of the fractional-order manipulator model, a finite-time controller is designed; Step 5: Apply the established adaptive estimation law of the unknown parameters of the fractional-order manipulator model and the designed finite-time controller to the fractional-order manipulator model to achieve finite-time vibration suppression.

2. The variable structure finite time vibration suppression method of the fractional order manipulator model according to claim 1 is characterized in that: The lumped parameter uncertainty ΔL(X, t) of the fractional-order manipulator model related to time t includes: model error, parameter fluctuation, unmodeled dynamics and external disturbance, and 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.

3. The variable structure finite time vibration suppression method of the fractional order manipulator model according to claim 1 is characterized in that: The fan-shaped nonlinear input φ(u) is continuous in the interval [δ1,δ2] and satisfies: δ1u 2 ≤uφ(u)≤δ2u 2 (5) Among them, δ1 is the first slope, and δ2 is the second slope.

4. The variable structure finite time vibration suppression method for the fractional order manipulator model according to claim 1 is characterized in that: Step 2 includes the following sub-steps: Step 21: Use PID control technology and sliding mode control theory, combined with the state variables in step 12, to establish the fractional order P(ID) α Terminal sliding surface: q1s+q2D α s+q3I α [(|s|+|D α s| σ )sgn(D α s)]=k p x1+k d D α x1+k i I α [(|x1|+|D α x1| ρ )sgn(D α x1)](6) Among them, s is the sliding surface variable, I α is the fractional integral operator, sgn() is the sign function, q1 is the third positive real number, q2 is the fourth positive real number, q3 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 fractional order P(ID) α Proportional coefficient of terminal sliding surface, k d is the fractional order P(ID) α Differential coefficient of terminal sliding surface, k i >0 is fractional order P(ID) α Integration coefficient of terminal sliding surface; Step 22: When the state trajectory of the fractional-order robot model reaches the sliding surface, s = D α s=D 2α s=0,logarithmic fractional order P(ID) α The terminal sliding surface obtains the α-order fractional derivative: k p D α x1+k d D 2α x1+k i (|x1|+|D α x1| ρ )sgn(D α x1)=0 (7) Step 23: Combine the fractional differential properties D 2α x1=D α (D α x1)=D α x2, we get the expected sliding mode equation:

5. The variable structure finite time vibration suppression method of the fractional order manipulator model according to claim 1 is characterized in that: The process of establishing the adaptive estimation law of the unknown parameters of the fractional-order manipulator model is as follows: Among them, D α is a 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=[x1,x2] T is the state vector of the fractional-order manipulator model, m>0 is the first estimation law gain, n>0 is the second estimation law gain, k d is the fractional order P(ID) α The differential coefficient of the terminal sliding surface, s is the sliding surface variable.

6. The variable structure finite time vibration suppression method of the fractional order manipulator model according to claim 1 is characterized in that: The finite time controller is: Where 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 structure variable, s is the sliding surface variable, sgn(D α s) is the symbol function, k d is the fractional order P(ID) α The differential coefficient of the terminal sliding surface, q1 is the third positive real number, q2 is the fourth positive real number, D α is a Caputo-type fractional calculus operator, q3 is the fifth positive real number, B is the viscous friction coefficient of the actuator, K e is the back electromotive force constant, x2 is the second state variable, J is the moment of inertia of the actuator, τ is the torque on the joint of the control manipulator, is the estimated value of the first unknown parameter γ, X=[x1,x2] T is the state vector of the fractional-order robot model, is the estimated value of the second unknown parameter λ, k i is the fractional order P(ID) α The integral coefficient of the terminal sliding surface, x1 is the first state variable, and ρ∈(0,1) is the second fixed value.

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