CN115202189A - Variable structure finite time vibration suppression method of fractional order mechanical arm model - Google Patents
Variable structure finite time vibration suppression method of fractional order mechanical arm model Download PDFInfo
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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
Technical Field
The invention relates to the technical field of mechanical arm model control, in particular to a variable structure finite time vibration suppression method of a fractional order mechanical arm model.
Background
The mechanical arm system can execute repetitive tasks and has higher accuracy in dangerous areas, so the mechanical arm system is very widely applied, for example, in the medical field, the mechanical arm is used for minimally invasive intervention and has very good treatment effect on the treatment of patients of neurosurgery, orthopedics, urology surgery and the like; currently, an enhanced robotic arm device has been used for limb rehabilitation, which can replace a physical therapist; in the field of aerospace, the mechanical arm plays an important role in the construction of a space station and space science experiments; in the field of industrial production, the mechanical arm can be competent for tasks which cannot be completed manually, and for personnel under special operation, the use of the mechanical arm greatly ensures safe production and life safety of people. Therefore, the design of a proper control law to realize the trajectory planning of the mechanical arm system has important practical significance.
Disclosure of Invention
In view of the above, the invention provides a variable structure finite time vibration suppression method of a fractional order mechanical arm model, which introduces a fractional order calculus theory into the construction of a sliding mode surface to establish P (ID) α The terminal sliding mode surface can well solve the problem of limited-time vibration suppression of the fractional order mechanical arm system, and fills the blank of the research result of the fractional order mechanical arm system.
In order to achieve the technical purpose, the invention is realized by the following technical scheme: a variable structure finite time vibration suppression method of a fractional order mechanical arm model comprises the following steps:
step 1, selecting state variables in a two-degree-of-freedom mechanical arm structure, converting a kinetic equation of a mechanical arm structure model into an integer order state space model of the mechanical arm structure model, and determining a fractional order mechanical arm model by combining a fractional order calculus theory;
step 2, establishing a state variable in the step 1 by utilizing a PID control technology and a sliding mode control theory and combining the state variableFractional order P (ID) α A terminal sliding mode surface, when the state track of the fractional order mechanical arm model in the step 1 reaches the fractional order P (ID) α When the terminal sliding mode is completed, fractional order P (ID) α The fractional order derivative of the terminal sliding mode surface is solved to obtain an expected sliding mode state equation;
step 3, considering uncertainty influence of lumped parameters of the fractional order mechanical arm model, and according to the fractional order P (ID) α Establishing a self-adaptive estimation law of unknown parameters of a fractional order mechanical arm model on a terminal sliding mode surface;
step 4, considering the influence of fan-shaped nonlinear input and combining with an adaptive estimation law of unknown parameters of a fractional order mechanical arm model, and designing a finite time controller;
and 5, applying the established self-adaptive estimation law of 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.
Further, step 1 comprises the following substeps:
step 11, obtaining an actual joint position theta, a rotational inertia J of an actuator, a viscous friction coefficient B of an actuating mechanism, an efficiency eta of a gearbox, a gear ratio upsilon and a torque constant K measured by an encoder from a two-degree-of-freedom mechanical arm structure t Counter electromotive force constant K e The resistance R, the moment tau on the manipulator joint and the armature voltage input u are controlled, and a dynamic equation of the manipulator structure model is established:
wherein,
is the first derivative of the value of theta and,
is the second derivative of θ;
step 12, taking theta as a first state variable x 1 ，
As a second state variable x 2 Converting a kinetic equation of the mechanical arm structure model into an integer order state space model of the mechanical arm structure model:
step 13, adopting Caputo type fractional calculus operator D to the integral order state space model of the mechanical arm structure model in the step 12 α Describing, considering the influence of the fan-shaped nonlinear input phi (u) and the influence of the lumped parameter uncertainty delta L (X, t) of the fractional order mechanical arm model relative to the time t, determining the fractional order mechanical arm model:
wherein alpha epsilon (0,1) is a fractional order, X = [ X ] 1 ,x 2 ] T Is the state vector of the fractional order mechanical arm model.
Further, the fractional order manipulator model lumped parameter uncertainty Δ L (X, t) related to time t includes: model errors, parameter fluctuations, unmodeled dynamics, and external disturbances, the Δ L (X, t) satisfying:
|ΔL(X,t)|≤γ||X||+λ (4)
where γ is the first unknown parameter and λ is the second unknown parameter.
Further, the fan-shaped nonlinear input phi (u) is in the interval [ delta ] 1 ,δ 2 ]Continuous, and satisfies:
δ 1 u 2 ≤uφ(u)≤δ 2 u 2 (5)
wherein, delta 1 Is a first slope, δ 2 And is the second slope, and u is the finite time controller.
Further, step 2 comprises the following sub-steps:
step 21Establishing a fractional order P (ID) by utilizing a PID control technology and a sliding mode control theory and combining the state variables in the step 12 α A 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)
wherein s is a sliding mode surface variable, I α For fractional order integration operators, sgn () is a sign function, q 1 Is a third positive real number, q 2 Is a fourth positive real number, q 3 Is a fifth positive real number, σ e (0,1) is a first fixed value, ρ e (0,1) is a second fixed value, k p Fractional order P (ID) greater than 0 α Coefficient of proportionality, k, of the sliding surface of the terminal d Is of fractional order P (ID) α Differential coefficient of terminal sliding mode surface, k i Fractional order P (ID) > 0 α Integral coefficient of the terminal sliding mode surface;
step 22, when the state track of the fractional order mechanical arm model reaches the sliding mode surface, s = D α s＝D 2α s =0, for fractional order P (ID) α And (3) solving an alpha fractional derivative from a 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 fractional order differential properties D 2α x 1 ＝D α (D α x 1 )＝D α x 2 And obtaining an expected sliding mode state equation:
further, the establishment process of the adaptive estimation law of the unknown parameters of the fractional order manipulator model is as follows:
wherein D is α In order to be a Caputo type fractional calculus operator,
is an estimate of the first unknown parameter y,
is an estimate of the second unknown parameter λ, X = [ X = 1 ,x 2 ] T Is the state vector of the fractional order mechanical arm model, m is more than 0 and is the first estimation law gain, n is more than 0 and is the second estimation law gain, k is d Is of fractional order P (ID) α And the differential coefficient of the terminal sliding mode surface, s, is a sliding mode surface variable.
Further, the finite time controller is:
wherein J is the moment of inertia of the actuator, R is the resistance, eta is the efficiency of the gearbox, upsilon is the gear ratio, K t In order to be a constant of the torque,
is the coefficient of the finite time controller, epsilon (t) the structure variable of the finite time controller, s is the variable of the sliding mode surface, sgn (D) α s) is a sign function, k d Is of fractional order P (ID) α Differential coefficient of terminal slip form surface, q 1 Is a third positive real number, q 2 Is a fourth positive real number, D α For Caputo type fractional calculus operators, q 3 Is a fifth positive real number, B is the viscous friction coefficient of the actuator, K e Is a back electromotive force constant, x 2 Is a second state variable, J is the rotational inertia of the actuator, tau is the moment on the manipulator joint,
x = [ X ] as an estimate of the first unknown parameter γ 1 ,x 2 ] T Is a state vector of the fractional order mechanical arm model,
is an estimate of a second unknown parameter, k i Is a fractional order P (ID) α Integral coefficient of terminal sliding mode surface, x 1 For the first state variable, ρ ∈ (0,1) is a second fixed value.
Compared with the prior art, the invention has the following beneficial effects:
(1) The variable structure finite time vibration suppression method of the fractional order mechanical arm model is combined with a terminal sliding mode surface designed by the traditional PID technology and the sliding mode control technology, has the advantages of strong robustness and adaptability of the traditional PID control, has the advantages of high convergence speed of the sliding mode control and the like, and simultaneously constructs a fractional order P (ID) α The terminal sliding mode surface contains a fractional calculus operator, the convergence time can be effectively adjusted by adjusting the calculus order, and the control effect is good;
(2) The variable structure finite time vibration suppression method of the fractional order mechanical arm model realizes finite time control through the fractional order mechanical arm model, researches the fractional order model of the mechanical arm structure for the first time, fully considers the uncertainty influence of lumped parameters in the fractional order mechanical arm model in the research process, designs a proper self-adaptive estimation law on the unknown upper bound of the uncertainty of the lumped parameters, and greatly improves the identification effect of unknown parameters;
(3) According to the variable structure finite time vibration suppression method of the fractional order mechanical arm model, the Lyapunov function in a proper form is selected to verify the finite time stability of an approach stage and a sliding mode stage, namely the approach stage and the sliding mode stage are both finite time convergence;
(4) The variable structure finite time vibration suppression method of the fractional order mechanical arm model fully considers the influence of fan-shaped nonlinear input on the fractional order mechanical arm model, designs the finite time controller, can well overcome the adverse influence caused by nonlinear input and lumped parameter uncertainty, and improves the robustness of a controlled system;
(5) The variable structure finite time vibration suppression method of the fractional order mechanical arm model can effectively suppress the vibration phenomenon in the fractional order mechanical arm model within a given time, obtain expected output and performance indexes, save control cost and improve economic benefit.
Drawings
FIG. 1 is a flow chart of a variable structure finite time vibration suppression method of a fractional order mechanical arm model according to the present invention;
FIG. 2 is a block diagram of a fan nonlinear input function.
Detailed Description
For a better understanding of the objects and advantages of the invention, reference should be made to the following detailed description taken in conjunction with the accompanying drawings.
Fig. 1 is a flowchart of a variable structure finite time vibration suppression method of a fractional order mechanical arm model according to the present invention, and the variable structure finite time vibration suppression method specifically includes the following steps:
step 1, selecting state variables in a two-degree-of-freedom mechanical arm structure, converting a kinetic equation of a mechanical arm structure model into an integer order state space model of the mechanical arm structure model, determining a fractional order mechanical arm model by combining a fractional order calculus theory, wherein a fractional order operator is particularly suitable for describing a system with memory characteristics, and the motion process of the mechanical arm relates to a large number of memory characteristic links, so that a more accurate effect can be obtained by adopting the fractional order calculus to perform mathematical modeling on the mechanical arm model; the method specifically comprises the following substeps:
step 11, obtaining an actual joint position theta, a rotational inertia J of an actuator, a viscous friction coefficient B of an actuating mechanism, an efficiency eta of a gearbox, a gear ratio upsilon and a torque constant K measured by an encoder from a two-degree-of-freedom mechanical arm structure t Counter electromotive force constant K e The resistance R, the moment tau on the manipulator joint and the armature voltage input u are controlled, and a dynamic equation of the manipulator structure model is established:
wherein,
is the first derivative of the value of theta,
is the second derivative of θ;
step 12, taking theta as a first state variable x 1 ，
As a second state variable x 2 Converting a kinetic equation of the mechanical arm structure model into an integer order state space model of the mechanical arm structure model:
step 13, adopting a Caputo type fractional calculus operator D to the integral order state space model of the mechanical arm structure model in the step 12 α Describing, considering the influence of the fan-shaped nonlinear input phi (u) and the influence of the lumped parameter uncertainty delta L (X, t) of the fractional order mechanical arm model relative to the time t, determining the fractional order mechanical arm model:
wherein alpha epsilon (0,1) is a fractional order, X = [ X ] 1 ,x 2 ] T Is the state vector of the fractional order mechanical arm model.
The lumped parameter uncertainty Delta L (X, t) of the fractional order mechanical arm model related to the time t comprises the following steps: model errors, parameter fluctuations, unmodeled dynamics, and external disturbances, to facilitate controller design derivation, assume Δ L (X, t) is bounded and satisfies:
|ΔL(X,t)|≤γ||X||+λ (4)
where γ is the first unknown parameter and λ is the second unknown parameter.
Referring to FIG. 2, the fan-shaped nonlinear input phi (u) of the present invention has a first slope delta 1 And a second slope delta 2 The sector area is non-linear and continuous, which is one of typical non-linear characteristics commonly encountered in the current controller implementation process, and such non-linear characteristics can seriously affect the performance of the controlled system, even cause instability of the mechanical arm structure model, and therefore, the design of the controller must be considered. The fan-shaped nonlinear input phi (u) satisfies the following conditions:
δ 1 u 2 ≤uφ(u)＜δ 2 u 2 (5)
wherein u is a finite time controller.
Step 2, establishing a fractional order P (ID) by utilizing a PID control technology and a sliding mode control theory and combining the state variables in the step 1 α A terminal sliding mode surface, when the state track of the fractional order mechanical arm model in the step 1 reaches the fractional order P (ID) α Terminal sliding mode surface, fractional order P (ID) α The fractional order derivative is obtained from the terminal sliding mode surface to obtain an expected sliding mode state equation, and the designed fractional order P (ID) α The terminal sliding mode surface combines the traditional PID technology and the sliding mode control technology, and has the advantages of high convergence speed, strong robustness and the like; the method specifically comprises the following substeps:
step 21, establishing a fractional order P (ID) by utilizing a PID control technology and a sliding mode control theory and combining the state variables in the step 12 α A 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)
wherein s is a sliding mode surface variable, I α For fractional order integration operators, sgn () is a sign function, q 1 Is a third positive real number, q 2 Is a fourthPositive real number, q 3 Is a fifth positive real number, σ e (0,1) is a first fixed value, ρ e (0,1) is a second fixed value, k p Fractional order P (ID) greater than 0 α Coefficient of proportionality, k, of the sliding surface of the terminal d Is of fractional order P (ID) α Differential coefficient of terminal sliding mode surface, k i Fractional order P (ID) > 0 α Integral coefficient of the terminal sliding mode surface;
step 22, when the state track of the fractional order mechanical arm model reaches the fractional order P (ID) α Terminal sliding mode surface, will follow fractional order P (ID) α And the terminal sliding mode surface performs sliding mode movement until the terminal sliding mode surface moves to the original point. To a fractional order P (ID) α S = D is satisfied after the sliding form surface of the terminal α s＝D 2α s =0, for fractional order P (ID) α And (3) solving an alpha-order fractional derivative from a 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 fractional order differential properties D 2α x 1 ＝D α (D α x 1 )＝D α x 2 And obtaining an expected sliding mode state equation:
selecting a Lyapunov function aiming at a sliding mode stage in the structural control process of the mechanical arm
To verify the finite time stability of the desired sliding mode state equation.
To V 1 (t) solving an alpha-order fractional order derivative, and obtaining:
let k i ＞k d Equation (11) is simplified to
Selecting an auxiliary function
Transform equation (12) to:
according to the results of the Stabilization of a class of case node switched systems with application to electronic systems, there is a constant η 1 > 0, such that
Obtaining:
according to the study of Graph-based fine-time synchronization of fractional digital networks, the fractional derivative of the energy function satisfies the system of formula (15), the sliding mode of which will be at the finite time t 2 Inner convergence to the origin:
wherein, t 1 To reach sliding mode surface time, Γ () is a gamma function.
Step 3, considering the uncertain influence of lumped parameters of the fractional order mechanical arm model, and according to the fractional order P (ID) α A terminal sliding mode surface is used for establishing a self-adaptive estimation law of unknown parameters of a fractional order mechanical arm model, so that the upper bound of the unknown parameters can be well predicted, and the robustness of the system is improved;the method comprises the following steps of:
wherein,
is an estimate of the first unknown parameter y,
the value of m and n directly determines the identification rate of the unknown parameter.
Step 4, considering the influence of fan-shaped nonlinear input and combining with a self-adaptive estimation law of unknown parameters of a fractional order mechanical arm model, designing a finite time controller, and inhibiting the vibration phenomenon generated by the system in a given time so as to improve the running reliability of the system; the finite time controller in the invention is:
wherein,
is a finite time controller coefficient, ε (t) is a finite time controller structure variable, sgn (D) α s) is a sign function, the value of which is equal to D α s is related to 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。
Combining the formula (5), obtaining:
due to the fact that
And the mechanical arm structure parameters are positive, further obtaining:
-sgn(D α s)φ(u)≥ε(t) (18)
multiplying the inequality of equation (18) on both sides simultaneously by | D α s | and according to | D α s|sgn(D α s)＝D α s, can obtain:
next, the approach phase finite time characteristics are verified:
selecting a Lyapunov function
Wherein,
estimation error, i.e. estimated value, for unknown upper bound parameter y
And the difference between the actual value y and the,
error of estimation for unknown upper bound parameter lambda, i.e. estimated value
And the difference between the actual value of x,
the α fractional derivative is obtained by calculating the α fractional derivative from equation (20) and combining equations (3), (6) and (9):
based on the designed finite time controller (10) and the relation (19) satisfied by the fan nonlinear input, the formula (21) is further simplified as:
similar to equation (12), the trajectory of the fractional order manipulator model will be within a finite time t 1 The inner part reaches the sliding mode surface and continues to move to the original point along the sliding mode surface, and the system vibration is effectively inhibited at the moment:
wherein eta is 2 > 0 is an auxiliary parameter.
In summary, the approach phase and the sliding mode phase are both verified to be finite-time stable, i.e. the whole control phase is finite-time stable.
And 5, applying the established self-adaptive estimation law of 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 finite time vibration suppression method of the fractional order mechanical arm model can realize the finite time vibration suppression of the fractional order mechanical arm model and realize the convergence of the state track of the fractional order mechanical arm model to the equilibrium state within finite time. Particularly, 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 mechanical arm model. Fractional order P (ID) compared to conventional sliding mode control techniques α The design of the terminal sliding mode surface can flexibly adjust the transition time by adjusting the fractional calculus orderAnd (3) removing the solvent.
The above is only a preferred embodiment of the present invention, and the scope of the present invention is not limited to the above embodiment, and any technical solutions that fall under the spirit of the present invention fall within the scope of the present invention. It should be noted that modifications and adaptations to those skilled in the art without departing from the principles of the present invention may be apparent to those skilled in the relevant art and are intended to be within the 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 by comprising the following steps:
step 1, selecting state variables in a two-degree-of-freedom mechanical arm structure, converting a kinetic equation of a mechanical arm structure model into an integer order state space model of the mechanical arm structure model, and determining a fractional order mechanical arm model by combining a fractional order calculus theory;
step 2, establishing a fractional order P (ID) by utilizing a PID control technology and a sliding mode control theory and combining the state variables in the step 1 α A terminal sliding mode surface, when the state track of the fractional order mechanical arm model in the step 1 reaches the fractional order P (ID) α When the terminal sliding mode is face, fractional order P (ID) α The fractional order derivative of the terminal sliding mode surface is solved to obtain an expected sliding mode state equation;
step 3, considering uncertainty influence of lumped parameters of the fractional order mechanical arm model, and according to the fractional order P (ID) α A terminal sliding mode surface is used for establishing a self-adaptive estimation law of unknown parameters of the fractional order mechanical arm model;
step 4, considering the influence of fan-shaped nonlinear input and combining with a self-adaptive estimation law of unknown parameters of a fractional order mechanical arm model, designing a finite time controller;
and 5, 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.
2. The variable structure finite time vibration suppression method of fractional order mechanical arm model according to claim 1, wherein the step 1 comprises the following sub-steps:
step 11, obtaining an actual joint position theta, a rotational inertia J of an actuator, a viscous friction coefficient B of an actuating mechanism, efficiency eta of a gearbox, a gear ratio upsilon and a torque constant K which are measured by an encoder from a two-degree-of-freedom mechanical arm structure t Counter electromotive force constant K e The resistance R, the moment tau on the manipulator joint and the armature voltage input u are controlled, and a dynamic equation of the manipulator structure model is established:
wherein,
is the first derivative of the value of theta and,
is the second derivative of θ;
step 12, taking theta as a first state variable x 1 ，
As a second state variable x 2 Converting a kinetic equation of the mechanical arm structure model into an integer order state space model of the mechanical arm structure model:
step 13, adopting a Caputo type fractional calculus operator D to the integral order state space model of the mechanical arm structure model in the step 12 α Describing, considering the influence of the fan-shaped nonlinear input phi (u) and the influence of the lumped parameter uncertainty delta L (X, t) of the fractional order mechanical arm model relative to the time t, determining the fractional order mechanical arm model:
wherein alpha epsilon (0,1) is a fractional order, X = [ X ] 1 ，x 2 ] T Is the state vector of the fractional order mechanical arm model.
3. The method for structure-variable finite-time vibration suppression of a fractional order mechanical arm model according to claim 2, wherein the time t-dependent fractional order mechanical arm model lumped parameter uncertainty Δ L (X, t) comprises: model errors, parameter fluctuations, unmodeled dynamics, and external disturbances, the Δ L (X, t) satisfying:
|ΔL(X，t)|≤γ||X||+λ (4)
where γ is the first unknown parameter and λ is the second unknown parameter.
4. The method for suppressing variable-structure finite-time vibration of a fractional order mechanical arm model according to claim 2, wherein the fan-shaped nonlinear input phi (u) is in an interval [ delta ] (u) 1 ，δ 2 ]Continuous, and satisfies:
δ 1 u 2 ≤uφ(u)≤δ 2 u 2 (5)
wherein, delta 1 Is a first slope, δ 2 Is the second slope, u is the finite time controller.
5. The method for suppressing variable structure finite time vibration of a fractional order mechanical arm model according to claim 2, wherein the step 2 comprises the following sub-steps:
step 21, establishing a fractional order P (ID) by utilizing a PID control technology and a sliding mode control theory and combining the state variables in the step 12 α A 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)
wherein s is a sliding mode surface variable, I α For fractional order integration operators, sgn () is a sign function, q 1 Is a third positive real number, q 2 Is a fourth positive real number, q 3 Is a fifth positive real number, σ e (0,1) is a first fixed value, ρ e (0,1) is a second fixed value, k p Fractional order P (ID) > 0 α Proportionality coefficient of terminal sliding mode surface, k d Is a fractional order P (ID) α Differential coefficient of terminal sliding mode surface, k i Fractional order P (ID) > 0 α Integral coefficient of the terminal sliding mode surface;
step 22, when the state track of the fractional order mechanical arm model reaches the sliding mode surface, s = D α s＝D 2α s =0, for fractional order P (ID) α And (3) solving an alpha-order fractional derivative from a 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 fractional order differential properties D 2α x 1 ＝D α (D α x 1 )＝D α x 2 And obtaining an expected sliding mode state equation:
6. the variable structure finite time vibration suppression method of the fractional order mechanical arm model according to claim 1, wherein the establishment process of the self-adaptive estimation law of the unknown parameters of the fractional order mechanical arm model is as follows:
wherein D is α In order to be a Caputo type fractional calculus operator,
is an estimate of the first unknown parameter y,
is an estimate of the second unknown parameter λ, X = [ X = 1 ，x 2 ] T Is the state vector of the fractional order mechanical arm model, m is more than 0 and is the first estimation law gain, n is more than 0 and is the second estimation law gain, k is d Is of fractional order P (ID) α And (4) the differential coefficient of the terminal sliding mode surface, wherein s is a sliding mode surface variable.
7. The variable structure finite time vibration suppression method of fractional order mechanical arm model according to claim 1, wherein the finite time controller is:
wherein J is the rotational inertia of the actuator, R is the resistance, eta is the efficiency of the gearbox, upsilon is the gear ratio, K t In order to be a constant of the torque,
is the coefficient of the finite time controller, epsilon (t) the structure variable of the finite time controller, s is the variable of the sliding mode surface, sgn (D) α s) is a sign function, k d Is of fractional order P (ID) α Differential coefficient of terminal sliding mode surface, q 1 Is a third positive real number, q 2 Is a fourth positive real number, D α As a Caputo type fractional calculus operator, q 3 Is a fifth positive real number, B is the viscous friction coefficient of the actuator, K e Is a back electromotive force constant, x 2 Is a second state variable, J is the rotational inertia of the actuator, tau is the moment on the manipulator joint,
x = [ X ] as an estimate of the first unknown parameter γ 1 ，x 2 ] T Is a state vector of the fractional order mechanical arm model,
is an estimate of a second unknown parameter, k i Is a fractional order P (ID) α Integral coefficient of terminal sliding mode surface, x 1 For the first state variable, ρ ∈ (0,1) is a second fixed value.
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| CN115586724A (en) * | 2022-10-27 | 2023-01-10 | 南京师范大学泰州学院 | Self-adaptive fractional order global sliding mode control method for gear inspection robot system |
| CN116394257A (en) * | 2023-05-18 | 2023-07-07 | 南通大学 | Mechanical arm vibration reduction control method based on fractional order feedback |
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| CN115586724A (en) * | 2022-10-27 | 2023-01-10 | 南京师范大学泰州学院 | Self-adaptive fractional order global sliding mode control method for gear inspection robot system |
| CN115586724B (en) * | 2022-10-27 | 2023-11-24 | 南京师范大学泰州学院 | Self-adaptive fractional order global sliding mode control method for gear inspection robot system |
| CN116394257A (en) * | 2023-05-18 | 2023-07-07 | 南通大学 | Mechanical arm vibration reduction control method based on fractional order feedback |
| CN116394257B (en) * | 2023-05-18 | 2024-09-20 | 南通大学 | Mechanical arm vibration reduction control method based on fractional order feedback |
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