CN114851196B - Trajectory tracking control method of manipulator based on fuzzy adaptive global sliding mode - Google Patents

Abstract

The invention discloses a mechanical arm track tracking control method based on a fuzzy self-adaptive global sliding mode, which comprises the following steps of: step 1, designing a real variable parameter sliding mode surface structure based on fuzzy control; step 2, designing a sliding mode variable s; step 3, designing the input of the fuzzy controller as an error q _e The output is a time-varying coefficient alpha ₁ (t)、α ₂ (t), and designing a corresponding membership function and fuzzy logic; step 4, designing a global sliding mode variable sigma of an integral form; and 5, designing a mechanical arm track tracking control law and a self-adaptive law based on a fuzzy self-adaptive sliding mode to realize track tracking control. The method adopts a Lagrange dynamics model of the mechanical arm, considers model parameter uncertainty and external environment interference of the mechanical arm, designs a time-varying parameter dynamic sliding mode surface and an integral global sliding mode surface based on fuzzy control, further designs a control law and a self-adaptive law, and realizes the track tracking control of the mechanical arm.

Description

Trajectory tracking control method of manipulator based on fuzzy adaptive global sliding mode

technical field

The invention belongs to the field of nonlinear system control, and relates to a trajectory tracking control algorithm of a mechanical arm, in particular to a trajectory tracking control method of a mechanical arm based on a fuzzy self-adaptive global sliding mode.

Background technique

The manipulator is a highly nonlinear system that is difficult to accurately model. In addition, there are other unknown external disturbances when the manipulator is working, which brings great difficulties to the trajectory tracking control of the manipulator.

Sliding mode control is widely used in the control of nonlinear systems. Once the traditional fixed parameter sliding mode surface structure is determined, the convergence process of the system on the sliding mode surface is a movement on a fixed plane, and it is difficult for the system to ensure that the error has a fast convergence speed in different positions and regions of the plane.

Considering the uncertainties of the model and external disturbances when the manipulator is working, the requirements for the robustness of the system are increased.

Contents of the invention

The purpose of the present invention is to provide a method for tracking control of manipulator trajectory based on fuzzy adaptive global sliding mode, which adopts the Lagrangian dynamics model of manipulator and considers the model parameter uncertainty and external environment disturbance of manipulator , design a time-varying parameter dynamic sliding surface based on fuzzy control and global sliding surface in integral form, and further design the control law and adaptive law to realize the trajectory tracking control of the manipulator.

The purpose of the present invention is achieved through the following technical solutions:

A method for tracking control of manipulator trajectory based on fuzzy adaptive global sliding mode, comprising the following steps:

Step 1. Aiming at the trajectory tracking control problem of a rigid manipulator considering model uncertainty and external disturbance, design a real variable parameter sliding surface structure based on fuzzy control:

s(t)=f(x(t),k(t));

k(t)=α(t)k;

Among them, α(t) is a time-varying coefficient, and k is a fixed parameter;

Step 2. For the time-varying parameter sliding mode surface structure in step 1, combined with the mathematical model of the manipulator, design the sliding mode variable s, whose parameters are adaptively adjusted by the output of the fuzzy controller. The specific form is as follows:

K ₁ (t) = α ₁ (t)K ₁ ;

K ₂ (t)=α ₂ (t)K ₂ ;

λ₂＞1，0＜α₁(t),α₂(t)＜1；q_e为机械臂的角位置误差；Among them, K ₁ , K ₂ >0,

λ ₂ >1, 0<α ₁ (t), α ₂ (t)<1; q _e is the angular position error of the mechanical arm;

Step 3. Design the input of the fuzzy controller as the error q _e , and the output as the time-varying coefficients α ₁ (t) and α ₂ (t), and design the corresponding membership function and fuzzy logic, where:

The fuzzy logic is: when the input q _e is large, the output α ₂ (t) is a large value, and α ₁ (t) is a small value; when the input q _e is moderate, the output α ₂ (t) is a moderate value , α ₁ (t) is a moderate value; when the input q _e is small, the output α ₂ (t) is a small value, and α ₁ (t) is a large value;

Step 4. For the sliding mode variable designed in step 2, design the global sliding mode variable σ in integral form:

Wherein, K ₃ >0, γ<1;

Step 5. For the sliding mode variable, fuzzy controller and global sliding mode variable designed in step 2, step 3 and step 4, design the trajectory tracking control law and adaptive law of the manipulator based on the fuzzy adaptive sliding mode to realize trajectory tracking Control, where:

The trajectory tracking control law of the manipulator based on fuzzy adaptive sliding mode is:

为自适应律。Among them, k>0, q _d is the expected trajectory of the manipulator,

for adaptive law.

The adaptive law is:

Wherein, λ ₀ , λ ₁ , λ ₂ >0, and q is the angular position of the mechanical arm.

Compared with the prior art, the present invention has the following advantages:

The present invention designs a kind of controller based on the fuzzy self-adaptive global sliding mode for the trajectory tracking problem of the manipulator, and the model of its control object is a rigid manipulator considering model uncertainty and external interference, adopts self-adaptive sliding mode control The control law and adaptive law are designed by the method, and the trajectory tracking control is realized. At the same time, fuzzy control is introduced, and the sliding mode parameters are adjusted in real time according to the distance from the state variable of the system to the equilibrium point, so that the entire convergence process of the system on the sliding mode surface has a faster convergence speed, and the waste of torque is reduced. The global sliding mode improves The robustness of the system enables the system to effectively overcome model uncertainty and external interference, and does not need to know the upper bound specific value of the uncertainty item, which is beneficial to the manipulator system under different aging degrees and different environments. normal work.

Description of drawings

Figure 1 is a block diagram of the trajectory tracking control system of the manipulator based on fuzzy adaptive sliding mode;

Fig. 2 is the membership function of fuzzy control input;

Fig. 3 is the membership function of fuzzy control output;

Figure 4 is the angular position error convergence curve when d=0.1;

Figure 5 is the angular velocity error convergence curve when d=0.1;

Fig. 6 is moment curve when d=0.1;

Figure 7 is the angular position error convergence curve when d=0.1 and d=0.2;

Figure 8 is the angular position error convergence curve when d=0.3 and d=0.4;

Fig. 9 is the angular velocity error convergence curve when d=0.1 and d=0.2;

Figure 10 is the angular velocity error convergence curve when d=0.3 and d=0.4;

Figure 11 is the angular position and angular velocity error convergence curves when a fixed disturbance is added to a sinusoidal disturbance;

Figure 12 is the angular position and angular velocity error convergence curve when the fixed disturbance is added to the white noise;

Figure 13 is the angular position and angular velocity error convergence curve of the fixed parameter sliding mode structure;

Figure 14 is the angular position and angular velocity error convergence curve of the time-varying parameter sliding mode structure;

Figure 15 shows the moment curves of the sliding mode structure with fixed parameters (left) and time-varying parameters (right).

Detailed ways

The technical solution of the present invention will be further described below in conjunction with the accompanying drawings, but it is not limited thereto. Any modification or equivalent replacement of the technical solution of the present invention without departing from the spirit and scope of the technical solution of the present invention should be covered by the present invention. within the scope of protection.

The invention provides a method for tracking and controlling the trajectory of a manipulator based on a fuzzy adaptive global sliding mode. The method includes the following steps:

Step 1. Aiming at the trajectory tracking control problem of a rigid manipulator considering model uncertainty and external disturbance, a real variable parameter sliding surface structure based on fuzzy control is designed.

In sliding mode control, the structure of the sliding surface will determine the motion of the system on the sliding surface. For the design of fixed parameter sliding surface structure, it can be written as follows:

s(t)=f(x(t),k);

Among them, s(t) is the sliding mode variable, x(t) is the state of the system, and k is the parameter of the sliding mode surface.

When the system state converges to the sliding surface s(t)=0=0, the dynamic equation of the system is:

f(x(t),k)=0.

will converge to an equilibrium point on a fixed sliding surface. The traditional fixed-parameter sliding mode surface structure often only guarantees that the system state has a faster convergence speed in a certain range, thus limiting the overall convergence process on the sliding mode surface.

When the fixed parameters of the sliding mode surface are changed, the structure of the sliding mode surface becomes:

s(t)=f(x(t),k(t));

k(t)=α(t)k;

Among them, α(t) is a time-varying coefficient, and k is a fixed parameter. The structure of the sliding surface changes continuously with the convergence of the system, thereby speeding up the convergence speed. The time-varying coefficient α(t) is output by the fuzzy controller.

Step 2. Aiming at the time-varying parameter sliding surface structure in step 1, combined with the mathematical model of the manipulator, design a specific sliding surface.

Design the sliding mode variable s, whose form is as follows:

K ₁ (t) = α ₁ (t)K ₁ ;

K ₂ (t)=α ₂ (t)K ₂ ;

λ₂＞1，0＜α₁(t),α₂(t)＜1为模糊控制器的输出；q_e为机械臂的角位置误差。Among them, K ₁ , K ₂ >0,

λ ₂ >1, 0<α ₁ (t), α ₂ (t)<1 is the output of the fuzzy controller; q _e is the angular position error of the mechanical arm.

When the system is on the sliding surface s=0, the dynamic equation of the system state is as follows:

对误差的收敛起到主导作用；当系统误差q_e接近于0时，

对误差的收敛起到主导作用。When the systematic error q _e is large,

Play a leading role in the convergence of the error; when the system error q _e is close to 0,

Play a leading role in the convergence of errors.

Step 3. Design a fuzzy controller for the time-varying parameters in step 2.

In step 2, the convergence dominant term of the system on the sliding surface is affected by the error _qe . The input of the designed fuzzy controller is the error q _e , the output is the time-varying coefficient α ₁ (t), α ₂ (t), and the corresponding membership function is designed, the specific expression is as follows:

Enter the membership function:

Output membership function:

Among them, i=1,2.

The fuzzy logic of the fuzzy controller is:

When the input q _e is large, the output α ₂ (t) is a large value, and α ₁ (t) is a small value; when the input q _e is moderate, the output α ₂ (t) is a moderate value, and α ₁ ( t) is a moderate value; when the input q _e is small, the output α ₂ (t) is a small value, and α ₁ (t) is a large value.

The above-mentioned fuzzy controller plays the role of further strengthening the dominant term and weakening the non-dominant term, so that the convergence speed of the system is accelerated. At the same time, because each parameter affects the output of the control torque, the waste of torque can also be reduced.

Step 4. Design global sliding mode variables for the time-varying parameter sliding mode surface structure designed in step 2.

In the sliding mode control system, the system convergence process is divided into two stages, one is the arrival stage, that is, the stage when the system reaches the sliding surface from the initial position; the second is the sliding stage, that is, the system is on the sliding surface, and the convergence Only limited by the sliding surface itself, it converges to the equilibrium point along the sliding surface. The global sliding mode makes the system on the sliding mode surface from the beginning, so that the original arrival segment also has the strong robustness of the sliding mode control.

Design global sliding mode variables:

Wherein, K ₃ >0, γ<1.

At the same time, due to the introduction of the integrator, the change of the sliding mode variable σ is more gentle than that of the variable s, which can reduce the chattering of the control torque when it is used as the control variable.

Step 5. For the sliding mode variables and fuzzy controllers designed in steps 2, 3 and 4, design the trajectory tracking control law and adaptive law of the manipulator.

The trajectory tracking control law of the manipulator based on fuzzy adaptive sliding mode is designed as follows:

为自适应律。Among them, k>0, q _d is the expected trajectory of the manipulator,

for adaptive law.

The adaptive law is designed as:

Wherein, λ ₀ , λ ₁ , λ ₂ >0, and q is the angular position of the mechanical arm.

Example:

For a general rigid manipulator, its Lagrangian dynamics model can be expressed by the following second-order nonlinear differential equation:

分别表示关节的角位置，角速度和角加速度；M(q)∈R^n×n，是一个对称正定矩阵，表示惯性；

表示离心力和科里奥利力转矩；G(q)∈Rⁿ，表示重力转矩；τ∈Rⁿ，表示控制转矩。in,

respectively represent the angular position, angular velocity and angular acceleration of the joint; M(q)∈R ^n×n is a symmetric positive definite matrix representing inertia;

Represents the centrifugal force and Coriolis force torque; G(q)∈R ⁿ represents the gravitational torque; τ∈R ⁿ represents the control torque.

Write the above equation in error form:

Wherein, q _d is the angular position of the desired joint, and q _e =qq _d is the error of the joint angular position.

动力学方程变为：Consider the model uncertainty of the manipulator, expressed as parameter uncertainty in Eq. Considering the external disturbance and friction torque of the manipulator, set τ _d and

The kinetic equation becomes:

G₀(q)为已知的名义部分，Δ_M(q)、

Δ_G(q)为模型不确定性，τ_d和

分别为固定扰动和时变扰动。Among them, M ₀ (q),

G ₀ (q) is the known nominal part, Δ _M (q),

Δ _G (q) is the model uncertainty, τ _d and

are fixed and time-varying disturbances, respectively.

This further simplifies to:

Among them, ρ ₀ , ρ ₁ , and ρ ₂ are constants greater than 0, and ||·|| represents the two-norm of the vector, which satisfies the assumption:

Considering the friction torque makes the following equation hold:

Among them, γ ₀ , γ ₁ , and γ ₂ are constants greater than 0, which satisfy the assumption:

The final form of writing the error is:

Design the sliding mode variable s, whose form is as follows:

λ₂＞1，0＜α₁(t),α₂(t)＜1为模糊控制器的输出；q_e为机械臂的角位置误差。Among them, K ₁ , K ₂ >0,

λ ₂ >1, 0<α ₁ (t), α ₂ (t)<1 is the output of the fuzzy controller; q _e is the angular position error of the mechanical arm.

When the system is on the sliding surface s=0, the dynamic equation of the system state is as follows:

对误差的收敛起到主导作用；当系统误差q_e接近于0时，

对误差的收敛起到主导作用。When the systematic error q _e is large,

Play a leading role in the convergence of the error; when the system error q _e is close to 0,

Play a leading role in the convergence of errors.

According to the logic idea of the above-mentioned dominant control, the two-norm ||q _e || of the system state error is used as the input of the fuzzy control to measure the distance between the system and the equilibrium point. The fuzzy controller is designed as follows:

The membership function of fuzzy control input ||q _e || is shown in Fig. 2, and the membership function of fuzzy control output α ₀ (t), α ₁ (t), α ₂ (t) is shown in Fig. 3. Fuzzy control The rules are shown in Table 1, and the center of gravity method is used for defuzzification.

Table 1 Fuzzy rules

||q_e|| α₁(t) α₂(t) S B S m m m B S B

Note: S, B, and M are fuzzy sets, representing small, large, and medium.

项对误差收敛起到主导作用，此时模糊自适应控制器输出系数α₂(t)较大，α₁(t)较小，以此增强该项的主导作用。同理，当系统离平衡点距离适中时，输出α₂(t)为适中值、α₁(t)为适中值，

和

共同作为主导项；当系统离平衡点距离较小时，输出α₂(t)为较小值、α₁(t)为较大值，增强

项的主导作用。通过模糊控制器增强不同阶段的主导项作用，以此提高收敛速度。由于这些参数直接影响控制力矩的大小，这种自适应控制的引入也可以减少控制力矩的浪费。Through fuzzy self-adaptive control, the parameters of the sliding surface can be adjusted adaptively according to the distance from the equilibrium point during the convergence process of the system. When the system is far away from the equilibrium point, ||q _e || is larger,

The term plays a leading role in the error convergence. At this time, the output coefficient of the fuzzy adaptive controller α ₂ (t) is larger, and α ₁ (t) is smaller, so as to enhance the leading role of this term. Similarly, when the system is at a moderate distance from the equilibrium point, the output α ₂ (t) is a moderate value, and α ₁ (t) is a moderate value,

and

Together as the dominant item; when the distance from the system to the equilibrium point is small, the output α ₂ (t) is a small value, α ₁ (t) is a large value, and the enhanced

item's dominance. The fuzzy controller is used to enhance the role of the leading term in different stages, so as to improve the convergence speed. Since these parameters directly affect the size of the control torque, the introduction of this adaptive control can also reduce the waste of control torque.

In order to further improve the robustness of the system and reduce the chattering of the control torque, the global sliding mode variable is designed as follows:

Wherein, K ₃ >0, γ<1.

Due to the introduction of the integrator, the system is on the sliding mode surface σ=0 from the beginning, and the system is always on (near) σ=0 by designing the control law, so that the sliding mode variable s moves according to the following formula:

Among them, ||ε|| is finite and very small.

According to the above method, the control law can be designed as follows:

为自适应律。Among them, k>0,

for adaptive law.

The adaptive law is designed as:

Wherein, λ ₀ , λ ₁ , λ ₂ >0, and |·| is a norm of the vector.

The following is the stability analysis of the designed control law and adaptive law:

From formula (7) can get:

Putting equations (11) and (12) into equation (6), the kinetic equation of the variable s can be obtained as:

From formula (9) can get:

Putting Equation (16) into Equation (15), the kinetic equation of variable σ can be obtained as:

Consider the Lyapunov function V, which has the following form:

It can be derived from:

Put (17) into (19) to get:

为反对称矩阵，进一步可得：by the nature of the robotic arm

As an anti-symmetric matrix, it can be further obtained:

In summary, the designed control law based on fuzzy adaptive global sliding mode can keep the system on the sliding mode surface σ=0, so that the system state error converges to the equilibrium point when moving on the sliding mode surface.

Select a two-degree-of-freedom rigid manipulator for simulation, and the mathematical model is shown in formula (21):

Among them, the form of each matrix is:

The expression of each parameter in the coefficient matrix is as follows:

Consider d times the model uncertainty parameters, namely: m ₁ =(1+d)m ₁₀ , m ₂ =(1+d)m ₂₀ , l ₁ =(1+d)l ₁₀ , l ₂ =(1 +d)l ₂₀ , J ₁ =(1+d)J ₁₀ , J ₂ =(1+d)J ₂₀ .

The physical parameters of the robotic arm are as follows:

Mass: m ₁₀ =0.5 kg, m ₂₀ =1.5 kg.

Length: l ₁₀ =1 m, l ₂₀ =0.8 m.

Moment of inertia: J ₁₀ =J ₂₀ =5kg·m ² .

Gravity acceleration: g=9.8N/kg.

Set desired trajectory:

The uncertainty interference in formula (4) is set as: τ _d =5,

The parameters for setting the sliding mode variables in formulas (7) and (9) are: K ₁ =5, K ₂ =2, K ₃ =10, λ ₁ =0.6, λ ₂ =2, γ=0.7.

Set the control law and adaptive law parameters shown in formulas (11) and (12) as: k=5, λ ₀ =λ ₁ =λ ₂ =1, and limit the maximum control torque to 150N·m.

The simulation results of d taking a value of 0.1 are shown in Fig. 4-6. It can be seen from Figures 4 to 6 that the designed controller can make the system converge to the equilibrium point in a short period of time, while controlling the torque within an appropriate range. In order to further illustrate the superiority of the present invention, the following comparative simulation experiments were carried out.

The convergence curves under different degrees of uncertainty are shown in Figures 7 to 10. It can be seen from Figures 7 to 10 that as the degree of uncertainty increases, the convergence curves of angular position and angular velocity errors do not change significantly. The speed is basically unchanged, which verifies the strong robustness of the system to model uncertainty.

Set the model uncertainty coefficient to 0.1 again, and conduct simulation experiments on different types of environmental disturbances. The results are as follows:

为固定扰动和正弦扰动的叠加，收敛曲线如图11所示。The external disturbance is set as the following form: τ _d =5,

For the superposition of fixed disturbance and sinusoidal disturbance, the convergence curve is shown in Figure 11.

为一均值为0，方差为1的高斯白噪声信号，为固定扰动加白噪声的叠加，收敛曲线如图12所示。从图12中可以看出，系统在不同环境干扰下，仿真曲线没有明显变化，收敛速度基本不变，验证了系统对环境干扰的强鲁棒性。The external disturbance is set as the following form: τ _d =5,

is a Gaussian white noise signal with a mean value of 0 and a variance of 1, which is a superposition of fixed disturbance and white noise, and the convergence curve is shown in Figure 12. It can be seen from Figure 12 that under different environmental disturbances, the simulation curve does not change significantly, and the convergence speed is basically unchanged, which verifies the strong robustness of the system to environmental disturbances.

Re-set the external disturbance as the superimposed form of fixed disturbance and sinusoidal disturbance, and set the output of the fuzzy controller as: α ₁ =α ₂ =0.55. At this time, the system will degenerate into a fixed parameter sliding mode structure. Compare it with the time-varying The results of the convergence process under the parametric sliding mode structure are shown in Figures 13-14.

The speed of convergence is directly affected by the magnitude of the control torque. While considering the convergence speed, the comparison torque curve is shown in Figure 15. From the convergence curve in Figure 15, it can be seen directly that the convergence speed under the time-varying parameter sliding mode structure is faster than that under the fixed parameter sliding mode structure, and the magnitude of the moment cannot be directly obtained from the curve. Define the average moment index of the convergence process t time as:

Among them, I(τ)∈R ⁿ , τ∈[τ ₁ τ ₂ … τ _n ] ^T ∈ R ⁿ .

时变参数滑模结构下各关节的平均力矩指标为：

可以得出时变参数滑模结构下，收敛过程中各关节的平均力矩均小于固定参数滑模结构，验证了基于模糊控制的时变参数滑模结构的优越性，即加快收敛速度、减少力矩浪费。The average moment index of each joint under the fixed parameter sliding mode structure is:

The average moment index of each joint under the time-varying parameter sliding mode structure is:

It can be concluded that under the time-varying parameter sliding mode structure, the average moment of each joint during the convergence process is smaller than the fixed parameter sliding mode structure, which verifies the superiority of the time-varying parameter sliding mode structure based on fuzzy control, that is, to speed up the convergence speed and reduce the torque waste.

Claims (3)

1. A mechanical arm track tracking control method based on a fuzzy self-adaptive global sliding mode is characterized by comprising the following steps:

step 1, aiming at the track tracking control problem of the rigid mechanical arm considering model uncertainty and external interference, designing a real variable parameter sliding mode surface structure based on fuzzy control:

s(t)＝f(x(t),k(t))；

k(t)＝α(t)k；

wherein x (t) is the state of the system, alpha (t) is a time-varying coefficient, and k is a fixed parameter;

step 2, aiming at the time-varying parameter sliding mode surface structure in the step 1, a mechanical arm mathematical model is combined to design a sliding mode variable s, and parameters of the sliding mode variable s are adaptively adjusted through the output of a fuzzy controller, wherein the specific form is as follows:

K ₁ (t)＝α ₁ (t)K ₁ ；

K ₂ (t)＝α ₂ (t)K ₂ ；

wherein, K ₁ ,K ₂ ＞0，

λ ₂ ＞1，0＜α ₁ (t),α ₂ (t)＜1；q _e Is the angular position error of the mechanical arm;

step 3, designing the input of the fuzzy controller as an error q _e The output is a time-varying coefficient alpha ₁ (t)、α ₂ (t) and designing corresponding membership function and fuzzy logic, wherein:

the specific expression of the membership function is as follows:

membership function of input:

output membership function:

wherein i =1,2;

the fuzzy logic is: when inputting q _e When larger, output alpha ₂ (t) is a large value, α ₁ (t) is a smaller value; when inputting q _e When the size is moderate, alpha is output ₂ (t) is the appropriate median value, α ₁ (t) is a moderate value; when inputting q _e When smaller, output alpha ₂ (t) is a smaller value, α ₁ (t) is greater;

step 4, aiming at the sliding mode variable designed in the step 2, designing a global sliding mode variable sigma of an integral form:

wherein, K ₃ ＞0，γ＜1；

And 5, aiming at the sliding mode variables, the fuzzy controller and the global sliding mode variables designed in the steps 2, 3 and 4, designing a mechanical arm track tracking control law and a self-adaptive law based on a fuzzy self-adaptive sliding mode to realize track tracking control.

2. The method for tracking and controlling the trajectory of the mechanical arm based on the fuzzy adaptive global sliding mode according to claim 1, wherein the trajectory tracking and controlling law of the mechanical arm based on the fuzzy adaptive sliding mode is as follows:

wherein k is more than 0,q _d In order to be the desired trajectory of the robotic arm,

is an adaptive law.

3. The mechanical arm trajectory tracking control method based on the fuzzy adaptive global sliding mode according to claim 1, characterized in that the adaptive law is as follows:

wherein λ is ₀ ,λ ₁ ,λ ₂ > 0,q is the angular position of the robotic arm.

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