CN116901061A - Robotic arm trajectory tracking control method based on preset performance - Google Patents

Abstract

The robot arm trajectory tracking control method based on preset performance belongs to the field of nonlinear system control. The present invention designs a controller based on a preset performance function at a specified time for the trajectory tracking problem of a robotic arm. Its control object is a rigid robotic arm that considers unknown system dynamics and external interference, and adopts preset performance control and conversion errors. This method is used to design the control law to achieve specified time trajectory tracking control. The convergence time can be directly given, the convergence accuracy is precise and controllable, and the transient performance of the system can also be specified in advance. At the same time, the radial basis function neural network is introduced to estimate the unknown system dynamics of the manipulator according to the state quantity of the system, so that the system can effectively overcome the unknown system dynamics and external interference without knowing the specific upper bound value of the external disturbance, which is beneficial to Robotic arm systems work normally under different degrees of aging and in different environments.

Description

Robotic arm trajectory tracking control method based on preset performance

Technical field

The invention belongs to the field of nonlinear system control, relates to a robot arm trajectory tracking control algorithm at a specified time, and specifically relates to a robot arm trajectory tracking control method based on preset performance.

Background technique

The manipulator system is a highly nonlinear control system with strong time-varying and strong coupling characteristics. Its nonlinear terms include gravity, Coriolis force and centrifugal force. Therefore, achieving high control accuracy has always been a problem.

Preset performance control technology refers to designing the controller so that the tracking error of the closed-loop system converges to a preset allowable range while ensuring that the convergence speed and overshoot meet the preset conditions, that is, transient and steady state requirements. Performance is met simultaneously to improve the performance of the control system.

In recent years, research on robotic arm trajectory tracking control has achieved very significant results. However, there are still certain problems with the system convergence time. It is impossible to directly give a value as the convergence time. Therefore, for the convergence time problem Awaiting further research.

Contents of the invention

The purpose of the present invention is to solve the problem that the convergence time cannot be given in the trajectory tracking control of the robotic arm, and to provide a designated time trajectory tracking control method for the robotic arm based on preset performance and RBF neural network. This method uses the robotic arm's The Lagrangian dynamics model considers the unknown system dynamics and external environmental interference of the manipulator, and designs an uncertainty estimator based on the RBF neural network. It adopts the idea of preset performance control and uses the RBF neural network to estimate the unknown system dynamics. It learns and processes external disturbances, and designs control laws and adaptive laws to achieve specified time trajectory tracking control of the robotic arm.

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

A robotic arm trajectory tracking control method based on preset performance, the method is:

Step 1: Design an unknown system dynamics estimator based on RBF neural network;

Step 2: Design a preset performance function at a specified time to constrain the tracking error;

Step 3: Use error conversion to convert the constraint problem of tracking error into a bounded problem;

Step 4: Design the sliding mode so that the error after conversion is bounded;

Step 5: Design the control law and adaptive law based on the preset performance and RBF neural network to achieve trajectory tracking control.

Further, the first step is specifically: considering the trajectory tracking control problem of the rigid manipulator with unknown system dynamics and external interference, designing a radial basis function neural network to estimate the unknown system dynamics, and its expression is:

in, is the unknown system dynamics, abbreviated as H, q=[q ₁ , q ₂ ,..., q _n ] ^T is the joint angular position, is the joint angular velocity, n is the number of joints of the manipulator, W ^* ∈R ^N×n is the expected weight of the neural network, R ^N×n is the matrix of N*n dimensions, N is the number of neural network nodes, S(X )=[s ₁ (X), s ₂ (X),..., s _N (X)] ^T ∈R ^N is the hidden layer output of the neural network, s _i (X) is the radial basis function, where The expression is s _i (X) = exp [-(X-ξ _i ) ^T (X-ξ _i )/η _i ² ], ξ _i is the center value, η _i is the width value, and R ⁿ is an n-dimensional column vector , W ^*T S(X)∈R ⁿ is the output of the neural network, and ζ is a small approximate error vector.

Further, step one is specifically: introducing a robotic arm dynamics model:

Among them, τ = [τ ₁ , τ ₂ ,..., τ _n ] ^T is the control torque provided by each joint actuator, q = [q ₁ , q ₂ ,..., q _n ] ^T is the joint angle Location, is the joint angular velocity,/> is the joint angular acceleration, M(q)∈R ^n×n is the inertia matrix,/> is the moment vector including centrifugal force and Coriolis force, G(q)∈R ⁿ is the gravity moment vector, n is the number of joints of the mechanical arm,/> is the friction moment, R ⁿ is an n-dimensional column vector;

There are various unknown factors such as unknown system dynamics and external interference in the robotic arm system. Here, auxiliary variables are introduced to simplify the uncertainty of the robotic arm, let X ₁ =q, Change the system to the following form:

make Represents unknown system dynamics;

The RBF neural network is used to compensate for unknown system dynamics. The specific expression is:

Among them, W ^* is the expected weight of the neural network, but the specific value is unknown. Therefore, its design adaptive law is estimated online, defined is the estimated value of W ^* ,/> is the estimation error of the weight, and its expression is:

The adaptive law of is:/>

Among them, Γ is a positive definite diagonal matrix, is the transposed signal of the linear sliding mode, γ is a normal value, used to improve the robustness of the adaptive law, select appropriate parameters and enough nodes, use/> able to cope with uncertainty is estimated, and the estimation error is bounded.

Further, the second step is specifically: according to the manipulator system error model, design a preset performance function at a specified time to constrain the trajectory tracking error and achieve the actual convergence requirements at the specified time. The preset performance function form at the specified time is for:

Among them, β _i is the specified time performance function, a _i is the initial value of the performance function, b _i is the final value of the performance function, T is the freely set specified time, and t is the time signal;

Define the expected signal q _d for the joint angle of the manipulator, and the tracking error is e=qq _d . The specified time performance function is used to constrain the tracking error as:

-F _i β _i ≤ e _i ≤ F _i β _i

F _i is a freely selectable constant, e _i is the tracking error of the i-th joint of the manipulator. Simplifying the above equation, we can get that when t>T, the tracking error constraint is:

-F _i b _i ≤ e _i ≤ F _i b _i

In the above constraints, b _i , T and F _i are all constant values that can be set freely. By selecting appropriate parameters, the system tracking error can converge to the limit of absolute value F _i b _i before the specified time T. Inside.

Further, the third step is specifically: in order to realize the constraint on the tracking error in step two, an error conversion method is introduced for processing, and the original tracking error constraint problem is converted into an equivalent unconstrained problem. The error conversion relationship is: for:

e _i =β _i T _i (ε _i )

The conversion error obtained by back-solving the error conversion relationship is:

Among them, T _i is the error conversion function. It can be seen that when ε _i →+∞, T _i =F _i , when ε _i →-∞, T _i =-F _i , it can be known from the relationship between e _i and ε _i , as long as it is proved that the converted error ε _i is bounded, the error constraint relationship can be satisfied, that is, the preset performance control at the specified time is completed.

Furthermore, the derivative of the conversion error ε _i is:

make Got:/>

Simplify this to:

After the conversion error and its derivative are calculated through the above steps, the control law can be further designed to achieve bounded control of the conversion error.

Further, the fourth step is specifically: for the conversion error obtained in step three, design a linear sliding mode to achieve the boundedness requirement of the conversion error ε _i . The designed sliding mode form is:

is the i-th conversion error signal of the two-joint manipulator, which is used in the previous explanation of preset performance control; and/> It is to write the two signals into a vector form, which will be used in the subsequent design of the controller;

Among them, Λ is a positive definite diagonal matrix. By designing a controller to prove the boundedness of the sliding modulus, the boundedness proof of the conversion error mentioned above can be completed.

Furthermore, by deriving and simplifying the sliding mode formula, we can get:

in,

Analyzing the sliding mode expression, we can get that: when the sliding modulus is bounded, the conversion error ε _i and its derivative are both bounded. Therefore, by designing the controller to prove the boundedness of the sliding modulus, the above can be completed. Proof of boundedness of conversion error is mentioned.

Further, the fifth step is specifically: based on the neural network calculation formula designed in steps one to four, the constraint formula for the tracking error, the error conversion relationship formula and the linear sliding modulus, design a specified time trajectory of the manipulator based on the preset performance Tracking control law and adaptive law realize specified time trajectory tracking control, where:

The specified time trajectory tracking control law of the manipulator based on the preset performance is:

Among them, K is a freely selectable diagonal matrix;

The adaptive law is:

Among them, Γ is a positive definite diagonal matrix, and γ is a positive constant value, which is used to improve the robustness of the adaptive law. Both can be set freely. By using the above control law and adaptive law, the specified time convergence of the manipulator tracking error can be achieved, and its transient performance can be guaranteed to be within a controllable range.

Compared with the existing technology, the present invention has the following advantages: for the trajectory tracking problem of the robotic arm, the present invention designs a controller based on a preset performance function at a specified time, and its control object is a controller that takes into account unknown system dynamics and external interference. The rigid manipulator adopts the method of preset performance control and conversion error to design the control law to achieve specified time trajectory tracking control. Its convergence time can be directly given, the convergence accuracy is precise and controllable, and the transient performance of the system can also be specified in advance. At the same time, the radial basis function neural network is introduced to estimate the unknown system dynamics of the manipulator according to the state quantity of the system, so that the system can effectively overcome the unknown system dynamics and external interference without knowing the specific upper bound value of the external disturbance, which is beneficial to Robotic arm systems work normally under different degrees of aging and in different environments.

Description of the drawings

Figure 1 is a block diagram of the robot arm's specified time trajectory tracking control system;

Figure 2 shows the joint angle 1 position tracking situation when the convergence time T=2s is specified;

Figure 3 shows the joint angle 2 position tracking situation when the convergence time T=2s is specified;

Figure 4 is a diagram showing the convergence of joint angle 1 position tracking error when the convergence time T=2s is specified;

Figure 5 is a diagram showing the convergence of joint angle 2 position tracking error when the convergence time T=2s is specified;

Figure 6 shows the convergence situation of the neural network weight W ₁ when the convergence time T=2s is specified;

Figure 7 shows the convergence situation of the neural network weight W ₂ when the convergence time T=2s is specified;

Figure 8 shows the control torque signal diagram when the convergence time T=2s is specified;

Figure 9 shows the joint angle 1 position tracking situation when the convergence time T=0.7s is specified;

Figure 10 shows the joint angle 2 position tracking situation when the convergence time T=0.7s is specified;

Figure 11 is a diagram showing the convergence of joint angle 1 position tracking error when the convergence time T=0.7s is specified;

Figure 12 is a diagram showing the convergence of joint angle 2 position tracking error when the convergence time T=0.7s is specified;

Figure 13 shows the convergence situation of the neural network weight W ₁ when the convergence time T=0.7s is specified;

Figure 14 shows the convergence situation of the neural network weight W ₂ when the convergence time T=0.7s is specified;

Figure 15 shows the control torque signal diagram when the convergence time T=0.7s is specified.

Detailed ways

The technical solution of the present invention will be further described below in conjunction with the accompanying drawings and examples, 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 shall be covered. within the protection scope of the present invention.

Example 1:

For a general rigid manipulator, consider the following n-link manipulator model in Lagrangian form:

Among them, τ = [τ ₁ , τ ₂ ,..., τ _n ] ^T is the control torque provided by each joint actuator, q = [q ₁ , q ₂ ,..., q _n ] ^T is the joint angular position, is the joint angular velocity,/> is the joint angular acceleration, M(q)∈R ^{n ×n} is the inertia matrix,/> is the moment vector including centrifugal force and Coriolis force, G(q)∈R ⁿ is the gravity moment vector, n is the number of joints of the mechanical arm,/> is the friction moment, R ⁿ represents an n-dimensional column vector. Introducing auxiliary variables X ₁ and X ₂ to simplify the robotic arm model

make Represents unknown system dynamics. From/> It can be seen from the expression that it is related to the q and /> of the system The signals are related, that is, the joint angular position signal and the joint angular velocity signal of the manipulator. These two signals are selected as the input signals of the RBF neural network. By designing an appropriate adaptive law for the weights, the estimation of the dynamics of the unknown system can be completed. .

The RBF neural network is used to compensate for unknown system dynamics. The specific expression is as follows:

^{Among them ,} [s ₁ (X),s ₂ (X),...,s _N (X)] ^T ∈R ^N is the hidden layer output of the neural network, s _i (X) is the radial basis function, and its expression for ξ _i is the center value, and eta _i is the width value.

Among them, W ^* is the expected weight, but the specific value is unknown, so its design adaptive law is estimated online. definition is the estimated value of W ^* ,/> is the estimation error of the weight, and its expression is:

The adaptive law is as follows:

Among them, Γ is a positive definite diagonal matrix, is the transposed signal of the linear sliding mode, γ is a small normal value, which is used to improve the robustness of the adaptive law. According to the expression of the radial basis function s _i (X), select the appropriate center value and width value, and select enough nodes, use/> To complete the analysis of uncertainty/> is estimated, and the estimation error is bounded.

Considering the above manipulator system, introduce the auxiliary variable Z ₁ =e, Convert the dynamic model into an error dynamics model, as shown in the following formula:

Among them, H is The abbreviation of M is the abbreviation of M(q),/> is the desired angular acceleration signal.

Next, introduce the preset performance function for a specified time:

The specified time performance function is used to constrain the robot arm tracking error as follows:

-F _i β _i ≤ e _i ≤ F _i β _i (8);

In view of the above error constraint problem, the problem is equivalent to the boundedness proof problem of the converted error in the form of error conversion, and its expression is as follows:

e _i =β _i T _i (ε _i ) (9);

T _i is the error conversion function. It can be seen that when ε _i →+∞,T _i =F _i , when ε _i →-∞,T _i =-F _i , from the relationship between e _i and ε _i , as long as It is proved that the converted error ε _i is bounded and can satisfy the error constraint relationship, that is, the preset performance control at the specified time is completed.

According to the inverse solution of the relationship, the conversion error can be obtained as:

Then, derivation of the conversion error ε _i can be obtained:

make have to:

Further, writing the above equation into vector form, we can get:

in,

As mentioned above, we need to design the controller so that the conversion error ε is bounded, and there are no requirements for convergence time and accuracy. Therefore, we consider using a simple linear sliding mode to achieve bounded stability for ε. The design slip modulus is as follows:

Taking its derivative, we can get:

Among them, Q is an intermediate variable, and its expression is

Then, the controller τ can be designed so that the sliding modulus E _s is bounded, and then the conversion error ε can be obtained to be bounded, that is, the specified time control of the tracking error is finally completed.

Considering the simplified robotic arm system obtained above, and the RBF neural network calculation formula obtained by estimating the dynamics of the unknown system and its weight adaptive law, the controller of the following formula is designed to complete the specified time control of the system.

Among them, K is a freely selectable diagonal matrix.

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

Bringing the controller τ into equation (16), and combining equations (3) and (4), we can get:

Consider the following Lyapunov function:

Taking its derivative we can get:

Will and/> Substitute to get:

Obviously, Then the above formula can be simplified to:

Next, these four items are scaled separately.

-E _s ^T KE _s ≤ -λ _min (K)||E _s || ² (23).

λ _min (K) is the minimum eigenvalue of matrix K;

is matrix/> The maximum eigenvalue;

Introduce Young’s inequality:

Among them, x and y are any real numbers, λ>0 can be freely selected. In the above formula, p and q are real numbers defined by the inequality, which have no practical significance. p>1, q>1 and satisfy (p-1) (q -1)＝1

By scaling E _s ^T ζ in the above formula, we can get:

α ₀ is a positive constant, and its value can be selected freely.

By definition of the trace of a matrix, Expand to get:

respectively matrix/> The i-th row and j-th column element of W ^* .

Among them, by Young’s inequality

Available:

Right now:

Comprehensive formulas (23), (24), (26), (30) can be obtained:

make Further simplifying the above formula we get:

make Available:

From the above formula, we can get α ₃ >0 by choosing appropriate K so that α ₁ >0. By integrating equation (33) we can get:

From equation (34), we can get the linear sliding mode E _s and the weight error of the neural network is bounded, and according to the definition of sliding modulus (14), we can get, ε and /> is bounded, that is, the conversion error ε _i is bounded, so that the tracking error satisfies equation (8). It can be seen that the tracking error is always maintained within a predefined set. By adjusting the parameters of the preset performance function, the tracking error can be It converges to a smaller domain within the specified time and completes the actual specified time convergence.

A two-degree-of-freedom manipulator model is used for simulation, and the manipulator model of equation (1) is considered:

in,

C ₂₁ =0,

G ₁ =(m ₁ +m ₂ )l ₁ gcos(q ₂ )+m ₂ l ₂ gcos(q ₂ +q ₂ ),

G ₂ =m ₂ l ₂ gcos(q ₂ +q ₂ ),

m _i is the mass of the connecting rod, l _i is the length of the connecting rod, J _i is the moment of inertia of the connecting rod, and its nominal value is selected as follows

The expected value of the joint angle is set as follows:

Assume a 20% uncertainty in the robot arm parameters:

In this simulation, the tracking error is required to satisfy the constraints shown in equation (8), where,

F _i =0.2, T is the convergence time, which can be set freely. The final result is that the tracking error will converge to an absolute value of 0.0002 before the specified T time.

The external disturbance is set in the following form: According to the control law given by Equation (17), the simulation results are shown in Figures 2 to 15 below.

Figures 2 to 8 show the situation where the set time is 2s. It can be seen that the joint angle of the manipulator can track the given expected signal before the set time of 2s, and the neural network weights used are bounded. Figures 9 to 15 show the case where the set time is 0.7s. Similarly, the robotic arm can track the desired signal before the set time of 0.7s, which proves that the proposed method can ensure that the robotic arm tracks before the freely set time. ability to receive desired signals.

Claims (9)

1. The mechanical arm track tracking control method based on the preset performance is characterized by comprising the following steps of: the method comprises the following steps:

step one: designing an unknown system dynamics estimator based on an RBF neural network;

step two: designing a preset performance function of the appointed time to restrict tracking errors;

step three: the constraint problem of tracking errors is converted into a bounded problem by adopting an error conversion mode;

step four: designing a sliding mode to limit the error after conversion;

step five: and designing a control law and a self-adaptive law based on preset performance and RBF neural network to realize track tracking control.

2. The method for tracking and controlling the trajectory of the mechanical arm based on the preset performance according to claim 1, wherein the method comprises the following steps: the first step is specifically as follows: the method comprises the steps of designing a radial basis function neural network to estimate the dynamics of an unknown system by considering the track tracking control problems of the rigid mechanical arm with the dynamics of the unknown system and external interference, wherein the expression is as follows:

wherein ,for unknown system dynamics, q= [ q ] ₁ ，q ₂ ，...，q _n ] ^T For joint angular position->The angular velocity of the joints, n is the number of joints of the mechanical arm, W ^* ∈R ^N×n R is the expected weight of the neural network ^N×n A matrix with dimension N X N, N is the number of nodes of the neural network, S (X) = [ S ] ₁ (X)，s ₂ (X)，...，s _N (X)] ^T ∈R ^N Is the hidden layer output of the neural network, s _i (X) is a radial basis function, the expression of which is +.>ξ _i Is the central value, eta _i Is of width value, R ⁿ Is an n-dimensional column vector, W ^*T S(X)∈R ⁿ Zeta is a small approximation error vector, which is the output of the neural network.

3. The method for tracking and controlling the trajectory of the mechanical arm based on the preset performance according to claim 2, wherein the method comprises the following steps: the first step is specifically as follows: introducing a mechanical arm dynamics model:

wherein τ= [ τ ] ₁ ，τ ₂ ，...，τ _n ] ^T For the control moment provided by the respective joint actuators, q= [ q ] ₁ ，q ₂ ，...，q _n ] ^T In order to achieve the angular position of the joint,for joint angular velocity>For angular acceleration of joint, M (q) ∈R ^n×n Is inertia matrix>To include moment vectors of centrifugal and Golgi forces, G (q) ∈R ⁿ For the moment vector, n is the number of joints of the mechanical arm, < ->R is friction moment ⁿ Is an n-dimensional column vector;

introducing auxiliary variable to simplify the uncertainty of the mechanical arm to make X ₁ ＝q，The system was changed to the following form:

order theRepresenting unknown system dynamics;

the unknown system dynamics are compensated by adopting an RBF neural network, and the specific expression is as follows:

wherein ,W^* For the expected weight of the neural network, the adaptive law is designed to estimate on line, and definition is carried outIs W ^* Estimated value of ∈10->The expression of the estimation error is as follows:

the adaptive law of (2) is: />

Wherein Γ is a positive diagonal matrix,for transposed signal of linear sliding mode, gamma is a normal value, using +.>Can complete the uncertainty->Is a function of the estimate of (2).

4. The robot arm trajectory tracking control method based on preset performances according to any one of claims 1 to 3, characterized by: the second step is specifically as follows: according to the error model of the mechanical arm system, a preset performance function of the appointed time is designed and used for restraining the track tracking error, so that the actual convergence requirement of the appointed time is realized, and the preset performance function of the appointed time is in the form of:

wherein ,β_i To specify a time performance function, a _i As initial value of performance function, b _i The final value of the performance function is T, which is a freely settable designated time, and T is a time signal;

definition of the arm joint Angle desired Signal q _d Tracking error is e=q-q _d The tracking error is constrained by using a specified time performance function as follows:

-F _i β _i ≤e _i ≤F _i β _i

F _i e is a freely selectable constant _i For the tracking error of the ith joint of the mechanical arm, the above formula is simplified, and when T is more than T, the constraint of the tracking error is as follows:

-F _i b _i ≤e _i ≤F _i b _i

in the above constraint, b _i ，T，F _i Are all constant values which can be freely set, namely, the system tracking error is converged to an absolute value F before the appointed time T by selecting proper parameters _i b _i Is within the limits of (2).

5. The method for tracking and controlling the trajectory of the mechanical arm based on the preset performance according to claim 4, wherein the method comprises the following steps: the third step is specifically as follows: the method is characterized in that an error conversion mode is introduced to process, the original tracking error constraint problem is converted into an equivalent unconstrained problem, and the error conversion relation is as follows:

e _i ＝β _i T _i (ε _i )

the inverse solution error conversion relation obtains the conversion error as follows:

wherein ,T_i As an error transfer function, when epsilon _i →+∞,T _i ＝F _i When epsilon _i →-∞,T _i ＝-F _i From e _i And epsilon _i As long as the converted error ε is proved _i Is bounded, and can meet the error constraint relation, namely, the preset performance control of the appointed time is completed.

6. The method for tracking and controlling the trajectory of the mechanical arm based on the preset performance according to claim 4, wherein the method comprises the following steps: for conversion error epsilon _i And (3) deriving:

order theObtaining: />

Further simplifying it into:

after the conversion error and the derivative thereof are calculated by the steps, a control law can be further designed to realize the bounded control of the conversion error.

7. The method for controlling the track following of the mechanical arm based on the preset performance according to claim 1, 5 or 6, wherein the method comprises the following steps: the fourth step is specifically as follows: aiming at the conversion error obtained in the step three, a linear sliding mode is designed to realize the conversion error epsilon _i Is designed into the form of sliding mode:

wherein Λ is a positive definite diagonal array, and the finite demonstration of the slip modulus can be completed by designing a controller.

8. The method for tracking and controlling the trajectory of the mechanical arm based on the preset performance according to claim 7, wherein the method comprises the following steps: the sliding mode formula is derived and simplified to obtain:

wherein ,

analysis of the sliding mode expression can result in: conversion error ε when slip modulus is bounded _i And its derivatives are bounded, the above-mentioned transition error can be accomplished by designing the controller to accomplish the bounded demonstration of slip modulus.

9. The mechanical arm track tracking control method based on preset performances according to any one of claims 1 to 8, characterized in that: the fifth step is specifically as follows: aiming at the neural network calculation formulas designed in the first to fourth steps, a constraint formula of tracking errors, an error conversion relation formula and a linear sliding modulus, a mechanical arm specified time track tracking control law and a self-adaptive law based on preset performances are designed, and specified time track tracking control is realized, wherein:

the mechanical arm specified time track tracking control law based on preset performance is as follows:

wherein K is a positive angular array which can be selected freely;

the self-adaptive law is:

wherein Γ is a positive diagonal matrix, γ is a positive constant value, which is used to improve the robustness of the adaptive law, and both can be freely set. Under the condition of using the control law and the self-adaptive law, the appointed time convergence of the tracking error of the mechanical arm can be realized, and the transient performance of the mechanical arm is ensured to be in a controllable range.