A fixed-time backstepping trajectory tracking control method for a robot arm with model uncertainty and external disturbance

Abstract

The application discloses a fixed-time backstepping trajectory tracking control method for a mechanical arm with model uncertainty and external disturbance, and belongs to the technical field of trajectory tracking control of mechanical arm systems.The application solves the problems of poor stability and low robustness of the existing fixed-time control method.The application firstly establishes a dynamics model of a multi-freedom mechanical arm in joint space by considering model uncertainty and external disturbance;then the designed adaptive radial basis function neural network is used to approximate the lumped disturbance of the model uncertainty and external disturbance of the mechanical arm, so that an estimated value of the lumped disturbance of the model uncertainty and external disturbance is obtained;finally, a fixed-time backstepping trajectory tracking controller is designed according to the established dynamics model and the estimated value of the lumped disturbance.The method can be used for controlling the mechanical arm with model uncertainty and external disturbance.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of trajectory tracking control of mechanical arm system, and particularly relates to a fixed-time backstepping trajectory tracking control method for a mechanical arm with model uncertainty and external disturbance. BACKGROUND

[0002] In recent years, mechanical arm trajectory tracking control has been widely applied in industrial automation, robot operation, medical equipment, scientific research and other fields. These application backgrounds require the mechanical arm to accurately and stably track a given trajectory, improve production efficiency, reduce labor cost, reduce operation risk, and help to achieve more accurate experiments and operations. Considering the nonlinearity of the system and the existence of complex external disturbances, it is difficult to obtain an accurate model of the controlled object and environmental parameters.

[0003] Backstepping control is a nonlinear control method, which aims to design a controller to realize the stability and performance of the system, can handle complex nonlinear systems, and can offset the uncertainty and external disturbance of the system through a nonlinear compensator to improve the robustness and stability of the system.

[0004] However, the traditional backstepping control method can only guarantee the gradual convergence of the system, and cannot meet the demand of fast and stable actual system. Finite-time backstepping control can guarantee the convergence of the system in a finite time, but the convergence time is limited by the initial state of the system, and the actual application effect is not ideal. Therefore, in recent years, fixed-time control has attracted the attention of researchers. The phenomenon of fixed-time convergence was first discovered by Andrieu, and the upper bound of its convergence time is independent of the initial state of the system. Polyakov designed a fixed-time stable controller for uncertain linear objects, realizing the robustness to matching uncertainty and external disturbance. Subsequently, researchers proposed a variety of fixed-time control algorithms based on backstepping control, but did not consider the convergence speed of the system and did not consider the model uncertainty problem.

[0005] In summary, since the existing fixed-time control method does not consider the influence of model uncertainty, the stability of the existing fixed-time control method is still poor, and the robustness is still low, and it is necessary to propose a fixed-time control method to solve the above problems. SUMMARY

[0006] The purpose of the present application is to solve the problem of poor stability and low robustness of the existing fixed-time control method, and a fixed-time backstepping trajectory tracking control method for a mechanical arm with model uncertainty and external disturbance is proposed.

[0007] The technical scheme adopted by the present application to solve the above technical problems is: a fixed-time backstepping trajectory tracking control method for a mechanical arm with model uncertainty and external disturbance, which specifically comprises the following steps:

[0008] Step one, a dynamics model of a multi-degree-of-freedom mechanical arm in joint space considering model uncertainty and external disturbance is established;

[0009] Step two, an adaptive radial basis function neural network is designed, and the adaptive radial basis function neural network is used to approximate the lumped disturbance of model uncertainty and external disturbance of the mechanical arm, so as to obtain an estimated value of the lumped disturbance of model uncertainty and external disturbance;

[0010] Step three, a fixed-time backstepping trajectory tracking controller is designed according to the dynamics model established in step one and the estimated value of the lumped disturbance in step two.

[0011] Further, the dynamics model of the multi-degree-of-freedom mechanical arm in joint space considering model uncertainty and external disturbance is:

[0012]

[0013] In the formula, q represents a vector composed of joint angles of the mechanical arm, represents a vector composed of joint angular velocities of the mechanical arm, represents a vector composed of joint angular accelerations of the mechanical arm, τ(t) is a vector composed of joint control input torques, and F(t) represents viscous friction torque. represents a real number, n represents the number of degrees of freedom of the mechanical arm, G(q) is a gravity torque matrix, M(q) is a mechanical arm system inertia matrix, is a centrifugal force and Coriolis force matrix.

[0014] M(q)=M0(q)+ΔM(q) (2)

[0015]

[0016] In the formula, M0(q) and C0(q) are nominal model parameter matrices, ΔM(q) and are uncertain parts of M0(q) and respectively.

[0017] The joint angle second-order system of the mechanical arm is considered: x1=q, The dynamics model of formula (1) is transformed into:

[0018]

[0019] In the formula, M0(q) is abbreviated as M0, the upper index -1 represents the inverse of the matrix, AM(q) is abbreviated as AM, C is abbreviated as C, G(q) is abbreviated as G, F(t) is abbreviated as F, and τ(t) is abbreviated as τ, In the formula, M0(q) is abbreviated as M0, the upper index -1 represents the inverse of the matrix, AM(q) is abbreviated as AM, C is abbreviated as C, G(q) is abbreviated as G, F(t) is abbreviated as F, and τ(t) is abbreviated as τ, is the first derivative of x1, is the first derivative of x2;

[0020] A second-order system of mechanical arm trajectory tracking error is defined as:

[0021] e1=x1-q r , e2=x2-α (5)

[0022] In the formula, e1 and e2 are error variables of the multi-freedom mechanical arm, q r is a joint position reference signal, and α is a vector composed of backstepping auxiliary variables corresponding to each joint;

[0023] and are respectively:

[0024]

[0025] wherein, is a lumped disturbance considering the model uncertainty of the multi-freedom mechanical arm and external disturbance, is the first derivative of e1, is the first derivative of e2, is a joint speed reference signal.

[0026] Further, the specific form of the adaptive radial basis function neural network is:

[0027]

[0028] In the formula, δ(Z i ) is the output of the adaptive radial basis function neural network, that is, the model uncertainty of the i-th joint and the lumped disturbance estimation value of the external disturbance received by the i-th joint, δ=[δ(Z1), δ(Z2), …, δ(Z n )], ω i * is the weight matrix of the i-th joint, is a radial basis function, Z=[z1,z2,z3] T is an input variable of the adaptive radial basis function neural network, z1=e1, z2=e2, and z3=α, e1, e2, and α are all n×1-dimensional vectors, Z i =[e 1i ,e 2i ,α i ]T ∈R 3×1 , R is a real number, e 1i , e 2i , α i is the i-th element in e1, e2, α, respectively, ε i is the estimation error of the i-th joint, and for any small positive number ε m , |ε i | < ε m , |·| represents taking the absolute value.

[0029] Further, the j-th component in the radial basis function is:

[0030]

[0031] In the formula, μ j is the center vector of the j-th hidden layer node, b j is the width vector of the j-th hidden layer node, j = 1, 2, …, k, k is the total number of hidden layer nodes, and the superscript T represents the transpose of the matrix.

[0032] Further, the backstepping auxiliary variable α i corresponding to the i-th joint is:

[0033]

[0034] In the formula, k 1i and k 2i are the power coefficients of the backstepping auxiliary variable corresponding to the i-th joint, l1 and l2 are the proportional coefficients of the backstepping auxiliary variable, and l1 > 0, l2 > 0, and are sign power transformation functions.

[0035] Further, the power coefficients k 1i and k 2i of the backstepping auxiliary variable corresponding to the i-th joint are:

[0036]

[0037] In the formula, m1 and m2 are switching constants, and m1 > 1,

[0038] Further, the specific process of step three is:

[0039]

[0040] In the formula, τ i ​Let be a fixed-time backstep trajectory tracking controller for the i-th joint, where λ is a constant greater than zero, and l3 and l4 are the proportional coefficients of the controller, with l3 > 0 and l4 > 0. and Let k be the sign power transformation function. 3i and k 4i Let a = diag[a1, a2, ..., a] be the power coefficient of the controller corresponding to the i-th joint. n ],a1,a2,...,a n All are constants greater than 0. For θ i *T The estimated value, θ i *T It is θ i * The transpose of θ i * It is the weight vector of the i-th joint.

[0041] Furthermore, the weight vector θ of the i-th joint i * for:

[0042]

[0043] In the formula, ||ω i * || 2 ω is the weight matrix of the i-th joint. i * The square of.

[0044] Furthermore, the power coefficient k of the controller corresponding to the i-th joint 3i and k 4i for:

[0045]

[0046] Furthermore, the estimated value for:

[0047]

[0048] In the formula, It is the adaptive law for the weight of the i-th joint. yes transpose, yes The first derivative, Γ i Γ is the proportional parameter of the adaptive law for the weight of the i-th joint. i >0, σ 1i and σ 2i It is the weight parameter of the adaptive law, σ1i > 0, sigma 2i > 0, e 2i is the velocity error of the i th joint of the mechanical arm, i.e. 2i is the i th element in e2.

[0049] The beneficial effects of the present application are:

[0050] The present application designs an adaptive radial basis function neural network to observe the lumped disturbance of system model uncertainty and external disturbance, and designs an adaptive backstepping trajectory tracking controller based on the lumped disturbance estimation result, so that the system can realize convergence within a fixed time range independent of the initial state, and at the same time, the stability and robustness of the multi-freedom mechanical arm system in the face of model uncertainty and external disturbance are improved.

[0051] The simulation results show that the method of the present application can make the output trajectory of the mechanical arm system have better dynamic performance and steady-state performance. BRIEF DESCRIPTION OF DRAWINGS

[0052] Figure 1 is a flow chart of a fixed-time backstepping trajectory tracking control method for a mechanical arm with model uncertainty and external disturbance of the present application;

[0053] Figure 2a is a tracking trajectory waveform one under traditional fixed-time adaptive backstepping control;

[0054] Figure 2b is a tracking trajectory waveform two under traditional fixed-time adaptive backstepping control;

[0055] Figure 3a is a tracking error waveform one under traditional fixed-time adaptive backstepping control;

[0056] Figure 3b is a tracking error waveform two under traditional fixed-time adaptive backstepping control;

[0057] Figure 4a is a control torque waveform one under traditional fixed-time adaptive backstepping control;

[0058] Figure 4b is a control torque waveform two under traditional fixed-time adaptive backstepping control;

[0059] Figure 5a is a tracking trajectory waveform one under fast fixed-time adaptive backstepping control;

[0060] Figure 5b is a tracking trajectory waveform two under fast fixed-time adaptive backstepping control;

[0061] Figure 6ais the tracking error waveform one under the fast fixed-time adaptive backstepping control;

[0062] Figure 6b is the tracking error waveform two under the fast fixed-time adaptive backstepping control;

[0063] Figure 7a is the tracking torque waveform one under the fast fixed-time adaptive backstepping control;

[0064] Figure 7b is the tracking torque waveform two under the fast fixed-time adaptive backstepping control;

[0065] Figure 8a is the tracking trajectory waveform one under the fixed-time backstepping trajectory tracking control of the mechanical arm system with model uncertainty and external disturbance of the application;

[0066] Figure 8b is the tracking trajectory waveform two under the fixed-time backstepping trajectory tracking control of the mechanical arm system with model uncertainty and external disturbance of the application;

[0067] Figure 9a is the tracking error waveform one under the fixed-time backstepping trajectory tracking control of the mechanical arm system with model uncertainty and external disturbance of the application;

[0068] Figure 9b is the tracking error waveform two under the fixed-time backstepping trajectory tracking control of the mechanical arm system with model uncertainty and external disturbance of the application;

[0069] Figure 10a is the control torque waveform one under the fixed-time backstepping trajectory tracking control of the mechanical arm system with model uncertainty and external disturbance of the application;

[0070] Figure 10b is the control torque waveform two under the fixed-time backstepping trajectory tracking control of the mechanical arm system with model uncertainty and external disturbance of the application. DETAILED DESCRIPTION

[0071] Detailed implementation one: combined with Figure 1 The application discloses a fixed-time backstepping trajectory tracking control method for a mechanical arm with model uncertainty and external disturbance. The method specifically comprises the following steps:

[0072] Step one, establish a dynamics model of a multi-degree-of-freedom (referring to the number of degrees of freedom being greater than or equal to 2) mechanical arm in joint space considering model uncertainty and external disturbance;

[0073] Step two, design an adaptive radial basis function neural network, use the adaptive radial basis function neural network to approximate the lumped disturbance of the model uncertainty and external disturbance of the manipulator, and obtain the estimated value of the lumped disturbance of the model uncertainty and external disturbance;

[0074] Step three, design a fixed-time backstepping trajectory tracking controller according to the dynamic model established in step one and the lumped disturbance estimate value in step two.

[0075] The embodiment is aimed at the trajectory tracking problem of a manipulator system with uncertainty, and proposes a new fixed-time backstepping controller based on an adaptive neural network. By designing an adaptive radial basis function neural network to compensate for the model uncertainty and external disturbance, and designing a corresponding backstepping intermediate variable to realize the rapid convergence of the controlled system within a fixed time range, it is ensured that the multi-degree-of-freedom manipulator system with uncertainty can accurately track the target output trajectory. Moreover, the system can converge within a fixed time independent of the initial state, and the stability of the system is proved by using Lyapunov stability theorem.

[0076] Specific implementation method two: different from the specific implementation method one, the multi-degree-of-freedom manipulator considering model uncertainty and external disturbance has a dynamic model in joint space as follows:

[0077]

[0078] In the formula, q represents a vector composed of joint angles of the manipulator, represents a vector composed of joint angular velocities of the manipulator, represents a vector composed of joint angular accelerations of the manipulator, τ(t) is a vector composed of joint control input torques, and F(t) represents viscous friction torque. represents a real number, n represents the number of degrees of freedom of the manipulator, G(q) is a gravity matrix, M(q) is an inertia matrix of the manipulator system, is a centrifugal force and Coriolis force matrix;

[0079] M(q) = M0(q) + ΔM(q) (2)

[0080]

[0081] In the formula, M0(q) and C0(q) are nominal model parameter matrices, ΔM(q) and are uncertain parts of M0(q) and , respectively;

[0082] Consider the joint angle second-order system of the manipulator: x1=q, The dynamic model of formula (1) is transformed into:

[0083]

[0084] In the formula, M0(q) is abbreviated as M0, the superscript -1 represents the inverse of the matrix, ΔM(q) is abbreviated as ΔM, and so on. Abbreviated as C, G(q) is abbreviated as G, F(t) is abbreviated as F, and τ(t) is abbreviated as τ. It is the first derivative of x1. It is the first derivative of x²;

[0085] Define a second-order system for robotic arm trajectory tracking error:

[0086] e1 = x1 - q r e2=x2-α (5)

[0087] In the formula, e1 and e2 are the error variables of the multi-degree-of-freedom robotic arm, and q r α is the joint position reference signal, and α is a vector composed of backstep auxiliary variables corresponding to each joint;

[0088] and They are respectively:

[0089]

[0090] in, To account for the uncertainties of the multi-degree-of-freedom robotic arm model and the lumped disturbances from external interference, It is the first derivative of e1. It is the first derivative of e². It is a joint velocity reference signal.

[0091] The other steps and parameters are the same as in Specific Implementation Method 1.

[0092] Specific Implementation Method Three: This implementation method differs from Specific Implementation Method One or Two in that the specific form of the adaptive radial basis function neural network is as follows:

[0093]

[0094] In the formula, δ(Z) i ) represents the output of the adaptive radial basis function neural network, i.e., the model uncertainty of the i-th joint and the lumped perturbation estimate of the i-th joint caused by external disturbances, δ=[δ(Z1),δ(Z2),…,δ(Z n )],ω i * It is the weight matrix of the i-th joint. These are radial basis functions (Gaussian radial basis functions are used). Z = [z1, z2, z3] T For the input variables of the adaptive radial basis function neural network, z1 = e1, z2 = e2, z3 = α, where e1, e2, and α are all n×1 dimensional vectors. i =[e 1i ,e 2i ,α i ] T ∈R 3×1 R is a real number, e 1i ,e 2i ,α i These are the i-th elements in e1, e2, and α, respectively, and ε i It is the estimation error of the i-th joint, and for arbitrarily small positive constants ε m ,|ε i |<ε m , |·| represents taking the absolute value.

[0095] Other steps and parameters are the same as in specific implementation method one or two.

[0096] Specific Implementation Method Four: This implementation method differs from Specific Implementation Methods One to Three in that the radial basis function... The j-th component for:

[0097]

[0098] In the formula, μ j Let μ be the center vector of the j-th hidden layer node. j It is a 3×1 column vector, b j It is the width vector of the j-th hidden layer node, j = 1, 2, ..., k, where k is the total number of hidden layer nodes, and the superscript T represents the transpose of the matrix.

[0099] The other steps and parameters are the same as those in one of the specific implementation methods one to three.

[0100] Specific Implementation Method Five: This implementation method differs from Specific Implementation Methods One to Four in that the backstepping auxiliary variable α corresponding to the i-th joint is... i for:

[0101]

[0102] In the formula, k 1i and k 2i l1 and l2 are the power coefficients of the backstepping auxiliary variables corresponding to the i-th joint, and l1 and l2 are the proportional coefficients of the backstepping auxiliary variables, where l1 > 0 and l2 > 0. and This is the sign power transformation function.

[0103] The other steps and parameters are the same as one of the first to fourth embodiments.

[0104] The sign power transformation function satisfies:

[0105] sig c x = [x1| c sign(x1),|x2| c sign(x2),...,|x m | c sign(x m )] T (10)

[0106] wherein, sign(·) is a sign function, and x1 is the first element in x.

[0107] The sixth embodiment is different from one of the first to fifth embodiments in that the power order coefficient k 1i and k 2i of the anti-windup auxiliary variable corresponding to the ith joint are:

[0108]

[0109] wherein, m1 and m2 are switching constants, and m1>1,

[0110] The other steps and parameters are the same as one of the first to fifth embodiments.

[0111] The seventh embodiment is different from one of the first to sixth embodiments in that the specific process of the step three is:

[0112]

[0113] wherein, τ i is the fixed-time anti-windup trajectory tracking controller of the ith joint, λ is a constant greater than zero, l3 and l4 are the proportional coefficients of the controller, and l3>0, l4>0, is a sign power transformation function, k 3i and k 4i are the power order coefficients of the controller corresponding to the ith joint, a = diag[a1, a2, …, a n ], a1, a2, …, a n are all constants greater than 0, is the estimated value of θ i *T i *T i * ​​​transpose of i * is the weight vector of the ith joint.

[0114] The other steps and parameters are the same as one of the first to sixth embodiments.

[0115] The embodiment is to achieve the control objective that the system output trajectory q tracks to a given value q r , and proposes a fixed-time backstepping trajectory tracking controller.

[0116] The eighth embodiment is different from one of the first to seventh embodiments in that the weight vector θ i * of the ith joint is:

[0117]

[0118] where ||ω i * || 2 is the square of the weight matrix ω i * of the ith joint.

[0119] The other steps and parameters are the same as one of the first to eighth embodiments.

[0120] The ninth embodiment is different from one of the first to eighth embodiments in that the power coefficients k 3i and k 4i of the ith joint corresponding controller are:

[0121]

[0122] The other steps and parameters are the same as one of the first to eighth embodiments.

[0123] The tenth embodiment is different from one of the first to ninth embodiments in that the estimated value is:

[0124]

[0125] where, is the adaptive law of the weight of the ith joint, is the transpose of , is the first derivative of , Γ i is the proportional parameter of the adaptive law of the weight of the ith joint, Γ i > 0, σ 1i and σ 2i are the weight parameters of the adaptive law, σ1i > 0, σ 2i > 0, e 2i is the velocity error of the i th joint of the robot arm, i.e. e 2i is the i th element of e2.

[0126] The other steps and parameters are the same as one of the first to ninth embodiments.

[0127] The effectiveness of the present application is verified by simulation results as follows. For a two-degree-of-freedom robot arm system, the reference trajectory is selected as q r = [sin(0.5t) cos(0.5t) -1] T , and the initial state is q0 = [-3 -2] T . In order to prove the effectiveness of the present application, the performances of the traditional fixed-time adaptive backstepping control method (i.e. method one), the fast fixed-time adaptive backstepping control method (i.e. method two) and the fixed-time backstepping control method proposed by the present application are analyzed, and the specific control parameters are shown in Table 1:

[0128] Table 1

[0129]

[0130] The controller of the traditional fixed-time adaptive backstepping control method is as follows:

[0131]

[0132] The controller of the fast fixed-time adaptive backstepping control method is as follows:

[0133]

[0134] Figure 2a , Figure 2b , Figure 5a , Figure 5b , Figure 8a and Figure 8b show the tracking trajectory waveforms of the two-degree-of-freedom robot arm under the three methods, Figure 3a , Figure 3b , Figure 6a , Figure 6b , Figure 9a and Figure 9b show the tracking error waveforms of the two-degree-of-freedom robot arm under the three methods, Figure 4a , Figure 4b , Figure 7a , Figure 7b , Figure 10a and Figure 10bThe control torque waveforms of the two-degree-of-freedom robot arm under three methods are shown. -3 From the results, it can be seen that the new fast backstepping fixed-time control method based on the radial basis function neural network realizes faster convergence speed and better convergence accuracy, the convergence errors are all within 4x10 -3 rad, and the effectiveness of the designed control algorithm is verified.

[0135] The above calculation examples of the present application are only to illustrate the calculation model and calculation process of the present application, and are not limited to the embodiments of the present application. For those skilled in the art, other different forms of changes or variations can be made on the basis of the above description, and all the embodiments cannot be exhausted here, and any obvious changes or variations derived from the technical solutions of the present application still fall within the protection scope of the present application.

Claims

1. A fixed-time backstepping trajectory tracking control method for a robotic arm with model uncertainty and external disturbances, characterized in that, The method specifically includes the following steps: Step 1: Establish a dynamic model of the multi-degree-of-freedom robotic arm in joint space, taking into account model uncertainties and external disturbances; Step 2: Design an adaptive radial basis function neural network to approximate the lumped disturbance of the robot arm model uncertainty and external disturbances, and obtain the estimated values ​​of the lumped disturbance of the model uncertainty and external disturbances. The specific form of the adaptive radial basis function neural network is as follows: In the formula, δ(Z) i ) represents the output of the adaptive radial basis function neural network, i.e., the model uncertainty of the i-th joint and the lumped perturbation estimate of the i-th joint caused by external disturbances, δ=[δ(Z1),δ(Z2),…,δ(Z n )],ω i * It is the weight matrix of the i-th joint. For radial basis functions, Z = [z1, z2, z3] T For the input variables of the adaptive radial basis function neural network, z1 = e1, z2 = e2, z3 = α, where e1, e2, and α are all n×1 dimensional vectors. i =[e 1i ,e 2i ,α i ] T ∈R 3×1 R is a real number, e 1i ,e 2i ,α i These are the i-th elements in e1, e2, and α, respectively, and ε i It is the estimation error of the i-th joint, and for arbitrarily small positive constants ε m ,|ε i |<ε m , |·| represents taking the absolute value; The radial basis function The j-th component for: In the formula, μ j Let b be the center vector of the j-th hidden layer node. j It is the width vector of the j-th hidden layer node, j = 1, 2, ..., k, where k is the total number of hidden layer nodes, and the superscript T represents the transpose of the matrix; The backstep auxiliary variable α corresponding to the i-th joint i for: In the formula, k 1i and k 2i l1 and l2 are the power coefficients of the backstepping auxiliary variables corresponding to the i-th joint, and l1 and l2 are the proportional coefficients of the backstepping auxiliary variables, where l1 > 0 and l2 > 0. and It is a sign power transformation function; Step 3: Design a fixed-time backstepping trajectory tracking controller based on the dynamic model established in Step 1 and the lumped disturbance estimate in Step 2; The specific process of step three is as follows: In the formula, τ i Let be a fixed-time backstep trajectory tracking controller for the i-th joint, where λ is a constant greater than zero, and l3 and l4 are the proportional coefficients of the controller, with l3 > 0 and l4 > 0. and Let k be the sign power transformation function. 3i and k 4i Let a = diag[a1, a2, ..., a] be the power coefficient of the controller corresponding to the i-th joint. n ],a1,a2,...,a n All are constants greater than 0. for The estimated value, yes transpose, It is the weight vector of the i-th joint.

2. The fixed-time backstepping trajectory tracking control method for a robotic arm with model uncertainty and external disturbances according to claim 1, characterized in that, The dynamic model of the multi-degree-of-freedom robotic arm in joint space, considering model uncertainties and external disturbances, is as follows: In the formula, q represents the vector composed of the angles of the joints of the robotic arm. This represents the vector composed of the angular velocities of the robotic arm's joints. Let τ(t) be the vector composed of the angular accelerations of each joint of the robotic arm, τ(t) be the vector composed of the control input torques of each joint, and F(t) represent the viscous friction torque. Let represent a real number, n represent the number of degrees of freedom of the robotic arm, G(q) be the gravity matrix, and M(q) be the inertia matrix of the robotic arm system. The matrix represents the centrifugal force and the Coriolis force. M(q)=M0(q)+ΔM(q) (2) In the formula, M0(q) and C0(q) are the nominal model parameter matrices, and ΔM(q) and ΔC0(q) are the model parameter matrices. M0(q) and The uncertain part; Consider a second-order system of robotic arm joint angles: x1 = q, The dynamic model of equation (1) is transformed as follows: In the formula, M0(q) is abbreviated as M0, the superscript -1 represents the inverse of the matrix, ΔM(q) is abbreviated as ΔM, and so on. Abbreviated as C, G(q) is abbreviated as G, F(t) is abbreviated as F, and τ(t) is abbreviated as τ. It is the first derivative of x1. It is the first derivative of x²; Define a second-order system for robotic arm trajectory tracking error: e1=x1-q r ,e2=x2-α (5) In the formula, e1 and e2 are the error variables of the multi-degree-of-freedom robotic arm, and q r α is the joint position reference signal, and α is a vector composed of backstep auxiliary variables corresponding to each joint; and They are respectively: in, To account for the uncertainties of the multi-degree-of-freedom robotic arm model and the lumped disturbances from external interference, It is the first derivative of e1. It is the first derivative of e². It is a joint velocity reference signal.

3. The fixed-time backstepping trajectory tracking control method for a robotic arm with model uncertainty and external disturbances according to claim 2, characterized in that, The power coefficient k of the backstep auxiliary variable corresponding to the i-th joint 1i and k 2i for: In the formula, m1 and m2 are switching constants, and m1 > 1.

4. The fixed-time backstepping trajectory tracking control method for a robotic arm with model uncertainty and external disturbances according to claim 2, characterized in that, The weight vector of the i-th joint for: In the formula, ||ω i * || 2 ω is the weight matrix of the i-th joint. i * The square of.

5. The fixed-time backstepping trajectory tracking control method for a robotic arm with model uncertainty and external disturbances according to claim 2, characterized in that, The power coefficient k of the controller corresponding to the i-th joint 3i and k 4i for:

6. The fixed-time backstepping trajectory tracking control method for a robotic arm with model uncertainty and external disturbances according to claim 2, characterized in that, The estimated value for: In the formula, It is the adaptive law for the weight of the i-th joint. yes transpose, yes The first derivative, Γ i Γ is the proportional parameter of the adaptive law for the weight of the i-th joint. i >0, σ 1i and σ 2i σ is the weight parameter of the adaptive law. 1i >0, σ 2i >0, e 2i It is the speed error of the i-th joint of the robotic arm, i.e., e 2i It is the i-th element in e2.

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