Flexible link robot arm fixed time adaptive neural network composite control method

Abstract

The application discloses a flexible connecting rod mechanical arm fixed time adaptive neural network composite control method, a flexible connecting rod mechanical arm model is decomposed into a slow subsystem and a fast subsystem based on a singular perturbation theory, and slow subsystem state variables and fast subsystem state variables are defined in the decomposition process, and the slow subsystem comprises unmodeled dynamics, and the control torque in the model is decomposed into a tracking controller for controlling the slow subsystem and a flexible suppression controller for controlling the fast subsystem. The technical scheme simultaneously reduces the elastic vibration of the flexible mode, realizes the tracking of the time-varying desired position of the mechanical arm, and also enables the mechanical arm to achieve fixed time stability.

Description

Technical Field

[0001] This invention belongs to the field of robotic arm control technology, specifically relating to a fixed-time adaptive neural network composite control method for a flexible linkage robotic arm. Background Technology

[0002] Flexible linkage robotic arms possess many advantages, including light weight, high speed, flexibility, and low energy consumption. However, they also exhibit more complex dynamic characteristics such as nonlinearity, underactuation, and multiple degrees of freedom. Controlling flexible linkage robotic arms requires addressing the following three technical challenges:

[0003] (1) Oscillations of flexible modes inevitably exist in flexible linkage robotic arm systems. How to avoid oscillations of flexible modes during high-precision trajectory tracking.

[0004] (2) Due to modeling errors, model simplification, external disturbances, and measurement noise, some dynamic characteristics of the system may be lost. These uncertainties are called unmodeled dynamics. How can the stability of the control system be guaranteed when unmodeled dynamics exist?

[0005] (3) Continuous work is an essential capability that a robotic arm needs to have. How to achieve fixed-time control of the control system?

[0006] However, in the existing technology, the control of flexible linkage robotic arms can generally only solve some of the above-mentioned technical problems, and the control effect is not good, as in references [1]-[4].

[0007] Document [1]: Boundary output constrained control for a flexible beamsystem with prescribed performance;

[0008] Document [2] "Uncalibrated visual servoing for a planar two link rigid-ffexible manipulator without jointspace-velocity measurement";

[0009] Document [3] "Singular perturbation-based adaptive integral sliding modecontrol for flexible joint robots";

[0010] Document [4] "Command filter-based adaptive control of flexible-jointmanipulator with input saturation and output constraints".

[0011] Reference [1] uses a differential equation-based modeling method to handle the oscillations of flexible modes in the system model. However, this method is not suitable for practical robotic arm systems due to the high order of the system and the complexity of its numerical characteristics.

[0012] Reference [2] developed an uncalibrated visual servo control scheme to solve the trajectory tracking and flexible vibration problems of a planar double-link rigid-flexible manipulator; Reference [3] constructed an adaptive integral sliding mode controller based on singular perturbation theory and two state observers to ensure the high tracking performance of the flexible joint manipulator. To ensure high-precision trajectory tracking of the manipulator, these methods deal with the oscillation control problem of the flexible mode of the manipulator in an ideal environment. For actual manipulator systems, due to modeling errors, model simplification, external disturbances and measurement noise, there will inevitably be unmodeled dynamics in the manipulator system. Therefore, when the system has unmodeled dynamics, the method will not achieve the expected control effect, or even lead to system instability.

[0013] Although reference [4] considers the oscillations of the flexible modes and the unmodeled dynamics in the system model, it is only applicable to asymptotic convergence control problems. When the system needs to achieve convergence in a fixed time, this method will no longer be applicable. Summary of the Invention

[0014] To overcome the shortcomings of the prior art, this invention provides a fixed-time adaptive neural network composite control method for flexible linkage robotic arms.

[0015] This invention is achieved through the following technical solution:

[0016] A fixed-time adaptive neural network composite control method for a flexible linkage robotic arm includes the following steps:

[0017] S1: Confirm Model of a flexible linkage robotic arm with multiple degrees of freedom. The value is a positive integer. Based on the singular perturbation theory, the model is decomposed into a slow subsystem and a fast subsystem. During the decomposition process, the state variables of the slow subsystem and the state variables of the fast subsystem are defined. The slow subsystem includes unmodeled dynamics. At the same time, the control torque in the model is decomposed into a tracking controller used to control the slow subsystem and a flexible suppression controller used to control the fast subsystem.

[0018] S2: For the slow subsystem, construct the slow subsystem error based on the slow subsystem state variables, the desired position of the robotic arm, and the first virtual controller obtained by the backstepping method.

[0019] S3: For the slow subsystem, construct the first Lyapunov function, which includes dynamic signals for unmodeled dynamics; based on the first Lyapunov function, the slow subsystem error, and the first set of radial basis function vectors, use the backstepping method to obtain the first virtual controller and the first adaptive law;

[0020] S4: For the slow subsystem, construct a second Lyapunov function. Based on the second Lyapunov function, the first virtual controller, the slow subsystem error, and the second set of radial basis function vectors, use the backstepping method to obtain the tracking controller and the second adaptive law.

[0021] S5: For the fast subsystem, construct the fast subsystem error based on the fast subsystem state variables and the second virtual controller subsequently obtained by the backstepping method;

[0022] S6: For the tachyon system, construct the third Lyapunov function, and based on the third Lyapunov function and the tachyon system error, use the backstepping method to obtain the second virtual controller;

[0023] S7: For the tachyon system, a fourth Lyapunov function is constructed. Based on the fourth Lyapunov function, the second virtual controller, the tachyon system error, and the third set of radial basis function vectors, the flexible suppression controller and the third adaptive law are obtained using the backstepping method.

[0024] S8: The slow subsystem is controlled by the tracking controller, and the fast subsystem is controlled by the flexible suppression controller; the first and second adaptive laws are used to assist the tracking controller in controlling the slow subsystem, and the third adaptive law is used to assist the flexible suppression controller in controlling the slow subsystem, thereby realizing the control of the robotic arm.

[0025] Further, in step S1, the model is:

[0026] (1);

[0027] Where, vector , It is the angular position vector of the robotic arm. , For the first The angular position vectors of each link. ; It is a flexible generalized coordinate vector. , For the first The first link A flexible module, , This represents the total number of flexible variables. Indicates the first The number of flexible variables in each link; The inertia matrix, It is the matrix of Coriolis force and centrifugal force; Represents the stiffness matrix. Indicates the input weighting matrix. To control the torque, The external disturbance torque is unknown, and ,in For unknown positive constants; , .

[0028] Furthermore, in step S1, based on singular perturbation theory, the model is decomposed into a model defined in the first time domain. The slow subsystem described in the second time domain is defined in the second time domain. The fast subsystem described in the text; the time-scale transformation form is as follows: ,in, For the first time domain Mid-time point, For the second time domain The time point in the middle, Small scale factor ,in It is the stiffness matrix The smallest non-zero stiffness element in the matrix; the decomposition process is as follows:

[0029] Equation (1) can be written in block form based on the rigid and flexible conditions:

[0030] (2);

[0031] in, , , and for Write the submatrix after dividing the matrix into blocks. , , and for Write the submatrix after dividing the matrix into blocks. for Write it as a submatrix after dividing it into blocks; for rigid systems It is the identity matrix. ;

[0032] definition:

[0033] (3);

[0034] in , , and for Write it as a submatrix after the block matrix;

[0035] From equations (2) and (3), we can derive:

[0036] (4);

[0037] Define the proportional stiffness matrix and new variables Then equation (4) can be reformulated as:

[0038] (5);

[0039] (6);

[0040] in For the tracking controller, The flexible suppression controller;

[0041] When the model approaches a rigid system , Then we have:

[0042] (7);

[0043] in, , , , , , , , , and respectively under rigid conditions , , , , , , , , and ;

[0044] From equation (7) and the properties of rigid robotic arms We can obtain:

[0045] (8);

[0046] Define the state variables of the slow subsystem. ,in, and , For the slow subsystem Corner positions, For the slow subsystem angular velocity, For rigid conditions Due to the matrix Uncertainty, assumption ,in For rigid conditions , The first nominal matrix, the first modeling error It is bounded;

[0047] The slow subsystem is obtained as follows:

[0048] (9);

[0049] in It is an unknown function that satisfies the local Lipschitz condition. For unmodeled dynamics, It is an unknown continuous nonlinear function. It is the first total disturbance that includes modeling errors and external disturbance torques;

[0050] Define the state variables of the fast subsystem as follows: ,in,

[0051] ,

[0052] ,

[0053] For the slow subsystem The first link The amplitude of each flexible module For the slow subsystem The first link The vibration velocity of each flexible module; due to the matrix Uncertainty, assumption ,in The second nominal matrix, the second modeling error It is bounded;

[0054] The fast subsystem is obtained as follows:

[0055] (10);

[0056] in It is an unknown continuous nonlinear function. For rigid conditions , It is the second total disturbance that includes modeling error.

[0057] Further, in step S2, the error of the slow subsystem is:

[0058] (11);

[0059] in, To track errors, For speed error, It is the first virtual controller. This is the desired position of the robotic arm. , It is the first The expected time-varying trajectory of each link angular position.

[0060] Furthermore, in step S3, the first Lyapunov function is:

[0061] (12);

[0062] in, and It is a positive number. It is the dynamic signal, , This represents the first estimation error and the first adaptive parameter. yes The estimated value;

[0063] The first virtual controller is:

[0064] (13);

[0065] The first adaptive law is:

[0066] (14);

[0067] in, , , , , , Is it only the first Line number The column element is 1 and all other elements are 1. of Dimensional array, , This is the first set of weight vectors. It is the first set of radial basis function vectors. , , , , and It is a positive number.

[0068] Furthermore, in step S4, the second Lyapunov function is:

[0069] (15);

[0070] in, It is a positive number. , This represents the second estimation error and the second adaptive parameter. yes The estimated value;

[0071] The tracking controller is:

[0072] (16);

[0073] The second adaptive law is:

[0074] (17);

[0075] in, The tracking controller's first Each component , , , , , , This is the second set of weight vectors. These are the second set of radial basis function vectors. , , , , and It is a positive number.

[0076] Further, in step S5, the error of the fast subsystem is:

[0077] (18);

[0078] in, For oscillation error, For vibration velocity error, It is the second virtual controller.

[0079] Furthermore, in step S6, the third Lyapunov function is:

[0080] (19);

[0081] in, To constrain amplitude The designable function is represented as ,parameter , and It is a positive number;

[0082] The second virtual controller is:

[0083] (20);

[0084] in, , , and It is a positive number.

[0085] Furthermore, in step S7, the fourth Lyapunov function is:

[0086] (twenty one);

[0087] in, It is a positive number. , This represents the third estimation error and the third adaptive parameter. yes The estimated value;

[0088] The flexible suppression controller is:

[0089] (twenty two);

[0090] The third adaptive law is:

[0091] (twenty three);

[0092] in, The first flexible suppression controller Each component , , , , , Is it only the first Line number The column element is 1 and all other elements are 1. of Dimensional array, , It is the third set of weight vectors. It is the third set of radial basis function vectors. , , , , and It is a positive number.

[0093] The beneficial effects that this invention can achieve are as follows:

[0094] (1) Compared with reference [1], this invention decomposes the model into slow subsystems and fast subsystems based on singular perturbation theory, which reduces the system order and alleviates the system complexity.

[0095] (2) Compared with references [2] and [3], this invention considers the influence of elastic vibration of flexible mode on the robotic arm and considers unmodeled dynamics, which enhances the anti-interference ability of the robotic arm as the controlled object and improves the robustness of the robotic arm.

[0096] (3) Compared with reference [4], the present invention effectively reduces the elastic vibration of the flexible mode and solves the unmodeled dynamics in the system, while successfully realizing the fixed time control of the robotic arm.

[0097] (4) The present invention can realize the tracking of the desired position of the robotic arm in a time-varying manner, and the tracking error of the angular position of the robotic arm can be stabilized at a fixed time.

[0098] (5) The proposed composite control method for robotic arms utilizes adaptive control technology, which has universal adaptability to uncertainties in the model and improves the robustness of the designed control method.

[0099] (6) The present invention takes into account the unmodeled dynamics that inevitably exist in the system, making the controller more in line with the actual situation. Attached Figure Description

[0100] Figure 1 The tracking controller in this embodiment of the invention The system angular position and the tracking curve of the desired time-varying trajectory.

[0101] Figure 2 The tracking controller in this embodiment of the invention The first and second adaptive parameter curves.

[0102] Figure 3 The tracking controller in this embodiment of the invention The first and second component curves.

[0103] Figure 4 This is the flexible suppression controller in the embodiments of the present invention. The amplitude of the first link and the first and second flexible modules.

[0104] Figure 5This is the flexible suppression controller in the embodiments of the present invention. The amplitude of the second link and the first and second flexible modules.

[0105] Figure 6 This is the flexible suppression controller in the embodiments of the present invention. The third adaptive parameter curve.

[0106] Figure 7 This is the flexible suppression controller in the embodiments of the present invention. The first and second component curves.

[0107] Figure 8 This is the flexible suppression controller in the embodiments of the present invention. The third and fourth component curves. Detailed Implementation

[0108] For a class of unmodeled dynamics ( , For a flexible linkage robotic arm with degrees of freedom (representing the set of positive integers), a fixed-time adaptive neural network composite control method is designed, including the following steps:

[0109] S1, Confirm The model of a degree-of-freedom flexible linkage manipulator is decomposed into a slow subsystem and a fast subsystem based on singular perturbation theory. During the decomposition process, the state variables of the slow subsystem and the fast subsystem are defined. The slow subsystem includes unmodeled dynamics. At the same time, according to the model decomposition, the control torque in the model is decomposed into a tracking controller to be designed to control the slow subsystem and a flexible suppression controller to control the fast subsystem.

[0110] A type The model of the flexible linkage robotic arm with degrees of freedom is as follows:

[0111] (1)

[0112] Where vector Including the angular position vector of the robotic arm and flexible generalized coordinate vector , For the first The angular position vectors of each link. , For the first The first link A flexible module, , This represents the total number of flexible variables. Indicates the first The number of flexible variables in each link; The inertia matrix, It is the matrix of Coriolis force and centrifugal force. and These represent the stiffness matrix and the input weighting matrix, respectively. and These are the control torque in the model and the disturbance torque of the unknown time-varying external environment, respectively. for of First derivative, for of First derivative, and These are the angular velocity and angular acceleration of the robotic arm, respectively. and They are respectively of order derivative and The first derivative.

[0113] Property 1: It is a non-singular positive definite symmetric matrix.

[0114] Property 2: It is a skew-symmetric matrix.

[0115] Assumption 1: External disturbance torque Meet the conditions ,in It is an unknown positive number.

[0116] According to singular perturbation theory, model (1) can be decomposed into a function defined in the first time domain. The slow subsystem in the second time domain is defined. The fast subsystem in the middle is constructed in the form of a time-scale transformation. ,in For the first time domain Mid-time point and For the second time domain The time point in the middle, Small scale factor ,in It is the stiffness matrix The smallest non-zero stiffness element in the matrix. The decomposition process is as follows:

[0117] Equation (1) can be written in block form according to the rigidity and flexibility conditions, as shown below:

[0118] (2)

[0119] in , , and For the unknown Write the unknown submatrix after dividing the matrix into blocks. , , and For the unknown Write the unknown submatrix after dividing the matrix into blocks. For matrix Write it as a submatrix after dividing the matrix into blocks. For rigid systems... It is the identity matrix. .

[0120] definition:

[0121] (3)

[0122] in , , and For matrix Write it as the unknown submatrix after dividing it into blocks.

[0123] Therefore, from equations (2) and (3), we can conclude that:

[0124] (4)

[0125] Define the proportional stiffness matrix and new variables Then (4) can be rewritten as:

[0126] (5)

[0127] After that, the subscript " "The marked variables and matrices belong to the slow subsystem, subscripts" "The marked variables and matrices belong to the fast subsystem. To achieve the dual control objectives, the control torque in the model is decomposed into:"

[0128] (6)

[0129] in This refers to the tracking controller that controls the slow subsystem. This refers to a flexible suppression controller that controls the tachy subsystem.

[0130] When equation (1) approaches a rigid system, the minimum stiffness element Therefore, take At that time, there were:

[0131] (7)

[0132] in, , , , , , , , , and respectively under rigid conditions , , , , , , , , and .

[0133] From equation (7) and the properties of rigid robotic arms We can obtain:

[0134] (8)

[0135] Define the state variables of the slow subsystem as belonging to the slow subsystem. ,in and , For the slow subsystem Corner positions, For the slow subsystem angular velocity, For rigid conditions Considering the matrix Uncertainty, assumption ,in For rigid conditions , The first nominal matrix and the first modeling error It is bounded. Considering the effects of unmodeled dynamics, the slow subsystem is obtained as follows:

[0136] (9)

[0137] in, It is an unknown function that satisfies the local Lipschitz condition. For unmodeled dynamics, It is an unknown continuous nonlinear function. It is the first total disturbance that includes modeling errors and external disturbance torques.

[0138] Define the state variables of the tachy subsystem as follows: ,in and , For the slow subsystem The first link The amplitude of each flexible module For the slow subsystem The first link The vibration velocity of each flexible module. Considering the matrix... Uncertainty, assumption ,in The second nominal matrix and the second modeling error It is bounded. From (5) and (7), the fast subsystem is obtained as follows:

[0139] (10)

[0140] in It is an unknown continuous nonlinear function. For rigid conditions , It is the second total disturbance that includes modeling error.

[0141] Assumption 2: Desired position of the robotic arm and its relation to time until The first derivative is known, smooth, and bounded, where It is the first The expected time-varying trajectory of each link angular position.

[0142] S2. For the slow subsystem, construct the slow subsystem error based on the slow subsystem state variables, the desired position of the robotic arm, and the first virtual controller obtained subsequently by the backstepping method:

[0143] (11)

[0144] in, To track errors, For speed error, It is the first virtual controller and will be built in subsequent design steps.

[0145] S3. For the slow subsystem containing unmodeled dynamics, the first Lyapunov function is constructed. At the same time, in order to solve the problem that the slow subsystem is difficult to control due to unmodeled dynamics, dynamic signals are introduced into the first Lyapunov function. Based on the constructed first Lyapunov function, the slow subsystem error and the neural network basis functions in the known radial basis function neural network, the first virtual controller and the first adaptive law are designed using the backstepping method.

[0146] S301. For the slow subsystem, the first Lyapunov function is constructed. Simultaneously, to address the problem of the slow subsystem being difficult to control due to unmodeled dynamics, a dynamic signal is introduced into the first Lyapunov function:

[0147] Constructing the first Lyapunov function:

[0148] (12)

[0149] in, and It is a positive number. It is a dynamic signal. , This represents the first estimation error and the first adaptive parameter. yes The estimated value, The representation will be given in subsequent design steps.

[0150] Dynamic signals It is necessary to satisfy Lemma 1 on the premise that Assumption 3 holds.

[0151] Assumption 3: In the slow subsystem The fact that the input state is actually stable indicates the existence of a Lyapunov function. , so that:

[0152] (13)

[0153] in, , and yes Class function, and It is a positive number.

[0154] Lemma 1: If Lyapunov function If assumption 3 is satisfied, then for conditions... arbitrary constant any initial time Arbitrary initial conditions and The value at the initial moment The conditions are met. any nonnegative continuous function Bounded time exists ,function and the dynamic signal described below :

[0155] (14)

[0156] Make and

[0157] (15)

[0158] S302. For the slow subsystem, based on the constructed first Lyapunov function, the slow subsystem error, and the neural network basis functions in the known radial basis function neural network, the first virtual controller and the first adaptive law are designed using the backstepping method.

[0159] Using equations (9) and (11), for Seeking information about The derivative can be obtained

[0160] (16)

[0161] Non-negative continuous functions in dynamic signals After using polynomial fitting Relationship: ,in It is a non-negative smooth function. It is a bounded fitting error, meaning there are unknown constants. , making Therefore, we can conclude that:

[0162] (17)

[0163] in, , , , and It is a positive number.

[0164] In existing technologies that consider the impact of unmodeled dynamics on robotic arms, some fail to introduce dynamic signals into the first Lyapunov function. Based on Lyapunov stability theory, this makes it impossible to explain the stability of the dynamic signals or quantify the boundedness of the unmodeled dynamics. Other technologies use non-negative continuous functions... Addressing the unknowns within a single joint leads to direct interference from data from other joints when estimating the unknowns of that joint using a neural network, making accurate estimation difficult. However, this technique addresses the unknowns of non-negative continuous functions. The processing method not only makes up for the shortcomings of existing technologies, but also combines them with... By associating the neural network with a non-negative smooth function, the unknowns can be distributed to each joint of the robotic arm while satisfying the requirement for the number of degrees of freedom. This allows the neural network to make more accurate estimates of the unknowns.

[0165] Based on Lemma 3 in the well-known existing technology, using neural networks To estimate Make it satisfy the following equation:

[0166] (18)

[0167] in, This is the first set of weight vectors. It is the first set of radial basis function vectors, and the estimation error is... satisfy , It is an unknown positive number.

[0168] Subsequently, using Lemma 7 from the well-known prior art, we can obtain

[0169] (19)

[0170] in, It is a positive number. , Is it only the first Line number The column element is 1 and all other elements are 1. of Dimensional array.

[0171] Substituting equation (19) into equation (17), we get

[0172] (20)

[0173] Based on equation (20), the first virtual controller is constructed. and the first adaptive law for

[0174] (twenty one)

[0175] (twenty two)

[0176] in , , , , , , , , and It is a positive number.

[0177] The first Lyapunov function constructed, after differentiation, satisfies the following based on Lemmas 5 and 6:

[0178] (twenty three)

[0179] in, , and , .

[0180] S4. For the slow subsystem, construct a second Lyapunov function. Based on the constructed second Lyapunov function, the designed first virtual controller, the slow subsystem error, and the neural network basis functions in the known radial basis function neural network, design a tracking controller and a second adaptive law using the backstepping method. The tracking controller uses the designed first and second adaptive laws as an aid to control the slow subsystem.

[0181] S401. For the slow subsystem, construct the second Lyapunov function:

[0182] (twenty four)

[0183] in, It is a positive number. , This represents the second estimation error and the second adaptive parameter. yes The estimated value, The representation will be given in subsequent design steps.

[0184] S402. Based on the constructed second Lyapunov function, the designed first virtual controller, the slow subsystem error, and the neural network basis functions in the known radial basis function neural network, a tracking controller and a second adaptive law are designed using the backstepping method.

[0185] Using equations (9) and (11), for Seeking information about The derivative can be obtained

[0186] (25)

[0187] in and .

[0188] Note 1: According to Assumption 1, external disturbances... It is bounded. Furthermore, due to modeling errors... Given the boundedness of the state, the limitations of the robotic arm, and the constraints of the task space, we can assume state-related perturbations. It is bounded, and its bound is an unknown positive constant. .

[0189] Therefore, using Lemma 4 from the well-known prior art, the following inequality holds:

[0190] (26)

[0191] Substituting equation (26) into equation (25), we get

[0192] (27)

[0193] in, , and .

[0194] Based on Lemma 3 in the well-known existing technology, using neural networks To estimate Make it satisfy the following equation:

[0195] (28)

[0196] in, This is the second set of weight vectors. It is the second set of radial basis function vectors, estimation error , It is an unknown positive number.

[0197] Subsequently, using Lemma 7 from the well-known prior art, we can obtain

[0198] (29)

[0199] in, It is a positive number. .

[0200] Substituting equation (29) into equation (27), we get

[0201] (30)

[0202] Based on equation (30), a tracking controller for controlling the slow subsystem is constructed. and the second adaptive law for

[0203] (31)

[0204] (32)

[0205] in, The tracking controller's first Each component , , , , , , , , and It is a positive number.

[0206] S403, the tracking controller uses the designed first and second adaptive laws to control the slow subsystem.

[0207] In this example, the second Lyapunov function constructed, after differentiation, satisfies, based on Lemmas 5 to 9 of the known prior art:

[0208] (33)

[0209] Its parameters are shown below. It can be seen that it satisfies the conditions of Lemma 2 in the well-known practical fixed-time stability theory. That is, based on assumptions 1-3, the proposed control scheme has a tracking controller, a first adaptive law, and a second adaptive law, which ensures the dynamic signal... and slow subsystem state variables In the first time domain The above is actually stable at a fixed time. Based on the dynamic signal satisfying equation (15), the unmodeled dynamic signal can be obtained. In the first time domain The above is actually stable at a fixed time, and thus the slow subsystem is stable in the first time domain. The above is actually stable at a fixed time, meaning that within a fixed time, all signals in the slow subsystem are bounded and It can accurately track the desired position of the robotic arm. .

[0210] , , , , , ,against Divided into and ,in and For positive integers that satisfy Unknown positive numbers for The upper realm, , , It is an unknown positive constant that makes , .

[0211] S5. For the fast subsystem, construct the fast subsystem error based on the fast subsystem state variables and the second virtual controller subsequently obtained by the backstepping method:

[0212] (34)

[0213] in, For oscillation error, For vibration velocity error, It is the second virtual controller and will be built in subsequent design steps.

[0214] S6. For the tachyon system, construct a third Lyapunov function. Based on the constructed third Lyapunov function and the error of the tachyon system, design a second virtual controller using the backstepping method.

[0215] S601. For the tachy subsystem, construct the third Lyapunov function:

[0216] (35)

[0217] in, To constrain amplitude The designable function is represented as ,parameter , and It is a normal number.

[0218] S602. For the tachy subsystem, based on the constructed third Lyapunov function and the tachy subsystem error, a second virtual controller is designed using the backstepping method and according to the actual fixed-time stability theory:

[0219] Using equations (10) and (34), for Seeking information about The derivative is:

[0220] (36)

[0221] Based on (36), construct a second virtual controller. for:

[0222] (37)

[0223] in , and and It is a positive number.

[0224] Note 2: To avoid when hour The singularity problem, in In this case, use a sufficiently small positive number. replace .

[0225] Substituting equation (37) into equation (36) yields

[0226] (38)

[0227] S7. For the fast subsystem, construct a fourth Lyapunov function. Based on the constructed fourth Lyapunov function, the designed second virtual controller, the fast subsystem error, and the neural network basis functions in the known radial basis function neural network, design a flexible suppression controller and a third adaptive law using the backstepping method. The flexible suppression controller uses the designed third adaptive law as an aid to control the slow subsystem.

[0228] S701. For the tachy subsystem, construct the fourth Lyapunov function:

[0229] (39)

[0230] in, It is a positive number. This represents the third estimation error and the third adaptive parameter. yes The estimated value, The representation will be given in subsequent design steps.

[0231] S702. For the tachyon system, based on the constructed fourth Lyapunov function, the designed second virtual controller, the tachyon system error, and the neural network basis functions in the known radial basis function neural network, a flexible suppression controller and a third adaptive law are designed using the backstepping method:

[0232] Using equations (10) and (34), for Seeking information about The derivative can be obtained

[0233] (40)

[0234] in and .

[0235] Note 3: Due to modeling errors Given the boundedness of the state, the limitations of the robotic arm, and the constraints of the task space, we can assume state-related perturbations. It is bounded, and its bound is an unknown positive constant. .

[0236] Therefore, using Lemma 4 from the well-known prior art, the following inequality holds:

[0237] (41)

[0238] Substituting equation (41) into equation (40), we get:

[0239] (42)

[0240] in, and .

[0241] Based on Lemma 3 in the well-known existing technology, using neural networks To estimate Make it satisfy the following equation:

[0242] (43)

[0243] in, It is the third set of weight vectors. It is the third set of radial basis function vectors, estimation error , It is an unknown positive number.

[0244] Then, using Lemma 7, we can obtain

[0245] (44)

[0246] in, It is a positive constant and , Is it only the first Line number The column element is 1 and all other elements are 1. of Dimensional array.

[0247] Substituting equation (44) into equation (42), we get:

[0248] (45)

[0249] Based on equation (45), a flexible suppression controller for controlling the fast subsystem is constructed. and the third adaptive law for:

[0250] (46)

[0251] (47)

[0252] in, The first flexible suppression controller Each component , , , , , , , , and It is a positive number.

[0253] The S703 flexible suppression controller uses the designed third adaptive law to control the slow subsystem.

[0254] In this example, the fourth Lyapunov function constructed, after differentiation, satisfies, based on the well-known lemmas 5 to 8 in the prior art:

[0255] (48)

[0256] Its parameters are shown below. It can be seen that it satisfies the conditions of Lemma 2 in the well-known practical fixed-time stability theory, that is, the proposed control scheme has a flexible suppression controller (46) and a third adaptive law (47), which ensures that the fast subsystem (10) is stable in the second time domain. The above is actually stable at a fixed time, that is, all signals in the fast subsystem are bounded within a fixed time.

[0257] , , , , , , , .

[0258] S8: The slow subsystem is controlled by a tracking controller, and the fast subsystem is controlled by a flexible suppression controller. First and second adaptive laws assist the tracking controller in controlling the slow subsystem, and a third adaptive law assists the flexible suppression controller in controlling the slow subsystem, thereby achieving control of the robotic arm. This effectively handles the oscillations of the flexible modes within the robotic arm while simultaneously tracking the desired time-varying trajectory and achieving actual fixed-time stabilization of the robotic arm.

[0259] To verify the effectiveness of the designed controller, we performed simulations using MATLAB. Taking a flexible linkage robotic arm with multiple degrees of freedom as an example, the parameters of equation (1) are: the number of flexible variables for each link is The length of each link is The mass of each link is The tip quality is Constant bending stiffness is The disturbance torque of the external environment is The desired position of the robotic arm is specified as follows: ,in .

[0260] For the slow subsystem, the unmodeled dynamics are: ,in and The dynamic signal is ,in The initial conditions are: The parameters are , , , , , , , .

[0261] For the tachy subsystem, the initial conditions are as follows: and The parameters are , , , , , , , , .

[0262] Figure 1 The system's angular position and the desired time-varying trajectory are displayed. It can be observed that the tracking error is very small. From... Figure 2 and Figure 3 It can be seen that the first adaptive parameter The second adaptive parameter and tracking controller The first and second components are bounded. Therefore, the proposed tracking controller for the slow subsystem achieves good tracking performance while being robust to system uncertainties, external disturbances, and unmodeled dynamics. Flexible suppression controller The first link and the first flexible module and the second flexible module And the second link and the first flexible module and the second flexible module The amplitude curve is as follows Figure 4 and Figure 5 As shown, it can be observed that the amplitude of the flexible module remains within the boundary range. From Figure 6 , Figure 7 and Figure 8 It can be seen that the third adaptive parameter and flexible suppression controller The first, second, third, and fourth components are bounded. Therefore, the flexible suppression controller designed for the tachyon system keeps the tachyon system stable within a fixed time.

[0263] The following are the known prior art techniques mentioned in this invention:

[0264] 1. Actual fixed-time stability theory:

[0265] Definition 1: Consider the following nonlinear system:

[0266] (14)

[0267] If it exists and And stable time Constrained by constants, i.e., there exists a non-negative constant. , making ,in If it is an upper bound of the steady time, then it has initial conditions. ( The solution of (14) for a nonlinear system (compact set containing the origin). It is known as semi-globally consistent fixed-time convergence.

[0268] Lemma 2: For a nonlinear system (14), if there exists a constant and and positive numbers , and , making

[0269]

[0270] in It is the Lyapunov function of the nonlinear system (14). express of Power of 1 express of The power of this is true. Therefore, the trajectory of the nonlinear system (14) is actually fixed-time stable, and the stable time is... Can be described as

[0271]

[0272] in Residual set Represented as

[0273]

[0274] 2. Radial Basis Function Neural Network

[0275] Lemma 3 (General Approximation Theorem): For constants and definition in compact set Any continuous function on A radial basis function neural network can be constructed. , making

[0276]

[0277] in Optimal weight vector It is a radial basis function vector. It is the approximation error.

[0278] 3. Scaling Inequality

[0279] Lemma 4: For any real variable and The following formula is true

[0280]

[0281] in , and are positive real numbers and .

[0282] Lemma 5: For any real variable Sum of positive real numbers The following inequalities hold.

[0283]

[0284] Lemma 6: For any nonnegative variable , For positive integers, the following formula holds true.

[0285]

[0286] Lemma 7: For any variable and ,have

[0287]

[0288] in , and It is a positive real number.

[0289] Lemma 8: For any real number Nonnegative variables The following formula is true

[0290]

[0291] Lemma 9: Consider sets ,in Let be a designable positive constant. For any variable . ,have .

Claims

1. A fixed-time adaptive neural network composite control method for a flexible linkage robotic arm, characterized by: Includes the following steps: S1: Confirm Model of a flexible linkage robotic arm with multiple degrees of freedom. The value is a positive integer. Based on the singular perturbation theory, the model is decomposed into a slow subsystem and a fast subsystem. During the decomposition process, the state variables of the slow subsystem and the state variables of the fast subsystem are defined. The slow subsystem includes unmodeled dynamics. At the same time, the control torque in the model is decomposed into a tracking controller used to control the slow subsystem and a flexible suppression controller used to control the fast subsystem. S2: For the slow subsystem, construct the slow subsystem error based on the slow subsystem state variables, the desired position of the robotic arm, and the first virtual controller obtained by the backstepping method. S3: For the slow subsystem, construct the first Lyapunov function, which includes dynamic signals for unmodeled dynamics; based on the first Lyapunov function, the slow subsystem error, and the first set of radial basis function vectors, use the backstepping method to obtain the first virtual controller and the first adaptive law; S4: For the slow subsystem, construct a second Lyapunov function. Based on the second Lyapunov function, the first virtual controller, the slow subsystem error, and the second set of radial basis function vectors, use the backstepping method to obtain the tracking controller and the second adaptive law. S5: For the fast subsystem, construct the fast subsystem error based on the fast subsystem state variables and the second virtual controller subsequently obtained by the backstepping method; S6: For the tachyon system, construct the third Lyapunov function, and based on the third Lyapunov function and the tachyon system error, use the backstepping method to obtain the second virtual controller; S7: For the tachyon system, a fourth Lyapunov function is constructed. Based on the fourth Lyapunov function, the second virtual controller, the tachyon system error, and the third set of radial basis function vectors, the flexible suppression controller and the third adaptive law are obtained using the backstepping method. S8: The slow subsystem is controlled by the tracking controller, and the fast subsystem is controlled by the flexible suppression controller; the first and second adaptive laws are used to assist the tracking controller in controlling the slow subsystem, and the third adaptive law is used to assist the flexible suppression controller in controlling the slow subsystem, thereby realizing the control of the robotic arm.

2. The fixed-time adaptive neural network composite control method for flexible linkage robotic arms according to claim 1, characterized in that: step In S1, the model is: （1）； Where, vector , It is the angular position vector of the robotic arm. , For the first The angular position vectors of each link. ; It is a flexible generalized coordinate vector. , For the first The first link A flexible module, , This represents the total number of flexible variables. Indicates the first The number of flexible variables in each link; The inertia matrix, It is the matrix of Coriolis force and centrifugal force; Represents the stiffness matrix. Indicates the input weighting matrix. To control the torque, The external disturbance torque is unknown, and ,in For unknown positive constants; , .

3. The fixed-time adaptive neural network composite control method for flexible linkage robotic arms according to claim 2, characterized in that: step In S1, based on singular perturbation theory, the model is decomposed into a function defined in the first time domain. The slow subsystem described in the second time domain is defined in the second time domain. The fast subsystem described in the text; the time-scale transformation form is as follows: ,in, For the first time domain Mid-time point, For the second time domain The time point in the middle, For small scale factors, ,in It is the stiffness matrix The smallest non-zero stiffness element in the matrix; the decomposition process is as follows: Equation (1) can be written in block form based on the rigid and flexible conditions: （2）； in, , , and for Write the submatrix after dividing the matrix into blocks. , , and for Write the submatrix after dividing the matrix into blocks. for Write it as a submatrix after dividing it into blocks; for rigid systems It is the identity matrix. ; definition: （3）； in , , and for Write the submatrix after the block matrix; From equations (2) and (3), we can derive: （4）； Define the proportional stiffness matrix and new variables Then equation (4) can be reformulated as: （5）； （6）； in For the tracking controller, The flexible suppression controller; When the model approaches a rigid system , Then we have: （7）； in, , , , , , , , , and respectively under rigid conditions , , , , , , , , and ; From equation (7) and the properties of rigid robotic arms We can obtain: （8）； Define the state variables of the slow subsystem. ,in, and , For the slow subsystem Corner positions For the slow subsystem angular velocity, For rigid conditions Due to the matrix Uncertainty, assumption ,in For rigid conditions , The first nominal matrix, the first modeling error It is bounded; The slow subsystem is obtained as follows: （9）； in It is an unknown function that satisfies the local Lipschitz condition. For unmodeled dynamics, It is an unknown continuous nonlinear function. It is the first total disturbance that includes modeling errors and external disturbance torques; Define the state variables of the fast subsystem as follows: ,in, ， ， For the slow subsystem The first link The amplitude of each flexible module For the slow subsystem The first link The vibration velocity of each flexible module; due to the matrix Uncertainty, assumption ,in The second nominal matrix, the second modeling error It is bounded; The fast subsystem is obtained as follows: （10）； in It is an unknown continuous nonlinear function. For rigid conditions , It is the second total disturbance that includes modeling error.

4. The fixed-time adaptive neural network composite control method for flexible linkage robotic arms according to claim 3, characterized in that: In step S2, the error of the slow subsystem is: （11）； in, To track errors, For speed error, It is the first virtual controller. This is the desired position of the robotic arm. , It is the first The expected time-varying trajectory of each link angular position.

5. The fixed-time adaptive neural network composite control method for flexible linkage robotic arms according to claim 4, characterized in that: In step S3, the first Lyapunov function is: （12）； in, and It is a positive number. It is the dynamic signal, , This represents the first estimation error and the first adaptive parameter. yes The estimated value; The first virtual controller is: （13）； The first adaptive law is: （14）； in, , , , , , Is it only the first Line number The column element is 1 and all other elements are 1. of Dimensional array, , This is the first set of weight vectors. It is the first set of radial basis function vectors. , , , , and It is a positive number.

6. The fixed-time adaptive neural network composite control method for flexible linkage robotic arms according to claim 5, characterized in that: In step S4, the second Lyapunov function is: （15）； in, It is a positive number. , This represents the second estimation error and the second adaptive parameter. yes The estimated value; The tracking controller is: （16）； The second adaptive law is: （17）； in, The tracking controller's first Each component , , , , , , This is the second set of weight vectors. It is the second set of radial basis function vectors. , , , , and It is a positive number.

7. The fixed-time adaptive neural network composite control method for flexible linkage robotic arms according to claim 6, characterized in that: In step S5, the error of the fast subsystem is: （18）； in, For oscillation error, For vibration velocity error, It is the second virtual controller.

8. The fixed-time adaptive neural network composite control method for flexible linkage robotic arms according to claim 7, characterized in that: step S6 In the text, the third Lyapunov function is: （19）； in, To constrain amplitude The designable function is represented as ,parameter , and It is a positive number; The second virtual controller is: （20）； in, , , and It is a positive number.

9. The fixed-time adaptive neural network composite control method for flexible linkage robotic arms according to claim 1, characterized in that: In step S7, the fourth Lyapunov function is: （21）； in, It is a positive number. , This represents the third estimation error and the third adaptive parameter. yes The estimated value; The flexible suppression controller is: （22）； The third adaptive law is: （23）； in, The first flexible suppression controller Each component , , , , , Is it only the first Line number The column element is 1 and all other elements are 1. of Dimensional array, , It is the third set of weight vectors. It is the third set of radial basis function vectors. , , , , and It is a positive number.

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