Flexible connecting rod mechanical arm fixed time adaptive neural network composite control method

Abstract

The invention discloses a flexible connecting rod mechanical arm fixed time adaptive neural network compound control method which comprises the following steps: decomposing a flexible connecting rod mechanical arm model into a slow subsystem and a fast subsystem based on a singular perturbation theory, and defining a slow subsystem state variable and a fast subsystem state variable in the decomposition process; and the slow subsystem comprises an unmodeled dynamic state, and meanwhile, the control moment in the model is decomposed into a tracking controller used for controlling the slow subsystem and a flexible suppression controller used for controlling the fast subsystem. According to the technical scheme, elastic vibration of the flexible mode is relieved, meanwhile, tracking of the time-varying expected position by the mechanical arm is achieved, and the mechanical arm is stable within the fixed time.

Description

TECHNICAL FIELD

[0001] The present application belongs to the technical field of mechanical arm control, and particularly relates to a flexible link mechanical arm fixed time adaptive neural network composite control method. BACKGROUND

[0002] The flexible link mechanical arm has many advantages such as light weight, high speed, flexibility and low energy consumption, and also has more complex dynamic characteristics such as nonlinearity, under-actuation and multi-freedom degree. For the control of the flexible link mechanical arm, the following three technical problems need to be solved: (1) The oscillation of flexible mode inevitably exists in the flexible link mechanical arm system, and how to avoid the oscillation of flexible mode in the process of high-precision trajectory tracking.

[0003] (2) Due to modeling errors, model simplification, external disturbances and measurement noise, etc., the system part of the dynamic characteristics will be lost, which is called unmodeled dynamics. When there are unmodeled dynamics, how to ensure the stability of the control system.

[0004] (3) Continuous operation is an essential ability that the mechanical arm needs to have, and how to realize the fixed time control of the control system.

[0005] However, in the prior art, for the control of the flexible link mechanical arm, only part of the above technical problems can be solved, and the control effect is not good, for example, documents 【1】-document 【4】.

[0006] Document 【1】: Boundary output constrained control for a flexible beam system with prescribed performance; Document 【2】“Uncalibrated visual servoing for a planar two link rigid-ffexible manipulator without joint space-velocity measurement”; Document 【3】“Singular perturbation-based adaptive integral sliding mode control for flexible joint robots”; Document 【4】“Command filter-based adaptive control of flexible-joint manipulator with input saturation and output constraints”.

[0007] Document 【1】 handles the oscillation of flexible modes in the system model by using the method based on differential equation modeling. This method is not suitable for application in actual manipulator system due to the high order of the system and the complex numerical characteristics.

[0008] Document 【2】 develops an uncalibrated visual servoing control scheme to solve the trajectory tracking and flexible vibration problems of planar double-link rigid-flexible manipulator; Document 【3】 constructs an adaptive integral sliding mode controller based on singular perturbation theory and two state observers to ensure high tracking performance of flexible joint manipulator. To ensure high precision trajectory tracking of the manipulator, these methods handle the oscillation control problem of flexible modes in the manipulator in an ideal environment. For actual manipulator system, due to modeling errors, model simplification, external disturbances and measurement noise, there will be unmodeled dynamics in the manipulator system. Therefore, when the system has unmodeled dynamics, the method will not achieve the expected control effect, and even lead to system instability.

[0009] Document 【4】 considers the oscillation of flexible modes in the system model and unmodeled dynamics, but is only applicable to asymptotic convergence control problem. When the system needs to achieve fixed time convergence, this method will no longer be applicable. SUMMARY

[0010] The present application provides a flexible link manipulator fixed time adaptive neural network composite control method to make up for the shortcomings of the prior art.

[0011] The present application is realized by the following technical solutions: A flexible link manipulator fixed time adaptive neural network composite control method comprises the following steps: S1: determining a flexible link manipulator model with degrees of freedom, is a positive integer; the model is decomposed into a slow subsystem and a fast subsystem based on singular perturbation theory, and slow subsystem state variables and fast subsystem state variables are defined in the decomposition process, and the slow subsystem includes 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; S2: for the slow subsystem, based on the slow subsystem state variables, the desired position of the manipulator and the first virtual controller obtained by backstepping method, a slow subsystem error is constructed; S3: for the slow subsystem, constructing a first Lyapunov function, the first Lyapunov function including a dynamic signal for unmodeled dynamics; based on the first Lyapunov function, a slow subsystem error and a first set of radial basis function vectors, obtaining the first virtual controller and a first adaptive law using backstepping method; S4: for the slow subsystem, constructing a second Lyapunov function, based on the second Lyapunov function, the first virtual controller, a slow subsystem error and a second set of radial basis function vectors, obtaining the tracking controller and a second adaptive law using backstepping method; S5: for the fast subsystem, constructing a fast subsystem error based on fast subsystem state variables and a second virtual controller subsequently obtained by backstepping method; S6: for the fast subsystem, constructing a third Lyapunov function, based on the third Lyapunov function and the fast subsystem error, obtaining the second virtual controller using backstepping method; S7: for the fast subsystem, constructing a fourth Lyapunov function, based on the fourth Lyapunov function, the second virtual controller, a fast subsystem error and a third set of radial basis function vectors, obtaining the flexibility suppression controller and a third adaptive law using backstepping method; S8: controlling the slow subsystem by the tracking controller and controlling the fast subsystem by the flexibility suppression controller; the first and second adaptive laws are used to assist the tracking controller to control the slow subsystem, and the third adaptive law is used to assist the flexibility suppression controller to control the slow subsystem, so as to realize the control of the robot arm.

[0012] Further, in step S1, the model is: (1) ; wherein the vector , is an angular position vector of the robot arm, , is an angular position vector of the first link, ; is a flexible generalized coordinate vector, , is the first flexible module of the first link, , denotes the total number of flexible variables, denotes the number of flexible variables of the first link; is an inertia matrix, is a Coriolis force and centrifugal moment matrix; denotes a stiffness matrix, denotes an input weighting matrix, is a control torque, is an unknown external disturbance torque, and wherein is an unknown constant; , .

[0013] Further, in step S1, based on singular perturbation theory, the model is decomposed into the slow subsystem defined in the first time domain and the fast subsystem defined in the second time domain ; a time scale transformation is constructed as wherein is a time point in the first time domain , is a time point in the second time domain , is a small scale factor, wherein is the smallest non-zero stiffness element in the stiffness matrix ; the decomposition process is as follows: Equation (1) is written in a block form according to the rigid condition and the flexible condition: (2); wherein , , and are submatrices after is written in a block matrix, , , and are submatrices after is written in a block matrix, is a submatrix after is written in a block matrix; for a rigid system is a unit matrix, ; Definition: (3); wherein , , and are submatrices after is written in a block matrix; From equation (2) and equation (3), we have: (4); Define the proportional stiffness matrix and the new variable Then equation (4) can be rewritten as: (5); (6); where is the tracking controller, is the flexible suppression controller; when the model approaches a rigid system, , then there is: (7); where , , , , , , , , and are the rigid condition , , , , , , , , and , respectively; from equation (7) and the rigid robot property , we have: (8); define the slow subsystem state variable where and , is the th angular position of the slow subsystem, is the th angular velocity of the slow subsystem, is the rigid condition ; due to the uncertainty of the matrix , assume where is the rigid condition , is the first nominal matrix, and the first modeling error is bounded; get the slow subsystem: (9); where is an unknown function satisfying a 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.

[0014] Further, 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.

[0015] Furthermore, 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 an estimate of the first virtual controller is (13); the first adaptive law is (14); wherein , , , , , is a row column matrix with 1 in the , is a first set of weight vectors, is a first set of radial basis function vectors, , , , , and are normal numbers.

[0016] Further, in step S4, the second Lyapunov function is (15); wherein is a normal number, , denotes a second estimation error, and a second adaptive parameter is an estimate of an estimate of the tracking controller is (16); the second adaptive law is (17); wherein , the component of the tracking controller , , , , , , is a second set of weight vectors, is a second set of radial basis function vectors, , , , , and is a positive constant.

[0017] Further, in step S5, the fast subsystem error is: (18); wherein, is the oscillation error, is the velocity error, is the second virtual controller.

[0018] Further, in step S6, the third Lyapunov function is: (19); wherein, is a designable function used to constrain the amplitude of the oscillation error, is a positive constant, , and are positive constants; the second virtual controller is: (20); wherein, , , and are positive constants.

[0019] Further, in step S7, the fourth Lyapunov function is: (21); wherein, is a positive constant, , denotes a third estimation error, and a third adaptive parameter is an estimated value of ; the flexible suppression controller is: (22); the third adaptive law is: (23); wherein, , a first component of the flexible suppression controller, , , , , , , is a first row and a first column of the matrix , Column elements are 1 and the rest of the elements are of a 3x3 matrix, , is a third set of weight vectors, is a third set of radial basis function vectors, , , , , and are normal numbers.

[0020] The beneficial effects that can be achieved by the present application are: (1) Compared with document 【1】, the present application decomposes the model into slow subsystem and fast subsystem based on singular perturbation theory, reduces the order of the system, and reduces the complexity of the system.

[0021] (2) Compared with document 【2】 and document 【3】, the present application considers the influence of elastic vibration of flexible mode on the robot arm, and considers the unmodeled dynamics, enhances the anti-interference ability of the robot arm as a controlled object, and improves the robustness of the robot arm.

[0022] (3) Compared with document 【4】, the present application successfully realizes the fixed time control of the robot arm while effectively reducing the elastic vibration of the flexible mode and solving the unmodeled dynamics in the system.

[0023] (4) The present application can realize the tracking of the robot arm to the time-varying desired position, and the tracking error of the angular position of the robot arm reaches the fixed time stability.

[0024] (5) The robot arm compound control method proposed in the present application uses adaptive control technology, which has universal adaptive ability to the uncertainty in the model, and improves the robustness of the designed control method.

[0025] (6) The present application considers the unmodeled dynamics that must exist in the system, so that the controller is more in line with the actual situation. BRIEF DESCRIPTION OF DRAWINGS

[0026] Figure 1 is the tracking curve of the system angular position and the desired time-varying trajectory of the tracking controller in the embodiment of the present application.

[0027] Figure 2 is the first and second adaptive parameter curve of the tracking controller in the embodiment of the present application.

[0028] Figure 3 is the first and second component curve of the tracking controller in the embodiment of the present application.

[0029] Figure 4 is the first link of the flexible suppression controller of the first and second flexible module amplitudes.

[0030] Figure 5 is the second link of the flexible suppression controller of the first and second flexible module amplitudes.

[0031] Figure 6 is the third adaptive parameter curve of the flexible suppression controller of the first and second component curves.

[0032] Figure 7 is the first and second component curves of the flexible suppression controller of the third and fourth component curves.

[0033] Figure 8 is the third and fourth component curves of the flexible suppression controller of the first and second component curves. DETAILED DESCRIPTION

[0034] A fixed-time adaptive neural network composite control method is designed for a class of flexible link manipulators with unmodeled dynamics ( , representing a set of positive integers) degrees of freedom, including the following steps: S1, determining a model of a flexible link manipulator with degrees of freedom, decomposing the model into a slow subsystem and a fast subsystem based on singular perturbation theory, and defining slow subsystem state variables and fast subsystem state variables in the decomposition process, and the slow subsystem includes unmodeled dynamics, and at the same time, according to the model decomposition, 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 which are to be designed subsequently.

[0035] A class of flexible link manipulator models with degrees of freedom is as follows: (1) wherein the vector contains the angle position vector of the manipulator and the flexible generalized coordinate vector , is the angle position vector of the th link, , is the th flexible module of the th link, , , N denotes the total number of flexible variables, N denotes the number of flexible variables of the th link; I is the inertia matrix, C is the Coriolis and centrifugal force matrix. K and denote the stiffness matrix and the input weighting matrix, respectively. and are the control torque in the model and the disturbance torque of the unknown time-varying external environment, respectively. is the th derivative of is the th derivative of and are the angular velocity and angular acceleration of the manipulator, respectively, and are the th derivative and the th derivative of

[0036] Property 1: K is a nonsingular positive definite symmetric matrix.

[0037] Property 2: C is a skew-symmetric matrix.

[0038] Assumption 1: The external disturbance torque satisfies the condition where is an unknown positive constant.

[0039] According to the singular perturbation theory, the model (1) can be decomposed into a slow subsystem defined in the first time domain and a fast subsystem defined in the second time domain , and the time-scale transformation form is where is the time point in the first time domain and is the time point in the second time domain , is a small-scale factor, where is the smallest non-zero stiffness element in the stiffness matrix . The decomposition process is as follows: Equation (1) is written in block form according to the rigidity condition and the flexibility condition as follows: (2) where , , and are unknown unknown sub-matrices after writing into block matrix, , , and are unknown unknown sub-matrices after writing into block matrix, is the matrix sub-matrices after writing into block matrix. For rigid system is the identity matrix, .

[0040] Definition: (3) where , , and is the matrix unknown sub-matrices after writing into block matrix.

[0041] Therefore, from equation (2) and equation (3), we have: (4) Definition of proportional stiffness matrix and new variable , equation (4) can be rewritten as: (5) After that, the variables and matrices marked with subscript "s" belong to slow subsystem, and the variables and matrices marked with subscript "f" belong to fast subsystem. In order to achieve the dual control objectives, the control torque in the model is decomposed as: (6) where denotes the tracking controller to control the slow subsystem, denotes the flexibility suppression controller to control the fast subsystem. When equation (1) approaches a rigid system, the minimum stiffness element . Therefore, when

[0042] , we have: (7) where, , , , , , , , , , , and respectively under rigid conditions , , , , , , , , and .

[0043] From equation (7) and the properties of rigid robotic arms We can obtain: (8) 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: (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.

[0044] 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 is the second nominal matrix and the second modeling error is bounded. From (5) and (7), the fast subsystem is obtained as follows: (10) where is an unknown continuous nonlinear function, is the , second total disturbance including the modeling error.

[0045] Assumption 2: The desired position of the manipulator and its up to time derivatives with respect to time are known, smooth and bounded, where is the desired time-varying trajectory of the link position.

[0046] S2, for the slow subsystem, the slow subsystem error is constructed based on the slow subsystem state variables, the desired position of the manipulator and the first virtual controller which will be constructed later by backstepping method: (11) where, is the tracking error, is the velocity error, is the first virtual controller and will be constructed in the following design steps.

[0047] S3, for the slow subsystem including the unmodeled dynamics, the first Lyapunov function is constructed, meanwhile, in order to solve the problem that the unmodeled dynamics make the slow subsystem difficult to be controlled, the dynamic signal is introduced in the first Lyapunov function, based on the constructed first Lyapunov function, the slow subsystem error and the neural network basis functions in the well-known radial basis function neural network, the first virtual controller and the first adaptive law are designed by using the backstepping method.

[0048] S301, for the slow subsystem, the first Lyapunov function is constructed, meanwhile, in order to solve the problem that the unmodeled dynamics make the slow subsystem difficult to be controlled, the dynamic signal is introduced in the first Lyapunov function: The first Lyapunov function is constructed as follows: (12) where, and are positive constants, is the dynamic signal, , denotes the first estimation error, the first adaptive parameter is the estimate of . The representation will be given in subsequent design steps.

[0049] Dynamic signals It is necessary to satisfy Lemma 1 on the premise that Assumption 3 holds.

[0050] Assumption 3: In the slow subsystem The fact that the input state is actually stable indicates the existence of a Lyapunov function. , so that: (13) in, , and yes Class function, and It is a positive number.

[0051] 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 : (14) Make and (15) 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. Using equations (9) and (11), for Seeking information about The derivative can be obtained (16) Non-negative continuous functions in dynamic signals After using polynomial fitting Relationship: ,in It is a non-negative smooth function. is a bounded fitting error, i.e. there exists an unknown constant such that . Therefore, we have (17) where , , , and are positive constants.

[0052] In the prior art considering the influence of unmodeled dynamics on the robot arm, part of the prior art does not introduce the dynamic signal in the first Lyapunov function, and based on the Lyapunov stability theory, the stability of the dynamic signal cannot be explained, and the boundedness of the unmodeled dynamics cannot be quantified. Another part puts the non-negative continuous function into a certain joint to deal with its unknownness, which leads to direct interference of the data of the remaining joints when estimating the unknownness of the joint by using the neural network, and it is difficult to accurately estimate. However, the processing method of the non-negative continuous function in the present technology not only makes up for the defects of the prior art, but also associates it with non-negative smooth functions, so that it meets the requirement of the robot arm for the number of degrees of freedom, disperses the unknownness to each joint of the robot arm, and makes the estimation of the unknownness by the neural network more accurate.

[0053] Based on the known lemma 3 in the prior art, the neural network is used to estimate so as to satisfy the following equation: (18) where is the first set of weight vectors, is the first set of radial basis function vectors, and the estimation error satisfies , is an unknown positive constant.

[0054] Subsequently, by using the known lemma 7 in the prior art, we have (19) where is a positive constant, , is a dimensional matrix whose only element in the row and the column is 1 and the rest of the elements are .

[0055] Substituting equation (19) into equation (17) has (20) Based on equation (20), the first virtual controller is constructed and the first adaptive law is (21) (22) where , , , , , , , , and are positive constants.

[0056] The first Lyapunov function is constructed, and by derivation, based on Lemma 5 and Lemma 6, it can be obtained that (23) where , and , .

[0057] S4, for the slow subsystem, a second Lyapunov function is constructed, based on the constructed second Lyapunov function, the designed first virtual controller, the slow subsystem error and the neural network basis function in the known radial basis function neural network, the tracking controller and the second adaptive law are designed by using the backstepping method; the tracking controller controls the slow subsystem with the aid of the designed first and second adaptive laws.

[0058] S401, for the slow subsystem, a second Lyapunov function is constructed: (24) where is a positive constant, , denotes the second estimation error, and the second adaptive parameter is an estimation value of , denotes will be given in the subsequent design steps.

[0059] S402, based on the constructed second Lyapunov function, the designed first virtual controller, the slow subsystem error and the neural network basis function in the known radial basis function neural network, the tracking controller and the second adaptive law are designed by using the backstepping method; by using equation (9) and equation (11), the derivative of with respect to the upper bound of the disturbance (25) where and .

[0060] Note 1: According to the assumption 1, the external disturbance is bounded. In addition, due to the boundedness of the modeling error , the limits of the manipulator and the constraints of the task space, it can be assumed that the state-dependent disturbance is bounded, whose bound is an unknown positive constant .

[0061] Therefore, by using the well-known Lemma 4 in the prior art, the following inequality holds: (26) Substituting equation (26) into equation (25), we have (27) where , and .

[0062] Based on the well-known Lemma 3 in the prior art, the neural network is used to estimate so that it satisfies the following equation: (28) where is the second set of weight vectors, is the second set of radial basis function vectors, the estimation error , is an unknown positive constant.

[0063] Subsequently, by using the well-known Lemma 7 in the prior art, we have (29) where is a positive constant, .

[0064] Substituting equation (29) into equation (27) gives (30) Based on equation (30), the tracking controller of the slow subsystem and the second adaptive law are constructed as (31) (32) where, , the first component of the tracking controller , , , , , , , , and are positive constants.

[0065] S403, the tracking controller controls the slow subsystem with the designed first and second adaptive laws.

[0066] In this example, the second Lyapunov function is constructed, and by derivation, based on the known prior art lemmas 5 to 9, it can be obtained that it satisfies: (33) The parameters of which are shown as follows, and it can be seen that it satisfies the condition of lemma 2 in the known practical fixed-time stability theory, that is, on the basis of assumptions 1-3, the proposed control scheme has a tracking controller, a first adaptive law and a second adaptive law, which ensures that the dynamic signal and the slow subsystem state variable are practically fixed-time stable in the first time domain , based on the fact that the dynamic signal satisfies equation (15), it can be obtained that the unmodeled dynamics is practically fixed-time stable in the first time domain , and further, the slow subsystem is practically fixed-time stable in the first time domain , that is, all signals in the slow subsystem are bounded in a fixed time and can accurately track the desired position of the robot arm .

[0067] , , , , , , for , it is divided into and , where and are positive integers satisfying , the unknown positive constant is the upper bound of , , , is an unknown positive constant such that ​, .

[0068] S5, for the fast subsystem, construct the fast subsystem error based on the fast subsystem state variables and the second virtual controller which will be constructed in the following step by backstepping method: (34) where, is the oscillation error, is the velocity error, is the second virtual controller and will be constructed in the following design step.

[0069] S6, for the fast subsystem, construct the third Lyapunov function, and based on the constructed third Lyapunov function and the fast subsystem error, design the second virtual controller by backstepping method.

[0070] S601, for the fast subsystem, construct the third Lyapunov function: (35) where, is a designable function used to constrain the amplitude , denoted as , parameters , and are normal numbers.

[0071] S602, for the fast subsystem, based on the constructed third Lyapunov function and the fast subsystem error, design the second virtual controller by backstepping method according to the actual fixed-time stability theory: By using equation (10) and equation (34), the derivative of with respect to can be obtained as: (36) Based on equation (36), the second virtual controller is constructed as: (37) where , and and are normal numbers.

[0072] Note 2: In order to avoid the singularity problem of when , in the case of , replace with a small enough normal number .

[0073] Substituting equation (37) into equation (36) gives (38) S7, for the fast subsystem, constructing 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 function in the known radial basis function neural network, using backstepping method, designing a flexible suppression controller and a third adaptive law; the flexible suppression controller controls the slow subsystem with the designed third adaptive law as auxiliary.

[0074] S701, for the fast subsystem, constructing a fourth Lyapunov function: (39) wherein, is a normal number, denotes a third estimation error, and a third adaptive parameter is an estimated value of , and denotes will be given in the subsequent design steps.

[0075] S702, for the fast subsystem, based on the constructed fourth Lyapunov function, the designed second virtual controller, the fast subsystem error and the neural network basis function in the known radial basis function neural network, using backstepping method, designing a flexible suppression controller and a third adaptive law: Using equation (10) and equation (34), the derivative of with respect to can be obtained as (40) wherein and .

[0076] Note 3: due to the boundedness of modeling error , the limitations of the robot arm and the constraints of the task space, it can be assumed that the state-dependent disturbance is bounded, and its bound is an unknown normal number .

[0077] Therefore, using the known existing technology Lemma 4, the following inequality holds: (41) Substituting equation (41) into equation (40) gives: (42) wherein, and .

[0078] Based on Lemma 3 in the known prior art, the neural network is estimated to satisfy the following equation: (43) wherein, is a third set of weight vectors, is a third set of radial basis function vectors, and the estimation error , is an unknown constant.

[0079] Subsequently, based on Lemma 7, it can be obtained that (44) wherein, is a constant and , is a dimensional matrix with only the element in the first row and the first column being 1 and the rest elements being . Substituting equation (44) into equation (42) has:

[0080] (45) Based on equation (45), a flexible suppression controller for controlling the fast subsystem is constructed and a third adaptive law is: (46) (47) wherein, is the first component of the flexible suppression controller , , , , , , , , , , and are constants.

[0081] S703, the flexible suppression controller controls the slow subsystem with the designed third adaptive law.

[0082] In this example, a fourth Lyapunov function is constructed, and after derivation, based on Lemmas 5 to 8 in the known prior art, it can be obtained that it satisfies: (48) ​The parameters of which are shown below, it can be seen that it satisfies the conditions of Lemma 2 in the known actual fixed-time stability theory, that is, the proposed control scheme has a flexible suppression controller (46), a third adaptive law (47), which ensures that the fast subsystem (10) is actually fixed-time stable in the second time domain The upper is actually fixed-time stable, that is, all signals in the fast subsystem are bounded within a fixed time.

[0083] , , , , , , , .

[0084] S8: Control the slow subsystem through the tracking controller and control the fast subsystem through the flexible suppression controller; the first and second adaptive laws are used to assist the tracking controller to control the slow subsystem, and the third adaptive law is used to assist the flexible suppression controller to control the slow subsystem, so as to realize the control of the manipulator. Achieve the tracking of the desired time-varying trajectory while effectively processing the oscillation of the flexible mode in the manipulator, and realize the actual fixed-time stability of the manipulator.

[0085] In order to verify the effectiveness of the designed controller, we carry out simulation verification through matlab. Take a flexible link manipulator with degrees of freedom as an example, the parameters of formula (1) are: the number of flexible variables of each link is , the length of each link is , the mass of each link is , the tip mass is , the constant bending stiffness is . The external disturbance torque is . The desired position of the manipulator is specified as , where .

[0086] For the slow subsystem, the unmodeled dynamics are , where and . The dynamic signal is , where . The initial condition is . The parameters are , , , , , , , .

[0087] For the fast subsystem, the initial condition is and . The parameters are , , , , , , , , .

[0088] Figure 1 The system angle position and the desired time-varying trajectory are shown. 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 the first and second components of the tracking controller are bounded. Therefore, the tracking controller proposed for the slow subsystem achieves good tracking performance while being robust to system uncertainties, external disturbances, and unmodeled dynamics. The amplitude curves of the first flexible module and the second flexible module of the first link and the first flexible module and the second flexible module of the second link under the flexible suppression controller are shown in Figure 4 and Figure 5 . It can be observed that the amplitudes of the flexible modules remain within the boundary range. From Figure 6 , Figure 7 and Figure 8 , it can be seen that the third adaptive parameter and the first, second, third, and fourth components of the flexible suppression controller are bounded. Therefore, the flexible suppression controller designed for the fast subsystem keeps the fast subsystem stable within a fixed time.

[0089] The following are the known prior art mentioned in the present invention: 1. Practical fixed-time stability theory: Definition 1: Consider a nonlinear system as follows: (14) If there exist and , and the stable time is limited by a constant, i.e., there exists a non-negative constant such that , where is the upper bound of the stable time, then the nonlinear system with initial condition ( The solution of (14) for a nonlinear system (compact set containing the origin). It is known as semi-globally consistent fixed-time convergence.

[0090] Lemma 2: For a nonlinear system (14), if there exists a constant and and positive numbers , and , making

[0091] 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

[0092] in Residual set Represented as

[0093] 2. Radial Basis Function Neural Network 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

[0094] in Optimal weight vector It is a radial basis function vector. It is the approximation error.

[0095] 3. Scaling Inequality Lemma 4: For any real variable and The following formula is true

[0096] in , and are positive real numbers and .

[0097] Lemma 5: For any real variable and positive real numbers , the following inequality holds

[0098] Lemma 6: For any non-negative variable , is a positive integer, the following holds

[0099] Lemma 7: For any variables and , we have

[0100] where , and are positive real numbers.

[0101] Lemma 8: For any real numbers and non-negative variable , the following holds

[0102] Lemma 9: Consider the set where is a designable normal number. For any variable , we have .

Claims

1. A flexible link robot fixed-time adaptive neural network composite control method, characterized by: The method comprises the following steps: S1: determine A model of a flexible link robot with degrees of freedom, is a positive integer; based on the singular perturbation theory, the model is decomposed into a slow subsystem and a fast subsystem, and the slow subsystem state variable and the fast subsystem state variable are defined in the decomposition process, and the slow subsystem includes unmodeled dynamics, and the control torque in the model is decomposed into a tracking controller for controlling the slow subsystem and a flexibility suppression controller for controlling the fast subsystem; S2: for the slow subsystem, a slow subsystem error is constructed based on a slow subsystem state variable, a desired position of the robot arm, and a first virtual controller obtained by a backstepping method subsequently; S3: for the slow subsystem, a first Lyapunov function is constructed, the first Lyapunov function comprising a dynamic signal for unmodeled dynamics; based on the first Lyapunov function, the slow subsystem error, and a first group of radial basis function vectors, the first virtual controller and a first adaptive law are obtained by the backstepping method; S4: for the slow subsystem, a second Lyapunov function is constructed, based on the second Lyapunov function, the first virtual controller, the slow subsystem error, and a second group of radial basis function vectors, the tracking controller and a second adaptive law are obtained by the backstepping method; S5: for the fast subsystem, a fast subsystem error is constructed based on a fast subsystem state variable and a second virtual controller obtained by a backstepping method subsequently; S6: for the fast subsystem, a third Lyapunov function is constructed, based on the third Lyapunov function and the fast subsystem error, the second virtual controller is obtained by the backstepping method; S7: for the fast subsystem, a fourth Lyapunov function is constructed, based on the fourth Lyapunov function, the second virtual controller, the fast subsystem error, and a third group of radial basis function vectors, the flexible suppression controller and a third adaptive law are obtained by 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, so as to realize control of the robot arm.

2. The fixed-time adaptive neural network composite control method of flexible link manipulator 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 of flexible link manipulator 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, Small scale factor ,in It is the stiffness matrix The smallest non-zero stiffness element in the matrix; the decomposition process is as follows: Equation (1) is written in a block form according to a rigid condition and a flexible condition: （2）； wherein , , and are submatrices after writing the matrix in block form, , , and are submatrices after writing the matrix in block form, is a submatrix after writing the matrix in block form; for rigid systems is the identity matrix, ; It is defined that: （3）； wherein , , and are submatrices after writing the matrix in block form Equations (2) and (3) are obtained: （4）； defining the proportional stiffness matrix and new variables Equation (4) can then be re-written as: （5）； （6）； wherein is the tracking controller, is the flexible suppression controller; When the model approaches a rigid system, , then there is: （7）； wherein , , , , , , , , and are the rigid-body conditions , , , , , , , , and ; From equation (7) and the rigid manipulator property we have: （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: （9）； wherein is an unknown function satisfying a local Lipschitz condition, is an unmodeled dynamics, is an unknown continuous nonlinear function, is a first total disturbance including modeling errors and external disturbance torques; defining said fast subsystem state variables as wherein, ， ， 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: （10）； wherein is an unknown continuous nonlinear function, is the , is a second total disturbance including modeling errors.

4. The flexible manipulator fixed-time adaptive neural network complex control method according to claim 3, characterized in that: In step S2, the slow subsystem error is: （11）； wherein, is a tracking error, is a velocity error, is a first virtual controller, is a desired position of the robot arm, , is a desired time-varying trajectory of a first link angle position.

5. The flexible manipulator fixed-time adaptive neural network complex control method of claim 4, wherein: In step S3, the first Lyapunov function is: （12）； wherein and are normal numbers, is the dynamic signal, , denotes a first estimation error, a first adaptive parameter is an estimated value of The first virtual controller is: （13）； The first adaptive law is: （14）； wherein , , , , , is an by matrix with the element in the first row and the first column being one and all other elements being , , , is a first set of weight vectors, is the first set of radial basis function vectors, , , , , and are normal numbers.

6. The flexible manipulator fixed-time adaptive neural network complex control method of claim 5, wherein: In step S4, the second Lyapunov function is: （15）； wherein is a normal number, , denotes a second estimation error, a second adaptive parameter is an estimated value of The tracking controller is: （16）； The second adaptive law is: （17）； wherein , the first component , , , , , , is a second set of weight vectors, is the second set of radial basis function vectors, , , , , and are normal numbers.

7. The flexible manipulator fixed-time adaptive neural network complex control method according to claim 6, characterized in that: In step S5, the fast subsystem error is: （18）； wherein, is the oscillation error, is the oscillation velocity error, is the second virtual controller.

8. The fixed-time adaptive neural network compound control method of the flexible link robot arm according to claim 7, characterized in that step S6 In the third Lyapunov function, the second virtual controller is: （19）； wherein, is a designable function to constrain the amplitude of the signal x(t) is represented as with parameters , and are normal numbers; In step S7, the fourth Lyapunov function is: （20）； wherein , , and are normal numbers.

9. The fixed-time adaptive neural network composite control method of the flexible link robot arm according to claim 1, characterized in that: The flexible suppression controller is: （21）； wherein is a normal number, , denotes a third estimation error, a third adaptive parameter is an estimated value of The third adaptive law is: （22）； ​ （23）； wherein , the first component , , , , , is a row column element is 1 and the rest are , dimensional matrix, , is a third set of weight vectors, is the third set of radial basis function vectors, , , , , and are normal numbers.

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