A neural network-based adaptive variable impedance control method for a flexible robot arm

Abstract

The application discloses a flexible mechanical arm adaptive variable impedance control method based on a neural network, relates to the technical field of flexible joint mechanical arm control, and can estimate and compensate model uncertainty by using the neural network, and can keep compliance when interacting with people or objects, so that the safety of people and equipment is ensured. In order to achieve the above object, the technical scheme of the application comprises the following steps: for a four-order nominal model of a two-degree-of-freedom flexible mechanical arm, the four-order nominal model is decomposed into two two-order subsystems by using singular perturbation. The model uncertain part including friction in the neural network estimation system is designed, and the neural network is trained by using an improved optimal bounded ellipse algorithm. For unknown high-order angular velocities, an observer is designed by using the training results of the neural network to obtain the observation values of the high-order angular velocities. A variable stiffness impedance control controller is designed for a slow subsystem, wherein the stiffness is adaptively adjusted according to the contact force. A sliding mode controller is designed for a fast subsystem, and finally, the overall controller design is given.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of flexible joint robot control, and particularly relates to a flexible robot adaptive variable impedance control method based on a neural network. BACKGROUND

[0002] Robots have been widely applied to manufacturing assembly and other scenarios, and many robots are also applied to daily life. Interaction between robots and people is inevitable, and it is very important to realize safe interaction between robots and people. A flexible joint robot is generally driven by a motor and an elastic element, and the joint movement is driven by the elastic element. The flexible joint robot has inherent flexibility in structural design, and is very suitable for application to collaborative robots and other products that frequently interact with people.

[0003] However, compared with a traditional rigid robot, the existence of the flexible joint increases the complexity of the system and increases the difficulty of designing a controller. Meanwhile, the flexible joint robot has strong nonlinearity, and the system parameters often change during movement and grasping, further increasing the complexity of controller design. A commonly used control method such as PID control is prone to produce severe chattering in a highly nonlinear and strongly coupled system such as the flexible joint robot, affecting the control performance of the robot. Therefore, how to design a suitable control strategy for the flexible joint robot with unknown nonlinearity and variable system parameters to achieve good control effect of the robot under the condition of only outputting measurable values is a problem to be solved in the prior art. In addition, in human-robot collaboration and other scenarios, the robot needs to interact with people or the environment. In order to ensure the safety of the interaction process, an impedance control method is often used. However, the traditional fixed stiffness impedance control has fixed controller parameters, and in actual application, either the reference trajectory can be quickly followed but the flexibility is poor, or the flexibility can be well performed but the performance of following the reference trajectory is poor.

[0004] Therefore, designing an impedance control method that can adaptively adjust the stiffness, and then enabling the robot to follow the reference trajectory faster when there is no external force contact and maintaining better flexibility when there is external force contact, so that the robot can better adapt to different working environments compared with the traditional fixed impedance control, is another technical problem to be solved in the prior art. SUMMARY

[0005] Therefore, the present application provides a flexible robot adaptive variable impedance control method based on a neural network, which can adaptively adjust the stiffness, and then enable the robot to follow the reference trajectory faster when there is no external force contact and maintain better flexibility when there is external force contact, so that the robot can better adapt to different working environments compared with the traditional fixed impedance control.

[0006] To achieve the above object, the technical scheme of the application is a flexible robot arm adaptive variable impedance control method based on neural network, comprising the following steps:

[0007] S1: for the four-order nominal model of the two-degree-of-freedom flexible joint robot arm, singular perturbation method is used to decompose it into two two-order subsystems of fast subsystem and slow subsystem for simplifying the controller design.

[0008] S2: the friction in the neural network estimation system and the unmodeled part of the model are designed, and the neural network is trained by using the improved optimal bounded ellipse algorithm.

[0009] S3: for the unknown system state, an observer is designed based on the training result of the neural network to observe it.

[0010] S4: for the slow subsystem, a variable stiffness stiffness controller is designed to change the stiffness.

[0011] S5: for the fast subsystem, a sliding mode controller is designed, and the controllers of the two subsystems are combined according to the singular perturbation method to give the overall controller design.

[0012] Further, in S1, for the four-order nominal model of the two-degree-of-freedom flexible joint robot arm, singular perturbation method is used to decompose it into two two-order subsystems of fast subsystem and slow subsystem for simplifying the controller design, specifically:

[0013] The flexible joint robot arm model is considered as shown below

[0014]

[0015] wherein, is the joint angle, is the driving motor angle, is the inertia matrix, is the Coriolis force and centrifugal force, is the gravity, is the external force torque, is the system friction, is the unmodeled dynamics of the system, is the control input, is the transmission ratio matrix of the motor and is a diagonal matrix, is the rotational inertia matrix of the motor rotor and is a diagonal matrix, is the elastic coefficient matrix of the joint elastic element and is a diagonal matrix; is a real set, and the dimension of the upper mark is.

[0016] The singular perturbation method is specifically: first, the system model is rewritten into the state space form, and the following system state is taken:

[0017]

[0018] where x1, x2, y1, y2 are four system states, 0 < ε « 1 is a pre-set parameter, B0= ε 2 K; meanwhile, the un-modeled dynamics of the system is combined with the friction and the non-linear terms of the system to form a lumped uncertainty The state space form is further:

[0019]

[0020] where a1= D -1 B0, a2= (J -1 N -2 + D -1 )B0, b1= J -1 N -1 .

[0021] Since 0 < ε « 1, the change speed of y1, y2 is faster than that of x1, x2 in the system, and the system states have two different time scales. The original four-order system is decomposed into two two-order subsystems by using the singular perturbation method.

[0022] Let the new lumped uncertainty term F * = D -1 F, the uncertainty compensation term and u = u a + u b , u b is the tracking control term to be designed, where is the estimated value of F * , then (3) becomes (4)

[0023]

[0024] where

[0025] Let ε = 0, the equilibrium point of the system is obtained

[0026]

[0027] where u bs is the control component of the slow subsystem when ε = 0; replace y1 in (2) with , and the slow subsystem is obtained as shown in (6)

[0028]

[0029] where

[0030] Define a new state variable Substituting (2) into (1) gives (7):

[0031]

[0032] Definition of the extended time variable And let ε = 0 in (4), then the fast subsystem is:

[0033]

[0034] where u bf is the control component of the slow-fast system when ε = 0, and satisfies u bs + u bf = u b .

[0035] Further, S2: design a neural network to estimate the friction and the model unmodeled part in the system, and use the improved optimal bounded ellipsoid algorithm to train the neural network, which is:

[0036] Considering the lumped uncertainty term F * in the slow subsystem (5), a neural network is designed to estimate the lumped uncertainty term F * :

[0037] F * = W *T Φ(Z) + ∈ (9)

[0038] Φ(Z) = [φ(z1), φ(z2), …, φ(z N )] T (10)

[0039] where ∈ is the estimation error, W * is the optimal weight from the hidden layer to the output layer of the neural network, Z = [z1, z2, …, z N ] T = Vx, is the input of the neural network, is the weight from the input layer to the hidden layer of the neural network, N is the number of nodes in the intermediate layer of the neural network, and φ(z i ) is the activation function of the neural network.

[0040] The selected activation function is as follows:

[0041]

[0042] where β t is a design parameter.

[0043] Using (9), (6) is rewritten as:

[0044]

[0045] Further, in S2, the neural network is trained to obtain the optimal weight W by using the following method * , which is specifically as follows:

[0046] First, in order to prevent noise interference and because the angular velocity is unknown, a second-order filter is used to filter the left and right sides of equation (12), and then the following equation is obtained:

[0047]

[0048] where ω n is the second-order filter parameter, ζ is the damping ratio, s is the complex frequency, x 2f is the filtered value of x2, is the filtered value of , and C 1f is the filtered value of C1, Φ f is the filtered value of Φ, and ∈ f is the filtered value of ∈.

[0049] After filtering the left and right sides of equation (9), the following equation is obtained:

[0050]

[0051] where is the filtered value of F f , is the ith column of , and ∈ fi is the ith column of ∈ f .

[0052] Consider the following identification model:

[0053]

[0054] where is the estimated value corresponding to .

[0055] Define the identification error as:

[0056]

[0057] The neural network is trained by using the improved optimal bounded ellipse method, and the specific expression is as follows:

[0058]

[0059] where

[0060]

[0061] g i = g i0 r i r pi (20)

[0062]

[0063] where is a gain coefficient satisfying i > 1; is an upper bound of the neural network estimation error; is a symmetric positive definite matrix, is P i to the lower bound p li I; is a fixed forgetting factor satisfying 0 < g i0 < 1; and are the lower and upper bounds of ||P i ||, respectively, satisfying p li < p ui , Q = Φ(Vx), Q f = Φ(Vx f ); the projection operator is given by

[0064]

[0065] where is a parameter, r pi is given by

[0066]

[0067] Further, S3: for the unknown system state, an observer is designed based on the training results of the neural network to observe it, including the observer for the slow subsystem, specifically:

[0068] For the unknown state x2, the slow subsystem dynamic equation is as follows:

[0069]

[0070] where

[0071] The total disturbance ∈ x of the system is extended into a new state variable x3, let v(t) represent the differential of x3, then the system equation becomes:

[0072]

[0073] and the neural network observer is designed as follows

[0074]

[0075] where β0is the bandwidth of the observer, β1= 3β0, is the estimate of x1, is the state estimation error; using the neural network state observer (26), the estimates of the unknown system states x2and the unknown total disturbance x3are obtained This prepares the conditions for the following controller design.

[0076] Further, S3: for the unknown system states, design an observer based on the training results of the neural network to observe them, which also includes the observer for the fast subsystem, specifically:

[0077] For the unknown state y2, consider the fast subsystem dynamic equation of formula (8), and expand the total disturbance ∈ y of the system into a new state variable y3, ι(t) represents the differential of y3, and the system equation becomes

[0078]

[0079] Further, design the neural network observer as follows

[0080]

[0081] where β0is the bandwidth of the observer, β1= 3β0 is the estimate of z1, is the state estimation error; using the neural network observer (28), the estimates of the unknown system states y2and the total disturbance y3are obtained and This prepares the conditions for the following controller design.

[0082] Further, S4: for the slow subsystem, design an adaptive variable impedance control controller, specifically:

[0083] Consider the slow subsystem formula (12), and the reference trajectory x r , write the error dynamic equation of the slow subsystem as follows:

[0084]

[0085] where is the second-order derivative of x r ; select two positive real auxiliary variables λ v1 and λ v2 , which satisfy:

[0086]

[0087] where To satisfy the impedance relationship as follows:

[0088]

[0089] where e = x1-x r is the tracking error.

[0090] Using λ v1 The sliding surface s s is designed as follows:

[0091] s s = λ v1 e1+e2 (33)

[0092] Using equations (32) and (33), a variable impedance controller is designed for the slow subsystem, which is expressed as follows:

[0093]

[0094] where |τ ext | is the absolute value of τ ext ; β s and α s are constant matrices.

[0095] Further, S5: for the fast subsystem, a sliding mode controller is designed, and the controllers of the two subsystems are combined according to the singular perturbation method to give the overall controller design, which is specifically:

[0096] Considering the fast subsystem equation (8), the sliding surface s y is defined as follows:

[0097] s y = λ y z1+y2 (35)

[0098] where is a design parameter.

[0099] Further, the derivative of s y with respect to the time variable t ∈ is:

[0100]

[0101] Then the fast subsystem controller can be designed as:

[0102]

[0103] where and are controller parameters.

[0104] By using singular perturbation method, the final overall controller is designed as:

[0105] u = u a + u bs + u bf (38)

[0106] wherein

[0107] Advantages:

[0108] Advantages:

[0109] 1. The application provides a flexible mechanical arm adaptive variable impedance control method based on a neural network, which can train the neural network under the condition that part of the system state is unknown, realize real-time online estimation of unknown nonlinear system, and design a state observer for unknown high-order angular velocity based on the neural network training results, so that the observation value of the unknown quantity can be well obtained by using the designed observer, so as to facilitate the design of the controller. Compared with the traditional fixed stiffness impedance control, the controller designed by the application can make the mechanical arm follow the reference trajectory faster when there is no external force contact, and can keep better compliance characteristics when there is external force contact, so that the adaptability of the control method to different working conditions and working scenes can be improved. BRIEF DESCRIPTION OF DRAWINGS

[0110] Figure 1 It is a step flow chart of the flexible mechanical arm adaptive variable impedance control method based on a neural network.

[0111] Figure 2 It is a block diagram of the flexible mechanical arm adaptive variable impedance control method based on a neural network.

[0112] Figure 3 It is a control output and angle tracking curve graph of the adaptive variable impedance control method based on a neural network compared with other methods in the case of tracking a reference sinusoidal signal.

[0113] Figure 4 It is a tracking error curve graph of the adaptive variable impedance control method based on a neural network compared with other methods in the case of tracking a reference sinusoidal signal. DETAILED DESCRIPTION

[0114] The application will be described in detail below with reference to the accompanying drawings and embodiments.

[0115] The application provides a kind of

[0116] As Figure 1A step flow chart of a neural network-based adaptive variable impedance control method for a flexible robot arm is shown, and specifically includes:

[0117] S1: Singular perturbation method is used to decompose the nominal fourth-order model of the flexible joint robot arm. Consider the flexible joint robot arm model as shown below

[0118]

[0119] wherein, is the joint angle, is the motor angle, is the inertia matrix, is the Coriolis force and centrifugal force, is the gravity, is the external force torque, is the system friction, is the system unmodeled dynamics, is the control input. is the transmission ratio matrix of the motor and is a diagonal matrix, is the rotational inertia matrix of the motor rotor and is a diagonal matrix, is the elastic coefficient matrix of the joint elastic element and is a diagonal matrix.

[0120] Rewrite the system model into the state space form, take the following system state:

[0121]

[0122] wherein 0 < ε << 1 is a man-made design parameter, B0 = ε 2 K. At the same time, consider the system unmodeled dynamics and the friction force and system nonlinear terms to be combined to write the lumped uncertainty and the state space form is:

[0123]

[0124] wherein a1 = D -1 B0, a2 = (J -1 N -2 + D -1 )B0, b1 = J -1 N -1 .

[0125] Because 0 < ε << 1, the change speed of y1, y2 in the system will be much faster than that of x1, x2, the system state has two different time scales, so the singular perturbation method can be used to decompose the original fourth-order system into two second-order subsystems.

[0126] Let F * = D -1 F, and u = u a + u b where is the estimate of F * , then (3) can be changed to (4)

[0127]

[0128] where

[0129] Let ε = 0, The equilibrium point of the system can be obtained as

[0130]

[0131] where u bs is the control component of the slow subsystem when ε = 0. Substitute y1 in (2) with , the slow subsystem can be obtained as (6)

[0132]

[0133] where

[0134] Define a new state variable Substitute it into (2) to obtain (7)

[0135]

[0136] Define an extended time variable and let ε = 0 in (4), then the fast subsystem is given as

[0137]

[0138] where u bf is the control component of the slow-fast system when ε = 0, and satisfies u bs + u bf = u b .

[0139] S2: Design a neural network to estimate the friction force and the model unmodeled part in the system, and train the neural network using the improved optimal bounded ellipsoid algorithm, specifically:

[0140] Consider the lumped uncertainty term F * in the slow subsystem (5), design a neural network to estimate the lumped uncertainty term

[0141] F * = W *TΦ(Z) + ε (9)

[0142] Φ(Z) = [φ(z1), φ(z2),..., φ(z N )] T (10)

[0143] where ε is the estimation error, W * is the optimal weight from the hidden layer to the output layer of the neural network, is the input of the neural network, is the weight from the input layer to the hidden layer of the neural network, Z = [z1, z2,..., z N ] T = Vx, N is the number of nodes in the intermediate layer of the neural network. φ(z i ) is the activation function of the neural network, and the activation function selected in the design of the present application is as follows:

[0144]

[0145] where β t is a design parameter.

[0146] Using equation (9), equation (6) can be rewritten as:

[0147]

[0148] In order to train the neural network to obtain the optimal weight W * , the following method is used to train the neural network in the design of the present application, which is as follows:

[0149] First, in order to prevent noise interference, and since the angular velocity is unknown, a second-order filter is used to filter the left and right sides of equation (12), and then the following equation is obtained:

[0150]

[0151] where ω n is the parameter of the second-order filter, ξ is the damping ratio, x 2f is the filtered value of x2, is the filtered value of , and C 1f is the filtered value of C1, Φ f is the filtered value of Φ, and ε f is the filtered value of ε.

[0152] After filtering the left and right sides of equation (9), the following equation is obtained:

[0153]

[0154] where is Fthe filtered value of f , is the i-th column of , fi is the i-th column of f .

[0155] Consider the following identification model:

[0156]

[0157] where is the estimated value of .

[0158] Define the identification error:

[0159]

[0160] The neural network is trained by using the improved optimal bounded ellipsoid method, and the specific expression is as follows:

[0161]

[0162] where

[0163]

[0164] g i = g i0 r i r pi (20)

[0165]

[0166] The gain coefficient λ i > 1 needs to be satisfied, is the upper bound of the neural network estimation error, is a symmetric positive definite matrix, is the distance from P i to the lower bound p li I, is a fixed forgetting factor, which needs to satisfy 0 < g i0 < 1, and are the lower bound and the upper bound of ||P i ||, respectively, which need to satisfy p li < p ui , Q = Φ(Vx), Q f = φ(V xf ). The projection operator is given by:

[0167]

[0168] in is a parameter, r pi It is given by:

[0169]

[0170] S3: For unknown system states, an observer is designed based on the training results of the neural network to observe them. Specifically:

[0171] For the unknown state x2, the dynamic equation of the slow subsystem is considered as follows:

[0172]

[0173] in

[0174] The total disturbance of the system ∈ x Expand it into a new state variable x3, let v(t) represent the differential of x3, then the system equation becomes

[0175]

[0176] Then design the neural network observer as shown below

[0177]

[0178] Where β0 is the bandwidth of the observer, β1=3β0 is the estimated value of x1, is the state estimation error. Using the neural network state observer (26), we can get the estimated values ​​of the unknown system state x2 and the unknown total disturbance x3 This prepares the conditions for the subsequent controller design.

[0179] For the unknown state y2, consider the fast subsystem dynamic equation of formula (8), and transform the total disturbance of the system ∈ y Expand it into a new state variable y3, let ι(t) represent the differential of y3, then the system equation becomes

[0180]

[0181] Then design the neural network observer as shown below

[0182]

[0183] Where β0 is the bandwidth of the observer, β1=3β0, is the estimated value of z1, is the state estimation error. With the neural network observer (28), the estimation values of the unknown system state y2 and the total disturbance y3 can be obtained and This prepares conditions for the subsequent controller design.

[0184] S4: For the slow subsystem, a self-adaptive variable impedance control controller is designed, specifically:

[0185] Considering the slow subsystem formula (12), and the reference trajectory x r , the error dynamic equation of the slow subsystem is written as follows:

[0186]

[0187] where is the second-order derivative of x r . Two positive real auxiliary variables λ v1 and λ v2 are selected, which satisfy:

[0188]

[0189] where is the impedance control parameter that satisfies the impedance relationship shown below:

[0190]

[0191] where e = x1-x r is the tracking error.

[0192] Using λ v1 , the sliding surface s s is designed as shown in the following formula:

[0193] s s = λ v1 e1+e2 (33)

[0194] Using formula (32) and (33), a variable impedance controller can be designed for the slow subsystem, and its expression is as follows:

[0195]

[0196] where |τ ext | is the absolute value of τ ext . β s and α s are constant matrices. As can be seen from formula (34), the stiffness of the method in the present application can be adaptively adjusted according to the external force of the mechanical arm interacting with the outside world. In the new variable impedance design method shown above, when the mechanical arm is not in contact with the obstacle or target object, the external force is 0, τs = β s , τ s s s = β s s s is very large, so the controller has a high feedback gain at this time, which can reduce the tracking error and ensure the tracking accuracy of the system. When the robot arm contacts the obstacle or target object, the external force is often large at this time, τ s is close to 0, so τ s s s is also close to 0. At this time, by selecting appropriate impedance parameters, the robot arm can exhibit more compliant characteristics when it contacts the environment, reducing the contact force and protecting the safety of equipment and personnel. In summary, the variable impedance control proposed in the application makes the robot arm better follow the reference trajectory when it does not contact the obstacle or target object, and can exhibit better compliance when it contacts the obstacle or target object.

[0197] S5: For the fast subsystem, a sliding mode controller is designed, and the controllers of the two subsystems are combined according to the singular perturbation method to give the overall controller design, specifically:

[0198] Considering the fast subsystem formula (8), the sliding surface s y is defined as follows:

[0199] s y = λ y z1+y2 (35)

[0200] wherein is a design parameter.

[0201] Further, s y about the time variable t ∈ is the derivative of:

[0202]

[0203] Then the fast subsystem controller can be designed as:

[0204]

[0205] wherein and are controller parameters.

[0206] Using the singular perturbation method, the final overall controller is designed as:

[0207] u = u a + u bs + u bf (38)

[0208] wherein

[0209] Figure 2 A block diagram of a flexible robot arm adaptive variable impedance control method based on a neural network is described, and the framework of the entire control system is described.

[0210] The simulation verification of the disclosed technical solution is as follows:

[0211] Step 1: Select the system parameters of a two-degree-of-freedom flexible joint robot arm and the parameter design of the controller.

[0212] The parameters in the robot arm model are:

[0213]

[0214] Selection of singular perturbation decomposition parameters

[0215] ε = 0.1

[0216] The variable stiffness controller design parameters are selected as follows:

[0217]

[0218] The sliding mode controller parameters are selected as follows:

[0219]

[0220] Step 2: Select the reference signal, design the sampling period and the simulation duration.

[0221] The reference signal is x r1 = x r2 = sin(0.2πt) + 0.1

[0222] For both reference trajectories, an obstacle is placed at x b = 0.9 to detect the flexibility of the control algorithm. The sampling period is set to dt = 0.0001s, and the simulation duration is designed to t stop = 10s.

[0223] Step 3: Compare the adaptive variable impedance control method proposed in the application with the traditional impedance control method through simulation experiments. The experimental results are shown in Figure 3 、 Figure 4 . Figure 3 The angular tracking curve of the adaptive variable impedance control method and other methods under the condition of tracking the reference sinusoidal signal. Figure 4The angle tracking error curve of the adaptive variable impedance control method in the case of tracking a reference sinusoidal signal is compared with other methods. As can be seen from the figure, compared with the traditional method, the method proposed in the application can keep sufficient compliance characteristics when encountering obstacles, can produce smaller impact, and can prevent safety problems caused by excessive torque of the mechanical arm and the object or the human body.

[0224] Based on the above technical content, a neural network-based adaptive variable impedance control method for a flexible robot arm can be obtained. The singular perturbation method is used to decompose the complex system, the neural network is used to estimate the lumped uncertainty in the system, the improved optimal bounded ellipse algorithm is used to train the neural network, and the improved variable stiffness impedance control is used to design the controller. The problems of gradient disappearance and gradient explosion in the traditional neural network training are solved, the state observer is established, the problem of unmeasurable high-order angular velocity is solved, and the problem that the fixed parameters in the traditional impedance control cannot be applied to various scenes and various working states is solved.

[0225] To sum up, the above is only a preferred embodiment of the application, and is not used to limit the protection scope of the application. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the application shall be included in the protection scope of the application.

Claims

1. A neural network-based adaptive variable impedance control method for a flexible robotic arm, characterized in that: The steps include: S1: For the fourth-order nominal model of a two-degree-of-freedom flexible joint manipulator, in order to simplify the controller design, the singular perturbation method is used to decompose it into two second-order subsystems: a fast subsystem and a slow subsystem; S2: Design a neural network to estimate the friction in the system and the unmodeled parts of the model, and train the neural network using the improved optimal bounded ellipse algorithm; S3: For unknown system states, an observer is designed based on the training results of the neural network to observe them; S4: Design a variable stiffness impedance controller that adaptively changes the stiffness for the slow subsystem; S5: Design a sliding mode controller for the fast subsystem and combine the controllers of the two subsystems using the singular perturbation method to give the overall controller design.

2. The method for adaptive variable impedance control of a flexible manipulator neural network based on output feedback according to claim 1, characterized in that: In S1, for the fourth-order nominal model of the two-degree-of-freedom flexible joint manipulator, in order to simplify the controller design, the singular perturbation method is used to decompose it into two second-order subsystems, a fast subsystem and a slow subsystem, specifically: Consider the following flexible joint robotic arm model in, is the joint angle, is the driving motor angle, is the inertia matrix, are the Coriolis force and the centrifugal force, is gravity, is the external force torque, is the system friction, For the unmodeled dynamics of the system, is the control input, is the transmission ratio matrix of the motor and is a diagonal matrix, is the moment of inertia matrix of the motor rotor and is a diagonal matrix, is the elastic coefficient matrix of the joint elastic element and is a diagonal matrix; is the set of real numbers, and the superscript is the dimension; The singular perturbation method is as follows: first, rewrite the system model into a state space form and take the following system state: Among them, x1, x2, y1, y2 are four system states, 0<ε<<1 is the preset parameter, B0=ε 2 K; At the same time, consider combining the unmodeled dynamics of the system with the friction and nonlinear terms of the system to write it as a lumped uncertainty Then the state space form is: where a1=D -1 B0, a2=(J -1 N -2 +D -1 )B0, b1=J -1 N -1 ; Because 0<ε<<1, the change speed of y1, y2 in the system is faster than that of x1, x2, and the system state has two different time scales. The singular perturbation method is used to decompose the original fourth-order system into two second-order subsystems; Let the new lumped uncertainty F * =D -1 F, uncertain compensation item and u=u a +u b ,u b is the tracking control item to be designed, where It's F * The estimated value of , then (3) becomes (4) in Let ε = 0, Get the system equilibrium point where u bs is the control component of the slow subsystem when ε = 0; Substituting y1 in (2), we get the slow subsystem shown in equation (6): in Defining new state variables Substituting it into (2) we get formula (7): Defining extended time variables And let ε = 0 in (4), and then give the fast subsystem as: in u bf is the control component of the slow-fast system when ε = 0, and satisfies u bs +u bf =u b .

3. The method for adaptive variable impedance control of a flexible manipulator neural network based on output feedback according to claim 1, characterized in that: S2: Design a neural network to estimate the friction in the system and the unmodeled parts of the model, and use the improved optimal bounded ellipse algorithm to train the neural network, specifically: Consider the lumped uncertainty term F in the slow subsystem equation (5) * , design a neural network to estimate the lumped uncertainty F * : F * =In *T Φ(Z)+∈ (9) Φ(Z)=[φ(z1),φ(z2),…,φ(z N )] T (10) Where ∈ is the estimation error, W * is the optimal weight from the hidden layer to the output layer of the neural network, Z=[z1,z2,…,z N ] T =Vx, is the neural network input, is the weight from the input layer to the hidden layer of the neural network, N is the number of nodes in the middle layer of the neural network, φ(z i ) is the neural network activation function; The selected activation functions are as follows: where β t is the design parameter; Using formula (9), formula (6) can be rewritten as: 。 4. The method for adaptive variable impedance control of a flexible manipulator neural network based on output feedback according to claim 3, characterized in that: In S2, the following method is used to train the neural network to obtain the optimal weight W * , as follows: First, in order to prevent noise interference, and because the angular velocity is unknown, a second-order filter is used Filter the left and right sides of equation (12) to obtain: where ω n is the second-order filter parameter, ζ is the damping ratio, the complex frequency s is the independent variable of the Laplace transform, x 2f is the filtered value of x2, yes The filtered value of C 1f is the filter value of C1, Φ f is the filtered value of Φ, ∈ f is the filtered value of ∈; After filtering on both sides of equation (9), we can obtain: in It's F f The filtered value of yes The i-th column of fi is∈ f Column i of Consider the following identification model: in It corresponds to estimated value of; Define identification error: The improved optimal bounded ellipse method is used to train the neural network. Its specific expression is as follows: in g i =g i0 r i r pi (20) in is the gain coefficient, satisfying λ i >1; is the upper bound of the neural network estimation error; is a symmetric positive definite matrix, It's P i To the lower bound p li I distance; is a fixed forgetting factor, satisfying 0<g i0 <1; and For ||P i ||Lower and upper bounds satisfy p li <p ui ,Q=Φ(Vx),Q f =Φ(Vx f ); projection operator It is given by: in is a parameter, r pi It is given by:

5. The method for adaptive variable impedance control of a flexible manipulator neural network based on output feedback according to any one of claims 1 to 4, characterized in that: S3: For unknown system states, an observer is designed based on the training results of the neural network to observe them, including an observer for the slow subsystem, specifically: For the unknown state x2, the dynamic equation of the slow subsystem is as follows: in The total disturbance of the system ∈ x Expand it into a new state variable x3, let v(t) represent the differential of x3, then the system equation becomes: Then design the neural network observer as shown below Where β0 is the bandwidth of the observer, β1=3β0, is the estimated value of x1, is the state estimation error; using the neural network state observer (26), we get the estimated values ​​of the unknown system state x2 and the unknown total disturbance x3 This prepares the conditions for the subsequent controller design.

6. The method for adaptive variable impedance control of a flexible manipulator based on neural network output feedback according to claim 5, characterized in that: S3: For unknown system states, an observer is designed based on the training results of the neural network to observe them, and also includes an observer for the fast subsystem, specifically: For the unknown state y2, consider the fast subsystem dynamic equation of formula (8), and transform the total disturbance of the system ∈ y Expanded into a new state variable y3, ι(t) represents the differential of y3, the system equation becomes Then design the neural network observer as shown below Where β0 is the bandwidth of the observer, β1=3β0, is the estimated value of z1, is the state estimation error; using the neural network observer (28), we can get the estimated values ​​of the unknown system state y2 and the total disturbance y3 as well as This prepares the conditions for the subsequent controller design.

7. The method for adaptive variable impedance control of a flexible manipulator arm neural network based on output feedback according to claim 5, characterized in that: S4: Design an adaptive variable impedance control controller for the slow subsystem, specifically: Consider the slow subsystem (12), and the reference trajectory x r , the error dynamic equation of the slow subsystem is written as follows: in is x r The second derivative of; choose two positive real auxiliary variables λ v1 and λ v2 , they satisfy: in To satisfy the impedance control parameters in the impedance relationship shown below: where e = x1 - x r is the tracking error; Using λ v1 Design the sliding surface s as shown below s : s s =λ v1 e1+e2 (33) Using equations (32) and (33), a variable impedance controller is designed for the slow subsystem, which is expressed as follows: in |τ ext | is τ ext The absolute value of β s and α s is a constant matrix.

8. The method for adaptive variable impedance control of a flexible manipulator arm using a neural network based on output feedback according to claim 6, wherein: S5: Design a sliding mode controller for the fast subsystem and combine the controllers of the two subsystems according to the singular perturbation method to give the overall controller design, specifically: Considering the fast subsystem (8), define the sliding surface s y as follows: s y =λ y z1+y2 (35) in is the design parameter; Then we can give s y About the time variable t ∈ The derivative of is: Then the fast subsystem controller can be designed as: in and All are controller parameters; Using the singular perturbation method, the final overall controller design is: in=in a +in bs +in bf (38) in

Images