Adaptive neural network control method for variable load manipulator system

Abstract

The present invention discloses an adaptive neural network switching control method for a variable-load manipulator system. First, the desired signal and error signal are used as inputs to a sliding mode function. The output of the sliding mode function is used to design the weight update laws for neural networks I and IV, thereby respectively estimating and compensating the system model and lumped unknown nonlinear terms. Secondly, an ideal controller is derived based on the outputs of neural networks II and IV, as well as the sliding mode function. Furthermore, neural networks II and III are used to estimate and compensate for the dead zone, respectively. Finally, the dead zone input signal is derived by combining the output of neural network III with the ideal control input signal. The dead zone output is , which is the actual control input for the manipulator system. Finally, under the action of a switching signal, the actual control input signal is transmitted to the actuator end of each subsystem, achieving desired trajectory tracking control for the closed-loop system.

Description

Technical field:

[0001] The present invention belongs to the technical field of trajectory tracking control of a robotic arm, and in particular relates to an adaptive neural network control method for a variable load robotic arm system. Background technology:

[0002] As a powerful aid in industrial production, robots play a vital role in promoting national economic development. As a key component of industrial robots, robotic arms feature high transmission accuracy, fast response speed, and the ability to operate in hazardous environments. Trajectory tracking control of robotic arms has been a hot topic of research. Ensuring safe and stable system operation in complex real-world situations remains a pressing challenge. Using different end effectors, the same manipulator can perform diverse tasks, such as assembly, welding, and handling. In automotive assembly, the same actuator can grasp different workpieces at different times, significantly improving the manipulator's efficiency. However, due to physical variations in workpiece shape and mass, a single controller may be unable to implement precise control, potentially reducing tracking accuracy or leading to system instability. Currently, numerous advanced control algorithms have been proposed for tracking control, such as sliding mode control, backstepping, neural networks, and reinforcement learning. However, these methods are primarily designed for controlling fixed-load robotic systems with a single controller. Therefore, designing a switching method for position tracking control of variable-load robotic arm systems is crucial. This approach not only improves the efficiency of the robotic arm system but also ensures tracking accuracy and stable operation under variable load conditions. Summary of the invention:

[0003] To address the trajectory tracking control problem of a variable-load, uncertain robotic arm system, effectively improve the robustness and motion quality of the servo drive system, and enhance the system's energy efficiency, this paper proposes a neural network-based adaptive switching control method for a variable-load robotic arm system. This method considers the dynamic model of the robotic arm system under variable load as a switching system with multiple modes. RBFNN is used to approximate the nonlinear terms containing modeling uncertainties in different modes to avoid the controller's reliance on precise models. Furthermore, the present invention utilizes a neural network to estimate and compensate for unknown dead zones and eliminates potential interference and estimation errors by designing a robust compensator.

[0004] In order to achieve the above-mentioned object, the present invention relates to an adaptive neural network switching control method for a variable load manipulator system, which specifically includes the following steps:

[0005] (1) First, the expected signal and error signal As the input of , the sliding mode function s is calculated based on formula (14);

[0006] The sliding surface variables are designed as

[0007]

[0008] Where s represents the sliding surface variable of the sliding mode control method adopted, e = q d -q means unknown tracking error, is the derivative of e, λ represents a positive definite diagonal matrix;

[0009] (2) Based on formulas (18) and (20), the sliding mode function s is used to design the weight update law of NNI, and then the system model F i (x) making estimates and compensation;

[0010] F i The estimated value of (x) can be expressed as

[0011]

[0012] The weight update law can be expressed as

[0013]

[0014] Among them, h fi (x) represents the ideal output signal F of the i-th subsystem i Gaussian basis function of (x), Ψ0 is the gain in the update law signal, s T is the transposed matrix of s, for The derivative of for The transposed matrix of

[0015] (3) Based on formulas (42) and (43), the sliding mode function s is used to design the NN IV weight update law, and then the lumped unknown nonlinear term T i (t) making estimates and compensation;

[0016] Let T i (t) = v i (t)-ε fi -τ di represents the unknown lumped perturbation and is approximated by NN IV, T i The estimated value of (t) can be expressed as

[0017]

[0018] The NN weight update law can be designed as

[0019]

[0020] Among them, h ti (x) represents the ideal output signal T of the i-th subsystem i (t), Ψ3 represents a positive constant diagonal matrix, express The derivative of for The transposed matrix of

[0021] (4) Secondly, based on the output value of NN I NNIV output value and K i , we get the ideal controller τ c ;

[0022]

[0023] in, represents the correction term τ s The estimated value of represents the lumped perturbation The estimated value of Indicates F i The estimated value of (x), K i =diag{k 1i ,...,k pi ,...,k ni} represents a positive definite matrix,

[0024] (5) Furthermore, based on formulas (23)-(27), NN II and NN III are used to estimate and compensate for the dead zone, respectively. Specifically, NN II is regarded as a dead zone estimator, while NN III is used to estimate the correction term τ s , to compensate for the dead zone, NN II acts as an auxiliary performance “observer” to adjust NN III for further compensation of the dead zone;

[0025] The ideal output value of the RBFNN precompensator is

[0026]

[0027] Among them, h si (τ c ) represents the ideal output signal τ of the i-th subsystem s Gaussian basis function, ε si (τ c ) represents the unknown bounded error and |ε si (τ c )|≤ε sM , W si Represents the ideal output signal τ sThe ideal weight of and unknown bound || W si || F ≤W sM , W si The transposed matrix of

[0028] τ s The estimated value of can be expressed as

[0029]

[0030] in, W si The estimated value of for The transposed matrix of

[0031] The weight update law is

[0032]

[0033] Where Ψ1 represents a positive definite diagonal constant matrix, express The derivative of h′ i (u) is h i The derivative of (u).

[0034] The ideal output value of RBFNN in the dead zone is

[0035]

[0036] Among them, h i (u) represents the Gaussian basis function of the i-th subsystem with respect to the ideal output signal D(u), ε i (u) represents the bounded error and |ε i (u)|≤ε M , W i Represents the ideal weight about the ideal output signal D(u), and has an unknown upper bound ||W i || F ≤W M , W i The estimated value of τ can be expressed as the transposed matrix of

[0037]

[0038] in, W i The estimated value of for The transposed matrix of , Equations (24) and (27) represent the estimation of the inverse of the modified dead zone and the dead zone, respectively.

[0039] (6) Then, combined with the output value of NN III and the ideal control input signal τ c According to formula (21), the input signal u of the dead zone D(u) is obtained, and the output of the dead zone D(u) is τ, which is the actual control input τ of the manipulator system.

[0040] The control input of the robotic arm system with dead zone is designed as

[0041]

[0042] (7) Finally, under the action of the switching signal σ, the actual control input signal τ is transmitted to the actuator end of each subsystem to achieve the desired trajectory tracking control of the closed-loop system.

[0043] Compared to existing technologies, this paper rationally models an industrial robot arm system with a switching load, establishes a switching model for the industrial robot arm, and uses RBFNN to adaptively adjust the subsystem models and sliding mode controller gains to reduce chattering and address the controller's reliance on precise models. Furthermore, based on the mean dwell time principle and the Lyapunov function method, this approach ensures consistent and ultimately bounded trajectory tracking error, enabling better tracking of the desired trajectory and reducing system energy consumption and chattering. Description of the drawings:

[0044] Figure 1 Schematic diagram of the variable load robotic arm system model structure.

[0045] Figure 2 Schematic diagram of the RBFNN pre-compensation control system structure.

[0046] Figure 3 This is a flow chart of the adaptive neural network control method for the variable load manipulator system of the present invention.

[0047] Figure 4 The position tracking curves of joint 1 obtained from simulation experiments of three control methods.

[0048] Figure 5 The position tracking curves of joint 2 obtained from simulation experiments of three control methods.

[0049] Figure 6 The position tracking error curves of joint 1 obtained from simulation experiments of three control methods.

[0050] Figure 7 The position tracking error curves of joint 2 obtained from simulation experiments of three control methods.

[0051] Figure 8 The control torque curves of joint 1 obtained from simulation experiments of three control methods.

[0052] Figure 9 The control torque curves of joint 2 obtained from simulation experiments of three control methods.

[0053] Figure 10 is the switching signal change curve. Specific implementation method:

[0054] The present invention will be further described below by way of examples.

[0055] Example 1:

[0056] In order to solve the trajectory tracking control problem of a variable load uncertain manipulator system, effectively improve the robustness and motion quality of the servo drive system, and enhance the energy efficiency of the system, the present invention proposes an adaptive switching control method based on a neural network. This method regards the dynamic model of the manipulator system under variable load as a switching system with multiple modes. RBFNN is used to approximate the nonlinear terms containing modeling uncertainties in different modes to avoid the controller's dependence on an accurate model. In addition, the present invention uses a neural network to estimate and compensate for unknown dead zones, and eliminates potential interference and estimation errors by designing a robust compensator. In order to solve the above technical problems, the technical solutions and specific steps provided are as follows:

[0057] Step 1: Establish a variable load robotic arm system model

[0058] For an n-joint robotic arm system with variable load, the Lagrangian dynamic model is established as follows:

[0059]

[0060] Where q=[q1,q2...q n ] T ∈R n is the joint position, q n Represents the joint displacement of the n-joint robotic arm system, R n represents an n-dimensional real vector; represents joint velocity; represents joint acceleration; H σ(t) (q)∈R n×n is the inertia matrix, R n×n Represents a real matrix 2 with n rows and n columns is the Coriolis force and centripetal force matrix; W σ(t) (q)∈R n represents the gravity term; τ dσ(t) ∈R nrepresents external disturbance and uncertainty; τ∈R n represents the control input torque; σ(t): [0, ∞) → Ω = {1, 2, ..., ι} represents the switching signal, ι represents the number of system modes, and Ω represents the set of different types of loads. For simplicity, Indicates that the i-th subsystem is activated, and the model parameters of the corresponding system can be written as: H σ(t) (q) = H i (q), G σ(t) (q) = G i (q), τ dσ(t) =τ di Here, we take the two-joint variable load manipulator system as an example, and its model is as follows: Figure 1 shown.

[0061] The industrial robot arm dynamics system model generally has the following characteristics:

[0062] Property 1: Yes H i (q) is bounded and has a constant value n 1i >0,n 2i >0 makes

[0063] n 1i ||x|| 2 ≤x T H i (q)x≤n 2i ||x|| 2 (2)

[0064] Property 2: Yes skew-symmetric matrix satisfy

[0065]

[0066] in, Indicates H i The derivative of (q).

[0067] Switch system description

[0068] A switching system is a special type of hybrid system, which consists of a series of continuous-time or discrete-time subsystems and switching rules between the subsystems. The mathematical model of a switching system can be described as

[0069]

[0070] Where x(t) represents a continuous state variable, is the derivative of x(t), t∈R + .

[0071] Definition 1

[0072] For the switching signal σ(t), Indicates the number of switching times in the time period (t1, t2). If the given constant N0>0, T σ >0, and satisfies

[0073]

[0074] Then T σ It is called the average residence time.

[0075] RBFNN has attracted much attention due to its simple structure and strong approximation ability, and is widely used to approximate nonlinear continuous functions f(x): R n →R m , R m is an m-dimensional real vector,

[0076] f(x)=W T h(x)+ε, (6)

[0077] Where W = [ω1, ...ω m ] T Represents the weight vector of RBFNN, W T represents the transposed matrix of W, h(x)=[h1,h2,...,h l ] T is the hidden layer activation function, and ε is a small positive number. The Gaussian function is selected as

[0078]

[0079] x=[x1,x2,...,x n ] T represents the input layer vector signal; m represents the number of hidden layers; c j =[c j1 ,...,c jn ] T represents the center vector of the jth neuron in the hidden layer; b j =[b j1 ,...,b jl ] T Here, the estimated value of the weight vector W is recorded as for The transposed matrix of f(x) is used to estimate f(x) using the RBF neural network, and the estimated value of f(x) is recorded as but

[0080]

[0081] in, It can be adaptively adjusted through the weight update law, and the error of the weight is recorded as

[0082] Mathematical description of unknown dead zone

[0083] Generally, the dead zone phenomenon in motion can be described as follows:

[0084]

[0085] Where u and τ represent the input and output of the dead zone, g(u) and h(u) represent reversible unknown continuous nonlinear functions, and d + and d - Represents an unknown positive scalar. Here, the RBFNN precompensator is introduced to compensate for the unknown dead zone characteristics of the system to reduce the impact on tracking accuracy. The RBFNN precompensation structure is as follows: Figure 2 In order to compensate for the dead zone, the inverse of the dead zone needs to be obtained to satisfy

[0086] D(D -1 (τ c ))=τ c . (10)

[0087] Considering that formula (9) is reversible, we can have

[0088]

[0089] Dead Zone Inverse D - 1(τ c ) is equivalent to

[0090] D -1 (τ c )=τ c +τ s , (12)

[0091] Among them, τ s It represents the improved inverse of the dead zone and is used as the correction signal. The specific form is as follows:

[0092]

[0093] As shown in formula (12), D -1 (τ c ) consists of two parts, namely the ideal input term τ c and the correction term τ s .

[0094] Step 2: Design of adaptive switching controller

[0095] For an n-joint robotic arm system, when the load changes, the traditional single adaptive controller may not be able to respond in a timely and accurate manner, which may reduce the tracking accuracy. Therefore, based on the average dwell time method, this paper designs an adaptive switching controller to improve the robot trajectory tracking accuracy. During the operation of the robotic arm, the actual trajectory is q and the expected trajectory is q d The sliding surface variables are designed as

[0096]

[0097] Where s represents the sliding surface variable of the sliding mode control method adopted, e = q d -q means unknown tracking error, is the derivative of e, and λ represents a positive definite diagonal matrix. Auxiliary variable q r Designed for

[0098]

[0099] From (14), we can see that The auxiliary equation is designed as

[0100]

[0101] in, represents the derivative of the sliding surface variable s, represents the nonlinear term related to the system model information, Indicates calculation F i (x) Required information. In the present invention, F i (x) is estimated by RBFNN, which can avoid the controller design from relying on the accurate system model. The control flow chart is as follows Figure 3 shown.

[0102] According to the basic principle of RBFNN, F i The ideal output value of (x) is

[0103]

[0104] Among them, h fi (x) represents the ideal output signal F of the i-th subsystem i Gaussian basis function of (x), ε fi represents the unknown bounded error and |ε fi |≤ε fM , W fi Represents the ideal output signal F i (x) has an ideal bounded weight and ||W fi || F ≤W M , Wfi The transposed matrix of .

[0105] F i The estimated value of (x) can be expressed as

[0106]

[0107] in, W fi The estimated value of for The transposed matrix of .

[0108] From formula (17) and formula (18), we can get

[0109]

[0110] The weight update law can be expressed as

[0111]

[0112] in, Ψ0 is the gain in the update law signal, s T is the transposed matrix of s.

[0113] The control input of the robotic arm system with dead zone is designed as

[0114]

[0115]

[0116] in, represents the correction term τ s The estimated value of represents the lumped perturbation The estimated value of Indicates F i The estimated value of (x), K i =diag{k 1i ,...,k pi ,...,k ni} represents a positive definite matrix.

[0117] Note: The present invention uses two neural networks to estimate and compensate for the unknown dead zone of the system. Specifically, NN II is regarded as the dead zone estimator, while NN III is used to estimate the correction term τ s , to compensate for the dead zone. Here, the output of NN III can intuitively affect the input u, while NN II is used to adjust NN III.

[0118] The ideal output value of the RBFNN precompensator is

[0119]

[0120] Among them, h si (τ c ) represents the ideal output signal τ of the i-th subsystem s Gaussian basis function, ε si (τ c ) represents the unknown bounded error and |ε si (τ c )|≤ε sM , W si Represents the ideal output signal τ s The ideal weight of and unknown bound || W si || F ≤W sM , W si The transposed matrix of τ. s The estimated value of can be expressed as

[0121]

[0122] in, W si The estimated value of for The transposed matrix of .

[0123] The weight update law is

[0124]

[0125] Where Ψ1 represents a positive definite diagonal constant matrix.

[0126] The ideal output value of RBFNN in the dead zone is

[0127]

[0128] Among them, h i (u) represents the Gaussian basis function of the i-th subsystem with respect to the ideal output signal D(u), ε i (u) represents the bounded error and |ε i (u)|≤ε M , W i Represents the ideal weight about the ideal output signal D(u), and has an unknown upper bound ||W i || F ≤W M , W i The estimated value of τ can be expressed as

[0129]

[0130] in, W i The estimated value of for Equations (24) and (27) represent the estimates of the modified dead zone inverse (13) and the dead zone (9), respectively.

[0131] The weight update law is

[0132]

[0133] Where Ψ2 represents a positive constant diagonal matrix, h i ′(u) represents h i The derivative of (u).

[0134] Combining equations (21) and (26), the dead zone output can be expressed as

[0135]

[0136] From formula (10) and formula (12), we can get

[0137] τ c =D(D -1 (τ c ))=D(τ c +τ s ). (30)

[0138] Substituting equations (23) and (29) into equation (30), we can obtain

[0139]

[0140] From formula (31), we can get

[0141]

[0142]

[0143]

[0144] From the first-order Taylor expansion of f(x), we can get

[0145] f(z)=f(z0)+f′(z0)(z-z0)+o(ξ), (35)

[0146] Among them, ξ∈(z0,z), represents the Lagrangian remainder, and ξ0 is a value between z0 and z.

[0147] From Taylor expansion (35) and combined with equations (33)-(34), we can get

[0148]

[0149] Among them, w i (t) as follows

[0150]

[0151] Combining equations (21), (24), (36) and (37), we can get

[0152]

[0153] From formula (29) and formula (38), we can get

[0154]

[0155] Among them, v i (t) is given as follows:

[0156]

[0157] Let T i (t) = v i (t)-ε fi -τ di represents the unknown lumped perturbation and is approximated by NN IV. The output value of NN IV can be expressed as

[0158]

[0159] Among them, h ti (x) represents the ideal output signal T of the i-th subsystem i Gaussian basis function of (t), ε ti represents a bounded error, and |ε ti |≤ε T , W ti Represents the ideal output signal T i The ideal weight of (t) with unknown bound || W ti || F ≤W M , W ti The transposed matrix of T. i The estimated value of (t) can be expressed as

[0160]

[0161] in, W tiThe estimated value of for The transposed matrix of .

[0162] The NN weight update law can be designed as

[0163]

[0164] Among them, Ψ3 represents a positive constant diagonal matrix, express The derivative of

[0165] Substituting equations (22) and (39) into equation (16), we can obtain

[0166]

[0167] in,

[0168] Note: Equation (10) couples the inherent information of NN II and NN III. In practice, compensating for the unknown influence (NN III) depends on what is observed (NN II); and what is observed about the unknown influence (NN II) depends on how the system is adjusted (NN III). Combining Equations (25) and (28), we can intuitively observe that the weight update laws of NN II and NN III are coupled.

[0169] Step 3: Analysis of the uniformly bounded nature of tracking error

[0170] In step 3, the uniform ultimate boundedness of the system tracking error is analyzed based on the average residence time method and the multiple Lyapunov function method.

[0171] For an n-joint manipulator system, if the control laws (21), (22), RBFNN precompensator (24), dead zone estimator (27), and NN weight update laws (20), (25), (28), and (43) are used, and the switching signal satisfies T σ >lnμ / 2λ0, the tracking error will remain consistent and eventually bounded.

[0172]

[0173]

[0174]

[0175] Among them, λ min {K i} represents the matrix K i The minimum eigenvalue of .

[0176] Specific proof: Lyapunov function is chosen as

[0177]

[0178]

[0179]

[0180] Among them, X -1 represents the inverse of X, such as represents the inverse of Ψ0, “tr(…)” represents the trace, and Make ||X(t)|| represents the 2-norm of X(t).

[0181] Taking the derivative of formula (48) and using formula (44), we can get

[0182]

[0183] Substituting equations (20), (25), (28) and (43) into equation (51), we can obtain

[0184]

[0185] According to the approximation characteristics of neural networks, we can know that: and 1>β i >0, so

[0186]

[0187] Then you can get

[0188]

[0189] Combining equations (51) and (54), we can get

[0190]

[0191] in, And β ip <k ip .

[0192] make Combining formula (46), if

[0193]

[0194] Then there is

[0195]

[0196] Then, we can get

[0197]

[0198] Right now

[0199]

[0200] make Combined with formula (45), if ||s||≥θ1, then

[0201]

[0202] Similarly, if μ satisfies formula (47), and

[0203]

[0204] but

[0205]

[0206] Further available

[0207]

[0208] Let θ = max{θ1,θ2}, if ||s|| ≥ θ, then equations (60) and (63) always hold. Combining equation (60), we can get

[0209]

[0210] Among them, t k Indicates the kth switching moment of the system within the running time interval.

[0211] Select any time point Indicates the switching time in the interval (t0, t). σ (t, t0)≤N0+(t-t0) / T σ , combined with formula (63), we can get

[0212]

[0213] Combined with formula (65), considering that the average residence time satisfies T σ >lnμ / 2λ0, state variable will reach the compact set Φ in a finite time θ ={X(t)|||s||≤θ}. Assume that the sliding mode variable s is at t=T f Time arrival Φ θ , if ||s|| continues to increase, it will reach Φ again at s θ Before, there is

[0214]

[0215] make Available

[0216]

[0217] Further available

[0218]

[0219] Therefore, the variable ||s|| is uniformly bounded. Combining formula (14), we can further know that the error variables e and is uniformly eventually bounded.

[0220] Simulation experiment verification

[0221] To demonstrate the rationality of the method of the present invention, use Figure 1 The 2-joint robotic arm shown in the figure is used as the simulation object. In addition, a set of objects with masses of 0kg, 3kg, and 6kg are selected as the loads of the robotic arm. The physical parameters of the robotic arm are as follows:

[0222]

[0223]

[0224]

[0225] Where i∈[1, 2, 3} represents the modal set. The initial state of the system is q(0)=[1.03, 0.8] T , The expected trajectory is q d1 =1+0.2sin(0.5πt),q d2 =1-0.2cos(0.5πt). Set λ0=1.5μ=200, T σ >0.18s; the dead zone related parameter is selected as d + =25,d - =-25, h(u)=u+d + , g(u)=u+d - ; The controller parameters are selected as Ψ0=15, Ψ1=600, Ψ2=0.1, Ψ3=600, λ=50I,K1=100I,K2=160I,K3=180I; the center points of NN I and NN IV are selected as c1=c4=[-1.5, -1, -0.5, 0, 0.5, 1, 1.5] respectively, and the center points of NN II and NN III are selected as c2=c3=[-60, -40, -20, 0, 20, 40, 60] respectively.

[0226] In order to demonstrate the feasibility and superiority of the method (M1) of the present invention, a comparison was made with the M2 and M3 methods, wherein M2 represents a control method without a switching mechanism, and M3 represents a state feedback switching control method under specific switching rules. Position tracking and control torque are important indicators for measuring control performance. In the simulation, position tracking, position tracking error and control torque were compared respectively. Figures 4-10 . Figure 4-Figure 7 The comparison curves of position tracking and tracking error of the three algorithms under the same load are shown. Figure 10 It indicates the switching signal and also indicates the load change and controller switching. It can be seen that the method of the present invention can make the actual trajectory quickly track the reference trajectory with smaller tracking error and faster convergence speed. Figure 8 and Figure 9 Represents the control torque of each joint under the three methods respectively. It can be seen that the M2 and M3 methods have larger driving torque and amplitude, as well as larger overshoot and jitter due to load changes. In addition, the present invention also compares the performance indicators of integral absolute error (IAE) and integral time multiplied absolute error (ITAE) to show the transient performance and steady-state performance of the robotic arm system, as shown in Table 1. It can be seen from Table 1 that under the action of the method of the present invention, the system has smaller IAE and ITAE. The formulas for integral absolute error and integral time multiplied absolute error are as follows:

[0227] IAE=∫||e(t)||dt, ITAE= / t||e(t)||dt

[0228] Table 1 Tracking error values ​​under three methods

[0229]

[0230]

[0231] in conclusion

[0232] This paper proposes a novel adaptive compensatory switching control method to solve the trajectory tracking problem of a robotic arm system with modeling uncertainty, transient loads, and unknown dead zones. First, a RBFNN is used to estimate the system model and dead zone, respectively, and robust compensation terms are used to offset the estimation errors. Then, incorporating the mean dwell time principle, an adaptive neural network-based compensatory switching control method is designed. The uniformly bounded nature of the tracking error is demonstrated using the multi-Lyapunov method. Finally, simulations verify the effectiveness and superiority of this method.

Claims

1. An adaptive neural network switching control method for a variable load manipulator system, characterized in that: The following steps are involved: (1) First, the expected signal q d , angular velocity signal Error signal e and angular velocity tracking error signal As input, the sliding mode function s is calculated based on formula (14), The sliding surface variables are designed as Where s represents the sliding surface variable of the sliding mode control method adopted, e = q d -q means unknown tracking error, is the derivative of e, λ represents a positive definite diagonal matrix; (2) Based on formulas (18) and (20), the sliding mode function s is used to design the weight update law of NN I, and then to i (x) making estimates and compensation; F i The estimated value of (x) can be expressed as The weight update law can be expressed as Among them, h fi (x) represents the i-th subsystem with respect to F i Gaussian basis function of (x), Ψ0 is the gain in the update law signal, s T is the transposed matrix of s, express The derivative of express The transposed matrix of (3) Based on formulas (42) and (43), the sliding mode function s is used to design the NN IV weight update law, and then the lumped unknown perturbation T i (t) make estimates and compensations; let T i (T) = v i (t)-ε fi -τ di , and approximated by NN IV, T i The estimated value of (t) can be expressed as The NN weight update law can be designed as Among them, h ti (x) represents the lumped unknown perturbation T of the i-th subsystem i (t), Ψ3 represents a positive constant diagonal matrix, express The derivative of for The transposed matrix of (4) Secondly, based on the output value of NN I Output value of NNⅣ and K i , we get the ideal controller τ c ; in, represents the correction term τ s The estimated value of represents the lumped unknown perturbation T i The estimated value of (t), Indicates F i The estimated value of (x), K i =diag{k 1i ,...,k pi ,...,k ni } represents a positive definite matrix; (5) Furthermore, based on formulas (23)-(27), NN II and NN III are used to estimate and compensate for the dead zone, respectively; specifically, NN II is regarded as a dead zone estimator, while NN III is used to estimate the correction term τ s , to compensate for the dead zone; NN II acts as an auxiliary performance "observer" to adjust NN III to further compensate for the dead zone; The ideal output value of the RBFNN precompensator is Among them, h si (τ c ) represents the ideal output signal τ of the i-th subsystem s Gaussian basis function, ε si (τ c ) represents the unknown bounded error and |ε si (τ c )|≤ε sM , W si Represents the ideal output signal τ s The ideal weight of and unknown bound || W si || F ≤W sM , W si The transposed matrix, τ s The estimated value of can be expressed as in, W si The estimated value of for The transposed matrix of The weight update law is Where Ψ1 represents a positive definite diagonal constant matrix, express The derivative of h′ i (u) is h i The derivative of (u); Dead zone is Among them, h i (u) represents the Gaussian basis function of the ith subsystem with respect to the dead zone D(u), ε i (u) represents the bounded error and |ε i (u)|≤ε M , W i Denotes the ideal weight of the dead zone D(u) with an unknown upper bound ||W i || F ≤W M , W i The estimated value of τ can be expressed as the transposed matrix of in, W i The estimated value of for The transposed matrix of , Equations (24) and (27) represent the estimation of the inverse of the modified dead zone and the dead zone respectively; (6) Then, combined with the output value of NN III and the ideal control input signal τ c According to formula (21), the input signal u of the dead zone D(u) is obtained, and the output of the dead zone D(u) is τ, which is the actual control input τ of the manipulator system. The control input of the manipulator system with dead zone is designed to be (7) Finally, under the action of the switching signal σ, the actual control input signal τ is transmitted to the actuator end of each subsystem to achieve the desired trajectory tracking control of the closed-loop system.

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