Flexible vibration control method for tendon-driven mechanical arm based on hybrid optimization strategy

Abstract

The invention provides a tendon-driven mechanical arm flexible vibration control method based on a hybrid optimization strategy. The tendon-driven mechanical arm flexible vibration control method comprises the steps that a dynamic model of a tendon-driven mechanical arm is established; constructing a trajectory optimization model for suppressing residual vibration; solving the trajectory optimization model for suppressing residual vibration by using a hybrid particle swarm optimization strategy, and obtaining an expected trajectory; and designing a sliding mode controller based on an RBF neural network to actively track the trajectory so as to suppress residual vibration. According to the invention, a hybrid particle swarm optimization algorithm is adopted to optimize a motion trajectory, so that residual vibration is reduced, an expected trajectory for trajectory tracking is obtained, active control is combined to further suppress the residual vibration, and the tail end vibration amplitude is reduced by more than 50%; the self-adaptive neural network all-drive system control method is designed by combining the RBF neural network, so that the tail end vibration is further inhibited while the mechanical arm completes trajectory tracking, and the tracking error is smaller than 0.5 degree.

Description

Technical Field

[0001] The present invention relates to the technical field, and in particular to a flexible vibration control method of a tendon-driven robotic arm based on a hybrid optimization strategy. Background Art

[0002] With the advancement of space technology, large spacecraft (such as solar power plants and giant space antennas) have become critical infrastructure for space resource utilization and long-term orbital presence. Their construction necessitates modular design, phased launches, and in-orbit assembly; single launches and integrated deployment are no longer feasible. Systems such as Canadarm-2 and CSSRMS utilize robotic arms for in-orbit assembly, but the bulky joints of traditional robotic arms limit their workspace, envelope, structural stiffness, and end-effector output force.

[0003] To overcome the limitations of traditional robotic arms, tendon-driven mechanisms have attracted considerable attention. In 2008, NASA proposed a cable-driven lunar manipulator for heavy-duty operations during lunar base construction. In 2014, NASA further investigated a tendon-driven space manipulator for asteroid capture missions. These studies demonstrated the advantages of tendon-driven manipulators, including light weight, long reach, and high torque. However, the flexible structure and cable elasticity of these manipulators can cause deformation and residual vibration when manipulating high-inertia targets. Therefore, suppressing this residual vibration, improving kinematic performance, and ensuring accurate trajectory tracking are key challenges.

[0004] To address the above issues, existing technologies use multi-point position feedback and active damping to achieve trajectory tracking and vibration suppression in flexible-link robotic arms. Although active vibration control methods are effective, optimizing controller parameters for highly nonlinear tendon-driven space robotic arms remains challenging. In addition, these methods usually require external sensors, which leads to complex controller structures in space applications.

[0005] Trajectory planning, as an open-loop control method, can effectively suppress residual vibration during point-to-point motion. Existing technologies use joint acceleration constraints and boundary conditions to minimize residual vibration. However, in practical applications, model uncertainties and time-varying disturbances often limit their effectiveness. Summary of the Invention

[0006] In response to the shortcomings of the existing technology, the present invention provides a flexible vibration control method for a tendon-driven robotic arm based on a hybrid optimization strategy. The present invention first uses a hybrid particle swarm optimization algorithm to optimize the trajectory of a tendon-driven spatial flexible robotic arm to reduce residual vibration; then, a neural network sliding mode controller is used as an active vibration control means to suppress residual vibration while ensuring accurate tracking of the end effector trajectory.

[0007] The technical solution of the present invention is: a method for controlling the flexible vibration of a tendon-driven manipulator based on a hybrid optimization strategy, comprising the following steps:

[0008] S1), establishing a dynamic model of the tendon-driven robotic arm;

[0009] S2), constructing a trajectory optimization model for suppressing residual vibration based on quintic polynomial;

[0010] S3), using a hybrid particle swarm optimization strategy to solve the trajectory optimization model for suppressing residual vibration and obtain the desired trajectory;

[0011] S4) Design a sliding mode controller based on RBF neural network using the full drive system method, use RBF neural network to estimate the unknown disturbance of the system, and realize trajectory tracking and residual vibration suppression through the sliding mode controller.

[0012] Preferably, in step S1), the dynamic model of the tendon-driven robotic arm takes into account the lateral vibration and tendon-driven characteristics of the flexible link, adopts the Euler-Bernoulli beam model to describe the flexible link, and is derived through the Lagrangian method.

[0013] Preferably, in step S1), the dynamic model of the tendon-driven robotic arm is expressed as:

[0014]

[0015] Where, is the inertia matrix; for Coriolis and centrifugal matrices; is a generalized coordinate variable; is the generalized coordinate variable velocity; is the generalized coordinate variable acceleration; T represents the transposed sign; θ k represents the kth joint angle; δ kj represents the j-th mode of the k-th flexible joint angle; n and m represent the joint angle dimension and mode dimension; is the stiffness matrix; Represents the motor current vector, J m Jacobian matrix representing the current-torque relationship.

[0016] Preferably, in step S2), the trajectory optimization model for suppressing residual vibration takes minimization of residual vibration as the optimization objective function, which is expressed as:

[0017]

[0018] Where, represents the total cost function; represents the terminal amplitude cost function; represents the penalty factor cost function; Respectively represent the trajectory function coefficients to be determined;

[0019]

[0020] is the inertia matrix; for Coriolis and centrifugal matrices; is a generalized coordinate variable; is the generalized coordinate variable velocity; is the generalized coordinate variable acceleration; T represents the transposed sign; θ k represents the kth joint angle; δ kj represents the j-th mode of the k-th flexible joint angle; n and m represent the joint angle dimension and mode dimension; is the stiffness matrix; Represents the motor current vector, J m Jacobian matrix representing the current-torque relationship.

[0021] Preferably, in step S3), the hybrid particle swarm optimization strategy includes a particle swarm optimization algorithm PSO and a genetic algorithm GA, and the optimization problems of formulas (2-11) and (2-12) are solved by the hybrid particle swarm optimization strategy.

[0022] Preferably, in step S4), the control law of the sliding mode controller is:

[0023]

[0024] Where, To switch control items; is the system lumped interference estimated by RBF neural network; λ is a non-negative constant; is a positive constant, is the switching gain; sign is the sign function; s represents the sliding surface; A0 and A1 are controller parameters; Joint angle tracking error; represents the joint angular velocity trajectory tracking error; is the unknown disturbance term.

[0025] The beneficial effects of the present invention are:

[0026] 1. The present invention optimizes the motion trajectory by using a hybrid particle swarm optimization algorithm to reduce residual vibration and obtain the desired trajectory for trajectory tracking. It also combines active control to further suppress the residual vibration, reducing the terminal vibration amplitude by more than 50%;

[0027] 2. By combining RBF neural networks, the present invention designs an adaptive neural network all-wheel drive system control method, which enables the robot arm to further suppress terminal vibration while completing trajectory tracking. In addition, the neural network sliding mode controller is highly robust to model uncertainty and disturbances, and the tracking error is less than 0.5°.

[0028] 3. The system of the present invention has good stability and is suitable for the complex environment of space. BRIEF DESCRIPTION OF THE DRAWINGS

[0029] Figure 1 Schematic diagram of the process of the present invention;

[0030] Figure 2 Schematic diagrams comparing simulation results in embodiments of the present invention, wherein (a) is a schematic diagram of the simulation of joint angle trajectories using the hybrid optimization strategy and cubic polynomial optimization of this embodiment; (b) is a schematic diagram of the simulation of joint angular velocity trajectories using the hybrid optimization strategy and cubic polynomial optimization of this embodiment; (c) is a schematic diagram of the simulation of joint angular acceleration trajectories using the hybrid optimization strategy and cubic polynomial optimization of this embodiment; (d) is a schematic diagram of the convergence curve of the fitness value of the hybrid optimization algorithm; and (e) is a schematic diagram of the change in vibration amplitude of the end effector using the hybrid optimization strategy and cubic polynomial optimization of this embodiment.

[0031] Figure 3 Schematic diagram comparing trajectory tracking of the method according to an embodiment of the present invention and that of the PD controller;

[0032] Figure 4 Schematic diagram comparing trajectory tracking errors of the method according to an embodiment of the present invention and the PD controller;

[0033] Figure 5 Schematic diagram of vibration of the end of the robotic arm of the method and PD controller according to an embodiment of the present invention. DETAILED DESCRIPTION

[0034] The specific embodiments of the present invention will be further described below with reference to the accompanying drawings:

[0035] like Figure 1 As shown, this embodiment provides a method for controlling flexible vibration of a tendon-driven manipulator based on a hybrid optimization strategy, comprising the following steps:

[0036] S1) Establish a dynamic model of the tendon-driven robotic arm; the details are as follows:

[0037] In this embodiment, the tendon-driven manipulator is converted into an Euler-Bernoulli beam model, and the lateral displacement of the flexible link can be expressed as the modal vibration function φ according to the assumed modal method. i (x) and the generalized hypothetical modal coordinate δ iThe linear combination of the flexible link relative to the position x in the rotating coordinate system and the time t gives the lateral displacement w(x,t), that is:

[0038]

[0039] Where m is the number of modal truncation terms of the flexible link;

[0040] Then, the Lagrangian method is used to obtain the dynamic model of the tendon-driven robotic arm:

[0041]

[0042] Where, is the inertia matrix; for Coriolis and centrifugal matrices; is a generalized coordinate variable; is the generalized coordinate variable velocity; is the generalized coordinate variable acceleration; T represents the transposed sign; θ k represents the kth joint angle; δ kj represents the j-th mode of the k-th flexible joint angle; n and m represent the joint angle dimension and mode dimension; is the stiffness matrix; Represents the motor current vector, J m Jacobian matrix representing the current-torque relationship.

[0043] S2) Constructing a trajectory optimization model for suppressing residual vibration based on a quintic polynomial; specifically comprising the following steps:

[0044] S21) Use a quintic polynomial function to represent the trajectory of each joint of the robotic arm, that is:

[0045]

[0046] Where θ d,i (t) represents the expected trajectory after optimization; t represents time; represents the coefficient of the trajectory function to be determined; N represents the number of joints of the tendon-driven flexible robotic arm;

[0047] S22) Construct the boundary conditions of the quintic polynomial function; namely:

[0048]

[0049] Where θ 0,i ,θ f,i They represent the initial value and the end point of the expected trajectory respectively; t f Indicates the total optimization time; They represent the first-order derivative and second-order derivative of the expected trajectory respectively; v 0,i 、vf,i They represent the initial velocity and terminal velocity of the desired trajectory respectively; a i represents the expected trajectory acceleration;

[0050] S23) Calculate the third-order derivative of equation (2-1), and according to the properties of the quadratic equation, obtain the expected trajectory acceleration a i The discriminant Δ i for:

[0051]

[0052] Case 1: When the discriminant Δ i ≤0, only t=0 and t=t f The acceleration constraint when , we can get:

[0053] t = 0 and t = t f Ensure the joint angular acceleration of the desired trajectory In the entire interval [0,t f ] satisfies the acceleration constraint; therefore, the necessary and sufficient conditions for satisfying the acceleration constraint are:

[0054]

[0055]

[0056] Therefore, the performance index of residual vibration is defined as:

[0057]

[0058] Where, represents the terminal amplitude cost function; is the position of the tendon-driven flexible manipulator end effector relative to the inertial coordinate system XOY at the corresponding moment; t s represents an optimized time span not less than the vibration period;

[0059] Case 2: When the discriminant Δ i >0, There is a maximum and a minimum value respectively;

[0060]

[0061] Where, t i,1 , t i,2 They represent extreme points respectively;

[0062] If the joint angular acceleration of the desired trajectory is The extreme value appears in [0,t f ] range, it will destroy its monotonic change condition. In order to meet the acceleration constraint, It must be set at t = 0, t = t f and two extreme points t i,1 , t i,2 The constraints are satisfied everywhere, so the following conditions must also be met:

[0063]

[0064] Where the function ψ(·) is defined as: Indicates the joint acceleration at the current time;

[0065] In addition, in order to prevent the acceleration extreme value from exceeding the limit, the following penalty function is introduced:

[0066]

[0067]

[0068] Where, Represents the penalty function cost; is the penalty factor. When the acceleration exceeds the limit, the penalty function value increases, thereby inhibiting the optimization algorithm from selecting such solutions.

[0069] S24), the expression of the optimization objective function of the trajectory optimization model for suppressing residual vibration is finally obtained:

[0070]

[0071] Where, represents the total cost function; represents the terminal amplitude cost function; represents the penalty factor cost function; denote the trajectory coefficients respectively; is the inertia matrix; for Coriolis and centrifugal matrices; is the generalized coordinate variable velocity; is the generalized coordinate variable acceleration; T represents the transposed sign; θ k represents the kth joint angle; δ kj represents the j-th mode of the k-th flexible joint angle; n and m represent the joint angle dimension and mode dimension; is the stiffness matrix; Represents the motor current vector, J m Jacobian matrix representing the current-torque relationship.

[0072] S3), using a hybrid particle swarm optimization strategy to solve the trajectory optimization model for suppressing residual vibration and obtain the desired trajectory;

[0073] The hybrid particle swarm optimization strategy includes the particle swarm optimization algorithm PSO and the genetic algorithm GA. The optimization problems of formula (2-11) and formula (2-12) are solved by the hybrid particle swarm optimization strategy. The search space is 2N dimensions, which is composed of the coefficients of all joints. and The updated position and velocity of the descendant particles are expressed as follows:

[0074]

[0075] Where ch(px) and ch(pv) represent the position and velocity of the offspring particle respectively; r is a random number between 0 and 1; pt1(px) and pt2(px) represent the positions of parent class 1 particles and parent class 2 particles respectively; pt1(pv) and pt2(pv) represent the velocities of parent class 1 particles and parent class 2 particles respectively.

[0076] S4) Designing a sliding mode controller based on an RBF neural network to actively track the trajectory to suppress the residual vibration, including the following steps:

[0077] S41) Using RBF neural network to estimate the total interference of the system, for the continuous function of RBF neural network The estimation is done in the following form:

[0078]

[0079] Where, Represents the estimated value of the RBF neural network output; x(t) represents the input state vector, is the RBF neural network estimation error, satisfying ||ε f ||≤ε N , ε N is any given positive constant; is the weight matrix; l represents the number of nodes in the RBF neural network; h(x(t) is the Gaussian function;

[0080] in:

[0081]

[0082] Where x(t) represents the state vector; c i represents the center of the i-th basis function; σ represents the standard deviation of the i-th mirror basis function;

[0083] S42) Define the inertia matrix, Coriolis matrix and centrifugal matrix as:

[0084]

[0085] Among them, M 11 ~M 22 is an element of M(Θ); C 11 ~C 22 for Elements of

[0086] The dynamic model of the tendon-driven robotic arm (1-2) is decomposed into:

[0087]

[0088] Where, Represent joint angular velocity and joint angular acceleration respectively; δ, Represent the modal coordinates and their first and second order derivatives respectively; τ s Indicates the elastic loss torque of the rope; K b represents the stiffness matrix; represents the motor current vector, j m The Jacobian matrix representing the current-torque relationship;

[0089] S43) Substitute formula (4-6) into formula (4-5) to obtain:

[0090]

[0091] S44) Define joint angle tracking error Δθ = θ - θ d , and substitute it into formula (4-7) to obtain the joint angular acceleration trajectory tracking error

[0092]

[0093] in, is the unknown disturbance term; θ represents the actual joint angle; θ d 、 They represent the expected joint angle, expected joint angular velocity, and expected joint angular acceleration respectively; represents the joint angular velocity trajectory tracking error;

[0094] S45) The sliding surface s of the sliding mode controller is designed as:

[0095]

[0096] Then, there is

[0097] Designed as equivalent control torque u eq (t):

[0098]

[0099] Where, s, Represent the switching function of the sliding surface and the derivative of the sliding surface respectively; A0 and A1 are controller parameters; dv is represented as the differential operator;

[0100] S46) have Expressed in state space form, we get:

[0101]

[0102] Where Δθ (2) represents the state vector related to the second-order derivative of Δθ; Φ represents the state transfer matrix; A 0～1 Represents the system matrix combination composed of A0 and A1; Δθ (0～1) represents the state vector composed of Δθ and its first-order derivative; I represents the identity matrix;

[0103] S47) The control law of the sliding mode controller is obtained as follows:

[0104]

[0105] Where, To switch control items; is the system lumped interference estimated by RBF neural network; λ is a non-negative constant; is a positive constant, is the switching gain; sign is the sign function.

[0106] The adaptive law of the RBF neural network is defined as Where γ is the adaptive gain.

[0107] This example verifies the effectiveness of the proposed controller through numerical simulation. The trajectory planning method is used to suppress the residual vibration of the flexible link manipulator, and the designed controller is used to track the trajectory to further achieve vibration suppression.

[0108] Based on the angular position and angular velocity conditions of the joints, a standard cubic polynomial widely used in trajectory planning is introduced for comparison; the parameters of the PSO algorithm are set as follows: population size is 50, maximum number of iterations is 50, penalty coefficient is 1000, and inertia weight is 0.9. The motion parameters of the flexible link manipulator in the two groups are set as follows: terminal time t f = 5s, initial joint angle θ0 = 0rad, terminal joint angle θ f =π / 2rad, initial joint angular velocity v0 = 0rad / s, terminal joint angular velocity v f =0rad / s, maximum joint angular acceleration a = πrad / s 2 The trajectory planning results are as follows: Figure 2As shown in (a)-(e) in the figure. Under the action of the optimized hybrid algorithm, the vibration amplitude of the end effector is significantly reduced, indicating the effectiveness of the trajectory planning method in suppressing residual vibration. The residual vibration performance index of the optimized trajectory is It is more than 40% lower than the traditional cubic polynomial.

[0109] According to the simulation results of trajectory optimization, the residual vibration of the flexible link manipulator is effectively suppressed; however, since the influence of the system structure damping is not considered, the residual vibration does not decay over time after the movement. In view of this, the traditional PD controller is introduced as a benchmark for comparison. Figure 3-5 shown.

[0110] from Figure 3-5 It can be seen that the joint angle tracking error of the method of this embodiment is significantly reduced, and the vibration amplitude of the end effector is also greatly reduced, which verifies the effectiveness of the method of this embodiment in suppressing residual vibration.

[0111] The above embodiments and descriptions are only for explaining the principles and best embodiments of the present invention. Without departing from the spirit and scope of the present invention, the present invention may be subject to various changes and improvements, which shall fall within the scope of the invention to be protected.

Claims

1. A flexible vibration control method for a tendon-driven manipulator based on a hybrid optimization strategy, characterized in that: The steps include: S1), establishing a dynamic model of the tendon-driven robotic arm; S2), constructing a trajectory optimization model for suppressing residual vibration based on quintic polynomial; S3), using a hybrid particle swarm optimization strategy to solve the trajectory optimization model for suppressing residual vibration and obtain the desired trajectory; S4) Design a sliding mode controller based on RBF neural network using the full drive system method, use RBF neural network to estimate the unknown disturbance of the system, and realize trajectory tracking and residual vibration suppression through the sliding mode controller.

2. The method for controlling flexible vibration of a tendon-driven manipulator based on a hybrid optimization strategy according to claim 1, characterized in that: In step S1), the dynamic model of the tendon-driven manipulator takes into account the lateral vibration and tendon driving characteristics of the flexible link, adopts the Euler-Bernoulli beam model to describe the flexible link, and is derived by the Lagrangian method to obtain the dynamic model of the tendon-driven manipulator: Where, is the inertia matrix; for Coriolis and centrifugal matrices; is a generalized coordinate variable; is the generalized coordinate variable velocity; is the generalized coordinate variable acceleration; T represents the transposed sign; θ k represents the kth joint angle; δ kj represents the j-th mode of the k-th flexible joint angle; n and m represent the joint angle dimension and mode dimension; is the stiffness matrix; represents the motor current vector, j m Jacobian matrix representing the current-torque relationship.

3. The method for controlling flexible vibration of a tendon-driven manipulator based on a hybrid optimization strategy according to claim 2, characterized in that: In step S2), a fifth-order polynomial function is used to represent the trajectory of each joint of the robotic arm, namely: Where θ d,i (t) represents the expected trajectory after optimization; t represents time; represents the coefficient of the trajectory function to be determined; N represents the number of joints of the tendon-driven flexible robotic arm; The boundary conditions of the quintic polynomial are: Where θ 0,i ,θ f,i They represent the initial value and the end point of the expected trajectory respectively; t f Indicates the total optimization time; They represent the first-order derivative and second-order derivative of the expected trajectory respectively; v 0,i 、v f,i They represent the initial velocity and terminal velocity of the desired trajectory respectively; a i represents the expected trajectory acceleration.

4. The method for controlling flexible vibration of a tendon-driven manipulator based on a hybrid optimization strategy according to claim 3, characterized in that: In step S2), the desired trajectory acceleration a is obtained by taking the third-order derivative of equation (2-1): i The discriminant Δ i for:

5. The method for controlling flexible vibration of a tendon-driven manipulator based on a hybrid optimization strategy according to claim 4, characterized in that: In step S2), when the discriminant Δ i ≤0, only t=0 and t=t f The acceleration constraint at the time of , the joint angle acceleration of the desired trajectory can be obtained t = 0 and t = t f Ensure the joint angular acceleration of the desired trajectory In the entire interval [0,t f ] satisfies the acceleration constraint; therefore, the necessary and sufficient conditions for satisfying the acceleration constraint are: Therefore, the performance index of residual vibration is defined as: Where, represents the terminal amplitude cost function; is the position of the tendon-driven flexible manipulator end effector relative to the inertial coordinate system XOY at the corresponding moment; t s Represents an optimization time span that is not less than the vibration period.

6. The method for controlling flexible vibration of a tendon-driven manipulator based on a hybrid optimization strategy according to claim 5, characterized in that: In step S2), when the discriminant Δ i >0, joint angular acceleration of the desired trajectory There is a maximum and a minimum value respectively; Where, t i,1 , t i,2 They represent extreme points respectively; Jerk of desired trajectory It must be set at t = 0, t = t f and two extreme points t i,1 , t i,2 The constraints are satisfied everywhere, so the following conditions must also be met: Where the function ψ(·) is defined as: Indicates the joint angular acceleration at the current time; In order to prevent the acceleration extreme value from exceeding the limit, the following penalty function is introduced: Where, Represents the penalty function cost; is the penalty factor. When the acceleration exceeds the limit, the penalty function value increases, thereby inhibiting the optimization algorithm from selecting such solutions.

7. The method for controlling flexible vibration of a tendon-driven manipulator based on a hybrid optimization strategy according to claim 6, characterized in that: In step S2), the trajectory optimization model for suppressing residual vibration takes minimizing residual vibration as the optimization objective function, and its expression is: Where, represents the total cost function; represents the terminal amplitude cost function; represents the penalty factor cost function; denote the trajectory coefficients respectively; is the inertia matrix; for Coriolis and centrifugal matrices; is the generalized coordinate variable velocity; is the generalized coordinate variable acceleration; T represents the transposed sign; θ k represents the kth joint angle; δ kj represents the j-th mode of the k-th flexible joint angle; n and m represent the joint angle dimension and mode dimension; is the stiffness matrix; Represents the motor current vector, J m Jacobian matrix representing the current-torque relationship.

8. The method for controlling flexible vibration of a tendon-driven manipulator based on a hybrid optimization strategy according to claim 7, characterized in that: In step S4), the RBF neural network is used to estimate the total interference of the system. For the continuous function of the RBF neural network The estimation is done in the following form: Where, Represents the estimated value of the RBF neural network output; x(t) represents the input state vector, is the RBF neural network estimation error, satisfying ||ε f ||≤ε N , ε N is any given positive constant; is the weight matrix; T represents the transpose operation; l represents the number of nodes in the RBF neural network; h(x(t)) is the Gaussian function.

9. The method for controlling flexible vibration of a tendon-driven manipulator based on a hybrid optimization strategy according to claim 8, characterized in that: In step S4), define the joint angle tracking error Δθ = θ - θ d , and the sliding surface s of the sliding mode controller is designed according to the tracking error Δθ: Design the equivalent control torque u according to the sliding surface s eq (t) is: Where s represents the sliding surface; A0 and A1 are controller parameters; dv represents the differential operator; is the unknown disturbance term; M 11 ~M 22 is an element of M(Θ); C 11 ~C 22 for Elements of represents the first-order derivative of the modal coordinate δ; τ s Indicates the elastic loss torque of the rope; K b represents the stiffness matrix; Joint angle tracking error; represents the joint angular velocity trajectory tracking error J m The Jacobian matrix representing the current-torque relationship; is the desired joint angular velocity.

10. The method for controlling flexible vibration of a tendon-driven manipulator based on a hybrid optimization strategy according to claim 9, characterized in that: In step S4), by introducing the switching control item u sw The control law u(t) of the sliding mode controller is obtained as: Where, To switch control items; is the system lumped interference estimated by RBF neural network; λ is a non-negative constant; is a positive constant, is the switching gain; sign is the sign function.

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