Position / Force Tracking Impedance Control for Robotic Systems with Uncertainties Based on Adaptive Jacobian and Neural Network

In this paper, an adaptive Jacobian and neural network based position/force tracking impedance control scheme is proposed for controlling robotic systems with uncertainties and external disturbances. To achieve precise force control performance indirectly by using the position tracking, the control scheme is divided into two parts: the outer-loop force impedance control and the innerloop position tracking control. In the outer-loop, an improved impedance controller, which combines the traditional impedance relationship with the PID-like scheme, is designed to eliminate the force tracking error quickly and to reduce the force overshoot effectively. In this way, the satisfied force tracking performance can be achieved when the manipulator contacts with environment. In the inner-loop, an adaptive Jacobian method is proposed to estimate the velocities and interaction torques of the end-effector due to the system kinematical uncertainties, and the system dynamical uncertainties and the uncertain term of adaptive Jacobian are compensated by an adaptive radial basis function neural network (RBFNN).Then, a robust term is designed to compensate the external disturbances and the approximation errors of RBFNN. In this way, the command position trajectories generated from the outer-loop force impedance controller can be then tracked so that the contact force tracking performance can be achieved indirectly in the forced direction. Based on the Lyapunov stability theorem, it is proved that all the signals in closed-loop system are bounded and the position and velocity errors are asymptotic convergence to zero. Finally, the validity of the control scheme is shown by computer simulation on a two-link robotic manipulator.


Introduction
The development of technology has aroused people's evergrowing interest on the safer and reliable, higher accuracy of robots.Many position tracking control methods have made some achievements in various application fields in recent years, such as cutting, grinding, welding, and so on [1][2][3]; the insufficient compliance for manipulator is always a key problem and has been given a lot of attention in robotics field, especially in complex and accurate tasks and even in collaboration with humans [4,5].Therefore, the applications of human-robot interaction (HRI) have become a new development domain and direction [6,7].It is necessary to develop an interaction control method that achieves position tracking and reliably adapts the force exerted on the environment in order to avoid damage both in the environment and in the manipulator itself [8].
The traditional force control methods proposed since 1980s can be mainly classified into two categories, namely, hybrid position/force control [9] and impedance control [10].Impedance control method is to apply the motion trajectory and contact force into one dynamic framework, which can avoid the separate control processes of positions and force.In general, compared with the hybrid position/force control, the impedance control method can reduce the complexity of controller design and has better adaptability and robustness in the complex dynamic tasks [11].Hence, in order to achieve the stable force tracking control, Chan and Yao [12] integrated sliding mode control into the impedance control method, which includes ideal impedance relationship in the sliding mode.Seul et al. [13,14] proposed a series of force tracking impedance control methods.In [13] the impedance control schemes were classified into torque-based and position-based impedance methods based on different implementations of impedance function.Then, the manipulator was controlled in the free space and in the contact space [14], where the impedance function was improved to achieve position and force tracking, and the force error could be converged to zero for any environmental stiffness by using an adaptive technique.The force tracking performance of these methods, however, greatly depends on the accurate environmental information and the precise robotic system model.
Recently, many researchers tried to introduce the intelligent control methods into the force control to improve the tracking performance and robustness of the robotic system [15][16][17][18][19][20].In [21], the neural network control technique was applied in impedance controller to compensate the uncertainties in an online manner.Li and Liu [22] designed an adaptive impedance hybrid controller, which could implement the desired contact force and track the command position in orthogonal subspace without precise environmental information.Jhan and Lee [23] proposed an adaptive fuzzy NN-based impedance control, where the adaptive fuzzy NN was used to approximate the robot dynamical model, so that the actual parameters of the manipulator need not be precisely known.In [24], a fuzzy logic system was applied as an approximator to estimate the unknown system dynamics, and then the proposed adaptive fuzzy back-stepping position/force control method could ensure all the signals of the close-loop system ultimately uniformly bounded.Duan et al. [25] presented an adaptive variable impedance control, which can adapt the environmental stiffness uncertainties.However, the contact force overshoot, which generated from the contact between robot and environment, was usually unconcerned in the above methods design.An improved impedance relationship was proposed in [26,27] to reduce the contact force overshoot and to achieve the direct accurate time-varying force tracking.Roveda et al. [28,29] calculated the parameters online in external controller to reduce the contact force overshoot.The methods can reduce the contact force overshoot provided that the model parameters of robotic systems are precisely known.
Since the model parameters of robotic systems can not be precisely obtained generally, lots of methods such as the adaptive Jacobian scheme, neural networks, back-stepping technology were employed to facilitate the tracking control of the manipulator with kinematic and dynamic uncertainties [30][31][32].Liang et al. [33] designed a task-space observer to estimate the task-space positions and velocities simultaneously while NNs were employed to further improve the control performance through approximating the modified robot dynamics.These methods designed in Cartesian space can avoid inverse kinematical solution; however, there were only the position control problems included.To obtain the force control in Cartesian space, Jung et al. [13] proposed the position-based impedance control method, which can introduce the position-based control methods into the impedance control schemes.Literatures [27,34,35] used two-loop controllers, which included the inner-loop position control and the outer-loop force control, to achieve the position and force tracking control.Based on these methods, Bonilla et al. [36] proposed an inverse dynamical control law based on impedance control to achieve the position path tracking in both free and constrained spaces, which mainly focused on developing the compliant control scheme for constrained path tracking; however, the force tracking was not involved.
Consider the above drawbacks, in this paper, an adaptive neural network position/force tracking impedance controller is proposed for controlling robotic system with uncertainties in both free and contact spaces, where the force tracking can be achieved by position-based control method and the force overshoot can also be reduced efficiently.Consequently, the main contributions of this paper are presented as follows: (1) Two-loop control architecture is presented to facilitate the position/force tracking impedance control for robotic systems, which are the outer-loop force impedance control and the inner-loop position tracking control.
(2) In the outer-loop, the improved impedance relationship based on PID-like scheme is proposed to reduce force overshoot when the end-effector contacts with environment.
(3) In the inner-loop, the adaptive Jacobian and RBFNN methods are employed to compensate the system model parameter and dynamical uncertainties.Then, the approximation errors of RBFNN and the external disturbances are restrained by a robust term.In this way, the desired contact force and the desired position trajectories can be tracked efficiently by the proposed intelligent position-based impedance control method.
(4) Based on the Lyapunov stability theorem, it is proved that all the signals in the closed-loop system are bounded, the position and velocity errors can be asymptotically converged to zero, and the contact force can be tracked to the desired force.
The reminder of this paper is organized as follows.In Section 2, some theoretical preliminaries are addressed, which consist of the mathematical notations, the RBFNN, and the details on dynamics of robotic system.In Section 3, the outer-loop position-based force tracking impedance control is designed to regulate interaction force when the robot contact with environment.Based on the reference trajectory generated from the outer-loop force impedance control, the adaptive neural network position tracking control scheme and its stability analysis are derived in Section 4. In Section 5, simulation results are presented to verify the effectiveness of the proposed method, and conclusions are drawn in Section 6.

Problem Formulation and Preliminaries
In this paper, standard notations are used.We denote R as the real number set, R  as the n-dimensional vector space, and R × represents the  ×  real matrix space.The norm of vector  ∈ R  and that of matrix  ∈ R × are defined as ‖‖ = √  T  and ‖‖ = √ tr( T ), respectively.If  is a scalar, let ‖‖ denote the absolute value. min () and  max () are the minimum and the maximum eigenvalue of matrix , respectively. × is the  ×  identity matrix.

Description of RBFNN.
In general, RBFNN has fast learning convergence speed and strong capability and has been proved in theory that RBFNN can approximate any nonlinear continuous function over a compact set to arbitrary accuracy [37].The RBFNN structure is described as follows: where  = [ where  = 1, .
where () is the RBFNN reconstruction error.Since the ideal weight  * is unknown, the estimated weight Ŵ is generally used to replace  * to approximate the unknown, continuous, nonlinear function; that is, where Ŵ is the estimated weight matrix and can be trained by a weight learning law.We assume that Ω  is existed, and ideal parameter is in the compact set Ω *  , which is defined as where   is a positive constant.
In this paper, to ensure the control performance of manipulator, the RBFNN is used as a compensator to eliminate the dynamical uncertainties and the uncertain term of adaptive Jacobian in robotic system.

Robotic Manipulator Dynamics and Properties.
Consider a general -degree of freedom (DOF) rigid robotic manipulators with uncertain dynamics, the dynamical model can be described as  () q +  (, q ) q +  () +   ( q ) +   =  −   (7) where , q , and q ∈ R  represent the joint angular position vectors, velocity vectors, and acceleration vectors of the manipulator, respectively; () ∈ R × is the positive definite and symmetric inertia matrix; (, q ) ∈ R × represents the effect of centrifugal and Coriolis forces; () ∈ R  is the gravity vector;   ( q ) ∈ R  is the friction effects;   ∈ R  denotes the bounded unknown disturbances including unknown payload dynamics and unstructured dynamics;  ∈ R  is the torque input vector;   ∈ R  is the interaction torque vector when the manipulator contacts with environment; () ∈ R × represents the Jacobian matrix from join space to task space; and   ∈ R  is the contact force at the end-effector.
The following properties and assumption are required for the subsequent development.
Property 2. The inertia matrix () is positive definite and symmetric, which is uniformly bounded and satisfies where   and   are some positive constants.
Assumption 5.The unknown disturbance term is bounded by ‖  ‖ ≤   where   is a positive constant.
Let  ∈ R  be the position vector of the end-effector in task space.The relation between task space and joint space can be described by forward kinematics as where (⋅) is the forward kinematics map, generally a nonlinear transformation between task space and joint space.The task-space velocities of end-effector Ẋ is related to joint velocities q as Assumption 6.In general, a manipulator should work in a finite task space.The matrix  −1 () ∈ R × is the inverse matrix of the Jacobian matrix () when  = .When  ̸ = , the inverse matrix of the () can be represented as where  + () denotes the generalized inverse matrix of ().Similar notations hold for the estimate Jacobian matrix Ĵ(, ) as detailed later in Section 4.

Complexity
In general, the uncertainties of the model parameters and the robot dynamics decreased the control performances of the robotic system directly.In this paper, an improved impedance relationship is designed to derive the reference trajectory planning scheme so that the reference position trajectories can be then generated.Then, an intelligent robust position-based impedance control scheme is proposed to achieve position trajectory tracking performance and the contact force tracking performance, where an adaptive Jacobian and RBFNN methods are designed to compensate the system uncertainties of robotic manipulator.

Design of Outer-Loop Force Tracking Control Based on Improved Impedance Relationship
Impedance control method regulates the relationship between the position and force by selecting suitable impedance parameters.According to literature [10], the traditional impedance relationship of the robotic system satisfies where   ∈ R  is the planned reference trajectory of endeffector for position control, which determined from environmental position and parameters, impedance parameters, and desired contact force; , , and  ∈ R × are the desired inertia, damping, and stiffness matrices, respectively;   ∈ R  is the force exerted on the environment at the endeffector.
In general, the contact force   is determined by environmental stiffness and environmental damping; therefore, a second-order nonlinear function is used to approximate the environmental model, which can be expressed as where   ∈ R × and   ∈ R × denote the diagonal symmetric positive definite environmental stiffness and damping matrices, respectively, and   ∈ R  is the environment position vector.Assume that the environment position is a constant, we have Ẋ = 0, then ( 16) can be rewritten as 3.1.Reference Trajectory Planning.The force tracking response cannot generally be achieved quickly by using the traditional impedance control method shown in (15); when the end-effector of manipulator contacts with environment, the force overshoot may result in task failures.Therefore, in this paper, a PID-like impedance relationship is designed to improve the force tracking performance, which can be expressed as where   ,   , and   ∈ R × are the diagonal symmetric positive definite parameter matrices.The introduced PIDlike force compensation can achieve a better expectation than the pure force in (15) so that the contact force generated at the end-effector can quickly converge to the desired value and reduce the force overshoot.
For convenience, we consider the force is exerted on one direction only.Replacing   , ,   ,   ,   , , , ,   ,   ,   ,   ,   by   , ,   ,   ,   , , , ,   ,   ,   ,   ,   , respectively, then the improved impedance function (18) can be rewritten as and the environment model (17) becomes Define   =   −   ; ( 19) and ( 20) can be rewritten as and Taking Laplace transform to ( 21) and ( 22) yields and Then, (24) can be rewritten as Substituting ( 25) into ( 23) yields Then, the force tracking    () in frequency domain can be obtained as where Then, steady state force tracking error can be obtained as To ensure the steady state force error   to be zero as the system approaches the stable equilibrium state, the reference position trajectory can be designed as According to (30), it is obvious that the planned reference position trajectory   () is a dynamic function including the desired force   (), the environmental position   , the environmental stiffness   and damping   , the impedance parameters , , , and the PID-like parameters   ,   ,   .Assumed that the parameters in the improved impedance relationship (18) have been selected properly and the environmental information is accurately obtained, the reference position trajectory   () can be generated by the desired input force   ().

Position-Based Impedance Control Scheme.
In general, the traditional impedance control was classified into two methods: the position-based and the torque-based impedance control [13].In the position-based impedance control, the outer-loop force impedance, and the inner-loop position tracking can be designed separately, where the force tracking performance mainly depends on the accuracy of the position tracking control in the inner-loop.Therefore, the position-based impedance control method has been widely applied in complex industrial systems such as servo control.
Denote the position command   as the control input of the inner-loop position tracking control, where  =   − .Based on the improved impedance equation (18), the desired relationship between  and   can be represented as Taking Laplace transform to (32) yields To achieve the force tracking performance, the contact force   is regulated to track the desired force   ; an intelligent-based robust position tracking controller will be designed in Section 4.
Remark 7. The improved impedance relationship combines the PID-like method with the traditional impedance relationship in outer-loop force control to improve the response speed and the performance of force tracking.By choosing the appropriate PID parameters, the force tracking errors and the force overshoot can be reduced effectively with fast convergence when the manipulator contacts with environment.In addition, the improved impedance method can be applied to track a constant force or a twice-differentiable time-varying force.
Remark 8. Assumed that the proposed inner-loop position tracking controller is "perfect", the manipulator can work well in the free and contact spaces based on the positionbased impedance control scheme [27,34,35].It means that the position command   satisfies (31) if the forced direction is considered only.In this way, the position tracking and the force tracking can be achieved based on the two-loop separation design method in the position-based impedance control scheme.

Design of Inner-Loop Position Tracking Control and Stability Analysis
In this section, an adaptive position tracking controller is proposed as the inner-loop in control system to track the command position trajectory   generated from the outerloop force impedance, where an adaptive Jacobian method is employed to approximate the task-space end-effector velocities and the interaction torques, and an adaptive RBFNN is designed to compensate the dynamic uncertainties of robotic systems and the uncertain Jacobian term.Based on Lyapunov theorem, the stability of the closed-loop robotic control system is then guaranteed.

Adaptive Jacobian and RBFNN Position Tracking Controller Design.
Define  = [ 1 , . . .,   ] T as the parameter vector in Jacobian matrix (); the task-space velocity of the end-effector and the robot interaction torque can be expressed as Ẋ =  () q =  (, q )  (34) where (, q ) ∈ R × and   (,   ) ∈ R × denote the velocity regressor matrix and the interaction torque regressor matrix, respectively.Note that the robot kinematic parameters uncertainties are always existed such as link length and mass so that the Jacobian matrix () cannot be known precisely.Therefore, define the estimated Jacobian matrix Ĵ(, θ) ∈ R × , the estimation ̂Ẋ of Ẋ and the estimation τ of   can be represented as ̂Ẋ = Ĵ (, θ) q =  (, q ) θ (36) where θ ∈ R  denotes the estimated parameters vector.
Then, the estimated task-space velocity error ̃Ẋ and estimated interaction torque error τ can be expressed as where θ =  − θ.Define a vector ] ∈ R  as where Λ = Λ T > 0 is a positive matrix and   =   −  is the position tracking error of the end-effector.Then, differentiating (40) with respect to time yields where Ė  is the velocity tracking error of the end-effector.Define a filtered tracking error as where ̂Ė  = Ẋ − ̂Ẋ is the estimated value of the velocity tracking error Ė  .Then, according to (36) and (40), we have Differentiating (43), we obtain where ̂Ë  and ̂Ẍ denote the derivative of ̂Ė  and ̂Ẋ, respectively.
In this paper, the unknown function () is approximated by using RBFNN, where () denotes the activation function of RBFNN,  * represents the ideal weight matrix, and () denotes the minimum reconstructed error vector.Then, the adaptive RBFNN position trajectory tracking control law can be given as where   > 0 and  V > 0 are the controller position and velocity gain matrices, respectively, and  rss denotes the robust compensation term which is used to compensate the approximation error of RBFNN and external disturbances.Substituting the control law (58) into (55), the closed-loop error equation of the robotic system is as follows: where W =  * − Ŵ denotes the weight estimated error.Then, substituting (39) and ( 60) into (59) results in By the projection algorithm, the adaptive updating law for the Jacobian matrix parameter θ and the weight matrix Ŵ of RBFNN are designed as follows: where Γ  and Γ  are both the positive matrices.Then, the filter error   , the task-space position error   , the adaptive Jacobian matrix parameter error θ, and the adaptive RBFNN weight matrix error W are all bounded and the contact force   can converge to desired force   as  → ∞.

Proof. Choose a Lyapunov function candidate as
Differentiating (66) with respect to time and substituting (61) yield According to the bounded modeling errors (62) and robust term (63) and the RBFNN adaptive law (65), considering the fact Ẇ = − Ẇ, we have and According to Property 3, substituting (68) and ( 69) into (67) yields Substituting ( 38), ( 39), (42), and ( 45) into (70) yields According to the adaptive updating law (64) and considering the fact θ = − θ , we have Substituting ( 72) into (71) yields According to (66) and ( 73), it can be concluded that the error signals   ,   , θ, and W are all bounded.That means θ and Ŵ are both bounded, and r = Ĵ(, θ)  is also bounded using (45).From (39),   is bounded since the contact force   is bounded.Then, according to (40) and ( 42), it can be concluded that ̂Ė  and ] are bounded as the position controller input   and Ẋ is bounded, and q ] is also bounded according to (47) as the estimate Jacobian matrix Ĵ(, ) is nonsingular.Therefore, q is bounded according to (46) which implies that Ẋ is bounded.Next, it can be concluded that Ė  = Ẋ − Ẋ is bounded and ] is also bounded from (41) as Ẍ is bounded.And ̂] , q ] are both bounded according to (51) and (52).In addition, according to Property 4, it can be concluded that ṙ  is bounded from (61) so that q is also bounded from (49), and then Ẍ is bounded, which means Ë  = Ẍ − Ẍ is bounded as Ẍ is bounded.Considering (44), ṙ  is bounded, which means ̂Ë  is also bounded.
Differentiating (73) with respect to time, we have where ̂Ë  denotes the derivative of ̂Ė  .Since the error signals   , Ė  , ̂Ė  , ̂Ë  are all bounded, V is uniformly continuous.
According to the above analysis, the block diagram of the whole closed-loop impedance control system is shown as Figure 1.
Remark 11.The position-based impedance control has a superior design feature that the pure position tracking control is designed separately as the inner-loop of the robotic control system.Based on the position trajectory command   generated from the outer-loop force impedance control (designed in Section 3), it allows the manipulator only uses a position trajectory tracking to achieve the desired contact force tracking and track the command position in orthogonal.If there is no robot−environment interaction, i.e.,   ≡ 0, the objective of impedance control is equivalent to the objective of unconstrained motion control in the task space.This design structure is more concise and convenient and easy to implement when the robotic system is complex and uncertainty exists.
Remark 12.In this paper, since the position trajectory command   is as the input of the inner-loop position trajectory tracking in Cartesian space rather than in joint space, the proposed task-space control scheme can effectively avoid the robot inverse kinematic solution so that the robustness of robotic system can be then improved.Eq.( 65) Eq.( 31) q, q q, q q, q Eq.( 36) The proposed closed-loop impedance control system.

Simulation Examples
To verify the theoretical results, simulations were conducted on a two-DOF robotic manipulator, which is described as, the Jacobian matrix is be given where  1 and  2 are the length of link 1 and link 2, respectively;  1 and  2 are the mass of link 1 and link 2, respectively;   denotes sin(  );   represents cos(  );   represents sin(  +  ), for ,  = 1, 2;   denotes cos(  +   ), for ,  = 1, 2;  is acceleration of gravity.

Design Procedure.
To summarize the analysis in Sections 3 and 4, the step-by-step procedures of the adaptive RBFNN impedance force/position tracking control based on impedance control for robotic system are outlined as follows: Step 1. Select the environmental stiffness and the damping parameters   = 2000 2×2 and   = 5 2×2 and the environmental position   = 1m.
Case 1. Assume that the desired force is constant   = [50, 0] T N and the contact force exserted on the end-effector occurred as  ≥   = 1 in x direction.The adaptive force tracking impedance control (AFTIC) in [14] is compared with the adaptive neural network position/force tracking impedance control (ANNPFTIC) proposed in this paper.Figures 2-7 show the results, where Figures 2(a In the simulation results in both Cases 1 and 2, we can see that the manipulator can work well both in free space and in contact space.When the manipulator is in free space, the position tracking (Figures 1 and 7) and velocity tracking (Figures 3 and 9) can be achieved both in x and y direction.When the manipulator is in contact space, the force tracking (Figures 5 and 11) can be achieved in x direction and the position tracking (Figures 1 and 7) and velocity tracking (Figures 3   and 9) can be obtained in y direction (x direction is the forced direction).From the simulation results, one can see that the position/force tracking errors of ANNPFTIC are smaller than those of AFTIC, where the ANNPFTIC can effectively reduce the force overshoot with fast response by choosing the appropriate PID-like parameters in improved impedance relationship.It means that the proposed ANNPFTIC can achieve better performance than AFTIC and can be applied on position/force controlling of robotic systems with uncertainties and disturbances efficiently.

Conclusions
In this paper, an adaptive neural network position/force tracking impedance control scheme is proposed for controlling robotic systems with uncertainties and external disturbances, where the robot can work in both free space and contact space.The control strategy is divided into the outer-loop force tracking impedance control and the innerloop position tracking control.In the outer-loop, a novel impedance relationship based on the PID-like scheme is proposed to improve the force tracking performance and to reduce the force overshoot, so that the good force tracking performance can be achieved when the manipulator contacts with environment.Next, according to the command position trajectory generated from the outer-loop force impedance control, an adaptive Jacobian and an adaptive RBFNN methods are designed in the inner-loop position tracking control to compensate the robotic system uncertainties so that the position and contact force tracking accuracy can be then improved, and a robust term is used to compensate the approximation error of RBFNN and the uncertain Jacobian

ComplexityFd
y axis (m) (b) Position tracking in the y direction
) and 2(b) are the desired and actual positions of end-effector in x and y directions of Cartesian space, respectively, Figures3(a) and 3(b) are the position errors between the desired and actual positions in x and y directions of Cartesian space, respectively, Figures 4(a) and 4(b) are the desired and actual velocities of end-effector in x and y directions of Cartesian space, respectively, Figures 5(a) and 5(b) are the velocity errors in x and y directions of Cartesian space, respectively, Figure 5(a) is the force exerted on the environment at endeffector in x direction, Figure 6(b) is the force error in x direction, and Figures 7(a) and 7(b) are the end-effector position trajectory and joint angular position, respectively.Case 2. Assume that the desired force is time-varying force   = [50 + 20 sin(2), 0] T N. Similar to Case 1, Figures 8-13 show the results, where Figures 8(a) and 8(b) are the desired and actual positions of end-effector in x and y directions of Cartesian space, respectively, Figures 9(a) and 9(b) are the position errors of end-effector in x and y directions of Cartesian space, respectively, Figures 10(a) and 10(b) are the desired and actual velocities of end-effector in x and y directions of Cartesian space, respectively, Figures 11(a) and 11(b) are the velocity errors of end-effector in x and y directions of Cartesian space, respectively, Figure 10(a) is the force exerted on the environment of end-effector in the x direction, Figure 12(b) is the force error in the x direction, and Figures 13(a) and 13(b) are the end-effector position trajectory and joint angular position, respectively.

Fd
Position error in the y direction

Fd
Velocity error in the y direction
Position tracking in the y direction

Fd
x axis (m/s) (a) Velocity error in the x direction Velocity error in the y direction

Figure 11 :FdFigure 12 :Figure 13 :
Figure 11: Velocity tracking errors for Case 2.Fd = 50+20sin(2t) N , the validity of the control scheme is shown by computer simulation on a two-DOF robotic manipulator.
1 , . . .,    ] T ∈ R   is the input vector, (, ) is the output vector,   is the control input dimension,   is the neuron node number,   is the output dimension,  = [ 1 , . . .,    ] T ∈ R   ×  is the weight matrix with   ∈ R   ,  = 1, . . .,   , () = [ 1 (), . . .,    ()] T ∈ R   is the RBFNN active function with hidden layer output function   (), and the Gaussian function is chosen as follows: . .,   ,  = [ 1 , . . .,    ] T ∈ R   is the center of the th neuron node, and   is the width of the th neuron.Numerous results indicate that for any continuous smooth function () : Ω  →  over a compact set Ω  ∈ R   , applying RBFNN (1) to approximate (), if   is sufficiently large, a set of ideal bounded weights  * exist, and we have