Dynamic Learning from Adaptive Neural Control of Uncertain Robots with Guaranteed Full-State Tracking Precision MinWang

A dynamic learning method is developed for an uncertain n-link robot with unknown system dynamics, achieving predefined performance attributes on the link angular position and velocity tracking errors. For a known nonsingular initial robotic condition, performance functions and unconstrained transformation errors are employed to prevent the violation of the full-state tracking error constraints. By combining two independent Lyapunov functions and radial basis function (RBF) neural network (NN) approximator, a novel and simple adaptive neural control scheme is proposed for the dynamics of the unconstrained transformation errors, which guarantees uniformly ultimate boundedness of all the signals in the closed-loop system. In the steadystate control process, RBF NNs are verified to satisfy the partial persistent excitation (PE) condition. Subsequently, an appropriate state transformation is adopted to achieve the accurate convergence of neural weight estimates. The corresponding experienced knowledge on unknown robotic dynamics is stored in NNs with constant neural weight values. Using the stored knowledge, a static neural learning controller is developed to improve the full-state tracking performance. A comparative simulation study on a 2-link robot illustrates the effectiveness of the proposed scheme.


Introduction
In the past decades, the force/position tracking control problem of robots has attracted wide attention in both theory and applications [1,2].In the early stage of robotic tracking control, the system model is usually assumed to be accurately known, and the corresponding model-based control method has been proposed in [3,4].Along with the diversity of robot working environment and the complexity of the robot's structure, the force/position tracking control has been studied in different kinds of uncertainties.Nowadays, ignoring uncertainties to simplify control design may cause the large steady-state errors or/and poor transient response [5].For the case of parametric uncertainties, adaptive control method has been presented in [6,7] to make robots adapt the changing control environment.In order to enhance system robustness on the uncertain parameters in the presence of external disturbances, sliding mode control [8,9] has been proposed to obtain the desired robotic tracking control performance.Owing to the universal approximation property [10][11][12][13][14][15][16][17], a great number of intelligent control schemes, such as adaptive neural/fuzzy control, have been developed for controlling robotic systems with uncertain nonlinearities [18][19][20][21][22].
Although intelligent control of robotic systems has attracted considerable attention in the past few years, relatively few robot control methods could achieve human-like performance in a dynamic and uncertain environment.It is well known that intelligent control was initially inspired by the learning and control abilities of human beings, thus intelligent control should at least possess the aforementioned properties "learning by doing" and "doing with the learned knowledge" [23][24][25][26].But most existing intelligent control schemes can only ensure the stability of closedloop systems without being able to achieve the information acquisition and storage and utilization of unknown system dynamics.This means that the existing intelligent control schemes do not solve the accurate convergence of estimated parameters, which usually needs to guarantee the exponential stability of the derived closed-loop system.However, it is extremely difficult for uncertain nonlinear systems to verify 2 Complexity the exponential stability of the derived closed-loop system.Recently, a deterministic learning method was proposed in [27] using RBF NNs for a second-order Brunovsky system, where the derived closed-loop system was described by a class of linear time-varying systems.By verifying that RBF NNs satisfied persistent excitation (PE) condition, the convergence of partial neural weights and accurate neural approximation of unknown system dynamics were guaranteed in [27] because of the exponential convergence of the derived linear time-varying closed-loop systems.The deterministic learning method was further extended to thorder Brunovsky systems with an unknown affine/nonaffine term [28,29], where the derivative of unknown affine terms was assumed to be bounded.By combining backstepping with a system decomposition strategy, an elegant dynamic learning method was proposed to cope with the learning and control problem of third-order strict-feedback systems [30].By combining dynamic surface control technology [31], the result in [30] was further extended to th-order strictfeedback systems.The deterministic learning method was also applied in many physical systems such as marine surface vessels [32,33] and robot manipulators [34].
Recently, the deterministic learning methods are mainly used to solve the learning and control problem for single input single output nonlinear systems without any constraint.In practice, there are many different kinds of constraints in most of physical systems, such as output or state constraints, tracking performance constraints.The violation of the constraints may cause severe performance degradation, safety problem, or system damage [35].Therefore, it is of great importance for solving the control and learning problem of constrained systems.Based on Lyapunov theorem, a barrier Lyapunov function (BLF) method has been presented to solve output constraints for strict-feedback nonlinear systems [36], output feedback nonlinear systems [37], flexible systems [38], and robotic manipulator [35,39].The BLF-based methods were also extended to solve state constraints [40][41][42].Although the aforementioned results on the output or state constraints can guarantee that system outputs or states converge to a predefined bounded set, the predefined performance requirements on the convergence rate, maximum overshoot, and steady-state error have not been studied fully.The predefined performance issue is an extremely challenging problem.Recently, adaptive neural prescribed performance controller was proposed in [43,44] for feedback linearizable nonlinear systems by means of transformation functions.The proposed method was also developed to deal with the constrained output tracking control problem in many applications such as robotic systems [45], nonlinear servo mechanisms [46], marine surface vessels [33], nonlinear stochastic large-scale systems with actuator faults [47], and switched nonlinear systems [48].To solve partial tracking error constraints, a fuzzy dynamic surface control design was developed in [49,50] for a class of strict-feedback nonlinear systems by transforming the state tracking errors into new virtual variables.However, the existing control schemes, such as [35][36][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51], can only guarantee the stability of closed-loop systems with different constraints, which are not capable of achieving the learning of unknown system dynamics.The main reason is that the derived closed-loop error system is extremely complex, such that its exponential convergence is difficult to be verified using the existing stability analysis tools.To solve this problem, a neural learning control with the output tracking error constraint was presented in [52,53] for a class of nonlinear systems.These methods proposed in [52,53] cannot be adopted to deal with the dynamic learning problem for multi-input and multioutput nonlinear systems with full-state tracking error constraints.
Based on the above discussions, this paper proposes a novel dynamic learning method for a multi-input and multioutput -link robot with full-state tracking error constraints and a mild assumption.To prevent the violation of the fullstate tracking error constraints, performance functions are firstly introduced to characterize the transient and steadystate performance of full-state tracking errors.And then, using a nonlinear transformation method, the constrained tracking control problem is effectively transformed into the stabilization problem of equivalent unconstrained transformation error systems.By combining backstepping and Lyapunov stability, a novel adaptive neural control scheme is proposed to guarantee the uniformly ultimate boundedness of all closed-loop signals and the prescribed full-state tracking error performance.It should be pointed out that the proposed control scheme is different from the traditional backstepping design.In our control design, the correlative interconnection term can not be compensated in next step of backstepping design because of the full-state tracking error constraints.To overcome the difficulty, two independent Lyapunov functions are constructed in each step of backstepping design.Based on the independent Lyapunov functions, the unconstrained transformation error of the link angular velocity tracking error is firstly proved to be bounded, and the corresponding link angular velocity tracking error is further proved to satisfy the prescribed performance.Invoking the boundedness of the link angular velocity tracking error, we backward derive the link angular position tracking error to satisfy the prescribed performance.By means of the tracking convergence in the steady-state control process, the regression subvector consisting of the RBFs along the recurrent tracking orbit satisfies the partial PE condition.Subsequently, an appropriate state transformation is introduced to transform the closed-loop system into a linear time-varying (LTV) system with small perturbed terms.With the perturbation theory of LTV system, the knowledge of the closed-loop robotic dynamics can be accurately stored by RBF NNs with constant weight values.The stored knowledge can be reused to develop a static neural learning controller for achieving the better control performance with smaller transient-state tracking errors, smaller control gains, and less computational time.
The rest of this paper is organized as follows.Section 2 introduces the system formulation, the full-state tracking error transformation, and some useful preliminaries about RBF networks.In Section 3, a novel adaptive neural control design is proposed for rigid robotic manipulators with constrained full-state tracking performances.Neural learning control is developed in Section 4, which achieves the knowledge acquisition, storage, and utilization of unknown robotic dynamics.In Section 5, simulation studies on 2-link robotic system are given to show the effectiveness of the proposed method.Section 6 includes the conclusions of the paper.
In this paper, we choose   =  1 ∈   as a recurrent reference trajectory of the angular position , which is generated by the following reference model: where   = [  1 ,   2 ]  ∈  2 is the state vector of the reference model, which is assumed to be recurrent signals, and   (  ) is a known smooth function.The reference orbit along the given initial condition   (0) is defined as   (  (0)).In this paper,   (  (0)) is assumed to be a recurrent motion.

Full
where  = 1, 2,  = 1, 2, . . ., , and   ,   are positive design constants and   () is a smooth, strictly positive, bounded, and decreasing function, which satisfies lim →∞   () > 0 and is called performance function.In this paper,   () is chosen as the following exponential performance function: where  0 ,  ∞ and   are positive design constants.
For any given initial condition   (0), these design constants  0 ,   and   can be chosen appropriately such that   (0) satisfies the predefined condition (3).From (3) and ( 4), the different selection of these design parameters,  0 ,  ∞ ,   ,   and   , can obtain different performance requirements on the tracking error   .
(1) Universal Approximation Lemma 2 (see [55]).Using sufficiently large node number , the RBF NNs   () can approximate any smooth function () : Ω  →   over a compact set Ω  to any arbitrary accuracy as where  * is the ideal weight vector of  and () is any small approximation error which satisfies |()| ≤  * .
(2) Spatially Localized Approximation Property.It should be noted that the radial basic function satisfies   (‖ −   ‖) → 0 when ‖ −   ‖ → ∞.This property shows that the network output is only locally affected by each basis function.Therefore, any smooth function () over a compact set Ω  can be approximated using a limited number of neurons, which are located in a local region along bounded input trajectory : where   () ∈    ,   <  is the subvector of (), composed of RBFs that are close to the trajectory ,  *  ∈    is the corresponding subvector of  * , and approximation errors   () = () −  *     (),  are defined as the region far away from the trajectory , which means ‖  ()‖ → 0. Therefore,   () is close to (), which is any small value.
(3) Partial PE Condition.Persistent excitation plays a key role in accurate convergence of neural weight estimator.To clearly show that RBF NNs satisfy the PE property, a definition of PE condition is given as follows: Definition 3 (see [55]).A continuous, uniformly bounded, vector-valued function  : [0, ∞] →   is said to satisfy the persistent excitation condition, if there exist positive constants  1 ,  2 , and  such that holds for every constant vector  ∈   , where  is a positive, ∑ -finite Borel measure on [0, ∞].
Lemma 4 (see [27]).Consider any recurrent orbit  ∈   , remaining in a compact set Ω  with Ω  ⊂   .Then, for the RBF network   () with centers placed on a regular lattice, which is large enough to cover the compact set Ω  , the regressor subvector   () in ( 6) (rather than the entire regressor vector ()) is persistently exciting.

Adaptive Neural Control with Full-State Tracking Error Constraints
In this section, a novel stable adaptive neural tracking control scheme will be developed for the system (1) with fullstate tracking error constraints (3) using nonlinear error transformations, independent Lyapunov functions, and backstepping.The proposed control scheme will guarantee that all the signals in the closed-loop system are ultimately bounded and the full-state tracking errors satisfy the prescribed performances (3).
Constructing the following Lyapunov function candidate: whose derivative along ( 16) yields Remark 5. From ( 18), the boundedness of the transformation error  1 depends on the boundedness of the link angular velocity tracking error  2 .In the traditional backstepping design, the term   1 Υ 1  2 is called the correlative interconnection term, which can be usually compensated in the next step of backstepping.However, in our control design, it is impossible to deal with   1 Υ 1  2 in Step 2. The main reason is that the full-state tracking error constraints are considered in this paper, which derives the transformation error  2 in Step 2, instead of the traditional error  2 .Therefore, it is difficult to construct the Lyapunov  2 with  2 to compensate for   1 Υ 1  2 ; see Step 2 for details.
Step 2. By adding and subtracting Υ −1 2   (, q ) 2 , the transformed error subsystem ( 14) can be rewritten as where  = [  , q  ,   2 ,   ]  , and It should be pointed out that the term Υ −1 2   (, q ) 2 introduced in (19) facilitates the stability analysis based on the property (1).Since (, q ), , and   (, q ) are unknown smooth function vectors, Φ() is also unknown and smooth, which can not directly be used to construct the controller.To solve this problem, the unknown dynamics Φ() are approximated by RBF NN (5) in this paper.Then, we have where the approximation error () ∈   satisfies ‖()‖ ≤  * and  * = [ * 1 , . . .,  *  ] ∈  × is the unknown optimal NN weight vector with NN node number  > 1, and () = [ 1 (),  2 (), . . .,   ()]  ∈   is a radial basis function vector.Define Ŵ as the estimated weight values of  * , and let W = Ŵ −  * be the corresponding estimated error.Then, design the adaptive neural control law as and construct the neural weight updated law as where  20 and  21 are positive design constants, Γ > 0 is positive diagonal matrix, and  > 0 is a small value, which is used to improve the robustness of the adaptive controller (22).Substituting ( 22) into ( 21), we have where  = () − ().Noting that the inertia matrix  is positive definite, thus we construct the following Lyapunov function candidate: In what follows, one of our main results will be shown by the following theorem.
Theorem 6 (stability and tracking).Consider the closedloop system consisting of the robotic system (1), the bounded reference trajectory (2), the full-state tracking performance condition (3), the transformed error (10), the proposed adaptive neural control law (22) with the virtual control law (15), and the weight updated law (23).Assume the given bounded initial conditions satisfy the condition (3) (this condition can be satisfied by choosing proper designed parameters   ,   , and  0 ).Then, we have that (1) all signals of the closed-loop system remain uniformly ultimately bounded (2) the constrained full-state tracking errors   satisfy the prescribed performance (3) and they converge to a small residual set of zero in a finite time .
Proof.Differentiating  2 in (25) with respect to time and using ( 18), (24) yields Using the Property 1 and substituting ( 23) into (26), the derivation of  2 is rewritten as Subsequently, using the appropriate inequality and combining Assumption 1, we have where  * 2 =  * +  * is a positive constant.Substituting (28) into (27) gives where From the definition (25) of  2 , the transformed error  2 and the neural weight error W are bounded.Noticing that  2 = [ 21 ,  22 , . . .,  2 ]  ∈   , and the error transformed relationship ( 8)-( 10), thus we obtain that the tracking error  2 is bounded and satisfies the prescribed tracking error constraints (3).This means ‖ 2 ‖ ≤  * ,  * is any small value depending on design parameters   ,   ,  20 , and  2∞ .Using the bounded property of  2 , we can obtain that Noting  1 =  10 +  11 , thus (18) can be rewritten as where Further, let  1 =  1 / 1 ; we have Since  1 =   1  1 /2, it follows from (33) that  1 is bounded.Similarly, we can obtain that the tracking error  1 is bounded and satisfies the prescribed tracking error constraints (3).By combining the boundedness of  1 ,  2 and the deigned bounded function Υ, Ψ 1 , it can be verified that  and  1 are bounded.Because of  2 = q −  1 , q is also bounded.Because the desired weight  * is bounded, the weight estimate Ŵ is also bounded.From (22), we can verify that  is also bounded.Hence, all closed-loop signals are uniformly ultimately bounded.

Complexity
Moreover, from ( 17), ( 25), (30), and (33), we can obtain the convergence region of the transformed error  1 ,  2 as follows: By choosing  1 > √2 Subsequently, the convergence region after a finite time  can be constructed as follows: where  can be adjusted to be arbitrarily small if we choose appropriate controller parameters  0 ,  1 , and .From (36), the transformed error   converges to a small residual set Ω  in a finite time .Owing to the convergence of   and the error transformed relationship ( 8)-( 10), the fullstate constrained tracking errors   () satisfy (3) in a finite time .From ( 3) and ( 4), the tracking error   exponentially converges to the interval [−   ∞ ,    ∞ ], which can be adjusted to be a small residual set of zero by choosing the appropriate design parameters  ∞ ,   ,   .
Remark 7. In order to verify the boundedness of transformation errors  1 and  2 , two independent Lyapunov functions  1 and  2 are constructed in Steps 1 and 2. Using the appropriate inequality technology, we firstly prove the transformation error  2 is bounded.Subsequently, the boundedness of the tracking error  2 is indirectly verified based on the error transformed relationship ( 8)- (10).As indicated by the Remark 5, the boundedness of  2 backward derives the boundedness of the transformation error  1 .Similarly,  1 can be verified to satisfy the predefined tracking performance (3).
Remark 8.In order to verify the boundedness of transformation errors  1 and  2 , two independent Lyapunov functions  1 and  2 are constructed in Steps 1 and 2. Using the appropriate inequality technology, we firstly prove the transformation error  2 is bounded.Although the boundedness of  2 depends on  * in (30) which may be large, the tracking error  2 can still satisfy the prescribed performance (3) based on the error transformed relationship (8)- (10).Subsequently, the boundedness of  2 backward derives the boundedness of the transformation error  1 .Similarly,  1 can be verified to satisfy the predefined tracking performance (3).

Neural Learning Control
Based on the stable adaptive neural control scheme developed in Section 3, this section will use the spatially localized approximation ability of RBF NNs to achieve the knowledge acquisition and storage of the unknown system dynamics Φ().And then, the stored knowledge will be reused to develop a neural learning controller so that the improved control performance of the robotic system (1) can be achieved for the same or a similar control task.
4.1.Knowledge Acquisition, Expression, and Storage.In this section, the regression subvector   () of RBF NN is firstly verified to satisfy the PE condition, which is key to achieve the exponential convergence of NN weight values Ŵ and the accurate NN approximation of the unknown system dynamics Φ().From Lemma 4, the NN input vector  = [  , q  ,   2 ,   ]  needs to be verified as recurrent signals, so that the regression subvector   () along the input orbit  satisfies partial PE condition.Based on Theorem 6, it can be obtained that the system output  converges to a small neighborhood of   for ∀ > .Since   is a recurrent signal,  is also recurrent.Since q =  1 +  2 ,  1 = − 1  1 +  2 + Ψ 1  1 ,  1 , and  1 are very small values, q is recurrent with the same period as  2 .In the steady-state control,  2 is small and recurrent.Noting that  = α 1 + Ψ 2  2 , which is a function of the variables   ,   ,   , and   , so it can be recurrently verified as a recurrent signal.According to Lemma 4, the regression subvector   () satisfies partial PE condition.
Using the spatially localized approximation ability of RBF NNs, the closed-loop system from ( 23) and ( 24) can be given by where which is a positive definite and symmetric matrix,   () is the subvector of (), which is composed of RBFs that are close to the reference orbit , Ŵ = [ Ŵ1 , Ŵ2 , . . ., Ŵ ] ∈  × is the corresponding estimated weight subvector with Ŵ ∈   and 0 <  < ,  denotes the region far away from the orbit , and   =  − W    () are the approximate errors along the reference orbit.Theorem 9. Consider the closed-loop system consisting of the robotic system (1) with Assumption 1, the bounded reference model (2), full-state tracking error constrained condition (3), the transformation error (10) and adaptive neural control law (22) with the virtual control law (15), and the NN updated law (23).Assume the given bounded initial conditions satisfy (3) and Ŵ (0) = 0, 1 ≤  ≤ .Then, for any recurrent orbit   (  ())| ≥ 0, we have that the NN weight estimates Ŵ exponentially converge to a small neighborhood of optimal values  *  after  ≥ , and the corresponding system dynamics Proof.Up to now, we have verified that   () satisfies the PE condition.Furthermore, in order to achieve the exponential convergence of neural weight estimates Ŵ , the closed-loop system (37) needs to be transformed into a class of linear time-varying (LTV) systems with small perturbations based on Lemma 4.6 given in [56].From (37), the perturbation term (Υ 2 , )  may be very large.The main reasons lie in the fact that the term   =  − W    () = () − () − W    () may be large with the possible large () and the term (Υ 2 , ) = Υ 2  −1 () may be also large due to the possible large Υ 2 and  −1 ().Noting (Υ 2 , ) is a positive definite and symmetric matrix, a state transformation  2 =  −1 (Υ 2 , ) 2 /  is introduced to obtain the following class of LTV systems: where  is abbreviation of (Υ 2 , ), and It is worth pointing out that   /  and −Γ  Ŵ are small perturbations by choosing large enough   and small enough .Therefore, the system (40) can be regarded as a class of LTV systems with very small perturbations.It has been shown in [28] that the nominal part of the system (40) can be guaranteed to be exponentially stable if the system (40) satisfies the following three conditions: (i) There exists a positive constant  such that, for all  ≥ 0, the bound max{‖()‖, ‖()/‖} ≤  is satisfied; (ii) There exist symmetric and positive matrices () and () such that   ()() + ()() + Ṗ () = −(); (iii)   () satisfies the PE condition.
From Section 3, all closed-loop signals remain uniformly ultimately bounded.Therefore, the satisfaction of condition (i) can be easily checked.Moreover, we have verified that   () satisfy the PE condition (see the above analysis of Theorem 9 for the details).Next, choose a matrix () = Υ 2 .Since Υ 2 and  are positive definite and symmetric, the matrix () is also symmetric and positive.Then, we have The inequality    +  + Ṗ < 0 holds when choosing the appropriate control parameter Therefore, we has verified that the nominal part of the system (40) is uniformly exponentially stable.Further, based on the perturbation theory given in Lemma 4.6 [56], the weight estimate error W converges exponentially to a small neighborhood of zero for ∀ ≥ .Noting that W = Ŵ −  *  , so Ŵ converges exponentially to a small neighborhood of the desired weight value  *  in a finite time , and the corresponding desired weight value can be stored by constant neural weight values   in (39).
Based on the spatially localized approximation property of RBF NN (6) and the constant weight values , the unknown system dynamics () could be expressed as where   and  are close to .From (43), neural networks   (), containing the experience knowledge , can be used to accurately approximate the unknown system dynamics Φ().Furthermore, the learned knowledge can be described as follows: for the experienced recurrent orbit   (  ()), there exist positive constants  and  * , which describe a local region where  * is close to  * .For a new task, once the NN inputs () enter the region Ω   , the trained RBF networks   () can accurately approximate the uncertain nonlinearity ().

Static Controller Design with Knowledge Utilization
. By invoking the stored weight values  (39), a static controller will be developed in this section to guarantee the prescribed performance of full-state tracking errors of the robotic system (1) for the same or a similar control task.Using the stored knowledge  in (39), a static control law without neural weight estimated parameter adjustment online, instead of the adaptive NN control law (22), is designed as follows: where  20 and  21 are positive design parameters and  2 and Υ 2 are defined in (12).Moreover, the virtual control law  is chosen the as same as Section 3; see (15) for the detail.Then, by combining ( 16), (19), and ( 45), we can obtain the following closed-loop system: where   = −().Subsequently, construct the following Lyapunov function candidate: Noting the condition (44) and applying the similar backstepping step in Section 3, we have the following results.
Theorem 10.Consider the closed-loop system consisting of the robotic system (1), the bounded reference trajectory (2), the full-state tracking performance condition (3), the transformed error (10), the static neural learning control law (45) with the stored constant weight  given in (39), and the virtual control law (15).Then, for the same or a similar recurrent reference orbit   (  ()) given in Theorem 6 and the initial conditions satisfying the prescribed performance (3), it can be guaranteed that all the closed-loop signals are uniformly ultimately bounded, and the constrained full-state tracking errors   satisfy the prescribed performance (3) and converge to a small residual set of zero.

Simulation Results
To demonstrate the effectiveness of the proposed dynamic learning scheme, we consider a 2-link robot manipulator which is shown in From the -link rigid robotic system (1),  = [ 1 ,  2 ]  denotes the angular position of each joint and  = [ 1 ,  2 ]  is the actuator input applied at the manipulator joints, respectively.Based on the system (1), the dynamic parameters of a 2-link robot manipulator are given by  ( q ) = [ 0.8 q where and   and   denote the length and the mass of link-,  = 1, 2, and  denotes the gravity acceleration.In this paper, these system parameters are chosen as  1 =  2 = 1 kg,  1 = 0.8 m,  2 = 2.3 m, and  = 9.8 m/s 2 , and the external disturbances  = [0.1 sin(), 0.1 cos()]  , which are bounded and satisfy Assumption 1.

ANC Results with Full-State Tracking Error Constraints.
According to Theorems 6 and 9, the main objective of this section is to use the proposed stable adaptive neural controller (22) with virtual control law (15) and neural weight adaptation law (23) such that the full-state tracking errors  1 and  2 satisfy the prescribed performance (51); the neural weight estimator Ŵ exponentially converges to the constant weight value ; the unknown system dynamics Φ() in ( 20) is accurately approximated by the constant RBF NNs   ().
In the simulation studies, the RBF network   () consists of 3375 neurons whose centers are evenly spaced on [−0.9, 0.9] × [−0.9, 0.9] × [−0.9, 0.9] × [−0.2-5 show the constrained joint angular position and velocity tracking error performances, respectively.From Figures 2-5, it can be clearly seen that good transient performances have been achieved by adjusting the performance function (53) and design parameters   ,   .The control input response is  given in Figure 6.The partial weight convergence of Ŵ = [ Ŵ1 , Ŵ2 ] is presented in Figures 7 and 8.It can be seen from Figures 7 and 8   full-state tracking performance constraints, the simulation comparison is given between the proposed method and the existing method without prescribed performances [30].For comparison purpose, the existing method [30] is also used to control the same 2-link robot manipulator with the same initial condition (0) = [−0.6,1]  , q (0) = [1.5, −1]  and the same reference trajectory (50).For clarity, the existing method without prescribed performance proposed in [30]   required to have the similar amplitude, which is shown in Figure 6.The simulation results for comparison are given in Figures 2-5.It is clearly showed from Figures 2-5 that the proposed adaptive neural control scheme with full-state tracking performance constraints achieves a better transient and steady-state tracking control performance with smaller overshoot, faster convergence rate, and smaller steady-state error.
Remark 11.It is well known that the tracking performance relies on the choice of the control parameters and structure of RBF neural networks.However, how to choose design parameters for achieving the good tracking performance is still an open problem.In this simulation, the RBF networks   () are constructed appropriately such that all neurons can cover the entire NN input trajectory () space.Moreover, these controller parameters are chosen with large enough  10 ,  20 ,  21 , and Γ and small enough .It should be pointed out that these design parameters are chosen in this simulation by a trial-and-error method.

Neural Learning Control Results
. By using the stored constant weight values   in (55), the objective of this section is to invoke a static neural learning controller (45) to achieve the improved control performance with full-state tracking error constraints for the same or similar control tasks.For comparison purpose, the plant controlled, the reference trajectory, and the constrained tracking error performance are chosen the same as Section 5.1, while the initial conditions and the neural network structures are unchanged.In the simulation, with the control gains selected as  10 = 3,  20 = 15, and  21 = 8, simulation results for static neural learning control (45) are shown in  From Figures 11-14, it can be seen that the smaller overshoot and the faster convergence are obtained using the learned knowledge   in (55), while full-state tracking errors satisfy the prescribed performance.It is worth pointing out that a smaller control signal is used in neural learning control to achieve the   aforementioned improved tracking control performance; see Figure 15 for the details.Moreover, because the proposed static learning control scheme avoids the online adjustment of the neural weight values, the running time saves nearly 1/2 for the same simulation time interval  = [0, 100] s and the same computer configuration.The static neural learning controller (45) especially avoids the trial-and-error process on control design parameters and NN parameters which is tuned in adaptive neural control process.Therefore, the proposed learning control scheme avoids effectively a great deal of time consumed by the adaptive neural control process.

Conclusions
This paper focused on the problem of full-state tracking error constraints for an -link rigid robot with unknown  system dynamics and external disturbances.The performance transformation method was employed to transform the constrained full-state tracking errors into the unconstrained ones.By combining backstepping design and two independent Lyapunov functions, a novel adaptive neural control scheme was presented to guarantee all the signals in the closed-loop system are uniformly ultimately bounded, while this control scheme achieves predefined transient and steady-state tracking control performances concerning the link angular position and velocity tracking errors.Particularly, in the steady-state control process, the proposed neural control scheme can acquire, express, and store the knowledge of unknown system dynamics.The stored knowledge was reused to complete the same or similar tasks, so that the improved control performance was achieved with the less computational burden and the better transient-state tracking performance.It should be pointed out that the considered -link rigid robot is a class of simple multi-input and multioutput nonlinear systems.Therefore, how to extend the proposed method to complex nonlinear systems, such as nonaffine nonlinear systems, switched nonlinear systems, and stochastic large-scale systems, presents a challenging opportunity for future work.

Figure 5 :
Figures 9 and 10 display unknown system dynamics Φ 1 () and Φ 2 (), along the periodic reference signals , and can be accurately approximated by the constant RBF NN   1 () and   2 ().To further show the improved transient and steady-state tracking performance for the proposed control method with
along recurrent signals  is approximated by the stored where   () approaches the desired error  *  , and the constant weight values   are chosen as   = mean ∈[  ,  ] Ŵ () = 1   −   ∫ with [  ,   ], 1 ≤  ≤ ,   >   >  representing a time segment in a steady-state stage.