# Neural Learning Control of Flexible Joint Manipulator with Predefined Tracking Performance and Application to Baxter Robot

Authors: Min Wang; Huiping Ye; Zhiguang Chen
Journal: Complexity
 (2017)

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## Abstract

This paper focuses on neural learning from adaptive neural control (ANC) for a class of flexible joint manipulator under the output tracking constraint. To facilitate the design, a new transformed function is introduced to convert the constrained tracking error into unconstrained error variable. Then, a novel adaptive neural dynamic surface control scheme is proposed by combining the neural universal approximation. The proposed control scheme not only decreases the dimension of neural inputs but also reduces the number of neural approximators. Moreover, it can be verified that all the closed-loop signals are uniformly ultimately bounded and the constrained tracking error converges to a small neighborhood around zero in a finite time. Particularly, the reduction of the number of neural input variables simplifies the verification of persistent excitation (PE) condition for neural networks (NNs). Subsequently, the proposed ANC scheme is verified recursively to be capable of acquiring and storing knowledge of unknown system dynamics in constant neural weights. By reusing the stored knowledge, a neural learning controller is developed for better control performance. Simulation results on a single-link flexible joint manipulator and experiment results on Baxter robot are given to illustrate the effectiveness of the proposed scheme.


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## Body

## 1. Introduction

Due to the great demands in industrial applications, the tracking control problem for flexible joint robot (FJR) manipulator has attracted much attention in recent years. Unlike rigid joint robot, the joint flexibility of FJR results in complex control situation, so that the control problem of FJR becomes much more difficult. In the past few decades, lots of efforts have been made on the research of FJR systems. Based on the model of FJR presented in [1], multifarious nonlinear control methods are presented such as backstepping method [2–4], sliding-mode control [5–8], switching control [9], fuzzy control [10], and neural network control [11, 12]. In consideration of the problem caused by the inherent structure of FJR under practical circumstance, such as friction, time delay, and variable stiffness, some researchers proposed effective strategies to solve such problem [13–15]. Moreover, the teleoperation control method is also widely used in robot research [16, 17].The backstepping control [18] is known as one of the popular method for designing the control scheme of FJR. Nevertheless, it should be pointed out that this method has a drawback called “explosion of complexity” [19]. This problem generally occurs in the design of neural networks (NNs) during backstepping procedure. To overcome this problem, some researchers used the intermediate variables as neural inputs to reduce the dimension of neural network input vector [20]. The method in [20] did work well but the problem remained unsolved. Then other researchers proposed a dynamic surface control (DSC) method by introducing a first-order filter at each step of the backstepping procedure [21]. Due to the property of DSC method, many researchers presented their control schemes combined with DSC method [22–27]. In [24], a new robust output feedback control approach for flexible joint electrically driven robots via the observer-based dynamic surface method was proposed, which only requires position measurement of the system.Besides, the transient and steady-state tracking performance constraints of system’s output are an important issue that needs to be taken into consideration [28, 29]. According to the practical operating environment, the manipulator is not only demanded to trace the reference trajectory accurately but also required to keep the tracking error within a specified range. To satisfy this condition, a performance function transformation was used to convert the “constrained” system into the “unconstrained” one [30]. Based on the idea in [30], further researches on prescribed performance for a variety of systems are proposed [31–36]. Authors in [31, 32] presented novel controllers for FJRs to achieve tracking control of link angles with any prescribed performance requirements. By combining neural learning control scheme, further results are given in [33–35]. In [36], an adaptive prescribed performance tracking control scheme is investigated for a class of output feedback nonlinear systems with input unmodeled dynamics based on dynamic surface control method.In addition, adaptive neural control of nonlinear system has been widely studied for decades, but most of the traditional works focus on the system stability through online adjustment of neural weights and less works discuss the knowledge acquisition, storage, and utilization of optimal neural weights. To achieve such learning ability, the key problem is to verify the persistent excitation (PE) condition. Thanks to the results in [37], a deterministic learning mechanism is proposed, which proved the satisfaction of PE condition for the localized radial basis function (RBF) NN centered in a neighborhood along recurrent orbits. The result was extended to nonlinear systems satisfying matching conditions [38–40]. By combining recursive design technologies such as backstepping control and the system decomposition strategy, the deterministic learning was also applied to solve learning problem of accurate identification of ocean surface ship and robot manipulation in uncertain dynamical environments [41–44]. However, due to the recursive property of backstepping control, the convergence of neural weights has to be recursively verified based on the system decomposition strategy. It would be a tedious and complex process since the intermediate variables grow drastically as the order of system increases. Therefore, it is difficult to prove all neural weights convergence for high-order system by existing works.This paper focuses on learning from adaptive neural control of flexible joint manipulator with unknown dynamics under the prescribed constraints. A performance function is introduced to transform the constrained tracking error into the unconstrained variable. To avoid the curse of dimensionality of RBF NN, first-order filters are introduced to reduce the number of NN approximators and decrease the dimension of NN inputs. The control law is constructed based on Lyapunov stability, which guarantees the closed-loop stability and the tracking error satisfying the prescribed performance during the transient process. Subsequently, due to the property of DSC and structure features of the considered manipulator, a system decomposition strategy is employed to decompose the stable closed-loop system into two linear time-varying (LTV) perturbed subsystems on the basis of the number of NNs in the whole system. Through the recursive design, the recurrent properties of NN input variables are easily proven. Consequently, with the satisfaction of the PE condition of RBF NNs, the convergence of partial neural weights is verified, and the unknown dynamics of system are approximated accurately in a local region along recurrent orbits. By utilization of the constant neural weights stored, a neural learning controller is developed to achieve the closed-loop stability and better control performance under the prescribed constraints for the same or similar control task. Compared with the existing neural learning results, the proposed neural learning control scheme not only achieves better control performance with specified transient and steady-state constraints but also reduces the dimension of NN inputs and the number of NNs significantly.This paper is organized as follows. In Section2, the problem formulation and preliminaries are stated before the control scheme design. In Section 3, a novel adaptive neural dynamic surface control scheme is proposed to guarantee that the constrained tracking error converges to a small neighborhood around zero with the prescribed performance in a finite time, and all the signals in the closed-loop system are uniformly ultimately bounded. Section 4 shows that the knowledge acquisition, expression, storage, and utilization of the manipulator’s unknown dynamics can be achieved after the steady-state control process. To verify the effectiveness of the proposed control scheme, simulation results on a single-link flexible joint manipulator and experiment results on Baxter robot are given in Section 5. Last but not least, the conclusions are drawn in Section 6.


## 2. Problem Formulation and Preliminaries

### 2.1. System Formulation

In this paper, we consider ann-link manipulator with flexible joints, whose model is described by [1](1)Mq1q¨1+Cq1,q˙1q˙1+gq1+Kq1-q2=0,Jq¨2-Kq1-q2=u,where q1∈Rn is the vector of links’ angle positions and q2∈Rn is the vector of motors’ angle positions. M(q1)∈Rn×n is the link inertia matrix and J∈Rn×n is the diagonal and positive definite motor inertia matrix. Moreover, C(q1,q˙1)∈Rn×n denotes the Coriolis and centrifugal matrix and g(q1)∈Rn represents the gravitational terms. K∈Rn×n is a diagonal and positive definite matrix of joint spring constants; thus K-1 is also positive definite. Finally, u∈Rn is the control input of system (1), and the output of system (1) is q1.Property 1 (see [2]).
The inertia matrixM(q1) is symmetric and positive definite; both M(q1) and M-1(q1) are uniformly bounded.Property 2 (see [2]).
The Coriolis and centrifugal matrixC(q1,q˙1) can be defined such that M˙(q1)-2C(q1,q˙1) is skew symmetric; that is, xT(M˙(q1)-2C(q1,q˙1))x=0,∀x∈Rn.The reference trajectory vectoryd∈Rn is generated by the following smooth and bounded reference model:(2)x˙di=xd,i+1,i=1,…,m-1,x˙dm=fdxd,yd=xd1,where xd=[xd1T,…,xdmT]T∈Rm·n and yd is the system outputs vector. fd(·) is a smooth known nonlinear function. Assume xdi∈Rn, 1≤i≤m are recurrent signals and the reference orbit (denoted as ψd(xd(0))) is a recurrent motion. Moreover, M-1(q1), C(q1,q˙1), g(q1), K, and J are assumed as unknown terms.Our goal is to design a neural learning controller, which forces the tracking error vector (i.e.,e1=q1-yd) converges to a small neighborhood around zero with prescribed performance in a finite time. Before the design of learning control (LC) scheme, a stable adaptive neural dynamic surface controller with prescribed performance is developed to verify the feasibility of ANC scheme. According to the deterministic learning theory, the unknown system dynamics are accurately approximated by localized RBF networks along the recurrent orbits of NN inputs. Then, based on the ANC scheme and the approximation of localized RBF networks, the knowledge on unknown system dynamics is stored in static neural weights, which is also reused to develop a neural learning controller. This neural learning controller is verified to achieve the closed-loop system stability and better control performance with prescribed constraints for the same or similar tasks.


### 2.2. Prescribed Tracking Performance

In this paper, the output error vector of system (1) is defined as e1=q1-yd=[e11,e12,…,e1n]∈Rn. To achieve the prescribed performance (i.e., overshoot, convergence rate, and convergence accuracy), each element in e1 is constrained into the following prescribed region: (3)-δ1kβt<e1k<δ2kβt,k=1,2,…,n,where δ1k and δ2k are positive design constants. β(t) is a bounded, smooth, strictly positive, and decreasing performance function. In addition, β(t) is chosen as the following form by setting limt→0⁡β(t)=β0 and limt→∞⁡β(t)=β∞: (4)βt=β0-β∞e-κt+β∞,where β0, β∞, and κ are positive constants. With (3) and (4), it can be concluded that the convergence rate of e1k(t) is constrained by the decreasing rate κ of β(t), while its maximum bound of overshoot at initial moment is constrained by -δ1kβ0 and δ2kβ0, and its steady error is constrained within a range from -δ1kβ∞ to δ2kβ∞.


### 2.3. RBF Neural Network

According to [45], RBF NN can approximate any continuous function vector F(Z)∈Rn over a compact set ΩZ∈Rm to any arbitrary accuracy as (5)FZ=W∗TSZ+ϵZ,∀Z∈ΩZ∈Rm,where W∗∈Rl×n is the ideal weights matrix, l is the NN node number, S(Z)=[s1(Z),s2(Z),…,sl(Z)]T∈Rl is the basis function vector with si(Z)(i=1,2,…,l) chosen as the Gaussian function si(Z)=exp⁡[-(Z-μi)T(Z-μi)/2ηi2], and ϵ(Z)=[ϵ1(Z),ϵ2(Z),…,ϵn(Z)]T∈Rn is the approximation error vector which satisfies ϵi(Z)≤ϵi∗,i=1,2,…,n, with constant ϵi∗>0.On the other hand, it has been shown in [46] that, for any bounded trajectory Z(t) over the compact set ΩZ, the continuous and smooth function F(Z) can be approximated to an arbitrary accuracy using the localized RBF NNs with a limited number of neurons located in a local region along the trajectory: (6)FZ=Wζ∗TSζZ+ϵζZ,where Sζ(Z)∈Rζl is the subvector of S(Z) and Wζ∗T∈Rζl with lζ<l. ϵζ(Z) is the approximation error which is close to ϵ(Z).Lemma 1 ((partial PE condition for RBF NNs) [46]).
Consider any continuous recurrent trajectoryZ(t). Assume that Z(t) is a continuous map from [0,∞) into Rq, and Z(t) remains in a bounded compact set ΩZ with ΩZ⊂Rq. Then, for the RBF NN WTS(Z) with centers placed on a regular lattice (large enough to cover the compact set ΩZ), the regression subvector Sζ(Z) consisting of RBFs with centers located in a small neighborhood of Z(t) is persistently exciting.


## 2.1. System Formulation

In this paper, we consider ann-link manipulator with flexible joints, whose model is described by [1](1)Mq1q¨1+Cq1,q˙1q˙1+gq1+Kq1-q2=0,Jq¨2-Kq1-q2=u,where q1∈Rn is the vector of links’ angle positions and q2∈Rn is the vector of motors’ angle positions. M(q1)∈Rn×n is the link inertia matrix and J∈Rn×n is the diagonal and positive definite motor inertia matrix. Moreover, C(q1,q˙1)∈Rn×n denotes the Coriolis and centrifugal matrix and g(q1)∈Rn represents the gravitational terms. K∈Rn×n is a diagonal and positive definite matrix of joint spring constants; thus K-1 is also positive definite. Finally, u∈Rn is the control input of system (1), and the output of system (1) is q1.Property 1 (see [2]).
The inertia matrixM(q1) is symmetric and positive definite; both M(q1) and M-1(q1) are uniformly bounded.Property 2 (see [2]).
The Coriolis and centrifugal matrixC(q1,q˙1) can be defined such that M˙(q1)-2C(q1,q˙1) is skew symmetric; that is, xT(M˙(q1)-2C(q1,q˙1))x=0,∀x∈Rn.The reference trajectory vectoryd∈Rn is generated by the following smooth and bounded reference model:(2)x˙di=xd,i+1,i=1,…,m-1,x˙dm=fdxd,yd=xd1,where xd=[xd1T,…,xdmT]T∈Rm·n and yd is the system outputs vector. fd(·) is a smooth known nonlinear function. Assume xdi∈Rn, 1≤i≤m are recurrent signals and the reference orbit (denoted as ψd(xd(0))) is a recurrent motion. Moreover, M-1(q1), C(q1,q˙1), g(q1), K, and J are assumed as unknown terms.Our goal is to design a neural learning controller, which forces the tracking error vector (i.e.,e1=q1-yd) converges to a small neighborhood around zero with prescribed performance in a finite time. Before the design of learning control (LC) scheme, a stable adaptive neural dynamic surface controller with prescribed performance is developed to verify the feasibility of ANC scheme. According to the deterministic learning theory, the unknown system dynamics are accurately approximated by localized RBF networks along the recurrent orbits of NN inputs. Then, based on the ANC scheme and the approximation of localized RBF networks, the knowledge on unknown system dynamics is stored in static neural weights, which is also reused to develop a neural learning controller. This neural learning controller is verified to achieve the closed-loop system stability and better control performance with prescribed constraints for the same or similar tasks.


## 2.2. Prescribed Tracking Performance

In this paper, the output error vector of system (1) is defined as e1=q1-yd=[e11,e12,…,e1n]∈Rn. To achieve the prescribed performance (i.e., overshoot, convergence rate, and convergence accuracy), each element in e1 is constrained into the following prescribed region: (3)-δ1kβt<e1k<δ2kβt,k=1,2,…,n,where δ1k and δ2k are positive design constants. β(t) is a bounded, smooth, strictly positive, and decreasing performance function. In addition, β(t) is chosen as the following form by setting limt→0⁡β(t)=β0 and limt→∞⁡β(t)=β∞: (4)βt=β0-β∞e-κt+β∞,where β0, β∞, and κ are positive constants. With (3) and (4), it can be concluded that the convergence rate of e1k(t) is constrained by the decreasing rate κ of β(t), while its maximum bound of overshoot at initial moment is constrained by -δ1kβ0 and δ2kβ0, and its steady error is constrained within a range from -δ1kβ∞ to δ2kβ∞.


## 2.3. RBF Neural Network

According to [45], RBF NN can approximate any continuous function vector F(Z)∈Rn over a compact set ΩZ∈Rm to any arbitrary accuracy as (5)FZ=W∗TSZ+ϵZ,∀Z∈ΩZ∈Rm,where W∗∈Rl×n is the ideal weights matrix, l is the NN node number, S(Z)=[s1(Z),s2(Z),…,sl(Z)]T∈Rl is the basis function vector with si(Z)(i=1,2,…,l) chosen as the Gaussian function si(Z)=exp⁡[-(Z-μi)T(Z-μi)/2ηi2], and ϵ(Z)=[ϵ1(Z),ϵ2(Z),…,ϵn(Z)]T∈Rn is the approximation error vector which satisfies ϵi(Z)≤ϵi∗,i=1,2,…,n, with constant ϵi∗>0.On the other hand, it has been shown in [46] that, for any bounded trajectory Z(t) over the compact set ΩZ, the continuous and smooth function F(Z) can be approximated to an arbitrary accuracy using the localized RBF NNs with a limited number of neurons located in a local region along the trajectory: (6)FZ=Wζ∗TSζZ+ϵζZ,where Sζ(Z)∈Rζl is the subvector of S(Z) and Wζ∗T∈Rζl with lζ<l. ϵζ(Z) is the approximation error which is close to ϵ(Z).Lemma 1 ((partial PE condition for RBF NNs) [46]).
Consider any continuous recurrent trajectoryZ(t). Assume that Z(t) is a continuous map from [0,∞) into Rq, and Z(t) remains in a bounded compact set ΩZ with ΩZ⊂Rq. Then, for the RBF NN WTS(Z) with centers placed on a regular lattice (large enough to cover the compact set ΩZ), the regression subvector Sζ(Z) consisting of RBFs with centers located in a small neighborhood of Z(t) is persistently exciting.


## 3. Adaptive Neural DSC Design with Predefined Tracking Performance

In this section, performance function is introduced for describing constraints of system (1). Then an adaptive neural DSC is developed, with the design of adaptive control law based on the transformed error. Meanwhile, RBF NN is used to approximate the unknown dynamics.Step 1.
Similarly to the traditional backstepping design, we set(7)e1=q1-yd=e11,e12,…,e1nT∈Rn.It should be pointed out that any errors set in previously traditional design are under unrestricted condition [20], while e1 is constrained to satisfy condition (3) in this paper, which is rewritten as (8)-δ1k<e1kβt<δ2k,k=1,2,…,n.It implies thate1 can not be used for design directly due to the limitation of the traditional design method. To solve the constrained tracking control problem, the constrained error should be transformed into the unconstrained one equivalently. Therefore, a new error vector is defined as z1=[z11,z12,…,z1n]T∈Rn called transformed error vector. Define a smooth transformed functions vector Φ(z1)=[ϕ(z11),ϕ(z12),…,ϕ(z1n)]T with ϕ(z1k) being chosen as (9)ϕz1k=δakarctan⁡z1k-tan⁡δbkδak+δbk,where δak=(δ2k+δ1k)/π, and δbk=(δ2k-δ1k)/2. Moreover, for the symmetric tracking error constraints -δkβ(t)<e1k<δkβ(t),k=1,2,…,n, the transformation function (9) can be constructed as ϕ(z1k)=2/πδkarctan⁡z1k. For clarity, Figure 1 illustrates the relationship between ϕ(z1k) and z1k.Figure 1
Transformed functionϕ(z1k).It can be concluded from (9) and Figure 1 that the transformation function ϕ(z1k) is smooth and strictly increasing while possessing the following properties: (10)-δ1k<ϕz1k<δ2k,∀z1k∈L∞,limz1k→+∞⁡ϕz1k=δ2k,limz1k→-∞⁡ϕz1k=-δ1k.By combining (8) with (10), e1k can be rewritten as (11)e1k=ϕz1kβt,k=1,2,…,n.Sinceϕ(·) is a strictly monotonic increasing function and β(t)≠0, the inverse function z1k of ϕ(·) exists, which can be expressed as (12)z1k=tan⁡e1kδakβt-δbkδak+tan⁡δbkδak.Noting thatz1=[z11,z12,…,z1n]T, its derivative can be presented as (13)z˙1=Υq˙1-y˙d-β˙tβte1,where Υ=diag⁡γ1,γ2,…,γn, with γk(k=1,2,…,n) being presented as (14)γk=1δakβt1+tan2⁡e1kδakβt-δbkδak.It is clear thatΥ is positive definite, which is helpful for the stability analysis. By introducing a new filter variable α1f∈Rn and noting z2=q˙1-α1f, then the virtual controller α1∈Rn is constructed as (15)α1=-Υ-1C1+Υz1+y˙d+β˙tβte1,where C1∈Rn×n is a diagonal and positive design matrix. Take α1 as the input of a first-order filter and α1f as the output of it, a differential equation is constructed as (16)τ1α˙1f+α1f=α1,α1f0=α10,where τ1 is the filter time constant and set y1=α1f-α1; then (13) can be rewritten as (17)z˙1=-C1+ΥTΥz1+Υz2+Υy1.Step 2.
Letz2=q˙1-α1f, its derivative can be obtained: (18)z˙2=M-1q1Kq2-q1-K-1Cq1,q˙1α1f-K-1gq1-K-1Mq1α˙1f-K-1Cq1,q˙1z2.Define the unknown dynamics in (18) as (19)H2Ψ2=-K-1Cq1,q˙1α1f-K-1gq1-K-1Mq1α˙1f,where (20)Ψ2=q1T,q˙1T,α˙1fTT∈R3n.According to the property of RBF NN,H2(Ψ2) can be approximated accurately by RBF NN and (19) is rewritten as (21)H2Ψ2=W2∗TS2Ψ2+ϵ2,where ϵ2∈Rn is the bounded approximation error vector which satisfies ϵ2≤ϵ2∗. Define W^2 as the estimate of W2∗ and set W~2=W^2-W2∗. Then a new filter variable α2f∈Rn is introduced and note z3=q2-α2f; (18) can be rewritten by combining (19) and (21): (22)z˙2=M-1q1Kz3+α2f-q1+W2∗TS2Ψ2+ϵ2-K-1Cq1,q˙1z2.Then the virtual controllerα2∈Rn is constructed as (23)α2=-Υz1-C2z2-W^2TS2Ψ2+q1,where C2∈Rn×n is a diagonal and positive design matrix. The updated law of NN weights is given by (24)W^˙2=Γ2S2Ψ2z2T-σ2W^2with diagonal matrix Γ2=Γ2T>0 and small value σ2>0 for enhancing the robustness of the controller (23).Takeα2 as the input of a first-order filter and α2f as the output of it, a differential equation is constructed as (25)τ2α˙2f+α2f=α2,α2f0=α20,where τ2 is the filter time constant and set y2=α2f-α2; then (22) can be rewritten as (26)z˙2=M-1q1K-Υz1-C2z2+z3+y2-W~2TS2Ψ2+ϵ2-K-1Cq1,q˙1z2.Step 3.
Definez3=q2-α2f, its derivative can be obtained: (27)z˙3=q˙2-α˙2f.Introduce a new filter variableα3f∈Rn and note z4=q˙2-α3f. Then the virtual controller α3∈Rn is constructed as (28)α3=-z2-C3z3+α˙2f,where C3∈Rn×n is a diagonal and positive design matrix. Take α3 as the input of a first-order filter and α3f as the output of it, a differential equation is constructed as (29)τ3α˙3f+α3f=α3,α3f0=α30,where τ3 is the filter time constant and set y3=α3f-α3; then (27) can be rewritten as (30)z˙3=-z2-C3z3+z4+y3.Step 4.
Letz4=q˙2-α3f; the following can be obtained: (31)z˙4=J-1u+Kq1-q2-Jα˙3f.Define the unknown dynamics of the system as (32)H4Ψ4=Kq1-q2-Jα˙3f,where (33)Ψ4=q1T,q2T,α˙3fTT∈R3n.According to the property of RBF NN,H4(Ψ4) can be approximated accurately by RBF NN and (32) is rewritten as (34)H4Ψ4=W4∗TS4Ψ4+ϵ4,where W4∗∈Rj×n is the ideal constant weight matrix with j being the NN node number, S4(Ψ4)∈Rj is the basis function vector, and ϵ4 is the bounded approximation error vector which satisfies ϵ4≤ϵ4∗. Define W^4 as the estimate of W4∗ and let W~4=W^4-W4∗. Then (31) can be rewritten by combining (32) and (34): (35)z˙4=J-1u+W4∗TS4Ψ4+ϵ4.Then the control inputu is constructed as (36)u=-z3-C4z4-W^4TS4Ψ4,where C4∈Rn×n is a diagonal and positive design matrix. The updated law of NN weights is given by (37)W^˙4=Γ4S4Ψ4z4T-σ4W^4with diagonal matrix Γ4=Γ4T>0 and small value σ4>0 for enhancing the robustness of the controller (36). Then (35) can be rewritten as (38)z˙4=J-1-z3-C4z4-W~4TS4Ψ4+ϵ4.Let us construct the following Lyapunov function candidate:(39)V=12∑i=14ziTKizi+12∑k=12trW~2kTΓ2k-1W~2k+12∑l=13ylTyl,where K1=I, K2=K-1M(q1), K3=I, and K4=J.Remark 2.
It should be pointed out that, in the adaptive neural backstepping design [20], the derivative α˙2 of the virtual control α2 in Step 2 is usually used to design the virtual control α3 in Step 3. However, according to (1) and (23), it can be seen clearly that α˙2 is not available because the unknown terms, such as M-1(q1), C(q1,q˙1), and g(q1), are included in α˙2. Therefore, a neural network has to be employed in Step 3 of backstepping to approximate the unknown dynamics in α˙2. However, too many neural networks employed make the control scheme implemented difficultly. To solve this problem, this paper introduces a new variable α2f in (25) to design virtual control α2 using a first-order filter. From (25), it is easy to calculate that α˙2f=(α2-α2f)/τ2=-y2/τ2. The advantage of the proposed approach is that the unknown dynamics in the previous step does not affect the design of virtual control in next step, so that the number of NNs employed can be greatly reduced. Moreover, the proposed method uses α˙if, instead of the intermediate variables used in [20], as neural input variable, so that there are only 3n neural input variables for neural networks W2∗TS2(Ψ2) and W4∗TS4(Ψ4), respectively, which reduces significantly the number of neural input variables used in [20], where 4n and 8n neural input variables are used in W2∗TS2(Ψ2) and W4∗TS4(Ψ4).Theorem 3.
Consider the manipulator model (1), the reference trajectory yd, the prescribed performance bounds (8), the state transformation (11), the adaptive NN control law (36), the weight updated law (24), (37), and Lyapunov function (39). Given any constant μ>0, for any bounded initial conditions satisfying the prescribed performance (8) and V(0)<μ, there exists design parameters C1, C2, C3, C4, Γ2, σ2, Γ4, σ4, τ1, τ2, and τ3, such that the proposed control scheme guarantees that (1) all the signals in the closed-loop system are uniformly ultimately bounded and (2) the tracking error converges to a small neighborhood around zero with the prescribed performance (8) in a finite time T1.Proof.
See AppendixA.


## 4. Dynamic Neural Learning

In this section, we will show the learning ability of RBF NNs for unknown system dynamicsH2(Ψ2) and H4(Ψ4) in the case of the manipulator with predefined tracking performance. Subsequently, the stored knowledge on the unknown dynamics will be utilized to design a neural learning controller achieving better control performance, while satisfying the prescribed performance.


### 4.1. Learning from Adaptive Neural DSC

According to Lemma1, it can be concluded that a recurrent orbit can make regression subvectors Sζ(Z) satisfy the partial PE condition, which is a key condition to ensure the accurate convergence of neural weights.Theorem 4.
Consider the closed-loop system consisting of the flexible joint manipulator model (1), the reference trajectory yd, the prescribed performance bounds (8), the state transformation (11), the adaptive NN control law (36), and the weight updated law (24) and (37). Then, for any recurrent orbit ψd(xd(t))|t≥0 and initial condition x(0)∈Ω0 (Ω0 is an appropriately chosen compact set) satisfying the prescribed performance (8) and W^2(0)=W^4(0)=0, we have that the neural weights W^i converge to small neighborhoods around optimal values Wi∗, and the locally accurate approximation of the system dynamics Hi(Ψi) is obtained by the stored knowledge W¯i: (40)W¯i=meant∈tai1,tbi2W^itwith i=2,4, T1≤ta22≤tb22≤T2≤ta41≤tb42. [ta21,tb22] and [ta41,tb42] are time segments after the steady-state control process.Proof.
From Theorem3, all the signals in the closed-loop system are uniformly ultimately bounded and the tracking error vector e1=q1-yd converges to a small neighborhood around zero with the prescribed performance in a finite time T1. Thus, the state q1 converges closely to the recurrent signals yd for all t≥T1. In addition, it can be obtained from the proof of Theorem 3 that the transformed error vector z1 converges exponentially to a small neighborhood around zero in a finite time T1. From (15), it can be concluded that virtual control α1 is recurrent with the same period as y˙d by combining the convergence of z1 and e1. Noting q˙1=z2+y1+α1 with z2 and y1 being close to a small neighborhood around zero based on Theorem 3, then q˙1 is a recurrent signal with the same period as α1. In addition, α1f is also a recurrent signal with the same period as α1 since y1=α1f-α1 and y1 is a small value. From (16), it can be obtained that α˙1f is a recurrent signal as well. Therefore, the NN inputs Ψ2=[q1T,q˙1T,α˙1fT]T are recurrent for all t≥T1, and a partial PE condition of S2(Ψ2) is satisfied according to Lemma 1. By combining the convergence of z1 and the localized RBF NN along the recurrent signals Ψ2(t)(t>T1), it can be obtained from (24) and (26) that (41)z˙2=M-1q1K-Υz1-C2z2+z3+y2-W~2ζTS2ζΨ2+ϵ2ζ-K-1Cq1,q˙1z2W~˙2ζ=Γ2ζS2ζΨ2z2T-σ2W^2ζ,(42)W~˙2ζ¯=Γ2ζ¯S2ζ¯Ψ2z2T-σ2W^2ζ¯,where the subscript ζ stands for the region near the orbits Ψ2(t), S2ζ(Ψ2) is the subvector of S2(Ψ2) consisting of the corresponding RBFs, and W^2ζ is the corresponding weight submatrix of W^2. Moreover, the subscript ζ- represents the region away from the orbits Ψ2(t), and ϵ2ζ=ϵ2(Ψ2)-W~2ζ¯TS2ζ¯(Ψ2) is the NN approximation error along the orbits Ψ2(t). Since W^2ζ¯TS2ζ¯(Ψ2) is small, ϵ2ζ is close to ϵ2(Ψ2).It would be shown that the perturbation termM-1(q1)Kϵ2ζ may be large, which will make the accurate convergence of neural weights become difficult. To solve this problem, a state transformation z2s=K-1M(q1)z2 is introduced to eliminate the influence of M-1(q1)Kϵ2ζ. Subsequently, set W~2ζ=[W~21ζ,W~22ζ,…,W~2nζ]∈Rζ×n and W^2ζ=[W^21ζ,W^22ζ,…,W^2nζ]∈Rζ×n; (41) can be transformed into the following form: (43)z˙2sW~˙21ζ⋮W~˙2nζ=A2tB2tC2t0z2sW~21ζ⋮W~2nζ+ϵ2ζ′-σ2Γ2ζW^21ζ⋮-σ2Γ2ζW^2nζ,where(44)A2t=-FM-1q1K∈Rn×nB2t=-diag⁡S2ζT,…,S2ζT∈Rn×ζnC2t=Γ-2ζS-2ζM-1q1K∈Rζn×nF=C2-K-1M˙q1-Cq1,q˙1∈Rn×nΓ-2ζ=diag⁡Γ2ζ,…,Γ2ζ∈Rζn×ζnS-2ζ=diag⁡S2ζ,…,S2ζ∈Rζn×nϵ2ζ′=Υz1+z3+y2+ϵ2ζ.According to Theorem3, ϵ2ζ′ is a small value and -σ2Γ2ζW^2iζ(i=1,2,…,n) can be made as a small value by choosing a small design parameter σ2. Therefore, system (43) can be considered as a linear time-varying (LTV) system with a small pertubation term. Choose P(t)=M-1(q1)K; then (45)P˙t+PtA2t+A2TtPt=M˙-1q1K-F1C2+F2M-1q1K,where F1=M-1(q1)K+KM-1(q1), F2=KM-1(q1)[M˙(q1)-C(q1,q˙1)]K-1-M-1(q1)[M˙(q1)-C(q1,q˙1)]. Choose C2 appropriately such that P˙(t)+P(t)A2(t)+A2T(t)P(t)<0. Subsequently, based on the pertubation theory from Lemma 4.6 in [47], both z2s and W~2ζ converge exponentially to a small neighborhood around zero in a finite time T1, and the size of neighborhood is determined by ϵ2ζ′ and -σ2Γ2ζW^2ζ, respectively. Noting W~2ζ=W^2ζ-W2ζ∗, it is clear that W^2ζ can converge to a small neighborhood of optimal weights W2ζ∗ in a finite time T1, and the constant weights W¯2 can be obtained from (40). According to the localization property of RBF NN, the system dynamics H2(Ψ2) can be described by (46)H2Ψ2=W^2ζTS2ζΨ2+ϵ2ζΨ2=W¯2TS2Ψ2+ϵ-2Ψ2,where both ϵ2ζ(Ψ2) and ϵ-2(Ψ2) are close to ϵ2(Ψ2) due to the convergence of W~2ζ.According to the above analysis and notingα2=-Υz1-C2z2-W^2TS2(Ψ2)+q1, there exists a constant T2>T1, such that the virtual control α2 can be rewritten as (47)α2=-Υz1-C2z2-W¯2TS2Ψ2+q1+ε2for all t>T2, where ε2=[W¯2-W^2]TS2(Ψ2) is a small value because of the convergence of W^2. According to Theorem 3, α2 is a recurrent signal. Since y2=α2f-α2 and y2 is a small value, α2f is also recurrent with the same period as α2. From (25), it can be obtained that α˙2f is also a recurrent signal as well. From (28), by combining the convergence of z2 and z3, virtual control α3 is a recurrent signal with the same period as α˙2f. Since y3=α3f-α3 and y3 is a small value, α3f is also recurrent with the same period as α3. From (29), it can be obtained that α˙3f is also a recurrent signal as well. Therefore, the NN inputs Ψ4=[q1T,q2T,α˙3fT]T are recurrent for all t≥T2, and a partial PE condition of S4(Ψ4) is satisfied according to Lemma 1. By combining the convergence of z4 and the localized RBF NN along the recurrent signals Ψ4(t)(t>T2), it can be obtained from (37) and (38) that(48)z˙4=J-1-z3-C4z4-W~4ζTS4ζΨ4+ϵ4ζW~˙4ζ=Γ4ζS4ζΨ4z4T-σ4W^4ζ,W~˙4ζ¯=Γ4ζ¯S4ζ¯Ψ4z4T-σ4W^4ζ¯.Similarly,S4ζ(Ψ4) is the subvector of S4(Ψ4) consisting of the RBFs near the orbits Ψ4(t), and W^4ζ is the corresponding weight submatrix of W^4. Moreover, ϵ4ζ=ϵ4(Ψ4)-W~4ζ¯TS4ζ¯(Ψ4) is the NN approximation error along the orbits Ψ4(t). Since W^4ζ¯TS4ζ¯(Ψ4) is small, ϵ4ζ is close to ϵ4(Ψ4).It would be shown that the perturbation termJ-1ϵ4ζ may be large, which will make the accurate convergence of neural weights become difficult. To solve this problem, a state transformation z4s=Jz4 is introduced to eliminate the influence of J-1ϵ4ζ. Subsequently, set W~4ζ=[W~41ζ,W~42ζ,…,W~4nζ]∈Rζ′×n and W^4ζ=[W^41ζ,W^42ζ,…,W^4nζ]∈Rζ′×n; (46) can be transformed into the following form:(49)z˙4sW~˙41ζ⋮W~˙4nζ=A4tB4tC4t0z4sW~41ζ⋮W~4nζ+ϵ4ζ′-σ4Γ4ζW^41ζ⋮-σ4Γ4ζW^4nζ,where (50)A4t=-C4J-1∈Rn×nB4t=-diag⁡S4ζT,…,S4ζT∈Rn×ζ′nC4t=Γ-4ζS-4ζJ-1∈Rζ′n×nΓ-4ζ=diag⁡Γ4ζ,…,Γ4ζ∈Rζ′n×ζnS-4ζ=diag⁡S4ζ,…,S4ζ∈Rζ′n×nϵ4ζ′=-z3+ϵ4ζ.Using the similar step and choosingP′(t)=J, we have (51)P˙′t+P′tA4t+A4TtP′t=-2JC4J-1<0with C4>0. Then, it can be proven that W^4ζ converges to a small neighborhood of optimal weights W4ζ∗ in a finite time T2, and the constant weights W¯4 can be obtained from (40). According to the localization property of RBF NN, the system dynamics H4(Ψ4) can be described by (52)H4Ψ4=W¯4TS4Ψ4+ϵ-4Ψ4,where ϵ-4(Ψ4) is close to ϵ4(Ψ4) due to the convergence of W~4ζ. Therefore, the dynamics Hi(Ψi),i=1,2, can be accurately approximated by the constant RBF NN W¯iTSi(Ψi) with the stored knowledge W¯i obtained in (40).


### 4.2. Neural Learning Control Using the Stored Knowledge

Since the locally accurate NN approximation can be achieved by the constant RBF NNW-iTSi(Ψi), for a similar control task, we will reuse the knowledge W-i to design a neural learning controller for system (1): (53)u=-z3-C4z4-W¯4TS4Ψ4with virtual controller(54)α1=-Υ-1C1+Υz1+y˙d+β˙tβte1α2=-Υz1-C2z2-W¯2TS2Ψ2+q1α3=-z2-C3z3+α˙2f,where W¯2TS2(Ψ2) and W¯4TS4(Ψ4) are the accurate neural approximators of the unknown system dynamics H2(Ψ2) and H4(Ψ4), respectively. W¯2 and W¯4 are constant weight matrices, which are obtained from the previous control process. Similar to the proof of Theorem 3, another Lyapunov function candidate is constructed as (55)V=12∑i=14ziTKizi+12∑l=13ylTyl,where K1=I, K2=K-1M(q1), K3=I, and K4=J.Theorem 5.
Consider the closed-loop system consisting of the manipulator model (1), the reference trajectory yd, the prescribed performance bounds (8), the state transformation (11), the neural learning controller (53) with the constant weights given by (40), and Lyapunov function (55). Then, for initial conditions xd(0) which generate the same or similar recurrent reference orbit ψd(xd(0)) as in Theorem 4, and initial conditions x(0) satisfying the prescribed performance (8) and V(0)≤μ with given μ>0, such that all the closed-loop signals remain uniformly ultimately bounded, and the tracking error converges to a small neighborhood around zero with the prescribed performance (8) when initial conditions x(0) are in a close vicinity of ψd(xd(0)).Proof.
See AppendixB.Remark 6.
For clarity, a block diagram of the proposed schemes is shown in Figure2. From Figure 2, the main difference between adaptive neural control and learning neural control lies in the adaptation of NN weights. The neural weights W^2 and W^4 are updated online in the adaptive neural control process, while the stored constant weights W-2 and W-4 are reused in the neural learning process for the same or similar control task. Without the repeat adjustment of the neural weights, the neural learning controller (53) and (54) can achieve the better control performance with faster tracking convergence rate and smaller tracking error.Figure 2
Block diagram of the proposed learning control scheme.


## 4.1. Learning from Adaptive Neural DSC

According to Lemma1, it can be concluded that a recurrent orbit can make regression subvectors Sζ(Z) satisfy the partial PE condition, which is a key condition to ensure the accurate convergence of neural weights.Theorem 4.
Consider the closed-loop system consisting of the flexible joint manipulator model (1), the reference trajectory yd, the prescribed performance bounds (8), the state transformation (11), the adaptive NN control law (36), and the weight updated law (24) and (37). Then, for any recurrent orbit ψd(xd(t))|t≥0 and initial condition x(0)∈Ω0 (Ω0 is an appropriately chosen compact set) satisfying the prescribed performance (8) and W^2(0)=W^4(0)=0, we have that the neural weights W^i converge to small neighborhoods around optimal values Wi∗, and the locally accurate approximation of the system dynamics Hi(Ψi) is obtained by the stored knowledge W¯i: (40)W¯i=meant∈tai1,tbi2W^itwith i=2,4, T1≤ta22≤tb22≤T2≤ta41≤tb42. [ta21,tb22] and [ta41,tb42] are time segments after the steady-state control process.Proof.
From Theorem3, all the signals in the closed-loop system are uniformly ultimately bounded and the tracking error vector e1=q1-yd converges to a small neighborhood around zero with the prescribed performance in a finite time T1. Thus, the state q1 converges closely to the recurrent signals yd for all t≥T1. In addition, it can be obtained from the proof of Theorem 3 that the transformed error vector z1 converges exponentially to a small neighborhood around zero in a finite time T1. From (15), it can be concluded that virtual control α1 is recurrent with the same period as y˙d by combining the convergence of z1 and e1. Noting q˙1=z2+y1+α1 with z2 and y1 being close to a small neighborhood around zero based on Theorem 3, then q˙1 is a recurrent signal with the same period as α1. In addition, α1f is also a recurrent signal with the same period as α1 since y1=α1f-α1 and y1 is a small value. From (16), it can be obtained that α˙1f is a recurrent signal as well. Therefore, the NN inputs Ψ2=[q1T,q˙1T,α˙1fT]T are recurrent for all t≥T1, and a partial PE condition of S2(Ψ2) is satisfied according to Lemma 1. By combining the convergence of z1 and the localized RBF NN along the recurrent signals Ψ2(t)(t>T1), it can be obtained from (24) and (26) that (41)z˙2=M-1q1K-Υz1-C2z2+z3+y2-W~2ζTS2ζΨ2+ϵ2ζ-K-1Cq1,q˙1z2W~˙2ζ=Γ2ζS2ζΨ2z2T-σ2W^2ζ,(42)W~˙2ζ¯=Γ2ζ¯S2ζ¯Ψ2z2T-σ2W^2ζ¯,where the subscript ζ stands for the region near the orbits Ψ2(t), S2ζ(Ψ2) is the subvector of S2(Ψ2) consisting of the corresponding RBFs, and W^2ζ is the corresponding weight submatrix of W^2. Moreover, the subscript ζ- represents the region away from the orbits Ψ2(t), and ϵ2ζ=ϵ2(Ψ2)-W~2ζ¯TS2ζ¯(Ψ2) is the NN approximation error along the orbits Ψ2(t). Since W^2ζ¯TS2ζ¯(Ψ2) is small, ϵ2ζ is close to ϵ2(Ψ2).It would be shown that the perturbation termM-1(q1)Kϵ2ζ may be large, which will make the accurate convergence of neural weights become difficult. To solve this problem, a state transformation z2s=K-1M(q1)z2 is introduced to eliminate the influence of M-1(q1)Kϵ2ζ. Subsequently, set W~2ζ=[W~21ζ,W~22ζ,…,W~2nζ]∈Rζ×n and W^2ζ=[W^21ζ,W^22ζ,…,W^2nζ]∈Rζ×n; (41) can be transformed into the following form: (43)z˙2sW~˙21ζ⋮W~˙2nζ=A2tB2tC2t0z2sW~21ζ⋮W~2nζ+ϵ2ζ′-σ2Γ2ζW^21ζ⋮-σ2Γ2ζW^2nζ,where(44)A2t=-FM-1q1K∈Rn×nB2t=-diag⁡S2ζT,…,S2ζT∈Rn×ζnC2t=Γ-2ζS-2ζM-1q1K∈Rζn×nF=C2-K-1M˙q1-Cq1,q˙1∈Rn×nΓ-2ζ=diag⁡Γ2ζ,…,Γ2ζ∈Rζn×ζnS-2ζ=diag⁡S2ζ,…,S2ζ∈Rζn×nϵ2ζ′=Υz1+z3+y2+ϵ2ζ.According to Theorem3, ϵ2ζ′ is a small value and -σ2Γ2ζW^2iζ(i=1,2,…,n) can be made as a small value by choosing a small design parameter σ2. Therefore, system (43) can be considered as a linear time-varying (LTV) system with a small pertubation term. Choose P(t)=M-1(q1)K; then (45)P˙t+PtA2t+A2TtPt=M˙-1q1K-F1C2+F2M-1q1K,where F1=M-1(q1)K+KM-1(q1), F2=KM-1(q1)[M˙(q1)-C(q1,q˙1)]K-1-M-1(q1)[M˙(q1)-C(q1,q˙1)]. Choose C2 appropriately such that P˙(t)+P(t)A2(t)+A2T(t)P(t)<0. Subsequently, based on the pertubation theory from Lemma 4.6 in [47], both z2s and W~2ζ converge exponentially to a small neighborhood around zero in a finite time T1, and the size of neighborhood is determined by ϵ2ζ′ and -σ2Γ2ζW^2ζ, respectively. Noting W~2ζ=W^2ζ-W2ζ∗, it is clear that W^2ζ can converge to a small neighborhood of optimal weights W2ζ∗ in a finite time T1, and the constant weights W¯2 can be obtained from (40). According to the localization property of RBF NN, the system dynamics H2(Ψ2) can be described by (46)H2Ψ2=W^2ζTS2ζΨ2+ϵ2ζΨ2=W¯2TS2Ψ2+ϵ-2Ψ2,where both ϵ2ζ(Ψ2) and ϵ-2(Ψ2) are close to ϵ2(Ψ2) due to the convergence of W~2ζ.According to the above analysis and notingα2=-Υz1-C2z2-W^2TS2(Ψ2)+q1, there exists a constant T2>T1, such that the virtual control α2 can be rewritten as (47)α2=-Υz1-C2z2-W¯2TS2Ψ2+q1+ε2for all t>T2, where ε2=[W¯2-W^2]TS2(Ψ2) is a small value because of the convergence of W^2. According to Theorem 3, α2 is a recurrent signal. Since y2=α2f-α2 and y2 is a small value, α2f is also recurrent with the same period as α2. From (25), it can be obtained that α˙2f is also a recurrent signal as well. From (28), by combining the convergence of z2 and z3, virtual control α3 is a recurrent signal with the same period as α˙2f. Since y3=α3f-α3 and y3 is a small value, α3f is also recurrent with the same period as α3. From (29), it can be obtained that α˙3f is also a recurrent signal as well. Therefore, the NN inputs Ψ4=[q1T,q2T,α˙3fT]T are recurrent for all t≥T2, and a partial PE condition of S4(Ψ4) is satisfied according to Lemma 1. By combining the convergence of z4 and the localized RBF NN along the recurrent signals Ψ4(t)(t>T2), it can be obtained from (37) and (38) that(48)z˙4=J-1-z3-C4z4-W~4ζTS4ζΨ4+ϵ4ζW~˙4ζ=Γ4ζS4ζΨ4z4T-σ4W^4ζ,W~˙4ζ¯=Γ4ζ¯S4ζ¯Ψ4z4T-σ4W^4ζ¯.Similarly,S4ζ(Ψ4) is the subvector of S4(Ψ4) consisting of the RBFs near the orbits Ψ4(t), and W^4ζ is the corresponding weight submatrix of W^4. Moreover, ϵ4ζ=ϵ4(Ψ4)-W~4ζ¯TS4ζ¯(Ψ4) is the NN approximation error along the orbits Ψ4(t). Since W^4ζ¯TS4ζ¯(Ψ4) is small, ϵ4ζ is close to ϵ4(Ψ4).It would be shown that the perturbation termJ-1ϵ4ζ may be large, which will make the accurate convergence of neural weights become difficult. To solve this problem, a state transformation z4s=Jz4 is introduced to eliminate the influence of J-1ϵ4ζ. Subsequently, set W~4ζ=[W~41ζ,W~42ζ,…,W~4nζ]∈Rζ′×n and W^4ζ=[W^41ζ,W^42ζ,…,W^4nζ]∈Rζ′×n; (46) can be transformed into the following form:(49)z˙4sW~˙41ζ⋮W~˙4nζ=A4tB4tC4t0z4sW~41ζ⋮W~4nζ+ϵ4ζ′-σ4Γ4ζW^41ζ⋮-σ4Γ4ζW^4nζ,where (50)A4t=-C4J-1∈Rn×nB4t=-diag⁡S4ζT,…,S4ζT∈Rn×ζ′nC4t=Γ-4ζS-4ζJ-1∈Rζ′n×nΓ-4ζ=diag⁡Γ4ζ,…,Γ4ζ∈Rζ′n×ζnS-4ζ=diag⁡S4ζ,…,S4ζ∈Rζ′n×nϵ4ζ′=-z3+ϵ4ζ.Using the similar step and choosingP′(t)=J, we have (51)P˙′t+P′tA4t+A4TtP′t=-2JC4J-1<0with C4>0. Then, it can be proven that W^4ζ converges to a small neighborhood of optimal weights W4ζ∗ in a finite time T2, and the constant weights W¯4 can be obtained from (40). According to the localization property of RBF NN, the system dynamics H4(Ψ4) can be described by (52)H4Ψ4=W¯4TS4Ψ4+ϵ-4Ψ4,where ϵ-4(Ψ4) is close to ϵ4(Ψ4) due to the convergence of W~4ζ. Therefore, the dynamics Hi(Ψi),i=1,2, can be accurately approximated by the constant RBF NN W¯iTSi(Ψi) with the stored knowledge W¯i obtained in (40).


## 4.2. Neural Learning Control Using the Stored Knowledge

Since the locally accurate NN approximation can be achieved by the constant RBF NNW-iTSi(Ψi), for a similar control task, we will reuse the knowledge W-i to design a neural learning controller for system (1): (53)u=-z3-C4z4-W¯4TS4Ψ4with virtual controller(54)α1=-Υ-1C1+Υz1+y˙d+β˙tβte1α2=-Υz1-C2z2-W¯2TS2Ψ2+q1α3=-z2-C3z3+α˙2f,where W¯2TS2(Ψ2) and W¯4TS4(Ψ4) are the accurate neural approximators of the unknown system dynamics H2(Ψ2) and H4(Ψ4), respectively. W¯2 and W¯4 are constant weight matrices, which are obtained from the previous control process. Similar to the proof of Theorem 3, another Lyapunov function candidate is constructed as (55)V=12∑i=14ziTKizi+12∑l=13ylTyl,where K1=I, K2=K-1M(q1), K3=I, and K4=J.Theorem 5.
Consider the closed-loop system consisting of the manipulator model (1), the reference trajectory yd, the prescribed performance bounds (8), the state transformation (11), the neural learning controller (53) with the constant weights given by (40), and Lyapunov function (55). Then, for initial conditions xd(0) which generate the same or similar recurrent reference orbit ψd(xd(0)) as in Theorem 4, and initial conditions x(0) satisfying the prescribed performance (8) and V(0)≤μ with given μ>0, such that all the closed-loop signals remain uniformly ultimately bounded, and the tracking error converges to a small neighborhood around zero with the prescribed performance (8) when initial conditions x(0) are in a close vicinity of ψd(xd(0)).Proof.
See AppendixB.Remark 6.
For clarity, a block diagram of the proposed schemes is shown in Figure2. From Figure 2, the main difference between adaptive neural control and learning neural control lies in the adaptation of NN weights. The neural weights W^2 and W^4 are updated online in the adaptive neural control process, while the stored constant weights W-2 and W-4 are reused in the neural learning process for the same or similar control task. Without the repeat adjustment of the neural weights, the neural learning controller (53) and (54) can achieve the better control performance with faster tracking convergence rate and smaller tracking error.Figure 2
Block diagram of the proposed learning control scheme.


## 5. Simulation and Experiment

In this section, to illustrate the effectiveness of the proposed approach, a single-link manipulator system with flexible joint is considered by the following form:(56)Iq¨1+MgLsin⁡q1+Kq1-q2=0Jq¨2-Kq1-q2=u,where M is the mass, g is the gravitational acceleration, and L is the length of link. Figure 3 illustrates the structure of a single-link flexible joint manipulator. In the simulation, the system parameter is chosen exactly as M=2.3, L=1, g=9.8, I=ML2, J=0.5, and K=15. Then the desired reference trajectory yd is generated from the following equation: (57)2y¨d+3y˙d+yd=4sin⁡1.5t.Figure 3
Single-link flexible joint manipulator.


### 5.1. Numerical Simulation Results

Based on Theorem3, the adaptive control law is chosen as (36), and the virtual controllers are chosen as (15), (23), and (28) with the virtual control laws (24) and (37). In the performance function, β0=1, β∞=0.05, κ=2, and δ1=δ2=1. W^2TS2(Ψ2) is constructed by 121 neurons whose centers evenly spaced on [-0.8,0.8]×[-2,2] and widths η21=0.2, η22=0.5. Similarly, W^4TS4(Ψ4) is constructed by 2783 neurons whose centers evenly spaced on [-0.8,0.8]×[-2,2]×[-11,11] and widths η41=0.2, η42=0.5, η43=1.25. Other design parameters are c1=2, c2=5, c3=2, c4=30, Γ2=5, Γ4=60, and σ2=σ4=0.0001. The initial states are [q1(0),q˙1(0),q2(0),q˙2(0)]T=[0.5,0,0.2,0]T and [yd(0),y˙d(0)]T=[1,0]T. The initial weights W^2(0) and W^4(0) are both set as zero vectors.According to (40), the constant neural weights W¯2 and W¯4 in (53) and (54) are obtained as W¯2=meant∈[200,300]W^2(t) and W¯4=meant∈[400,500]W^4(t). In order to be compared with the adaptive neural DSC results, in this simulation, the parameters in performance function and the initial states are set the same as the values set in adaptive neural DSC, while the control gain are set as c1=1, c2=4, c3=2, and c4=5.The related simulation results are shown in Figures4–10. Figure 4 illustrates that, for the proposed adaptive neural DSC scheme, the tracking error e1 can ultimately converge to a small neighborhood of zero besides satisfying the prescribed performance; for the proposed learning control scheme, not only does the tracking error satisfy the prescribed performance, but also the convergence rate is faster under the similar control input amplitude shown in Figure 7. Moreover, the time consumption is decreased by 2/3 in contrast with adaptive neural DSC scheme. Figure 5 shows that the output x1 of system (56) tracks to the reference trajectory yd quickly. Figure 6 shows that other state variables are bounded for the proposed scheme. Figure 8 gives the NN approximation ability for the unknown dynamics. Figures 9 and 10 show that the NN weights converge to certain values along with the updated laws of NN weights.Figure 4
Tracking error: LC (-) and ANC (- -).Figure 5
System output:x1 (-) and yd (- -).Figure 6
State variablesx2 (-), x3 (- -), x4 (-.-).Figure 7
Control input: LC (-) and ANC (- -).Figure 8
Function approximation:W^iTSi(Ψi) (- -) and hi(Ψi) (-) (i=2,4).Figure 9
The convergence of partial NN weightsW^2.Figure 10
The convergence of partial NN weightsW^4.Remark 7.
In order to compare with the difference of tracking performance between different parameter selection in (3), these parameters in (3) are chosen as β0=1, β∞=0.05, κ=2, and δ1=δ2=1; the others are set as β0′=1, β∞′=0.05, κ′=1, and δ1′=δ2′=2. Define the tracking error of the former as e and the latter one as e′. Figure 11 illustrates two different tracking performances of the system’s output by selecting different parameters of constraints. It is evident that the tracking performance is affected by the parameter selection of performance function. And as the range of permissible error gets smaller, the tracking error is forced to converge faster and the tracking accuracy gets higher.Figure 11
Tracking error:e (-) and e′ (-.-).


### 5.2. Experiment Results on Baxter Robot

Moreover, in order to validate the effectiveness of the proposed control scheme, the Baxter bimanual robot is used in the experiment, as shown in Figure12. It is of two 7-DOF arms and advanced sensing technologies, including position, force, and torque sensors and control at every joint. The resolution for the joint sensors is 14 bits with 360 degrees (0.022 degrees per tick resolution), while the maximum joint torques that can be applied to the joints are 50 Nm (the first four joints) and 15 Nm (the last three joints). To compare with the simulation’s results, the desired reference trajectory in the experiment is the same as one in the simulation, which is also generated from (57). In the experiment, one of the Baxter robot’s links (such as the wrist’s link of robot’s right arm) is commanded to track the desired reference trajectory, and the tracking error between link’s angle position and reference trajectory is forced to converge to a small neighborhood around zero with prescribed performance in a finite time, while other links of the robot stay still. Figure 13 illustrates the desired motion of the robot’s link.Figure 12
Overview of the experimental platform.Figure 13
The desired motion of the robot’s link.To verify the effectiveness of neural learning control scheme for Baxter robot, the constant weightsW¯2 and W¯4 in Section 5.1 are reused in experiment. For comparison, the parameters in performance function and the initial states are set the same as these values in neural learning control, while the control gains are set as c1=1.5, c2=3, c3=2, and c4=4.The related results are shown in Figures14–16. Figure 14 shows the difference between LC of simulation and LC of Baxter robot. Although the vibration of the robot affects the tracking performance, it is evident that the tracking error of Baxter robot can also ultimately converge to a small neighborhood of zero besides satisfying the prescribed performance. Figure 15 show that the output of Baxter robot’s link (i.e., q1 of system (56)) tracks to the reference trajectory yd quickly, and other state variables are bounded for the proposed learning control scheme. Figure 16 shows that the control input of Baxter robot is also bounded with small overshoot and small mechanical vibration.Figure 14
Tracking error: LC (-) of Baxter robot and LC (- -) of simulation.Figure 15
State variablesq˙1 (-), q2 (- -), and q˙2 (-.-) of Baxter robot.Figure 16
Control input: LC (-) of Baxter robot and LC (- -) of simulation.


## 5.1. Numerical Simulation Results

Based on Theorem3, the adaptive control law is chosen as (36), and the virtual controllers are chosen as (15), (23), and (28) with the virtual control laws (24) and (37). In the performance function, β0=1, β∞=0.05, κ=2, and δ1=δ2=1. W^2TS2(Ψ2) is constructed by 121 neurons whose centers evenly spaced on [-0.8,0.8]×[-2,2] and widths η21=0.2, η22=0.5. Similarly, W^4TS4(Ψ4) is constructed by 2783 neurons whose centers evenly spaced on [-0.8,0.8]×[-2,2]×[-11,11] and widths η41=0.2, η42=0.5, η43=1.25. Other design parameters are c1=2, c2=5, c3=2, c4=30, Γ2=5, Γ4=60, and σ2=σ4=0.0001. The initial states are [q1(0),q˙1(0),q2(0),q˙2(0)]T=[0.5,0,0.2,0]T and [yd(0),y˙d(0)]T=[1,0]T. The initial weights W^2(0) and W^4(0) are both set as zero vectors.According to (40), the constant neural weights W¯2 and W¯4 in (53) and (54) are obtained as W¯2=meant∈[200,300]W^2(t) and W¯4=meant∈[400,500]W^4(t). In order to be compared with the adaptive neural DSC results, in this simulation, the parameters in performance function and the initial states are set the same as the values set in adaptive neural DSC, while the control gain are set as c1=1, c2=4, c3=2, and c4=5.The related simulation results are shown in Figures4–10. Figure 4 illustrates that, for the proposed adaptive neural DSC scheme, the tracking error e1 can ultimately converge to a small neighborhood of zero besides satisfying the prescribed performance; for the proposed learning control scheme, not only does the tracking error satisfy the prescribed performance, but also the convergence rate is faster under the similar control input amplitude shown in Figure 7. Moreover, the time consumption is decreased by 2/3 in contrast with adaptive neural DSC scheme. Figure 5 shows that the output x1 of system (56) tracks to the reference trajectory yd quickly. Figure 6 shows that other state variables are bounded for the proposed scheme. Figure 8 gives the NN approximation ability for the unknown dynamics. Figures 9 and 10 show that the NN weights converge to certain values along with the updated laws of NN weights.Figure 4
Tracking error: LC (-) and ANC (- -).Figure 5
System output:x1 (-) and yd (- -).Figure 6
State variablesx2 (-), x3 (- -), x4 (-.-).Figure 7
Control input: LC (-) and ANC (- -).Figure 8
Function approximation:W^iTSi(Ψi) (- -) and hi(Ψi) (-) (i=2,4).Figure 9
The convergence of partial NN weightsW^2.Figure 10
The convergence of partial NN weightsW^4.Remark 7.
In order to compare with the difference of tracking performance between different parameter selection in (3), these parameters in (3) are chosen as β0=1, β∞=0.05, κ=2, and δ1=δ2=1; the others are set as β0′=1, β∞′=0.05, κ′=1, and δ1′=δ2′=2. Define the tracking error of the former as e and the latter one as e′. Figure 11 illustrates two different tracking performances of the system’s output by selecting different parameters of constraints. It is evident that the tracking performance is affected by the parameter selection of performance function. And as the range of permissible error gets smaller, the tracking error is forced to converge faster and the tracking accuracy gets higher.Figure 11
Tracking error:e (-) and e′ (-.-).


## 5.2. Experiment Results on Baxter Robot

Moreover, in order to validate the effectiveness of the proposed control scheme, the Baxter bimanual robot is used in the experiment, as shown in Figure12. It is of two 7-DOF arms and advanced sensing technologies, including position, force, and torque sensors and control at every joint. The resolution for the joint sensors is 14 bits with 360 degrees (0.022 degrees per tick resolution), while the maximum joint torques that can be applied to the joints are 50 Nm (the first four joints) and 15 Nm (the last three joints). To compare with the simulation’s results, the desired reference trajectory in the experiment is the same as one in the simulation, which is also generated from (57). In the experiment, one of the Baxter robot’s links (such as the wrist’s link of robot’s right arm) is commanded to track the desired reference trajectory, and the tracking error between link’s angle position and reference trajectory is forced to converge to a small neighborhood around zero with prescribed performance in a finite time, while other links of the robot stay still. Figure 13 illustrates the desired motion of the robot’s link.Figure 12
Overview of the experimental platform.Figure 13
The desired motion of the robot’s link.To verify the effectiveness of neural learning control scheme for Baxter robot, the constant weightsW¯2 and W¯4 in Section 5.1 are reused in experiment. For comparison, the parameters in performance function and the initial states are set the same as these values in neural learning control, while the control gains are set as c1=1.5, c2=3, c3=2, and c4=4.The related results are shown in Figures14–16. Figure 14 shows the difference between LC of simulation and LC of Baxter robot. Although the vibration of the robot affects the tracking performance, it is evident that the tracking error of Baxter robot can also ultimately converge to a small neighborhood of zero besides satisfying the prescribed performance. Figure 15 show that the output of Baxter robot’s link (i.e., q1 of system (56)) tracks to the reference trajectory yd quickly, and other state variables are bounded for the proposed learning control scheme. Figure 16 shows that the control input of Baxter robot is also bounded with small overshoot and small mechanical vibration.Figure 14
Tracking error: LC (-) of Baxter robot and LC (- -) of simulation.Figure 15
State variablesq˙1 (-), q2 (- -), and q˙2 (-.-) of Baxter robot.Figure 16
Control input: LC (-) of Baxter robot and LC (- -) of simulation.


## 6. Conclusion

In this paper, we studied learning from adaptive neural dynamic surface control for a class of flexible joint manipulator with unknown dynamics under the prescribed constraint. A novel error transformed function was utilized to transform the constrained tracking problem into the the equivalent unconstrained one so as to facilitate the controller design. Furthermore, by combining DSC method, which was used to reduce the number of NN approximators and decrease the dimension of NN inputs, a novel adaptive neural control scheme was proposed to guarantee the prescribed performance during the transient process. Then, the closed-loop stability and control performance were achieved according to the construction of the Lyapunov function. After the stable control process, since two NNs were used in the controller design, the recurrent property of the NN input variables and the partial PE condition of RBF NNs were proved recursively. Therefore, the locally accurate approximations of unknown system dynamics by RBF NNs were achieved, and the proposed control scheme was verified to be capable of storing the learned knowledge in constant RBF NNs. Finally, the stored knowledge was reused to develop the neural learning controller for the same system model and the same or similar control task, so that the closed-loop stability and better control performance were achieved under the prescribed constraint. Simulation results for a single-link flexible joint manipulator and experiment results for Baxter robot were presented to prove the effectiveness of the proposed control scheme.



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*Source: 7683785-2017-10-31.xml*