# Weighted Multiple-Model Neural Network Adaptive Control for Robotic Manipulators with Jumping Parameters

Authors: Jiazhi Li; Weicun Zhang; Quanmin Zhu
Journal: Complexity
 (2020)

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## Abstract

This study addresses the tracking control issue forn-link robotic manipulators with largely jumping parameters. Based on radial basis function neural networks (RBFNNs), we propose weighted multiple-model neural network adaptive control (WMNNAC) approach. To cover the variation ranges of the parameters, different models of robotic are constructed. Then, the corresponding local neural network controller is constructed, in which the neural network has been used to approximate the uncertainty part of the control law, and an adaptive observer is implemented to estimate the true external disturbance. The WMNNAC strategy with improved weighting algorithm is adopted to ensure the tracking performance of the robotic manipulator system when parameters jump largely. Through the Lyapunov stability theory and the method of virtual equivalent system (VES), the stability of the closed-loop system is proved. Finally, the simulation results of a two-link manipulator verify the feasibility and efficiency of the proposed WMNNAC strategy.


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## Body

## 1. Introduction

Robotic manipulators are highly coupled, time-varying, and multivariable nonlinear dynamic systems. Due to the complexity of the system, the control problem has been extensively studied [1–4] in recent years.It should be noted that dynamic uncertainties in robotic system models are unavoidable due to the unknown load, mass, etc., and such modeling uncertainties may lead to a degradation on the control accuracy or even cause instability of the robotic system. The modeling uncertainties can be divided as structured uncertainties and unstructured uncertainties. For structured uncertainties, adaptive control has been widely used as an effective method for controlling complex systems in the field of robotic control [5–12] in recent years. In order to deal with the unstructured uncertainties of robotic manipulator systems, various learning-based control methods [13–17], including neural network (NN) and fuzzy systems, have been proposed to overcome them. As an effective method for approximating arbitrary nonlinear continuous functions with arbitrary precision, NN technology is very mature in modeling complex processes and compensating for unstructured uncertainties [14, 16, 18–22].However, most of the abovementioned references only consider the case that the robotic dynamic parameters are fixed or changed slowly. In real applications, robots often pick up or lay down different loads abruptly, which is known as large parameter uncertainties of robotic dynamics. Thus, the control performance of the abovementioned methods did not consider the jumping parameters which may deteriorate. Moreover, the system identification rate of conventional adaptive control is comparatively slow, which makes the transient performance of robotic systems poor in this situation [23]. As the robotic system with uncertain parameters is very complex, some researchers combine multiple-model strategy with neural networks to improve the control performance in recent years [24, 25].Multiple-model control strategy is a solution for controlling complex systems with jumping parameters [26, 27]. The goal of multiple-model control is to determine the most appropriate controller at any time based on identification of errors and appropriate criteria. In the past few years, switching multiple-model adaptive control [28–30] has been widely used to deal with the robotic system with jumping parameters. However, the switching between the controllers may make the control performance nonsmooth. In order to make the transition process smoother, we introduce weighted multiple-model adaptive control (WMMAC) approach [31–35]. Through the WMMAC method, the control range can cover the variation range of system parameter change, which can solve the control problem of the complex nonlinear system well when parameters change or jump unexpectedly. Moreover, it enhances the robustness of the system, effectively reduce the model identification time, and decrease the system transient error. As we all know, it is rather difficult to prove the closed-loop stability of the WMMAC system. Fortunately, in [36–38], some new results are presented on the system stability based on VES theory in recent years.Aiming at stabilizing the two-link robotic manipulator with largely jumping parameters, uncertainties, and external disturbance, a WMMAC scheme combining multiple NN-based controllers is proposed. The main contributions of this paper are summarized as follows:(i)
A WMNNAC scheme is proposed for robotic manipulators to deal with dynamic uncertainties and largely jumping parameters. It can improve the transient performance of the system.(ii)
The disturbance observer is constructed based on the RBFNNs. By online estimation, the error of observation is reduced.(iii)
In this paper, a modified weighting algorithm is given under the structure of robotic manipulators. Through VES theory, the stability and convergence of the robotic system are analyzed.The rest of this paper is organized as follows. Section2 gives the preliminaries. Section 3 is divided into three parts which successively describe the design process of the local controller and global controller and stability analysis of the controller against the jumping parameters in practice. Section 4 gives the simulation results to verify the feasibility of the developed control method. Finally, Section 5 concludes the paper.Notations.
Throughout this paper, the superscriptT stands for matrix transposition. ℝ denotes the space of real numbers; ℝn denotes the n-dimensional Euclidean space with the vector norm ⋅; and ℝn×m is the set of all n×m real matrices. λmin⋅ and λmax⋅ are the minimum and maximum eigenvalues of matrix ⋅, respectively. ∗ denotes taking the absolute values of all the elements in the vector ∗.


## 2. Preliminaries

### 2.1. Model of Robotic Manipulators

Considering the disturbance, the dynamics model of then-link rigid robotic can be drawn by the Lagrange equation as(1)Mqq¨+Cq,q˙q˙+Gq=u−d,where q,q˙,q¨∈ℝn represent the vector of the joint angular position, corresponding velocity, and acceleration, respectively; u∈ℝn is the applied joint torque; Mq∈ℝn×n is the inertia matrix; Cq,q˙∈ℝn×n denotes the Coriolis and centrifugal force; Gq∈ℝn is the gravity items; and d∈ℝn is the external disturbance. In this study, the following properties of system (1) with revolute joints are available [39, 40].Property 1.
M
q is symmetric and positive definite and is bounded by the following inequalities: ma≤Mq≤mb, where ma and mb are positive constants.Property 2.
1
/
2
M
˙
q
−
C
q
,
q
˙ is skew-symmetric, i.e., xTM˙q−2Cq,q˙x=0, ∀x∈ℝn.
The following assumption is used in this paper.Assumption 1.
We assume that the disturbancedt is continuous and uniformly bounded, i.e., there exists a constant d¯>0 such that dt≤d¯, ∀t∈0,∞ and a constant d¯′>0 such that d˙t≤d¯′, ∀t∈0,∞.


### 2.2. RBFNN

RBFNN is a special NN architecture with some advantages. In the field of robotic control, RBFNN has been widely used owing to the powerful approximation ability to nonlinear functions. RBFNN has a simple structure with input, output, and hidden layers. The input layer is simply a fan-out layer, and the input vector is donated asZ=Z1,Z2,…,ZqT∈ΩZ⊂ℝq. In the hidden layer, Gaussian kernel functions are selected as activation functions, that is,(2)hj=exp−Z−μjTZ−μjbj2,j=1,2,…,m,where μj=μj1,μj2,…,μjq represents the center value of the jth node, NN node number m>1, and bj is the value of width [41].The output layer computes the output value through the vector of connection weights. In this paper, RBFNN is utilized to approximate the continuous function, where the weight vectorWi∈ℝm. There exist an ideal constant weight vector Wi∗ and the bounded approximation error εiZ, i.e., εiZ<ε¯i with ε¯i>0 for all input Z∈ΩZ. This is computed as(3)fiZ=Wi∗THiZ+εiZ,∀Z∈ΩZ⊂ℝq,i=1,2,…,n,where Hi=h1,h2,…,hmT.


### 2.3. Useful Mathematic Tools

Definition 1.
sgn
a
∈
ℝ
n is defined as(4)sgna=signa1,signa2,…,signanT,where a is an n-dimensional vector and sign⋅ is the standard signum function.Definition 2.
b
t
∈
ℝ
n is defined as(5)bt=b1tsignb1,b2tsignb2,…,bntsignbnT,where b is an n-dimensional vector.Definition 3.
The operator “⊙” is defined as(6)c⊙d=c1d1,c2d2,…,c2d2T,where c and d are two n-dimensional vectors.


## 2.1. Model of Robotic Manipulators

Considering the disturbance, the dynamics model of then-link rigid robotic can be drawn by the Lagrange equation as(1)Mqq¨+Cq,q˙q˙+Gq=u−d,where q,q˙,q¨∈ℝn represent the vector of the joint angular position, corresponding velocity, and acceleration, respectively; u∈ℝn is the applied joint torque; Mq∈ℝn×n is the inertia matrix; Cq,q˙∈ℝn×n denotes the Coriolis and centrifugal force; Gq∈ℝn is the gravity items; and d∈ℝn is the external disturbance. In this study, the following properties of system (1) with revolute joints are available [39, 40].Property 1.
M
q is symmetric and positive definite and is bounded by the following inequalities: ma≤Mq≤mb, where ma and mb are positive constants.Property 2.
1
/
2
M
˙
q
−
C
q
,
q
˙ is skew-symmetric, i.e., xTM˙q−2Cq,q˙x=0, ∀x∈ℝn.
The following assumption is used in this paper.Assumption 1.
We assume that the disturbancedt is continuous and uniformly bounded, i.e., there exists a constant d¯>0 such that dt≤d¯, ∀t∈0,∞ and a constant d¯′>0 such that d˙t≤d¯′, ∀t∈0,∞.


## 2.2. RBFNN

RBFNN is a special NN architecture with some advantages. In the field of robotic control, RBFNN has been widely used owing to the powerful approximation ability to nonlinear functions. RBFNN has a simple structure with input, output, and hidden layers. The input layer is simply a fan-out layer, and the input vector is donated asZ=Z1,Z2,…,ZqT∈ΩZ⊂ℝq. In the hidden layer, Gaussian kernel functions are selected as activation functions, that is,(2)hj=exp−Z−μjTZ−μjbj2,j=1,2,…,m,where μj=μj1,μj2,…,μjq represents the center value of the jth node, NN node number m>1, and bj is the value of width [41].The output layer computes the output value through the vector of connection weights. In this paper, RBFNN is utilized to approximate the continuous function, where the weight vectorWi∈ℝm. There exist an ideal constant weight vector Wi∗ and the bounded approximation error εiZ, i.e., εiZ<ε¯i with ε¯i>0 for all input Z∈ΩZ. This is computed as(3)fiZ=Wi∗THiZ+εiZ,∀Z∈ΩZ⊂ℝq,i=1,2,…,n,where Hi=h1,h2,…,hmT.


## 2.3. Useful Mathematic Tools

Definition 1.
sgn
a
∈
ℝ
n is defined as(4)sgna=signa1,signa2,…,signanT,where a is an n-dimensional vector and sign⋅ is the standard signum function.Definition 2.
b
t
∈
ℝ
n is defined as(5)bt=b1tsignb1,b2tsignb2,…,bntsignbnT,where b is an n-dimensional vector.Definition 3.
The operator “⊙” is defined as(6)c⊙d=c1d1,c2d2,…,c2d2T,where c and d are two n-dimensional vectors.


## 3. Control Design

In this study, the control objective is that the robotic manipulators can still track the desired trajectory well with good transient performance when the system parameters jump. In order to achieve this objective, we develop the WMNNAC. The design process is given as follows: firstly, according to the variation ranges of parameters, we create multiple submodels to built a model set. Secondly, corresponding local NN-based controllers are designed by Lyapunov theory, and stable local closed-loop systems can be obtained. Finally, an improved weighting algorithm is proposed for the characteristics of the robotic system. Based on the new algorithm, the weights corresponding to each local controller are obtained at each moment. It is used for fusing local controllers to generate the global control signals.Obviously, the controller design is divided into two parts: the local controller and the global controller. The concise block diagram of the proposed WMNNAC system is shown in Figure1, where ut is the global control signal, yt is the output of robotic manipulators, yξt is the output of the local model, NNCξ is the ξth local NN-based controller, and uξt and pξt are the corresponding control signal and weight, ξ=1,2,…,N.Figure 1
Concise block diagram of the WMNNAC system.


### 3.1. Local Controller Design Based on RBFNNS

We first consider the design of the local controller. When the parameters of the robotic are known, the control objectives can be well achieved. However, the model of the robotic includes uncertainties and disturbance, and the accurate values are unknown in the actual system. Considering the complexity of manipulator dynamics and the effective learning mechanism of NN, the RBFNN adaptive controller is developed to eliminate the influence of uncertainties. Furthermore, we give a generalized error to speed up the recovery after system instability. Meanwhile, we design an adaptive observer to compensate for external disturbances.Letx1=q and x2=q˙, and the robotic dynamics model (1) is described as(7)x˙1=x2,x˙2=M−1u−d−Cx2−G.Define a tracking errorz1 and a second error by introducing a virtual control α1=−K1z1+x˙d, K1∈ℝn is a gain matrix:(8)z1=x1−xd,z2=x2−α1.Differentiatingz1 and taking the time derivative of z2 lead to(9)z˙1=z2+α1−x˙d=−K1z1+z2,z˙2=M−1u−d−Cx2−G−α˙1.Then, we propose the first Lyapunov function candidate as(10)V1=12z1Tz1.DifferentiatingV1 yields(11)V˙1=−z1TK1z1+z1Tz2.Choosing a generalized error yields(12)s=z1+1βz2φ,where β=diagβ1,β2,…,βn with βi>0 and φ satisfies the following condition: 1<φ<2. Then, s˙ can be written as(13)s˙=z˙1+φ1βz˙2⊙z2φ−1.Consider a new Lyapunov function candidate as follows:(14)V2=12z1Tz1+12sTMs.DifferentiatingV2 with respect to time, we obtain(15)V˙2=V˙1+sTMs˙+12sTM˙s=−z1TK1z1+z1Tz2+sTCs+sTMz˙1+φ1βz˙2⊙z2φ−1=−z1TK1z1+z1Tz2+sTCs+Mz˙1+sTφ1βu−d−Cx2−G−Mα˙1⊙z2φ−1.According to Assumption1, we can obtain the upper boundary of the disturbance. However, the upper boundary of external disturbance is usually unknown and cannot be measured. We construct a disturbance observer d^ to solve these problems. Define the estimation error d˜=d−d^. Then, a new Lyapunov function is given as follows:(16)V3=V2+12d˜TΓ1−1d˜.Its time derivative can be written as(17)V˙3=V˙2+d˜TΓ1−1d˙−d^˙≤−z1TK1z1+z1Tz2+sTCs+Mz˙1+sTφ1βu−d^−Cx2−G−Mα˙1⊙z2φ−1−∑i=1nd˜iΓ1i−1Γ1isiTφ1βi⊙z2iφ−1+d^˙i+d˜TΓ1−1d¯′.WhenM, C, and G are known, the controller can be designed as(18)u0=Cx2+G+Mα˙1+d^−φ−1βCs+Mz˙1⊙z21−φ−φ−1βss2z1Tz2+ηssigns⊙z21−φ.Also, the controller can be designed as(19)d^˙i=−Γ1isiTφ1βi⊙z2iφ−1+δ1id^i,where Γ1 is the constant matrix and δ1 is a positive constant. Plugging (18) and (19) into (17) yields(20)V˙3≤−z1TK1z1−ηs2+∑i=1nd˜iΓ1i−1d1i′+∑i=1nd˜iδ1id^i≤−z1TK1z1−ηs2−12∑i=1nδ1i−Γ1i−1d˜i2+12∑i=1nΓ1i−1d′¯i2+12∑i=1nδ1id¯i2≤−ρ1V3+c1,where ρ1 and c1 are two positive constants given as(21)ρ1=min2λminK1,2ηλmaxM,mini=1,2,…,nδ1i−Γ1i−1λmaxΓ1−1,c1=12∑i=1nΓ1i−1d′¯i2+12∑i=1nδ1id¯i2.To guaranteeρ1>0, the control gain Γ1i and δ1i is selected to satisfy (22)mini=1,2,…,nδ1i−Γ1i−1>0.Since uncertainties sometimes unavoidably exist inM, C, and G, the aforementioned control law, which is built on the assumption that the exact parameters can be obtained, is not applicable for the robotic system. Thus, we utilize RBFNNs to approximate the uncertainties by the online estimation.(23)W∗THZ=−Cx2−G−Mα˙1⊙z2φ−1+φ−1βCs+Mz˙1−εZ,where Z=q,q˙,s,α˙1 are the input variables and εZ∈ℝn is the approximation error.We propose the RBFNN adaptive control as(24)u=−φ−1βss2z1Tz2+ηssigns⊙z21−φ−W^THZ⊙z21−φ+d^.The weight adaptation law is designed as(25)Wi^˙=Γ2isiT1βiφHiZ−δ2iW^i,where Γ2 is the constant matrix and δ2 is a positive constant.Theorem 1.
Considering the robotic system described by (1) with bounded initial conditions, under the control law (24), the NN adaption law (25), and the disturbance observer adaption law (19), if Assumption 1 holds, we can obtain the semiglobal uniform boundedness stability of the closed-loop system. In addition, the closed-loop error signals z1, s, d˜, and W˜ will remain within the compact sets Ωz1, Ωs, Ωd˜, and ΩW˜, respectively, defined by(26)Ωz1=z1∈ℝnz1≤D,Ωs=s∈ℝns≤DλminM,Ωd˜=d˜∈ℝnd˜≤DλminΓ1i−1,Ωw˜=W˜∈ℝnW˜≤DλminΓ2i−1,where D=2V0+c/ρ and ρ and c are two positive constants.Proof.
Consider the overall Lyapunov function candidate(27)V=12z1Tz1+12sTMs+12d˜TΓ1−1d˜+12∑i=1nW˜iTΓ2i−1W˜i.
Differentiating (27) and substituting control law (24) into it yield(28)V˙≤−z1TK1z1−ηs2−12∑i=1nδ1i−Γ1i−1d˜i2+12∑i=1nΓ1i−1d′¯i2+12∑i=1nδ1id¯i2+sTφ1βε−W˜THZ+∑i=1nW˜iTΓ2i−1W^˙i.
Substituting weight adaptation law (25) into (28) and simplifying it yield(29)V˙≤−z1TK1z1−sTηI−φ12βs−12∑i=1nδ1i−Γ1i−1d˜i2−12∑i=1nW˜iTδ2iW˜i+12∑i=1nΓ1i−1d′¯i2+12∑i=1nδ1id¯i2+φ12βε¯2+12∑i=1nWi∗Tδ2iWi∗≤−ρV+c,where(30)ρ=min2λminK1,2λminηI−1/2βφλmaxM,mini=1,2,…,nδ1i−Γ1i−1λmaxΓ1−1,mini=1,2,…,nδ2iλmaxΓ2−1,c=12∑i=1nΓ1i−1d′¯i2+12∑i=1nδ1id¯i2+12βφε¯2+12∑i=1nWi∗Tδ2iWi∗.
To guaranteeρ>0, the control gains η, Γ1i, and δ1i are selected to satisfy(31)λminηI−12βφ>0,mini=1,2,…,nδ1i−Γ1i−1>0.
From the above analysis, we know that the signalsz1 and s, as well as the approximation errors d˜ and W˜ are bounded.
Multiplying both sides byeρt in (29) and integrating it, we obtain(32)Vt≤V0−cρe−ρt+cρ≤V0+cρ.
Then, the following inequalities hold(33)12z1Tz1≤V0+cρ.
Hence,z1 converges to the compact set. Bounds for s, d˜ and W˜ can be similarly shown, and this concludes the proof.


### 3.2. Global Controller Design Based on Weighted Multiple-Model Approach

Great progress has been made in solving uncertainties by NN, yet it cannot be handled well when the parameters jump largely. WMNNAC, as a method combining conventional adaptive control and prior knowledge, can improve the trajectory tracking performance of robotic with largely jumping parameters. The design process includes the construction of the model set and controller set, as well as the weighting algorithm.Several local models with fixed parameter are established to form a model set, which is expressed asΛ=Mξξ=1,2,…,N, where Mξ is the ξth local model.The design process of the local controller has been given based on Lyapunov direct method. It is capable of achieving good trajectory tracking of the robotic with uncertainties and disturbances. Corresponding to each local model, we design a local NN-based controller by this method. Thus, we can establish the controller setΨ=Cξξ=1,2,…,N.Based on [42], a new weighting update algorithm is proposed for the structure of the robotic manipulators. It is employed to coordinate local NN-based controllers to generate the global control signals in real time. The key to the new algorithm is to define a performance index based on the model output error.For clarity, the algorithm is expressed in the discrete-time form, and a zero-order holder can be adopted to obtain continuous signals.Since this paper studies then-link robotic manipulators, we should consider that single error has the approximately same influence on model output errors, and we define the model output errors as(34)eξ=qξ−q=eξ1,eξ2,…,eξn,ξ=1,2,…,N,eξ2k=θ1eξ12k+θ2eξ22k+⋯+θneξn2k,where θi>0 is the error weight of the ith link of robotic in the model set.The performance index is designed as(35)lξ′k=α+eξ2k,where α is a small value.Then, we give the following weighting algorithm:(36)lmin′k=minlξ′k,(37)gk=lmin′klξ′k,(38)lξk=lξk−1,gk=1,lξk−1gkceil1/1−gk,gk<1,(39)pξk=lξk∑ξ=1Nlξk.From above all, the global control law is defined as(40)uck=∑ξ=1Npξkuξk,where the value of pξk lies between 0 and 1, and ∑ξ=1Npξk=1,∀k≥0.


### 3.3. Convergence Analysis of the WMNNAC System

We have the following convergence result for the WMNNAC system.Theorem 2.
Respecting the WMNNAC system structured in Figure1, it owns the following properties:(1)
The time interval of parameter jump in the plant model is long enough (for details, see description in section Simulation)(2)
Various stages of the plant model can be approximated by model setΛ(3)
Each local NN-based controller is well designed such that the closed-loop system is stable, and the output of system is tracking the reference signal(4)
Mξ∈Λ is the model closest to the current plant in the following sense with probability one for ∀k≥k¯(41)∑r=1Tj+keμ2r<∑r=1Tj+keξ2r,∀k≥k¯,μ≠ξ,limk⟶∞1k∑r=1Tj+keμ2r=Rμ;limk⟶∞1k∑r=1Tj+keξ2r=Rξ,Rμ<Rξ,μ≠ξ,where k¯ is an unknown finite time instant, Rμ is a constant value, Rξ may be constant value or infinity, and Tj,j=0,1,2,… is the time sequence for jumping parameters.Then, the global controller converges to the most appropriate local controller at each stage of parameter jumps, and the WMNNAC system is stable.Proof.
Firstly, it is not difficult to see that algorithms (36)–(39) together with properties (2) and (4) in Theorem 2 guarantee with probability one that(42)lmin′k=lξ′k,lmin′klξ′k=1,lmin′klμ′k<1,∀k≥k¯,μ≠ξ.
Moreover, if there is(43)limk⟶∞gk=limk⟶∞lmin′klξ′k=1.
Then, from (38), we have(44)limk⟶∞gkceil1/1−gk=1e<1.
It can always guarantee that(45)lξk⟶lξ0;lμk⟶lμ0,ξ≠μ.
Thus, from (39), we obtain(46)limk⟶∞pξk=1;limk⟶∞pμk=0,ξ≠μ.
That means the weighting algorithm is convergent.
Further, we know that the global controller of the WMNNAC system will converge to the most appropriate local controller corresponding to the local modelMξ∈Λ. Besides, the NN-based local controllers are established in the same way. Then, the WMNNAC system can be described as VES in the input-output sense, as shown in Figure 2.
M
ξ
∈
Λ is the model closest to the true plant, Cξ is the corresponding NN-based controller, Δuk is the equivalent control error, Δuk=uk−uξk, and e′k is the equivalent output error, e′k=yk−yξk.
For the NN-based controller, we have rigorously proved the stability of the local closed-loop system based on Lyapunov theory, and the influence of other fixed controllers is included inΔuk. According to property (3), the stability can be obtained for a short time period.
Finally, since the jumps cannot be infinitely fast, the WMNNAC system is composed of “slow” switching between these stable local systems. By switching system theory [43], the WMNNAC system of the robotic manipulator with jumping parameters is stable.Figure 2
VES of the WMNNAC system.


## 3.1. Local Controller Design Based on RBFNNS

We first consider the design of the local controller. When the parameters of the robotic are known, the control objectives can be well achieved. However, the model of the robotic includes uncertainties and disturbance, and the accurate values are unknown in the actual system. Considering the complexity of manipulator dynamics and the effective learning mechanism of NN, the RBFNN adaptive controller is developed to eliminate the influence of uncertainties. Furthermore, we give a generalized error to speed up the recovery after system instability. Meanwhile, we design an adaptive observer to compensate for external disturbances.Letx1=q and x2=q˙, and the robotic dynamics model (1) is described as(7)x˙1=x2,x˙2=M−1u−d−Cx2−G.Define a tracking errorz1 and a second error by introducing a virtual control α1=−K1z1+x˙d, K1∈ℝn is a gain matrix:(8)z1=x1−xd,z2=x2−α1.Differentiatingz1 and taking the time derivative of z2 lead to(9)z˙1=z2+α1−x˙d=−K1z1+z2,z˙2=M−1u−d−Cx2−G−α˙1.Then, we propose the first Lyapunov function candidate as(10)V1=12z1Tz1.DifferentiatingV1 yields(11)V˙1=−z1TK1z1+z1Tz2.Choosing a generalized error yields(12)s=z1+1βz2φ,where β=diagβ1,β2,…,βn with βi>0 and φ satisfies the following condition: 1<φ<2. Then, s˙ can be written as(13)s˙=z˙1+φ1βz˙2⊙z2φ−1.Consider a new Lyapunov function candidate as follows:(14)V2=12z1Tz1+12sTMs.DifferentiatingV2 with respect to time, we obtain(15)V˙2=V˙1+sTMs˙+12sTM˙s=−z1TK1z1+z1Tz2+sTCs+sTMz˙1+φ1βz˙2⊙z2φ−1=−z1TK1z1+z1Tz2+sTCs+Mz˙1+sTφ1βu−d−Cx2−G−Mα˙1⊙z2φ−1.According to Assumption1, we can obtain the upper boundary of the disturbance. However, the upper boundary of external disturbance is usually unknown and cannot be measured. We construct a disturbance observer d^ to solve these problems. Define the estimation error d˜=d−d^. Then, a new Lyapunov function is given as follows:(16)V3=V2+12d˜TΓ1−1d˜.Its time derivative can be written as(17)V˙3=V˙2+d˜TΓ1−1d˙−d^˙≤−z1TK1z1+z1Tz2+sTCs+Mz˙1+sTφ1βu−d^−Cx2−G−Mα˙1⊙z2φ−1−∑i=1nd˜iΓ1i−1Γ1isiTφ1βi⊙z2iφ−1+d^˙i+d˜TΓ1−1d¯′.WhenM, C, and G are known, the controller can be designed as(18)u0=Cx2+G+Mα˙1+d^−φ−1βCs+Mz˙1⊙z21−φ−φ−1βss2z1Tz2+ηssigns⊙z21−φ.Also, the controller can be designed as(19)d^˙i=−Γ1isiTφ1βi⊙z2iφ−1+δ1id^i,where Γ1 is the constant matrix and δ1 is a positive constant. Plugging (18) and (19) into (17) yields(20)V˙3≤−z1TK1z1−ηs2+∑i=1nd˜iΓ1i−1d1i′+∑i=1nd˜iδ1id^i≤−z1TK1z1−ηs2−12∑i=1nδ1i−Γ1i−1d˜i2+12∑i=1nΓ1i−1d′¯i2+12∑i=1nδ1id¯i2≤−ρ1V3+c1,where ρ1 and c1 are two positive constants given as(21)ρ1=min2λminK1,2ηλmaxM,mini=1,2,…,nδ1i−Γ1i−1λmaxΓ1−1,c1=12∑i=1nΓ1i−1d′¯i2+12∑i=1nδ1id¯i2.To guaranteeρ1>0, the control gain Γ1i and δ1i is selected to satisfy (22)mini=1,2,…,nδ1i−Γ1i−1>0.Since uncertainties sometimes unavoidably exist inM, C, and G, the aforementioned control law, which is built on the assumption that the exact parameters can be obtained, is not applicable for the robotic system. Thus, we utilize RBFNNs to approximate the uncertainties by the online estimation.(23)W∗THZ=−Cx2−G−Mα˙1⊙z2φ−1+φ−1βCs+Mz˙1−εZ,where Z=q,q˙,s,α˙1 are the input variables and εZ∈ℝn is the approximation error.We propose the RBFNN adaptive control as(24)u=−φ−1βss2z1Tz2+ηssigns⊙z21−φ−W^THZ⊙z21−φ+d^.The weight adaptation law is designed as(25)Wi^˙=Γ2isiT1βiφHiZ−δ2iW^i,where Γ2 is the constant matrix and δ2 is a positive constant.Theorem 1.
Considering the robotic system described by (1) with bounded initial conditions, under the control law (24), the NN adaption law (25), and the disturbance observer adaption law (19), if Assumption 1 holds, we can obtain the semiglobal uniform boundedness stability of the closed-loop system. In addition, the closed-loop error signals z1, s, d˜, and W˜ will remain within the compact sets Ωz1, Ωs, Ωd˜, and ΩW˜, respectively, defined by(26)Ωz1=z1∈ℝnz1≤D,Ωs=s∈ℝns≤DλminM,Ωd˜=d˜∈ℝnd˜≤DλminΓ1i−1,Ωw˜=W˜∈ℝnW˜≤DλminΓ2i−1,where D=2V0+c/ρ and ρ and c are two positive constants.Proof.
Consider the overall Lyapunov function candidate(27)V=12z1Tz1+12sTMs+12d˜TΓ1−1d˜+12∑i=1nW˜iTΓ2i−1W˜i.
Differentiating (27) and substituting control law (24) into it yield(28)V˙≤−z1TK1z1−ηs2−12∑i=1nδ1i−Γ1i−1d˜i2+12∑i=1nΓ1i−1d′¯i2+12∑i=1nδ1id¯i2+sTφ1βε−W˜THZ+∑i=1nW˜iTΓ2i−1W^˙i.
Substituting weight adaptation law (25) into (28) and simplifying it yield(29)V˙≤−z1TK1z1−sTηI−φ12βs−12∑i=1nδ1i−Γ1i−1d˜i2−12∑i=1nW˜iTδ2iW˜i+12∑i=1nΓ1i−1d′¯i2+12∑i=1nδ1id¯i2+φ12βε¯2+12∑i=1nWi∗Tδ2iWi∗≤−ρV+c,where(30)ρ=min2λminK1,2λminηI−1/2βφλmaxM,mini=1,2,…,nδ1i−Γ1i−1λmaxΓ1−1,mini=1,2,…,nδ2iλmaxΓ2−1,c=12∑i=1nΓ1i−1d′¯i2+12∑i=1nδ1id¯i2+12βφε¯2+12∑i=1nWi∗Tδ2iWi∗.
To guaranteeρ>0, the control gains η, Γ1i, and δ1i are selected to satisfy(31)λminηI−12βφ>0,mini=1,2,…,nδ1i−Γ1i−1>0.
From the above analysis, we know that the signalsz1 and s, as well as the approximation errors d˜ and W˜ are bounded.
Multiplying both sides byeρt in (29) and integrating it, we obtain(32)Vt≤V0−cρe−ρt+cρ≤V0+cρ.
Then, the following inequalities hold(33)12z1Tz1≤V0+cρ.
Hence,z1 converges to the compact set. Bounds for s, d˜ and W˜ can be similarly shown, and this concludes the proof.


## 3.2. Global Controller Design Based on Weighted Multiple-Model Approach

Great progress has been made in solving uncertainties by NN, yet it cannot be handled well when the parameters jump largely. WMNNAC, as a method combining conventional adaptive control and prior knowledge, can improve the trajectory tracking performance of robotic with largely jumping parameters. The design process includes the construction of the model set and controller set, as well as the weighting algorithm.Several local models with fixed parameter are established to form a model set, which is expressed asΛ=Mξξ=1,2,…,N, where Mξ is the ξth local model.The design process of the local controller has been given based on Lyapunov direct method. It is capable of achieving good trajectory tracking of the robotic with uncertainties and disturbances. Corresponding to each local model, we design a local NN-based controller by this method. Thus, we can establish the controller setΨ=Cξξ=1,2,…,N.Based on [42], a new weighting update algorithm is proposed for the structure of the robotic manipulators. It is employed to coordinate local NN-based controllers to generate the global control signals in real time. The key to the new algorithm is to define a performance index based on the model output error.For clarity, the algorithm is expressed in the discrete-time form, and a zero-order holder can be adopted to obtain continuous signals.Since this paper studies then-link robotic manipulators, we should consider that single error has the approximately same influence on model output errors, and we define the model output errors as(34)eξ=qξ−q=eξ1,eξ2,…,eξn,ξ=1,2,…,N,eξ2k=θ1eξ12k+θ2eξ22k+⋯+θneξn2k,where θi>0 is the error weight of the ith link of robotic in the model set.The performance index is designed as(35)lξ′k=α+eξ2k,where α is a small value.Then, we give the following weighting algorithm:(36)lmin′k=minlξ′k,(37)gk=lmin′klξ′k,(38)lξk=lξk−1,gk=1,lξk−1gkceil1/1−gk,gk<1,(39)pξk=lξk∑ξ=1Nlξk.From above all, the global control law is defined as(40)uck=∑ξ=1Npξkuξk,where the value of pξk lies between 0 and 1, and ∑ξ=1Npξk=1,∀k≥0.


## 3.3. Convergence Analysis of the WMNNAC System

We have the following convergence result for the WMNNAC system.Theorem 2.
Respecting the WMNNAC system structured in Figure1, it owns the following properties:(1)
The time interval of parameter jump in the plant model is long enough (for details, see description in section Simulation)(2)
Various stages of the plant model can be approximated by model setΛ(3)
Each local NN-based controller is well designed such that the closed-loop system is stable, and the output of system is tracking the reference signal(4)
Mξ∈Λ is the model closest to the current plant in the following sense with probability one for ∀k≥k¯(41)∑r=1Tj+keμ2r<∑r=1Tj+keξ2r,∀k≥k¯,μ≠ξ,limk⟶∞1k∑r=1Tj+keμ2r=Rμ;limk⟶∞1k∑r=1Tj+keξ2r=Rξ,Rμ<Rξ,μ≠ξ,where k¯ is an unknown finite time instant, Rμ is a constant value, Rξ may be constant value or infinity, and Tj,j=0,1,2,… is the time sequence for jumping parameters.Then, the global controller converges to the most appropriate local controller at each stage of parameter jumps, and the WMNNAC system is stable.Proof.
Firstly, it is not difficult to see that algorithms (36)–(39) together with properties (2) and (4) in Theorem 2 guarantee with probability one that(42)lmin′k=lξ′k,lmin′klξ′k=1,lmin′klμ′k<1,∀k≥k¯,μ≠ξ.
Moreover, if there is(43)limk⟶∞gk=limk⟶∞lmin′klξ′k=1.
Then, from (38), we have(44)limk⟶∞gkceil1/1−gk=1e<1.
It can always guarantee that(45)lξk⟶lξ0;lμk⟶lμ0,ξ≠μ.
Thus, from (39), we obtain(46)limk⟶∞pξk=1;limk⟶∞pμk=0,ξ≠μ.
That means the weighting algorithm is convergent.
Further, we know that the global controller of the WMNNAC system will converge to the most appropriate local controller corresponding to the local modelMξ∈Λ. Besides, the NN-based local controllers are established in the same way. Then, the WMNNAC system can be described as VES in the input-output sense, as shown in Figure 2.
M
ξ
∈
Λ is the model closest to the true plant, Cξ is the corresponding NN-based controller, Δuk is the equivalent control error, Δuk=uk−uξk, and e′k is the equivalent output error, e′k=yk−yξk.
For the NN-based controller, we have rigorously proved the stability of the local closed-loop system based on Lyapunov theory, and the influence of other fixed controllers is included inΔuk. According to property (3), the stability can be obtained for a short time period.
Finally, since the jumps cannot be infinitely fast, the WMNNAC system is composed of “slow” switching between these stable local systems. By switching system theory [43], the WMNNAC system of the robotic manipulator with jumping parameters is stable.Figure 2
VES of the WMNNAC system.


## 4. Simulation

In this section, considering a two-link robotic manipulator shown in Figure3, simulations are carried out to verify the performance of the controller designed in Section 3. It is assumed to move on the Cartesian space and then the position vector q can be rewritten as(47)q=q1q2.Figure 3
Schematic of the two-link robot manipulator model.For kinematics model (1), the parameters can be written as(48)Mq=M11M12M21m2lc22+I2,(49)Cq,q˙=−m2l1lc2q˙2sinq2C12m2l1lc2q˙1sinq20,(50)Gq=G1m2lc2gcosq1+q2,,(51)M11=m1lc12+m2l12+lc12+2l1lc2cosq2+I1+I2,(52)M12=M21=m2lc22+l1lc2cosq2+I2,(53)C12=−m2l1lc2q˙1+q˙2sinq2,(54)G1=m1lc2+m2l1gcosq1+m2lc2gcosq1+q2,(55)l1=2lc1,l2=2lc2.System parameter values of robotic are given as(56)m1=2kg,m2=1kg,l1=0.35m,l2=0.31m,I1=61.25×10−3kgm2,I2=20.42×10−3kgm2.The states of robotic are initialized atq10=q20=0 and q˙10=q˙20=0, and the original configuration of the NN weights is zero. The target trajectory tracking a circular path is given as qd=q1d,q2dT=sint+cost,sint+costT, where t∈0,tf and tf=20s. The external disturbance is given as d=sint+1,2cost+0.5T.There are three parts encompassed in the following simulations. Case 1 presents the good control effects of the proposed local NN-based controller with external disturbance. Case 2 is used for comparison to show the poor performance of the local NN-based controller when largely jumping parameters occur in the robotic system. Case 3 shows the advancement of the proposed WMNNAC method for robotic in the case of parameters jumping.Case 1.
Local NN-based controller.
For the NN-based control system, the constant parameters are chosen to build model asK1=100I2×2, η=28, 1/β=27I2×2, φ=1.182, Γ1=0.5I2×2, and σ2=100I2×2, which satisfy the conditions in (31). For the RBFNN, it owns 256 nodes, and its parameters are given as Γ2=100I256×256 and σ2=0.2I2×2. Moreover, the initial weights of RBFNN are 0.
The simulation results of the NN-based controller are shown in Figures4–7. They can be stated that the angle positions can track desired trajectory gratifyingly, and the tracking errors are convergent and fluctuate gently around zero. The values of the controller are presented in Figure 6. From Figure 7, we can notice that the approximation weights of RBFNN remain within bound.Figure 4
Tracking performance ofq1 and q2 with the single NN-based controller.Figure 5
Tracking error under the single NN-based controller.Figure 6
Single NN-based control inputs.Figure 7
Estimation weights with the single NN-based controller.Case 2.
Single NN-based controller for the robotic manipulators with jumping parameters.
Meanwhile, we consider that the value ofm2 and l2 in the robotic system is mutative, and other parameters are the same as above. When the mass of link 2 is changed from 1 kg to 7 kg and the length of link 2 is changed from 0.31 m to 0.2 m at t=6s, we can obtain the results shown in Figures 8 and 9. Compare to Figure 5, they show that the tracking error fluctuates more intense. Obviously, single NN-based controller cannot meet the tracking effect and even bring about system failures when model parameters jump largely.Figure 8
Tracking performance ofq1 and q2 of the single NN-based controller with jumping parameters.Figure 9
Tracking error under the single NN-based controller with jumping parameters.Case 3.
WMNNAC for the robotic manipulators with jumping parameters.
The implementation method of each local system is similar to Case 1. We just set suitable parameters to each local system for better performances. According to (48)–(55), the different robotic manipulator system is built by different model parameters. In this simulation, m2 and l2 are different in each model. Under this circumstance, the parameters of the model set and controller set are shown in Table 1. The other parameters of the three local system are the same as Case 1. It is assumed that the manipulator parameter jumps twice, respectively, from model 1 to model 2 at t=3s and from model 2 to model 3 at t=6s.
The simulation results are shown in Figures10–13. Compared to Case 2, it can be seen that the angle positions of the robotic can still follow the desired trajectory well, and the errors fluctuate and remain within a small range even when the parameter jumps largely. Moreover, the weights of three local controllers are demonstrated in Figure 12. It can be seen that the weights converge well. Besides, Figure 13 shows the global control signals, and the results indicate that the system can choose suitable control inputs rapidly to adapt the abrupt changes of parameters by the weighting algorithm.
Finally, we analyze the transient performance of the robotic system by comparing the error of case 2 and case 3 when the parameters jump. By partial enlargement, the comparisons are shown in Figure14. As we can see, the greater error fluctuations and longer adjustment times based on the single NN-based controller are shown in Figure 14(a). Encouragingly, the WMNNAC can improve the transient performance shown in Figure 14(b).
By the superiority of fusing multiple controllers, the WMNNAC method is effective to avoid instability and poor transient performance caused by largely jumping parameters.Table 1
The designed parameters of the model set and the controller set.
Model set
Model parameters
Corresponding controller parameters
Model 1
m
2
=
1
kg, l2=0.31kg
η
=
28, 1/β=27I2×2
Model 2
m
2
=
3
kg, l2=0.25kg
η
=
39, 1/β=10I2×2
Model 3
m
2
=
7
kg, l2=0.20kg
η
=
72, 1/β=15I2×2Figure 10
Tracking performance ofq1 and q2 with WMNNAC.Figure 11
Tracking error under the WMNNAC.Figure 12
The weights for the local controllers of the WMNNAC system.Figure 13
Global control signals for the manipulator with jumping parameters.Figure 14
Transient performance of (a) single NN-based controller and (b) WMNNAC.
(a)
(b)


## 5. Conclusion

In this paper, we developed a WMNNAC method for robotic manipulators with jumping parameters. The local NN-based controller has been clearly deduced and has good performance in compensating for the influence of uncertainties and external disturbance. By combining multiple NN-based controllers and the WMMAC method based on the improved weighting algorithm, the robotic manipulators can track the desired trajectory well when parameters jump largely. Moreover, global stability analysis and proof have been given based on VES theory. Finally, the simulation results have demonstrated the efficiency of the developed control approach. In the future, research will focus on methods for establishing the model set and the practical application.



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*Source: 3172431-2020-01-22.xml*