# Adaptive Fuzzy Controller Design for Uncertain Robotic Manipulators Subject to Nonlinear Dead Zone Inputs

Authors: Hua Zhang
Journal: Computational Intelligence and Neuroscience
 (2022)

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## Abstract

This paper comes up with the adaptive fuzzy scheme for multi-degree robotic manipulators with dead zone. Fuzzy logic system is employed to approximate unknown nonlinear functions and input nonlinear functions with dead zones that are introduced. The adaptive fuzzy technique is utilized to deal with the problems of multi-joint robotic manipulators with unknown nonlinear functions in dead zones. Based on Lyapunov criterion, all the states and signals involved in the system are maintained in a certain bounded region and the tracking error converges to a small domain of origin. Finally, a simulation example of robotic manipulators with two joints is structured to confirm the practicability of the raised scheme.


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## Body

## 1. Introduction

In recent decades, the robotic manipulators are not only the most widely used automated mechanical device in the field of robotics technology but also have an extensive range of applications in daily life. For example, in the medical field, robotic manipulators can easily adapt to the existing operating environments to offer better control for surgeons and ultimately improve surgical outcomes. Besides, the emergence of robotic manipulators can take the place of heavy labor to achieve mechanization and automation of production and operate in harmful environments to ensure personal safety. With the exception of the abovementioned, it can also be applied in industrial manufacturing, medical care, entertainment services, military, semiconductor manufacturing, and so on. Therefore, due to its extensive application and practical value, the research of robotic manipulators is more meaningful.Since the robotic manipulator is a relatively sophisticated multiple-input-multiple-output (MIMO) nonlinear kinematics system with time-varying, highly coupling, and nonlinear dynamic characteristics, the control of robotic manipulators has been studied extensively and a number of methods have been proposed. In [1], for a redundant manipulator, an adaptive PID fast terminal sliding mode was proposed. A sliding mode and a state observer-based controller were designed to make flexible robotic manipulators perform remote rehabilitation missions in [2]. There are also some other approaches dealing with the robotic manipulators such as sliding mode control [1–6], robust control [3, 7], adaptive fuzzy [8–12], PID controller [13–15]. However, there is a prerequisite for the literature discussed above whose models are all known. To the best of our knowledge, the model equation of the robotic manipulators is uncertain and difficult to establish accurately due to some unavoidable uncertainties such as parameter perturbation, external disturbances, and unmodeled dynamics. Compared with other existing processing methods, adaptive fuzzy control has been widely studied due to its good approximation effect for dealing with a class of unknown and uncertain systems. In [16], the adaptive fuzzy controller was designed to settle multilateral cooperative teleoperation of multiple robotic manipulators under random network-induced delays. A decentralized adaptive fuzzy scheme was proposed for robotic manipulators combining the genetic algorithm and the gradient method in [17]. The adaptive fuzzy control was raised for a robotic manipulator with asymptotic tracking performance in [18]. However, it is noteworthy that the discussion of the aforementioned literature [17, 18] is not involved in the dead zone.In fact, dead zone is ubiquitous in many industrial systems which affect system operation, especially in the robotic manipulator’s system. In [19], a robot manipulator with the dead zone was solved by using the adaptive neural network tracking control method. Backstepping funnel control was used to handle the prescribed performance of robotic manipulators with unknown dead zones in [20]. For flexible manipulators with input dead zones, adaptive neural command filter tracking control was proposed in [21]. However, the models studied in [19–21] were unknown and uncertain, which makes their research methods more consistent with the actual system.Motivated by the above discussion, the proposed adaptive fuzzy controller for multi-degree robotic manipulators with dead zones will be an interesting and challenging topic for us. In comparison with the available literature, the main contributions of this work are listed as follows:(1)
The proposed adaptive fuzzy controller design can be utilized to approximate a large class of uncertain robotic manipulator systems and unknown nonlinear functions under actual conditions. In addition, even though the system model is fully unknown, our controller still works effectively.(2)
The uncertain multi-joint robotic manipulator system with dead-zone considered is addressed by the adaptive fuzzy control approach, and the proposed method is valid for other high-order nonlinear systems with dead-zone inputs.In the end, this paper is settled as below. Section2 brings in fuzzy logic system. The models of multi-joint robotic manipulators and input nonlinearity are raised in Section 3. Section 4 provides a fuzzy controller in the light of the universal approximation criterion for the sake of dealing with robotic manipulators. In Section 5, a simulation case of robotic manipulators with two joints is offered to confirm the feasibility of the proposed method. Besides, conclusions are given to the end.


## 2. Description of the Fuzzy Logic System

Here, we make a short presentation of the fuzzy logic system (FLS). As is known, the FLS is a formidable nonlinear function approximation containing fuzzy IF-THEN rules which are listed as [8].RK: if x1 is P1K and … and xn is PnK, then f^ is fK, K=1, 2,…, N,where N is fuzzy rules, P1K, P2K, … and PnK are a collection of fuzzy, fK is the output of the fuzzy singleton in the Kth rule. In accordance with applying the fuzzifier, product inference, and defuzzifier, the output of the FLS is constructed as [22, 23](1)f^x¯=∑i=1Nfi∏j=1nμPjixj∑i=1N∏j=1nμPjixj=ΘTRx¯,where μPJixj represents the membership grade which xj is part of Pji, ΘT=f1,f2,…,fN lies on the following parameters, and RT=R1R2…RN, where(2)Rix¯=∏j=1nμPjixj∑i=1N∏j=1nμPjixj.Once that fuzzy basis function has been determined, there always exists more than one rule i.e.∑i=1n∏j=1nμPjixj>0 [24].


## 3. Dynamic Model of Robotic Manipulators

In this paper, we will consider a more general multi-degree robotic manipulators whose control inputs contain dead zone. The model can be written as(3)Mqq¨+Cq,q˙q˙+Dq˙+gq=Ψu,where q=q1,…,qnT∈Rn represents joints position, u∈Rn is a control signal with dead zones, Mq is a inertia matrix, Cq,q˙q˙ is centrifugal pull, and Dq˙ and gq are resistance and power of gravity, respectively. Mq with respect to t total derivative is M˙, D is a semi-definite symmetric matrix [8, 25].Lety=q, x=q˙, (2) can be converted into(4)y¨=M−1yΨu−Cy,x−D−gy.In the light of (4), the n joints robotic manipulators with dead zones can be written as the following compact formula:(5)y¨=Fx+GxΨu,where Fx=−M−1yD+gy+Cy,x, and Gx=M−1y.There exists some necessary assumptions that need to be given.Assumption 1.
Presuming bounded referable signal isyd=yd1,…,ydnT and x=x1,…,xnT is known and measurable. Besides, yd is a known and continuous vector function in a compact set Ωyd.Assumption 2.
The functionG is a positive definite matrix, namely, G⩾σ0In×n, with σ0≥0.
In accordance with the dynamic model of robotic manipulators system (5), the fuzzy adaptive control approach is designed as follows.
To begin with, the tracking errors are defined by(6)ei=yi−ydi,i=1,2,…,n,and the filter tracking error is given as(7)S=S1,…,SnT,with(8)Si=λi,1eie˙i.
In order to express more clearly, we rewrite (8) into the form of a matrix. Hence, the vector S can be expressed as(9)S=CTE,where(10)CT=λ1100⋯0000λ21⋯00⋮⋮⋮⋮⋮⋮0000…λn1n×2n,(11)E=e1,e˙1,…,en,e˙n2n×1T.
To proceed given the dynamic ofSi as the form of a matrix, we can also gain(12)S˙i=0,λieie˙i+e¨i.
Hence, the dynamic ofS can be given as(13)S˙=CrTE+e¨,where(14)CrT=0λ100⋯00000λ2⋯00⋮⋮⋮⋮⋮⋮0000…0λnn×2n,with(15)e¨=y¨−y¨d.
Then, substituting (15) into (13) yields(16)S˙=CrTE+y¨−y¨d.


### 3.1. Input Nonlinearity

Noting that in the robotic manipulators (3), the dead zone input exists. In this work, this input nonlinearity Ψiui can be expressed as [25](17)Ψiui=Qi+uiui−uiT,0,Qi−uiui+uit,ui>uiT,−uit⩽ui⩽uiT,ui<−uit.To facilitate the transformation of the dead zone input, here, we suppose that it has some particular properties, i.e., there are two positive constantsMi+∗ and Mi−∗ such that the below conditions can be constructed as [16](18)ui−uiTΨiui⩾Mi+∗ui−uiT2,ui>uiT,ui+uitΨiui⩾Mi−∗ui+uit2,ui<−uit,with βi=minMi+∗,Mi−∗.


## 3.1. Input Nonlinearity

Noting that in the robotic manipulators (3), the dead zone input exists. In this work, this input nonlinearity Ψiui can be expressed as [25](17)Ψiui=Qi+uiui−uiT,0,Qi−uiui+uit,ui>uiT,−uit⩽ui⩽uiT,ui<−uit.To facilitate the transformation of the dead zone input, here, we suppose that it has some particular properties, i.e., there are two positive constantsMi+∗ and Mi−∗ such that the below conditions can be constructed as [16](18)ui−uiTΨiui⩾Mi+∗ui−uiT2,ui>uiT,ui+uitΨiui⩾Mi−∗ui+uit2,ui<−uit,with βi=minMi+∗,Mi−∗.


## 4. The Construction of AFC

In what follows, how to construct a reasonable controller to control the robotic manipulator system (5) is given as follows.First of all, by substituting (5) into (16), we can obtain(19)S˙=CrTE+Fx+GΨu−yd..Now posingG^1=G−1, we have(20)G^1S˙=G1^CrTE−yd.+Fx+Ψu.In order to design the optimal controller, (20) is conducted as(21)G^1S˙=G^1CrTE−yd.+Fx+Ψu=κx,m+Ψu,where κx,m=G^1m+Fx, and m=CrTE−yd..Assumption 3.
It is supposed thatκix,m is a bound continuous function satisfying κix,m⩽βκ¯ix, with β=minβi.Remark 1.
The establishment of Assumption3 is not limited mainly due to the upper limit βκ¯ix is unknown, in addition, κix,m is a continuous function in the interval Ωx such that βκ¯ix always exists [25].
Here, in light of description of the FLS,κ¯ix can be depicted as(22)κ¯^ix,Θ=ΘiTRix,where Rix is the FBF which is given by the decision maker and Θi can be tuned based on requirements of the FLS.
We define(23)Θi∗=argminΘisupx∈Ωxκ¯ix−κ¯^ix,Θi,as the optimal estimate values of Θi. It is worth mentioning that artificial constant Θi∗ is introduced only for analysis purposes in the whole process.
Let the estimate error be(24)Θ˜i=Θi−Θi∗,and the fuzzy approximation error is given as(25)εix=κ¯ix−κ¯^ix,Θi∗,where κ¯^ix,Θi∗=Θi∗TRix.
It must be emphasized that the input function of FLS must be contained in the scope ofΩx, otherwise the system cannot operate normally. Assume that εix∈Ωx is bounded, that is to say, εix⩽ε¯i, ε¯i∈R, one can obtain(26)κ¯^ix,Θi−κ¯ix=κ¯^ix,Θi−κ¯^ix,Θi∗+κ¯^ix,Θi∗−κ¯ix=κ¯^ix,Θi−κ¯^ix,Θi∗−εix=Θ˜iTRix−εix.
In what follows, let us consider a suitable adaptive fuzzy controller(27)ui=−QitsignSi−uit,0,−QitsignSi+uiT,Si>0,Si=0,Si<0,with Qit=z0i+z1iSi+ΘiTRix, and(28)z˙0i=−ζ0iσ0iz0i+ζ0iSi,z0i0⩾0,Θ˙i=−ζ1iσ1iΘi+ζ1iSiRix,Θij⩾0,where ζ0i, ζ1i, σ0i, σ1i, z1i>0 are marked up to decision makers, z0i and Θi are estimated values of z0i∗=ε¯i and Θi∗, respectively.Remark 2.
Withz0i0⩾0 and Θi0⩾0, it is known from that solutions of adaptive laws (28) hold z0it⩾0 and Θit⩾0, for t>0.
Based on Assumption3 and multiplying both sides of equation (21) by 1/βST, one can obtain(29)1βSTG^1xS˙=1βSTκx,m+1βSTu,⩽∑i=1nSiκ¯ix+1βSTu.
From (26) and (29), one gets(30)1βSTG1^xS˙⩽∑i=1nSiκ¯ix+1βSTu⩽−∑i=1nSiz˜0i−∑i=1nSiΘ˜iTRix+∑i=1nSiz0i+∑i=1nSiΘiTRix+1βSTu,with Θ˜i=Θi−Θi∗ and z˜0i=z0i−z0i∗=z0i−ε¯i.Theorem 1.
When Assumptions1–3 are satisfied, for the robotic manipulator system (5) with control law (27) and adaptive laws (28), all the states are remained as a bounded domain and the error function of the closed-loop system tends to be within a small variable range of origin via using adaptive fuzzy approach.Proof.
We define(31)V=12βSTG^1S+12∑i=1n1ζ0iz˜0i2+12∑i=1n1γ1iΘ˜iTΘ˜i,(32)V˙=1βSTG^1S˙+∑i=1n1ζ0iz˜0iz˙0i+∑i=1n1γ1iΘ˜iTΘ˙i,with G1^.=0. Thus, from (28) and (41), we can obtain the following important inequality. When Si>0,(33)ui+uitΨiui=−QitsignSiΨiui⩾Mi−∗Qi2t⩾βQi2t,and when Si<0,(34)ui−uiTΨiui=−QitsignSiΨiui⩾Mi+∗Qi2t⩾βQi2t.
Hence, we conclude that for allSi holding(35)−QitsignSiΨiui⩾βQ2t.
Owing toSi2>0 and SisignSi=Si, (35) becomes(36)−QitSi2signSiΨiui⩾βQi2tSi2=βQi2tSi2.
For allSi, we have(37)SiΨiui≤−βQitSi.
Then, substituting (28), (30), and (37) into (32), we obtain(38)V˙⩽∑i=1nSiz0i+∑i=1nSiΘiTRix+1β∑i=1nSiΨiui−∑i=1nσ0iz˜0iz0i−∑i=1nσ1iΘ˜iTΘi⩽∑i=1nSiz0i+∑i=1nSiΘiTψix+∑i=1n−zitSi−∑i=1nσ0iz˜0iz0i−∑i=1nσ1iΘ˜iTΘi=−∑i=1nz1iSi2−∑i=1nσ0iz˜0iz0i−∑i=1nσ1iΘ˜iTΘi.
Obviously, we have(39)−σ0iz˜0iz0i⩽−σ0i2z˜0i2+σ0i2z0i∗2,−σ1iΘ˜iTΘi⩽−σ1i2Θ˜i‖2+σ1i2Θi∗‖2.
Then, (38) becomes(40)V˙⩽−∑i=1nz1iSi2−∑i=1nσ0i2z˜0i2−∑i=1n−∑i=1nσ1i2Θ˜i2+∑i=1nσ0i2z0i∗2+∑i=1nσ1i2Θi∗2
SinceG^1⩾σg0In, then we have(41)STG−1S=STG^1S⩽1σg0S2.
From (40) and (41), one has(42)V˙⩽−μV+τ,where(43)τ=∑i=1nσ0i2z0i∗2+∑i=1nσ1i2Θi∗2,ω=minmini2ησg0z1i,miniζ0iσ0i,miniζ1iσ1i.
Multiplying (42) by eωt yields(44)ddtVeωt⩽τeωt.
Then, integrating over the interval [0, t] of the (44), we can get(45)0⩽Vt⩽τω+V0−τωe−ωt.
By utilizing (31), V0 is built as(46)V0=12βS0TG1^S0+12∑i=1n1ζ0iz0i0−z0i∗2+12∑i=1n1ζ1iΘi0−Θi∗TΘi0−Θi∗,where G1^⩾σg1In, σg1>0.
In the light of (31) and (45), one can achieve(47)Si⩽2βσg1τω+V0−τωe−ωt1/2.
From the above inequality, it can be concluded that the exponential solution ofSi gradually tends to be bounded, where ΩSi=SiSi⩽2η/σg1τ/ω1/2, and the proof of the theorem is finished.Remark 3.
It is worth mentioning that there is a special exampleΨiui=ui, that is, when the system model does not contain nonlinear input functions with dead zones, the controller we structured is still applicable to system (5), and the tracking performance are shown in Figures 1 and 2.Figure 1
The tracking performance of two-link robotic manipulators systems without dead zones (withα=1). (a) Tracking of y1 and sint. (b) Tracking of y2 and sint.
(a)(b)Figure 2
Simulation results (withα=1). (a) The controller u1. (b) The controller u2.
(a)(b)Remark 4.
(1)
IfuiT=uit=uii are established, (27) can be transformed into(48)ui=−uii+QitsignSi,whereQit=z0i+z1iSi+ΘTRix.(2)
It is worth highlighting that the approximation effect of the robotic manipulator is significantly better when the functionsign. is substituted for a smooth function arcsin., arctan., and so on.


## 5. Simulation Results

With the aim of checking the control efficiency of the proposed scheme, let us consider robotic manipulators with two single joints without disturbance.(49)q¨1q¨2=M11M12M21M22−1Ψ1u1Ψ2u2−C11C12C21C22q˙1q˙2−g11g21,where M11=b1+4b4cosq2, M12=b2+b4cosq2, M21=b1+b4cosq2, M22=b3, C11=−b4q˙2sinq2, C12=−b4sinq2q˙1+q˙2, C21=b4q˙1sinq2, C22=0, g11=c1cosq1+c2cosq1+q2, g21=c2cosq1+q2 and the parameters are designed below without disturbance: b1=200.03, b2=23.7, b3=122.8, b4=25.3, c1=784.9, c2=245.35.Let us denotey=y1,y2T=q1,q2T, x=x1,x2,x3,x4T=q1,q2,q˙1,q˙2T, and Ψu=Ψ1u1,Ψ2u2, thus, (49) is depicted as(50)y¨=Fx+GxΨu.The input nonlinearitiesΨ1u1 and Ψ2u2 are arranged as(51)Ψ1u1=u1−21−0.3sinu1,u1>2,0,−2⩽u1⩽2,u1+20.8−0.3cosu1,u1<−2,and Ψ2u2 is supposed to be(52)Ψ2u2=u2−51−0.3sinu2,u2>5,0,−5≤u2≤5,u2+50.8−0.3cosu2,u2<−5.In the simulation process, a fuzzy logic system is used, which contains four identical Gaussian functions. Each Gaussian function is distributed in the interval−10,10 and their standard deviation is selected as 0.9. The input vector of the fuzzy system ΘTRx¯ is x=x1,x2,x3,x4T. Here, we set the step size as 5 so that there exists 5×5×5×5=625 rules in the FLS. The parameters in the simulation process are designed as ζ01=31, ζ02=79, ζ11=ζ12=4000, σ01=σ02=0.001, σ11=σ12=0.0005, λ1=λ2=2, and z11=z12=2. Simulation initial parameters of adaptive laws and control law are chosen as z01=z02=0, Θ1j=Θ2j, j=1,…,625, and u1T=u2T=2, u1t=u2t=5. Original values of the robotic manipulator system are designed as x0=1,2,3,4T and y0=1,1T. Here, one can see that signSi is a piecewise discontinuous function that can take the place of tanhzsiSi with zsi=20, i=1, 2. It is noteworthy that our target is to make the robotic manipulator system joints q1 and q2 to track the desired given trajectories yd1=yd2=sint.Finally, it is apparent that presents the good tracking performances of the two-link robotic manipulators with dead zones in Figure3. Figure 4 shows transient behaviors of the tracking error, sliding mode, controllers, and nonlinear functions with dead zones. In terms of Remark 4, it is noticeable that the tracking effect in Figure 1 is the same as Figure 3 and the range of variation of controllers with different parameter values are shown in Figure 2.Figure 3
The tracking performance of two-link robotic manipulators systems with dead zones (withα=1). (a) Tracking of y1 and sint. (b) Tracking of y2 and sint.
(a)(b)Figure 4
Simulation results (withα=1). (a) The tracking error e1 and e2. (b) The sliding mode S1 and S2. (c) The controller u1 and u2. (d) Nonlinear functions Ψ1u1 and Ψ2u2.
(a)(b)(c)(d)


## 6. Conclusion

This paper adopts the adaptive fuzzy control approach for uncertain multi-degree robotic manipulators with a dead zone. The fuzzy logic system has been presented to dispose of unknown and uncertain nonlinear functions of the robotic manipulator. It has been proven in accordance with Lyapunov criterion that the proposed adaptive fuzzy control scheme can guarantee that the states and signals of the whole closed-loop system tend toward bounded and the tracking errors are close to small regions of origin gradually. The practicability of the proposed approach is affirmed by an example of robotic manipulators with two joint. The deficiency of this paper is that the tracking error observed from the image is relatively large, how to reduce the tracking error will be our future research direction.



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