# Fuzzy Adaptive Variable Structure Control of Second-Order Robotic Manipulators with Dead Zones

Authors: Songyu Wang; Xianmin Hou
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this study, with respect to certain second-order robotic systems with dead zones, a fuzzy adaptive variable structure controller (VSC) is implemented. Some suitable adaptive fuzzy systems are used to estimate uncertain functions. Based on Lyapunov stability theorems, parameter adaptive laws are designed, and it is proven that all signals involved will remain bounded and the stability of the controlled system is also guaranteed. Our controller is effective for the system with or without sector nonlinearity. Finally, a simulation example is presented to illustrate the correctness of the theoretical derivation.


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## Body

## 1. Introduction

In the real world, almost all systems are multivariate in essence, and the multivariate control theory system can be immediately applied to various problems (aerospace technology, electric motors, and robot technology) [1–4]. Because of the coupling between control inputs and outputs, how to design an effective controller for multivariable systems is a challenging work. When multivariable systems are subject to system uncertainties, the control problems become more complicated [5, 6]. In this field, theoretical results and constructive procedures for designing satisfactory controllers are constantly evolving, and only a few works have been reported up to now.In recent years, fuzzy logic system (FLS) has been shown to be one of the powerful tools in functional approximately. Fuzzy control mainly utilizes the basic ideas of fuzzy mathematics and theoretical control methods. With respect to a traditional control method, the accuracy of the dynamic mode of the control system is the most important factor affecting the control performance. The more detailed the system dynamic information is, the more precise the control can be achieved [7–13]. But, with respect to a complicated system, because there are a lot of system variables, it is very hard to accurately model the system, so scholars utilize kinds of methods to describe dynamics to achieve certain purposes. That is, the traditional control theory has a strong control ability for known systems, but it is powerless for systems that are too complicated or difficult to accurately describe. So scholars try to deal with these control problems via FLS. In [9], an adaptive fuzzy control (AFC) method was proposed for a chaotic system with unknown control direction. In [11], an AFC backstepping method was proposed, where it is shown that the FLS has a very powerful approximation ability. An adaptive fuzzy observer was proposed in [13]. For more related works about the FLC, one can refer to [4, 14–19] and the references therein.Since the 1980s, the control of robots has become a very meaningful work with a wide range of practical applications and has been increasingly studied by experts and scholars in various fields. The robot model is highly coupled and complex. In the early research on robot control, researchers often need to know the specific model of the robot. For some robots that are easy to model, many control methods have been proposed, such as active control, inversion based control, and backstepping control [20–23]. However, in real life, robot systems often contain many uncertainties, such as modeling estimation errors, system external disturbances, and parameter measurement errors. If the unknowns of these systems cannot be handled properly, it will affect the stability of the system and even make the robot system uncontrollable. Consequently, FLSs are used to tackle the uncertainties in the robotic systems. For example, in [8, 24, 25], FLSs were used to model the robotic manipulators, and the model errors are satisfactory. More results can be referred to [24, 26–29]. However, the stability of the complicated robotic systems is hard to be theoretically discussed in the above literature.There are two main reasons that derive the investigation of this work. First, although many control strategies have been proposed for robotic manipulators, the dead-zone input has been rarely considered. Secondly, in the traditional VSC method, the chattering phenomenon does exist, and how to solve this problem is a challenging work. Inspired by the above discussion, in this paper, an adaptive fuzzy VSC method is proposed for ann-link robotic manipulator with dead zone. The main contributions of this work are listed as follows: (1) to handle the dead zone in the robotic manipulator, an adaptive fuzzy VSC method is proposed, and the VSC can estimate the dead-zone model, and (2) the stability of the controlled robotic manipulator is proven strictly based on Lyapunov stability criteria.


## 2. Preliminaries and Problem Description

Consider the followingn-link robotic manipulator:(1)Mξξ¨+Cξ,ξ˙ξ˙+Gξ=Φu+at.From system (1), we can get the following formula:(2)ξ¨=−M−1ξCξ,ξ˙ξ˙−M−1ξGξ+M−1ξΦu+M−1ξat.So, the robot system (2) is rewritten as(3)y¨=Fx+GxΦu+Dt,where(4)Fx=−M−1ξCq,ξ˙ξ˙−M−1ξGξ,Gx=M−1ξ,Dt=M−1ξat.F.,Φ.,D.∈Rn,G.∈Rn×n,y=y1,y2,…,ynT=ξ1,ξ2,…,ξnT∈Rn is the output, x=y1,y˙1,…,yn,y˙nT=ξ1,ξ˙1,…,ξn,ξ˙nT∈R2n is the measurable state, u=u1,u2,…,unT∈Rn denotes the controller to be designed, fix is unknown, gij,i,j=1,2,…,n is an unknown constant control gain, Φu=Φ1u1,Φ2u2,…,ΦnunT means dead zone, and dit is an external disturbance.Let us denote(5)y¨=y¨1,y¨2,…,y¨nT,Fx=f1x,f2x,…,fnxT,Dt=d1t,…,dntT.Now, we need the following assumptions.Assumption 1.
We assume that the desired trajectory vectorxd=yd1,y˙d1,…,ydn,y˙dn,y¨d1,…,y¨dnT is continuous and bounded and can be obtained from the measurement. So xd∈Ωxd⊂R2n+n, where Ωxd is an open compact set with a finite radius.Assumption 2.
(i)
Without loss of generality, one assumes thatGx is symmetric and positive definite. Therefore, there is an unknown positive constant δg0, such that Gx≥δg0En, where En is the identity matrix.(ii)
We assume that this internal disturbancedit is bounded and satisfies the following property: dit≤di¯, ∀i=1,2,…,n, where di¯ is an unknown positive constant.Remark 1.
It should be noted that, for a robotic system, Assumption2 is reasonable, and this assumption guarantees the controllability of system (3).
The main purpose is to find a way to implementu so that the output vector y can better follow the desired tracking vector yd=yd1,…,ydnT∈Rn, and we assume that the signals involved are bounded.
Define(6)e=y−yd,S=γe+e˙,where γ>0. Here, S is a sliding surface variable. Then, we have(7)S˙=γe˙+e¨=γe˙+y¨−y¨d.
Thereafter, (7) will be used in the controller design.


### 2.1. Description of the Fuzzy Logic System

The basic mechanism of a fuzzy logic system includes a fuzzy inference engine and a fuzzifier, some fuzzy IF-THEN rules, and a defuzzifier.Thei-th fuzzy rule is in the following form: Ri: if x1 is B1i and … and xn is Bni, then f^isfi, where B1i,B2i,…,Bni are fuzzy sets and fi denotes the singleton for the output in the i-th rule. The FLS can be rewritten as follows:(8)f^x¯=∑i=1lfi∏j=1nνBjixj∑i=1l∏j=1nνBjixj=θTϱx¯,where νBjixj is the degree of membership, θT=f1,…,fl can be adjusted online, and ϱT=ϱ1,…,ϱl, with(9)ϱix¯=∏j=1nνBjixj∑i=1l∏j=1nνBjixj.Throughout the paper, we always assume that∑i=1l∏j=1nνBjixj>0.


### 2.2. Input Nonlinearity

Suppose thatΦiui satisfies(10)Φiui=ϕi+uiui−ui+,ui>ui+,0,−ui−≤ui≤ui+,ϕi−uiui+ui−,ui<−ui−,where ϕi+ui>0 and ϕi−ui>0 are functions with respect to ui and ui+>0,ui−>0.Suppose thatΦiui satisfies(11)ui−ui+Φiui≥li+∗ui−ui+2,ui>ui+,ui+ui−Φiui≥li−∗ui+ui−2,ui<−ui−,where li+∗ and li−∗ are constants called “gain reduction tolerances”. We can write ζi=minli+∗,li−∗.In order to facilitate the study of the characteristics of this input nonlinear function in the control problems, we give the following assumption.Assumption 3.
Suppose thatli+∗ and li−∗ are unknown, that is, ζi is also unknown. The explicit mathematical expression of the nonlinear function Φiui is uncertain, but this property (11) and constants ui+ and ui− are known.Remark 2.
One knows from (10) and (11) that, for ui+=ui−=0, the input Φiui is reduced to the special sector nonlinear function. We can make the following restriction: the gain reduction tolerances li+∗ and li−∗ and their minimum ζi=minli+∗,li−∗ are known.


## 2.1. Description of the Fuzzy Logic System

The basic mechanism of a fuzzy logic system includes a fuzzy inference engine and a fuzzifier, some fuzzy IF-THEN rules, and a defuzzifier.Thei-th fuzzy rule is in the following form: Ri: if x1 is B1i and … and xn is Bni, then f^isfi, where B1i,B2i,…,Bni are fuzzy sets and fi denotes the singleton for the output in the i-th rule. The FLS can be rewritten as follows:(8)f^x¯=∑i=1lfi∏j=1nνBjixj∑i=1l∏j=1nνBjixj=θTϱx¯,where νBjixj is the degree of membership, θT=f1,…,fl can be adjusted online, and ϱT=ϱ1,…,ϱl, with(9)ϱix¯=∏j=1nνBjixj∑i=1l∏j=1nνBjixj.Throughout the paper, we always assume that∑i=1l∏j=1nνBjixj>0.


## 2.2. Input Nonlinearity

Suppose thatΦiui satisfies(10)Φiui=ϕi+uiui−ui+,ui>ui+,0,−ui−≤ui≤ui+,ϕi−uiui+ui−,ui<−ui−,where ϕi+ui>0 and ϕi−ui>0 are functions with respect to ui and ui+>0,ui−>0.Suppose thatΦiui satisfies(11)ui−ui+Φiui≥li+∗ui−ui+2,ui>ui+,ui+ui−Φiui≥li−∗ui+ui−2,ui<−ui−,where li+∗ and li−∗ are constants called “gain reduction tolerances”. We can write ζi=minli+∗,li−∗.In order to facilitate the study of the characteristics of this input nonlinear function in the control problems, we give the following assumption.Assumption 3.
Suppose thatli+∗ and li−∗ are unknown, that is, ζi is also unknown. The explicit mathematical expression of the nonlinear function Φiui is uncertain, but this property (11) and constants ui+ and ui− are known.Remark 2.
One knows from (10) and (11) that, for ui+=ui−=0, the input Φiui is reduced to the special sector nonlinear function. We can make the following restriction: the gain reduction tolerances li+∗ and li−∗ and their minimum ζi=minli+∗,li−∗ are known.


## 3. Main Results

In this article, a fuzzy adaptive VSC scheme is proposed for a class of two-link robotic manipulator systems described in (3).By bringing (3) into (7), the dynamic equation of S can become(12)S˙=γe˙+Fx+GxΦu+Dt−y¨d.PosingG1x=G−1x and Dht=G−1xDt, we have(13)G−1xS˙=γG−1xe˙+G−1xFx+Φu+G−1xDt−G−1xy¨d,Then,S can be expressed by(14)G1xS˙=G1xγe˙+Fx−y¨d+Φu+Dht=βx,ω+Φu+Dht,where βx,ω=β1x,ω,…,βnx,ωT=G1xω+Fx and ω=γe˙−y¨d.Assumption 4.
One can find a functionβ¯ix such that(15)βix,ω≤ζβ¯ix,∀x∈Ωx⊂R2n,where ζ=miniζi.Remark 3.
Assumption4 is reasonable because(i)
ζ=miniζiaζβ¯ix is unknown, where is ζ is a positive scalar(ii)
Nothing thatω is a function of x,xd, xd∈L∞, and βix,ω is a continuous function; therefore, such a function β¯ix always exists
Multiplying (14) by ST and using Assumption 4, we have(16)1ζSTG1xS˙=1ζSTβx,ω+1ζSTΦu+1ζSTDht=1ζ∑i=1nSiβix,ω+1ζSTΦu+1ζSTDht≤1ζ∑i=1nSiβix,ω+1ζSTΦu+1ζSTDht≤∑i=1nSiβi¯x+1ζSTΦu+1ζSTDht.
On a compact setΩx, the unknown continuous nonlinear function β¯ix can be approximated by the fuzzy system (8) as follows:(17)βi¯^x,θi=θiTϱix.
Next, we can define the optimal parameter as(18)θi∗=argminθisupx∈Ωxβi¯x−βi¯^x,θi.
Define(19)θi˜=θi−θi∗,as the parameter estimation error, with(20)εix=βi¯x−βi¯^x,θi∗,being the approximation error, with βi¯^x,θi∗=θiT∗ϱix.
As in the literature, the fuzzy systems used on the compact setΩx do not violate this universal approximation property. So, we can assume that(21)εix≤εi¯,∀x∈Ωx,with εi¯ being unknown.
Then, one has(22)βi¯^x,θi−βi¯x=βi¯^x,θi−βi¯^x,θi∗+βi¯^x,θi∗−βi¯x=βi¯^x,θi−βi¯^x,θi∗−εix=θiTϱix−θi∗Tϱix−εix=θi˜Tϱix−εix.
From (16) and (22), we can get(23)1ζSTG1xS˙≤∑i=1nSiβi¯x+1ζSTΦu+1ζSTDht=∑i=1nSiθiTϱix−θ˜iTϱix+εix+1ζSTΦu+1ζSTDht≤∑i=1nSiθiTϱix−∑i=1nSiθ˜iTϱix+∑i=1nSiεix+1ζSTΦu+1ζSTDht≤∑i=1nSiθiTϱix−∑i=1nSiθ˜iTϱix+∑i=1nSiεi¯+1ζSTΦu+∑i=1nSidhi¯≤∑i=1nSiθiTϱix−∑i=1nSiθ˜iTϱix+1ζSTΦu−∑i=1nSiκ0i˜+∑i=1nSiκ0i,where θi˜=θi−θi∗ and κ0i˜=κ0i−κ0i∗=κ0i−εi¯−dhi¯. Note that dhi¯ are unknown positive constants which satisfy ∑i=1ndhi¯Si≥STDht/ζ.
In order to meet the control goal, we propose a suitable fuzzy adaptive VSC as follows:(24)ui=−ϖitsignSi−ui−,Si>0,0,Si=0,−ϖitsignSi+ui+,Si<0,with(25)ϖit=κ0i+κ1iSi+θiTϱix,∀i=1,2,…,n,(26)κ˙0i=−χ0iδ0ik0i+χ0iSi,κ0i0≥0,(27)θ˙i=−χ1iδ1iθi+χ1iSiϱix,θij0≥0,where χ0i,χ1i,δ0i,δ1i,κ1i>0, and κ0i and θi are the online estimates of the uncertain terms κ0i∗=εi¯+dhi¯ and θi∗, respectively.
From (11) and (24), we can easily get that ui<−ui− for Si>0,(28)ui+ui−Φiui=−ϖitsignSiΦiui≥li−∗ui+ui−2=li−∗ϖi2t≥ζϖi2t,and ui>ui+ for Si<0,(29)ui−ui+Φiui=−ϖitsignSiΦiui≥li+∗ui−ui+2=li+∗ϖi2t≥ζϖi2t.
Then, forSi>0 and for Si<0, we have(30)−ϖitsignSiΦiui≥ζϖi2t.
Next, we can useSi2≥0 and SisignSi=Si, and we get(31)−ϖitSi2signSiΦiui≥ζϖi2tSi2=ζϖi2tSi2.
Finally, for allSi (i.e., for Si>0, Si=0, and Si<0), we get(32)SiΦiui≤−ζϖitSi.Theorem 1.
For the robotic manipulator (3) with Assumptions 1–4, the proposed adaptive fuzzy VSC (24)–(27) guarantee that the tracking error converges to an adjustable region.Proof.
Let(33)V=12ζSTG1xS+12∑i=1n1χ0iκ0i2˜+12∑i=1n1χ1iθiT˜θi˜.
Its time derivative is given by(34)V˙=12ζSTG1xS+∑i=1n1χ0iκ0i˜κ˙0i+∑i=1n1χ1iθiT˜θ˙i,with G˙1x=0.
Using expressions (16), (26), (27), and (32), (34) can become(35)V˙≤∑i=1nSiκ0i+∑i=1nSiθiTϱix+1ζ∑i=1nSiΦiui−∑i=1nδ0iκ0i˜κ0i−∑i=1nδ1iθiT˜θi≤∑i=1nSiκ0i+∑i=1nSiθiTϱix+∑i=1n−ϖitSi−∑i=1nδ0iκ0i˜κ0i−∑i=1nδ1iθiT˜θi=−∑i=1nκ1iSi2−∑i=1nδ0iκ0i˜κ0i−∑i=1nδ1iθiT˜θi.
We can easily verify that(36)−δ0iκ0i˜κ0i≤−δ0i2κ0i2˜+δ0i2κ0i∗2,−δ1iθiT˜θi≤−δ1i2θi˜2+δ1i2θi∗2.
Using the previous inequalities, (35) becomes(37)V˙≤−∑i=1nκ1iSi2−∑i=1nδ0i2κ˜0i2+∑i=1nδ0i2κ0i∗2−∑i=1nδ1i2θi˜2+∑i=1nδ1i2θi∗2.
SinceGx≥δg0En (recall that the matrix Gx is positive definite and symmetric), then(38)STG−1xS=STG1xS≤1δg0S2.
From (37) and (38), we have(39)V˙≤−ωV+τ,where(40)τ=∑i=1nδ0i2κ0i∗2+∑i=1nδ1i2θi∗2,ω=minmini2ζδg0κ1i,miniχ0iδ0i,miniχ1iδ1i.
Multiplying (39) by eωt yields(41)ddtVeωt≤τeωt.
Integrating (41) over 0,t, we have(42)0≤Vt≤πω+V0−τωe−ωt.
So, all signalsκ0i,θi,Si, and x are bounded.
From (33), V0 can be written as follows:(43)V0=12ζS0TG1xS0+12∑i=1n1χ0iκ0i0−κ0i∗2+12∑i=1n1χ1iθi0−θi∗Tθi0−θi∗.
SinceG1x is symmetric and positive definite (i.e., it exists an unknown positive constant δg1, such that G1x≥δg1En), from (33) and (42), we have(44)Si≤2ζδg1τω+V0−τωe−ωt1/2.
Finally, the solution ofSi exponentially converges to a bounded region ΩSi=Si|Si≤2ζ/δg1τ/ω1/2.Remark 4.
IfΦiui=ui (or when ui+=ui−=0, and ϕi+ui=ϕi−ui=1), i.e., there are no dead-zone nonlinear function terms in the input function, we can clearly show that the proposed controller is still suitable for such two-link robotic manipulator systems.Remark 5.
In the case whereui+=ui−=ui0, (24) is reduced to the following expression:(45)ui=−ϖit+ui0signSi,where ϖit=κ0i+κ1iSi+θiTϱix.


## 4. Simulation Results

In this part, a simulation example is given to show the effectiveness of the proposed adaptive fuzzy controller. System (3) is given by(46)ξ¨=M−1ξ−Cξ,ξ˙ξ˙−Gξ+Φu+Dt.And the details of system model (46) are given as follows:(47)Mξ=M11M12M12M22,Cξ,ξ˙ξ˙=−12m2l1l22ξ˙1ξ˙2+ξ˙22sinq212m2l1l2ξ˙12sinξ2,Gξ=12m1+m2gl1cosξ2+12m2l2gcosξ1+ξ212m2gl2cosξ1+ξ2,Dt=sintsint.The input nonlinearitiesΦ1u1 are described by(48)Φ1u1=u1−21−0.3sinu1,u1>2,0,−2≤u1≤2,u1+20.8−0.3cosu1,u1<−2,and the input nonlinearity Φ2u2 is supposed to be(49)Φ2u2=u2−51−0.3sinu2,u2>5,0,−5≤u2≤5,u2+50.8−0.3cosu2,u2<−5.Then, we haveM11=1/4m1+m2l12+1/4m2l22+m2l1l2cosq2, M12=1/4m2l22+1/2m2l1l2cosq2, and M22=1/4m2l22, m1andm2 are the mass, and l1andl2 denote the length. In the simulation, m1=4,m2=2,l1=2,l2=1,andg=9.8.Definey=y1,y2T=ξ1,ξ2T, u=u1,u2T, Φu=Φ1u1,Φ2u2T, Dt=d1t,d2tT, and x=x1T,x2TT∈ℛ4, where x1t=ξ1,ξ2Tandx2t=ξ˙1,ξ˙2T. The initial conditions of the system are taken as x0=y10,y20,y˙10,y˙20T=0,0,0,0T. The desired signal is yd=sin2t,sin2tT, and the external disturbances are selected as follows: d1t=d2t=sint.The adaptive fuzzy systemsθiTϱix, with i=1,2, have the vector x=y1,y2,y˙1,y˙2T as input, and we define three triangular membership functions uniformly distributed on the intervals −5,5, so we will use 3×3×3×3=81 fuzzy rules in the simulation. The design parameters are chosen as follows: χ01=χ02=10,χ11=χ12=5,δ01=δ02=0.001,δ11=δ12=0.0005,γ1=γ2=2,andκ11=κ12=2. The initial condition is κ010=κ020=0,θ1j=θ2j=0,j=1,2,…,81.The simulation results are shown in Figures1–7. Figure 1 clearly shows the simulation results of the output vector y tracking the expected vector yd, which shows that the tracking performance is very good. Figures 2(a) and 2(b) show the tracking errors and their derivatives of the two-link robotic manipulator system, i.e., ei and e˙i, for i=1,2. Figure 3 presents the control input vector u1 and u2, and the boundary of the control input signal is bounded. It is worth noting that, in all these simulation results, the assumptions of the two-link robotic manipulator system model are completely unknown, but for some functions, i.e., Φiui,i=1,2 and Gx, their basic properties are already known. Figure 4 shows the control input function of the system. Figure 5 shows the estimated value of the control input function. The adaptive fuzzy parameters of the fuzzy system are shown in Figures 6 and 7. Through these simulation results, we can see the boundedness of these control signals in closed-loop systems, and through the proposed fuzzy adaptive control scheme, internal disturbances in two-link robotic manipulator system and uncertain nonlinear input functions can be effectively resolved. Finally, we can get that, through the control signal, these input vectors can quickly converge to the tracking vector at time t=0.Figure 1
Simulation results for (a)y1t and (b) y2t.
(a)(b)Figure 2
Tracking errors for (a)e1 and (b) e2.
(a)(b)Figure 3
Control input signal foru1 and u2.Figure 4
Control input function forΦ1u1 and Φ2u2.Figure 5
Estimation of input function forf1x and f2x.Figure 6
Fuzzy parameter forθ1.Figure 7
Fuzzy parameter forθ2.


## 5. Conclusions

In this study, a fuzzy adaptive VSC for uncertain robotic manipulators system with dead zones has been proposed. It is proven that the proposed VSC can well handle the dead zone. The FLSs are used to model the robotic manipulators, and it is also shown that the model error is small enough by using the proposed adaptation law. Meanwhile, the VSC can maintain the good tracking performance even in the presence of dead zones. How to extend the proposed method to tackle the input saturation is one of our research directions.



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