# Adaptive Fuzzy Integral Terminal Sliding Mode Control of a Nonholonomic Wheeled Mobile Robot

Authors: Shuying Peng; Wuxi Shi
Journal: Mathematical Problems in Engineering
 (2017)

---

## Abstract

In this paper, the trajectory tracking problem is investigated for a nonholonomic wheeled mobile robot with parameter uncertainties and external disturbances. In this strategy, combining the kinematic model with the dynamic model, the actuator voltage is employed as the control input, and the uncertainties are approximated by a fuzzy logic system. An auxiliary velocity controller is integrated with an adaptive fuzzy integral terminal sliding mode controller, and a robust controller is employed to compensate for the lumped errors. It is proved that all the signals in the closed system are bounded and the auxiliary velocity tracking errors can converge to a small neighborhood of the origin in finite time. As a result, the tracking position errors converge asymptotically to zeros with faster response than other existing controllers. Simulation results demonstrate the effectiveness of the proposed strategy.


---

## Body

## 1. Introduction

A wheeled mobile robot (WMR) is an uncertain nonlinear MIMO dynamic system. When the WMR constrains the wheel’s “pure rolling without slipping,” it is also a typical kind of nonholonomic systems characterized by kinematic constraints. Such constraints are not integrable and can not be eliminated from the model equations. Given so many characteristics that are hard to handle, there has been tremendous research on the nonholonomic WMR (NWMR) in the past few decades.The trajectory tracking problem is one of the most popular problems on the WMR. With an assumption of “perfect velocity tracking,” the initial kinematic controller for the NWMR was designed in [1, 2]. However, such an assumption is difficult to hold in practice for the dynamic model of the NWMR is neglected. Considering the kinematic model and the dynamic model of the NWMR together, based on backstepping technique, Fierro and Lewis [3] presented a dynamical extension that combines a kinematic controller with a torque controller. In this method, it is assumed that the dynamic structure of the NWMR and the parameters are completely known. However, in practical WMRs, there exist parameter uncertainties and external disturbances. In addition, wheel skidding and slipping may happen. To overcome these difficulties, the torus shaped rear wheels were used for three WMRs in [4, 5]. The modeling and analysis were investigated to design the controller for the WMR in [6]. Meanwhile, a variety of nonlinear control techniques have been used by many researchers, such as adaptive control [7–11], robust adaptive control [12–14], adaptive fuzzy logic control [15–18], adaptive neural network control [19, 20], and sliding mode control [21–23], and several kinds of the aforementioned methodologies are integrated to solve this problem [24, 25].One idea of some proposed literatures related to the trajectory tracking problem of the NWMR is that an auxiliary velocity controller is designed for the kinematic model of the NWMR to make the tracking position errors converge asymptotically to zeros, and a dynamic controller is designed for the dynamic model of the NWMR to make the auxiliary velocity tracking errors as small as possible. Meanwhile, a robust controller is employed to compensate the total uncertainties. For instance, by virtue of the universal approximation property of the fuzzy logic system (FLS) [26–31], a control structure combining a kinematic controller with a dynamic controller plus a fuzzy compensator was proposed in [15]. A complete control law based on a kinematic controller and an adaptive fuzzy sliding mode controller was developed for a NWMR in the presence of dynamic uncertainties as well in [25]. These dynamic controllers share a common idea of choosing the wheel torque as the control input. However, as stated in [17], the wheel is driven by the actuator in reality. Hence, the resulting electrically driven mobile robot (the robot kinematics, robot dynamics, and wheel actuator dynamics) is represented as a third-order system. So most of existing torque controllers designed with respect to the second-order, that is, the wheel actuator dynamics, have been neglected and might degrade the performance of the tracking control. Therefore, it is more reasonable to use the actuator voltage as the control input. For realizing the trajectory tracking of the NWMR with high performance, the wheel actuator dynamics are combining with the dynamics of the NWMR and the actuator voltage is employed as the control input in [8, 16–18]. All these dynamic controllers can guarantee that the auxiliary velocity tracking errors converge to an adjustable neighborhood of the origin as time goes to infinity. However, the finite time convergence of the auxiliary velocity tracking errors can not be guaranteed.The terminal sliding mode control (TSMC), which was first proposed in [32, 33], is an effective scheme to guarantee the finite time convergence of the auxiliary velocity tracking errors. However, the initial TSMC may cause the singularity problem around the equilibrium [34], which would result in an unbounded control signal. In order to avoid this problem, a nonsingular terminal sliding mode control (NTSMC) was developed in [35–38]. The continuous nonsingular terminal sliding mode [36] has been extended into a class of MIMO nonlinear systems [39]. Furthermore, using integral operation, an integral terminal sliding mode control (ITSMC) was presented in [40, 41] for a class of first-order systems. Apart from finite time convergence and nonsingularity, in the ITSMC design, the system can also start on the integral terminal sliding mode surface from the initial time instant. Therefore, the reaching time to the sliding mode surface is eliminated.Based on the previous results, this paper addresses the trajectory tracking problem for the NWMR with parameter uncertainties and external disturbances. Combining the kinematic model with the dynamic model, a control strategy is proposed which integrates an auxiliary velocity controller with an adaptive fuzzy integral terminal sliding mode controller. In this control strategy, using the universal approximation property of the FLS, the uncertainties are approximated by a fuzzy logic system and a robust controller is employed to compensate for the lumped errors. Meanwhile, instead of the wheel torque, the actuator voltage is employed as the control input. The main originality of the proposed control strategy is that the adaptive fuzzy integral terminal sliding mode controller can guarantee the finite time convergence of the auxiliary velocity tracking errors. It is proved that all the signals in the closed system are bounded and the auxiliary velocity tracking errors converge to a small neighborhood of the origin in finite time. Therefore, the tracking position errors converge asymptotically to zeros with faster response than other existing controllers. Simulation results demonstrate the effectiveness of the proposed strategy.The remainder of this paper is organized as follows. Section2 reviews some basics of the model of the NWMR, the ITSMC, and the FLS. By use of the ITSMC and the FLS, a control strategy is proposed which integrates an auxiliary velocity controller with an adaptive fuzzy integral terminal sliding mode controller in Section 3. Section 4 gives simulation results to illustrate our results. Conclusions are given in Section 5.


## 2. Preliminaries

In this section, we will review some basics of the model of the NWMR, the ITSMC, and the FLS briefly.


### 2.1. Model of the Nonholonomic Wheeled Mobile Robot

We consider a typical example of the WMR, which is called Type (2,0) WMR in [6]. Such a WMR is composed of two deriving wheels and one passive wheel. The two deriving wheels are controlled independently by two actuators to achieve the motion and orientation, and the passive wheel prevents the robot from tipping over as it moves on a plane. Figure 1 describes the posture of the WMR in Cartesian coordinates. Both driving wheels with the same radius r are mounted on the same axis and separated by 2R. The center of mass of the WMR is located at C, and P is located in the midpoint of the two driving wheels of the WMR. The distance between P and C is d. When the electrical part of the actuator is taken into account, the kinematic equation and the dynamic equation of the NWMR can be written as follows from [16, 19]:(1)q˙=sqϑ,(2)M¯qϑ˙+V¯q,q˙ϑ+F¯ϑ+τ¯d=NKTRaB¯u-N2KTKbRaB¯Xϑ,where(3)Sq=cos⁡θ-dsin⁡θsin⁡θdcos⁡θ01,M¯q=m00I-md2,V¯q,q˙=0000,B¯=1r11R-R,X=B¯T.q=[x,y,θ]T,  C(x,y) is the coordinate of C in the global coordinate frame XOY, and θ is the orientation of the local coordinate frame XCCYC attached on the WMR platform measured from X axis and is also called the heading angle of the WMR. ϑ=[ν,ω]T,  where ν and ω are the linear velocity of the point P along the robot axis and angle velocity, respectively. M¯(q) is the inertia matrix, V¯(q,q˙) is the centripetal and Coriolis matrix, F¯(ϑ)∈R2×1 is the surface friction, and τ¯d∈R2×1 denotes bounded unknown disturbances including unstructured unmodeled dynamics. N is the gear ration, KT is the motor torque constant, Kb is the counter electromotive force coefficient, and Ra is the electric resistance. u=[u1,u2]T is the actuator voltage input vector.Figure 1
A wheeled mobile robot.Several properties of the NWMR are given as follows [19].Property 1.
The matrixM¯(q) is symmetric and positive definite.Property 2.
The matrixM¯(q) is bounded; that is, there exist positive constants m1 and m2 satisfying m1x2≤xTM¯(q)x≤m2x2, for all x∈R2.Property 3.
The matrixM¯(q)-2V¯(q,q˙) is skew symmetric resulting in the following characteristic: xT(M¯(q)-2V¯(q,q˙))x=0 for all x∈R2.In view of the dynamic model of the NWMR, (2) is a first-order system; the ITSM can be utilized so that the finite time convergence of the auxiliary velocity tracking errors of the NWMR is obtained.


### 2.2. Integral Terminal Sliding Mode

Now, a new form of the integral terminal sliding mode is defined as(4)s=xt-x0+β∫0txτγsignxτdτ,where x∈R is the system state variable, β>0, and 0<γ<1, which generalizes the integral terminal sliding mode [41](5)s=xt+β∫0txq/pτdτ,where β>0 and  p and q are odd integers satisfying p>q>0.Remark 1.
It is worthwhile to notice that the range of the powerγ is larger than that of the power q/p. Meanwhile, by means of the basic theorem of differential and integral calculus [42], the integral terminal sliding mode (4) is continuous and differentiable although the absolute and signum operators are involved. Besides these properties, from (4), it is obvious that s(0)=0 without the prior knowledge of the parameter β. This implies that the system starts on the integral terminal sliding mode surface (4) from the initial time instant much easily.Furthermore, on the sliding surface,s=0, which results in(6)x˙t+βxtγsignxt=0.The finite time ts that is taken from x(0)≠0 to x(ts)=0 is given by(7)ts=1β1-γx01-γ.As we stated previously, there exist parameter uncertainties and unknown disturbances in practical WMR. Taking these factors into account, an unknown nonlinear function is contained in the model of the NWMR. We will use the FLS to approximate this function.


### 2.3. Fuzzy Logic Systems

In this section, the FLS is discussed briefly. The basic configuration of an FLS consists of four components: fuzzifier, fuzzy rule base, fuzzy inference engine, and defuzzifier. The fuzzy rule base is a collection of IF-THEN rules and thelth fuzzy rule is written asRl: IF x1 is F1l and…and xn is Fnl, THEN y is Gl,where Fil and Gl are fuzzy sets, associating with fuzzy membership functions μFil(xi) and μGl(y), respectively, i=1,…,n,  l=1,…,m,  m is the number of rules.Based on these fuzzy IF-THEN rules, the FLS performs a mapping from an input vectorx=[x1,…,xn]T∈Rn to an output variable y∈R. If we use the strategy of singleton fuzzifier, product inference, and center-average defuzzifier, the output of the FLS can be defined as follows:(8)yx=∑l=1myl∏i=1nμFilxi∑l=1m∏i=1nμFilxi,where yl is the point in Gl at which μGl(y) obtains its maximum value 1.For simplicity,y(x) can be written in the following compact form:(9)yx=θ^Tξx≔f^x∣θ^,where θ^=[y1,…,ym]T is called the unknown parameter vector which is to be updated and ξ(x)=[ξ1(x),…,ξm(x)]T is called the fuzzy basis function vector, ξl(x)=∏i=1nμFil(xi)/∑l=1m∏i=1nμFil(xi),  l=1,2,…,m.Lemma 2 (see [31]).
Letf(x) be a continuous function defined on a compact set Ω. Then, for any constant ε>0, there exists a fuzzy system (9) such that supx∈Ωf(x)-f^(x∣θ^)≤ε.


## 2.1. Model of the Nonholonomic Wheeled Mobile Robot

We consider a typical example of the WMR, which is called Type (2,0) WMR in [6]. Such a WMR is composed of two deriving wheels and one passive wheel. The two deriving wheels are controlled independently by two actuators to achieve the motion and orientation, and the passive wheel prevents the robot from tipping over as it moves on a plane. Figure 1 describes the posture of the WMR in Cartesian coordinates. Both driving wheels with the same radius r are mounted on the same axis and separated by 2R. The center of mass of the WMR is located at C, and P is located in the midpoint of the two driving wheels of the WMR. The distance between P and C is d. When the electrical part of the actuator is taken into account, the kinematic equation and the dynamic equation of the NWMR can be written as follows from [16, 19]:(1)q˙=sqϑ,(2)M¯qϑ˙+V¯q,q˙ϑ+F¯ϑ+τ¯d=NKTRaB¯u-N2KTKbRaB¯Xϑ,where(3)Sq=cos⁡θ-dsin⁡θsin⁡θdcos⁡θ01,M¯q=m00I-md2,V¯q,q˙=0000,B¯=1r11R-R,X=B¯T.q=[x,y,θ]T,  C(x,y) is the coordinate of C in the global coordinate frame XOY, and θ is the orientation of the local coordinate frame XCCYC attached on the WMR platform measured from X axis and is also called the heading angle of the WMR. ϑ=[ν,ω]T,  where ν and ω are the linear velocity of the point P along the robot axis and angle velocity, respectively. M¯(q) is the inertia matrix, V¯(q,q˙) is the centripetal and Coriolis matrix, F¯(ϑ)∈R2×1 is the surface friction, and τ¯d∈R2×1 denotes bounded unknown disturbances including unstructured unmodeled dynamics. N is the gear ration, KT is the motor torque constant, Kb is the counter electromotive force coefficient, and Ra is the electric resistance. u=[u1,u2]T is the actuator voltage input vector.Figure 1
A wheeled mobile robot.Several properties of the NWMR are given as follows [19].Property 1.
The matrixM¯(q) is symmetric and positive definite.Property 2.
The matrixM¯(q) is bounded; that is, there exist positive constants m1 and m2 satisfying m1x2≤xTM¯(q)x≤m2x2, for all x∈R2.Property 3.
The matrixM¯(q)-2V¯(q,q˙) is skew symmetric resulting in the following characteristic: xT(M¯(q)-2V¯(q,q˙))x=0 for all x∈R2.In view of the dynamic model of the NWMR, (2) is a first-order system; the ITSM can be utilized so that the finite time convergence of the auxiliary velocity tracking errors of the NWMR is obtained.


## 2.2. Integral Terminal Sliding Mode

Now, a new form of the integral terminal sliding mode is defined as(4)s=xt-x0+β∫0txτγsignxτdτ,where x∈R is the system state variable, β>0, and 0<γ<1, which generalizes the integral terminal sliding mode [41](5)s=xt+β∫0txq/pτdτ,where β>0 and  p and q are odd integers satisfying p>q>0.Remark 1.
It is worthwhile to notice that the range of the powerγ is larger than that of the power q/p. Meanwhile, by means of the basic theorem of differential and integral calculus [42], the integral terminal sliding mode (4) is continuous and differentiable although the absolute and signum operators are involved. Besides these properties, from (4), it is obvious that s(0)=0 without the prior knowledge of the parameter β. This implies that the system starts on the integral terminal sliding mode surface (4) from the initial time instant much easily.Furthermore, on the sliding surface,s=0, which results in(6)x˙t+βxtγsignxt=0.The finite time ts that is taken from x(0)≠0 to x(ts)=0 is given by(7)ts=1β1-γx01-γ.As we stated previously, there exist parameter uncertainties and unknown disturbances in practical WMR. Taking these factors into account, an unknown nonlinear function is contained in the model of the NWMR. We will use the FLS to approximate this function.


## 2.3. Fuzzy Logic Systems

In this section, the FLS is discussed briefly. The basic configuration of an FLS consists of four components: fuzzifier, fuzzy rule base, fuzzy inference engine, and defuzzifier. The fuzzy rule base is a collection of IF-THEN rules and thelth fuzzy rule is written asRl: IF x1 is F1l and…and xn is Fnl, THEN y is Gl,where Fil and Gl are fuzzy sets, associating with fuzzy membership functions μFil(xi) and μGl(y), respectively, i=1,…,n,  l=1,…,m,  m is the number of rules.Based on these fuzzy IF-THEN rules, the FLS performs a mapping from an input vectorx=[x1,…,xn]T∈Rn to an output variable y∈R. If we use the strategy of singleton fuzzifier, product inference, and center-average defuzzifier, the output of the FLS can be defined as follows:(8)yx=∑l=1myl∏i=1nμFilxi∑l=1m∏i=1nμFilxi,where yl is the point in Gl at which μGl(y) obtains its maximum value 1.For simplicity,y(x) can be written in the following compact form:(9)yx=θ^Tξx≔f^x∣θ^,where θ^=[y1,…,ym]T is called the unknown parameter vector which is to be updated and ξ(x)=[ξ1(x),…,ξm(x)]T is called the fuzzy basis function vector, ξl(x)=∏i=1nμFil(xi)/∑l=1m∏i=1nμFil(xi),  l=1,2,…,m.Lemma 2 (see [31]).
Letf(x) be a continuous function defined on a compact set Ω. Then, for any constant ε>0, there exists a fuzzy system (9) such that supx∈Ωf(x)-f^(x∣θ^)≤ε.


## 3. Controller Design

It is easy to see that the posture of the NWMRq(t)=xt,yt,θtT satisfies the following equations from (1):(10)x˙=νcos⁡θ-ωdsin⁡θ,y˙=νsin⁡θ+ωdcos⁡θ,θ˙=ω.It is assumed that the reference trajectoryqr(t)=xrt,yrt,θrtT is generated by a reference NWMR with the kinematic equation as (10):(11)x˙r=νrcos⁡θr-ωrdsin⁡θr,y˙r=νrsin⁡θr+ωrdcos⁡θr,θ˙r=ωr.The objective of the trajectory tracking control is to design a strategy such thatq(t) converges asymptotically to qr(t), while all signals in the derived closed-loop system remain bounded. In this study, an auxiliary velocity controller ϑc is designed for the kinematic model (1) to meet the control objective. Then, the actuator voltage control input u is designed for the dynamic model (2) such that ϑ converges to ϑc which is designed at the first step in finite time.Remark 3.
As pointed out in [18], the classical auxiliary velocity controller [1] adopted in [3, 16, 18, 19, 25] can only guarantee that q(t) converges asymptotically to qr(t) when d equals zero. However, d does not equal zero in general. Therefore, the reference point of the practical NWMR is not in accordance with the desired point of the reference NWMR, which results in incomplete tracking of the posture. In this paper, we modify the kinematic model of the reference NWMR as (11) and adopt another auxiliary velocity controller [43].


### 3.1. An Auxiliary Velocity Controller Design

We define the tracking position errors as the difference between the center of massC of the NWMR and the desired point of the reference NWMR as follows [16, 19]:(12)ep=e1e2e3=cos⁡θsin⁡θ0-sin⁡θcos⁡θ0001xr-xyr-yθr-θ.The first derivative of the error yields(13)e˙1e˙2e˙3=-1e20-e1-d0-1νω+νrcos⁡e3-ωrdsin⁡e3νrsin⁡e3+ωrdcos⁡e3ωr.Therefore, the objective of this study becomes the design of an auxiliary velocity controller to make the tracking position errors asymptotically converge to zeros. In this study, according to [43], the auxiliary velocity controller is designed as(14)ϑc=νcωc=νrcos⁡e3+k1e1+d-dcos⁡e3ωr+k3νre2-dsin⁡e3+k2sin⁡e3,where k1,k2,k3>0 are design parameters.Substituting (14) into (13), the closed-loop kinematic equation can be written as(15)e˙1e˙2e˙3=-1e20-e1-d0-1νrcos⁡e3+k1e1+d-dcos⁡e3ωr+k3νre2-dsin⁡e3+k2sin⁡e3+νrcos⁡e3-ωrdsin⁡e3νrsin⁡e3+ωrdcos⁡e3ωr.Assumption 4 (see [8]).
The reference velocitiesϑr=[νr,ωr]T and ν˙r are bounded.Lemma 5.
For the kinematic model (1) of the NWMR satisfying Assumption 4, the auxiliary velocity controller (14) will ensure that the tracking position errors converge asymptotically to zeros.Proof.
Consider the following Lyapunov function candidate(16)V1=12e1+d-dcos⁡e32+12e2-dsin⁡e32+1-cos⁡e3k3.
DifferentiatingV1 with respect to time, we have(17)V˙1=e1+d-dcos⁡e3e˙1+de˙3sin⁡e3+e2-dsin⁡e3e˙2-de˙3cos⁡e3+1k3e˙3sin⁡e3.
Replacing (15) into (17) and after some manipulations, one obtains(18)V˙1=-k1e1+d-dcos⁡e32-k2k3sin2⁡e3≤0.Therefore, the tracking position error eP=[e1,e2,e3]T is bounded. With Assumption 4, ϑc and e˙P=[e˙1,e˙2,e˙3]T are bounded. So V¨1 is bounded and V˙1 is uniformly continuous accordingly. By Barbalat’s lemma [44], V˙1→0 as t→∞, which implies that e1→0 and e3→0 as t→∞.
From (15), one obtains(19)e˙3=-k3νre2-dsin⁡e3-k2sin⁡e3.Using Barbalat’s lemma again, e˙3→0 as t→∞, which implies that e2→0 as t→∞.
Hence,eP→0 as t→∞; that is, the tracking position errors converge asymptotically to zeros.Now, it remains to design the actuator voltage control input so that the desired velocitiesϑc can be obtained in finite time.


### 3.2. Adaptive Integral Terminal Sliding Mode Controller Design

In this study, the auxiliary velocity tracking error is defined as(20)eϑ=eϑ1,eϑ2T=ϑc-ϑ.Consequently, the dynamic equation (2) of the NWMR can be rewritten as(21)M¯qe˙ϑ=-V¯q,q˙eϑ+N2KTKbRaB¯Xϑ+M¯qϑ˙c+V¯q,q˙ϑc+F¯ϑ+τ¯d-NKTRaB¯u.A continuous nonsingular integral terminal sliding mode is defined as in the form (4):(22)S=s1s2=eϑ1-eϑ10+β1∫0teϑ1γ1signeϑ1dτeϑ2-eϑ20+β2∫0teϑ2γ2signeϑ2dτ,where βi>0,  0<γi<1,  i=1,2.Denote(23)Λ=diagβ1,β2,sigeϑγ=eϑ1γ1signeϑ1,eϑ2γ2signeϑ2T.S can be rewritten as follows:(24)S=eϑ-eϑ0+Λ∫0tsigeϑγdτ.UtilizingS and its derivative with respect to time, (21) can be arranged as follows:(25)M¯qS˙=-V¯q,q˙S+N2KTKbRaB¯Xϑ+fx-NKTRaB¯u,where(26)fx=f1x,f2xT=M¯qϑ˙c+M¯qΛsigeϑγ+V¯q,q˙ϑc-V¯q,q˙eϑ0+V¯q,q˙Λ∫0tsigeϑγdτ+F¯ϑ+τ¯dand x=[ϑcT,ϑ˙cT,ϑT]T.Iff(x) is known, let the actuator voltage control input(27)u=RaNKTB¯-1N2KTKbRaB¯Xϑ+fx+K1S+K2sigSρ,where K1=diag(k11,k12),  K2=diag(k21,k22),  k1i>0,  k2i>0,  i=1,2, and  0<ρ<1.Substituting (27) into (25), the closed-loop dynamic equation can be written as(28)M¯qS˙=-V¯q,q˙S-K1S-K2sigSρ.MultiplyingST by (28) yields(29)STM¯qS˙=-STV¯q,q˙S-STK1S-STK2sigSρ.Theorem 6.
For the dynamic model (2) of the NWMR with a known function f(x) (26), if the integral terminal sliding mode is chosen as (22) and the actuator voltage control input is designed as (27), then the integral terminal sliding mode S and the auxiliary velocity tracking error eϑ will converge to zeros in finite time.To prove Theorem6, we introduce two lemmas.Lemma 7 (see [36]).
Supposea1,a2,…,an and 0<p<2 are all positive numbers, then the following inequality holds:(30)a12+a22+⋯+an2p≤a1p+a2p+⋯+anp2.Lemma 8 (see [36]).
An extended Lyapunov description of finite time stability can be given with the form of fast terminal sliding mode as(31)V˙x+αVx+βVγx≤0,α,β>0,0<γ<1,and the settling time can be given by(32)tr≤1α1-γlnαV1-γx0+ββ.Proof of Theorem6.
Consider the following Lyapunov function candidate:(33)V2=12STM¯qS.
DifferentiatingV2 with respect to time and using (29) yields(34)V˙2=STM¯qS˙+12STM¯˙qS=12STM¯˙q-2V¯q,q˙S-STK1S-STK2sigSρ.
From Property3, which makes the first term zero, V˙2 becomes(35)V˙2=-STK1S-STK2sigSρ.
Denoteλi=min{ki1,ki2},  i=1,2; the following inequality holds:(36)V˙2≤-λ1∑i=12si2-λ2∑i=12siρ+1.
Applying Lemma7 into (36), one obtains(37)V˙2≤-λ12m2∑i=12m22si2-λ22m2ρ+1/2∑i=12m2si22ρ+1/2.
Utilizing Property2, we have(38)V˙2≤-λ12m2V2-λ22m2ρ+1/2V2ρ+1/2,or(39)V˙2+λ12m2V2+λ22m2ρ+1/2V2ρ+1/2≤0.
From Lemma8, it follows that S will converge to zero in finite time(40)tr≤1λ12/m21-ρ+1/2ln⁡λ12/m2V21-ρ+1/20+λ22/m2ρ+1/2λ22/m2ρ+1/2.
Moreover, on the sliding mode surface, according to (7),(41)tsi=1βi1-γieϑi01-γi,i=1,2.
Therefore, the auxiliary velocity tracking erroreϑ will converge to zero in finite time t=tr+maxts1,ts2.Due to the fact thatf(x) contains all the mobile robot parameters (such as mass, moment of inertia, and friction coefficients) and external disturbances, in the following, we assume that fi(x),  i=1,2, can be approximated by the following FLS:(42)f^ix∣θ^fi=θ^fiTξfix,where ξfi(x) is the fuzzy basis function vector and θ^fi is the parameter vector of each fuzzy system designed later.Define the optimal approximation parametersθ^fi∗ as follows: (43)θ^fi∗=argminθ^fi∈Ωfisupx∈Ufix-f^ix∣θ^fi,where Ωfi is the compact set of allowable controller parameters. Moreover, the parameter error and the minimum approximation error are defined as θ~fi=θ^fi∗-θ^fi and ωfi(x)=fi(x)-f^i(x∣θ^fi∗), respectively.Assumption 9.
Fori=1,2,  ωfi is bounded. That is, there exists an unknown constant ω¯fi such that ωfix<ω¯fi.
Denote(44)f^x∣θ^f=f^1x∣θ^f1,f^2x∣θ^f2T,f^x∣θf∗=f^1x∣θf1∗,f^2x∣θf2∗T.
By using the fuzzy approximationf^(x∣θ^f) instead of f(x), the following control law from (27) is obtained:(45)u=RaNKTB¯-1N2KTKbRaB¯Xϑ+f^x∣θ^f+K1s+K2sigsρ+ur,where ur=[ur1,ur2]T is a robust controller, which is designed as(46)ur=ε^f+σss,s≠0,0,s=0,ε^f is the estimate of ∑i=12ω¯fi,  ε~f=ε^f-∑i=12‍ω¯fi, and σ is a positive constant.Remark 10.
It is noticed that the robust controllerur (46) is similar to that in [37]. However, in [37], σ is required no less than the estimation error of the unknown function. Whereas, in practice, it is difficult to determine such an estimation error. In this paper, σ is relaxed to be an arbitrary positive number.
Substituting (45) into (25), the closed-loop dynamic equation can be rewritten as(47)M¯qS˙=-V¯q,q˙S-f^x∣θ^-K1S-K2sigSρ-ur+fx.
MultiplyingST to (47) and after some manipulations, we can get(48)STM¯qS˙=-STV¯q,q˙S-STK1S-STK2sigSρ-STur+∑i=12siωfix+∑i=12siθ~fiTξfix.
We use the following adaptation laws to adjust the unknown parametersθ^fi and ε^f:(49)θ^˙fi=-μfisiξfix,ε^˙f=ηs,where μfi>0,  i=1,2, and η>0.The properties of the proposed adaptive fuzzy ITSMC law is summarized by the following theorem.Theorem 11.
For the dynamic model (2) of the NWMR with an unknown function f(x) (26), if the integral terminal sliding mode is chosen as (22), the actuator voltage control input with dynamic robust controller ur (46) is designed as (45), and the adaptation laws are (49); then (1)
all the signals in the closed system are bounded;(2)
the sliding variableS will converge to the neighborhood of the integral terminal sliding mode S=0 as S≤δ=min{δ1,δ2} in finite time, where(50)δ1=f^x∣θf∗-f^x∣θ^f+ε~fλ1,δ2=f^x∣θf∗-f^x∣θ^f+ε~fλ21/ρ.Moreover, the auxiliary velocity tracking error eνi will converge to the region eνi≤2δ+eνi(0),  i=1,2 in finite time.Proof.
Consider the following Lyapunov function candidate:(51)V3=V31+V32,where(52)V31=12STM¯qS,V32=12∑i=121μfiθ~fiTθ~fi+1ηε~f2.
(
1
) Differentiating V31 with respect to time, using (48) and Property 3, we have(53)V˙31=-STK1S-STK2sigSρ-STur+∑i=12siωfix+∑i=12siθ~fiTξfix.
Note that(54)∑i=12siωfix≤S∑i=12ω¯i.
Substituting (46) and (54) into (53), the following inequality holds:(55)V˙31≤-STK1S-STK2sigSρ-Sε^f+σ+S∑i=12ω¯i+∑i=12siθ~fiTξfix=-STK1S-STK2sigSρ-Sσ-Sε~f+∑i=12siθ~fiTξfix.
There results(56)V˙31≤-STK1S-STK2sigSρ-Sε~f+∑i=12siθ~fiTξfix.
DifferentiatingV32 with respect to time yields(57)V˙32=∑i=121μfiθ~fiTθ~˙fi+1ηε~fε~˙f.
Combining (56) with (57), we can get(58)V˙31+V˙32≤-STK1S-STK2sigSρ+∑i=12θ~fiTsiξfix+1μfiθ~˙fi+ε~f1ηε~˙f-S.
That is,(59)V˙31+V˙32≤-STK1S-STK2sigSρ+∑i=12θ~fiTsiξfix-1μfiθ^˙fi+ε~f1ηε^˙f-S.
Applying the adaptation laws (49) into (59), one has(60)V˙31+V˙32≤-STK1S-STK2sigSρ.
Clearly,V˙3=V˙31+V˙32≤0, it is concluded that all the signals si,  eνi,  θ^fi, and ε^f are bounded.
(
2
) According to (56), one gets(61)V˙31≤-STK1S-STK2sigSρ-Sε~f+STf^x∣θf∗-f^x∣θ^f=-STK1S-STK2sigSρ+STf^x∣θf∗-f^x∣θ^f-ε~fSS,which can be further changed into the following two forms:(62)V˙31<-STK1-diagf^x∣θf∗-f^x∣θ^f-ε~fSSdiag-1SS-STK2sigSρ,(63)V˙31<-STK1S-STK2-diagf^x∣θf∗-f^x∣θ^f-ε~fSSdiag-1sigSρsigSρ.
Denote(64)f^x∣θf∗-f^x∣θ^f-ε~fSS=f~1,f~2T.
For (62), if λ1>f~i/si,  i=1,2, which means the matrix K1-diag[f^(x∣θf∗)-f^(x∣θ^f)-ε~f/SS]diag-1(S) is positive definite, the similar structure as (35) is kept. Hence, finite time stability is guaranteed. Otherwise, si≤f~i/λ1,  i=1,2. We can conclude that(65)S2=s12+s22≤f~12+f~22λ12=f^x∣θf∗-f^x∣θ^f-ε~f/SS2λ12≤f^x∣θf∗-f^x∣θ^f+ε~f2λ12;that is,(66)S≤f^x∣θf∗-f^x∣θ^f+ε~fλ1=δ1.Therefore, the region S≤δ1 can be reached in finite time.
For (63), if λ2>f~i/siρ,  i=1,2, which means the matrix K2-diag[f^(x∣θf∗)-f^(x∣θ^f)-ε~f/SS]diag-1(Sρ) is positive definite, the similar structure as (35) is kept. Hence, finite time stability is guaranteed. Otherwise, siρ≤f~i/λ2,  i=1,2. We can conclude from Lemma 7 that(67)S4ρ=s12+s222ρ≤s12ρ+s22ρ2≤f~12+f~22λ222=f^x∣θf∗-f^x∣θ^f-ε~f/SS4λ24≤f^x∣θf∗-f^x∣θ^f+ε~f4λ24;that is,(68)S≤f^x∣θf∗-f^x∣θ^f+ε~fλ21/ρ=δ2.Therefore, the region S≤δ2 can be reached in finite time.
By virtue of (66) and (68), the region S≤δ=min{δ1,δ2} can be reached in finite time.
WhenS≤δ, for i=1,2,  si≤δ. The integral terminal sliding mode (22) can be changed into the following form:(69)si=eνi-eνi0+βi∫0teνiγisigneνidτ=ϕi,ϕi≤δ,or the equivalent form(70)eνi-eνi0+βi-ϕi∫0teνiγisigneνidτ∫0teνiγisigneνidτ=0.
Ifβi>ϕi/∫0teνiγisigneνidτ, (70) is kept in the form of the integral terminal sliding mode. Hence, finite time convergence is guaranteed. Otherwise, ∫0teνiγisigneνidτ≤ϕi/βi≤δ/βi; the region(71)eνi≤βi∫0teνiγisigneνidτ+δ+eνi0≤2δ+eνi0,can be reached in finite time.Remark 12.
Both the control law (27) and the adaptive fuzzy control law (45) contain a nonlinear term K1S+K2sigSρ with the form of fast terminal sliding mode, which assures the boundedness of the signals in the closed system and the finite time convergence of the auxiliary velocity tracking error.Remark 13.
According to (66) and (68), the parameters λ1 and λ2 can be chosen large enough to make the boundary δ small. However, increasing the parameters λ1 and λ2 will increase the level of control input and will cause implementation problem.Remark 14.
The relationship between the auxiliary velocity tracking erroreνi and the width of the boundary layer δ surrounding the integral terminal sliding mode surface S=0 is given by (69) and (71).


## 3.1. An Auxiliary Velocity Controller Design

We define the tracking position errors as the difference between the center of massC of the NWMR and the desired point of the reference NWMR as follows [16, 19]:(12)ep=e1e2e3=cos⁡θsin⁡θ0-sin⁡θcos⁡θ0001xr-xyr-yθr-θ.The first derivative of the error yields(13)e˙1e˙2e˙3=-1e20-e1-d0-1νω+νrcos⁡e3-ωrdsin⁡e3νrsin⁡e3+ωrdcos⁡e3ωr.Therefore, the objective of this study becomes the design of an auxiliary velocity controller to make the tracking position errors asymptotically converge to zeros. In this study, according to [43], the auxiliary velocity controller is designed as(14)ϑc=νcωc=νrcos⁡e3+k1e1+d-dcos⁡e3ωr+k3νre2-dsin⁡e3+k2sin⁡e3,where k1,k2,k3>0 are design parameters.Substituting (14) into (13), the closed-loop kinematic equation can be written as(15)e˙1e˙2e˙3=-1e20-e1-d0-1νrcos⁡e3+k1e1+d-dcos⁡e3ωr+k3νre2-dsin⁡e3+k2sin⁡e3+νrcos⁡e3-ωrdsin⁡e3νrsin⁡e3+ωrdcos⁡e3ωr.Assumption 4 (see [8]).
The reference velocitiesϑr=[νr,ωr]T and ν˙r are bounded.Lemma 5.
For the kinematic model (1) of the NWMR satisfying Assumption 4, the auxiliary velocity controller (14) will ensure that the tracking position errors converge asymptotically to zeros.Proof.
Consider the following Lyapunov function candidate(16)V1=12e1+d-dcos⁡e32+12e2-dsin⁡e32+1-cos⁡e3k3.
DifferentiatingV1 with respect to time, we have(17)V˙1=e1+d-dcos⁡e3e˙1+de˙3sin⁡e3+e2-dsin⁡e3e˙2-de˙3cos⁡e3+1k3e˙3sin⁡e3.
Replacing (15) into (17) and after some manipulations, one obtains(18)V˙1=-k1e1+d-dcos⁡e32-k2k3sin2⁡e3≤0.Therefore, the tracking position error eP=[e1,e2,e3]T is bounded. With Assumption 4, ϑc and e˙P=[e˙1,e˙2,e˙3]T are bounded. So V¨1 is bounded and V˙1 is uniformly continuous accordingly. By Barbalat’s lemma [44], V˙1→0 as t→∞, which implies that e1→0 and e3→0 as t→∞.
From (15), one obtains(19)e˙3=-k3νre2-dsin⁡e3-k2sin⁡e3.Using Barbalat’s lemma again, e˙3→0 as t→∞, which implies that e2→0 as t→∞.
Hence,eP→0 as t→∞; that is, the tracking position errors converge asymptotically to zeros.Now, it remains to design the actuator voltage control input so that the desired velocitiesϑc can be obtained in finite time.


## 3.2. Adaptive Integral Terminal Sliding Mode Controller Design

In this study, the auxiliary velocity tracking error is defined as(20)eϑ=eϑ1,eϑ2T=ϑc-ϑ.Consequently, the dynamic equation (2) of the NWMR can be rewritten as(21)M¯qe˙ϑ=-V¯q,q˙eϑ+N2KTKbRaB¯Xϑ+M¯qϑ˙c+V¯q,q˙ϑc+F¯ϑ+τ¯d-NKTRaB¯u.A continuous nonsingular integral terminal sliding mode is defined as in the form (4):(22)S=s1s2=eϑ1-eϑ10+β1∫0teϑ1γ1signeϑ1dτeϑ2-eϑ20+β2∫0teϑ2γ2signeϑ2dτ,where βi>0,  0<γi<1,  i=1,2.Denote(23)Λ=diagβ1,β2,sigeϑγ=eϑ1γ1signeϑ1,eϑ2γ2signeϑ2T.S can be rewritten as follows:(24)S=eϑ-eϑ0+Λ∫0tsigeϑγdτ.UtilizingS and its derivative with respect to time, (21) can be arranged as follows:(25)M¯qS˙=-V¯q,q˙S+N2KTKbRaB¯Xϑ+fx-NKTRaB¯u,where(26)fx=f1x,f2xT=M¯qϑ˙c+M¯qΛsigeϑγ+V¯q,q˙ϑc-V¯q,q˙eϑ0+V¯q,q˙Λ∫0tsigeϑγdτ+F¯ϑ+τ¯dand x=[ϑcT,ϑ˙cT,ϑT]T.Iff(x) is known, let the actuator voltage control input(27)u=RaNKTB¯-1N2KTKbRaB¯Xϑ+fx+K1S+K2sigSρ,where K1=diag(k11,k12),  K2=diag(k21,k22),  k1i>0,  k2i>0,  i=1,2, and  0<ρ<1.Substituting (27) into (25), the closed-loop dynamic equation can be written as(28)M¯qS˙=-V¯q,q˙S-K1S-K2sigSρ.MultiplyingST by (28) yields(29)STM¯qS˙=-STV¯q,q˙S-STK1S-STK2sigSρ.Theorem 6.
For the dynamic model (2) of the NWMR with a known function f(x) (26), if the integral terminal sliding mode is chosen as (22) and the actuator voltage control input is designed as (27), then the integral terminal sliding mode S and the auxiliary velocity tracking error eϑ will converge to zeros in finite time.To prove Theorem6, we introduce two lemmas.Lemma 7 (see [36]).
Supposea1,a2,…,an and 0<p<2 are all positive numbers, then the following inequality holds:(30)a12+a22+⋯+an2p≤a1p+a2p+⋯+anp2.Lemma 8 (see [36]).
An extended Lyapunov description of finite time stability can be given with the form of fast terminal sliding mode as(31)V˙x+αVx+βVγx≤0,α,β>0,0<γ<1,and the settling time can be given by(32)tr≤1α1-γlnαV1-γx0+ββ.Proof of Theorem6.
Consider the following Lyapunov function candidate:(33)V2=12STM¯qS.
DifferentiatingV2 with respect to time and using (29) yields(34)V˙2=STM¯qS˙+12STM¯˙qS=12STM¯˙q-2V¯q,q˙S-STK1S-STK2sigSρ.
From Property3, which makes the first term zero, V˙2 becomes(35)V˙2=-STK1S-STK2sigSρ.
Denoteλi=min{ki1,ki2},  i=1,2; the following inequality holds:(36)V˙2≤-λ1∑i=12si2-λ2∑i=12siρ+1.
Applying Lemma7 into (36), one obtains(37)V˙2≤-λ12m2∑i=12m22si2-λ22m2ρ+1/2∑i=12m2si22ρ+1/2.
Utilizing Property2, we have(38)V˙2≤-λ12m2V2-λ22m2ρ+1/2V2ρ+1/2,or(39)V˙2+λ12m2V2+λ22m2ρ+1/2V2ρ+1/2≤0.
From Lemma8, it follows that S will converge to zero in finite time(40)tr≤1λ12/m21-ρ+1/2ln⁡λ12/m2V21-ρ+1/20+λ22/m2ρ+1/2λ22/m2ρ+1/2.
Moreover, on the sliding mode surface, according to (7),(41)tsi=1βi1-γieϑi01-γi,i=1,2.
Therefore, the auxiliary velocity tracking erroreϑ will converge to zero in finite time t=tr+maxts1,ts2.Due to the fact thatf(x) contains all the mobile robot parameters (such as mass, moment of inertia, and friction coefficients) and external disturbances, in the following, we assume that fi(x),  i=1,2, can be approximated by the following FLS:(42)f^ix∣θ^fi=θ^fiTξfix,where ξfi(x) is the fuzzy basis function vector and θ^fi is the parameter vector of each fuzzy system designed later.Define the optimal approximation parametersθ^fi∗ as follows: (43)θ^fi∗=argminθ^fi∈Ωfisupx∈Ufix-f^ix∣θ^fi,where Ωfi is the compact set of allowable controller parameters. Moreover, the parameter error and the minimum approximation error are defined as θ~fi=θ^fi∗-θ^fi and ωfi(x)=fi(x)-f^i(x∣θ^fi∗), respectively.Assumption 9.
Fori=1,2,  ωfi is bounded. That is, there exists an unknown constant ω¯fi such that ωfix<ω¯fi.
Denote(44)f^x∣θ^f=f^1x∣θ^f1,f^2x∣θ^f2T,f^x∣θf∗=f^1x∣θf1∗,f^2x∣θf2∗T.
By using the fuzzy approximationf^(x∣θ^f) instead of f(x), the following control law from (27) is obtained:(45)u=RaNKTB¯-1N2KTKbRaB¯Xϑ+f^x∣θ^f+K1s+K2sigsρ+ur,where ur=[ur1,ur2]T is a robust controller, which is designed as(46)ur=ε^f+σss,s≠0,0,s=0,ε^f is the estimate of ∑i=12ω¯fi,  ε~f=ε^f-∑i=12‍ω¯fi, and σ is a positive constant.Remark 10.
It is noticed that the robust controllerur (46) is similar to that in [37]. However, in [37], σ is required no less than the estimation error of the unknown function. Whereas, in practice, it is difficult to determine such an estimation error. In this paper, σ is relaxed to be an arbitrary positive number.
Substituting (45) into (25), the closed-loop dynamic equation can be rewritten as(47)M¯qS˙=-V¯q,q˙S-f^x∣θ^-K1S-K2sigSρ-ur+fx.
MultiplyingST to (47) and after some manipulations, we can get(48)STM¯qS˙=-STV¯q,q˙S-STK1S-STK2sigSρ-STur+∑i=12siωfix+∑i=12siθ~fiTξfix.
We use the following adaptation laws to adjust the unknown parametersθ^fi and ε^f:(49)θ^˙fi=-μfisiξfix,ε^˙f=ηs,where μfi>0,  i=1,2, and η>0.The properties of the proposed adaptive fuzzy ITSMC law is summarized by the following theorem.Theorem 11.
For the dynamic model (2) of the NWMR with an unknown function f(x) (26), if the integral terminal sliding mode is chosen as (22), the actuator voltage control input with dynamic robust controller ur (46) is designed as (45), and the adaptation laws are (49); then (1)
all the signals in the closed system are bounded;(2)
the sliding variableS will converge to the neighborhood of the integral terminal sliding mode S=0 as S≤δ=min{δ1,δ2} in finite time, where(50)δ1=f^x∣θf∗-f^x∣θ^f+ε~fλ1,δ2=f^x∣θf∗-f^x∣θ^f+ε~fλ21/ρ.Moreover, the auxiliary velocity tracking error eνi will converge to the region eνi≤2δ+eνi(0),  i=1,2 in finite time.Proof.
Consider the following Lyapunov function candidate:(51)V3=V31+V32,where(52)V31=12STM¯qS,V32=12∑i=121μfiθ~fiTθ~fi+1ηε~f2.
(
1
) Differentiating V31 with respect to time, using (48) and Property 3, we have(53)V˙31=-STK1S-STK2sigSρ-STur+∑i=12siωfix+∑i=12siθ~fiTξfix.
Note that(54)∑i=12siωfix≤S∑i=12ω¯i.
Substituting (46) and (54) into (53), the following inequality holds:(55)V˙31≤-STK1S-STK2sigSρ-Sε^f+σ+S∑i=12ω¯i+∑i=12siθ~fiTξfix=-STK1S-STK2sigSρ-Sσ-Sε~f+∑i=12siθ~fiTξfix.
There results(56)V˙31≤-STK1S-STK2sigSρ-Sε~f+∑i=12siθ~fiTξfix.
DifferentiatingV32 with respect to time yields(57)V˙32=∑i=121μfiθ~fiTθ~˙fi+1ηε~fε~˙f.
Combining (56) with (57), we can get(58)V˙31+V˙32≤-STK1S-STK2sigSρ+∑i=12θ~fiTsiξfix+1μfiθ~˙fi+ε~f1ηε~˙f-S.
That is,(59)V˙31+V˙32≤-STK1S-STK2sigSρ+∑i=12θ~fiTsiξfix-1μfiθ^˙fi+ε~f1ηε^˙f-S.
Applying the adaptation laws (49) into (59), one has(60)V˙31+V˙32≤-STK1S-STK2sigSρ.
Clearly,V˙3=V˙31+V˙32≤0, it is concluded that all the signals si,  eνi,  θ^fi, and ε^f are bounded.
(
2
) According to (56), one gets(61)V˙31≤-STK1S-STK2sigSρ-Sε~f+STf^x∣θf∗-f^x∣θ^f=-STK1S-STK2sigSρ+STf^x∣θf∗-f^x∣θ^f-ε~fSS,which can be further changed into the following two forms:(62)V˙31<-STK1-diagf^x∣θf∗-f^x∣θ^f-ε~fSSdiag-1SS-STK2sigSρ,(63)V˙31<-STK1S-STK2-diagf^x∣θf∗-f^x∣θ^f-ε~fSSdiag-1sigSρsigSρ.
Denote(64)f^x∣θf∗-f^x∣θ^f-ε~fSS=f~1,f~2T.
For (62), if λ1>f~i/si,  i=1,2, which means the matrix K1-diag[f^(x∣θf∗)-f^(x∣θ^f)-ε~f/SS]diag-1(S) is positive definite, the similar structure as (35) is kept. Hence, finite time stability is guaranteed. Otherwise, si≤f~i/λ1,  i=1,2. We can conclude that(65)S2=s12+s22≤f~12+f~22λ12=f^x∣θf∗-f^x∣θ^f-ε~f/SS2λ12≤f^x∣θf∗-f^x∣θ^f+ε~f2λ12;that is,(66)S≤f^x∣θf∗-f^x∣θ^f+ε~fλ1=δ1.Therefore, the region S≤δ1 can be reached in finite time.
For (63), if λ2>f~i/siρ,  i=1,2, which means the matrix K2-diag[f^(x∣θf∗)-f^(x∣θ^f)-ε~f/SS]diag-1(Sρ) is positive definite, the similar structure as (35) is kept. Hence, finite time stability is guaranteed. Otherwise, siρ≤f~i/λ2,  i=1,2. We can conclude from Lemma 7 that(67)S4ρ=s12+s222ρ≤s12ρ+s22ρ2≤f~12+f~22λ222=f^x∣θf∗-f^x∣θ^f-ε~f/SS4λ24≤f^x∣θf∗-f^x∣θ^f+ε~f4λ24;that is,(68)S≤f^x∣θf∗-f^x∣θ^f+ε~fλ21/ρ=δ2.Therefore, the region S≤δ2 can be reached in finite time.
By virtue of (66) and (68), the region S≤δ=min{δ1,δ2} can be reached in finite time.
WhenS≤δ, for i=1,2,  si≤δ. The integral terminal sliding mode (22) can be changed into the following form:(69)si=eνi-eνi0+βi∫0teνiγisigneνidτ=ϕi,ϕi≤δ,or the equivalent form(70)eνi-eνi0+βi-ϕi∫0teνiγisigneνidτ∫0teνiγisigneνidτ=0.
Ifβi>ϕi/∫0teνiγisigneνidτ, (70) is kept in the form of the integral terminal sliding mode. Hence, finite time convergence is guaranteed. Otherwise, ∫0teνiγisigneνidτ≤ϕi/βi≤δ/βi; the region(71)eνi≤βi∫0teνiγisigneνidτ+δ+eνi0≤2δ+eνi0,can be reached in finite time.Remark 12.
Both the control law (27) and the adaptive fuzzy control law (45) contain a nonlinear term K1S+K2sigSρ with the form of fast terminal sliding mode, which assures the boundedness of the signals in the closed system and the finite time convergence of the auxiliary velocity tracking error.Remark 13.
According to (66) and (68), the parameters λ1 and λ2 can be chosen large enough to make the boundary δ small. However, increasing the parameters λ1 and λ2 will increase the level of control input and will cause implementation problem.Remark 14.
The relationship between the auxiliary velocity tracking erroreνi and the width of the boundary layer δ surrounding the integral terminal sliding mode surface S=0 is given by (69) and (71).


## 4. Simulation Results

In this section, a simulation will be provided to show the effectiveness of the proposed control strategy.Referring to [18], the parameters of the NWMR and its actuators are chosen as m=10kg,  I=5kg·m2,  R=0.12m,  r=0.067m,  d=0.3m,  N=50,  Kb=0.026,  KT=0.026Nm/A, and  Ra=3.5Ω. The surface friction and the external disturbance are generated by F¯(ϑ)+τ¯d=[5sin⁡(2t),5sin⁡(2t)]T. In this simulation, the initial posture and velocity of the practical NWMR are taken as q(0)=[0.1,0.2,0]T and ϑ=[ν,ω]T=[0,0]T, respectively.The reference linear velocity and angular velocity are defined asνr(t)=1 and ωr(t)=1. The trajectory of the reference NWMR is defined as(72)x˙r=νrcos⁡θr-ωrdsin⁡θr,y˙r=νrsin⁡θr+ωrdcos⁡θr,θ˙r=ωr.The initial posture of the reference NWMR is taken as qr(0)=[0,0,0]T.The objective of the trajectory tracking control is to design a strategy such thatq(t) converges asymptotically to qr(t), while all signals in the derived closed-loop system are able to remain bounded. In the proposed control strategy, an auxiliary velocity controller ϑc is designed for the kinematic model to meet the control objective. Then, the actuator voltage control input u is designed for the dynamic model such that ϑ converges to ϑc which is designed at the first step in finite time.In the actuator voltage control inputu, the nonlinear function(73)fx=f1x,f2xT=M¯qϑ˙c+M¯qΛsigeϑγ+V¯q,q˙ϑc-V¯q,q˙eϑ0+V¯q,q˙Λ∫0tsigeϑγdτ+F¯ϑ+τ¯dis contained, where x=[ϑcT,ϑ˙cT,ϑT]T.We suppose that there is no prior information of the robot parameters such as mass, moment of inertial, friction coefficients, and the external disturbance; that is, the nonlinear functionfi(x),  i=1,2, is assumed to be completely unknown. Two fuzzy systems in the form of (9) are used to approximate f1(x) and f2(x). The fuzzy systems have x1=νc,  x2=ωc,  x3=ν˙c,  x4=ω˙c,  x5=ν, and x6=ω as inputs; the fuzzy membership functions for each variable xi,  i=1,2,…,6, are chosen as(74)μFi1xi=exp-12xi+1.250.62,μFi2xi=exp-12xi0.62,μFi3xi=exp-12xi-1.250.62.The initial values of the estimated parameters θ^fi(0),  i=1,2, and ε^f(0) are all set to 0.01.Referring to [1, 31, 36], the parameters of the control law are chosen as k1=1,  k2=20,  k3=10,  β1=β2=1,  γ1=γ2=0.5,  k11=k12=k21=k22=20,  ρ=0.3,  μf1=μf2=0.5,  η=0.001, and  σ=0.1.Using our control strategy to control the NWMR, the simulation results are shown in Figures2–5. Figure 2 is the trajectory tracking process in X-Y plane of the NWMR, Figure 3 is the actuator voltage control input, Figure 4 is the auxiliary velocity tracking errors, and Figure 5 is the tracking position errors, respectively. From Figure 4, it can be observed that the auxiliary velocity tracking error eϑ1 converges to eϑ1≤5×10-3 in finite time t=1.389s and the auxiliary velocity tracking error eϑ2 converges to eϑ2≤7×10-3 in finite time t=3.052s, respectively.Figure 2
Reference trajectory (-) and actual trajectory(⋯).Figure 3
Actuator voltagesu1 (-) and u2 (- -).Figure 4
Auxiliary velocity tracking errorseϑ1 (-) and eϑ2 (- -).Figure 5
Tracking position errorse1 (-), e2 (- -), and e3(⋯).In order to compare the proposed integral terminal sliding mode with the integral sliding mode adopted in [25], we use integral sliding mode instead of the integral terminal sliding mode in our control strategy while other design parameters are the same as the corresponding design parameters used in the above simulation. The corresponding simulation results are shown in Figures 6–9. From Figure 8, it can be observed that the auxiliary velocity tracking error eϑ1 converges to eϑ1≤3×10-2 in finite time t=13.47s and the auxiliary velocity tracking error eϑ2 converges to eϑ2≤4×10-2 in finite time t=14.05s, respectively.Figure 6
Reference trajectory (-) and actual trajectory (- -).Figure 7
Actuator voltagesu1 (-) and u2 (- -).Figure 8
Auxiliary velocity tracking errorseϑ1 (-) and eϑ2 (- -).Figure 9
Tracking position errorse1 (-), e2 (- -), and e3(⋯).It is observed that the actual velocity can track the auxiliary velocity in less time using the control strategy proposed in this paper from Figures4 and 8. As a result, the practical NWMR can track the reference NWMR asymptotically with faster response from Figures 2 and 6. Meanwhile, this favorable performance was obtained with no prior information of the robot parameters such as mass, moment of inertial, friction coefficients, and the external disturbance.


## 5. Conclusions

In this paper, a control strategy has been proposed for the trajectory tracking problem of the NWMR with parameter uncertainties and external disturbances. In this study, we take the wheel actuator dynamics into system dynamics and choose the actuator voltage as the control input. The FLS is adopted to estimate the unknown function coming from parameter uncertainties and external disturbances. An adaptive fuzzy integral terminal sliding mode controller is integrated with an auxiliary velocity controller. It has been shown that all the signals in the closed system are bounded and the auxiliary velocity tracking errors converge to a small neighborhood of the origin in finite time. Hence, the tracking position errors converge asymptotically to zeros with faster response than other existing controllers. Simulation results have been provided to show the feasibility of the proposed control strategy. However, wheel skidding and slipping are unavoidable due to tire deformation and other reasons in real environments. In the future, we will extend our results to the trajectory tracking control of the WMR with wheel skidding and slipping.



---

*Source: 3671846-2017-04-05.xml*