adaptive fuzzy finite-time coordination control for networked nonlinear bilateral teleoperation...

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Adaptive Fuzzy Finite-time Coordination Control forNetworked Nonlinear Bilateral Teleoperation System

Yana Yang, Changchun Hua, Xinping Guan

Abstract—The master-slave control design problem is con-sidered for networked teleoperation system with friction andexternal disturbances. A new finite-time synchronization controlmethod is proposed with the help of adaptive fuzzy approxima-tion. We develop a new nonsingular fast terminal sliding mode(NFTSM) to provide faster convergence and higher precisionthan the linear hyperplane sliding mode and the classic terminalsliding mode (TSM). Then, the adaptive fuzzy-logic systemis employed to approximate the system uncertainties and thecorresponding adaptive fuzzy NFTSM controller is designed.By constructing Lyapunov function, the stability and finite timesynchronization performance are proved with the new controllerin the presence of system uncertainties and external disturbances.Compared with the traditional teleoperation design method, thenew control scheme achieves better transient-state performanceand steady-state performance. Finally, the simulations are per-formed and the comparisons are shown among the proposedmethod, the P+d method, the PD+d method, the DFF methodand the classic TSM, FTSM. The simulation results furtherdemonstrate the effectiveness of the proposed method.

Index Terms—Teleoperation system; Finite time control; Adap-tive fuzzy control; Time delay.

I. INTRODUCTION

Teleoperation system was first conceived to allow a humanoperator’ ability to perform dangerous tasks on a remotesite while the operator remains at a safe place. A typicalteleoperation is composed of five parts: human operator,master robot, communication channel, slave robot and externalenvironment [1]. Compared with the full-automatic system, thejoin of human operator makes the system more flexible. Today,applications of master-slave telerobotic systems can be foundin many areas from micro to macro scales, for example, spaceoperation, undersea exploration, handling hazardous materials,telesurgery (see [1] and the references cited therein).

Many methods have been proposed to deal with the teloeper-ation system design problem. The passivity-based methodwas developed to overcome the instability of teleoperationsystem caused by constant time delay in [2]. Latter, [3]introduced the wave variables following the former scatteringtransformation. Subsequent schemes, building on the abovetwo approaches, have been suggested in [4] and [5]. Afterthat, the passivity-based architecture was extended in [6] to

This paper is partially supported by the Science Fund for DistinguishedYoung Scholars of Hebei Province (F2011203110), Doctoral Fund of Ministryof Education of China (20121333110008), the National Natural ScienceFoundation of China (60934003, 61290322, 61273222).

The authors are with the Institute of Electrical Engineering, YanshanUniversity, Qinhuangdao City, 066004, China. X. Guan is also with theDepartment of Automation, Shanghai Jiao Tong University, Shanghai 200240,China. [email protected]

guarantee state synchronization of master/slave without usingscattering transformation. Following this line, in [7], a globalstable P+d controller was proposed, and the stability conditionswere developed for the constant time delay case without thedelayed derivative action. In addition, [8] proposed the delaydependent linear matrix inequality (LMI) stability criteriafor the closed-loop teleoperation system with time-varyingdelays. In [9], a kinematic control framework for asymmetricteleoperation systems was proposed. Considering the casethat the velocity signals can not be obtained directly, [10]proposed a velocity observer-based new controller. Moreoverthe exponential synchronization performance was achieved.However, in the above works, the asymptotic/exponentialsynchronization performances were achieved. It means thatthe synchronization error converges to zero when time goes toinfinity. It is well known that for an ideal teleoperation system,we expect the slave can follow the master quickly and thesynchronization performance could be achieved in finite time.

The finite-time control problem will be considered in thispaper by using sliding mode control (SMC) method. SMChas been widely used because of their relative simplicityof implementation and their robustness against both plantuncertainties and external disturbances. However, the classiclinear hyperplane-based sliding mode can only guarantee theasymptotic error convergence [11-13]. Thus the finite-timecontrol theory appeared to get a higher convergence speed.Terminal sliding mode control (TSMC) is an effective finite-time control approach, and it was first proposed in [14].Similar to the linear sliding mode technique, strong robustnesswith respect to uncertain dynamics can be obtained. Moreover,the tracking error converges to zero in finite time. However, thepremier papers about TSMC encountered singularity problem.To deal with the singularity problem, a nonsingular termi-nal sliding manifold was proposed in [15]. Then a globallycontinuous nonsingular terminal sliding mode control methodwas presented in paper [16]. Afterwards, fuzzy logic systemand a continuous nonsingular TSM were combined to con-trol a manipulator system in [17]. Different from the directmethod, an indirect method was proposed in [18] to avoidthe singularity problem, and the modified terminal slidingmanifold could produce a smoother switch. Motivated by[18], a Chebyshev neural network-based nonsingular TSMcontroller was presented in paper [19] for spacecraft control. In[20], some new forms of fast TSM strategies were presented,and comparisons about the convergence speed were madebetween the different fast TSMC.

It is well known that robotic manipulator is complicated, dy-



namically coupled, highly time-varying and highly nonlinearsystem. Moreover, with the limitation of modeling method andthe complex external task environment, teleoperation system isinevitably subject to structured and unstructured uncertainties.Fuzzy Logical system (FLs) was introduced by Zadeh in1965, as one of the most popular intelligent computationapproaches, which was proved to be an essential tool forsolving some various classes of engineering problems. Toapproximate the system uncertainties and external disturbance,the fuzzy logic system was widely used in robotic control [21,22]. Recently, the adaptive fuzzy approximation approach wasalso used in controlling MIMO nonlinear system and nonlineartime-delay system [23-30]. [31] proposed a fuzzy outputfeedback controller combining the backstepping method andthe stochastic small-gain approach. The adaptive sliding-modecontrollers for the Takagi-Sugeno (T-S) fuzzy system withmismatched uncertainties and exogenous disturbances wereproposed in [32]. A way of estimating learning errors for directadaptive fuzzy control systems was proposed in [33]. Theadaptive fuzzy method was used in controlling teleoperationsystem under time-varying delay in [34] and [35]. In thispaper, we will use the adaptive fuzzy method to deal withthe uncertainties in the master-slave model.

We propose a new teleoperation design scheme for net-worked teleoperation system with uncertainties. A novel TSMcontroller is presented at first. Then, by employing the adaptivefuzzy approximation method, we design master-slave con-trollers. With the new controllers, the finite-time synchroniza-tion performance is achieved. The main contributions of thispaper are stated as follows: (i) A new FTSM is proposed.Compared with TSM control method, the faster transient-stateand higher-precision control performances are both achieved.(ii) The FLs and parameter adaptive method are combined todeal with the system uncertainties and external disturbance,and the system stability is rigourously proved with the de-signed adaptive fuzzy controllers. (iii) The precise reachingtime and the sliding time can be computed with system initialstates, sliding mode parameters and controller parameters.

This paper is organized as follows. Section 2 presentssome preliminary knowledge for the dynamics of teleoperationsystem with the related properties and the knowledge offuzzy logical system. In section 3, the new controllers for theteleoperator are presented. The simulation results are presentedin section 4. Finally, section 5 concludes with a summary ofthe obtained results.

II. SYSTEM FORMULATION

A. The dynamics of Teleoperator

Consider a master-slave system given by the followingmodel

Mm (qm)··qm + Cm (qm, qm) qm + Fmqm

+fcm(qm) + τdm + gm (qm) = τm + Fh

Ms (qs)··qs + Cs (qs, qs) qs + Fsqs

+fcs(qs) + τds + gs (qs) = τs + Fe

(1)

where qm (t), qs (t) ∈ Rn×1 are the vectors of the jointdisplacements; qm (t), qs (t) ∈ Rn×1 are the vectors of jointvelocities; qm (t), qs (t) ∈ Rn×1 are the vectors of jointaccelerations; Mm (qm), Ms (qs) ∈ Rn×n are the positivedefinite inertia matrices; Cm (qm, qm), Cs (qs, qs) ∈ Rn×n arethe matrices of centripetal and coriolis torques; Fm,Fs denotethe viscous friction coefficients; fcm(qm), fcs(qs) ∈ Rn×1

are the Coulomb friction coefficients; τdm, τds ∈ Rn×1

denote the bounded unknown external disturbances; gm (qm),gs (qs) ∈ Rn×1 are the gravitational torques; Fh, Fe ∈ Rn×1

are the torques applied by the human operator and the remoteenvironment, respectively; τm, τs ∈ Rn×1 are the appliedtorques.

The well known properties for robot system are revisitedhere:

Property 1: The inertia matrix Mi (qi) is a symmetricpositive definite function and there exist positive constants mi1

and mi2 such that mi1I ≤ Mi (q) ≤ mi2I , with i = m, s.Property 2: The matrix Mi (qi) − 2Ci (qi, qi) is skew-

symmetric.Property 3: For all qi, x, y ∈ Rn×1, there exists a positive

scalar ai such that

∥Ci(qi, x)y∥ ≤ ai ∥x∥ ∥y∥

where ∥·∥ denotes the Euclidean norm of a vector and thecorresponding induced matrix norm.

Property 4: The gravity vector gi(qi) is bounded as itconsists of sinusoidal function qi. There exists a positive scalarµgi such that ∥gi(qi)∥ ≤ µgi .

Remark 1: In this paper, both the viscous friction and theCoulomb friction are considered in the master robot and theslave robot, respectively. Here, the Coulomb friction functionfci(qi) is a bounded and piecewise continuous function.

B. Fuzzy Logic Systems

During the past years, fuzzy logic systems (FLs) have beenextensively used as universal approximators for controllersdesign of dynamic systems with precise model unknown. Afuzzy system is a collection of fuzzy IF-THEN rules of theform:

R(j) : IF z1 (t) is Aj1 and · · · and zn (t) is Aj

n

THEN y (t) is Bj .

By using the strategy of singleton fuzzification, productinference and center-average defuzzification, the output of thefuzzy system is

y (z (t)) =

∑lj=1 y

j(∏n

i=1 µAji(zi (t))

)∑l

j=1

∏ni=1 µAj

i(zi (t))

, (2)

where µAji(zi (t)) is the membership function of linguistic

variable zi (t) , and yj is the point in R at which µBj achievesits maximum value (assume that µBi

(yi)= 1).

Introducing the concept of the fuzzy basic function vectorς (z (t)) gives



y (z (t)) = θ (t)Tς (z (t)) , (3)

where

θ (t) =(y1 (t) , y2 (t) , · · · , yl (t)

)T,

ς (z (t)) = (ς1 (z (t)) , ς2 (z (t)) , · · · , ςl (z (t)))T ,

and ςj (z (t)) is defined as

ςj (z (t)) =

∏ni=1 µAj

i(zi (t))∑l

j=1

∏ni=1 µAj

i(zi (t))

.

Based on the universal approximation theorem, there existsthe optimal approximation parameter θ∗ such that θ∗T ς (z (t))can approximate a nonlinear function G (z (t)) to any desireddegree over a compact set Ωz . The parameter θ∗ is defined asfollows

θ∗ = arg minθ∈Ωθ

(supz∈Ωz

∣∣θT (t) ς (z (t))−G (z (t))∣∣) ,

where Ωθ and Ωz denote the sets of suitable bounds on θ (t)and z (t) , respectively. The minimum approximation errorsatisfies

G (z (t)) = θ∗T ς (z (t)) + ϵ(z(t)) (4)

where ∥ϵ(z(t))∥ ≤ ϵ∗ over z (t) ∈ Ωz, and ϵ∗ is a positivescalar.

Actually, because of the existence of friction and backlash,the dynamic functions of manipulators are piecewise contin-uous functions possibly. Make a supposition that G′ (z (t)) isa piecewise function, which can be expressed as G′ (z (t)) =G1 (z (t))+G2 (z (t)), where G1 (z (t)) is the continuous partand G2 (z (t)) is the bounded piecewise term. Therefore, wehave

G′ (z (t)) = θ∗T ς (z (t)) + ϵ(z(t)) +G2 (z (t)) (5)

= θ∗T ς (z (t)) + ϵ(z(t))

where ϵ(z(t)) = ϵ(z(t)) + G2 (z (t)) with ∥ϵ(z(t))∥ ≤ ϵ∗ isthe piecewise function approximation error and ϵ∗ is an upperbound of the approximation error.

III. CONTROLLER DESIGN

A. TSM Manifold

In this section, a new nonsingular fast terminal slidingmanifold is used for dealing with the problem of positiontracking between the master and the slave. To simplify theexpression, the next definition and lemmas are needed.

Definition 1:

sig(ξ)α = [|ξ1|α1 sign(ξ1), |ξ2|α2 sign(ξ2) · · · |ξn|αn sign(ξn)]T

(6)where ξ = [ξ1, ξ2, · · · , ξn]T ∈ Rn, α1, a2, · · · , an > 0 andsign(·) being the standard signum function.

Lemma 1 (sliding time) [20]: Choose the terminal slidingmode as follows

s = e+ αsig(e)γ1 + βsig(e)γ2 (7)

where α > 0, β > 0, γ1 ≥ 1 and 0 < γ2 < 1. if s = 0, theconvergence time T1 of e is about

T1 =

∫ |ei0|

0

1

αeγ1 + βeγ2de =

|ei0|1−γ1

1− γ1α(1−γ1)/γ1 (8)

× F (1,γ1 − 1

γ1 − γ2;2γ1 − γ2 − 1

γ1 − γ2;−βα−1 |ei0|γ2−γ1)

where ei0 = ei(0), F (·) denotes Gauss’ Hypergeometricfunction [36], and the exact form of F (·) changes with theinvolved parameters. For example

F (1, 1; 2;Z) = −Z−1 ln(1−Z);F (1

2, 1;

3

2;−Z2) = Z−1 arctan(Z)

i ∈ 1, 2, · · · , n.Lemma 2 (reaching time) [16]: Consider the dynamics

model x = f(x), f(0) = 0 and x ∈ Rn. If there is a positivedefinite scalar function V (x) such that

V (x) ≤ −αV (x)− βV (x)δ (9)

where α, β > 0, 0 < δ < 1, then the system is finite-timestable. Furthermore, the settling time is given by

T ≤ 1

α(1− δ)ln

αV 1−δ(x0) + β

β(10)

Lemma 3 [16]: Assume a1 > 0, a2 > 0 and 0 < c < 1,the following inequality holds:

(a1 + a2)c ≤ ac1 + ac2 (11)

The NFTSM proposed in this paper is given as follows

sm = em + αm1sig(em)γm1 + αm2

βm(em) = em + λm(em)(12)

ss = es + αs1sig(es)γs1 + αs2βs(es) = es + λs(es)

where αm1 , αs1 , αm2 and αs2 are positive constants, γm1 > 1,γs1 > 1; em = qs(t − Ts) − qm and es = qm(t − Tm) −qs are the position synchronization errors between the mastermanipulator and the slave manipulator; em = qs(t−Ts)− qmand es = qm(t−Tm)− qs are the velocity tracking errors; Tm

represents the signal transmission time delay from the masterside to the slave side and Ts stands for the transmission timedelay from the slave side to the master side. The time delayTm and Ts are not known in advance. The βm(em) and βs(es)are defined as follows

βm(em) =

sig(em)γm2 if sm = 0 or sm = 0, |em| > µm

km1em + km2sign(em)e2m if sm = 0, |em| ≤ µm

(13)and



βs(es) =

sig(es)

γs2 if ss = 0 or ss = 0, |es| > µs

ks1es + ks2sign(es)e2s if ss = 0, |es| ≤ µs

(14)where 0 < γm2 < 1 and 0 < γs2 < 1; sm = em +αm1sig(em)γm1 +αm2sig(em)γm2 , ss = es+αs1sig(es)

γs1 +

αs2sig(es)γs2 ; km1 = (2 − γm2)µ

γm2−1m , km2 = (γm2 −

1)µγm2−2m , ks1 = (2 − γs2)µ

γs2−1s , ks2 = (γs2 − 1)µ

γs2−2s ;

µm, µs are small positive constants.Remark 2: Compared with the linear sliding mode (LSM),

the system with the TSM can get faster convergence and highercontrol precision. However, the classic TSM faces with thesingularity problem. To deal with the singularity problem, [18]proposed a new TSM by using the switching idea. However,the convergence speed in [18] was in a relatively slow ratewhen the system states are far away from the equilibriumpoints. Considering the speed problem, the comparisons be-tween the different TSM were made in [20]. Motivated by [18]and [20], we proposed a modified NFTSM. The new NFTSMnot only can avoid the singularity problem, but also can get ahigher convergence speed.

Remark 3: The new NFTSM is composed of three parts,When the system states stay at a distance from equilibriumpoints, αsig(e)γ1 dominates over βsig(e)γ2 , thus a fast con-vergence rate can be guaranteed; when the system states areclose to the region, the dominant term βsig(e)γ2 determinesfinite time convergence.

B. Controller Design

With the NFTSM, the teleoperation dynamics can be rewrit-ten as the following forms

Mm (qm) sm + Cm (qm, qm) sm= G′

m(Zm)− τm − Fh + τdmMs (qs) ss + Cs (qs, qs) ss= G′

s(Zs)− τs − Fe + τds

(15)

where Zm = [qTs (t − Ts), qTs (t − Ts), q

Ts (t − Ts), q

Tm, qTm]T ,

Zs = [qTm(t − Tm), qTm(t − Tm), qTm(t − Tm), qTs , qTs ]

T ,G′

m(Zm) and G′s(Zs) are defined as

G′m(Zm) = Mm(qm)(qs(t− Ts) + λm(em)) (16)

+ Cm(qm, qm)(qs(t− Ts) + λm(em))

+ Fmqm + fcm(qm) + gm(qm)

and

G′s(Zs) = Ms(qs)(qm(t− Tm) + λses) + Cs(qs, qs) (17)

× (qm(t− Tm) + λses) + Fsqs + fcs(qs) + gs(qs)

Based on FLs approximation property, we use the functionsG′

m(Zm) and G′s(Zs) to approximate the functions G′

m(Zm)and G′

s(Zs) with

G′m(Zm) = θTmςm(Zm), G′

s(Zs) = θTs ςs(Zs) (18)

where i = m, s; θi is a matrix of the fuzzy adaptionparameters; ςi(Zi) is a vector denoting a known fuzzy basisfunction.

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Fig. 1. The teleoperation with the new controller

Additionally, to streamline the presentation, we give thefollowing definitions

G′m(Zm) = G′

m(Zm)− G′m(Zm) (19)

= (θ∗Tm − θTm)ςm(Zm) + ϵm(Zm)

= θTmςm(Zm) + ϵm(Zm)

and

G′s(Zs) = G′

s(Zs)− G′s(Zs) (20)

= (θ∗Ts − θTs )ςs(Zs) + ϵs(Zs)

= θTs ςs(Zs) + ϵs(Zs)

Because of the existence of piecewise continuous functionsfcm(qm) and fcs(qs), ϵi(Zi) = ϵi(Zi)+ fci(qi), where i = m,s.

The adaptive fuzzy control scheme is shown in Fig.1 andthe controllers are as follows

τm = G′m(Zm) +Km1sm +Km2sig(sm)ρ (21)

+ wmsign(sm) + ξmsign(sm)

τs = G′s(Zs) +Ks1ss +Ks2sig(ss)

ρ

+ wssign(ss) + ξssign(ss)

where wm and ws are used to estimate the upper bound of thesum of fuzzy modeling error and the bounded disturbances wm

and ws, i.e. ∥ϵm(Zm) + τdm∥ ≤ wm and ∥ϵs(Zs) + τds∥ ≤ws. In addition, Km1 , Km2 , Ks1 and Ks2 are positive diagonalmatrices; ξm and ξs are positive scalars, ξm and ξs will bedefined; 0 < ρ < 1.

Remark 4: For the practical teleoperation design, thesynchronization accuracy and synchronization time are veryimportant. For example, the telesurgery can minimize healthcare cost and make specialist doctors available throughout theworld saving people’s life and improving health care systems.Meanwhile, the telesurgery requires a higher performance thanother applications. Because the work performance influences



the patient’ health status, even his life directly. However, theexisting teleoperation control design methods can guaranteethat the synchronization error converges to zero asymptot-ically. It is well known that the state variables reach zerowhen t → ∞ for the asymptotical stability. Therefore, thesynchronization performance can not suit the demand fortelesurgery. The proposed control scheme (21) can guaranteethe synchronization performance in the finite-time T shown in(29) which can be tuned by choosing the design parameters,and then the high-precision and fast synchronization perfor-mance are both realized.

C. Performance Analysis

With the controllers (21) for system (1), we have thefollowing two theorems. In theorem 1, the boundedness ofall the signals is achieved. And the master-slave finite timesynchronization performance is obtained in the theorem 2.

Theorem 1: Consider the teleoperator system (1) with thenew controller (21) in free motion, and the FLs adaptive lawsare given as

·θm = Λmςm(Zm)sTm,

·θs = Λsςs(Zs)s

Ts (22)

where Λm and Λs are positive definite matrices, the adaptivetuning laws are

·wm = ∥sm∥ ,

·ws = ∥ss∥ (23)

where wm (0) ≥ 0, ws (0) ≥ 0, then all the signals of theteleoperation system are bounded.

Proof:Let us propose the following Lyapunov function candidate

V =1

2sTmMm(qm)sm +

1

2sTs Ms(qs)ss +

1

2trace(θTmΛ−1

m θm)

(24)

+1

2trace(θTs Λ

−1s θs) +

1

2(wm − wm)2 +

1

2(ws − ws)

2

evaluating V along the system trajectories, yields

V = sTmMm(qm)sm +1

2sTmMm(qm)sm + sTs Ms(qs)ss

(25)

+1

2sTs Ms(qs)ss − trace(θTmΛ−1

m

·θm)− trace(θTs Λ

−1s

·θs)

+ (wm − wm)·wm + (ws − ws)

·ws

using the property 2 of the robot manipulators with thecontroller (21) and the FLs adaptive laws (22), yields

V = sTm(G′m(Zm)− G′

m(Zm)−Km1sm −Km2sig(sm)ρ

(26)

− wm ∥sm∥ − ξm ∥sm∥+ τdm) + sTs (G′s(Zs)− G′

s(Zs)

−Ks1ss −Ks2sig(ss)ρ − ws ∥ss∥ − ξs ∥ss∥+ τds)

− trace(θTmςm(Zm)sTm)− trace(θTs ςs(Zs)sTs )

+ (wm − wm)·wm + (ws − ws)

·ws

then with the equations (19), (20) and the adaptive tuning laws(23), we can obtain that

V = −sTmKm1sm − sTmKm2sig(sm)ρ − sTs Ks1ss (27)

− sTs Ks2sig(ss)ρ + sTm(ϵm(Zm) + τdm) + sTs (ϵs(Zs) + τds)

− wm ∥sm∥ − ws ∥ss∥+ (wm − wm) ∥sm∥+ (ws − ws) ∥ss∥ − ξm ∥sm∥ − ξs ∥ss∥

Substituting the upper bound wm of ∥ϵ∗m + τdm∥ and theupper bound ws of ∥ϵ∗s + τds∥ into (27), yields

V ≤ −sTmKm1sm − sTmKm2sig(sm)ρ (28)

− sTs Ks1ss − sTs Ks2sig(ss)ρ

The above inequality implies that sm, ss, θm and θs arebounded. Meanwhile, considering (12), we can obtain thatthe em, em, es and es are all bounded. Furthermore, theboundedness of qm, qm, qs and qs can be achieved. As a result,the sm and ss are bounded. Therefore, all the signals of theclosed-loop system are bounded. The proof is completed.

Next, the finite time convergence performance will beproved.

Theorem 2: Consider the teleoperator system (1) with thenew controller (21) in free motion, with the estimation errorsθTmςm(Zm) and θTs ςs(Zs), and the adaptive approximate errorwm − wm and ws − ws, if the design parameter ξm is setas ξm ≥

∥∥∥θTmςm(Xm)∥∥∥ + ∥wm − wm∥ and ξs is set as

ξs ≥∥∥∥θTs ςs(Xs)

∥∥∥ + ∥ws − ws∥; the system trajectory willconverge to sm = 0 and ss = 0 in finite time, furthermore,the synchronization errors em and es will converge to zero infinite time. And the exact convergence time is

T = T1 + T2 (29)

T1 =1

α(1− δ)ln

αU1−δ(x0) + β

β(30)

where α = min(λmin(Km1)2

mm2, λmin(Ks1)

2ms2

), β =

min(λmin(Km2)(2

mm2)(1+ρ)/2, λmin(Ks2)(

2ms2

)(1+ρ)/2) andδ = (1 + ρ)/2; λmin(Kij ) denotes the minimum eigenvalueof matrix Kij .

T2 = max |em(T1)|1−γm1

1− γm1

α(1−γm1 )/γm1m1 × F (1,

γm1 − 1

γm1 − γm2

;

(31)2γm1 − γm2 − 1

γm1 − γm2

;−αm2α−1m1

|em(T1)|αm2−αm1 ),

|es(T1)|1−γs1

1− γs1α(1−γs1 )/γs1s1 × F (1,

γs1 − 1

γs1 − γs2;

2γs1 − γs2 − 1

γs1 − γs2;−αs2α

−1s1 |es(T1)|αs2−αs1 ),

Proof:Select another Lyapunov function candidate



U =1

2sTmMmsm +

1

2sTs Msss (32)

Differentiating the Lyapunov function U along the systemtrajectories, yields

U = sTmMmsm +1

2sTmMmsm + sTs Msss +

1

2sTs Msss (33)

with the property 2 of the robot manipulators with the con-troller (21), we have

U ≤ −sTmKm1sm − sTmKm2sig(sm)ρ − sTs Ks1ss (34)

− sTs Ks2sig(ss)ρ + sTmθTmςm(Zm) + sTs θ

Ts ςs(Zs)

+ ∥sm∥ (wm − wm) + ∥ss∥ (ws − ws)

− ξm ∥sm∥ − ξs ∥ss∥

Then with∥∥∥sTmθTmςm(zm)

∥∥∥ ≤ ∥sm∥∥∥∥θTmςm(zm)

∥∥∥ and∥∥∥sTs θTs ςs(Zs)∥∥∥ ≤ ∥ss∥

∥∥∥θTs ςs(Zs)∥∥∥, we can obtain that

U ≤ −sTmKm1sm − sTmKm2sig(sm)ρ − sTs Ks1ss (35)

− sTs Ks2sig(ss)ρ + ∥sm∥

∥∥∥θTmςm(zm)∥∥∥

+ ∥ss∥∥∥∥θTs ςs(Zs)

∥∥∥+ ∥sm∥ (wm − wm)

+ ∥ss∥ (ws − ws)− ξm ∥sm∥ − ξs ∥ss∥

With the definitions of ξm and ξs, one has

U ≤ −sTmKm1sm + sTmKm2sig(sm)ρ (36)

− sTs Ks1ss − sTs Ks2sig(ss)ρ

With the manipulator property 1, we have

U ≤ −λmin(Km1)2

mm2

(1

2sTmMmsm) (37)

− λmin(Km2)(2

mm2

)(1+ρ)/2(1

2sTmMmsm)(1+ρ)/2

− λmin(Ks1)2

ms2

(1

2sTs Msss)

− λmin(Ks2)(2

ms2

)(1+ρ)/2(1

2sTs Msss)

(1+ρ)/2

With the Lemma 3, we have

U ≤ −minλmin(Km1)2

mm2

, λmin(Ks1)2

ms2

U (38)

−minλmin(Km2)(2

mm2

)(1+ρ)/2,

λmin(Ks2)(2

ms2

)(1+ρ)/2Uδ

= −αU − βU δ

By using Lemma 2, the reaching time can be computedaccurately as showed in equation (30). Moreover, with theLemma 1, the accurate sliding time also can be computed asshown in equation (31). This completes the proof.

Remark 5: Because switching idea is used to avoid thesingularity problem, when the terminal sliding mode smi andssi reach zero, two possible cases for synchronization errorsemi and esi approaching zero will occur. Take the smi asexample, at the first case, |emi | ≥ µm as the terminal slidingmode reaches zero, the convergence time can be computedwith the lemma 2. For the second case |emi | ≤ µm, trackingerror will approach zero along the general sliding manifold.Because of the enough small µm, the convergence time canbe neglected.

Remark 6: The NFTSM methods have been used in [17],[20] and other papers. This paper presents a new nonsingu-lar fast finite-time sliding mode surface, which is differentfrom the existing ones. The faster transient-state and higher-precision control performances are both achieved. Moreover,when sliding mode surfaces presented in [17] and [20] are usedto design controller. The term of diag(|e|) will be composedin the reaching time, so the finite reaching time could notbe computed exactly. In [20], the system uncertainties anddisturbance were not considered.

Remark 7: The control design parameters are chosen suchthat the system is stable and has good transient performance.With the sliding mode surface designed in (12), we choose thepositive parameters Km1 , Km2 , Ks1 , Ks2 , then the synchro-nization error system is stable. Next, let’s analyze the transientperformance. The settling time is T = T1+T2 shown in (29).Based on (30), one can choose big parameters Km2 , Ks2 andsmall parameters Km1 , Ks1 to achieve small T1. Based on(31), the sliding mode surface parameters αm1 , αm2 , αs1 andαs2 are chosen to be big to obtain small T .

IV. SIMULATIONS

In order to verify the effectiveness of our main results,in this section, the simulations are performed on 2-DOFmanipulators:

Mm (qm)··qm + Cm (qm, qm) qm + Fmqm

+fcm(qm) + gm (qm) + τdm = τm − Fh

Ms (qs)··qs + Cs (qs, qs) qs + Fsqs

+fcs(qs) + gs (qs) + τds = τs + Fe

, (39)

the friction functions are as follows

Fiqi + fci(qi) = F qi + fc(qi), for i = m, s (40)

with

F q + fc(q) = [fd1q1 + k1sgn(q1)fd2q2 + k2sgn(q2)

] (41)

For simulation, we choose the parameters m1 = 0.5kg,m2 = 1kg, l1 = 1m, l2 = 0.8m, g = 9.81m/s2, fd1 = 3,

fd2 = 4, k1 = 5, k2 = 4, τdi = [0.3q1 q1 sin t0.3q2 q2 sin t

]. The

controller (21) with the parameters as Km1 = Ks1 =



diag(1, 0.8), Km2 = Ks2 = diag(0.4, 0.6), Λm = Λs =diag(1, 1), γm1 = γs1 = 4, γm2 = γs2 = 5/7,ρ = 9/11, αm1 = αs1 = 10, αm2 = αs2 = 5,µm = µs = 0.0001 and Tm = Ts = 0.6 is adopted.The initial states are chosen as qm (0) =

[0.2π 0.1π

]Tqm (0) =

[0 0

]T, qs (0) =

[0.1π 0.12π

]T, qs (0) =[

0 0]T . The initial values of the fuzzy system is chosen as

[0.01, 0.01, 0.02, 0.01, 0.02, 0.001, 0.01, 0.02, 0.03, 0.01] andthe membership function is µA1 (z) = e−

(z−4)2

2 , µA2 (z) =

e−(z−2)2

2 , µA1 (z) = e−z2

2 , µA1 (z) = e−(z+2)2

2 and µA1 (z) =

e−(z+4)2

2 . The human input force is showed in Fig.2.To illustrate the effectiveness of the Fuzzy-NFTSM con-

troller, some comparisons with the P+d, PD+d, DFF will bemade. And it should be noticed that the controller parametersof P+d, PD+d and DFF are chosen to get the same maximumcontrol torques with the NFTSM.

First we carried out the comparison with the commonly usedP+d control approach proposed in paper [4]. The P+d controllaws are given by

τm = −Lmem −Nmqm +Gm (42)τs = −Lses −Nsqs +Gs

where Lm = Ls = 30 and Nm = Ns = 30. Fig.3 andFig.4 illustrate the position tracking errors at the master sideand the slave side, respectively. The master and slave controltorques are showed in Fig.5. It can be seen that the slavecompleted the tracking successfully, and after a transient dueto initial errors condition, the position tracking errors tend tozero. Furthermore, a faster response is achieved in comparisonwith the P+d control approach.

After the comparison with P+d controller, the comparisonwith the PD+d control laws proposed in paper [3] is alsoconducted. The PD+d control laws are given as follows:

τm = −Zmem − Vmem − Imqm +Gm (43)τs = −Zses − Vses − Isqs +Gs

where Zm = Zs = 30, Vm = Vs = 30 and Im = Is = 30.Simulation results are showed in Fig.6-Fig.8. It can be

clearly seen that the proposed finite-time nonsingular fastsliding mode controller obtains a much faster transient per-formance over the PD+d controller.

Moreover, the comparison with the direct force feedback(DFF) control laws proposed in paper [4] is also carried out.The DFF control laws are showed as follows:

τm = −τs(t− Ts)−Bmqm (44)τs = −Dses − Fsqs

where Bm = 30, Ds = 30 and Fs = 30.The simulation results are illustrated in Fig.9-Fig.11. It can

be seen that the proposed NFTSM controller also produces amuch higher synchronization speed over the DFF controller.

0 1 2 3 4 5 60

2

4

6

8

10

12

14

16

18

20

Time(s)

For

ce(N

)

Fig. 2. The human insert force

0 1 2 3 4 5 6−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

Time(s)

Pos

ition

err

or(r

ad)

P+d1P+d2NFTSM1NFTSM2

Fig. 3. The position tracking errors at master side

0 1 2 3 4 5 6−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Time(s)

Pos

ition

err

or(r

ad)

P+d1P+d2NFTSM1NFTSM2

Fig. 4. The position tracking errors at slave side



0 1 2 3 4 5 6−10

0

10

20

30

40

50

Time(s)

Con

trol

torq

ue(N

.m)

mtau1mtau2stau1stau2

Fig. 5. The master/slave control torques

0 1 2 3 4 5 6−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

Time(s)

Pos

ition

err

or(r

ad)

PD+d1PD+d2NFTSM1NFTSM2

Fig. 6. The position tracking error at master side

0 1 2 3 4 5 6−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Time(s)

Pos

ition

err

or(r

ad)

PD+d1PD+d2NFTSM1NFTSM2

Fig. 7. The position tracking error at slave side

0 1 2 3 4 5 6−10

0

10

20

30

40

50

Time(s)

Con

trol

torq

ue(N

.m)



0 1 2 3 4 5 6−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

Time(s)

Pos

ition

err

or(r

ad)

DFF1DFF2NFTSM1NFTSM2

Fig. 9. The position tracking error at master side

As a result, we can conclude that the proposed finite-timenonsingular fast terminal sliding control scheme for teleoper-ation system will produce a faster transient performance andhigher precision synchronization performance over P+d, PD+dand DFF control schemes.

To further illustrate the effectiveness of the control schemesproposed in this paper, the comparisons between the followingtwo sliding mode surfaces with our NFTSM are also imple-mented.

s = e+ α2sig(e)γ2 (45)

s = e+ α1e+ α2sig(e)γ2 (46)

The sliding mode surfaces (45) and (46) were proposed in[16]. We know that the finite-time sliding mode surfaces (45)



0 1 2 3 4 5 6−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Time(s)

Pos

ition

err

or(r

ad)

DFF1DFF2NFTSM1NFTSM2

Fig. 10. The position tracking error at slave side

0 1 2 3 4 5 6−20

−10

0

10

20

30

40

50

Time(s)

Con

trol

torq

ue(N

.m)



and (46) will result in the singular problem. To avoid thesingular problem, the switch idea used in our paper is alsoemployed for the cases of (45) and (46). First, the comparisonbetween the sliding mode surface (45) and our NFTSMis presented. The simulation results are showed in Fig.12and Fig.13. As we can see, the teleoperation system withNFTSM has a faster convergence speed. Next the comparisonbetween the sliding mode surface (46) and our NFTSM isalso presented. The simulation results are given in in Fig.14and Fig.15. The faster convergence performance also can beachieved with the NFTSM. From the simulation results, wecan see the proposed method is effective and can achieve fasterconvergence performance.

The values of nonsingular fast sliding mode surface areshowed in Fig.16. The values of sliding mode can reach zero

0 1 2 3 4 5 6−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

Time(s)

Pos

ition

err

or(r

ad)

NTSM1NTSM2NFTSM1NFTSM2


0 1 2 3 4 5 6−0.5

0

0.5

1

1.5

2

Time(s)

Pos

ition

err

or(r

ad)

NTSM1NTSM2NFTSM1NFTSM2


0 1 2 3 4 5 6−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

Time(s)

Pos

ition

err

or(r

ad)

NFTSM’1NFTSM’2NFTSM1NFTSM2




0 1 2 3 4 5 6−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Time(s)

Pos

ition

err

or(r

ad)

NFTSM’1NFTSM’2NFTSM1NFTSM2


0 1 2 3 4 5 6−10

−5

0

5

10

15

20

25

Time(s)

Val

ues

of s

lidin

g m

ode

sm1sm2ss1ss2

Fig. 16. The values of sliding mode surface

in finite-time. And we also can see that the reaching time issmaller than the system total convergence time. It shows thecorrectness of the theorem 2. In Fig.17 and Fig.18, the controltorques of master and slave are given. The control torques forthe master and the slave are bounded.

The Fuzzy tuning parameters are showed in Fig.19 andFig.20 for the master and slave, respectively. From thesetwo figures, we can see that the tuning parameters convergeto constants. The adaptive approximation upper bounds areshowed in Fig.21 and Fig.22 for the master and slave, respec-tively. The Fig.19-Fig.22 have verified the correctness of thetheorem 1. Overall, from the simulation results obtained in thissection, one can see that the networked bilateral teleoperationsystem with the new adaptive fuzzy finite-time controller canachieve good steady-state performance and good transient-state performance in the presence of system uncertainties andexternal disturbance.

0 1 2 3 4 5 6−15

−10

−5

0

5

10

15

Time(s)

Mas

ter

cont

rol t

orqu

e(N

.m)

mtau1mtau2

Fig. 17. The control torque of master

0 1 2 3 4 5 6−30

−20

−10

0

10

20

30

40

50

Time(s)

Sla

ve c

ontr

ol to

rque

(N.m

)

stau1stau2

Fig. 18. The control torque of slave

V. CONCLUSIONS

This paper addresses the control design problem for net-worked teleoperation system in the presence of system uncer-tainties and external disturbance. To obtain faster convergencespeed and higher precision performance, a new NFTSM isproposed. With the new NFTSM, a fuzzy logic based adaptivecontroller has been designed for the bilateral teleoperation.The finite time stability of the closed-loop system is proved.Moreover, the exact reaching time and the sliding time canbe deduced. In the simulation section, the comparisons aredone with the P+d, PD+d, DFF and classic TSM, NTSMcontrollers. The simulation results show the system with theproposed controller can get a faster convergence speed andhigher convergence precision.



0 1 2 3 4 5 6−1

−0.5

0

0.5

1

1.5

2

Time(s)

Val

ues

of s

tam

Fig. 19. The values of fuzzy tuning parameters at master side

0 1 2 3 4 5 6−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Time(s)

Val

ues

of s

tas

Fig. 20. The values of fuzzy tuning parameters at slave side

0 1 2 3 4 5 60

2

4

6

8

10

Time(s)

Upp

er b

ound

wm1wm2

Fig. 21. The upper bound at master side

0 1 2 3 4 5 60

2

4

6

8

10

12

14

Time(s)

Upp

er b

ound

ws1ws2

Fig. 22. The upper bound at slave side

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Yana Yang received her B.Sc. degree in electricalengineering from Yanshan University, Qinhuangdao,China, in 2011. She is currently working toward thePh.D. degree in electrical engineering. Her researchinterests are in nonlinear control systems and tele-operation systems design.

Changchun Hua received the Ph. D degree inelectrical engineering from Yanshan University, Qin-huangdao, China, in 2005. He was a research Fellowin National University of Singapore from 2006 to2007. From 2007 to 2009, he worked in CarletonUniversity, Canada, funded by Province of OntarioMinistry of Research and Innovation Program. From2009 to 2010, he worked in University of Duisburg-Essen, Germany, funded by Alexander von Hum-boldt Foundation. Now he is a full Professor inYanshan University, China. He is the author or

coauthor of more than 80 papers in mathematical, technical journals, andconferences. He has been involved in more than 10 projects supported bythe National Natural Science Foundation of China, the National EducationCommittee Foundation of China, and other important foundations. His re-search interests are in nonlinear control systems, control systems design overnetwork, teleoperation systems and intelligent control.

Xinping Guan received the B.S. degree in math-ematics from Harbin Normal University, Harbin,China, and the M.S. degree in applied mathematicsand the Ph.D. degree in electrical engineering, bothfrom Harbin Institute of Technology, in 1986, 1991,and 1999, respectively. He is currently a Profes-sor of Department of Automation, Shanghai JiaoTong University, Shanghai 200240, China. He is the(co)author of more than 200 papers in mathematical,technical journals, and conferences. He is CheungKong Scholars Programme Special appointment pro-

fessor. His current research interests include functional differential and differ-ence equations, robust control and intelligent control for time-delay systems,chaos control and synchronization, and congestion control of networks.

adaptive fuzzy finite-time coordination control for networked nonlinear bilateral teleoperation...

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