Sliding mode-based impedance control scheme forbilateral teleoperation under time delay and

super-twisting observer

N. Gonzalez Fonsecaa, J. de Leon Moralesa, S. Rosalesb, V. Parra Vegab

aFIME, Universidad Autonoma de Nuevo Leon. Avenida Universidad s/n, San Nicolas delos Garza, NL. CP 66451, Mexico

bDivision de Robotica y Manufactura Avanzada, CINVESTAV, Ramoz Arizpe, CO. CP25900, Mexico

Abstract

Teleoperated systems are widely used in several applications, in which accu-racy, precision and haptic features are required. Sliding-mode control basedon impedance techniques gives a good transparency sense to human operator.In order to implement the proposed sliding mode control, speed and acceler-ation measures are required. This paper presents a new sliding-mode controland sliding-mode observer scheme for a force-reflecting teleoperation system un-der time-delay. First, an impedance control is presented for the master robot.Second, an impedance control based on sliding mode techniques is presented,to guarantee robust tracking under unknown constant time delay. Moreover,a nonlinear observer designed via the super twisting algorithm is presented inorder to estimate velocity and acceleration in the presence of constant time de-lay. Special emphasis is given to experimental results which are presented anddiscussed, revealing the effectiveness of the proposed observer with the sliding-mode control. It is shown that despite of presence of time delay, the proposedscheme is stable.

1. Introduction

Teleoperation systems are very useful and in widespread use over the in-ternet, despite evident stability problems, mainly due to the communicationtime-delay. It is posible to find many applications in which a proper solutionfor the time-delay problem is critical, some as surgery [1] [2], quadrotors control[3] [4], and many others teleoperated systems.

Basically, a bilateral teleoperation system is understood as a system com-posed of five interconnected elements: the human operator maneuver the master

Email addresses: nicolasgzz@gmail.com (N. Gonzalez Fonseca), drjleon@gmail.com(J. de Leon Morales), wildsarm@gmail.com (S. Rosales), vparra@cinvestav.mx(V. Parra Vega)

Preprint submitted to Elsevier September 21, 2012

robot in order to generate the position, velocity and force commands that aresent, through the communication channel, to the slave robot in order to beperformed. The slave robot, when interacting with the environment, sends orreflects the contact force to the master robot so that, with a proper control law,the human operator may be able to perceive, with some degree of fidelity, theremote environment.

It is well-known that in teleoperation systems stability is mainly affectedby communication time-delays. However stability against time delays is notenough, but it is necessary to provide certain level of transparency between thehuman operator and the remote environment. In bilateral teleoperation, trans-parency refers to the matching degree between the impedance perceived by theoperator and the environment impedance. Along with stability, transparency isa major objective of bilateral teleoperation system design, under any operatingconditions and for any possible environment.

Thus, a complete analysis of bilateral teleoperation should include time-delayed systems; with this objective several schemes have been proposed, mostof them for linear systems and some for the nonlinear case. Particularly, asignificant interest on sliding-mode control techniques has become attractive inthe control research community worldwide. One of the most interesting aspectsof sliding-mode is the discontinuous nature of the control action, whose primaryfunction is to switch between two different structures such that a new typeof system motion, called sliding-mode, exists in a manifold. New results onsigh order sliding-modes allows to design chattering-free control to enforce aninvariant manifold. However, it requires velocity measurement to build suchmanifold.

State of Art. Sliding-mode controls has been used extensively in roboticsto deal with parameters uncertainty, model perturbations and system distur-bance. There are several applications on which sliding-mode control based onimpedance scheme is a viable solution. For example in [5] a robust sliding-mode control for time-delay systems with mismatched parametric uncertainsis proposed. In [6] an enhanced sliding mode control for a pneumatic master-slave teleoperation systems is discussed, a bilateral teleoperated system withan observer application in robots is also discussed in [7] in which a high ordersliding-mode control is implemented to avoid chattering. The work presented in[8] deals with the problem of a master-slave teleoperation robotic system witha particular application to robot surgery, and propose a sliding mode controlwhich guarantees robustness. Nevertheless, some works ensure that robustnessagainst time delay and transparency are two conflictive issues, hard to concili-ate. Therefore, almost all works that deal with transparency assume that thedelay is null or negligible [9] [10].

Observers has been used to deal in control schemes with time-delays. In [11]a delay identification scheme based on variable structure observers is proposed,an observer-based control for a class of triangular nonlinear systems is proposedin [12], where it is shown that a parameterized linear feedback that uses theobserver states can stabilize the system whatever the size of the delay. The caseof adaptive observer for time-delay in nonlinear systems in triangular form is

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treated in [13]. Recently, nonlinear observers based on super twisting sliding-mode techniques became an interesting option to be used in several kinds ofsystems. Thanks to the robustness properties and the finite-time convergenceof velocity estimation errors this technique is very attractive. Several works havebeen published using this technique [14]. In the observer design based on slidingmode methodology, the systems trajectories are constrained to reach and stay,after a finite time, on a given sliding manifold for which the output error is zero.The sliding motion provides an estimation of the system state, asymptotically orin finite time. The finite time convergence property of sliding mode observers isoften desirable in the framework of observation, and particularly for the purposeof observer-based controller design for nonlinear systems. The observer can bedesigned separately from the controller and the separation principle does notneed to be proved [15], which is an interesting property useful to exploit inteleoperation systems. This algorithm has probe to have good performance formechanical systems as shown in [16] or in electromechanical systems such asmotors [17].

Contribution . In this paper, a second order sliding mode impedance con-trol for the slave system is designed to track the time-delay master trajectories,of the nonlinear bilateral teleoperation system. We propose a sliding modeobserver-controller scheme, which guarantees the tracking trajectory generatedby the master system under an unknown constant time-delay and without usingvelocity and acceleration measurements of the slave system, avoiding expensiveand bulky sensors. The proposed teleoperation scheme guarantees transparencyunder the assumption that time delay is null. However, in the presence of timedelay, the same scheme ensure stability with a certain degree of transparency.Furthermore, experimental results are obtained in order to illustrate the feasi-bility of this scheme in presence of sensor noise, communication and hardwarecommune issues. The effects of the time delay introduced in the teleoperationsystem are discussed and analyzed experimentally for different time-delays.

Paper structure . The paper is organized as follows. In Section 2, thedynamical models of the bilateral teleoperated system is introduced. Next, inSection 3, the proposed second order sliding mode impedance control for theslave system is designed to track the master trajectories. Then, to implementthis controller an observer is given in Section 4 in order to estimate the ve-locity and the acceleration of the slave system. Experimental results for thecontrol-observer bilateral teleoperation scheme are presented in Section 5. Fi-nally, concluding remarks are given in Section 6.

2. Teleoperation System

In a teleoperation general setting, the human imposes a force on the mastermanipulator which in turn results in a displacement that is transmitted to theslave that mimics that movement during free-motion. If the slave possess forcesensors, then it can reflects to the master reaction forces and position duringcontact task being performed in a force master control, then the teleoperator issaid to be controlled bilaterally. Although reflecting the encountered forces back

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Figure 1: A block diagram of bilateral teleoperation

to the human to stimuli tactile senses along with visual senses or instrumentsto increase awareness in teleoperation, it may cause instability in the system ifdelays are present in the communication media [22].

The dynamics of the general nonlinear master/slave systems are the followingEuler-Lagrange system equations.

Mm(qm)qm +Cm(qm, qm)qm + g = h + um (1)

Ms(qs)qs +Cs(qs, qs)qs + g = us e (2)where qi, qi, qi Rn are the joint generalized positions, velocities and accel-erations; Mi(qi) Rnn stands for the inertia matrices; Ci(qi, qi) Rnnis the Coriolis and centrifugal effects, which are defined using the Christoffelsymbols of the first kind; g Rn represents the gravity force; and ui Rn arethe control signals where subscript m and s denote the master and the slave; fhand fe are the force applied at the master by the human operator, and the forceexerted on the slave by the environment, respectively. It is assumed that fhis upper bounded, and none mathematic expression is consider for the humanoperator. It is also assumed that the manipulators are composed by actuatedrevolute joints and that gravitational forces and friction are present, in contrastto [23]. In this dynamical models the inertia matrix is lower and upper bounded,i.e. 0 < m{Mi}I Mi(qi) M{Mi}I < . The Coriolis and inertia arebounded as |Ci(qi, qi)qi| kCi |qi|2qi, with qi, qi Rn and kCi R>0. Thisbilateral teleoperation system scheme is given in Figure 1. In this diagram, theslave block contains the slave manipulator, observer and controller. Also, thediagram shows how the position and force of the master are transmitted to theslave and the contact force of the slave is sent to the master through the com-munication channel, which introduces a time delay in the signals. Then, thesignals from and to the channel are related as

qdm(t) := qm(t T1), qdm(t) := qm(t T1)fdh(t) := fh(t T1), fde(t) := fe(t T2)

where qdm, qdm, and f

dh are the position and velocity of the master, and the

force exerted by a human operator, respectively, which are reflected to the slavethrough the communication channel; fde is the external force at the slave re-flected to the master through the communication channel; T1 is a time delay

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of the signal flowing from master to the slave, and T2 is in the opposite direction.

It is convenient to scaled up or down the communication signals by somefactors depending on teleoperation tasks. Using the scale factors, the pos-tion/velocity command to the slave and the force signal to the master are mod-ified such that

qds = Kpqdm, fh = Kf f

de

where Kp and Kf are diagonal matrices representing position and force scale

factors, respectively; and qds and fde are the position of the slave and the force

reflected by the environment, respectively, which are reflected to the masterthrough the communication channel.

3. Impedance controllers

In this section, an impedance controller and a sliding-mode-based impedancecontroller are designed for the master and the slave, respectively. By using animpedance control, the desired characteristics between the human force and theexternal force can be selected appropriately. The force-controller master devicereflects to a human operator the contact forces between the slave and its envi-ronment while the position-controlled slave follows the trajectory commandedfrom the master.

3.1. Impedance controller for the master

A distinction between impedance control and the more conventional ap-proaches to manipulator position control is that the controller attempts to im-plement a dynamic relation between manipulator variables such as end-pointposition and force rather than just control these variables independently. Itmight then be conclude that what is required in general is the control of a vec-tor of interaction forces, such as in this case are fe and fh.

Consider the following general master control structure

um = fh +Cm(qm, qm)qm +Mm(qm)M1m (qm) = fh Kf fde Cmqm Kmqm

(3)

and Mm, Cm, Km > 0 are desired inertia, damping coefficient, and stiffness,respectively, of a desired impedance, with Mm a diagonal matrix.

Substituting (3) into (1), it follows that the closed-loop master system isgiven by

M(qm)qm + C(qm, qm)qm + Kmqm = fh Kf fde (4)As it can be seen (3) imposes a desired impedance dynamics in the master

teleoperator by means of canceling the physical dynamics of master system and

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placing a new one given by M(qm) and C, between the speed of the masterand the linear combination of the human force and the delayed contact force.In practice, these parameters should be selected according with the physicalproperties of the robot manipulator to be controlled.

3.2. Sliding-mode based impedance controller for the slave

Under a similar rationale as the master controller, consider the slave con-trol design based on second order sliding mode approach to produce a desiredimpedance behavior modulated by the environmental contact forces, robust tounknown time-delay. Thus, a slave controller based on sliding mode control tech-niques will be designed [7]. To this end, consider the desired slave impedance

Ms(qs)qs + Cs(qs, qs) qs + Ksqs = fe (5)where Ms, Cs, Ks > 0 are the desired inertia, damping, and stiffness, respec-tively; and

qs := qs Kpqdm, qs := qs Kpqdm, qs := qs Kpqdmare the slave tracking errors for acceleration, velocity and position, respectively.Now, in order to design a slave controller such that the resulting closed loopsystem has the same closed loop dynamical behavior of (5), then we introducethe following sliding surface

=

t0

Ie()d +Ki

t0

0

sign(Ie())dd (6)

where

Ie = Ms(qs)qs + Cs(qs, qs) qs + Ksqs + fe = 0 (7)

Ki > 0 is a diagonal matrix.Then, a stabilizing slave controller us attains the desired slave impedance is

given by

us =Ms(qs)KpM1m Ms(qs)M1s (qs)

+ fe +Cs(qs, qs)qs Kg(8)

where,

= Cs(qs, qs)[qs Kpqdm] + Ksqs + fe

= Cm(qdm, qdm)qdm Kmqdm + udm Kpfdde+Ki

t0sign(Ie())d

(9)

and fdde = fe(t 2T ), the superscript dd stands for the round trip delay 2T ,Kg > 0, and sign() is the discontinuous signum function. Where term Kg isadded to achieve stability as will be seen afterwards.

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Notice that (6) requires acceleration measurement because Ie depends onacceleration and terms depending on the time-delay signals of the master system.To deal with this inconvenience, as will be seen in section 4, acceleration andvelocity will be estimated by means of super twisting observer.

3.3. Stability analysis

Now, we analyze the stability of the slave system in closed-loop with theslave controller (8). For that, the resulting closed-loop system obtained from theslave system (2) in closed-loop with the controller (8), and after straightforwardcalculation, it follows that

= MsM1s Kg (10)Now, defining the following candidate Lyapunov function

V () = 12> (11)

Then, taking the time derivative along (10), we have

V () = > = >MgKg (12)

where Ms,Kg are definite positive matrices. Let = min{MsKg}, then

V () > 2V () (13)Thus, we obtain

V ((t)) V ((t0))e2(tt0)

which proves that the trajectories of closed-loop slave system converge expo-nentially to = 0. Hence Ie = 0, which satisfies the desired slave impedance.This can be summarized as follows.

Proposition 4.1. Consider the slave system (2) in closed-loop with thecontrol (8). Then, the trajectories of slave system converges exponentially i.e.qs 0.

Remark 1. The robust impedance controller can be achieved by designinga sliding-mode controller such that the desired impedance model becomes ex-act on the sliding surface. The proposed controller (8) uses a robust propertyof a sliding control against uncertainties such as parametric uncertainty andunmodelled dynamics. The controller (8) can be modified using

us =Ms(qs)KpM1m Ms(qs)M1s (qs)

+ fe +Csqs Kgsign()(14)

Then, the proof of the proposition is

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V () > || (15)If the sliding condition is satisfied, then the slave system trajectories tends tothe sliding surface , where is a strictly positive constant

4. Observer design

Most of the control schemes proposed in the literature assume that velocitiesand accelerations can be measurable. However, to implement the impedancecontrol law (8), it is necessary to have access to velocity and acceleration of theslave system. Then, in order to avoid expensive and bulky sensors which mayadd noise to the system, an observer is proposed, which allows to avoid usingsensors measuring velocity and acceleration.

Several observer design for estimating the state of a nonlinear system havebeen reported in the literature. However, one the most attractive approach isthose based on sliding-mode approach thanks to its robustness and convergencein finite-time.

In this paper, we propose an observer based on sliding-mode approach toestimate the velocities and accelerations to solve the problem of controlling ateleoperation system (see [14], [24]).

Let us consider a nonlinear system in triangular form

:

{xj = xj+1, for j=1,..., n-1.xn = (x) + (x)u

(16)

where x = [x1, , xn]T Rn is the state vector, y = x1 R is the outputvector and u R is the unknown input. (x) and (x) are bounded smoothscalar functions. Now lets assume that the state of the system is uniformlybounded, i.e. t > 0, |xi(t)| < di, and t > 0. This means the state and itsderivatives are bounded. Then, the following system O is an observer for system(16)

O :

x1 = x2 + 1 |e1|1/2 sign(e1)x2 = 1sign(e1)x2 = E1

[x3 + 2 |e2|1/2 sign(e2)

]...

xn1 = En3n2sign(en2)xn1 = En2

[xn + n1 |en1|1/2 sign(en1)

]xn = En2n1sign(en1)xn = En1

[ + n |en|1/2 sign(en)

] = En1nsign(en)

(17)

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where ei = xixi for i = 1, . . . , n; with x1 = x1 and[x, ]T

=[x1, x2, . . . , xn,

]Tis the output of the observer. For i = 1, . . . , n 1; the scalar functions Ei aredefined as

Ei = 1 if |ej | = |xj xj | ,j i; else Ei = 0 (18)where is a small positive constant and i and i > 0; are the observer gains.The convergence of the state observation error is obtained in (n 1) steps infinite time.

The recursive scheme based on differentiator given in equations (17) is usedto reconstruct the non-measurable variables. The sliding mode differentiatorprovided an exact differentiation using a recursive method with a finite-timeconvergence. The proof of convergence in finite time of this observer follows thesame steps given in [15]. Next, following the same ideas we extend the aboveobserver design for the class of mechanical systems considered in this paper.

Let us consider the differential equation of slave system (2). Introducing thefollowing change of coordinates Xs2 = qs; Xs2 = qs, then the slave system canbe rewritten in the following state space representation

s :

{Xs1 = Xs2Xs2 = M1s (Xs1) {Cs(Xs1,Xs2) + us fh}

(19)

In order to design a sliding mode observer for slave system (19) estimating thevelocity and the acceleration, equations can be written in a canonical form suchthat achieve conditions presented in (16).

s :

{Xs1 = Xs2Xs2 = F(Xs1,Xs2, fh) +

(20)

with

F(Xs1,Xs2, fh) = M1s {Cs(Xs1,Xs2)X2 + fh} (21)

and

= M1s {us} (22)Suppose that the system states are assumed to be bounded, then there ex-

ists a constant f such that the inequality F(x1,x2, fh) < f holds for anystate. Furthermore, is considered as an uncertain term depending on thetime-delayed signals which are assumed uniformly bounded in a compact set,i.e. . It is clear that the slave system is observable.

The aim in this section is to design an observer for the slave system swhich converges in finite-time to the actual states of the system, when only theposition and the force exerted on the slave by the environment are available.

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The proposed super twisting observer has the following form

O :

Xs1 = Xs2 + 1

|Xs1 Xs1|sign(Xs1 Xs1)

Xs2 = 1sign(Xs1 Xs1)Xs2 = E1[1 + 2(

|Xs2 Xs2|)sign(Xs2 Xs2)]

1 = E22sign(Xs2 Xs2)

(23)

where X1 and X2 are the state estimated of the state vectors X1 and X2,1 = diag{1,1, ..., n,1}, 2 = diag{1,2, ..., n,2} and 1 = diag{1,1, ..., n,1}and 2 = diag{1,2, ..., n,2} are the gains of the observer,|Xs1 Xs1| = diag{

|xs1,1 xs1,1|, ...,|xs1,n xs1,n|},sign(Xs1 Xs1) = diag{sign(xs1,1 xs1,1), ..., sign(xs1,n xs1,n)}, for i =1, . . . , n, with Xs1 = Xs1.

Observer O estimates velocity xs and acceleration xs of the slave system,which are very difficult to measure due to noises, using only measures of positionxs and environment force fe. Notice that the time-delay signals appear in thesystem in such a way that it can be concentrated in a term which can be boundedby a constant. Now, we can establish the following result about the convergenceof the observer.

Proposition 4.2 Consider slave system (19) and suppose that only posi-tion Xs1 and environment force fe are available for measurement. Under theassumption of the time-delayed signals are bounded and the time-delay is con-stant but unknown, the system (23) is an observer for system (19), where thestates of the observer converge in finite-time to the states of the system (19).

Proof. The proof of convergence of this observer, taking into account thetime-delay present signals sent by the master system to slave system, can bestraightforward proved following the same procedure given in [14].

The following theorem ensures the stability of the closed-loop system usingthe observer-controller sliding mode scheme.

Theorem 4.1. Consider the slave system (19) in closed-loop with the impedancecontrol (8) and using estimated states given by the sliding mode observer (23).Then, the closed-loop slave system is exponentially stable, under time-delay sig-nals of the master slave. Furthermore, the slave system tracks the time-delayedtrajectories of the master system.

Proof. Since the finite-time convergence of the observer allows to designthe observer and the control law separately. Then, the separation principle issatisfied. Furthermore, since the slave control has been designed to stabilize theslave system. Then, the stability of the closed-loop system is proved.

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5. Simulation Results

Now, in order to illustrate the proposed methodology, and to simplify thesimulation analysis, consider a 1-dof system described by

mmqm + csqm = fh + um (24)

msqs + csqs = us fe (25)Now, simulation results are shown in order to illustrate the performance of

the proposed observer-controller scheme, assuming that the time delay is con-stant and unknown is shown. Then, the following master and slave controls aredesigned:

Master Control

um = fh + cmqm+mmmm

{fh kffde cmqm kmqm

}(26)

Slave Control

us = msms

{cs qs + ksqs + fe + ki

t0

sign(Ie())d

}+msmm

kp{fdh kffdde cmqdm kmqdm

}+ fe + csqs kg

(27)

Now, writing system (2) is in a state-space form. Then, the sliding mode ob-server for system (2) is given by

O :

x1 = x2 + 1

|x1 x1|sign(x1 x1)x2 = 1sign(x1 x1)x2 = E1[ + 2(

|x2 x2|)sign(x2 x2)] = E12sign(x2 x2)

(28)

The parameters applied in the simulation for master and slave system, arepresented in Table 1. The delay in the communication block between these twosystems are considered is equal to T1 = 1 s. Thus, in the case of double delayused in master system for control its value for simulation is T1 + T2 = 2 s. Inthe same way parameters applied in control laws at master and slave, as wellas in observer are presented in Table 2. Furthermore, in practical situationshuman operator applied a non continuous force to master systems as reference,so the force applied by the human in simulation is supposed to be a pulse signalwith amplitude 5 and a period of 20 seconds, and the environment torque fe isa sinusoidal. Moreover, the initial conditions used for the system and observerare given in Table 3, E1 is selected according to condition (18) and choosing = 0.01.

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Figure 2: Slave system under time delay, T = 1 s

Figure 3: Observer to slave system

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Simulations results can be seen in the following figures. In figure 2 is posibleto see how slave system is following the trajectories of master with a time-delayof 1 second as expected, the tracking error increase when a fast dynamic isgenerated by the operator fh. Although scheme fails in produced a perfecttransparency, the proposed scheme achieves to provide a good approximationof remote environment, under a constant and unknown time-delay. Moreover,the control shows a good performance using the state information provided bythe sliding modes observer, which in a practical experiment will allow us avoidthe use of some sensors. The figure 3 shows specifically how observer rapidlyreach slaves states when it stabilize. Since there is no signal at the beginningon the control during simulations because of time-delay, us presents large over-shoots during the transient. After the transient has finished, control leads slavesystem to converge to master trajectories. Based on further simulations we canshow that the amplitude of this transient period that appears in the observerestimated velocity is related with the magnitude of the time-delay affecting thebilateral system. Finally, controls signals can be seen in figure 4 where it can beobserved no chattering appears, in contrast to classical sliding mode controllers.

Table 1: Master and Slave system parameters

Master Slave Unitsmm 1.7 ms 7 kgcm 0.4 cs 0.9 N s m1mm 1.9 ms 0.3 kg m2cm 2 cs 0.5 N s m1km 0.01 ks 15

Nm

kf 0.9 - - N m

Table 2: Control and Observer parameters

Variableskp = 10.69 ki= 1 kg = 501= 10 2= 10 1 = 202 = 1

Table 3: Initial Conditionxm1 = 1 x1 = 1.1 xm2 = 0x2 = 1 xs1 = 0 x2 = 2

xs2 = 1 = 1

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6. Experimental Results

In this section, we present some experimental results obtained when thepropose observer-controller scheme is implemented in a teleoperation platformconsisting of two 1-DOF robot manipulators connected to a PC through dataacquisition hardware as is shown in Figure (5). In platform, an NI 6033E ac-quisition card is used with LabView to generate the algorithm environment. Acouple of servopacks SGDH-02BE Yaskawa 100V AC, together with two servo-motors with embedded encoders SGMAH-02F41 and a JR3-67M25A force andtorque sensor.

It is worth mentioning that these results can be extended to a robot manip-ulator with a higher number of DOF, i.e. the multi-variable case. Experimentalresults are obtained using a robot of 1-DOF, showing how the proposed schemeis implementable in practice and robust under uncertainties. Position encoderare used on each motor for position measurements, but no velocity sensor inthe slave side. We assume that in master side, a sensor for velocity is available.In practical, many master robots in teleoperation are more often equipped withsensors than remote stations.

To implement the proposed observer-controller scheme (26) (27) (28), theparameters used for the 1-DOF experiment are presented in Table 4. For thecomplete experiment, the sample time was fixed to T1 = 0.001 seconds. Fur-thermore, the noise introduced by the encoder can be compensate considering asuitable selection of gains in the observer, so a correct selection of the observersgain is very important. The observer gain parameters were chosen experimen-tally to ensure a good state estimation, this step can be partially done by simu-lation but with a data vector obtained from real robot manipulator movement.Furthermore, impedance desired parameters were chosen to be similar to thosepresented by the robot manipulator. This is important, since a wrong selectionof impedance desired parameters can make unstable the complete scheme pre-sented. Observer gains and controls parameters used in this case are presentedin Table 5. Moreover, E1 is selected according to condition (18) and choosing1 = 0.01 and 2 = 0.05.

Table 4: Master and Slave system parameters

Master Slave Unitsmm 0.0006 ms 0.0006 kgcm 0.0658 cs 0.0658

N sm

mm 0.0005 ms 0.0005 kg m2cm 0.2 cs 0.15

N sm

km 0.9 ks 1.5Nm

kf 0.1 - - N m

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Figure 4: Control signal for each systems. Where it can be observed no chattering appears,in contrast to classical sliding mode controllers.

Figure 5: Teleoperation platform on CINVESTAV, which consist of two robots of 1-DOF.

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Several tests have been done taking into account different values of timedelay. Experimental results are obtained considering a constant time-delay of2s. The scale use in all figures were chosen arbitrary for best the compressionof system and observer behavior.

Figure 6: Position and Velocity of master and slave systems under a time delay of T = 2s s

First, the implementation of the proposed scheme, for tracking a desiredreference generated by fh applied manually to the master robot, is shown infigure [6]. Furthermore, a cell force was used to measure the contact forcebetween slave system and the object to manipulate or environment. The forcefe affects the trajectory of the master which is send to the operator that feelthe remote force 8. This is clearly seen in the form of the positive part of slaveposition, it never reach position of master since remote system is sensing a valueof fe.

Experiments show that if contact with the object is too strong, it couldgenerate a bouncing at masters sides if operator is not applying any force atthe moment fe is received, specially under high time delay. In figure can be seen

Table 5: Control and Observer parameters

Variableskp = 1 ki= 1 kg = 0.0011= 30 2= 75 1 = 2502 = 150

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how slave [6] system tracks the reference generated by the master, reference isreached with the expected constant delay of T1 = 2 s. Moreover, the proposedcontrol shows a good performance using the state information provided by thesliding mode observer, which allows avoiding the use of velocity sensors. Controllaws behaviors of the master and slave systems can be seen in figure 7.

In Figure 9 the performance of the ST observer is shown, where we can seean important the velocity error, which is related to a poor measure of positionand acquisition time, as expected observer dynamics filter this noisy effects,increasing this apparent error.

To reenforce this results, other experiments with different parameters arepresented. First consider figure 10 were a human operator applies a sinusoidalforce fh to the system. In this case a time-delay of T1 = 250ms is applied.Then consider case when fh is completely a random signal with a time delay ofT1 = 250ms, result can be seen in figure 11.

7. Results Discussion

From experimental results, some observations can be made. The proposedscheme can be implemented in a teleoperated system within the assumption pre-sented in theory of section 4. Weve find high order sliding-mode control witha good performance even considering parametric uncertains. For the properwork of impedance control is important to select adequately the values of desireimpedance, this parameters cant be too much different from physical parame-ters, if so a good behavior of the scheme is not ensured.

The observer had shown to have good performance and robustness to para-metric uncertainties, moreover if gains are selected properly a fast convergenceis ensured. Furthermore, experiments also shown that sliding modes observerprovides a good estimation of the state even at relative high speeds for a proto-type robot manipulator but a high disturbance could make the scheme unstable,if it is operated at a relative high speed. Special attention must be given to makea good selection of gains, if not estimation of the super twisting are not goingto be good enough to make the complete scheme to work.

We also find out that complete scheme gives a good transparency to humanoperator depending on the magnitude of the time delay. Since there is nosignal at the beginning on the control during simulations because of delay, uspresents large overshoots during the transient. After the transient has finished,control leads sleave system to converge to master trajectories as expected undersimulations. Furthermore, the performance of the scheme prove to be stableunder sensor noise, a common situation under a practical environment, sinceencoders and cell force add noise to sliding mode observer were full schemestability was not affected.

8. Conclusions

In this paper, a nonlinear control-scheme for a n-DOF nonlinear bilateralteleoperated system, based on sliding mode techniques in presence of constants

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Figure 7: Control signal for master and slave system. Impedance base control implements anew dynamics in the system. Second order sliding based control reduce the chattering effect.

Figure 8: Force fe applied to the slave system.

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time delays in the communication channel, has been presented. The proposedscheme combines a second sliding mode control based on impedance to trackthe time-delayed signals sent by master system and a sliding mode observe,for estimating the no measurable components of the state of the slave systemwithout using any velocity sensor.

Furthermore, experimental results have illustrated the good performanceand transparency of the proposed control-observer scheme under time delaysand sensor noise.

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Figure 9: Velocity estimation error.

Figure 10: Position and velocity estimation error, for a sinusoidal signal with a time-delay ofT1 = 250ms.

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Figure 11: Position and velocity estimation error, for a random signal with a time-delay ofT1 = 500ms.

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