LIMITS OF HUMAN INTERACTION IN DYNAMICALLY SIMILAR

TELEOPERATION SYSTEMS

Under Unknown Constant Time Delay with Impedance Control

V. Parra-Vega, V. P. Garc´ıa-Alvizo, E. Olgu´ın-D´ıaz

Robotics and Advance Manufacturing Group, CINVESTAV, Carr. Saltillo-Monterrey, Km 13.5, Saltillo, M´exico

L. G. Garc´ıa-Valdovinos

Centro de Ingenier´ıa y Desarrollo Industrial, CIDESI, Quer´etaro, M´exico

Keywords:

Teleoperation, Haptic.

Abstract:

Bilateral teleoperation system are prone to instability coming out from the time-delay introduced by the inde-

terministic communication channel. This problem has been subject of intensive research under the assumption

of non-equal master-slave teleoperators, however, what are the implications of dynamically similar teleopera-

tion system (DSTS), is there simpler stability relationship and trade offs among several involved system and

feedback parameters? When we consider a linear DSTS system, there arises the question whether there is an-

alytically any advantage, as it was observed heuristically in several experiments (Cho and Park, 2002). In this

paper, the stability analysis of such system is reported under an impedance control scheme (Garcia-Valdovinos,

2006) when the delay is considered constant but unknown. by applying the Llewellyn’s and Raisbeck’s crite-

ria, it is found and explicit and straightforward relationships between the dynamic and kinematic scaling and

the stability of the system. This result explicitly suggests clearly guidelines among key factors, such as time

delay, desired velocities and feedback gains in terms of the scaling parameters, arises a clear advantage when

dealing with dynamically similar systems. This explains why the transparency of the teleoperation system

is improved by augmenting/reducing the dynamic/kinematic scaling factor, for given desired frequency, time

delay and feedback gains. Simulations and preliminary experimental results illustrate different cases subject

to a number of conditions, which can be very useful to design a physical teleoperation system. A preliminary

mechatronics design is presented.

1 INTRODUCTION

A bilateral teleoperation system is composed of a

master robot and a slave robot, with a human operator

commanding the master robot in order to produce the

desired position and contact force trajectories

for the

remotely located slave robot. In turns, the slave robot

follows these trajectories so as to produce contact

forces to be sent to the master robot as desired force

trajectories. In this way, a force/position-force con-

trol system is implemented in the master/slave station

with a communication channel introducing delayed

position and forces signals (Hokayem and Spong,

1984). It is well known that the source of instability

of bilateral teleoperation system appears because the

time-delay introduced by the indeterministic commu-

nication channel is not passive (Q.W. Deng, 2007).

Depending whether is in contact or not.

Then, the limits to achieve human operator stable in-

teraction with the slave robot, placed at a remote lo-

cation, through the master robot, becomes an issue, in

particular successful teleoperation requires a certain

degree of transparency

and/or telepresence

. Stable

interaction is intuitive and easier when mechanical

teleoperators are alike? What are the limits of sta-

ble interaction for dynamically similar teleoperation

system (DSTS) subject to time-delay in the commu-

nication channel?

Furthermore, when we consider a linear DSTS,

there arises the question whether there is analytically

any advantage for being the master and slave robot

dynamically similar, through the kinematic and dy-

Manipulation of the slave robot without any dynamics

involved in between.

Sensation of being physically in the remote environ-

ment.

102

García-Alvizo V., Parra-Vega V., Olguín-Díaz E. and García-Valdovinos L. (2009).

LIMITS OF HUMAN INTERACTION IN DYNAMICALLY SIMILAR TELEOPERATION SYSTEMS - Under Unknown Constant Time Delay with Impedance

Control.

In Proceedings of the 6th International Conference on Informatics in Control, Automation and Robotics - Robotics and Automation, pages 102-109

DOI: 10.5220/0002216201020109

 SciTePress

Figure 1: Basic Bilateral Teleoperation Scheme.

namic scaling parameters. It seems DSTS is preferred

to carry out over dissimilar teleoperation systems, as

it was observed heuristically in several experiments

(Cho and Park, 2002).

On one hand, telepresence depends of the degree

of transparency, which can be understood as the si-

multaneous convergence of the position and force er-

ror between the master and slave robot. However it

also depends on the subjective cognition of the hu-

man operator of being there, which depends, among

other aspects, on visual aids of the remote environ-

ment, kinesthetic coupling, the ability to deal with

delayed signals and the man-machine interface. How

does these factors are related in DSTS to guarantee

stable teleoperation?

In this paper, motivated by the empirical observa-

tion that a DSTS is easier to handle, it is argued that

dynamic and kinematic similarity introduce a clear

trade-off of some of these aspects, so it is reason-

able to expect a simpler a trade-off. Analytical re-

sults are found based on an impedance control scheme

(Garcia-Valdovinos, 2006), when the delay is consid-

ered constant but unknown.

1.1 Motivation

When the master and slave teleoperated robots are re-

lated linearly by scaling factors of position and force,

as well as scaling factors on dynamic and kinematic

parameters, a relationship between geometry, power

and perception arise to give to the operator a linear

relationship behavior. Such scaling factors might give

to humans the ability to increase their commanding,

perceptual and cognitive skills in different teleopera-

tion tasks, depending on the task undergoing. It has

been observed experimentally that by tuning prop-

erly these scaling factors a human operator improves

his ability to better teleoperate such system (Cho and

Park, 2002). Can DSTS yield teleoperation tasks with

greater dexterity? What are the trade-offs? It is of in-

terest to understand deeper this phenomena using for-

mal dynamical system tools to analyze properties of

stability of dynamically similar teleoperation system.

1.2 Contribution and Organization

Our basic hypothesis is that as long as the human per-

ceives linear correlated variations in both teleoperated

robots, he can improve the command of the closed-

loop bilateral teleoperation system since spatial and

temporal attributes of the visual remote location and

kinesthetic coupling will vary linearly without distor-

tion. So cognitively, the human can quickly learn to

command the task with greater dexterity. Addition-

ally in this paper we deal with unknown time delay

so we design a novel controller to deal with unknown

constant time delay (Garcia-Valdovinos, 2006), (Cho

and Park, 2002). A computed-torquecontroller is em-

ployed in the master station and a computed torque

second order sliding mode controller in the slave sta-

tion is proposed to produce a desired impedance in

closed loop. Then, absolute stability theory and pas-

sivity is used to analyze the closed-loop stability prop-

erties and therefore the limits of human-teleoperation

stability and thus we found the stability trade-offs. To

this end, a review is presented in Section 2. Then, in

Section 3 the dynamically similar coupled system is

presented, while controllers are explained in Section

4. With this result at hand, absolute stability using

Llewelyn criteria (Llewellyn, 1952), and passivity us-

ing Raisbeck criteria (Raisbeck, 1994), are analyzed

in Sections 5 and 6 respectively. The Llewellyn’s

analysis reveals that a good choice for dynamic scal-

ing factors give us the opportunity for greater bounds

on position and force scaling to execute tasks of high

performance. A quality criterium for transparency

analysis is also presented in Section 7. Simulations

on a 1 DoF teleoperation systems are shown to illus-

trate how this dynamic scaling factor improve the per-

formance of the system, shown in Section 8 to better

understand the numerical performance. Final conclu-

sions are given in Section 9.

LIMITS OF HUMAN INTERACTION IN DYNAMICALLY SIMILAR TELEOPERATION SYSTEMS - Under

Unknown Constant Time Delay with Impedance Control

103

2 DYNAMICALLY SIMILAR

DELAYED TELEOPERATION

SYSTEM

A dynamically similar teleoperation system has con-

stant scaling factors which relate kinematic and dy-

namic parameters of the master and slave robots

(Goldfarb, 1999). This similarity between the sys-

tems is poorly understood so far, since there is not

theoretical apparent evident advantage to work out

with bilateral system, despite some analysis reported

in (Li and Lee, 2003), where the advantages has not

been addressed properly in terms of explicit trade-

offs of feedback gains, system parameters, desired

trajectories and time delay. We surmise in this pa-

per that DSTS improvessigniﬁcantly the ability of hu-

mans operating the master teleoperator to carry out ef-

ﬁciently teleoperation tasks at remote environments,

when there is an unknown time delay involved in the

communication channel.

Impedance control has been explored in (Cho and

Park, 2002) to enforce a desired impedance dynam-

ics in closed loop in order to program arbitrarily the

desired impedance parameters. This closed-loop lin-

ear dynamic allows to model the entire system as a

2-port network to relate the force and ﬂows of input

and output, respectively, by an impedance matrix or

an hybrid matrix. This matrix can be used to describe

the stability of the entire system using the Absolute

Stability Theory (K. Hashtrudi-Zaad, 2000), where a

tight relationship between output scaling factors and

impedance parameters can be found to give sufﬁcient

conditions on stability. However, when dynamic scal-

ing factors are introduced, the master and slave robot

dynamics are related by constants, either in kinematic

and/or dynamic parameters, thus, a sort of advantages

can emerge from this relation since a single feedback

parameter appears. In this paper, we offer an analy-

sis in terms of both the Llewelyn’s criteria and Rais-

beck’s criteria, and verify its real time performance,

which demonstrates a clear and intuitive trade off in

terms of scaling factors of the DSTS.

3 DYNAMICALLY SIMILAR

SYSTEM

Consider the dynamics of a linear teleoperation sys-

tem consisting of two n-DoF manipulators decoupled

systems as follows

¨x

+ B

˙x

+ K

= F

+ F

(1)

¨x

+ B

˙x

+ K

= F

+ F

(2)

where ¨x

and ˙x

denote acceleration and velocity of the

robot i,respectively; F

, F

are the control force in-

puts and F

, F

are external forces to the master and

slave systems, respectively; and M

, B

and K

with

i = m, s are the inertia, dampness and stiffness pos-

itive coefﬁcients of the systems. Let K > 0 be the

kinematic scaling factor that relates both master and

slave conﬁguration spaces such that

(3)

This system is said dynamically similar after the co-

ordination (3) if there exists a scalar ζ > 0 such that

¨x

+ B

˙x

+ K

¨x

+ B

˙x

+ K

(4)

Thus, the dynamic parameters of (1)-(2) are related

linearly by

ζM

= M

, ζB

= B

, ζK

= K

(5)

The apparent advantage of this dynamic relationship

has not been well explored in the context of tele-

operation with unknown constant time delay, though

there are a lot of heuristical intuition which leads us

to conclude that these systems would allow greater

kinesthetic coupling with greater manipulability dex-

terity as consequence. To this end, it is introduced an

impedance control system (Garcia-Valdovinos, 2006)

to enforce a 2-port closed-loop desired linear system.

4 IMPEDANCE CONTROL LAW

4.1 1 DoF Teleoperation System

Similarly to (1)-(2), let a 1 DoF master/slave teleop-

eration system be modeled as a mass-spring-damper

system, where external master force F

on the master

is nothing but the human commanding force F

and

the external slave force force F

stands as the environ-

mental contact force F

, then (1)-(2) becomes

¨x

+ B

˙x

+ K

= F

+ F

(6)

¨x

+ B

˙x

+ K

= F

− F

(7)

where negative sign appears in F

due to the positive

convention of the inertial frame axis.

4.2 Impedance Control Law for the

Master

For completeness, the control law (Garcia-

Valdovinos, 2006) is introduced here. Consider

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104

the following master controller

= −F

+ B

˙x

+ K

−

˙x

−

) (8)

Eq. (8) into (6) gives rise to the following desired

impedance equation for the master robot

¨x

˙x

= F

− K

(9)

where positive

are the desired inertia,

dampness and stiffness for the master robot, respec-

tively, and F

= F

(t − T

), being T

the delay from

the slave to the master station. That is, the master

impedance control law enforces a desired impedance

(9) in closed-loop, whose parameters are chosen by

the user depending of a speciﬁc task, such that:

• when the slave robot is not touching the environ-

ment, F

= 0, then (9) becomes a mass-spring-

damper system driven solely by the human force

, notice that in this case the controller is in po-

sition impedance mode..

• when the slave robot is touching the environment,

> 0, then (9) becomes a mass-spring-damper

system driven by force error F

− K

. In this

case, actuators in the master station makes the

human perceives a contact force equal to K

while the human virtually recreates, cognitively,

the surface of the object according to this vec-

tor, through kinesthetic sensations of the scaled

and delayed slave contact force, which arise nor-

mal at the contact slave point and the visual im-

age coming from the slave station. Notice that

when the slave is contact, the master control is in

impedance force control mode.

To achieve such effects, it is necessary to control the

slave robot in impedance position and force control

modes, according to the contact regime.

4.3 Impedance Control Law for the

Slave (Garcia-Valdovinos, 2006)

Similarly to the master controller, the objective in the

slave station is to impose a desired impedance to the

slave robot

··

˜x

= −F

(10)

where positive

are the desired inertia,

dampness and stiffness for the slave robot, respec-

tively. The position tracking error ˜x

is expressed as

follows

˜x

= x

− K

(11)

where x

= x

(t − T

), being T

the delay from the

master to the slave station. Now, let the following

control law for the slave robot be

= −

(

˙x

+ F

+ K

σ)

−1



− K

−

˙x

−



+ F

+ B

˙x

− K

Ω (12)

σ =

sgn(I

(τ))dτ (13)

where F

= F

(t − T

). Notice that the feedforward

term F

allows control without any measurement of

the time delay. In any case, notice that F

is available

for measurement at any time.

Notice that the gain K

is a new control variable

that weights the extended error variable Ω. The pro-

posed sliding surface I

is proposed naturally out of

(10), that is we want (10) to be the attractive conver-

gent manifold, then the extended error manifold is

¨x

˙x

+ F

(14)

Then we can build a high order sliding surface Ω as a

function of the sliding surface I

as follows

Ω =





(τ)dτ +

sgn(I

(τ))dσdτ





(15)

Finally, substituting (12)-(13) into (7) gives rise to the

closed-loop error equation for the slave robot:

Ω = −βΩ (16)

where β =

> 0 is Lipschitz. Consequently, all

closed-loop signals in the slave station are bounded,

enforcing exponential convergence of Ω → 0. There-

fore, this chain of implications means that a second

order sliding mode is enforced, and a sliding mode

arises, at I

= 0, which means that (10) arises in ﬁnite-

time.

A closer analysis shows that the slave impedance

control law enforces a desired impedance in closed-

loop whose parameters are chosen by the user de-

pending of a speciﬁc task, such that:

• when the slave robot is not touching the environ-

ment, F

= 0, F

= 0 and (10) becomes an un-

forced mass-spring-damper system such that ˜x

→

0 and the slave tracks the desired delayed position

and velocities of the master. Notice that when the

slave is not in contact, the slave control is in posi-

tion impedance control mode.

Notice that if (14) converge to zero, then (9) appears

and the human would perceive the desired impedance to

control at will the slave robot.

LIMITS OF HUMAN INTERACTION IN DYNAMICALLY SIMILAR TELEOPERATION SYSTEMS - Under

Unknown Constant Time Delay with Impedance Control

105

• when the slave robot is touching the environment,

> 0 and (10) becomes a mass-spring-damper

system driven by the slave contact force F

. In

this case, actuators in the slave station make that

the slave robot maintains contact (| ˜x

|> 0) while

stays around F

. Notice that when the slave is

in contact, the slave control is in impedance force-

position control mode.

With this result, it is now important to analyze the ab-

solute stability properties to ﬁnd the conditions under

which this result is valid.

5 ABSOLUTE STABILITY

ANALYSIS

With the desired impedance imposed by the con-

trollers (8) and (12)-(13), the closed-loop dynamics

(9) and (10) can be modeled as a 2-port network.

Transforming this dynamic into the frequency domain

and doing some algebra, closed-loop system can be

represented as





= H



−F



(17)

where H is the so called Hybrid Matrix. Using the

relationship (5) the hybrid matrix is built from ele-

ments depending of function of the desired master

impedance parameter as follows

H =

−T

(18)

which is fundamental to carry out the implications

of a unique dynamic scaling factor. To proceed, it

is useful to give the following deﬁnition on 2-port

systems:

Deﬁnition: Absolute Stability Criteria for 2-port

Systems: A two-port system (17)-(18) is absolute sta-

ble if it does not exist a set of impedances for which

the entire system become unstable. If the network is

not absolutely stable, it is potentially unstable. By the

conditions of the llewellyn’s criteria a 2-port network

is absolutely stable if and only if

1. h

and h

have no poles in the right half plane

2. Any poles of h

and h

on the imaginary axis are

simple with real and positives residues

3. For all real values of ω

• Re{h

} ≥ 0

• Re{h

} ≥ 0

• 2Re{h

}Re{h

} − Re{h

} − |h

| ≥ 0

Notice that since the human operator is physically

holding with his hand the master robot, it is imper-

ative to ensure stable behavior, thus it is required to

guarantee the fulﬁllment of previous Deﬁnition. To

this end, notice that conditions 1. and 2. are trivially

satisﬁed with positive impedance parameters. The

third condition, when using (18), becomes:

A. [cos(T

+ T

)ω− 1]K

+ 2ζν ≥ 0

B. K

≤ ζν

(19)

where

ν =

(

ω)

(

−

)

+ (

ω)

(20)

Inequality (19), necessary for the absolute stabil-

ity of the system, shows that the scaling factors of

position, force, and dynamical similarity ζ are critical

for the design and performance of the teleoperation

system.

Therefore, the consequences of introducing a dy-

namic similar system in teleoperation, from the point

of view of Absolute Stability, are:

1. A unique similarity factor ζ is introduced, which

offers a simpler analysis and easy to tune system.

2. The similarity factor ζ allows to derive simpler

conditions of absolute stability.

3. The similarity factor ζ improves the design

methodology of teleoperators based in impedance

controllers. That is, there is a clear trade-off of all

important parameters of the system, depending on

the desired performance (ω), impedance parame-

ters

, position scaling K

and force scal-

ing K

and time delays, a ζ can be found.

4. From (18), the scaling factor ζ allows bigger mar-

gin on other parameters, thus the opportunity to

improve performance based on the physical struc-

ture of the teleoperation system.

5. Due to the fact that both master and slave

impedance parameters are related by this factor,

the whole set of parameters can be expressed in

terms of each other, which minimize the number

of parameters implied in the design process mak-

ing it easier to establish a performance limit.

6 PASSIVITY ANALYSIS

Passivity is a powerful criteria to analysis the ener-

getic coupling of a closed loop system, a more conser-

vative implication in comparison to Lyapunov stabil-

ity criteria, however since the human operator is phys-

ically coupled with a typically mechanical system in

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106

closed-loop, it is important to analyze the passivity of

the closed-loop system.

A two-port network is said to be passive if for

all inputs of energy, the output energy is equal

or less than the input energy. If the network is

not passive, it is active. Raisbeck’s passivity cri-

terion is used to determine the passivity of the system.

Deﬁnition: Raisbeck Passivity Criterion for 2-port

Systems: It is said that a 2 port-network is passive if

and only if

1. The parameters of the hybrid matrix H have no

poles in the right half plane

2. Any poles of the elements of the hybrid matrix on

the imaginary axis are simple and their residues

satisfy the following conditions, for all real values

of ω,

• r

≥ 0, r

≥ 0

• r

− r

≥ 0

• 4Re{h

}Re{h

} − [Re{h

} + Re{h

}]

−[Im{h

} − Im{h

}]

≥ 0

where r

denotes the residue of h

In a similar way, the ﬁrst two items are satisﬁed

with positive impedance parameters, and the third is

satisﬁed if the following inequality is fulﬁlled

+ K

− 2K

cos(T

+ T

)ω ≤ 4ζν (21)

Then absolute stability is a more relaxed stability cri-

teria than passivity. The passivity condition is nec-

essary to assure a complete energetic stability per-

formance of the closed loop system. Thus, we can

choose the dynamic scaling factor in order to have

a greater upper and lower bounds to vary the other

factors of position and force without affect passivity,

because it is of primary interest to maintain passivity

since the human is physically holding the mechanical

master robot.

7 TRANSPARENCY ANALYSIS

In order to determine the transparency of the system,

a quality criterion in teleoperation systems based on

the impedance matrix is derived from equations (9)

and (10),











(22)

The elements of the impedance matrix are in function

of desired impedance parameters and scaling factors

as follows



1+ ζK

−(T



= −

−T

= −

where

ξ =

The ideal transparency is reached by the system when

the (input) environmentalimpedance Z

is equal to the

output human impedance Z

, so that, we have

= Z

(23)

From (22), we have

= Z

+ Z

(24)

where Z

, Z

are the elements of the

impedance matrix Z. Expressing the elements of Z in

terms of the master impedance and substituting them

in (24), it gives

= ξ



1+ ζK

−(T





−(T



− ξs

(25)

Now, we can analyze the transparency of the system

in terms of free motion regime, which means that Z

0 ideally, and constrained motion regime Z

= ∞, in

the worst case, then we have the following:

1. When Z

→ 0, Z

becomes (25), such that when

the dynamic scaling factor ζ is greater, Z

→ 0,

or smaller desired impedance parameters is tuned,

the higher transparency is obtained in free motion

due to

→ 0 ⇒ Z

→

2. In contact tasks Z

→ ∞, ideally so does the output

impedance. In this case, the transparency relation

(24) becomes

→ Z



1+ ζK

−(T



Notice that the dynamic scaling factor ζ is directly

proportional to Z

, hence, the greater it is the better

the transparency is.

LIMITS OF HUMAN INTERACTION IN DYNAMICALLY SIMILAR TELEOPERATION SYSTEMS - Under

Unknown Constant Time Delay with Impedance Control

107

8 SIMULATION

In this section the effect of the dynamic scaling is

shown for a 1 DoF teleoperation system. Simulations

were made in 3 cases:

1. All dynamic, kinematic and force scaling factors

are the unit.

2. The scaling factors of position and force are K

2 and K

= 0.01, respectively, and the dynamic

scaling factor is ζ = 0.1.

3. zeta is increased to 10 and the rest of the parame-

ters are preserved as in case 2.

A smooth force proﬁle was introduced arbitrarily

as the force exerted by the human. This force trajec-

tory was designed in such a way that the teleoperator

goes from an initial position to the contact point arriv-

ing softly with null velocity. This is in order to avoid

large spikes due to hard contact. Once the slave robot

is in the contact point, the human begin to apply an

intermittently force on the constraint 3.

0 200 400 600 800 1000 1200

0.005

0.01

0.015

0.02

0.025

Time (mS)

Position (mts)

Slave robot position

Master robot position

Figure 2: Position error of the master (dashed line) and

slave (dotted line) for Case 1.

0 200 400 600 800 1000 1200

−0.02

0.02

0.04

0.06

0.08

0.1

Time (mS)

Force (N)

Human force

Enviroment force

Figure 3: Human force error (master/dashed line) and con-

straint force error (slave/dotted line) for Case 1.

In Case 1, the chosen scaling factors and the

impedance parameters give us an acceptable perfor-

mance and a stable behavior. The position error be-

tween master and slave in Figure 2 shows that both

robots follow the same constrained trajectory. Due

to the impedance programmed for the master robot

a slight movement toward the constraint is allowed.

However, the slave robot stands along the physi-

cal constraint. With this stable response under con-

0 200 400 600 800 1000 1200

−0.005

0.005

0.01

0.015

0.02

0.025

Time (mS)

Position (mts)

Slave robot position

Master robot position

Figure 4: Position error of the master (dashed line) and

slave (dotted line) for Case 2.

0 200 400 600 800 1000 1200

0.02

0.04

0.06

0.08

0.1

0.12

Time (mS)

Force (N)

Human force

Enviroment force

Figure 5: Human force error (master/dashed line) and con-

straint force error (slave/dotted line) for Case 2.

strained operation, the scaling factor of position is in-

creased to obtain a larger workspace with slave robot

and the force scaling factor is tuned in order to apply

a greater force proﬁle on the constraint and protect the

human to receive a large reﬂected force that could be

potentially dangerous.

In Case 2, the change described in the scaling fac-

tors make the system unstable. The slave robot go

away from its position as can be seen in Figure 4,

while the force at the constraint (see Figure 5) dis-

appears.

In order to handle this behavior, the dynamic scal-

ing factor is increased (Case 3). Then the slave robot

can reach the scaled position and force as shown in

Figures 6 and 7.

The results in simulation show that in case 1, we

obtain a acceptable performance and a stable behavior

in the system but when we try to increase the perfor-

mance changing the values of the others factors the

0 100 200 300 400 500 600

−0.08

−0.06

−0.04

−0.02

0.02

0.04

Time (mS)

Position (mts)

Slave robot position

Master robot position

Figure 6: Position error of the master (dashed line) and

slave (dotted line) for Case 3.

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108

0 100 200 300 400 500 600

−0.2

0.2

0.4

0.6

0.8

1.2

1.4

1.6

1.8

Time (mS)

Force (N)

Human force

Enviroment force

Figure 7: Human force error (master/dashed line) and con-

straint force error (slave/dotted line) for Case 3.

stability of the system is affected (Case 2). By mak-

ing a new choice on this dynamic scaling factor we

can preserve the scaling factors already chosen for an

speciﬁc task without affect the stability

9 CONCLUSIONS

Using a novel impedance controller and advanced sta-

bility tools, precise conditions to guarantee stability,

even in harsh conditions, is proposed for dynamically

similar bilateral teleoperation robotic system. In this

case, this system depends on a constant parameter,

which relates explicitly and clearly a trade off be-

tween stability, passivity and transparency. The con-

troller enforces convergence in ﬁnite time due to the

sliding surface, which is nothing but the impedance

equation, thus the closed-loop system dynamics is en-

tirely governed by the desired controlled equation.

This yields useful boundariesto varyimpedance, scal-

ing parameters and frequency,in terms of the bounded

ime delay, which in turn allows to introduce a desired

performance criteria in terms of surrounding physical

conditions. This result seemingly allows to establish

a simpler methodology to design dynamically simi-

lar teleoperators with a given desired performance in

realistic conditions.

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LIMITS OF HUMAN INTERACTION IN DYNAMICALLY SIMILAR TELEOPERATION SYSTEMS - Under

Unknown Constant Time Delay with Impedance Control

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