558 IEEE TRANSACTIONS ON SYSTEMS,MAN,AND CYBERNETICS—PART A:SYSTEMS AND HUMANS,VOL.42,NO.3,MAY
2012
Adaptive Haptic Control for Telerobotics
Transitioning Between Free,Soft,
and Hard Environments
Dean Richert,Chris J.B.Macnab,and Jeff K.Pieper
Abstract
—This paper presents an adaptive haptic control for
a one degreeoffreedom master–slave teleoperated device.The
aim is to reduce excessive collision forces that occur when there
are signiﬁcant time delays in master–slave communication.The
control design also allows the operator to move the slave in
free space and in a soft medium.Previous approaches to haptic
teleoperation typically design for either movement in a medium
or constrained contact with a solid surface;then,it is up to
the operator to avoid collisions or precisely anticipate collisions.
The proposed control runs on the slave side inner loop,with no
time delay,and tracks commanded forces from the outer loop.A
Lyapunovstable backsteppingwithtuningfunctions design pro
vides a way to ensure smooth forces are applied that guarantee
stability in the presence of unmodeled environmental stiffness and
viscosity.Experiments using a Phantom hand controller inter
acting with simulated environment show that collision forces are
substantially reduced compared to two other control methods.In
collisionfree operation,the performance is comparable to other
methods.
Index Terms
—Adaptive control,force control,haptic interfaces,
Lyapunov methods,radial basis function networks.
I.I
NTRODUCTION
I
N MASTER–SLAVE teleoperation,an operator uses a
masterhuman interface to control a slave robot using visual
feedback,an approach found to be advantageous in robot
assisted surgery [1],[2],space [3]–[5],deep sea exploration
[6],[7],and the construction industry [8],[9].In many ap
plications the system is more effective when operators feel
slave/environment contact forces,
haptic feedback
,for example
delicate surgeries [10].Typical control designs assume the tele
operation is occurring in either an unrestricted soft environment
or in contact with a hard surface that constrains motion.In
this paper,we examine the control problem encountered when
transitioning froma soft (or free) environment to a hard surface
in the particularly difﬁcult,yet common,scenario where time
delays occur in the teleoperation loop [11].
Manuscript received June 4,2010;revised March 1,2011;accepted June 28,
2011.Date of publication October 28,2011;date of current version April 13,
2012.This paper was recommended by Associate Editor W.A.Gruver.
D.Richert and C.J.B.Macnab are with the Department of Electrical and
Computer Engineering,Schulich School of Engineering,University of Calgary,
Calgary,AB T2N 1N4,Canada (email:cmacnab@ucalgary.ca).
J.K.Pieper is with the Department of Mechanical and Manufacturing
Engineering,Schulich School of Engineering,University of Calgary,Calgary,
AB T2N 1N4,Canada.
Color versions of one or more of the ﬁgures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identiﬁer 10.1109/TSMCA.2011.2170066
In haptic teleoperation,the operator typically wishes to
achieve a desired force in addition to a position or velocity
(
desired state
).Although many approaches have been proposed
for the control (see survey in [12]),usually a designer chooses
either the force or state to be reference signals to a closedloop
control.In this type of
4channel
,or 4ch,control [13],a signal
that is not a reference becomes either an openloop command or
an openloop feedback,in which case,it is up to the operator to
close the loop and achieve the desired result.Often,it is force
that is communicated back in a oneway fashion,and it is up
to the user to make sue of this information when commanding
a position,for example [14],although it is not necessarily a
trivial task for a human to combine visual and force infor
mation [15].Both force and state can be reference signals to
the computer control in more advanced designs,for example
[16].The 4ch description lends itself to linear control design
and stability analysis [17],[18].Ensuring closedloop stability
requires knowledge of both operator and environment stateto
force mapping (
impedances
),yet the environment impedance is
typically unknown,variable,and nonlinear;collisions provide
a particular difﬁculty.One solution is to assume that the range
of environment stiffness is limited when designing the control
and that the system can be shut down in case of collision [19],
[20].Another approach introduces gain scaling terms which
effectively tradeoff transparency for stability,widening the
range of allowable environment stiffness [21].Passivitybased
control,resulting in
inputoutput stability
or
unconditional
stability
,ensures stable operation in the presence of arbitrary
passive environments (for example [22]).To prove stability,one
shows each interconnected element is itself passive.Assuming
both the (trained,professional) operator and environment are
passive,one concludes a passive control law is sufﬁcient [23].
Systems typically retain their passivity in the presence of time
delay,yet collisions with hard surfaces may still result in
excessive forces in practice which can easily damage expensive
force sensors.
The constrained motion of the slave robot when it is in
contact with a hard surface can also present stability problems
[24],[25].Modeling the surface as a very stiff spring provides
a worst case scenario,in which case the system becomes
nonminimum phase violating the passivity condition,and high
frequency dynamics may become excited [26].Filtering out the
high frequency content of the control signal or force sensor
measurement may work in practice [27],[28],but stability
analysis of the nonlinear systembecomes difﬁcult.
10834427/$26.00 ©2011 IEEE
RICHERT
et al.
:HAPTIC CONTROL FOR TRANSITIONING BETWEEN FREE,SOFT,AND HARD ENVIRONMENTS 559
This paper proposes a control scheme for the slave side that
will track timedelayed commands fromthe haptic master con
trol transitioning through three scenarios:moving through a soft
medium,moving in free space,and maintaining contact with a
solid surface.The goal is to achieve a smooth transition from
free space to a hard medium,while still providing acceptable
performance in all scenarios and other transitions.
Three ideas form the basic contributions of this work.First,
tracking is based on an augmented error which includes force
error and velocity,eliminating any need to switch between con
trol laws in different scenarios.Second,the Lyapunov backstep
ping technique provides a way to ﬁlter the control signal while
including the effects of the ﬁltering in the stability analysis.
Finally,adaptive neural networks estimate unknown nonlinear
environmental forces,so that an unmodeled environment will
not cause instability.An adaptive tuningfunction design pro
vides neural weight updates,rather than an overparameterized
design,so that adaptation can occur quickly without need of
repetitive training.
II.B
ACKGROUND
In this section,we ﬁrst introduce basic notation for neural
network approximation,and then analyze the ﬁltering effect of
adaptive backstepping procedures using neural networks.These
methods then form the basis of a control scheme for the slave
side of a teleoperation system.
A.Radial Basis Function (RBF) Networks
A universal approximator (UA) can effectively approximate
an unknown and nonlinear function with a degree of accuracy
such that the approximation error is bounded [29].An RBF
network is a special case of a UA and is a weighted sum of
functions,each of which are centered at various points in the
domain
D
,and their sumspans the whole of
D
.Given
n
inputs
in column vector
q
and
m
basis functions in row vector
!
(
q
)
,
the output is
ˆ
o
=
!
(
q
)
ˆ
w
=
m
!
i
=1
!
i
(
q
) ˆ
w
i
(1)
where
ˆ
w
contains the weights of the neural network.Normal
ized Gaussian kernel functions are a common choice
s
i
(
q
) = exp
"
!
(
q
!
c
i
)
2
2
"
2
#
$
m
!
j
=1
exp
"
!
(
q
!
c
j
)
2
2
"
2
#
(2)
where vector
c
i
contains Gaussian centers and constant
"
determines the shape (
width
) of the basis functions.Assuming
uniform approximation capabilities implies an ideal set of
RBFN weights
w
and ideal output
o
that would minimize the
square error between the actual function and the RBFN output
in
D
.Expressing actual weights
ˆ
w
and actual output
ˆ
o
gives
error terms
˜
w
=
w
!
ˆ
w
,
˜
o
=
o
!
ˆ
o.
(3)
Using RBF ideal weights to model nonlinear function
f
(
q
)
"
R
with uniformapproximation error
d
(
q
)
can be expressed as
o
(
q
,
w
) +
d
(
q
) =
f
(
q
)
(4)
where,by deﬁnition of uniform approximation,positive con
stant
d
max
exists such that

d
(
q
)
#
d
max
,
$
q
"
D
.
B.Using Backstepping for Smoothing Actuation Signal
We propose using the adaptive backstepping method as a
way to ﬁlter an actuator signal,introduced here with the simple
example
˙
q
=
f
(
q
(
t
)) +
b#
(
t
)
where
b
is a positive constant and the objective is to make state
q
follow desired trajectory
q
d
,
˙
q
d
.Both
f
(
q
)
and
b
are unknown.
Given actuator signal
#
,its derivative is treated as the control
signal
u
in the design process
˙
x
=
f
(
q
) +
b#
!
˙
q
d
(5)
˙
#
=
u
(6)
where
x
=
q
!
q
d
.
Adaptive backstepping uses an additional error
z
,which is the
difference between virtual control
$
and actuation signal
#
z
=
#
!
$.
Appropriate virtual control and control for overparameterized
backstepping (see Appendix A) are
$
=
!
G
1
x
!
ˆ
o
1
(
q,
˙
q
d
,
ˆ
w
1
)
(7)
u
=
!
x
!
G
2
z
+ ˆ
o
2
(
q,q
d
,
˙
q
d
,
¨
q
d
,
ˆ
w
2
)
(8)
where
G
1
,
G
2
are positive control gains.If
d
max
= 0
,stable
weight updates are
˙
ˆ
w
1
=
%
!
T
1
(
q,
˙
q
d
)
x,
˙
ˆ
w
2
=
%
!
T
2
(
q,q
d
,
˙
q
d
,
¨
q
d
)
z.
(9)
To examine ﬁltering from a linearsystems perspective,ﬁrst
expand (8)
˙
#
=
!
(
q
!
q
d
)
!
G
2
(
#
!
$
) + ˆ
o
2
(
q,q
d
,
˙
q
d
,
¨
q
d
,
ˆ
w
2
)
.
(10)
When the motion is simply a lowamplitude oscillation of the
NN inputs (relative to the widths of the Gaussians) about an
equilibriumpoint,the neural network acts like an integrator
ˆ
o
2
=
%
!
1
%
!
T
1
x
%
c
%
x
(11)
560 IEEE TRANSACTIONS ON SYSTEMS,MAN,AND CYBERNETICS—PART A:SYSTEMS AND HUMANS,VOL.42,NO.3,MAY
2012
where
c
is a constant.A linear analysis occurs by taking the
Laplace transformof (10)
s
!(
s
) =
!
[
Q
(
s
)
!
Q
d
(
s
)]
!
G
2
[!(
s
)
!
A
(
s
)]
+
c
s
[
Q
(
s
)
!
cQ
d
(
s
)]
!(
s
) =
1
s
(
s
+
G
2
)
[(
!
s
+
c
)
Q
(
s
)+(
s
!
c
)
Q
d
(
s
)+
G
2
A
(
s
)]
which implies there is a ﬁrstorder lowpass ﬁlter effect on
all signals before they affect the actuation signal,with cutoff
frequency
G
2
rad/s.
When using tuning functions [30],assuming
˙
ˆ
b
= 0
allows
a linear analysis similar to the overparameterized case.From
Appendix A,the virtual control is
$
= [
ˆ
b
]
!
1
(
!
G
1
x
!
ˆ
o
(
q,
ˆ
w
) + ˙
q
d
)
(12)
and the control (74) has the form
u
=
!
ˆ
bx
!
G
2
z
+
g
1
(
r
)
q
+
g
2
(
r
)
q
d
+
g
3
(
r
) ˙
q
d
+
g
4
(
r
)¨
q
d
+
g
5
(
r
)
#
+
g
6
(
r
)ˆ
o
(13)
where
r
= [
ˆ
w
T
ˆ
b q q
d
˙
q
d
¨
q
d
]
T
and
ˆ
b
is a single adaptive
parameter.Using the same assumption of small oscillations
about the equilibrium point,all terms
g
i
(
r
)
can be linearized
about the equilibrium point,resulting in constants
&
i
,and the
neural network is expressed
ˆ
o
%
&
6
%
x
+
&
7
%
z
=
&
6
%
(
q
!
q
d
) +
&
7
%
(
#
!
$
)
.
Taking the Laplace transformof (13) and solving for
!(
s
)
gives
!=
1
(
s
2
+(
G
2
!
&
5
)
s
!
&
7
)
&'
(
&
1
!
ˆ
b
)
s
+
&
6
(
Q
+
'
&
4
s
3
+
&
3
s
2
+(
ˆ
b
+
&
2
)
s
!
&
6
(
Q
d
+(
G
2
s
!
&
7
)
A
)
.
(14)
Constants
&
5
and
&
7
can be estimated from (74) by again as
suming the oscillations are small compared to Gaussian widths
&
5
=
ˆ
b
'$
'q
%
ˆ
b
(
!
G
1
)
(15)
&
7
=
'$
'q
%!
G
1
.
(16)
Thus,examining (14) leads to the conclusion that desired
trajectory information is not lowpass ﬁltered,but the other
signals
q
(
t
)
,
$
(
t
)
are lowpass ﬁltered,with cutoff frequency
depending on
G
1
and
G
2
.
C.System Description
This section introduces the teleoperation systemunder study.
Consider a onedegreeoffreedom (DOF) slave robot with
position
x
(
t
)
,mass
M
,friction force
D
r
˙
x
touching an envi
ronment with nonlinear environment stiffness
K
(
x
)
and linear
environment friction/damping
D
e
˙
x
(Fig.1).The slave actuators
Fig.1.One DOF slave robot plus remote environment.
Fig.2.Typical teleoperation setup;unlabeled signals are combinations of
force,position,and velocity.
produce control force
F
c
,and a force sensor measures
F
e
which
is the result of environmental effects
F
e
(
x
) =
K
(
x
)
x
+
D
e
˙
x.
(17)
Since the backstepping technique will treat the derivative of
force as the control signal,ﬁltering the forcesensor output
may be unnecessary.Moreover,allowing the operator to feel
high frequencies [31] and noise [32] can actually improve the
quality the haptic performance.Dynamically,the slave plus
environment becomes
M
¨
x
=
!
D
r
˙
x
!
D
e
˙
x
!
K
(
x
)
x
+
F
c
.
(18)
By assumption the haptic master device performs correct
impedance transformations and is light enough its dynamics
can be neglected.We assume we can measure the position and
velocity of both master and slave device.Since the method is
essentially a force tracking method,it would be best if a force
sensor was available on the master device,yet we can simply
interpret the position as force command using a ﬁctitious spring
if need be (which is exactly what we do in the experiment).The
remaining problembecomes designing a slave control to follow
reference signal(s) (Fig.2).
III.C
ONTROLLER
D
ESIGN
A.Augmented Tracking Error
In many teleoperation implementations,the slave robot
tracks human commanded position,while measured force is
reﬂected back to the human through the master haptic device.
It is up to the human to achieve the desired force.Concerns
about stability drive this choice for position reference signal.
However,during a collision,the operator could command a
position beyond the solid surface,possibly causing damage
(particularly to sensitive and expensive force sensors).The
problem compounds when communication delays the haptic
feedback to the operator,in which case,the operator may need
to anticipate any collisions to avoid damage.
RICHERT
et al.
:HAPTIC CONTROL FOR TRANSITIONING BETWEEN FREE,SOFT,AND HARD ENVIRONMENTS 561
For these reasons,the proposed control tracks human com
manded force
F
h
,and the force error is deﬁned
(
=
F
e
!
F
h
.
In the presence of a delay of
T
seconds,the controller will track
the last commanded force
F
h
(
t
!
T
)
.Thus,the controller will
naturally stop the robot when it feels a large collision force.
We also propose adding a velocity penalty to the force tracking
error,resulting in augmented error
s
="
(
+ ˙
x
(19)
where
"
is a positive constant and
˙
x
is the slave velocity.The
goals of adding the velocity penalty termare to:
1) prevent excessive overshoot after losing stiff contact or
puncturing through;
2) damp highfrequency oscillations that occur after a colli
sion with a stiff surface;
3) provide a way to control the slave in free space.
To understand point 3 above,consider that when
F
e
= 0
,
then
s
&
0
implies
˙
x
="
F
h
.
(20)
Thus,in free space ideally the velocity will be proportional to
commanded force.
When in contact with a material driving
s
&
0
implies
˙
x
=
!
"
(
(21)
and using the systemdynamics reveal
˙
x
=
!
"
F
e
+"
F
h
(22)
=
!
"
K
(
x
)
x
!
"
D
e
˙
x
+"
F
h
(23)
=
!
"
K
(
x
)
1 +
D
e
x
+
"
1 +
D
e
F
h
(24)
which is a (nonlinear) openloop stable system.
It is wellestablished that a discontinuous robustcontrol term
will drive
s
to zero in the presence of unknown nonlinear
terms.We do not implement such a term,so the above analysis
with
s
&
0
merely justiﬁes the choice of augmented error.
Our adaptive control will result in uniform ultimate bounds
on
s
.Note that even ensuring
s
&
0
may not be suitable for
performing precise manipulation tasks,which would require
additional position tracking.The proposed control handles a
collision scenario well,while still providing a stable haptic
control in other situations.
Since typical haptic hand controllers do not have builtin
force sensor,a position deﬂection multiplied by a ﬁctitious hap
tic spring constant
K
h
provides a virtual force measurement,
and the total commanded force becomes
F
h
=
K
h
(
x
h
!
x
h
,
0
) +
F
e
.
(25)
Here,
x
h
is the position of the haptic device and
x
h
,
0
the
nominal position.In the absence of time delay,the force control
becomes a scaled position control problem.However,a time
delay of
T
seconds results in human commanded force
F
h
(
t
) =
K
h
(
x
h
(
t
)
!
x
h
,
0
) +
F
e
(
t
!
T
)
(26)
and the force error at the controller becomes
(
(
t
) =
F
e
(
t
)
!
F
h
(
t
!
T
)
(27)
=
F
e
(
t
)
!
F
e
(
t
!
2
T
)
!
K
h
(
x
h
(
t
!
T
)
!
x
h
,
0
)
.
(28)
Note that providing an openloop position signal back to the
operator,i.e.an
x
h
,
0
that depends on time could provide a more
natural feel to the haptic device.We leave this to future work.
B.Backstepping Adaptive Control
For the haptic control,the actuator signal is
F
c
(
t
)
and thus
˙
F
c
(
t
)
will be designed as the control using backstepping.
Consider the adaptive control Lyapunov function
V
1
(
s,
˜
w
1
,
˜
w
2
) =
1
2
s
2
+
1
2
%
1
˜
w
T
1
˜
w
1
+
1
2
%
2
˜
w
2
2
.
(29)
The time derivative of
V
1
is
˙
V
1
=
s
˙
s
+
1
%
1
˜
w
T
1
d
dt
(
w
1
!
ˆ
w
1
) +
1
%
2
˜
w
2
d
dt
(
w
2
!
ˆ
w
2
)
(30)
=
s
'
"(
˙
F
e
!
˙
F
h
) + ¨
x
(
!
1
%
1
˜
w
T
1
˙
ˆ
w
1
!
1
%
2
˜
w
2
˙
ˆ
w
2
.
(31)
Using the simpliﬁed environment dynamics (17) gives
˙
V
1
=
s
"
"
*
d
dt
(
K
(
x,t
)
x
) +
D
e
¨
x
!
˙
F
h
+
+ ¨
x
#
!
1
%
1
˜
w
T
1
˙
ˆ
w
1
!
1
%
2
˜
w
2
˙
ˆ
w
2
.
(32)
and using (18) for
¨
x
gives
˙
V
1
=
s
,
("
D
e
+1)
M
!
1
(
!
D
r
˙
x
!
D
e
˙
x
!
K
(
x,t
)
x
+
F
c
)
+"
*
d
dt
(
K
(
x,t
)
x
)
!
˙
F
h
+

!
1
%
1
˜
w
T
1
˙
ˆ
w
1
!
1
%
2
˜
w
2
˙
ˆ
w
2
.
(33)
The ﬁrst approximator
!
1
(
x,F
h
,s
)
ˆ
w
1
estimates nonlinear
terms
!
1
(
x,F
h
,s
)
w
1
= ("
D
e
+1)
M
!
1
(
!
D
r
˙
x
!
D
e
˙
x
!
K
(
x,t
)
x
)
+"
d
dt
(
K
(
x,t
)
x
) +
d
1
(
x,F
h
,s
)
(34)
where
d
1
is a bounded approximation error
(

d
1
#
d
1
,
max
)
.A
single adaptive parameter
ˆ
w
2
models the unknown parameter
w
2
=
M
!
1
("
D
e
+1)
(35)
562 IEEE TRANSACTIONS ON SYSTEMS,MAN,AND CYBERNETICS—PART A:SYSTEMS AND HUMANS,VOL.42,NO.3,MAY
2012
The virtual control
$
will represent a desired value of
F
c
,with
resulting error signal
z
=
F
c
!
$
.Substituting
w
2
= ˜
w
2
+ ˆ
w
2
and
F
c
=
$
+
z
into (33) gives
˙
V
1
=
s
'
!
"
˙
F
h
+
!
1
w
1
+
d
1
+ ˜
w
2
(
!
F
e
+
F
c
)
(
+ ˆ
w
2
(
!
F
e
+
$
+
z
)
!
1
%
1
˜
w
T
1
˙
ˆ
w
1
!
1
%
2
˜
w
2
˙
ˆ
w
2
.
(36)
Note the righthand side of (34) explicitly depends on
x
and
˙
x
,however,instead of providing
x
,
F
h
,and
s
=
s
(
x,
˙
x,F
h
)
as RBF inputs will prove to be convenient.Choosing virtual
control (desired
F
c
) as
$
=
F
e
+ ˆ
w
!
1
2
("
˙
F
h
!
!
1
ˆ
w
1
!
G
1
s
)
(37)
results in Lyapunov timederivative
˙
V
1
=
!
G
1
s
2
+
sd
1
+
s
ˆ
w
2
z
+
1
%
1
˜
w
T
1
'
!
T
1
s
!
˙
ˆ
w
1
(
+
1
%
2
˜
w
2
'
(
!
F
e
+
F
c
)
s
!
˙
ˆ
w
2
(
.
(38)
Rather than choose weight and parameter update laws at this
point (overparameterized design),deﬁning tuning functions
)
1
=
!
T
1
s
(39)
)
2
=(
!
F
e
+
F
c
)
s
(40)
allows the updates to be designed at the next stage of backstep
ping.The second stage starts with
V
2
(
s,z,
˜
w
1
,
˜
w
2
,
˜
w
3
) =
V
1
+
1
2
z
2
+
1
2
%
3
˜
w
T
3
˜
w
3
(41)
with time derivative
˙
V
2
=
!
G
1
s
2
+
s
ˆ
w
2
z
+
z
(
˙
F
c
!
˙
$
) +
1
%
1
˜
w
T
1
(
)
1
!
˙
ˆ
w
1
)
+
1
%
2
˜
w
2
(
)
2
!
˙
ˆ
w
2
)
!
1
%
3
˜
w
T
3
˙
ˆ
w
3
.
(42)
Choice of control signal
(
˙
F
c
)
as
u
=
!
G
2
z
+
˙
ˆ
$
!
s
ˆ
w
2
(43)
where
˙
ˆ
$
contains only the known components of
˙
$
,results in
˙
V
2
=
!
G
1
s
2
+
sd
1
!
G
2
z
2
+
z
(
˙
ˆ
$
!
˙
$
) +
1
%
1
˜
w
T
1
(
)
1
!
˙
ˆ
w
1
)
+
1
%
2
˜
w
2
(
)
2
!
˙
ˆ
w
2
)
!
1
%
3
˜
w
T
3
˙
ˆ
w
3
.
(44)
Analytic differentiation of
$
w.r.t.time results in
˙
$
=
˙
F
e
!
˙
ˆ
w
2
ˆ
w
!
2
2
("
˙
F
h
!
!
1
ˆ
w
1
!
G
1
s
)
+ ˆ
w
!
1
2
"
"
¨
F
h
!
d
dt
[
!
1
(
x,F
h
,s
)
ˆ
w
1
]
!
G
1
˙
s
#
=
˙
F
e
!
˙
ˆ
w
2
ˆ
w
!
2
2
("
˙
F
h
!
!
1
ˆ
w
1
!
G
1
s
)
+ ˆ
w
!
1
2
"
"
¨
F
h
!
'
'x
[
!
1
] ˙
x
ˆ
w
1
!
'
'F
h
[
!
1
]
˙
F
h
ˆ
w
1
!
'
's
[
!
1
] ˙
s
ˆ
w
1
!
!
1
˙
ˆ
w
1
#
!
ˆ
w
!
1
2
G
1
˙
s.
The implementable components of
˙
$
are
˙
ˆ
$
=
!
!
3
ˆ
w
3
!
˙
ˆ
w
2
ˆ
w
!
2
2
("
˙
F
h
!
!
1
ˆ
w
1
!
G
1
s
)
+ ˆ
w
!
1
2
"
"
¨
F
h
!
'
'x
[
!
1
] ˙
x
ˆ
w
1
!
'
'F
h
[
!
1
]
˙
F
h
ˆ
w
1
!
!
1
˙
ˆ
w
1
#
!
"
ˆ
w
!
1
2
G
1
+
'
's
[
!
1
]
ˆ
w
1
#
˙
ˆ
s
(45)
where an RBF network was used to approximate
!
˙
F
e
!
3
(
x,
˙
x,F
c
)
w
3
=
!
˙
F
e
+
d
3
(
x,
˙
x,F
c
)
(46)
with
d
3
a bounded approximation error.Likewise,
˙
ˆ
s
contains
the implementable components of
˙
s
.From previous analysis
[see Eqs.(30)–(32)]
˙
s
=
!
"
˙
F
h
+
!
1
w
1
+
w
2
(
!
F
e
+
F
c
)
(47)
implying
˙
ˆ
s
=
!
"
˙
F
h
+
!
1
ˆ
w
1
+ ˆ
w
2
(
!
F
e
+
F
c
)
.
(48)
Finally,
˙
ˆ
$
=
!
!
3
ˆ
w
3
+
d
3
!
˙
ˆ
w
2
ˆ
w
!
2
2
("
˙
F
h
!
!
1
ˆ
w
1
!
G
1
s
)
+ ˆ
w
!
1
2
"
"
¨
F
h
!
'
'x
[
!
1
] ˙
x
ˆ
w
1
!
'
'F
h
[
!
1
]
˙
F
h
ˆ
w
1
!
!
1
˙
ˆ
w
1
#
!
"
ˆ
w
!
1
2
G
1
+
'
's
[
!
1
]
ˆ
w
1
#
'
!
"
˙
F
h
+
!
1
ˆ
w
1
+ ˆ
w
2
(
!
F
e
+
F
c
)
(
.
(49)
Note that although
˙
F
h
,
¨
F
h
are required,
F
h
is a ﬁctitious
force  following from the haptic device’s measured position
in (25).Thus,
˙
F
h
follows from hapticdevice velocity and
¨
F
h
from acceleration.Using the digital encoders in the haptic
device makes these terms reasonable to estimate.The dif
ference
˙
ˆ
$
!
˙
$
is
˙
ˆ
$
!
˙
$
=
!
3
˜
w
3
+
d
3
+
"
ˆ
w
!
1
2
G
1
+
'
's
[
!
1
]
ˆ
w
1
#
(
!
1
˜
w
1
+ ˜
w
2
(
!
F
e
+
F
c
))
.
RICHERT
et al.
:HAPTIC CONTROL FOR TRANSITIONING BETWEEN FREE,SOFT,AND HARD ENVIRONMENTS 563
Substituting the above difference back into the control
Lyapunov function results in
˙
V
2
=
!
G
1
s
2
+
sd
1
!
G
2
z
2
+
zd
3
+
1
%
1
˜
w
T
1
"
)
1
+
!
T
1
*
ˆ
w
!
1
2
G
1
+
'
's
[
!
1
]
ˆ
w
1
+
z
!
˙
ˆ
w
1
#
+
1
%
2
˜
w
2
"
)
2
+
*
ˆ
w
!
1
2
G
1
+
'
's
[
!
1
]
ˆ
w
1
+
(
!
F
e
+
F
c
)
z
!
˙
ˆ
w
2
#
+
1
%
3
˜
w
T
3
'
!
T
3
z
!
˙
ˆ
w
3
(
(50)
leading to choice of RBF update laws
˙
ˆ
w
1
=
%
1
"
)
1
+
&!
T
1
*
ˆ
w
!
1
2
G
1
+
'
's
[
!
1
]
ˆ
w
1
+
z
+
*
1
ˆ
w
1
#
(51)
˙
ˆ
w
2
=
%
2
proj
.
)
2
+
&
*
ˆ
w
!
1
2
G
1
+
'
's
[
!
1
]
ˆ
w
1
+
(
!
F
e
+
F
c
)
z
+
+
( ¯
w
2
!
ˆ
w
2
)
/
(52)
˙
ˆ
w
3
=
%
3
0
!
T
3
z
+
*
3
ˆ
w
3
1
.
(53)
where
&
,
+
,and
*
are positive constants.The
*
–multiplied terms
provide
"
–modiﬁcation (leakage),ensuring bounded weights.
The scaling factor
&
"
[0
,
1]
provides a separate weighting
for updating the terms that depend on
z
[33],[34].Otherwise,
these terms tend to dominate and
)
1
(
x
)
,
)
2
(
x
)
have little
effect.More even (equal) contribution from the
x
and
z
terms
results in better performance and smoother control.For clarity,
the remainder of the stability analysis here uses
&
= 1
,and
Appendix B addresses the case
&
'
= 1
.The
+
–multiplied term
acts in a supervisory manner trying to avoid the projection
limits,and constant
¯
w
2
is selected
a priori
as an orderof
magnitude estimate [35].The projection operator
proj
[
∙
] =
2
0
,
ˆ
w
2
<
(
w
2
(
min
and
∙
<
0
∙
,
otherwise
(54)
ensures that
w
2
remains invertible.Note that
(
w
2
(
min
#
w
2
necessarily bounds
˜
w
2
fromthe bottom.
The result is
˙
V
2
=
!
"
T
*
G
1
0
0
G
2
+
"
!
˜
w
T
3
4
1
*
1
0 0
0
+
0
0 0
1
*
3
5
6
˜
w
+
"
T
d
+
˜
w
T
3
4
1
*
1
0 0
0
+
0
0 0
1
*
3
5
6
w
+
+
¯
w
2
˜
w
2
(55)
where
"
= [
s z
]
T
,
d
= [
d
1
d
3
]
T
,and
w
=
[
w
1
w
2
w
3
]
T
.Thus,
V
2
is bounded by
˙
V
2
#!
G
(
"
(
2
+
d
(
x
i
(!
*
(
˜
w
(
2
+
*
(
˜
w
(
(
(
w
(
+

¯
w
2

+/*
)
(56)
TABLE I
E
XPERIMENT
S
CENARIOS
where
G
= min(
G
1
,G
2
)
,
d
= max(
d
1
,
max
,d
3
,
max
)
,and
*
=
min(
*
1
,+,*
3
)
.By completing the square,one can establish
˙
V
2
<
0
when either
(
"
(
>,
!
or
(
˜
w
(
>,
w
where
,
!
=
d
2
G
+
7
d
2
4
G
2
+
*
(
(
w
(
+
+
¯
w
2
+/*
)
2
4
G
(57)
,
w
=
(
(
w
(
+
+
¯
w
2
/*
)
2
+
7
d
2
4
G*
+
(
(
w
(
+
+
¯
w
2
/*
)
2
4
G
(58)
Thus,a uniform ultimate bound is given by the level set
(Lyapunov surface) on the
(
(
"
(
,
(
˜
w
(
)
plane
V
2
(
(
"
(
,
(
˜
w
(
) =
V
2
(
,
!
,,
w
)
assuming that the uniform approximation region of the RBF
network includes this surface.One question that may arise is
if
F
c
is bounded.Since
z
=
F
c
!
$
and
z
is bounded,
F
c
is
bounded if all terms in
$
are.The only term in
$
(36) that
cannot be bounded is
˙
F
h
since it is supplied by the user.Thus,
F
c
is bounded under the condition that the user’s commanded
velocity is bounded.
IV.R
ESULTS
A.Scenarios
To test the utility of the proposed method,we examine the
performance transitioning from soft medium to free space,and
then to a hard surface.To make the scenario a challenging test
of performance and stability,a stiff spring with no damping
models the hard surface,and signiﬁcant communication delays
affect the signal.Speciﬁcally,in the 1DOF experiment,com
munication lags delay the signals by
T
= 200
ms (100 ms in
each direction).The stiffness of the simulated hard medium is
ﬁve orders of magnitude greater than the soft medium(Table I),
and the damping coefﬁcient of the soft medium is 0.1 Ns/m.
We refer to the transition from soft medium to free space as a
puncture
and the transition from free space to the hard surface
as a
collision
.
We do not present comprehensive human testing as the
qualities of force control are well known.The focus of this
paper is on solving the particular problem of collision force.
Excessive collision force is a commonly encountered problem
with current haptic systems that typically use positiontracking
controls in the inner loop.During collision,a positiontracking
system tries to drive the system “through the surface” until the
operator feels the effect (after 100 ms in our scenario) and
commands the robot to pull back (another 100 ms delay in
564 IEEE TRANSACTIONS ON SYSTEMS,MAN,AND CYBERNETICS—PART A:SYSTEMS AND HUMANS,VOL.42,NO.3,MAY
2012
our scenario).For this reason,position tracking is well known
to result in signiﬁcant collision forces,even with smaller time
delays.A forcefeedback systemin the inner loop,on the other
hand,automatically reacts to a collision force by trying to
drive the measured force back to the operator’s (much smaller)
commanded force during this period of time delay,after
which the operator can ﬁne tune the desired force to their liking.
Thus,the particular source of force command is irrelevant
during the 200 ms period of time delay when the collision
forces may be excessive.We provide both a human operator
and a ﬁltered proportionalintegral (PI) control as the sources of
force commands in this paper to verify that regardless of how
the force is commanded,the proposed method will reduce the
collision force automatically during the timedelay period.The
particular contribution of this paper is not in pointing out that
force control can reduce collision forces,rather it is developing
a force control with guaranteed stability and able to function in
other scenarios without need of a switching control law.
We wish to show the proposed control results in small forces
in collision yet still avoids excessive overshoot during a punc
ture;thus two scenarios are appropriate.In the ﬁrst the slave
moves through a simulated soft (viscous and elastic) medium
and then collides with a simulated solid (stiff elastic) surface
in a purely elastic collision.In the second scenario,the slave
moves through a simulated soft medium and then encounters
simulated free space (i.e.a puncture or loss of contact).Both
scenarios start at
x
= 0
with collision or puncture occurring
at
x
t
= 10
cm,where stiffnesses change (Table I).In the ﬁrst
test,a human operator uses a physical haptic device,and the
slave device and environment are simulated on a computer.
The ﬁrst test simply veriﬁes it is possible for a human to
control the system using this method.The second test takes
place entirely in simulation with the operator plus master device
replaced by a ﬁltered PI control producing a desired force.The
second test veriﬁes the expected performance of the inner con
trol in a repeatable experiment.All parameters remain constant
during different scenarios and experiments (Table II).Please
note our tests do not emulate any particular physical device,and
thus the parameters do not followfromany particular identiﬁca
tion scheme.We chose parameters appropriate for a device that
moves a few centimeters per second,i.e.precision movements
visible to the naked eye,shown at the bottomof the page.
We compare the proposed method to two other control
strategies,demonstrating the proposed method achieves a much
smaller collision force,and less collision oscillations,than the
other methods.The two other control strategies are
H
2
linear
control and a passivitybased outputfeedback control.The
H
2
controller uses a simple gain scheduling routine,and the control
signal is lowpass ﬁltered at the input to the slave actuator.Our
H
2
control tracks force only,and as a result,the control gains
TABLE II
C
ONSTANT
P
ARAMETERS
Fig.3.Master haptic device used for experiments:SensAble Technologies
PHANTOMOmni.
become inﬁnite in the absence of a slave force measurement
(that is,free space).For this reason only,the collision test uses
the
H
2
control.The total controller is the sum of a feedback
linearization
u
fbc
=
D
r
˙
x
+
F
m
+
K
!
1
e
M
¨
F
d
(59)
and
H
2
state feedback control.The augmented matrix for the
H
2
control is
K
2
=
*
A
+
BF
+
CL L
F
0
+
(60)
where
A
,
B
,and
C
are from the system describing the dy
namics of
e
=
F
m
!
F
d
after implementing
u
fbc
.They turn out
to be
A
=
*
0 1
0 0
+
B
=
*
0
K
e
M
!
1
+
C
= [ 1 0]
and the gains of the
H
2
controller are
L
= [
!
1
!
1 ]
T
F
[
!
10
!
1 ]
The outputfeedback controller implements the passivity
based control proposed in [36].A passivity observer observes
Operator/Task
Collision
Puncture
Human Test
1
/
Scenario
1
Test
1
/
Scenario
2
Filtered PI Test
2
/
Scenario
1
Test
2
/
Scenario
2
RICHERT
et al.
:HAPTIC CONTROL FOR TRANSITIONING BETWEEN FREE,SOFT,AND HARD ENVIRONMENTS 565
Fig.4.Human Operator:Proposed controller tracks force and achieves stable elastic collision.
Fig.5.Smaller
G
2
results in less oscillations after collision,due to the lowpass ﬁlter effect from backstepping.Bottom graphs show fast Fourier transform of
top graphs.
the dissipated energy of the system
E
(
n
) =
)
!
k
f
(
k
)
v
(
k
)
where in our case,
f
(
k
)
is the control at time
t
=
k
#
T
,and
v
(
k
)
is the slave velocity.When the dissipation
E
(
n
)
becomes
negative,the controller simply removes energy fromthe system
by implementing the control
F
c
=
F
h
!
E
(
n
)
2#
Tv
(
n
)
.
In this method,there is nominally no automatic control applied
per se
and the slave actuator force is the timedelayed human
commanded force.An innerloop control signal appears only
when active (nonpassive) behavior is observed,determined by
the condition
t
%
0
˙
x
(
)
)
F
c
(
)
)
d) <
0
.
The term
output feedback
describes this approach in [37].
B.Test 1:Human Operator
A PHANTOM Omni haptic device manufactured by
SensAble (Fig.3) provides the master device.The joints have
optical encoders.The device provides 3DOF force feedback
566 IEEE TRANSACTIONS ON SYSTEMS,MAN,AND CYBERNETICS—PART A:SYSTEMS AND HUMANS,VOL.42,NO.3,MAY
2012
Fig.6.Human Operator:Proposed controller collides with signiﬁcantly less measured force.
Fig.7.Human Operator:Proposed controller may puncture with less position overshoot than the output feedback controller.
in Cartesian
X
,
Y
,
Z
and measurements of 6DOF position
(
X
,
Y
,
Z
,roll,pitch,yaw).Cartesian resolution is 0.055 mm.
The available workspace is 160 mm
*
120 mm
*
70 mm.
The stylus has an apparent mass of 45 g.The Omni exerts
at most 3.3 N of force continuously,and the motors have
a backdrive friction force of 0.26 N.The device exhibits a
maximum stiffness of 2310 N/m.An IEEE1394 FireWire port
provides fast communication to a PC.Quanser’s QuaRCcontrol
RICHERT
et al.
:HAPTIC CONTROL FOR TRANSITIONING BETWEEN FREE,SOFT,AND HARD ENVIRONMENTS 567
Fig.8.PIModel Operator:Proposed controller collides with signiﬁcantly less measured force.
Fig.9.PIModel Operator:Proposed controller may puncture with slightly less position overshoot than the output feedback controller.
568 IEEE TRANSACTIONS ON SYSTEMS,MAN,AND CYBERNETICS—PART A:SYSTEMS AND HUMANS,VOL.42,NO.3,MAY
2012
software solution includes a PHANTOM Omni blockset for
MATLAB’s Simulink environment.The Omni Simulink block
sends force commands to the Omni’s motors and receives
6DOF positions.Communication occurs at sample frequencies
up to 1000 Hz,as used for this work.
For the 1DOF experiment,two proportional controllers lock
the haptic device onto the
(
X,
0
,
0)
line.In the collision sce
nario,the human operator attempts to achieve a constant veloc
ity before collision and then tries to apply a constant force once
in contact with the stiff surface.The proposed method tracks the
commanded force within 0.2 N,and after collision,elastic os
cillations disappear in approximately 1 s (Fig.4).The proposed
control results in a collision with about 4 N of force,80 N less
than
H
2
control and 20 N less than outputfeedback control in
the same situation (Fig.6).Practicality of the method stems
from the apparently smoothness of the slave control force.Re
peating the experiment with a much larger (inappropriate)
G
2
results in signiﬁcant frequency content (Fig.5),demonstrating
how backstepping ﬁlters the control in the ﬁrst experiment.
In the puncture scenario,the operator tries to achieve a
constant velocity in the soft environment and then tries to stop
after puncturing through into free space.The proposed control
does not do worse than outputfeedback after a puncture and
still operates reliably in free space (Fig.7).
In the puncture scenario,the desired velocity again begins
at constant
˙
x
desired
= 0
.
03
m/s in the soft medium.After
puncture,the ﬁrst time
x
(
t
)
goes past
x
t
,the desired velocity
is set to zero.The proposed control does not do worse after
puncturing than output feedback (Fig.9).
C.Test 2:Filtered PI Control Replaces Human
Eliminating the human operator as a variable results in a
repeatable experiment that can test the innerloop control.We
do not claim to have an accurate mathematical model of the
decision making of the human operator,rather we eliminate
the human and haptic device as variables in the experiment to
isolate the effect of the proposed control law.A proportional
velocitytracking term followed by a ﬁrstorder ﬁlter replaces
the operator and haptic device
F
h
(
s
) =
"
8 +
2
s
#"
20
s
+20
#
(
v
(
s
) +
F
e
(
s
)
(61)
where
(
v
(
t
) = ˙
x
desired
!
˙
x
is the slave velocity error.
In the collision scenario,the desired velocity begins at
constant
˙
x
desired
= 0
.
03
m/s in the soft medium.After the
collision,the ﬁrst time measured force rises above a threshold
of 5 N,the desired velocity is set to zero.The proposed control
collides with approximately 5 N of force,90 N less than
H
2
control,and 35 N less than output feedback (Fig.8).
V.C
ONCLUSION
This work addresses the problem of haptic teleoperation
when communication time delays between master and slave
may result in excessive collision forces.The proposed slave
side control uses an augmented output error that includes both
force error and velocity penalty.Using force error naturally
reduces collision force in the presence of time delay,unlike
position or tracking error.The velocity penalty term ensures
damping after collision,prevents excessive overshoot after a
puncture,and provides a way to control the slave in free space
when no force is measured.The proposed control follows from
an adaptive backstepping design.The adaptation allows stabil
ity guarantees when interacting with unknown environments,
and the backstepping ensures a smooth applied force.Exper
iments with a human operator interacting with a simulated
environment demonstrate that collision force is dramatically
reduced compared to using other control methods and that the
performance is comparable in other operations.Replacing the
human operator with a PI control produces similar results.
A
PPENDIX
A
Overparameterized Backstepping Design:
Overparameter
ized adaptive backstepping design for (5) and (6) would typi
cally use total Lyapunov function
V
=
1
2
b
x
2
+
1
2
z
2
+
˜
w
T
˜
w
/%.
Neuralnetworks
o
1
=
!
1
w
1
and
o
2
=
!
2
w
2
model nonlinear
functions as
o
1
+
d
1
=
f
(
q
)
/b
+
q
d
/b
and model the time derivative of the virtual control as
o
2
+
d
2
=
!
˙
$.
Using the fact
#
=
z
+
$
in the analysis gives
˙
V
=
x
˙
x/b
+
z
˙
z
!
˜
w
T
˙
ˆ
w
/%
(62)
=
x
(
f
(
q
)
/b
+
z
+
$
!
q
d
/b
) +
z
(
u
!
˙
$
)
!
˜
w
T
˙
ˆ
w
/%
(63)
=
x
(
o
1
+
d
1
+
z
+
$
) +
z
(
u
+
o
2
+
d
2
)
!
˜
w
T
˙
ˆ
w
/%
(64)
=
x
(ˆ
o
1
+
d
1
+
z
+
$
) +
z
(
u
+ ˆ
o
2
+
d
2
)
+
˜
w
T
1
'
!
T
1
x
!
˙
ˆ
w
1
/%
(
+
˜
w
T
2
'
!
T
2
z
!
˙
ˆ
w
2
/%
(
.
(65)
Thus,assuming
d
1
= 0
,
d
2
= 0
the (virtual) controls (7) and (8)
and weight updates (9) ensure negative deﬁnite
˙
V
=
!
G
1
x
2
!
G
2
z
2
.
TuningFunction Backstepping Design:
The adaptive tuning
function design for (5) and (6) uses
V
=
1
2
x
2
+
1
2
z
2
+
˜
w
T
˜
w
/%
+
˜
b
2
/%
b
.
Using neural network approximation
o
+
d
=
f
(
q
)
and adaptive parameter
ˆ
b
,the time derivative is
˙
V
=
x
[
f
(
q
) +
b
(
z
+
$
)
!
q
d
] +
z
(
u
!
˙
$
)
!
˜
w
T
˙
ˆ
w
/%
!
˜
b
˙
ˆ
b/%
b
=
x
&
o
+
d
+
˜
b#
+
ˆ
b
(
z
+
$
)
!
q
d
)
+
z
(
u
!
˙
$
)
+
˜
w
T
˙
ˆ
w
/%
!
˜
b
˙
ˆ
b/%
b
=
x
&
ˆ
o
+
d
+
ˆ
b
(
z
+
$
)
!
q
d
)
+
z
(
u
!
˙
$
)
+
˜
w
T
(
!
x
!
˙
ˆ
w
/%
) +
˜
b
(
x#
!
˙
ˆ
b/%
b
)
.
(66)
RICHERT
et al.
:HAPTIC CONTROL FOR TRANSITIONING BETWEEN FREE,SOFT,AND HARD ENVIRONMENTS 569
Expanding the time derivative of virtual control (12) gives
˙
$
=
'$
'q
˙
q
+
'$
'q
d
˙
q
d
+
'$
'
˙
q
d
¨
q
d
+
'$
'
ˆ
b
˙
ˆ
b
+
'$
'
ˆ
w
˙
ˆ
w
.
(67)
The ﬁrst termcan further be expanded
'$
'q
˙
q
=
'$
'q
(
f
(
x
) +
b#
)
(68)
=
'$
'q
(
o
+
d
+
b#
)
(69)
=
'$
'q
(ˆ
o
+
d
+
ˆ
b#
) +
'$
'q
(˜
o
+
˜
b#
)
.
(70)
Substituting (12),(67),(70) into (66) and assuming
d
= 0
results in
˙
V
=
!
G
1
x
2
+
x
ˆ
bz
+
z
"
u
!
'$
'q
(ˆ
o
+
ˆ
b#
)
!
'$
'q
d
˙
q
d
!
'$
'
˙
q
d
¨
q
d
!
'$
'
ˆ
b
˙
ˆ
b
!
'$
'
ˆ
w
˙
ˆ
w
#
+
˜
w
T
"
!
x
+
'$
'q
z
!
˙
ˆ
w
/%
#
+
˜
b
"
x#
+
'$
'q
#z
!
˙
ˆ
b/%
b
#
(71)
resulting in choice of weight and parameter updates
˙
ˆ
w
=
%
"
!
x
+
'$
'q
z
#
(72)
˙
ˆ
b
=
%
b
"
x#
+
'$
'q
#z
#
(73)
and control
u
=
!
ˆ
bx
!
G
2
z
+
'$
'q
(ˆ
o
+
ˆ
b#
)
+
'$
'q
d
˙
q
d
+
'$
'
˙
q
d
¨
q
d
+
'$
'
ˆ
b
˙
ˆ
b
+
'$
'
ˆ
w
˙
ˆ
w
(74)
in which case,
˙
V
=
!
G
1
x
2
!
G
2
z
2
is negative deﬁnite.
A
PPENDIX
B
This section proves systemstability for
&
'
= 1
.Begin with the
control Lyapunov function at the second stage of backstepping
V
2
=
V
1
+
&
2
z
2
+
1
2
%
3
˜
w
T
3
˜
w
3
.
(75)
For clarity,and due to space considerations,the following
analysis assumes perfect RBF network modeling
(
d
1
=
d
3
=
0)
.For a robust redesign of this method in the case of ap
proximations errors and disturbances,the reader is referred to
[33],[34].Continuing with the analysis as in Section IIIB,the
Lyapunov derivative becomes
˙
V
2
=
!
G
1
s
2
!
G
2
z
2
+(1
!
&
)
s
ˆ
w
2
z
+ ˜
w
2
"
)
2
+
&
(
!
F
e
+
F
c
)
"
'!
1
's
ˆ
w
1
+
G
1
#
ˆ
w
!
1
2
z
!
1
%
2
˙
ˆ
w
2
#
+
˜
w
T
1
"
)
1
+
&!
T
1
*
'!
1
's
ˆ
w
1
+
G
1
+
ˆ
w
!
1
2
z
!
1
%
1
˙
ˆ
w
1
#
+
˜
w
T
3
"
!
T
3
z
!
1
%
3
˙
ˆ
w
3
#
.
(76)
After implementing the RBFN update laws (51)–(53),the
result is
˙
V
2
=
!
G
1
s
2
!
G
2
z
2
+(1
!
&
)
s
ˆ
pz
+
!
+
˜
p
(¯
p
!
ˆ
p
)
+
*
1
˜
w
T
1
ˆ
w
1
+
*
2
˜
w
T
2
ˆ
w
2
.
(77)
The RBFN weight errors have already been proved stable,thus
consider only
˙
V
2
=
!
G
1
s
2
!
G
2
z
2
+(1
!
&
)
s
ˆ
w
2
z.
(78)
Since this control Lyapunov candidate is easily observable,we
are free to implement
&
subject to
&
=
2
1
,
if
!
G
1
s
2
!
G
2
z
2
+(1
!
&
)
s
ˆ
w
2
z >
0
&
design
,
otherwise
where
&
design
is any user desired value.The negative deﬁnite
ness of
˙
V
2
is thus proven.
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Dean Richert
received the B.S.and M.S.degrees in
electrical engineering at the University of Calgary,
Calgary,AB,Canada,in 2008 and 2010,respec
tively.His master’s thesis work concerned neural
adaptive haptic control of robot manipulators.He is
currently working toward the Ph.D.degree at the
University of California,San Diego,La Jolla,in
the Department of Mechanical and Aerospace Engi
neering where his research interests are distributed
control of robot networks and cooperative control
of UAVs.
Chris J.B.Macnab
received the B.Eng.degree in
engineering physics fromthe Royal Military College
of Canada,Kingston,ON,Canada,in 1993,and
the Ph.D.degree from the University of Toronto,
Toronto,ON,in 1999,where he attended the Institute
for Aerospace Studies and investigated stable neural
adaptive control of ﬂexiblejoint robots.
He worked at Dynacon Systems and at CRS
Robotics (now Thermo CRS Ltd.) in Toronto.He
is an Assistant Professor at the Department of
Electrical and Computer Engineering,University of
Calgary,Calgary,AB,where his current research interests include adaptive,
fuzzy,and neuralnetwork control applied to ﬂexiblejoint robots,helicopters,
haptic teleoperation,and biped running robots.
Jeff K.Pieper
received the B.Sc.,M.S.,and Ph.D.
degrees in mechanical engineering from Queen’s
University,Kingston,ON,Canada,and the Univer
sity of California at Berkeley.
He has held positions with the National Research
Council Institute for Aerospace Research,Carleton
University and Alcan Research.He is an Associate
Professor with the Department of Mechanical and
Manufacturing Engineering,University of Calgary,
Calgary,AB,where his research interests include
mechatronics,sliding mode control,and robust con
trol along with magnetic bearings,mobile robotics,and hydrokinetic turbines.
He is an Associate Editor for the IEEE Control Systems Society Conference
Editorial Board,past Chair of the IEEE Southern Alberta Section,General
Chair of the Canadian Society of Mechanical Engineers Forum2006,and Chair
of the 2003 CSME Symposiumon Mechatronics.
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