An Experimental Study of Dynamic VisualFeedback Control with a Fixed Camera
Toshiyuki Murao1, Hiroyuki Kawai2 and Masayuki Fujita1
1 Department of Electrical and Electronic Engineering, Kanazawa University
Kodatsuno 2-40-20, Kanazawa 920-8667, Japan
2 Information Technology Research Center, Hosei University
Fujimi 2-17-1, Chiyoda-ku, Tokyo 102-8160, Japanfujita@t.kanazawa-u.ac.jp
Abstract: This paper deals with the control and the estimation of dynamic visual feedback
systems with a ﬁxed camera. The model of the visual feedback system with four coordinate frames
is established by using the homogeneous representation and the adjoint transformation. Secondly
we derive the passivity of the dynamic visual feedback system by combining the manipulator
dynamics and the visual feedback system. Based on the passivity, stability and L2 -gain performance
analysis are discussed. Finally experimental results on SICE–DD arm are reported to conﬁrm theeﬀectiveness of the visual feedback control law.
Keywords: Visual Feedback Control, SICE–DD Arm, Passivity, Lyapunov Stability, L2 –Gain
Performance Analysis

1. Introduction
Vision based control of robotic systems involves the fusion of robot kinematics, dynamics, and computer vision system to control the position of the robot endeﬀector in an eﬃcient manner. The combination of mechanical control with visual information, so-called visual feedback control or visual servo, should become
extremely important, when we consider a mechanicalsystem working under dynamical environments1, 2) . Recently, the autonomous injection of biological cells has
been discussed using visual feedback control3) and ﬁelds
which need visual feedback control are increasing.
Our previous works4, 5) deal with a robot motion control with visual information in the eye-in-hand conﬁguration which has only three coordinate frames. However, the system can not be represented in this conﬁguration, such as the autonomous injection of biologicalcells.
This paper deals with the control and the estimationof dynamic visual feedback systems with a ﬁxed camera which have the four coordinate frames. The main
contribution of this paper is that the dynamic visualfeedback system with SICE–DD arm6) is constructed inorder to conﬁrm the eﬀectiveness of the visual feedback
control law.

2. Passivity-Based Visual Feedback System
2.1 Fundamental Representation for Visual Feedback System
We consider the ﬁxed-camera robotic system1) is shownin Fig. 1, where the coordinate frame Σw represents

Camera

Image

gch

Sc

gwc
gco

Sh

gwh

gho

=g

gwo
Target Object So

Sw
Robot
Manipulator

Figure 1: Visual feedback system

the world frame, Σh represents the hand (end-eﬀector)
frame, Σc represents the camera frame, Σo represents
the object frame, respectively. Let pco ∈ R3 andeξ̂θco ∈ R3×3 denote the position vector and the rotation matrix from the camera frame Σc to the object
frame Σo . Then, the relative rigid body motion fromΣc to Σo can be represented by gco = (pco , eξ̂θco ), andgho = (pho , eξ̂θho ) from Σh to Σo , and gch = (pch , eξ̂θch )from Σc to Σh . Similarly, we will deﬁne the rigid body
motion gwh = (pwh , eξ̂θwh ) from Σw to Σh , and gwc =(pwc , eξ̂θwc ) from Σw to Σc , and gwo = (pwo , eξ̂θwo ) fromΣw to Σo , respectively, as in Fig. 1. The homogeneous
representation of gab = (pab , eξ̂θab ) and the notation ‘∧’(wedge) are given in reference7) .
The objective of the visual feedback control is to bringthe actual relative rigid body motion gho = (pho , eξ̂θho )to a given reference gd = (pd , eξ̂θd ). Our goal is todetermine the robot motion using the visual information
for this purpose. The reference gd = (pd , eξ̂θd ) for the
rigid body motion gho = (pho , eξ̂θho ) is assumed to beconstant throughout this paper, because the end of thehand can track the moving target object in this case.
In this subsection, let us derive a fundamental representation for the three coordinate frames of the visual−1
feedback system. Using the notation gab
as the inverseof gab , the relative rigid body motion gco = (pco , eξ̂θco )of the target object, relative to the camera frame Σc in
Fig. 1, is given by−1
gco = gwc
gwo

(1)

which is obtained from the composition rule for rigid
body transformations7) .
The relative rigid body motion involves the velocity ofeach rigid body. To this aid, let us consider the velocity
of a rigid body as described in reference7) . Now, wedeﬁne the body velocity of the camera relative to thebTTworld frame Σw as Vwc= [vwcωwc] ∈ R6 . Similarly,
the body velocity of the target object relative to ΣwbTTwill be denoted as Vwo= [vwoωwo] ∈ R6 .
Diﬀerentiating (1) with respect to time, the fundamental representation of the relative rigid body motion
gco is described as follows4, 5) .
b
b
b
Vco
= −Ad(gco−1 Vwc + Vwo)

The visual feedback control task should require the information of the relative rigid body motion gco . Since
the measurable information is only the image information in the visual feedback systems, we consider a nonlinear observer in order to estimate the relative rigid
body motion from the image information.
In the case of the ﬁxed camera conﬁguration,bi.e. Vwc= 0, the fundamental representation of the relative rigid body motion gco can be rewritten as(3)

We shall consider the following model which is reproduced from the fundamental representation (3) just as
Luenberger observer for linear systems.

ˆ(5)−ξ̄ˆθ̄co(pco − p̄co ) and eξ̂θee =in other words, pee = eˆ−ξ̄ˆθ̄coiﬀe−ξ̄θ̄co eξ̂θco . Note that pco = p̄co and eξ̂θco = egee = I4 , i.e. pee = 0 and eξ̂θee = I3 . Using the notationeR (eξ̂θ ), the vector of the estimation error is given by
ee := [pTee eTR (eξ̂θee )]T . Hence, ee = 0 iﬀ pee = 0 andeξ̂θee = I3 . Therefore, if the vector of the estimation
error is equal to zero, then the estimated relative rigid
body motion ḡco equals the actual relative rigid body
motion gco . The vector of the estimation error ee canbe obtained from the actual image information and theestimated one4, 5) .
Next, we will derive an estimation error system. The
estimation error system will be derived in the same way
as the fundamental representation for the visual feedback system. Diﬀerentiating (5) with respect to time,and combining (3) and (4), we can obtainbbVee= −Ad(gee−1 ue + Vwo .
)(6)

Eq. (6) represents the estimation error system.

2.3 Control Error System

2.2 Nonlinear Observer and Estimation
Error System

b
V̄co
= ue

−1
gee = ḡco
gco ,

(2)

b
T
T T
where Vco:= [vcoωco] . The notation Ad(gab ) means
the adjoint transformation associated with gab 7) .

b
b
Vco
= Vwo
.

In order to establish the estimation error system, wedeﬁne the estimation error gee = (pee , eξ̂θee ) between theestimated value ḡco and the actual relative rigid body
motion gco as(4)b
are the estimated value
where ḡco = (p̄co , eξ̄θ̄co ) and V̄coof the relative rigid body motion and the estimatedbody velocity, respectively. The new input ue is to be
determined in order to converge the estimated value tothe actual relative rigid body motion. Because the design of ue needs a property of the whole visual feedback
system, we will propose ue in Section 3.2.

Let us derive the control error system in the same way
as the estimation error system in order to establish thevisual feedback system. Similarly to (1), the relative
rigid body motion gho = (pho , eξ̂θho ) of the target object, relative to the hand frame Σh in Fig. 1, is givenby−1
gho = gch
gco .

(7)

Because gco can not be obtained directly, we represent
the relative rigid body motion gho with the estimatedone ḡco as−1ḡho = gchḡco .

(8)−1
Here gch = gwc
gwh can be obtained directly, because
the rigid body motion gwc = (pwc , eξ̂θwc ) from Σw toΣc and gwh = (pwh , eξ̂θwh ) from Σw to Σh is known
by the structure of the system and the angle of themanipulator. It is supposed that the relative rigid body
motion gch can be measured exactly. Since the problemof the camera calibration is one of important research
topics and good solutions to it are reported in some
papers (see, e.g. reference8) ), we will not consider theerror of the camera calibration in this paper.
Then, the fundamental representation of the relative
rigid body motion ḡho will be obtained in the same way
as (2).
bbbV̄ho= −Ad(ḡ−1 ) Vch+ V̄cohob= −Ad(ḡ−1 ) Vwh
+ ue
ho

(9)

b
b
where we exploit (4) and Vch= Vwh
which is derived−1
from gch = gwc gwh . Here we deﬁne the control error
between the estimated value ḡho and the reference ofthe relative rigid body motion gd as

gec = gd−1 ḡho .

b
Vwo
OMFC

gco

(10)
ue

It should be remarked that the estimated relative rigid
body motion equals the reference one if and only if thecontrol error is equal to the identity matrix in matrixˆ
form, i.e. pd = p̄ho and eξ̂θd = eξ̄θ̄ho iﬀ gec = I4 . Using
the notation eR (eξ̂θ ), the vector of the control error isdeﬁned as ec := [pTec eTR (eξ̂θec )]T . Note that ec = 0 iﬀ
pec = 0 and eξ̂θec = I3 . Similarly to (6), the control
error system can be obtained asbb= −Ad(ḡ−1 ) Vwh+ ue .
Vec

(11)

ho

This is dual to the estimation error system.

Combining (6) and (11), we construct the visual feedback system as follows.
 b   
−Ad(ḡ−1 )
I
Vec
0
b
ho
=
Vwo
(12)
uce +
b
Vee
I
0
−Ad(gee−1)b T) uTe ]T denotes the control input
where uce := [(Vwhfor the visual feedback system. Let us deﬁne the errorTvector of the visual feedback system as e := eTc eTe
which contains of the control error vector ec and theestimation error vector ee . It should be noted that if
the vectors of the control error and the estimation error are equal to zero, then the estimated relative rigid
body motion ḡho equals the reference one gd and the estimated one ḡco equals the actual one gco , respectively.
Moreover, the error and the error vector between ḡho
and gho can be also represented as gee and ee by (5),(7) and (8), while gee and ee are deﬁned as the error andthe error vector between ḡco and gco in (5). Therefore,
the actual relative rigid body motion gho tends to thereference one gd when e → 0.
Now, we show an important lemma concerning a relation between the input and the output of the visualfeedback system.
b
Lemma 1 If Vwo= 0, then the visual feedback system(12) satisﬁes

 T0

uTce νcedτ ≥ −βce , ∀T > 0

where νce is deﬁned asνce :=−AdT(g−1 )dAd(e−ξ̂θec )and βce is a positive scalar.

0
−I

(13)


e

(14)

f
gco , fEstimated
OMFC

gco

uce
HMFC
b
Vwh

gch

Image Jacobian
JO

ee

+

^ ec)ec
Ad(e xq

gco
+

gho

nce

DesiregdgchTAd(gd-1)ec

Figure 2: Block diagram of the visual feedback system

Lemma 1 is proved by considering the following positive deﬁnite function
Vce (e) =2.4 Property of Visual Feedback System

Visual Feedback System

11pec 2 + φ(eξ̂θec ) + pee 2 + φ(eξ̂θee )(15)22where φ(eξ̂θ ) := 12 tr(I − eξ̂θ ) is the error function of therotation matrix (see, e.g. reference9) ).
The block diagram of the visual feedback system isshown in Fig. 2. In Fig. 2, let us take uce as the inputand νce as its output. Then, Lemma 1 would suggestthat the visual feedback system (12) is passive from the
input uce to the output νce.

3. Passivity-based Control of Dynamic Visual Feedback System
3.1 Property of Dynamic Visual Feedback System
The manipulator dynamics can be written as
M (q)q̈ + C(q, q̇)q̇ + g(q) = τ + τd(16)

where q, q̇ and q̈ are the joint angles, velocities andaccelerations, respectively. τ is the vector of the input
torques and τd represents a disturbance input.
b
The body velocity of the hand Vwh
is given byb= Jb (q)q̇
Vwh

(17)

where Jb (q) is the manipulator body Jacobian7) . We deﬁne the reference of the joint velocities as q̇d := Jb† (q)ud
where ud represents the desired body velocity of thehand.
Let us deﬁne the error vector with respect to the joint
velocities of the manipulator dynamics as ξ := q̇ − q̇d .
Here, we deﬁne the weight matrices Wc := diag{wpcI3 ,wrc I3 } ∈ R6×6 and We := diag{wpe I3 , wre I3 } ∈ R6×6
where wpc, wrc, wpe , wre ∈ R are positive. Now, we
consider the passivity–based dynamic visual feedback
control law as follows.
τ = M (q)q̈d + C(q, q̇)q̇d + g(q)+JbT (q)AdT(g−1 ) Wc ec + uξ . (18)dThe new input uξ is to be determined in order to achieve
the control objectives.
Using (12), (16) and (18), the visual feedback systemwith manipulator dynamics (we call the dynamic visualfeedback system) can be derived as follows⎤⎡⎤ ⎡−M −1 Cξ + M −1 JbT AdT(g−1 ) Wc ecξ˙d⎥⎣ Vb ⎦= ⎢−Ad(ḡ−1 ) Jb ξ
⎦
⎣
ec
ho
b
Vee
0
⎡ −1 ⎤
⎡ −1
⎤

M
0
0
M
0 
τd
⎣
⎦
⎦
−1
I
u
+
+ ⎣ 0 −Ad(ḡho
0
0
)
b
Vwo
0 I
0
0
−Ad(gee
−1
)


εx
z=
(19)
ρu
where x := [ξ T eTc eTe ]T and u := [uTξ uTd uTe ]T . εand ρ are weight matrices for the state and the input,
respectively. We deﬁne the disturbance of the dynamic


b T T
visual feedback system as w := τdT (Vwo)
. Beforeconstructing the dynamic visual feedback control law,
we derive an important lemma.

where Kξ :=diag{kξ1 , · · · , kξn } denotes the positive gain
matrix for each joint axis. Kc := diag{kc1 , · · · , kc6 } andKe := diag{ke1 , · · · , ke6} are the positive gain matricesof x, y and z axes of the translation and the rotation
for the control error and the estimation error, respectively. The result with respect to asymptotic stabilityof the proposed control input (23) can be established asfollows.
Theorem 1 If w = 0, then the equilibrium point x = 0
for the closed-loop system (19) and (23) is asymptotic
stable.
2(Proof) Omitted due to space limitations.

The block diagram of the dynamic visual feedback
system is shown in Fig. 3.
T

Ad(gd-1)Wcecb
Vwh(=Jb(q)q)_ud

Manipulator
Dynamics

ux

x

ue

VisualFeedback
Systemnce

Lemma 2 If w = 0, then the dynamic visual feedback
system (19) satisﬁes
 TuT νdτ ≥ −β, ∀T > 0(20)

Figure 3: Block diagram of the dynamic visual feedback
systemwhere

3.3 L2 -gain Performance Analysis for
Dynamic Visual Feedback System

0

⎤
I
0
0
T
⎥
⎢
ν := N x, N := ⎣ 0 −Ad(gd−1 ) Wc 0 ⎦0 Ad(e−ξ̂θec ) Wc −We⎡(21)

and β is a positive scalar.

Next, we consider L2 -gain performance analysis for the
dynamic visual feedback system. Now, let us deﬁne
P := N T KN −1
1
1
W − ε2 − ρKN 2 (24)
2
2γ
2
2

Lemma 2 is proved by considering the following positive deﬁnite function

where γ ∈ R is positive and W := diag{I, 0, We2 }. Then
we have the following theorem.

1 T
1ξ M ξ + wpcpec 2 + wrc φ(eξ̂θec )2
2
1+ wpe pee 2 + wre φ(eξ̂θee ). (22)2Theorem 2 Given a positive scalar γ and consider thecontrol input (23) with the weight matrices ε, ρ, Wc and
We and the gains Kξ , Kc and Ke such that the matrix
P is positive semi-deﬁnite, then the closed-loop system(19) and (23) has L2 -gain ≤ γ.

V (x) =The visual feedback system (12) satisﬁes the passivity
property as described in (13). It is well known that the
manipulator dynamics (16) also has the passivity. Thesepassivity properties are connected by the manipulator
Jacobian (17). In Lemma 2, the inequality (20) would
suggest that the dynamic visual feedback system (19) is
passive from the input u to the output ν.

3.2 Stability Analysis for Dynamic Visual Feedback System
It is well known that there is a direct link between passivity and Lyapunov stability. Thus, we propose thefollowing control input.
⎡
⎤
Kξ 0
0
u = −Kν = −KN x, K := ⎣ 0 Kc 0 ⎦ (23)00 Ke

(Proof) By diﬀerentiating the positive deﬁnite function V deﬁned in (22) along the trajectory of the closedloop system and completing the squares, it holds thatγ21V̇ + z2 − w2 ≤ −xT P x ≤ 0
2
2

(25)

if P is positive semi-deﬁnite. Integrating (25) from 0 to
T and noticing V (T ) ≥ 0, we have T
 Tz2 dt ≤ γ 2
w2 dt + 2V (0), ∀T > 0. (26)00This completes the proof.
2
γ represents a disturbance attenuation level for the
dynamic visual feedback system. Theorem 1 and 2 canbe proved using the energy function (22) as a Lyapunov
function and a storage function, respectively.

50

zh

Camera Frame

xh

5w
yw

xw

-100

xo

5o
yo

0.1

0.2

0.45
xo [m]( 0 < t < 4 [s] )

0.3

0.2

-300

0

4.0
8.0
time [s]

12.0

0

4.0
8.0
time [s]

12.0

100
0

-100

-50

-200

0

4.0
8.0
time [s]

12.0

-300

Gain A : Kc = diag{10, 10, 5, 5, 5, 10},
Wc = I, We = I

0.1

0.6

12.0

is chosen as Ke = 15I and Kξ = diag{10, 10}. Kc andthe weight matrices Wc and We are chosen as follows

yo [m]

0yo [m]

0

4.0
8.0
time [s]Figure 6: Estimated feature points(top: Gain A, bottom: Gain B)

Figure 4: Experimental arm
-0.1

0

0

-100

-0.1

-200

50

yh

zo
Object Frame

-100

-50

5h

Hand Frame

fy1 [pixel]

yc

zw

0

fy1 [pixel]

5c

100

0

fx1 [pixel]

xc

World Frame

fx1 [pixel]

zc

0.6

0.45
xo [m]

0.3( 4 < t < 9.6 [s] )Figure 5: Trajectory of the target object

4. Experimental Case Study
The manipulator use in the study, known as SICE-DD
arm (see Fig. 4), is controlled by a digital signal processor(DSP) from DSPACE Inc., which utilize a powerPC
750 running at 480 MHz. Control problem is written
in MATLAB and SIMULINK, and implemented on the
DSP using the Real-Time Workshop and dSPACE Software which includes ControlDesk, Real-Time Interface
and so on. A PULNiX TM-7EX camera is mounted on[0.47 0.10 0.19]T [m]. The video signals is acquired bya frame graver board PicPort-Stereo-H4D and a image
processing software HALCON.
We deﬁne the four coordinates which are describedin Fig. 4. Let the target object have four feature points
which are projected on the display. The object moves
along a straight line (0 ≤ t < 4) and a “Figure 8”
motion (4 ≤ t ≤ 9.6) as depicted in Fig. 5.
The experimental tests were carried out with thefollowing initial conﬁguration: ec (0) = ee (0) = 0,
q1 (0) = 30◦, q2 (0) = −30◦ , and q̇1 (0) = q̇2 (0) = 0. The
objective of the visual feedback control is to bring theactual relative rigid body motion gho = (pho , eξ̂θho ) toa given reference gd = (pd , eξ̂θd ). In this study, we set areference of position and rotation as pd = [0 0 − 0.81]T ,eξ̂θd = I.
Firstly, we conﬁrm the eﬀectiveness of the weight matrices Wc and We . The control gain of the manipulator

Gain B : Kc = diag{100, 100, 50, 50, 50, 100},
Wc = 0.1I, We = I.
Fig. 6 presents one of the four estimated featurepoints. The top graph and the bottom one show thecase of Gain A and Gain B, respectively. In this ﬁgure,the dashed lines denote the feature points obtained by
the actual image information and the solid lines denotethe feature points obtained by the estimated one. The
estimation error of the feature points can be decreasedby using the weight matrices as Gain B. Thus, we consider that the weight matrices play the role of the design
parameter for the estimation.
Next, we verify the stability and L2 -gain performanceof the dynamic visual feedback system. We design theweight matrices concerning controlled output as ε =diag{0.4, 0.4, 1, 1, 0.25, 0.25, 0.25, 1, 2.5, 2.5, 0.025,
0.025, 0.025, 2.5}, ρ = diag{0.02, 0.02, 2, 2, 1, 1, 1, 2,
10, 10, 0.1, 0.1, 0.1, 10} × 10−3 . Also we select Gain C
as follows
Gain C : Kc = diag{240, 240, 120, 120, 120, 240},
Ke = 30I, Kξ = diag{10, 10}, Wc = 0.1I, We = I
Then, the closed-loop system (19) and (23) with Gain
B has γ = 0.269 and with Gain C has γ = 0.225.
Fig. 7 and 8 present the control error vectors ec andthe estimation error vectors ee , the top graph, the middle one and the bottom one show the error of the translation of the x axis, the error of the translation of they axis and the error of the rotation of the z axis, respectively. In these ﬁgures, the left graphs denote theerror of the case of γ = 0.269 and the right ones denotethe error of the case of γ = 0.225. In Fig. 9, the top
graph and the bottom one show the norm of z in thecase of γ = 0.269 and γ = 0.225, respectively. In the

8.0

12.0

4.0

8.0

12.0

e^xqecz [rad]8.0e^xqecz [rad]

4.0

4.0
8.0
time [s]4.0
8.0time [s]12.0

Figure 7: Control errors(left: Gain B, right: Gain C)
0.02
0.01
0
-0.01
-0.02
0
12.0
0.04
0.02
0
-0.02
-0.04
12.0
0
0.1
0.05
0
-0.05
-0.1
12.0
0

4.0

8.0

12.0

4.0

8.0

12.0

4.0
8.0
time [s]12.0e^xqeez [rad]8.0e^xqeez [rad]

4.0

4.0
8.0
time [s]

0.1
0.05
0
0

2.0

4.0

6.0
time [s]

8.0

10.0

12.0

2.0

4.0

6.0
time [s]

8.0

10.0

12.0

0.2
0.15
0.1
0.05
0
0

and L2 -gain performance of the dynamic visual feedback system.

peey [m]8.0peey [m]

4.0

0.25

0.2
0.15

Figure 9: Euclid norms of z
(top: Gain B, bottom: Gain C)

peex [m]peex [m]

0.02
0.01
0
-0.01
-0.02
0
0.04
0.02
0
-0.02
-0.04
0
0.1
0.05
0
-0.05
-0.1
0

norm || z || (g =0.269)

4.0pecy [m]8.0pecy [m]

4.0

0.25

norm || z || (g =0.225)

0.01
0
-0.01
-0.02
-0.03
12.0
0
0.04
0.02
0
-0.02
-0.04
12.0
0
0.1
0.05
0
-0.05
-0.1
12.0
0
pecx [m]pecx [m]

0.01
0
-0.01
-0.02
-0.03
0
0.04
0.02
0
-0.02
-0.04
0
0.1
0.05
0
-0.05
-0.1
0

Figure 8: Estimation errors(left: Gain B, right: Gain C)

case of the static target object, i.e. after t = 9.6 [s],all errors in Fig. 9 tend to zero. It can be concluded
that the equilibrium point is asymptotically stable if
the target object is static. In the case of γ = 0.225,
the performance is improved as compared to the case ofγ = 0.269. After all, the experimental results show thatL2 -gain is adequate for the performance measure of the
dynamic visual feedback control.

5. Conclusion
This paper dealt with the control and the estimationof dynamic visual feedback systems with a ﬁxed camera. Firstly the fundamental representation of the relative rigid body motion and the nonlinear observer aredescribed in order to derive the dynamic visual feedback system. Secondly we proposed the dynamic visual feedback control law which is based on passivity.
Stability and L2 -gain performance analysis for the dynamic visual feedback system has been discussed using
the energy function. Finally experimental results on
SICE–DD arm have been shown to verify the stability

References[1] S. Hutchinson, G. D. Hager and P. I. Corke, “A
Tutorial on Visual Servo Control,” IEEE Trans.
Robotics and Automation, Vol. 12, No. 5, pp. 651–
670, 1996.
[2] Mini Special Issue, “Visual Servoing,” Journal of
SICE, Vol. 40, No. 9, 2001.
[3] S. Yu and B. J. Nelson, “Autonomous Injection of
Biological Cells Using Visual Servoing,” In:D. Rus
and S. Singh (Eds.), Experimental Robotics VII,
Springer–Verlag, pp. 169–178, 2001.
[4] H. Kawai, T. Murao and M. Fujita, “An Experimental Study of Dynamic Visual Feedback Control
on SICE-DD Arm,” Proc. of the 42nd SICE Annual
Conference, pp. 686-691, 2003.
[5] H. Kawai and M. Fujita, “Passivity-based Dynamic
Visual Feedback Control for Three Dimensional
Target Tracking:Stability and L2 -gain Performance
Analysis,” Proc. of the 2004 American Control
Conference, 2004(to appear).
[6] H. Kobayashi et al., Practical Control of Robot Manipulators, Corona Publishing Co., 1997.
[7] R. Murray, Z. Li and S. S. Sastry, A Mathematical
Introduction to Robotic Manipulation, CRC Press,
1994.
[8] B. E. Bishop and M. W. Spong, “Adaptive Calibration and Control of 2D Monocular Visual Servo
Systems,” Control Engineering Practice, Vol. 7,
No. 3, pp. 423–430, 1999.
[9] F. Bullo and R. Murray, “Tracking for Fully Actuated Mechanical Systems: a Geometric Framework,” Automatica, Vol. 35, No. 1, pp. 17–34, 1999.