Passivity-based Control of Visual Feedback Systems
with Dynamic Movable Camera Configuration
Toshiyuki Murao, Hiroyuki Kawai and Masayuki Fujita
Abstract— This paper deals with control of visual feedback
systems with a dynamic movable camera configuration. Thisconfiguration consists of a robot manipulator and a camera thatis attached to the end-effector of another robot manipulator.
Firstly the brief summary of the dynamic visual feedback
system with an eye-in-hand configuration is given with thefundamental representation of a relative rigid body motion.
Secondly we construct the visual feedback system with adynamic movable camera configuration by combining the handcontrol error system. Next, we derive the passivity of the
dynamic visual feedback system. Based on the passivity, stability
and L2 -gain performance analysis are discussed. Finally thevalidity of the proposed control law can be confirmed bycomparing the experimental results.

I. I NTRODUCTION
Robotics and intelligent machines need many information to behave autonomously under dynamical environments.
Visual information is undoubtedly suited to recognize unknown surroundings. Vision based control of robotic systems
involves the fusion of robot kinematics, dynamics, andcomputer vision to control the motion of the robot in an
efficient manner. The combination of mechanical control
with visual information, so-called visual feedback control orvisual servoing, should become extremely important, when
we consider a mechanical system working under dynamical
environments [1][2].
In classical visual servoing, many practical methods are reported by two well known approaches with two camera configurations, i.e., position-based visual feedback control andimage-based one with an eye-in-hand configuration or a fixedcamera one (see, e.g. [1]). Specifically, several approaches
which originate from the classical visual servoing have been
proposed to overcome the drawbacks which are found froman experimental point of view. In [3], the 2 1/2-D visual
servoing which includes the advantages of both positionbased and image-based visual servoing is proposed in orderto guarantee the robustness with respect to calibration errors.
The partitioned visual servoing is considered to remain allfeatures in the image [4]. Recently, the switching approach
among position-based visual servoing and backward motion
is investigated for dealing with the field of view problem [5].
However, classical visual servoing algorithms assume that the
manipulator dynamics is negligible and do not interact withthe visual feedback loop. This assumption is invalid for highspeed tasks, while it holds for kinematic control problems.
T. Murao and M. Fujita are with the Department of Mechanical andControl Engineering, Tokyo Institute of Technology, Tokyo 152-8552,
JAPAN fujita@ctrl.titech.ac.jp
H. Kawai is with the Faculty of Engineering, Kanazawa Institute ofTechnology, Ishikawa 921-8501, JAPAN

CameraCameraManipulator

Work
Manipulator

5c
gch

5h
gzh

gwc
gco

gwh
gwz
gwo

5w
World Frame

gho
5z
5o

Target Object

Fig. 1.
ration.

Visual feedback system with a dynamic movable camera configu-

5c

gwc

World
Frame

5w

gwo

gwz
5z

Fig. 2.

gwh
gzh

gco
gch
gho

5o
Target Object

5h

The upper half of the human body.

Kelly et al. [6] considered a simple image-based controller
for dynamic visual feedback system in the three dimensional(3D) workspace under the assumption that the objects’
depths are known. Zergeroglu et al. developed an adaptive
control law for the position tracking and the camera calibration problems of the dynamics visual feedback system withparametric uncertainties in [7]. Cowan et al. [8] addressed
the problems of the field of view for the 3D dynamic visualfeedback system by using the navigation functions. Although
the good solutions to the set-point problems are reported inthose papers, few results have been obtained for the tracking
problems of the moving target object in the full 3D dynamic
visual feedback system. Additionally, most of the previous
works are discussed for the camera configurations in detail,while the position-based visual feedback control and theimage-based one are developed in some issues [3][4][5].
In this paper, we discuss the visual feedback control for thetarget tracking problem with a dynamic movable camera configuration as in Fig. 1. This configuration consists of a robot
manipulator (we call a work manipulator) and a camera thatis attached to the end-effector of another robot manipulator

(we call a camera manipulator). While the objective of thissystem is obviously to control the work manipulator, we also
control the camera manipulator in order to enlarge the fieldof view. In terms of controlling the camera manipulator(eye)and the work manipulator(hand), this system can be regarded
as the upper half of the human body as depicted in Fig. 2.
Moreover, the dynamic visual feedback system with an eyein-hand configuration [9] and a fixed camera one [10] canbe interpreted as the special case of this system. Then, we
can derive that the visual feedback system with a dynamic
movable camera configuration preserves the passivity of thevisual feedback system which is obtained in our previous
works [9][10][11][12]. The validity of the proposed control
law is confirmed by comparing the experimental results.
Throughout this paper, we use the notation eξ̂θab ∈ R3×3
to represent the change of the principle axes of a frame Σb
relative to a frame Σa . The notation ‘∧’ (wedge) is the skewˆ = ξ ×θ for the vector crosssymmetric operator such that ξθ
product × and any vector θ ∈ R3 . The notation ‘∨’ (vee)denotes the inverse operator to ‘∧’: i.e., so(3) → R3 . ξab ∈R3 specifies the direction of rotation and θab ∈ R is theangle of rotation. Here ξ̂θab denotes ξ̂ab θab for the simplicity
of notation. We use the 4 × 4 matrixeξ̂θab pab
gab =
(1)
0
1

has three coordinate frames, i.e. the world frame Σw , thecamera frame Σc and the target object frame Σo . Besides,
control objective is to bring gco to gcd .
The relative rigid body motion from Σc to Σo can be led
by using the composition rule for rigid body transformations([13], Chap. 2, pp. 37, eq. (2.24)) as follows

as the homogeneous representation of gab = (pab , eξ̂θab ) ∈SE(3) which is the description of the configuration of aframe Σb relative to a frame Σa . The adjoint transformation
associated with gab is denoted by Ad(gab ) [13]. Let us define
the vector form of the rotation matrix as eR (eξ̂θab ) :=sk(eξ̂θab )∨ where sk(eξ̂θab ) denotes 12 (eξ̂θab − e−ξ̂θab ).

B. Estimation Error and Camera Control Error Systems

II. DYNAMIC PASSIVIT Y- BASED V ISUAL F EEDBACKS YSTE M WITH AN E YE - IN - HAND CONFIGURATI ON
A. Fundamental Representation for Visual Feedback System
In this paper, we consider the visual feedback systemwith a dynamic movable camera configuration as shown in
Fig. 1. Thus, the whole system has five coordinate frames
which consist of a world (base of the camera manipulator)
frame Σw , a camera (end-effector of the camera manipulator)
frame Σc , a base frame of the work manipulator Σz , ahand (end-effector of the work manipulator) frame Σh anda target object frame Σo as in Fig. 1. Then, gwc denotes
the rigid body motion from Σw to Σc . Similarly, the rigid
body motions gwz , gwh and gwo , and the relative rigid body
motions gch , gco , gzh and gho are represented, respectively,
as shown in Fig. 1.
The objective of the visual feedback control is to bringthe actual relative rigid body motions gco and gho to givenreferences gcd and ghd , respectively. Our goal is to determine
the camera’s motion and the hand’s motion using the visual
information for this purpose.
Firstly, we construct the dynamic visual feedback systemwith an eye-in-hand configuration in order to enlarge thefield of view. The visual feedback system with an eye-inhand configuration uses only the camera manipulator which−1
gco = gwc
gwo .

(2)

The fundamental representation of the relative rigid body
motion involves the velocity of each rigid body. To this aid,
let us consider the velocity of a rigid body as describedin [13]. Now, we define the body velocity of the camerabT
T T
relative to the world frame Σw as Vwc= [vwcωwc] , where
vwc and ωwc represent the velocity of the origin and theangular velocity from Σw to Σc , respectively ([13] Chap. 2,
eq. (2.55)).
Differentiating (2) with respect to time, the fundamental
representation of the relative rigid body motion gco is described as follows [9].
b
b
b
Vco
= −Ad(gco−1 V) wc + Vwo(3)bwhere Vwois the body velocity of the target object relativeto Σw . Roughly speaking, the relative rigid body motion
gco will be derived from the difference between the camerabbvelocity Vwc
and the target object velocity Vwo.

Here the brief summary of our prior work in [9] is given.
The visual information f(gco ) which includes the relative
rigid body motion can be exploited, while the relative rigid
body motion gco can not be obtained directly. In orderto bring the actual relative rigid body motion gco to agiven reference gcd in Fig. 1, we consider the control andestimation problems in the visual feedback system. Firstly,
we shall consider the following model which just comes fromthe fundamental representation (3).
b
b
−1 V
V̄co
= −Ad(ḡco) wc + ue(4)b
are the estimated value of the relative
where ḡco and V̄corigid body motion and the estimated body velocity from Σcto Σo , respectively. ue is the input in order to converge theestimated value to the actual relative rigid body motion. Next,
the estimation error of the relative rigid body motion fromΣc to Σo , i.e. the error between ḡco and gco , is defined as−1
gee = ḡco
gco ,

(5)

which is called the estimation error. Using the notationeR (eξ̂θ ), the vector of the estimation error is given by
ee := [pTee eTR (eξ̂θee )]T . Note that ee = 0 iff pee = 0
and eξ̂θee = I3 . Then, the estimation error vector ee can beobtained by using image information f(gco ). The estimation
error system is represented bybbVee= −Ad(gee−1 ue + V
wo .
)

(6)

Similarly, we define the error between gcd and ḡco , whichis called the camera control error, as follows−1
gec = gcd
ḡco .

(7)

The vector of the camera control error is defined as ec :=[pTec eTR (eξ̂θec )]T . The camera control error system is described by
b
b
b
Vec
= −Ad(ḡco−1 Vwc + ue − Ad −1 Vcd)(gec )(8)b
where Vcd
is the desired body velocity of the relative rigid
body motion gco .

C. Dynamic Passivity-based Visual Feedback System with an
Eye-in-hand Configuration
The manipulator dynamics of the camera manipulator (wecall the camera manipulator dynamics) can be written as
Mc (qc )q̈c + Cc (qc , q̇c)q̇c + gc(qc ) = τc + τcd(9)where Mc ∈ Rnc ×nc is the inertia matrix, Cc ∈ Rnc ×nc is
the Coriolis matrix, gc ∈ Rnc is the gravity terms, and qc ,q̇c and q̈c are the joint angles, velocities and accelerationsof the camera manipulator, respectively. τc is the vector of
the input torques and τcd represents a disturbance input. The
body velocity of the camera is given byb
Vwc= Jcb (qc )q̇c(10)where Jcb (qc ) is the manipulator body Jacobian [13]. We de†(qc )ucd
fine the reference of the joint velocities as q̇cd := Jcb
where ucd represents the desired body velocity of the camera.
bThus, Vwc
in (8) should be replaced by ucd .
Let us define the error vector with respect to the joint
velocities of the camera manipulator as ξc := q̇c − q̇cd . Now,
we consider the passivity–based dynamic visual feedback
control law as follows
τc = Mc (qc )q̈cd + Cc (qc, q̇c)q̇cd + gc (qc)T(qc)AdT(g−1 ) ec + uξc .
+Jcb

(11)

cd

The new input uξc is to be determined in order to achieve
the control objectives.
Using (6) and (8)–(11), the visual feedback system withthe camera manipulator dynamics (we call the dynamic
visual feedback system with an eye-in-hand configuration)
can be derived as follows

3
3 2
T AdT
−Mc−1 Cc ξc + Mc−1 Jcb
−1 ec
ξ˙c
)
(g
cd
7
b 5=6
4 Vec
5
4
−Ad(ḡ−1 ) Jcb ξc
co
b
Vee
0
2
32
3 2 −1 3
Mc−1
0
0
uξc
Mc 0 » τ –6 0 −Ad −1 I
74
cd
b 5+4
u
+
Ad
V
+4
0 05 b .
5
cd(gcd ) cd(ḡco )
Vwoue
00 −Ad (g−1 )
0 I
ee
(12)
2

The details are omitted due to space limitations, dynamic
visual feedback system with an eye-in-hand configuration(12) is passive from the input to a appropriate output ([9]Lemma 2).
III. PASSIVIT Y- BASED V ISUAL F EEDBACK C ONTROLWITH A DYNAMIC M OVABLE C AMERA C ONFIGURAT ION

A. Hand Control Error System
In this section, the work manipulator is considered inaddition to the camera manipulator, as depicted in Fig. 1, In
other words, we determine the hand’s motion of the work

manipulator (we call only the hand) to bring the actual
relative rigid body motion gho to a given reference ghd , inaddition to bring gco to gcd .
Because gco can not be obtained directly, we represent therelative rigid body motion from Σh to Σo with the estimatedone ḡco as−1ḡho = gchḡco .

(13)

−1
gwc
gwz gzh

Here gch =can be obtained directly, because
the rigid body motions gwc , gzh and gwz is known by theangle of the manipulator and the structure of the system. It
is supposed that the relative rigid body motion from Σc toΣh , i.e. gch , can be measured exactly. Since the problem of
the camera calibration in our approach is treated in [11], we
will not consider the error of the camera calibration in thispaper.
Here we define the hand control error between the estimated value ḡho and the reference of the relative rigid body
motion ghd as−1
geh = ghd
ḡho .

(14)

Using the notation eR (eξ̂θ ), the vector of the hand control
error is defined as eh := [pTeh eTR (eξ̂θeh )]T .
Similarly to (6) and (8), the hand control error system canbe obtained as
b
b
b
Veh
= −Ad(ḡ−1 ) Vwh+ ue − Ad(g−1 ) Vhd
ho

eh

(15)

b
where Vhd
is the desired body velocity of the relative rigid
body motion gho .

B. Passivity-based Visual Feedback System with a Dynamic
Movable Camera Configuration
Similarly the camera manipulator dynamics, the manipulator dynamics of the work manipulator (we call the work
manipulator dynamics) can be written as
Mh (qh )q̈h + Ch (qh , q̇h )q̇h + gh (qh ) = τh + τhd (16)where Mh ∈ Rnh ×nh is the inertia matrix, Ch ∈ Rnh ×nh is
the Coriolis matrix, gh ∈ Rnh is the gravity terms, and qh ,q̇h and q̈h are the joint angles, velocities and accelerationsof the work manipulator, respectively. τh is the vector of the
input torques and τhd represents a disturbance input. Using
b
b
Vzh
= Vwh
, the body velocity of the hand is given byb
Vwh= Jhb (qh )q̇h(17)where Jhb (qh ) is the manipulator body Jacobian of the work
manipulator. We define the reference of the joint velocities†
as q̇hd := Jhb(qh )uhd where uhd represents the desired body
velocity of the hand.
In addition, we define the error vector with respect to thejoint velocities of the hand manipulator as ξh := q̇h − q̇hd .
The passivity–based dynamic visual feedback control law is
proposed as follows
τh = Mh (qh )q̈hd + Ch (qh , q̇h )q̇hd + gh (qh )T(qh )AdT(g−1 ) eh + uξh .
+Jhb

(18)

hd

Using (12), (15)–(18), the visual feedback system withthe camera and the work manipulator dynamics (we callthe visual feedback system with a dynamic movable camera
configuration, or the dynamic visual feedback system forshort) can be derived as follows
2
3
T AdT
3
−Mc−1 Cc ξc + Mc−1 Jcb
−1 ec
)
(gcd
ξ̇c
6
7
T AdT
6 ξ̇
7 6 −Mh−1 Ch ξh + Mh−1 Jhb−1 eh 76 h 7 67(ghd)
6 Vb 7=67−Ad (ḡ−1 ) Jcb ξc
6 ec 7 67co
7
4 Vb 5 6
ee
4
5
0
b
Veh
−Ad −1 J ξ
2

2

Mc−1

6 0
6
6
+6 0
6 0
4
0

(ḡho ) hb h

3
2 −1
3
0
0
0
0
Mc
0 0
−1
7
Mh
0
0
0
−1
7 6 0 Mh 0 7
7 6
7
I
0
0 −Ad (ḡ−1 )7u+6 0
0 07w
co
7 4
5
0
0
0
−Ad (g−1 )
0
0 I
5
ee
0
0 0
0
0
I
−Ad(ḡ−1 )
ho

(19)
b T
where u := [uTξc uTξh (ucd + Ad(gcd ) Vcd) uTe (uhd +b T TAd(ghd ) Vhd ) ] . We define the disturbance of the dynamic
 T T

b T T
visual feedback system as w := τcd
τhd (Vwo).
Before constructing the dynamic visual feedback control
law, we derive an important lemma.
Lemma 1: If w = 0, then the dynamic visual feedback
system (19) satisfies
 T
uT νdτ ≥ −β, ∀T > 0(20)0where ν is defined as
2

3
I 0
0
0
0
3
2
ξc
60I
7
0
0
0
6
7
6 ξh 7
6 0 0 −AdT −1
700
77 , x:=6(gcd )
ν:=N x, N:=6
6 ec 76
7
4 e 5
6 0 0 Ad(e−ξ̂θec ) −I Ad −ξ̂θeh 7
e
(e
) 5
4
eh
00
0
0 −Ad T −1
(ghd )and β is a positive scalar.
Due to space limitations, the proof is only sketched. By
using the following positive definite function, the proof canbe completed.
V =

1 T
1 T
Mh ξh + E(gec ) + E(gee) + E(geh). (21)ξ Mc ξ c + ξ h2 c2where E(g) := 12 p2 + φ(eξ̂θ ) and φ(eξ̂θ ) := 12 tr(I −eξ̂θ ) which is the error function of the rotation matrix (seee.g. [14]).
Lemma 1 would suggest that the dynamic visual feedback
system is passive from the input u to the output ν as in thedefinition in [15].
b= 0, the desired
Remark 1: If the camera velocity Vwc
camera velocity ucd = 0, the relative rigid body motion
gwz = I, and the camera control error ec and the camera
manipulator is not considered, then the dynamic visualfeedback system (19) represents the dynamic visual feedback
system with a fixed camera configuration [10]. From section
II, it is obvious that the dynamic visual feedback system withan eye-in-hand configuration is included. Thus, the dynamic
visual feedback system with an eye-in-hand configuration
and a fixed camera one are regarded as the special cases ofthe system (19). This is one of main contributions of thiswork.

C. 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 the following
control input.
u = −Kν = −KN x, K := diag{Kξc , Kξh , Kc, Ke , Kh }(22)
where Kξc := diag{kξc 1 , · · · , kξc nc } and Kξh :=diag{kξh1 , · · · , kξhnh } denote the positive gain matrix foreach joint axis of the camera manipulator and work
one, respectively. Kc := diag{kc1 , · · · , kc6}, Ke :=diag{ke1, · · · , ke6 } and Kh := diag{kh1 , · · · , kh6 } are thepositive gain matrices of x, y and z axes of the translationand the rotation for the camera control error, the estimation
one and the hand control one, respectively. The result withrespect to asymptotic stability of the proposed control input(22) can be established as follows.
Theorem 1: If w = 0, then the equilibrium point x = 0
for the closed-loop system (19) and (22) is asymptotic stable.
The proof is omitted here due to space limitations. Theorem 1 can be proved using the energy function (21) asa Lyapunov function. It is interesting to note that stability
analysis is based on the passivity as described in (20).
D. L2 -gain Performance Analysis for Dynamic Visual Feedback System
Based on the dissipative systems theory, we consider L2 gain performance analysis for the dynamic visual feedback
system (19) in one of the typical problems, i.e. the disturbance attenuation problem. Here, we reconstruct the dynamic
visual feedback system as the generalized plant as follows.
2
3
T AdT
3
−Mh−1 Ch ξh + Mh−1 Jhb
−1 eh
)
(ghd
ξ̇h
6
7
T AdT
6 ξ̇c 7 6 −Mc−1 Cc ξc + Mc−1 Jcb−1 ec 7
6
7 6
7
)
(gcd
6 Vb 7 =67−Ad (ḡ−1 ) Jhb ξh6 eh 7 6
7
7
4 Vb 5 6
ho
ec
4
5
−Ad
J
ξ
−1
c
cb
b
(ḡco )
Vee
0
3
2 −1
2 −1
3
0
0
0
0
Mh
Mh
0 0
7
6 0 Mc−1000
7
6 0 Mc−1 07
6
7
6
7
6
I
0 −Ad (g−1 ) 0+6 0
7uL +6 0
0 07w
eh
7
6
4
5
0
0 −Ad (g−1 )
I
0
0 0
5
4 0
ec
0
0
I
0
0
0
0
−Ad (g−1 )
2

ee

(23)
b T
where uL := [uTξh uTξc (Ad(g−1 ) uhd + Vhd) (Ad(g−1 ) ucd +
hd
cd
b T
T T
Vcd ) ue ] .
For the generalized plant of the dynamic visual feedback
system in (23), we consider the following input

uL = −KL NL xL
where KL := diag{Kξh , Kξc , Kh , Kc, Ke },

(24)

3
3
2
I 0
0
0
0
ξh
7
60I
0
0
0
6 ξc 7
7
6
7
00
−I
0
0 7 , xL:=6NL:=66 eh 7
7
6
4 e 5
0
−I
0 5
400
c
0 0 Ad −ξ̂θeh Ad(e−ξ̂θec ) −I
ee
2

(e

)

In order to derive simple gain conditions, we redefine
Kc = kc I and Ke = ke I where kc and ke are positive
scalars.

Theorem 2: Given a positive scalar γ and consider thecontrol input (24) with the gains Kξh , Kξc , Kh , kc and ke
such that the following inequalities (25)–(29) are satisfied,then the closed-loop system (23) and (24) has L2 -gain ≤ γ
1
1
Kξh −
I− I>0
2γ 2
2
1
1
Kξc −
I− I >02γ 2
2
1(2kc − 1)l
Kh − I −I>02
2kc − 1 − 2l
1kc − − l > 0
2
1
1
ke −
− >0
2γ 2
2

where
l=

(25)
(26)
(27)
(28)
(29)

ke (γ 2 + 1).
2
γ (2ke − 1) − 1(30)

Proof: Differentiating the positive definite function V
defined in (21) along the trajectory of the closed-loop system,
it can be verified that the inequality1γ2xL 2 −
w2
2
2
1
1
T
≤ −xT
W xL 2 + xL2
L NL K L NL x L +
2γ 2
2
= −xT
L P xL ≤ 0

V̇ +

(31)

holds if P := NLT KL NL − 2γ1 2 W − 12 I is positive semidefinite, where W := diag{I, I, 0, 0, I}. Integrating (31)from 0 to T and noticing V (T )≥0, we have T
 TxL2 dt ≤ γ 2
w2 dt + 2V (0), ∀T > 0. (32)00From the Schur complement,
2
6
6
6
P =6
6
6
4

0
0
Kξh − 2γ12 I − 12 I
0Kξc − 2γ12 I − 12 I000
Kh + ke I − 12 I00ke Ad(eξ̂θec ) Ad −ξ̂θeh
(e

0
0
0
ke Ad ξ̂θeh Ad(e−ξ̂θec ))` (e´kc + ke − 12 I
−ke Ad(e−ξ̂θec )−ke Ad −ξ̂θeh
(e
)
3

0

0
0
−ke Ad ξ̂θeh(e)−ke Ad(eξ̂θec )“”ke − 2γ12 − 12 I

7
7
7
7>0
7
5

)

(33)

can be modified as the conditions (25)–(29).
The conditions (25)–(29) can be regarded as an extensionof the ones for the disturbance attenuation of the robot
motion control which are described in Proposition 3.1 [16],although these are only sufficient conditions.
The L2 -gain performance analysis of the dynamic visualfeedback system is discussed via the dissipative systems theory. In H∞ -type control, we can consider some problems by
establishing the adequate generalized plant. This paper has
discussed L2 -gain performance analysis for the disturbanceattenuation problem. The proposed strategy can be extended
for the other-type of generalized plants of the dynamic visualfeedback systems.

xc

zc

World Frame

zw

xw

5w
yw

yc 5c

Camera Frame

xo

zo
5o

Object Framexhzhyh 5 hBase Frame

xz

zz
5z
yz

Hand FrameyoFig. 3. Experimental arm. (Left: the camera manipulator, Right: the work
manipulator)

IV. E XPERIMENTAL C ASE S TUDY
The experimental results on the same kind of two degreeof-freedom manipulators are shown in order to understandour proposed method simply, though it is valid for 3D visualfeedback systems. The experimental arms are depicted inFig. 3. The left side is the camera manipulator and the rightside is the work one. These manipulators are controlled by adigital signal processor(DSP) from dSPACE Inc., which utilizes 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 Sony XC-HR57 camera is attached at the tipof the camera manipulator(see Fig. 3). The video signals are
acquired by a frame graver board PicPort-Stereo-H4D and aimage processing software HALCON.
Due to space limitations, we present experimental results
for the L2 -gain performance analysis in the case of a moving
target object. Specifically, we compare the performance inthe case of a dynamic movable camera system and a fixedcamera system discussed in [10].
The target object which is projected on the display, hasfour feature points and moves for t = 9.6 [s] along a straightline (0 ≤ t < 4) and a “Figure 8” motion (4 ≤ t ≤ 9.6)
as depicted in Fig.3 [10]. The experiment is carried out
with the initial condition x = 0, pwc = [0.4732 0.1 0]T ,ξθwc = [0 0 0]T , pwz = [0 0 1.16]T , ξθwz = [0 0 0]T ,pwh = [0.4732 0.1 1.16]T , ξθwh = [0 0 0]T .
The objective of the visual feedback control is to bringthe actual relative rigid body motions gco and gho to givenreferences gcd and ghd , respectively. We use the references of
the relative rigid body motions as constant values, i.e. pcd =[0 0 − 0.84]T , ξθcd = [0 0 0]T , phd = [0 0 − 2]T , ξθhd =bb[0 0 0]T , Vcd= 0, Vhd= 0.
Here, we show a design procedure in order to assign thegains for the disturbance attenuation problem of the dynamic
visual feedback system.
Step 1: The hand control gain Kh and the camera control
gain kc are suitably selected.
Step 2: The estimation gain ke satisfying the conditions(27)–(29) is decided.
Step 3: The hand velocity gain Kξh and the camera
velocity gain Kξc satisfying the conditions (25) and(26) are chosen for a given γ.
Based on the design procedure, the following gains wereselected in order to confirm the adequacy of the L2 -gain

|| x || (C = 0.388)

0.25

V. C ONCLUSIONS

0.2
0.15
0.1
0.05
0
0

2

4

2

4

6

8

9.6

6

8

9.6

|| x || (C = 0.241)

0.25

Fig. 4.

0.2
0.15
0.1
0.05
0
0

time [s]Norm of the state. (top: with Gain A, bottom: with Gain B)

fx1 [pixel]

0-50

R EFERENCE S

-100
-150
0

2

4

2

4

6

8

9.6

6

8

9.6

fy1 [pixel]

100
0

-100
-200
0

This paper dealt with the control of visual feedback
systems with a dynamic movable camera configuration.
Moreover, this system has been included the dynamic visualfeedback system with an eye-in-hand configuration [9] anda fixed camera configuration [10]. Then, we derived that the
dynamic visual feedback system preserves the passivity ofthe visual feedback system which is obtained in our previous
works [9][10][11][12]. Stability and L2 -gain performance
analysis for the dynamic visual feedback system have been
discussed based on passivity with the energy function. The
validity of the proposed control law was confirmed bycomparing the experimental results. In our future work, webbhave to consider that the reference velocities Vcd
and Vhd
will play a role in the trajectory planning of the dynamic
visual feedback system.

time [s]Fig. 5. One of feature points. (Dynamic movable camera system: solid,
Fixed camera system: dashed)performance for the dynamic visual feedback system.
Gain A : γ = 0.388, Kξh = diag{5, 5}, Kξc = diag{5, 5},
Kh = diag{20, 20, 10, 10, 10, 20}, kc = 10, ke = 20
Gain B : γ = 0.241, Kξh = diag{10, 10}, Kξc = diag{10, 10},
Kh = diag{50, 50, 25, 25, 25, 50}, kc = 30, ke = 30

In Fig. 4, the top graph and the bottom one show the normof x in the case of γ = 0.388 and γ = 0.241, respectively.
It can be verified that the tracking performance is improvedin the case of the smaller value of γ from Fig. 4. Afterall, the experimental results show that L2 -gain is adequate
for the performance measure of the dynamic visual feedback
control.
Next, we compare the performance of the proposed control
law and the previous one discussed in [10]. Fig. 5 presents
one of the four feature points. In Fig. 5, the solid lines denotethe feature point of the case of the proposed control law, andthe dashed lines denote the feature point of the case of theprevious one. We can verify that the variation of the featurepoint with the proposed control law is less than with theprevious one. This result states that the target object almost
exists in the center of the camera, and the camera can move
not to miss the moving target object. Thus, we consider thatthe visual feedback control with a dynamic movable camera
configuration can enlarge the field of view, in comparison
with a fixed camera [10].

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