Passivity-based Control and Estimation of Dynamic VisualFeedback Systems with a Fixed Camera
Hiroyuki Kawai*, Toshiyuki Murao** and Masayuki Fujita**
Abstract— This paper deals with the control and the estimation of dynamic visual feedback systems with a fixed camera.
Specifically, we consider the target tracking problem of dynamic visual feedback systems in the three dimensional(3D)
workspace. Firstly the visual feedback system with four
coordinate frames is established by using the homogeneous
representation and 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 simulation results are shown toverify the stability and L2 -gain performance of the dynamic
visual feedback system.

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]. Recently, technological fields which
need visual feedback control are undoubtedly increasing,such as the autonomous injection of biological cells [3], thelaparoscopic surgery [4] and so on. Control will be more
important for intelligent machines as future applications.
Classical visual servoing algorithms assume that the
manipulator dynamics is negligible and do not interact withthe visual feedback loop. However, as stated in [5], thisassumption is invalid for high speed tasks, while it holds
for kinematic control problems. Kelly [5] considered the setpoint problems with a static target for the dynamics visualfeedback system which includes the manipulator dynamics.
In [6], Bishop et al. proposed an inverse dynamics based
control law for the position tracking and the camera calibration problems of the dynamics visual feedback system.
Recently, 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]. Although these control laws
guarantee the stability of the system based on the Lyapunov
method and are effective for the dynamics visual feedback*Information Technology Research Center, Hosei University, Tokyo
102–8160, JAPAN**Department of Electrical and Electronic Engineering, Kanazawa University, Kanazawa 920–8667, JAPAN fujita@t.kanazawa-u.ac.jp

system, robot manipulators are unfortunately limited to theplanar type.
On the other hand, Kelly et al. [8] considered a simple
image-based controller for dynamic visual feedback systemin the three dimensional(3D) workspace under the assumption that the objects’ depths are known. Cowan et al. [9]addressed the problems of the target tracking and the fieldof view for the 3D dynamic visual feedback system byusing the navigation functions. More recently, the authors
proposed the passivity-based dynamic visual feedback control for the 3D target tracking problem with the Eye-inHand configuration in [13]. However, this configuration
has only three coordinate frames, while visual feedback
systems typically use four coordinate frames which consistof a world frame Σw , a target object frame Σo , a camera
frame Σc and a hand (end-effector) frame Σh as in Fig. 1.
Because the camera is attached to the end-effector of robots,
the camera frame represents the hand one in Eye-in-Handconfiguration.
Image

Camera

gch

5c

gwc
gco

5h

gwh

gho
gwo

5w
World Frame

Target Object 5oFig. 1.

Visual feedback system with a fixed camera

In this paper, we deal with the control and the estimation
problems for dynamic visual feedback systems with the four
coordinate frames. Extending the number of the coordinate
frames from three to four, we propose the dynamic visualfeedback systems with a fixed camera in the 3D workspace,based on our previous works [10], [11], [12], [13]. In thisframework, we can design the control gain and the observer
gain separately from each other, while the control problem
and the estimation problem of the visual feedback system
are considered in the same strategy. Moreover, we can derive that the dynamic visual feedback system preserves thepassivity of the visual feedback system which is obtained

in our previous works. Stability and L2 -gain performance
analysis for the dynamic visual feedback system will be
discussed based on passivity with an energy function.
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
skew-symmetric operator such that ξ̂θ = ξ×θ for the vector
cross-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 the angle of rotation. Here ξ̂θab denotes ξ̂ab θab for thesimplicity of notation. We use the 4 × 4 matrixeξ̂θab pab
gab =
(1)
0
1
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 ) [14]. Let us define
the vector form of the rotation matrix as eR (eξ̂θab ) :=sk(eξ̂θab )∨ where sk(eξ̂θab ) denotes 12 (eξ̂θab − e−ξ̂θab ).
II. PASSIVITY- BAS ED V ISUAL F EEDBACK S YSTEM
A. Fundamental Representation for Visual Feedback System
Visual feedback systems typically use four coordinate
frames which consist of a world frame Σw , a target object
frame Σo , a camera frame Σc and a hand (end-effector)
frame Σh as in Fig. 1. Then, gwh = (pwh , eξ̂θwh ) ∈ SE(3),gwc = (pwc, eξ̂θwc ) ∈ SE(3) and gwo = (pwo , eξ̂θwo ) ∈SE(3) denote the rigid body motion from Σw to Σh , fromΣw to Σc and from Σw to Σo , respectively. Similarly,
the relative rigid body motion from Σc to Σh , from Σcto Σo and from Σh to Σo can be represented by gch =(pch , eξ̂θch ) ∈ SE(3), gco = (pco , eξ̂θco ) ∈ SE(3) and
gho = (pho , eξ̂θho ) ∈ SE(3), respectively. as shown in
Fig. 1.
The objective of the visual feedback control is to bringthe actual relative rigid body motion gho = (pho , eξ̂θho ) to agiven reference gd = (pd , eξ̂θd ). Our goal is to determine thehand’s motion using the visual information for this purpose.
Hence we will extend the passivity-based approach which
is proposed for the visual feedback system with Eye-inHand configuration in [13] to the visual feedback systemwith the four coordinates. In other words, we consider
the relative rigid body motion from Σc to Σo , i.e. gco =(pco , eξ̂θco ), and the control problem of the relative rigid
body motion from Σh to Σo , i.e. gho = (pho , eξ̂θho ).
However, the relative rigid body motion gco =(pco , eξ̂θco ) can not be immediately obtained in the visual feedback system, because the target object velocity
is unknown and furthermore can not be measured directly.
Hence, we consider the estimation problem of the relative
rigid body motion gco .
The relative rigid body motion from Σc to Σo can be led
by using the composition rule for rigid body transformations([14], Chap. 2, pp. 37, eq. (2.24)) as follows−1
gwo .
gco = gwc

(2)

The relative rigid body motion involves the velocity of eachrigid body. To this aid, let us consider the velocity of a rigid
body as described in [14]. Now, we define the body velocityof the camera relative to the world frame Σw asω̂wc vwc
vwc
b−1bV̂wc = gwc ġwc =
Vwc =
(3)
0
0
ωwc
where vwc and ωwc represent the velocity of the origin
and the angular velocity from Σw to Σc , respectively ([14]Chap. 2, eq. (2.55)). Similarly, the body velocity of thetarget object relative to Σw will be denoted asω̂wo vwo
vwo
b
−1
b
Vwo =
V̂wo = gwo ġwo =
(4)
0
0
ωwo
where vwo and ωwo are the velocity of the origin and theangular velocity from Σw to Σo , respectively.
bThen, the fundamental representation Vcofor the threecoordinate frames Σw , Σc and Σo is described as follows
[13].
b
b
b
Vco
= −Ad(gco−1 Vwc + Vwo
)

(5)

b
T
T T
b
−1
where Vco:= [vcoωco] and V̂co:= gcoġco . The notation
Ad(gab ) means the adjoint transformation associated withgab [14]. In the case of the fixed camera configuration,bbi.e. Vwc= 0, the fundamental representation Vco
can berewritten as
b
b
Vco
= Vwo
.

(6)

Roughly speaking, if both the camera and the target object
move, then the relative rigid body motion gco = (pco , eξ̂θco )will be derived from the difference between the camerabbvelocity Vwc
and the target object velocity Vwo. Hence,b
the fundamental representation Vco equals the target objectbvelocity Vwo, in the case of the fixed camera configuration.
B. Nonlinear Observer and Estimation Error System
To estimate the relative rigid body motion gco using
visual information provided by a computer vision system.
The visual information f is defined in [13] which includes the relative rigid body motion can be exploited,while the relative rigid body motion gco can not be obtained
directly in the visual feedback system. Hence, we consider
a nonlinear observer in order to estimate the relative rigid
body motion from the image information.
We shall consider the following model which just comesfrom the actual fundamental representation (6).
b
V̄co
= ue

(7)

b
T
T T
b
−1 ˙
where V̄coḡco mean the:= [v̄coω̄co] and V̄ˆco:= ḡcoˆ
estimated body velocity. Here, ḡco = (p̄co , eξ̄θ̄co ) denotes
the estimated relative rigid body motion. The new input ue
is to be determined in order to converge the estimated value

to the actual relative rigid body motion. Because the designof ue needs a property of the whole visual feedback system,
we will propose ue in Section III-B
In order to establish the estimation error system, we
define the estimation error between the estimated value ḡcoand the actual relative rigid body motion gco as−1
gee = ḡco
gco ,

(8)

Σh , i.e. gch , can be measured exactly. Since the problem of
the camera calibration is one of important research topics
and good solutions to it are reported in some papers (see,e.g., [6], [7]), we will not consider the error of the camera
calibration in this paper.
Then, the relative rigid body motion ḡho will be obtainedin the same way as (5).
b
b
b
V̄ho
= −Ad(ḡ−1 ) Vch+ V̄co−ξ̄ˆθ̄co(pco − p̄co ) and eξ̂θee =in other words, pee = eˆˆe−ξ̄θ̄co eξ̂θco . Note that pco = p̄co and eξ̂θco = e−ξ̄θ̄co iff
gee = I4 , i.e. pee = 0 and eξ̂θee = I3 .
Using the notation eR (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 . Therefore, if the vector of theestimation error is equal to zero, then the estimated relative
rigid body motion ḡco equals the actual relative rigid body
motion gco .
We can derive the vector of the estimation error ee fromimage information f and the estimated value of the relativeˆrigid body motion (p̄co , eξ̄θ̄co ),ee = J † (ḡco )(f − f¯)(9)where J(ḡco ) and f¯ are the image Jacobian and the estimated image information, and † denotes the pseudo-inverse[13]. Therefore the estimation error ee can be exploited inthe 3D visual feedback control law using image information
f obtained from the camera. Hence, the estimation input ue
which can be determined from ee in (9) with an estimation
gain in Section III-B.
The estimation error system will be derived in the same
way as the fundamental representation for the visual feedback system. The estimation error system is described asfollows [13].
bb−1 ue + V
Vee
= −Ad(gee
wo .
)

(10)

This system should be noted that if the vector of theestimation error is equal to zero, then the estimated relative
rigid body motion ḡco equals the actual one gco .
C. Control Error System
Let us derive the control error system. Similarly to (2),the relative rigid body motion from Σh to Σo is describedas−1
gho = gch
gco .

(11)

Because gco can not be obtained directly, we represent therelative rigid body motion from Σh to Σo with the estimatedone ḡco as−1ḡho = gchḡco .

(12)−1
Here gch = gwc
gwh can be obtained directly, because therigid body motion gwc = (pwc , eξ̂θwc ) from Σw to Σcand gwh = (pwh , eξ̂θwh ) from Σw to Σh is known by the
structure of the system and the angle of the manipulator. It
is supposed that the relative rigid body motion from Σc tohob= −Ad(ḡ−1 ) Vwh
+ ue

(13)

ho

b
b
where we exploit (7) and Vch= Vwh
which is derived from−1
gch = gwc gwh . Here we define the control error between
the estimated value ḡho and the reference of the relative
rigid body motion gd as

gec = gd−1 ḡho .

(14)

It should be remarked that the estimated relative rigid body
motion equals the reference one if and only if the control
error is equal to the identity matrix in matrix form, i.e. pd =ˆ
p̄ho and eξ̂θd = eξ̄θ̄ho iff gec = I4 . Using the notationeR (eξ̂θ ), the vector of the control error is defined as ec :=[pTec eTR (eξ̂θec )]T . Note that ec = 0 iff pec = 0 and eξ̂θec =I3 .
Similarly to (10), the control error can be obtained as
b
b
b
Vec
= −Ad(gec−1 V) d + V̄hobb
= −Ad(ḡ−1 ) Vwh+ ue − Ad(gec
−1 V
) d

(15)

ho

where Vdb := [vdT ωdT ]T and V̂db := gd−1 ġd .
D. Property of Visual Feedback System
Combining (10) and (15), we construct the visual feedback system as follows b  b
−Ad(ḡ−1 )
I
Vwh
Vec
ho
=
b
Vee
ue
0
−Ad(gee
−1
)

 

−Ad(gec
−1
0
b
)
+
Vwo
. (16)Vdb +I
0Using the relation of the adjoint transformation, i.e.
Ad(gec−1 = Ad −1 Ad(ḡ ) , the above equation (16) can bed)(ḡho )
rewritten as  b  −Ad(ḡ−1 )
I
0
Vec
b
ho
=
Vwo
(17)
uce +
b
Vee
I
0
−Ad(gee
−1
)
where


uce :=b+ Ad(gd ) Vdb
Vwh
ue


(18)

denotes the control input for the visual feedback system.
Let us definetheT error vector of the visual feedback system
as e := eTc eTe
which contains of the control error vector
ec and the estimation 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 estimatedone ḡco equals the actual one gco , respectively. Moreover,the error and the error vector between ḡho and gho can bealso represented as gee and ee by (8), (11) and (12), while
gee and ee are defined as the error and the error vector
between ḡco and gco in (8). Therefore, the actual relative
rigid body motion gho tends to the reference one gd whene → 0.
Now, we show an important lemma concerning a relation
between the input and the output of the visual feedback
system.
b
Lemma 1: If Vwo= 0, then the visual feedback system(17) satisfies
 T
uTceνce dτ ≥ −βce , ∀T > 0(19)

Using (17), (21) and (23), 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
0
0
u
+
+ ⎣ 0 −Ad(ḡho
)
b
Vwo
0 I
−1
0
0
−Ad(gee
)


εx
z=
(24)
ρu
where

0

where νce is defined asνce :=−AdT(g−1 )dAd(e−ξ̂θec )

0
−I


e

(20)

and βce is a positive scalar.
This proof is not difficult using the same approach as in[13], and is omitted here due to space limitations.
Let us take uce as the input and νce as its output. Then,
Lemma 1 would suggest that the visual feedback system(17) is passive from the input uce to the output νce justformally as in the definition in [15].

ε and ρ are weight matrices for the state and the input,
respectively. We define the disturbance of dynamic visual


b T T
)
. Before constructfeedback system as w := τdT (Vwoing the dynamic visual feedback control law, we derive an
important lemma.
Lemma 2: If w = 0, then the dynamic visual feedback
system (24) satisfies
 T
uT νdτ ≥ −β, ∀T > 0
(25)
0

⎤
I
0
0
T
⎥
⎢
ν := N x, N := ⎣ 0 −Ad(gd−1 ) Wc 0 ⎦ .
0 Ad(e−ξ̂θec ) Wc −We
Proof: Consider the following positive definite func-

A. Dynamic Visual Feedback System
The manipulator dynamics can be written as(21)

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
is given by
The body velocity of the hand Vwhb= Jb (q)q̇
Vwh

⎤
⎡
⎤
uξ
ξ
x := ⎣ ec ⎦ , u := ⎣ ud + Ad(gd ) Vdb ⎦ .
ee
ue

where

III. DYNAMIC V ISUAL F EEDBACK C ONTROL

M (q)q̈ + C(q, q̇)q̇ + g(q) = τ + τd

⎡

(22)

where Jb (q) is the manipulator body Jacobian [14]. We
define the reference of the joint velocities as q̇d := Jb† (q)ud
where ud represents the desired body velocity of the hand.
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{wpc I3 , 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 asfollows.
τ = M (q)q̈d + C(q, q̇)q̇d + g(q)+JbT (q)AdT(g−1 ) Wc ec + uξ . (23)d

The new input uξ is to be determined in order to achieve
the control objectives.

⎡tion
V (x) =1 T
1
ξ M ξ + wpc pec2 + wrc φ(eξ̂θec )2
2
1+ wpe pee 2 + wre φ(eξ̂θee ). (26)2where φ(eξ̂θ ) := 12 tr(I − eξ̂θ ) is the error function of
the rotation matrix and has the following properties (seee.g. [16]). Differentiating (26) with respect to time yields
1 Tξ Ṁ ξ
2 ⎡
⎤⎡ ⎤
M (q)
0
0
ξ˙
b ⎦ .(27)⎦⎣Vec
0+xT ⎣ 0 Wc Ad(eξ̂θec )b00
We Ad(eξ̂θee ) VeeV̇ =Observing that the skew-symmetry of the matrices p̂ec andp̂ee , i.e., pTecp̂ec e−ξ̂θd ωwh = −pTec (e−ξ̂θd ωwh )∧ pec = 0,
pTee p̂ee ωwe = −pTee ω̂we pee = 0, the above equation along
the trajectories of the system (24) can be transformed into
⎡
⎤
I
0
0
V̇ = xT ⎣ 0 −Wc Ad(gd−1 ) Wc Ad(eξ̂θec ) ⎦ u. (28)00−WeIntegrating (28) from 0 to T , we can obtain T
uT νdτ = V (x(T )) − V (x(0))
0≥ −V (x(0)) := −β(29)where β is the positive scalar which only depends on theinitial states of ξ, gec and gee .
Remark 1: Similarly to Lemma 1, Lemma 2 would suggest that the dynamic visual feedback system is passivefrom the input u to the output ν just formally. From Lemma
2, we can state that the dynamic visual feedback system (24)preserves the passivity of the visual feedback system (17).
This is one of main contributions of this work.
B. 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 ⎦ (30)00 Ke
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 forthe control error and the estimation error, respectively. The
result with respect to asymptotic stability of the proposed
control input (30) can be established as follows.
Theorem 1: If w = 0, then the equilibrium point x =0 for the closed-loop system (17) and (30) is asymptotic
stable.
Proof: In the proof of Lemma 2, we have alreadyderived that the time derivative of V along the trajectoryof the system (24) is formulated as (28). Using the control
input (30), (28) can be transformed intoV̇ = −xT N T KN x

(31)

This completes the proof.
Considering the manipulator dynamics, Theorem 1 shows
the stability via Lyapunov method for the full 3D dynamic
visual feedback system. It is interesting to note that stability
analysis is based on the passivity as described in (25).

Theorem 2: Given a positive scalar γ and consider thecontrol input (30) 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 (24)and (30) has L2 -gain ≤ γ.
Proof: By differentiating the positive definite function
V defined in (26) along the trajectory of the closed-loopsystem and completing the squares, it holds that1γ2V̇ + z2 − w2 ≤ −xT P x ≤ 0(32)
2
2
if P is positive semi-definite. Integrating (32) from 0 to T
and noticing V (T ) ≥ 0, we have
 T
 T
2
2
z dt ≤ γw2 dt + 2V (0), ∀T > 0. (33)00This completes the proof.
Theorem 2 can be proved using the energy function (26)
as a storage function for L2 -gain performance analysis. γ
represents a disturbance attenuation level for the dynamic
visual feedback system. Theorem 1 and 2 can be proved
using the energy function (26) as a Lyapunov function and astorage function, respectively. Therefore, the passivity of the
dynamic visual feedback system is particularly important in
our framework.
IV. S IMULATIONThe simulation results on the two degree-of-freedom
manipulator as depicted in Fig. 2 are shown in order tounderstand our proposed method simply, though it is valid
for 3D visual feedback systems. The target object has fourfeature points and moves for t = 4.8 [s] moves alonga straight line (0 ≤ t < 2) and a “Figure 8” motion(2 ≤ t < 4.8) as depicted in Fig. 3 and Fig. 4, respectively.
Specifically, we use the reference of the relative rigid body
motion as a constant value, i.e. pd = [0 0 −0.81]T , eξ̂θd = I
and Vdb = 0, for the tracking problems in the simulation.
Camera Frame
zc

5c

xc

yc
gwc
gco
zo

C. 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 (24) in one of the typical problems, i.e. the disturbance attenuation problem. Now, let us define
P := N T KN −1
1
1
W − ε2 − ρKN 2
2γ 2
2
2

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

Gwo

Hand Frame
zh
gch
5h
Gwh(=q 1+q 2)
xh

gho

yh
gwo

xo

5o

yo
Target Object Frame

gwh

World Frame
zw
5w
q1
q2
xw
yw
2DOF Robot
Manipulator

Fig. 2. Coordinate frames for dynamic visual feedback system with twodegree of freedom manipulator

Firstly, we design the weight matrices concerning controlled output as ε = diag{0.4, 0.4, 1, 1, 0.25, 0.25, 0.25,

00yo [m]-0.1

yo [m]

-0.1

0.1

0.1

0.2
0.6

0.45
xo [m]

0.3

Fig. 3. Trajectory of target object
along the straight line in 0 ≤ t < 2

0.2
0.6

0.45
xo [m]

0.3

Fig. 4. Trajectory of target object
along the “Figure 8” in 2 ≤ t < 4.8

0.25

can design the control gain and the observer gain separatelyfrom each other, while the control problem and the estimation problem of the visual feedback system are consideredin the same strategy. Stability and L2 -gain performance
analysis for the dynamic visual feedback system have been
discussed based on passivity with the energy function. The
experimental results is omitted due to space limitations, thereader is referred to [17] for more details.
In our future work, we will establish the more generalframework by combining Eye-in-Hand configuration withthe four coordinate frames. Additionally, we consider thatthe reference velocity Vdb will play a role in the trajectory
planning of the dynamic visual feedback systems.
R EFERENCE S

0.2

norm || z ||

0.15

0.1

0.05

0
0

Fig. 5.

1.0

2.0

3.0
time [s]

4.0

5.0

6.0

Euclid norms of z(Gain A: dashed, Gain B: solid)

1, 2.5, 2.5, 0.025, 0.025, 0.025, 2.5}, ρ = diag{200, 200,
2, 2, 1, 1, 1, 2, 0.1, 0.1, 0.001, 0.001, 0.001, 0.1} × 10−3 .
Also, we select the weight matrices for control design as
Wc = 0.1I and We = I. Control gain of manipulator ischosen as Kξ = diag{20, 20} and gains Kc and Ke arechosen as follows
Gain A : Kc = diag{100, 100, 50, 50, 50, 100}, Ke = 25I
Gain B : Kc = diag{2, 2, 1, 1, 1, 2} × 102 , Ke = 50I.
Then, the closed-loop system (24) and (30) with gain A has
γ = 0.229 and with gain B has γ = 0.159.
In Fig. 5, dashed line and solid line show the norm of z
in the case of γ = 0.229 and γ = 0.159, respectively. In thecase of static target object, i.e. after t = 4.8 [s], all errors in
Fig. 5 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 = 0.159, the performance is improved ascompared to the case of γ = 0.229. After all, the simulation
results show that L2 -gain is adequate for the performancemeasure of the dynamic visual feedback control.
V. C ONCLUSIONSThis paper dealt with the control and the estimation of
dynamic visual feedback systems with a fixed camera. The
main contribution of this work is that the dynamic visualfeedback system with four coordinate frames is constructedin order to extend our previous works. In this framework, we[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] H. I. Christensen and P. Corke, “Editorial: Special Issue on Visual
Servoing,” The Int. Journal of Robotics Research, Vol. 22, No. 10–11,
pp. 779–780, 2003.
[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] K. Omote et al., “Self-Guided Robotic Camera Control for Laparoscopic Surgery Compared with Human Camera Control,” The
American Journal of Surgery, Vol. 177, No. 4, pp. 321–324, 1999.
[5] R. Kelly, “Robust Asymptotically Stable Visual Servoing of Planar
Robots,” IEEE Trans. Robotics and Automation, Vol. 12, No. 5,
pp. 759–766, 1996.
[6] 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.
[7] E. Zergeroglu, D. M. Dawson, M. S. de Queiroz and A. Behal, “Vision-Based Nonlinear Tracking Controllers With Uncertain Robot-Camera Parameters,” IEEE/ASME Trans. Mechatronics,
Vol. 6, No. 3, pp. 322–337, 2001.
[8] R. Kelly, R. Carelli, O. Nasisi, B. Kuchen and F. Reyes, “Stable
Visual Servoing of Camera-in-Hand Robotic Systems,” IEEE Trans.
Mechatronics, Vol. 5, No. 1, pp. 39–48, 2000.
[9] N. J. Cowan, J. D. Weingarten and D. E. Koditschek, “Visual
Servoing via Navigation Functions,” IEEE Trans. Robotics andAutomation, Vol. 18, No. 4, pp. 521–533, 2002.
[10] A. Maruyama and M. Fujita, “Visual Feedback Control of Rigid
Body Motion based on Dissipation Theoretical Approach,” Proc. ofthe 38th IEEE Conf. on Decision and Control, pp. 4161-4166, 1999.
[11] M. Fujita, A. Maruyama and H. Kawai, “Observer Based Dynamic
Visual Feedback Control for Nonlinear Robotics Systems,” Proc. of
the 15th IFAC World Congress on Automatic Control, Paper Number
1637, 2002.
[12] H. Kawai, S. Izoe and M. Fujita, “A Passivity Approach to Visionbased Dynamic Control of Robots with Nonlinear Observer,” In:A.
Bicchi, H. Christensen and D. Prattichizzo (Eds), Control Problemsin Robotics, Springer–Verlag, pp. 199-213, 2003.
[13] 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, pp. 1522-1527, 2004.
[14] R. Murray, Z. Li and S. S. Sastry, A Mathematical Introduction to
Robotic Manipulation, CRC Press, 1994.
[15] A. van der Schaft, L2 -Gain and Passivity Techniques in Nonlinear
Control (2nd ed.), Springer–Verlag, 2000.
[16] F. Bullo and R. Murray, “Tracking for Fully Actuated Mechanical
Systems: a Geometric Framework,” Automatica, Vol. 35, No. 1,
pp. 17–34, 1999.
[17] T. Murao, H. Kawai and M. Fujita, “An Experimental Study of
Dynamic Visual Feedback Control with a Fixed Camera,” Proc. of
the 43rd SICE Annual Conference, pp. 181-186, 2004.