Image-based Dynamic Visual Feedback Control via Passivity Approach
Hiroyuki Kawai† , Toshiyuki Murao‡ and Masayuki Fujita‡

Abstract— This paper deals with the image-based dynamic
visual feedback control via the passivity approach which isinvestigated in the position-based one. We construct an energyfunction using the error in the image plane. The passivity ofthe visual feedback system is derived from the energy function.
We show passivity of the image-based dynamic visual feedback
system by combining the passivity of both the visual feedback
system and the manipulator dynamics which allows us to provestability in the sense of Lyapunov. The L2 -gain performance
analysis, which deals with the disturbance attenuation problem,is then considered via dissipative systems theory. This paper
suggests that the two types of classical visual servoing, i.e.
the image-based visual servoing and the position-based visual
servoing, can be discussed with the same strategy.

I. I NTRODUCTION
Visual feedback control of robotic systems involves thefusion of robot kinematics, dynamics, and computer vision tocontrol the motion of the robot in an efficient manner. Visualfeedback control is classified into two groups, positionbased control and image-based control [1]. In position-based
control, the references are given in the three-dimensional
Cartesian space. The control objective is to bring a relativepose, which is the pose from a camera to a target orfrom a hand to a target, to a desired pose by using image
information. In image-based control, the references are given
in an image plane.
The 2 1/2-D visual servoing which incorporates the advantages of both position-based and image-based visual
servoing is proposed in order to guarantee robustness withrespect to calibration errors [2]. Partitioned visual servoing is
considered in [3] in order to guarantee that all features remainin the image. More recently, an approach based on switching
between position-based visual servoing and backward motion
is investigated for dealing with the field of view problem [4].
However, classical visual servoing algorithms assume that the
manipulator dynamics is negligible and does not interact withthe visual feedback loop. This assumption, while it holds forkinematic control problems, is invalid for high speed tasks.
Maruyama et al. [5] discussed image-based robust control
for the set-point problems of the dynamic visual feedback
systems that includes the manipulator dynamics. Although
the control law guarantees Lyapunov stability and is effective
for the dynamic visual feedback system, it is restricted toplanar manipulators. For the problem of 3D visual feedback control, Kelly et al. [6] considered an image-based
controller under the assumption that the object’s depths are† Department of Robotics, Kanazawa Institute of Technology, Ishikawa
921-8501, Japan hiroyuki@neptune.kanazawa-it.ac.jp‡ Department of Mechanical and Control Engineering, Tokyo Institute ofTechnology, Tokyo 152-8550, Japan

CameraSc

Robot
Manipulator

Image
gco=
gwc=
^ wc)
(pco,exq^co)(pwc,exq^
xq
Sw
So gwo=(pwo,e wo)
Target Object
World Frame
Fig. 1.

Eye-in-hand visual feedback system.

known. Although good solutions to the set-point problem
are reported in those papers, few results have been obtained
for the tracking problem of moving target objects in thefull 3D dynamic visual feedback system. For the target
tracking problem, the authors have proposed position-based
control via passivity approach [7], [8], [9]. However, imagebased control is suitable than position-based control for some
applications. For example, in recent applications of visualfeedback system, laparoscopic surgery [10] is investigated byimage-based control, because surgeons need a field of viewof the surrounding target in the image plane. The objectiveof image-based visual feedback control is to bring the image
feature points f to desired image feature points fd .
This paper deals with image-based dynamic visual feedback control for a moving target object in 3D workspacewith the eye-in-hand configuration as depicted in Fig. 1.
Our proposed image-based control method does not need
the assumption that the object’s depths are known, because
they are estimated by the same strategy in the positionbased dynamic visual feedback control. In Section II, we
provide an image dynamics with a brief summary of ourprevious work [7]. The visual feedback system is constructedin Section III and in Section IV, we show passivity of theimage-based dynamic visual feedback system combined withthe manipulator dynamics and present stability and L2 -gain
performance analysis by using an energy function. In Section
V we present simulation results for image-based dynamic
visual feedback control on a 2DOF manipulator. Finally, weoffer some conclusions in Section VI.
II. B ACKGROUND
A. Notation and Definition
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 skewsymmetric operator such that âb = a×b for the vector crossproduct × and any vector a, b ∈ R3 , i.e., â is a 3 × 3 skew-

xi

symmetric matrix. 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 ξˆab θab for the simplicity of notation. We
use the 4 × 4 matrixeξ̂θab pab
gab =
(1)
0
1

fyi
zo

−1
gwo .
gco = gwc

(2)

The basic 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 described in [11].
Now, we define the body velocity of the camera relativeb
T
T T= [vwcωwc] , where
to the world frame Σw as Vwc
vwc and ωwc represent the velocity of the origin and theangular velocity from Σw to Σc , respectively ([11] Chap. 2,
eq. (2.55)).
Differentiating (2) with respect to time, the basic representation of the relative rigid body motion gco is describedas follows [7].
bbb= −Ad(gco−1 V
Vco) wc + Vwo(3)bwhere Vwois the body velocity of the target object relativeto Σw . Equation (3) is a standard formula for the relation
between the body velocities of three coordinate frames ([11]Chap. 2, pp. 59, Proposition 2.15). Roughly speaking, therelative rigid body motion gco will depend on the differencebbetween the camera velocity Vwc
and the target objectbvelocity Vwo .

fi

poi

yi

xo

xw
yw

Target Object

Fig. 2.

zw

Image Plane
yo

So

Sc
yc

fxi

P

)∈
as the homogeneous representation of gab = (pab , e
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 ) [11].

zc

l

g

ξ̂θab

B. Basic Representation for Visual Feedback System
The visual feedback system considered in this paper has
the camera mounted on the robot’s end-effector as depicted
in Fig. 1. Thus, the whole system has three coordinate frames
which consist of a world (base of the manipulator) frame Σw ,a camera (end-effector of the manipulator) frame Σc and atarget object frame Σo . Then, the relative rigid body motion
from Σc to Σo can be represented by gco . Similarly, gwc and
gwo denote the rigid body motions from the world frame Σw
to the camera frame Σc and from the world frame Σw to theobject frame Σo , respectively, as shown in Fig. 1.
The relative rigid body motion from Σc to Σo can be led
by using the composition rule for rigid body transformations([11], Chap. 2, pp. 37, eq. (2.24)) as follows

pci

Camera Frame
xc

Sw

World Frame

Perspective projection in a simple camera model.

where pci and poi should be regarded, with a slight abuse ofnotation, as [pTci 1]T and [pToi 1]T via the well-known homogeneous coordinate representation in robotics, respectively(see, e.g., [11]).
The perspective projection of the i-th feature point onto
the image plane gives us the image plane coordinate fi :=[fxi fyi ]T ∈ R2 as


λ
xci
(5)
fi =
zci yci
where pci = [xci yci zci ]T . It is straightforward to extend
this model to m image points by simply stacking the vectorsof the image plane coordinate, i.e.,
T T
f(gco ) := [f1T · · · fm
] ∈ R2m(6)

and pc := [pTc1 · · · pTcm ]T ∈ R3m . We assume that multiple
point features on a known object are given.
Differentiating (4) and (5), yieldsbf˙i = Ji (gco )Vco

(7)

where Ji (gco ) is the image Jacobian defined by
⎤
⎡
λxci
λ
0 − 2 ⎥⎢ zci
zci ⎥ ξ̂θcoJi (gco ) = ⎢I3 −p̂oi⎣
λλyci ⎦ e
0
− 2
zci
zci

.(8)

Using (6) and (7), the time derivative of the image feature
vector can be denoted bybf˙ = J(gco )Vco

(9)

where
T
J(gco ) = J1T (gco ) · · · Jm
(gco )T.

(10)

C. Image Dynamics
The pinhole camera model with a perspective projection
is shown in Fig. 2. Let λ be a focal length, poi ∈ R3 andpci ∈ R3 be the position vectors of the target object’s i-thfeature point relative to Σo and Σc , respectively. Using atransformation of the coordinates, we have

Hereafter we assume that the matrix J(gco ) is full column
rank for all gco ∈ SE(3) and the target object has three
feature points, i.e. rank(J(gco )) = 6 and m = 3. This
assumption is required for technical reasons in stability andL2 -gain performance analyses.
The image dynamics is obtained by substituting (3) into(9)

pci = gco poi ,

b
b
−1 V
f˙ = −J(gco )Ad(gco) wc + J(gco )Vwo .

(4)

(11)

Then, we have an important lemma about the relation between the camera body velocity and the image features.
b
Lemma 1: If the target object is static, i.e. Vwo= 0, thenthe following inequality holds for the image dynamics (11).
T
0

b T
(Vwc
) (νf )dt ≥ −βf(12)T
where νf := −AdT(gco−1 J (gco )f and βf is a positive scalar.
)
Proof: Consider the positive definite function
1
Vf = f2 .
(13)
2
Differentiating (13) with respect to time yields
b
b T
−1 V
V̇f = f T f˙ = −f T J(gco )Ad(gco) wc = (Vwc ) νf . (14)Integrating (14) from 0 to T , we obtain

Equation (20) represents the estimation error system.
Remark 1: Generally, the image Jacobian expresses therelation between the motion of the feature point in the image
plane and one in the workspace. As mentioned in [7], therelation between the estimation error in the image plane andone in the workspace can be connected by the same image
Jacobian.
III. I MAGE - BASED V ISUAL F EEDBACK S YSTEM
Let us consider the image error dynamics in order toestablish the image-based visual feedback system. First, we
define the image error as follows:
fe = f¯ − fd(21)

where fd ∈ R6 denotes the desired image feature vector.
Then, the image error dynamics can be represented as

T
0

b T
(Vwc
) νf dt = Vf (T ) − Vf (0) ≥ −Vf (0) := −βf (15)where βf is a positive scalar that only depends on the initialstates of f.
D. Nonlinear Observer and Estimation Error System
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. Firstly, we
shall consider the following model which just comes fromthe basic representation (3).
bbV̄co= −Ad(ḡco−1 Vwc + ue .
)

(16)

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 = ḡcogcowhere p̄ci := [x̄ci ȳci z̄ci ]T . f¯(ḡco ) := [f¯1T f¯2T f¯3T ]T ∈ R6
means 3 image points case. The relation between the actual
image information and the estimated one can be given by(19)

where J(ḡco ) has the same form as the image Jacobian
defined in (10). Similarly to J(gco ), we assume that thematrix J(ḡco ) is full column rank for all ḡco ∈ SE(3).
Differentiating (17) with respect to time and using (3) and(16), we obtainbb−1 ue + V
Vee
= −Ad(gee
wo .
)

b
−1 V
= −J(ḡco )Ad(ḡco) wc + J(ḡco )ue .

(20)

(22)

Combining (20) and (22), we construct the image-based
visual feedback system as follows:

 

 
−1
−J(ḡco )Ad(ḡco0f˙e) J(ḡco )
b
=
u
+
Vwo
fe
b
−1
0
−Ad(gee
I
Vee
)
(23)
b T
where ufe := [(Vwc) uTe ]T . We define the state of theimage-based visual feedback system as xfe := [feT eTe ]T .
It should be noted that if the vectors of the image error
and the estimation error are equal to zero, then the image
feature vector f, the estimated image feature vector f¯, andthe desired image feature vector fd coincide. Therefore, theimage feature vector f tends to the desired fd when xfe → 0.
b
Lemma 2: If the target object is static, i.e. Vwo= 0, thenthe following inequality holds for the image-based visualfeedback dynamics (23).

(17)

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 andeξ̂θee = I3 . Similarly to (4) and (5), the estimated image
feature point f¯i (i = 1, 2, 3) is defined as


λ
x̄ci
¯
p̄ci = ḡco poi , fi =(18)z̄ci ȳci

f − f¯ = J(ḡco )ee ,bf˙e = J(ḡco )V̄coT
0

uTfe νfe ≥ −βfe

(24)

where
νfe := Nfe xfe , Nfe :=T
0−AdT(ḡco−1 J(ḡco ))TJ(ḡco )−I(25)

and βfe is a positive scalar.
Proof: Consider the positive definite function
Vfe =

1
1
fe 2 + pee 2 + φ(eξ̂θee )
2
2

(26)

where φ(eξ̂θ ) := 12 tr(I − eξ̂θ ) is the error function of
the rotation matrix (see e.g. [12]). Differentiating (26) withrespect to time yields



I
0
f˙e
T
T
V̇fe = fe eeb0 Ad(eξ̂ θee )
Vee


−1
−J(ḡco )Ad(ḡcoJ(ḡco )TT)= fe eeufe
0−I= uTfe νfe(27)

Integrating (27) from 0 to T , we obtainT0

uTfe νfe dt = Vfe (T ) − Vfe (0) ≥ −Vfe (0) := −βfe(28)where βfe is a positive scalar that only depends on the initialstates of fe and gee .
Let us consider ufe as the input and νfe as its output. Then,
Lemma 2 says that the image-based visual feedback system(23) is passive from the input ufe to the output νfe .
IV. I MAGE - BASED DYNAMIC V ISUAL F EEDBACK
C ONTROL
A. Image-based Dynamic Visual Feedback System
The dynamics of n-link rigid robot manipulators can be
written as
M (q)q̈ + C(q, q̇)q̇ + g(q) = τ + τd(29)

where q, q̇ and q̈ are the joint angle, velocity and acceleration,
respectively, τ is the vector of the input torque, and τdrepresents a disturbance input [13]. Equation (29) possesses
several important properties which will be used in the sequel.
manipulator dynamics (29) is passive from τ to q̇, that is
The
T T
τ
q̇dt ≥ −βm where βm is a positive scalar. Moreover,0Ṁ (q) − 2C(q, q̇) is skew-symmetric by defining C(q, q̇)using the Christoffel symbols. The visual feedback system(23) and the manipulator dynamics (29) will be connectedby the passivity.
Now, we will construct an image-based dynamic visualfeedback system by connecting the image-based visual feedback system (23) and the manipulator dynamics (29). Since
the camera is mounted on the end-effector of the manipulator
in the eye-in-hand configuration, the body velocity of thebcamera Vwc
is given byb= Jb (q)q̇
Vwc

(30)

where Jb (q) is the body manipulator Jacobian [11].
Next, we propose the control law for the manipulator asτ = M (q)q̈d + C(q, q̇)q̇d + g(q)T
+Jb (q)T AdT(ḡco−1 J(ḡco ) fe + ur .
)(31)

where q̇d and q̈d represent the desired joint velocity andacceleration, respectively. The first term in (31) is the compensation of the nonlinear effects.
Here, let us define the error vector with respect to the joint
velocity of the manipulator dynamics asr := q̇ − q̇d .

(32)

From the above discussion, we can obtain the following error
dynamics
T
M (q)ṙ + C(q, q̇)r − Jb (q)T AdT(ḡco−1 J(ḡco ) fe = ur + τd .
)(33)
Moreover, we design the reference of the joint velocity
based on the relation between the camera velocity and thejoint one (30) as follows:q̇d := Jb† (q)ud(34)

where ud is the desired body velocity of the camera which
will be obtained from the image-based visual feedback
system.
Using (23) and (33), the visual feedback system withmanipulator dynamics (we call the image-based dynamic
visual feedback system) can be derived as follows:
“
”3
3 2
ṙ
−M (q)−1 C(q, q̇)r + Jb (q)T AdT(ḡ−1 ) J (ḡco )T feco7
4 f˙e 5 = 6
4
5−J (ḡco )Ad(ḡco−1 Jb (q)r
)
b
Vee
0
3
3
2
2
0
0
M (q)−1
M (q)−1 0
J (ḡco ) 5 u + 4−1
0 05 w(35)+ 4 0 −J (ḡco )Ad(ḡco
)
−1
0 I
0
0
−Ad(gee)2

where u := [uTr uTd uTe ]T . We define the state and the
disturbance of image-based dynamic visual feedback systemb T T
as x := [r T feT eTe ]T and w := [τdT (Vwo) ] , respectively.
Before constructing the image-based dynamic visual feedback control law, we derive an important lemma.
Lemma 3: If w = 0, then the image-based dynamic visualfeedback system (35) satisfies
T
0

where

uT νdt ≥ −β, ∀T > 0

(36)
⎤

⎡

I
0
0
T
T
0 ⎦
−1 J (ḡco )ν := N x, N := ⎣ 0 −Ad(ḡco)−I0J (ḡco )T(37)

and β is a positive scalar.
Proof: Consider the positive definite function
11
1 T
r M (q)r + fe 2 + pee 2 + φ(eξ̂θee ) (38)222Differentiating (38) with respect to time yields
V =1bV̇ = r T M (q)ṙ + r T Ṁ (q)r + feT f˙e + eTe Ad(eξ̂θee ) Vee21 = r T Ṁ (q) − 2C(q, q̇) r + r T ur
2



−1
−J(ḡco )Ad(ḡcoud) J(ḡco )+ feT eTeue
0−I
= uT ν.

(39)

Integrating (39) from 0 to T , we obtainT0uT νdt = V (T ) − V (0) ≥ −V (0) := −β(40)where β is a positive scalar that only depends on the initialstates of r, fe and gee .
Lemma 3 shows that the image-based dynamic visual feedback system preserves the passivity of both the image-based
visual feedback system and the manipulator dynamics.
B. Dynamic Visual Feedback Control and Stability Analysis
We now propose the following control input for theinterconnected system:u = −Kν = −KN x, K :=

Kr
0
0

0
Kf
0

0
0
Ke

(41)

where Kr := diag{kr1 , · · · , krn } the positive gain matrix
for each joint axis. Kf := diag{kf1 , · · · , kf6 } and Ke :=diag{ke1, · · · , ke6 } are the positive gain matrix for image
errors and the positive gain matrix of x, y and z axesof the translation and the rotation for the estimation error,
respectively.
Theorem 1: If w = 0, then the equilibrium point x = 0
for the closed-loop system (35) and (41) is asymptotic stable.
Proof: In the proof of Lemma 3, we have alreadyderived that the time derivative of V along the trajectoryof the system (35) is formulated as (39). Using the control
input (41), (39) can be transformed intoV̇ = −x N KN x.
T

T

(42)

This completes the proof.
Theorem 1 shows the stability via Lyapunov method. It isinteresting to note that stability analysis is based on thepassivity as described in (36).
C. L2 -Gain Performance Analysis
The motion of the target object will be regarded as anexternal disturbance in our approach. Hence, we consider
L2 -gain performance analysis for the image-based dynamic
visual feedback system (35) in one of the typical problems
based on the dissipative system theory, i.e. the disturbance attenuation problem. Here, we define the controlled controlled
output as z = [(εx)T (ρu)T ]T where ε and ρ are positive
definite matrices to weight the state x and the auxiliary input
u directly. Now, let us define
P := N T KN −1
1
1
W − ε2 − ρKN 2 (43)
2γ 2
2
2

where γ ∈ R is positive and W := diag{I, 0, I}. Then we
have the following theorem.
Theorem 2: Given a positive scalar γ and consider thecontrol input (41) with the gains Kr , Kf and Ke such thatthe matrix P is positive semi-definite, then the closed-loopsystem (35) and (41) has L2 -gain ≤ γ.
Proof: Differentiating the positive definite function V
defined in (38) along the trajectory of the closed-loop systemyields
2

2

1
γ
1
1
γ
w2 − z2 −w2 + εx2 + ρu2
2 2
2
2
2
2 2
3
3
I
0
I
0
0
T 4
5 w.
0
0
−1 J (ḡco )5 u + x+ xT 40 −J (ḡco )Ad(ḡco)
0 Ad(eξ̂θee )
0
0
−I

V̇ =

By completing the squares, we haveγ21z2 −
w2
2
2
‚
»
– ‚2
‚
I 0
0
γ2 ‚
1
2
‚w − 1=−x‚‚‚ + 2 εx
2
0
0
Ad
−
ξ̂θ
2
γ
ee )
(e
‚»
– ‚2
‚
I 0
0
1 ‚
1
T
T
2
‚
x‚
+ 2 ‚ 0 0 Ad
‚ + x N u + 2 ρu2γ(e−ξ̂θee )

V̇ +

≤

1
1
1
W x2 + xT N T u + εx2 + ρu2 .
2γ 2
2
2

(44)

Substituting the control input (41) into (44), we obtain
V̇ +

1
γ2
z2 −
w2
2
2≤ −xT N T KN x +1
1
1
W x2 + εx2 + ρKN x2. (45)
2γ 2
2
2

It can be verified that the inequality1γ2V̇ + z2 − w2 ≤ −xT P x ≤ 0(46)
2
2
holds if P is positive semi-definite. Integrating (46) from 0
to T and noticing V (T ) ≥ 0, we haveT0z2 dt ≤ γ 2T0w2 dt + 2V (0), ∀T > 0. (47)This completes the proof.
We have discussed L2 -gain performance analysis for a
disturbance attenuation problem. In other words, L2 -gain
performance analysis is exploited to evaluate the tracking
performance of the control scheme in the presence of amoving target.
V. S IMULATIONThe simulation results on the two degree-of-freedom manipulator as depicted in Fig. 3 are shown in order to understand our proposed method simply, though it is valid for 3D
visual feedback systems. The target object has three featurepoints and moves for t = 4.8 [s] moves along a straightline (0 ≤ t < 2) and a “Figure 8” motion (2 ≤ t < 4.8) as
depicted in Fig. 4 and Fig. 5, respectively. The desired image
feature points are set to fd = [−61 − 61 − 61 61 61 − 61]T
pixels.
Camera Framezc

qwc(=q 1+q 2)

5c
xc

yc
zo

qwo
xo

gwc
gco
gwo

World Frame
zw
5w
q
q2
1
xw
yw
2DOF Robot
Manipulator

5oyoTarget Object Frame
Fig. 3. Coordinate frames for image-based dynamic visual feedback systemwith two degree of freedom manipulator.

Firstly, we design the weight matrices concerning controlled output as ε = diag{0.05, 0.05, 0.5, 0.5, 0.5, 0.5, 0.5,
0.5, 1.5, 1.5, 0.25, 0.25, 0.25, 1.5}, ρ = diag{0.02, 0.02, 2,
2, 2, 2, 2, 2, 10, 10, 0.1, 0.1, 0.1, 10} × 10−3 . Gains Kr ,Kf and Ke are chosen as follows
Gain A : Kr = diag{5, 5}, Kf = 10I, Ke = 10I
Gain B : Kr = diag{20, 10}, Kf = 30I, Ke = 30I
Then, the closed-loop system (35) and (41) with gain A has
γ = 0.912 and with gain B has γ = 0.381.

00peex [m]

-0.1

0.1

0.4

0.3

fx1 [pixel]

Fig. 4. Trajectory of target object
along the straight line in 0 ≤ t < 2

-40

-50

-50

-60

-60

-70

-70
2

3

50

0

0

-50

-50

-100

-100

-150

-150

2 3 4 4.8
time [s]xw [m]

0.4

0.3

Fig. 7.

4 4.8 -80 0 1

50

0 1

0.5

Fig. 5. Trajectory of target object
along the “Figure 8” in 2 ≤ t < 4.8-40-80 0 1fy1 [pixel]

0.2
0.6

qeez [rad]xw [m]

0 1

2

3

4 4.8

2 3 4 4.8
time [s]-0.2This paper dealt with the image-based dynamic visualfeedback control for three dimensional target tracking withthe eye-in-hand configuration. Our proposed image-based
control method does not need object’s depths, because they
are estimated by the same strategy in the position-based
dynamic visual feedback control. The main contribution ofthis paper is to show that image-based control and positionbased control can be discussed with the same passivity
approach. In future work we aim to combine image-based
control and position-based control in order to incorporate
the advantages of both methods.
R EFERENCE S[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] E. Malis, F. Chaumette and S. Boudet, “2-1/2-D Visual Servoing,”
IEEE Trans. on Robotics and Automation, Vol. 15, No. 2, pp. 238–
250, 1999.

3

-0.05
4 4.8
0 1
0.2

2

3

4 4.8

2

3

4 4.8

0
2

0
0 1

2 3 4 4.8
time [s]

-0.2

0 1

2 3 4 4.8
time [s]Estimation error with Gain A(left side) and Gain B(right side).
1.2
0.8
0.4
0

0

1

2

0

1

2

3

4

4.8

3

4

4.8

1.2
0.8
0.4

Fig. 8.

VI. C ONCLUSIONS

30

Fig. 6. Image feature point in the image plane with Gain A(left side) andGain B(right side).

Fig. 6 shows one of the image feature. The dashed lines
are the desired position. The estimation error is depicted inFig. 7. In both figures, the errors in the case of Gain A
and Gain B are shown in the left side and the right one,respectively. Fig. 8 shows the norm of z in the case ofγ = 0.912 and γ = 0.381. In the case of γ = 0.381,
the performance is improved as compared to the case ofγ = 0.912. After all, the simulation results show that L2 -gain
is adequate for the performance measure of the image-based
dynamic visual feedback control.

2

-0.01
4 4.8
0 1
0.05

0

|| z || (g = 0.912)

0.5

0

-0.05
0 1
0.2

|| z || (g = 0.381)

0.2
0.6

0.01

0

peey [m]

0.1

0.01

-0.01
0 1
0.05

yw [m]

yw [m]-0.1

0time [s]Euclid norms of z(Gain A: top, Gain B: bottom)[3] P. I. Corke and S. A. Hutchinson, “A New Partitioned Approach to
Image-Based Visual Servo Control,” IEEE Trans. on Robotics andAutomation, Vol. 17, No. 4, pp. 507–515, 2001.
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[5] A. Maruyama and M. Fujita, “Robust Control for Planar Manipulatorswith Image Feature Parameter Potential,” Advanced Robotics, Vol. 12,
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[6] R. Kelly, R. Carelli, O. Nasisi, B. Kuchen and F. Reyes, “Stable
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[7] H. Kawai and M. Fujita, “Passivity-based Dynamic Visual Feedback
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[8] M. Fujita, H. Kawai and M. W. Spong, “Passivity-based Dynamic Visual Feedback Control for Three Dimensional Target Tracking:Stability and L2 -gain Performance Analysis,” IEEE Transactions
on Control Systems Technology(submitted).
[9] T. Murao, H. Kawai and M. Fujita “Passivity-based Control of VisualFeedback Systems with Dynamic Movable Camera Configuration,”
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