Passivity-based Dynamic Visual Feedback Control with
Uncertainty of Camera Coordinate Frame
Hiroyuki Kawai*, Toshiyuki Murao** and Masayuki Fujita**
Abstract— In this paper, we consider the dynamic visualfeedback control with the uncertainty of the camera coordinate
frame based on the passivity. Firstly the brief summary of thenominal visual feedback systems with a fixed camera is given
with the fundamental representation of a relative rigid body
motion. Secondly we construct the visual feedback system withuncertainty which is not be limited to the orientation aroundthe optical axis. Next, 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 thevalidity of the proposed control law can be confirmed bycomparing the simulation results.

I. I NTRODUCTION
Robotics and intelligent machines need many information
to behave autonomously under dynamical environments.
One suitable way to recognize unknown surroundings is
to use visual information. Vision based control of robotic
systems involves the fusion of robot kinematics, dynamics,
and computer vision to control the motion of the robot inan 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].
Classical visual servoing algorithms assume that the manipulator dynamics is negligible and do not interact with thevisual feedback loop. However, this assumption is invalid
for high speed tasks, while it holds for kinematic control
problems [3]. Additionally, as mentioned in [1], camera
calibration problems are long standing research issues inthe visual feedback systems with a fixed camera as depicted
in Fig. 1. Kelly [3] considered the set-point problems forthe dynamic visual feedback system with the uncertaintyof the camera orientation. In [4], Bishop et al. proposed aninverse dynamics based control law for the position tracking
and the camera calibration problems of the dynamic visualfeedback system. Recently, Zergeroglu et al. developed anadaptive control law for the position tracking and the camera
calibration problems of the dynamic visual feedback systemwith parametric uncertainties in [5]. Although these control
laws guarantee the stability of the system based on the
Lyapunov method and are effective for the dynamic visualfeedback system, robot manipulators are unfortunately limited to the planar type. Moreover, the uncertainty of the*Information Technology Research Center, Hosei University, Tokyo
102–8160, JAPAN, hiroyuki@i.hosei.ac.jp**Graduate School of Natural Science and Technology, Kanazawa
University, Kanazawa 920–8667, JAPAN

Image

CameraRobot
Manipulator

gch

5c

gwc
gco

5h
gwh

gho
5o

gwo

5w
World Frame

Target Object

Fig. 1.

Visual feedback system with a fixed camera.

camera coordinate frame is limited to the orientation aroundthe optical axis.
In this paper, we consider the dynamic visual feedback
control with the uncertainty of the camera coordinate frame.
The uncertainty will not be limited to the orientation aroundthe optical axis, although we use a simple camera model.
In this work, our previous research [9] is extended to thecase of uncertain visual feedback systems. Hence, we canderive that the dynamic visual feedback system preserves
the passivity of the visual feedback system by the same
strategy in our previous works [6]-[9]. Stability and L2 gain performance analysis for the dynamic visual feedback
system will be discussed based on passivity with an energyfunction. Comparing the simulation results, the validity ofthe proposed control law can be confirmed.
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 ) [10]. 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 , gwc and gwo denote therigid body motions from Σw to Σh , from Σw to Σc andfrom Σw to Σo , respectively. Similarly, the relative rigid
body motions from Σc to Σh , from Σc to Σo and from Σhto Σo can be represented by gch , gco and gho , 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([10], Chap. 2, pp. 37, eq. (2.24)) as follows−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 [10]. Now, we define the body velocity of the camera
relative to the world frame Σw asω̂wc vwc
vwc
b
−1
b
Vwc =
V̂wc = gwc ġwc =
(3)
0
0
ωwc

where pci and poi should be regarded as [pTci 1]T and[pToi 1]T via the well-known representation in robotics,
respectively (see, e.g. [10]).
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 follows


λ
xci
fi =
(7)
zci yci
where pci := [xci yci zci ]T . It is straightforward to extend
this model to the m image points case by simply stacking the vectors of the image plane coordinate, i.e. f :=T T
[f1T · · · fm
] ∈ R2m . We assume that multiple point
features on a known object can be used.
C. Visual Feedback System with Fixed Camera
Here the brief summary of our prior work in [9] is given.
In order to bring the actual relative rigid body motion gho
to a given reference gd in Fig. 1, we consider the control
and estimation problems in the visual feedback systems.
The following dynamic model which just comes from thefundamental representation of the actual relative rigid body
motion (5) is considered.
b
V̄co
= ue

(8)

where vwc and ωwc represent the velocity of the origin
and the angular velocity from Σw to Σc , respectively ([10]Chap. 2, eq. (2.55)).
Then, the fundamental representation of the relative rigid
body motion gco is described as follows [8].

b
T
T T
b
−1 ˙
ḡco mean the:= [v̄coω̄co] and V̄ˆco:= ḡcowhere V̄coˆestimated body velocity. Here, ḡco = (p̄co , eξ̄θ̄co ) denotes
the estimated relative rigid body motion. Then, the estimation error of the relative rigid body motion from Σc to Σo ,i.e. the error between ḡco and gco , is defined as

b
b
b
−1 V
= −Ad(gco
Vco) wc + Vwo−1gee = ḡco
gco ,

(4)

b
where Vwois the body velocity of the target object. The
notation Ad(gab ) means the adjoint transformation associated with gab [10]. Roughly speaking, if both the camera
and the target object move, then the relative rigid body
motion gco will be derived from the difference between thebbcamera velocity Vwc
and the target object velocity Vwo. In
the case of the fixed camera configuration, the fundamental
representation of the relative rigid body motion gco can be
rewritten as
b
b
Vco
= Vwo
,

(5)

b
= 0 in the case of the
because the camera is static, i.e. Vwc
fixed camera configuration.

B. Camera Model(9)which is called the estimated object error. Using the notationeR (eξ̂θ ), the vector of the estimated object error is given by
ee := [pTee eTR (eξ̂θee )]T . The estimated object error system
is represented byb−1b−1 ue + V
Vee
= (geeġee )∨ = −Ad(gee
wo
)

(10)

where ue is the input in order to converge the estimatedvalue to the actual relative rigid body motion.
Similarly, we define the error between gd and ḡho , whichis called the control error, as follows
gec = gd−1 ḡho ,(11)where ḡho is the estimated relative rigid body motion fromΣh to Σo and obtained from−1ḡho = gchḡco .

(12)

To control the relative rigid body motion using visual
information provided by a computer vision system, wederive the model of a pinhole camera with a perspective
projection. Let λ be a focal length, poi ∈ R3 and pci ∈R3 be coordinates of the target object’s i-th feature point
relative to Σo and Σc , respectively. Using a transformationof the coordinates, we have

Here, we assume that gch is calculated by using the known
motion, i.e. gwc and gwh , exactly. The vector of the control
error is defined as ec := [pTec eTR (eξ̂θec )]T . The control error
system is described by

pci = gco poi ,bwhere Vwh
is the body velocity of the hand relative to Σw .

(6)

b
−1
b
Vec
= (gecġec )∨ = −Ad(ḡ−1 ) Vwh
+ ue
ho

(13)

Combining (10) and (13), the visual feedback systemwith a fixed camera is constructed as follows b   
−Ad(ḡ−1 )
I
Vec
0
b
ho
=
u
+
Vwo
(14)
ce
b
Vee
I
0
−Ad(gee
−1
)
where


uce :=

b
Vwh
ue



0

νce :=−AdT(g−1 )dAd(e−ξ̂θec )

0
−I



5h

gch

5c 0
Camera Frame(nominal)
gc0o

gco

gwc0

gc0h
gwc

gwo
5o
Target Object Frame

5w
World Frame

Fig. 2. Coordinate frames for the visual feedback system with uncertaintyof the camera frame.

where gwc0 is the rigid body motion from Σw to Σc0 . gc0 o
and gc0 h are the relative rigid body motions from Σc0 to
Σo and Σh , respectively.
From (12), we notice that gho may not be estimatedexactly as follow−1ḡ = (gδ gch )−1 ḡco = gchḡco (= ḡho ),gc−1
0 h co(19)even if the vector of the estimated object error ee equals tozero. Consequently, gho will not converge to gd owing to
the uncertainty of the camera coordinate frame, neverthelessece tends to zero in the visual feedback system (14) withthe passivity-based control law proposed in [9].
B. Estimated Hand Error System

ece

(17)

and βce is a positive scalar.
Let us take uce as the input and νce as its output. Then,
Result 1 would suggest that the visual feedback system (14)is passive from the input uce to the output νce just formallyas in the definition in [12].

Here we will consider the observer for estimating therelative rigid body motion gch in order to reduce the effectof uncertainty of the camera coordinate frame.
Based on the fundamental representation of the relative
rigid body motion described as (4), the relation among Σw ,Σc and Σh can be expressed asbbbb= −Ad(g−1 ) Vwc
+ Vwh
= Vwh
,
Vch
ch

III. V ISUAL F EEDBACK S YSTEM WITH U NCERTAI N
C OORDINATE F RAME
A. Uncertainty of Camera Coordinate Frame

(18)

(20)

because the camera is static in the fixed camera configuration. We shall consider the following model from (20)
b
V̄ch
= uh

In the visual feedback system with a fixed camera, camera
calibration problems are long standing research issues [1].
In our approach, the uncertainty of the camera coordinate
frame can be regarded as one of the camera calibration
problems. Fig. 2 shows the coordinate frames for the visualfeedback system with uncertainty of the camera coordinate
frame. Let Σc0 be the nominal camera frame which can be
known a priori, while Σc denotes the actual one which is
unknown. The uncertainty of the camera coordinate frame
is denoted as−1gδ = gwc
g
0 wc

gwh

gho

(15)

denotes the input for the visual feedback system.
Let us define the errorT vector of the visual feedback
system as ece := 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 theestimation error are equal to zero, then the estimated relative
rigid body motion ḡho equals the reference one gd and theestimated one ḡco equals the actual one gco , respectively.
Moreover, the error between ḡho and gho can be also
represented as gee , while gee are defined as the error
between ḡco and gco in (9). Therefore, the actual relative
rigid body motion gho tends to the reference one gd whenece → 0.
Now, we show the relation between the input and theoutput of the visual feedback system.
b= 0, then the visual feedback
Result 1: [9] If Vwo
system (14) satisfies
 T
uTceνce dτ ≥ −βce , ∀T > 0(16)
where νce is defined as

Camera Frame(real)
5c
Hand Frame(21)

where uh is the new input for reducing the estimation error
between gch and ḡch which represents the estimated relative
rigid body motion from Σc to Σh .
Similarly to the camera model and the image information
mentioned in II-B, we can discuss the image information
concerned with gch by replacing the target object frame
Σo with the hand frame Σh . Hence, we define the image
information of the hand and the estimated one as fh and f¯h ,respectively. It is assumed that both the image informationof the target object and of the hand, i.e. f and fh , areavailable from a single camera.

In order to compose the estimation error system betweenḡch and gch , we call the estimated hand error system inthis paper, the estimated hand error between ḡch and gch isdefined as−1
geh = ḡch
gch .

(22)

Using the notation eR (eξ̂θ ), the vector of the estimatedhand error is given by eh := [pTeh eTR (eξ̂θeh )]T . Then, therelationship between fh and f¯h can be given byfh − f¯h = J(ḡch )eh ,eheh ch ch cheh−1b= −gehûh geh + V̂wh
.

ch

ch ch

(24)

Furthermore, using the property concerning the adjoint
transformation, the above equation can be transformed into
the following
b
b
Veh
= Vwh
− Ad(g−1 ) uh .
eh

(25)

Eq. (25) represents the estimated hand error system.
C. Property of Visual Feedback System with Uncertain
Coordinate Frame
If gco and gch are measured, gho can be obtained without
the effect of uncertainty of the camera coordinate frame.
In order to reduce the effect of uncertainty, the estimatedrelative rigid body motion from Σh to Σo described in (12)should be replaced by−1ḡho = ḡchḡco .

(26)

Then, the control error system is transformed intobVec
= ue − Ad(ḡ−1 ) uh .
ho

(27)

Moreover, we redefine the estimated object error as−1
gee = ḡco
gco .

(28)

Where the estimated object error system is same as (10).
Combining (10), (25) and (27), the visual feedback
system with a fixed camera is constructed as follows⎤⎡ b ⎤ ⎡⎡⎤0I
−Ad(ḡ−1 )
Vec
0
ho
⎥
b ⎦ ⎢ 0 −Ad −1
b ⎦
⎣ Vee
0
(29)
=⎣
⎦ uceh + ⎣Vwo
(gee )b
Veh0
I0−Ad(g−1 )
eh

where

⎡

⎤
b
Vwh
uceh := ⎣ ue ⎦
uh

0

(23)

where J(ḡch ) plays the role of well-known image Jacobianand is defined in [8].
−1Differentiating (22) and multiplying it by geh, we canobtaing−1 ġ = −g−1 V̄ˆ b ḡ−1 g + g−1 ḡ−1 g V̂ beh

if the vectors of the control error, the estimated object error
and the estimated hand error are equal to zero, then ḡho , ḡcoand ḡch equal gd , gco and gch , respectively. Therefore, gho
tends to gd when e → 0, while the visual feedback system
has the uncertainty of the camera coordinate frame.
b= 0, then the visual feedback system
Lemma 1: If Vwo(29) satisfies
 T
uTceh νcehdτ ≥ −βceh , ∀T > 0
(31)

(30)

denotes the input for the visual feedback system.
Let us define the error vector
T of the visual feedback
system (29) as e := eTc eTe eTh . It should be noted that

where νceh is defined as⎡0⎢νceh := ⎣ Ad(e−ξ̂θec )−AdT(g−1 )d⎤0 Ad(e−ξ̂θeh )
⎥
−I
0
⎦e
0
−I

(32)

and βceh is a positive scalar.
Proof: Consider the following positive definite function
Vceh = E(gec) + E(gee ) + E(geh )(33)

where 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. [11]). Differentiating (33) with respect to time yields⎡⎤⎡⎤bAd(eξ̂θec )
0
0
Vec
⎢
⎥
b ⎦
0
Ad(eξ̂θee )0V̇ceh = eT ⎣(34)⎦⎣ VeebVeh00Ad(eξ̂θeh )where we use the property φ̇(eξ̂θ ) := eξ̂θ ω. Observing theskew-symmetry of the matrices p̂ec, p̂ee and p̂eh , the above
equation along the trajectories of the system (29) can betransformed into⎤⎡0Ad(eξ̂θec ) −Ad(g−1 )d⎦ uceV̇ceh = eT ⎣0−I0Ad(eξ̂θeh )0−I= uTceh νceh .

(35)Integrating (35) from 0 to T , we can obtain T
0

uTceh νcehdτ ≥ −Vceh (0) := −βceh(36)where βceh is the positive scalar which only depends on theinitial states of gec , gee and geh .
Remark 1: In the visual feedback system, pTecω̂wc pec =0, pTee ω̂ue pee = 0, pTeh ω̂uh peh = 0 holds. This skewsymmetric property is analogous to the one of the robot
dynamics, i.e. xT (Ṁ − 2C)x = 0, ∀x ∈ Rn (where M ∈Rn×n is the manipulator inertia matrix and C ∈ Rn×n is
the Coriolis matrix [10]). Thus, Lemma 1 suggests that thevisual feedback system (29) is passive from the input uceh
to the output νceh as in the definition in [12].

IV. DYNAMIC V ISUAL F EEDBACK C ONTROLDue to space limitations, the proof is only sketched. By
using the following positive definite function, the proof canbe completed by invoking Lemma 1

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

1 Tξ M ξ + E(gec ) + E(gee ) + E(geh ). (42)2Remark 2: 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 (40)preserves the passivity of the visual feedback system (29).
This is one of main contributions of this work.
V (x) =

M (q)q̈ + C(q, q̇)q̇ + g(q) = τ + τd(37)

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 VwhbVwh= Jb (q)q̇(38)

where Jb (q) is the manipulator body Jacobian [10]. We
define the reference of the joint velocities as q̇d := Jb† (q)ud
where ud represents the desired body velocity of the hand.
bThus, Vwhin (30) should be replaced by ud .
Let us define the error vector with respect to the joint
velocities of the manipulator dynamics as ξ := q̇ − q̇d . Now,
we consider the passivity-based dynamic visual feedback
control law as follows.
τ = M (q)q̈d + C(q, q̇)q̇d + g(q) + uξ
+JbT (q) AdT(g−1 ) ec + Ad(e−ξ̂θeh ) eh .

(39)

d

The new input uξ is to be determined in order to achieve
the control objectives.
Using (29), (37) and (39), the visual feedback systemwith manipulator dynamics (we call the dynamic visualfeedback system) can be derived as follows⎤⎡⎤ ⎡ −M −1 Cξ + M −1 JbT AdT −1 ec(gd )ξ˙+M −1 JbT Ad(g−1 ) eh ⎥⎢ Vb ⎥ ⎢⎥eh⎢ ec ⎥ = ⎢⎥⎢−Ad(ḡ−1 ) Jb ξ
b ⎦
⎣ Vee
⎦
⎣
ho
0
b
Veh
−Jb ξ
⎡ M −1 0
⎤
⎡
⎤
0
0
⎢
+⎣

I

−Ad(ḡ−1 )

0

0

0
0

0 −Ad (g−1 )0eeI0−Ad(g−1 )hoM −1 00⎦
I w (40)0 0⎥
0⎦u + ⎣ 0ehwhere x := [ξ T eT ]T and u := [uTξ uTd uTe uTh ]T . We
define the disturbance of dynamic visual feedback system


b T T
)
. Before constructing the dynamic
as w := τdT (Vwovisual feedback control law, we derive an important lemma.
Lemma 2: If w = 0, then the dynamic visual feedback
system (40) satisfies
 T
uT νdτ ≥ −β, ∀T > 0
(41)
0

where

⎡

I
⎢ 0

0
0

0
0
0 Ad −ξ̂θehν := N x, N := ⎢⎣ 0 Ad(e−ξ̂θec ) −I
0 −Ad T −1

(gd )

0

(e

0
−I

⎤
)

⎥
⎥.
⎦

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 K

0
0

0

0 Kh

0
0

u = −Kν = −KN x, K := ⎣ 0 0c Ke 0 ⎦ (43)0where Kξ := diag{kξ1 , · · · , kξn} denotes the positive gain
matrix for each joint axis. Kc := diag{kc1 , · · · , kc6 },Ke := diag{ke1 , · · · , ke6 } and Kh := diag{kh1 , · · · , kh6 }are the positive gain matrices of x, y and z axes of the translation and the rotation for the control error, the estimatedobject one and the estimated hand one, respectively. The
result with respect to asymptotic stability of the proposed
control input (43) can be established as follows.
Theorem 1: If w = 0, then the equilibrium point x =0 for the closed-loop system (40) and (43) is asymptotic
stable.
It can be proved by the energy function (42) as a Lyapunov
function. We omit the proof due to space limitations.
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 (41).
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 (40) in one of the typical problems, i.e. the disturbance attenuation problem. Now, let us define
P := N T KN −

1
1
W− I
2
2γ
2

(44)

where γ ∈ R is positive and W := diag{I, 0, I, 0}. Then
we have the following theorem.
Theorem 2: Given a positive scalar γ and consider thecontrol input (43) with the gains Kξ , Kc , Ke and Kh suchthat the matrix P is positive semi-definite, then the closedloop system (40) and (43) has L2 -gain ≤ γ.
The proof is omitted due to space limitations. Theorem
2 can be proved using the energy function (42) as a

zo
Gwo
xo

Gwh(=q 1+q 2)
xh

gho

yh

xw

gwh

gwo

5o
yo
Target Object Frameyw
2DOF Robot
Manipulator

Fig. 3. Coordinate frames for dynamic visual feedback system with twodegree of freedom manipulator.

storage function for L2 -gain performance analysis. The L2 gain performance analysis of the dynamic visual feedback
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
visual feedback systems.
V. S IMULATIONThe simulation results on the two degree-of-freedom
manipulator as depicted in Fig. 3 are shown in order tounderstand our proposed method simply, though it is valid
for 3D visual feedback systems. We use the reference ofthe relative rigid body motion as a constant value, i.e. pd =[0 0 − 0.81]T and eξ̂θd = I. Specifically, we consider
the set point problem, i.e. the target object is static, inorder to compare the performance of the proposed control
law and the previous one discussed in [9] clearly. The
uncertainty of the camera coordinate frame is chosen as
T ∧gδ = ([0.05 − 0.02 0.04]T , e([−π/24 π/25 π/18] ) ).
Fig. 5 shows the error between gho and gd which isdefined as ger := gd−1 gho . The errors for the control
objective with the proposed control law tend to zero in bothcases, while the previous control law with the uncertainty
can not achieve the control objective. Hence, the proposed
control law is valid for the uncertainty of the camera
coordinate frame. In the case of the tracking problem withthe moving target object, it can be confirmed by the same
simulation in [9].
VI. C ONCLUSIONSThis paper dealt with the dynamic visual feedback control
with the uncertainty of the camera coordinate frame, which
is not limited to the orientation around the optical axis.
We can derive that the dynamic visual feedback systempreserves the passivity of the visual feedback system byperx [m]

0
0.1
0
-0.1
-0.2
-0.3
0
0

0.2 0.4 0.6 0.8 1.0

pery [m]World Frame
zw
5w
q1
q2

0.15
0.1
0.05
0
-0.05

0.2 0.4 0.6 0.8 1.0e^NGerz [rad]

gwc
gco

Hand Frame
zh
gch
5h

e^NGerz [rad]ycpery [m]

5c

xc

0.15
0.1
0.05
0
-0.05

perx [m]Camera Frame
zc

-0.2
-0.4
0

0.2 0.4 0.6 0.8 1.0
time [s]

0
0.1
0
-0.1
-0.2
-0.3
0

0.2 0.4 0.6 0.8 1.0

0.2 0.4 0.6 0.8 1.0

0
-0.2
-0.4
0

0.2 0.4 0.6 0.8 1.0
time [s]Fig. 4. Errors for the control objective with the proposed control law(solid) and with the previous control law in [9] (dashed). Left side: without
the uncertainty, right side: with the uncertainty.

the same strategy in our previous works [6]-[9]. Stability
and L2 -gain performance analysis for the dynamic visualfeedback system are discussed based on passivity with theenergy function. The validity of the proposed control law
is confirmed by comparing the simulation resluts.
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