PASSIVITY-BASED DYNAMIC VISUALFEEDBACK CONTROL WITH A MOVABLECAMERA
Toshiyuki Murao ∗ Hiroyuki Kawai ∗∗Masayuki Fujita ∗∗Graduate School of Natural Science and Technology,
Kanazawa University, Kanazawa 920-8667, JAPAN∗∗
Information Technology Research Center,
Hosei University, Tokyo 102-8160, JAPAN

Abstract: This paper deals with the dynamic visual feedback control with a
movable camera instead of a ﬁxed camera in the ﬁxed camera conﬁguration. Firstly
the brief summary of the visual feedback system with a ﬁxed camera is given
with the fundamental representation of a relative rigid body motion. Secondly weconstruct the new error system in order to consider the camera ﬁeld of view. Next,
we derive the passivity of the dynamic visual feedback system by combining themanipulator dynamics and the visual feedback system. Based on the passivity,
stability and L2 -gain performance analysis are discussed. Finally the validity ofthe proposed control law can be conﬁrmed by comparing the simulation results.
c
Copyright 2005 IFAC
Keywords: Robot Control, Visual Servoing, Lyapunov Stability, Mechanical
Systems, Passivity

1. INTRODUCTION
The combination of mechanical control with visual information, so-called visual feedback control
or visual servoing, should become extremely important, when we consider a mechanical system
working under dynamical environments (Hutchinson
et al., 1996), (Christensen and Corke, 2003). Classical visual servoing algorithms assume that the
manipulator dynamics is negligible and do not interact with the visual feedback loop. However, thisassumption is invalid for high speed tasks, while itholds for kinematic control problems. Kelly considered the set-point problems with a static target
for the dynamic visual feedback system which
includes the manipulator dynamics (Kelly, 1996).
Bishop et al. proposed an inverse dynamics based
control law for the position tracking and the camera calibration problems of the dynamic visualfeedback system (Bishop and Spong, 1999). Recently, Zergeroglu et al. developed an adaptive
control law for the position tracking and the camera calibration problems of the dynamic visualfeedback system with parametric uncertainties(Zergeroglu et al., 2001). More recently, the authors proposed the passivity-based dynamic visualfeedback control for the 3D target tracking (Kawaiand Fujita, 2004), (Kawai et al., 2004). Althoughthese control laws guarantee the stability for the
dynamic visual feedback system, the problems ofthe camera ﬁeld of view (Chaumette, 1998) are
not dealt with in these issues. However, it is important to consider the problems of the camera
ﬁeld of view in order to enlarge the available
workspace for the robot hand with the conﬁguration separated a camera frame and a hand (end-ImageCameraRobot
Manipulator

gch

5c

gwc
gco

5h
gwh

gho
5o

gwo

5w
World Frame

Target Object

Fig. 1. Visual feedback system in the ﬁxed camera
conﬁguration.
eﬀector) one, we call the ﬁxed camera conﬁguration, as shown in Fig. 1.
In this paper, we discuss the dynamic visual feedback control with a movable camera instead of aﬁxed camera in the ﬁxed camera conﬁguration inorder to increase the available workspace for the
robot hand. This can be regarded as a solution for
one of the problems of the camera ﬁeld of view inthe proposed framework which is based on ourprevious works (Fujita et al., 2002), (Kawai and
Fujita, 2004), (Kawai et al., 2004). Moreover, we
can derive that the dynamic visual feedback system preserves the passivity of the visual feedback
system which is obtained in our previous works.
Throughout this paper, we use the notationeξ̂θab ∈ 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 crossproduct × and any vector θ ∈ R3 . The notation‘∨’ (vee) denotes the inverse operator to ‘∧’: i.e.,so(3) → R3 . ξab ∈ R3 speciﬁes the direction ofrotation and θab ∈ R is the angle of rotation. Hereξ̂θab denotes ξ̂ab θab for the simplicity of notation.
We use the 4 × 4 matrix


ξ̂θab
pab
gab = e(1)01
as the homogeneous representation of gab =(pab , eξ̂θab ) ∈ SE(3) which is the description of the
conﬁguration of a frame Σb relative to a frame Σa .
The adjoint transformation associated with gab isdenoted by Ad(gab ) (Murray et al., 1994). Let us
deﬁne the vector form of the rotation matrix aseR (eξ̂θab ) := sk(eξ̂θab )∨ where sk(eξ̂θab ) denotes
1 ξ̂θab− e−ξ̂θab ).
2 (e2. PASSIVITY-BASED VISUAL FEEDBACK
SYSTEM IN THE FIXED CAMERACONFIGURATION
2.1 Fundamental Representation for Visual Feedback
SystemVisual feedback systems typically use four coordinate frames which consist of a world frame Σw , atarget object frame Σo , a camera frame Σc and ahand (end-eﬀector) frame Σh as in Fig. 1. Then,gwh , gwc and gwo denote the rigid body motions
from Σw to Σh , from Σw to Σc and from Σw toΣo , respectively. Similarly, the relative rigid body
motions from Σc to Σh , from Σc to Σo and fromΣh to Σo can be represented by gch , gco and gho ,respectively, as shown in Fig. 1.
The relative rigid body motion from Σc to Σo canbe led by using the composition rule for rigid bodytransformations ((Murray et al., 1994), 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 eachrigid body. To this aid, let us consider the velocity
of a rigid body as described in (Murray et al.,1994). Now, we deﬁne the body velocity of thecamera relative to the world frame Σw asω̂wc vwc
vwc
b
−1
b
Vwc =
(3)
V̂wc = gwc ġwc =0 0
ωwc
where vwc and ωwc represent the velocity ofthe origin and the angular velocity from Σw toΣc , respectively ((Murray et al., 1994) Chap. 2,
eq. (2.55)).
Then, the fundamental representation of the relative rigid body motion gco is described as follows(Kawai and Fujita, 2004).
bbb= −Ad(gco−1 Vwc + Vwo
Vco
)

(4)

b
is the body velocity of the target object
where Vwo
relative to Σw . Roughly speaking, the relative
rigid body motion gco will be derived from theb
diﬀerence between the camera velocity Vwc
andb
the target object velocity Vwo.

2.2 Estimation Error and Control Error Systems
Here the brief summary of our prior works in(Kawai and Fujita, 2004) and (Kawai et al., 2004)
are given. The visual information f 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.
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 system. Firstly, we shallconsider the following model which just comesfrom the fundamental representation (4).
bb−1 V= −Ad(ḡcoV̄co) wc + ue(5)where ḡco is the estimated relative rigid body
motion from Σc to Σo and ue is the input inorder to converge the estimated value to the actual
relative rigid body motion. Next, the estimation
error of the relative rigid body motion from Σc toΣo , i.e. the error between ḡco and gco , is deﬁned
as−1gee = ḡco
gco ,

(6)

which is called the estimation error. Using thenotation eR (eξ̂θ ), the vector of the estimation
error is given by ee := [pTee eTR (eξ̂θee )]T . Then,the estimation error vector ee can be obtained byusing image information f. The estimation error
system is represented bybbVee= −Ad(gee−1 ue + V
wo .
)

(7)

Similarly, we deﬁne the error between gd and ḡho ,which is called the control error, as follows
gec = gd−1 ḡho ,(8)where ḡho is the estimated relative rigid body
motion from Σh to Σo and obtained from ḡho =−1gchḡco . 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
b
b
b
Vec
= −Ad(ḡ−1 ) Vwh+ ue − Ad(gec
−1 Vd
)

(9)

ho

b
where Vwh
and Vdb are the body velocity of thehand relative to Σw and the desired body velocityof the relative rigid body motion gho , respectively.

Combining (7) and (9), the visual feedback systemin the ﬁxed camera conﬁguration is constructed asfollows 
 b  −Ad(ḡ−1 )
I
0
Vec
ho
=
+
V b (10)ub
I wo
0−Ad(g−1 ) ce
Vee

uration, we call the camera ﬁeld error system,in order to increase the available workspace forthe robot hand. Here we deﬁne the camera ﬁeld
error between the estimated value ḡco and a givenreference gcd for the camera motion as−1ḡco .
gev = gcd

If ḡco is equal to gcd , then the target object
can be kept in the center of the camera ﬁeldof view. Using the notation eR (eξ̂θ ), the vectorof the camera ﬁeld error is deﬁned as ev :=[pTev eTR (eξ̂θev )]T . Note that ev = 0 iﬀ pev = 0
and eξ̂θev = I3 .
Similarly to (7) and (9), the camera ﬁeld error
system can be obtained as
b
b
b
−1 V
−1 V
= ue − Ad(ḡcoVev) wc − Ad(gev) cd , (13)bis the desired body velocity of thewhere Vcd
relative rigid body motion gco .

3.2 Property of Visual Feedback System
Combining (7), (9) and (13), we construct thevisual feedback system with a movable camera in
the ﬁxed camera conﬁguration as follows b  −Ad −1
 
I
0
(ḡ
)
Vec
0
ho
b
b
ucev + I Vwo
0 −Ad(g−1 )
0
=
Vee
b
Vev


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


(11)

denotes the input for the visual feedback system.
Let us deﬁne the error vector
ofT the visual feedback system as ece := eTc eTe
which containsof the control error vector ec and the estimation
error vector ee . Here, we deﬁne the output of thevisual feedback system (10) asνce :=−AdT −1(gd )

0Ad (e−ξ̂θec ) −I

ece ,then the visual feedback system (10) satisﬁes
T Tuceνcedτ ≥ −βce where βce is a positive scalar0(Kawai et al., 2004). This would suggest that thevisual feedback system (10) is passive from the
input uce to the output νce just formally as in thedeﬁnition in (van der Schaft, 2000).

3.1 Camera Field Error System
In this section, we construct the error system of
the movable camera in the ﬁxed camera conﬁg-

0

co

(14)

where


ucev :=b
Vwh+ Ad(gd ) Vdb
ue
b
b
Vwc
+ Ad(gcd ) Vcd(15)

denotes the input for the visual feedback system.
Let us deﬁne the error vectorof the visual feedTback system (14) as e := eTc eTe eTv . It should
be noted that if the vector of the estimation error
is equal to zero, not only ḡco equals gco but also ḡhoequals gho . Moreover, if the vectors of the control
error and the camera ﬁeld error are equal to zero,then ḡho and ḡco equal gd and gcd , respectively.
Thus, when e → 0, gho and gco tend to gd and
gcd , respectively. This states that the control objective can be achieved, in addition, the available
workspace for the robot hand will be increased bymoving of the camera ﬁeld of view.
b
Lemma 1. If Vwo= 0, then the visual feedback
system (14) satisﬁes
T0

3. VISUAL FEEDBACK SYSTEM WITH A
MOVABLE CAMERAee

I −Ad (ḡ−1 )

0

ee

where

(12)

uTcev νcev dτ ≥ −βcev , ∀T > 0

where νcev is deﬁned as

T


−Ad

−1
)
(gd

0

0




νcev :=  Ad(e−ξ̂θec ) −I Ad(e−ξ̂θev )  e
00 −Ad T −1
(gcd )and βcev is a positive scalar.

(16)

(17)

PROOF. Consider the following positive deﬁnite
function
Vcev = E(gec) + E(gee ) + E(gev )(18)

where E(g) := 12 p 2 + φ(eξ̂θ ) and φ(eξ̂θ ) :=1
tr(I − eξ̂θ ) which is the error function of the ro2
tation matrix (see e.g. (Bullo and Murray, 1999)).
Diﬀerentiating (18) with respect to time yieldsb Ad(eξ̂θec )
0
0
Vec
T 
b
 Vee
0
Ad(eξ̂θee )0V̇cev = e
(19)
0

0

b
Vev

Ad(eξ̂θev )where we use the property φ̇(eξ̂θ ) := eξ̂θ ω. Observing the skew-symmetry of the matrices p̂ec , p̂ee
and p̂ev , the above equation along the trajectoriesof the system (14) can be transformed intoV̇cev = eT −Ad (g−1 ) Ad(eξ̂θec )
0
0

d

0

 ucev0Ad(eξ̂θev ) −Ad(g−1 )−I

cd= uTcev νcev .

(20)

Let us deﬁne the error vector with respect to thejoint velocities of the manipulator dynamics asξ := 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)+JbT (q)AdT(g−1 ) ec + uξ . (24)d

The new input uξ is to be determined in order toachieve the control objectives.
Using (14), (22) and (24), the visual feedback
system with manipulator dynamics (we call the
dynamic visual feedback system) can be derived
as follows  −M −1 Cξ + M −1 J T AdT
−1 ec
b
ξ˙
(gd
)
b

 Vec
 
−Ad(ḡ−1 ) Jb ξVb = 
ho


ee
0
b
Vev



0
−1

M
0
0
I
 0 −Ad(ḡ−1 )
ho
+
 0
0
−Ad (g−1 )
0

I

0

ee

 

M −1 0  0 0
u+
w

0 I 

0
0
0
−Ad(ḡ−1 )

0

0

co

(25)

Integrating (20) from 0 to T , we can obtain
where

T
0

uTcev νcev dτ ≥ −Vcev (0) := −βcev(21)

Lemma 3. If w = 0, then the dynamic visualfeedback system (25) satisﬁesT0Iν := N x, N :=  0 Ad



04.1 Dynamic Visual Feedback System
The manipulator dynamics can be written as(22)

where q, q̇ and q̈ are the joint angles, velocities
and accelerations, respectively. τ is the vector of
the input torques and τd represents a disturbance
input. The body velocity of the hand is given byb= Jb (q)q̇VwhuT νdτ ≥ −β, ∀T > 0

0
T
0
−Ad
−1


4. DYNAMIC VISUAL FEEDBACK
CONTROL(23)

where Jb (q) is the manipulator body Jacobian(Murray et al., 1994). We deﬁne the reference ofthe joint velocities as q̇d := Jb† (q)ud where ud
represents the desired body velocity of the hand.
bThus, Vwhin (15) should be replaced by ud .

(26)

and x := [ξ T eT ]T . We deﬁne the disturbanceof dynamic visual feedback system as w :=

 T
b T T
)
. Before constructing the dynamicτd (Vwovisual feedback control law, we derive an important lemma.

where

M (q)q̈ + C(q, q̇)q̇ + g(q) = τ + τduξ
 ud + Ad(gd ) Vdb u := 
,uebb
Vwc + Ad(gcd ) Vcd

where βcev is the positive scalar which only depends on the initial states of gec , gee and gev .
Remark 2. In the visual feedback system, pTec(e−ξ̂θdωec)∧ pec = 0, pTee ω̂ee pee = 0, pTev (e−ξ̂θcd ωev )∧ pev =0 hold. This skew-symmetric 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 (Murray et al., 1994)). Thus,
Lemma 1 suggests that the visual feedback system(14) is passive from the input ucev to the output
νcev as in the deﬁnition in (van der Schaft, 2000).



(gd )

(e−ξ̂θec )

0

0
0

0
0

(27)




−I Ad(e−ξ̂θev )  .
0−AdT −1(gcd )

Due to space limitations, the proof is only
sketched. By using the following positive deﬁnite
function, the proof can be completed.
1
V = ξ T M ξ + E(gec) + E(gee ) + E(gev ). (28)2Remark 4. Similarly to Lemma 1, Lemma 3 would
suggest that the dynamic visual feedback systemis passive from the input u to the output ν justformally. From Lemma 3, we can state that the
dynamic visual feedback system (25) preserves thepassivity of the visual feedback system (14). Thisis one of main contributions of this work.

4.2 Stability Analysis for Dynamic Visual Feedback
System
It is well known that there is a direct link between passivity and Lyapunov stability. Thus, we
propose the following control input.


Kξ

0

0

0

0

0 Kv

0

 0 K 0 0 
u = −Kν = −KN x, K :=  0 0c K 0 (29)ewhere Kξ := diag{kξ1 , · · · , kξn } denotes the positive gain matrix for each joint axis. Kc :=diag{kc1 , · · · , kc6 }, Ke := diag{ke1, · · · , ke6} andKv := diag{kv1 , · · · , kv6} are the positive gain
matrices of x, y and z axes of the translation andthe rotation for the control error, the estimation
one and the camera ﬁeld one, respectively. The
result with respect to asymptotic stability of the
proposed control input (29) can be established asfollows.
Theorem 5. If w = 0, then the equilibrium pointx = 0 for the closed-loop system (25) and (29) is
asymptotic stable.
Theorem 5 can be proved using the energy function (28) as a Lyapunov function. The proof isomitted here due to space limitations. Considering
the manipulator dynamics, Theorem 5 shows thestability via Lyapunov method for the full 3D
dynamic visual feedback system. It is interesting
to note that stability analysis is based on thepassivity as described in (27).

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
disturbance attenuation problem. The proposed
strategy can be extended for the other-type ofgeneralized plants of the dynamic visual feedback
systems.
5. SIMULATIONThe simulation results on the two degree-offreedom manipulator as depicted in Fig. 2 areshown in order to understand our proposed
method simply, though it is valid for 3D visualfeedback systems. The target object has four feature points and moves for t = 4.8 [s] along a
straight line (0 ≤ t < 2) and a “Figure 8” motion (2 ≤ t < 4.8). Speciﬁcally, we compare theperformance in the case of the movable camera
system and the ﬁxed camera system discussedin (Kawai et al., 2004). We use the reference ofthe relative rigid body motion as constant values,
i.e. pd = [0 0 − 0.81]T , eξ̂θd = I, pcd = [0 0 − 2]T ,beξ̂θcd = I, Vdb = 0 and Vcd= 0, for the tracking
problems in the simulation.
Camera Frame
zc

5c

xc

yc
gwc
gco
zo

4.3 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 (25) in one of thetypical problems, i.e. the disturbance attenuation
problem. Now, let us deﬁne
P := N T KN −

1
1
W− I
2γ 2
2

(30)

where γ ∈ R is positive and W := diag{I, 0, I, 0}.
Then we have the following theorem.
Theorem 6. Given a positive scalar γ and consider
the control input (29) with the gains Kξ , Kc , Ke
and Kv such that the matrix P is positive semideﬁnite, then the closed-loop system (25) and (29)
has L2 -gain ≤ γ.
The proof is omitted due to space limitations,
Theorem 6 can be proved using the energy function (28) as a storage function for L2 -gain performance analysis. The L2 -gain performance analysisof the dynamic visual feedback system is discussed

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 visualfeedback system with two degree of freedom
manipulator.
Fig. 3 shows the error between gho and gd which isdeﬁned as ger := gd−1 gho for the control objective,
the left graphs denote the error of the case of
the proposed control law, and the right ones
denote the error of the case of the previous one(Kawai et al., 2004). In this ﬁgure, we focus
on the errors of the translations of x and y
and the rotation of z, because the errors of thetranslation of z and the rotations x and y are
zeros ideally on the deﬁned coordinates in Fig. 2.
Clearly, the proposed control law achieves thecontrol objective, not causing the performance
deterioration in comparison with the previous one.
Omitted due to space limitations, stability andL2 -gain performance analysis are veriﬁed as sameas the simulation in (Kawai et al., 2004). Fig. 4

0.01
perx [m]perx [m]0.010pery [m]

0.05

0

2.0

4.0

0

-0.05
0

2.0

4.0

-0.01

0

2.0

4.0

6.0

-0.05
0

2.0

4.0

6.0

0.05

6.0

e^NGerz [rad]0.1e^NGerz [rad]6.0

pery [m]

-0.01

0

0
-0.1
0

2.0
4.0
time [s]00.1

REFERENCES

0

Bishop, B. E. and M. W. Spong (1999). Adaptive
calibration and control of 2d monocular visual servo systems. Control Engineering Practice 7, 423–430.
Bullo, F. and R. Murray (1999). Tracking forfully actuated mechanical systems: a geometric framework. Automatica 35, 17–34.
Chaumette, F. (1998). Potential problems of stability and convergence in image-based andposition-based visual servoing. In: The Confluence of Vision and Control (D. J. Kriegman, G. D. Hager and A. S. Morse, Eds.).
pp. 66–78. Springer–Verlag. London.
Christensen, H. I. and P. Corke (2003). Editorial:
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Fujita, M., A. Maruyama and H. Kawai (2002).
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12, 651–670.
Kawai, H. and M. Fujita (2004). Passivitybased dynamic visual feedback control forthree dimensional target tracking:stability
and l2 -gain performance analysis. In Proc.
of the 2004 American Control Conference,
pp. 1522–1527.
Kawai, H., T. Murao and M. Fujita (2004).
Passivity-based control and estimation of dynamic visual feedback systems with a ﬁxed
camera. In Proc. of the 43rd IEEE Conf. on
Decision and Control, pp. 4022–4027.
Kelly, R. (1996). Robust asymptotically stablevisual servoing of planar robots. IEEE Trans.
Robotics and Automation 12, 759–766.
Murray, R., Z. Li and S. S. Sastry (1994). A Mathematical Introduction to Robotic Manipulation. CRC Press. London.
van der Schaft, A. (2000). L2 -Gain and Passivity
Techniques in Nonlinear Control (2nd ed.).
Springer–Verlag. London.
Zergeroglu, E., D. M. Dawson, M. S. de Queiroz
and A. Behal (2001). Vision-based nonlinear tracking controllers with uncertain robotcamera parameters. IEEE/ASME Trans.
Mechatronics 6, 322–337.

-0.1

6.0

0

2.0
4.0
time [s]6.0

Fig. 3. Error for the control objective. (Left side:
Movable camera system, Right side: Fixedcamera system)
fx1 [pixel]

50
30
10
-10
-30
0

1.0

2.0

3.0

4.0

5.0

6.0

1.0

2.0

3.0
time [s]

4.0

5.0

6.0

fy1 [pixel]

100
50
0
-50
0

in our previous works (Fujita et al., 2002), (Kawaiand Fujita, 2004), (Kawai et al., 2004). Stability
and L2 -gain performance analysis for the dynamic
visual feedback system have been discussed basedon passivity with the energy function. The validityof the proposed control law was conﬁrmed bycomparing the simulation results.

Fig. 4. One of feature points. (Movable camera
system: solid, Fixed camera system: dashed)presents one of the four feature points. In thisﬁgure, the solid lines denote the feature pointof the case of the proposed control law, and thedashed lines denote the feature point of the case of
the previous one. We can verify that the change ofthe feature point with the proposed control law isless than with the previous one. This result statesthat the target object almost exists in the centerof the camera, and the camera can move not tomiss the moving target object. Thus, we consider
that the dynamic visual feedback control with a
movable camera can enlarge the camera ﬁeld ofview, not causing the performance deterioration,in comparison with a ﬁxed camera (Kawai etal., 2004).
6. CONCLUSIONSThis paper dealt with the dynamic visual feedback
control with a movable camera instead of a ﬁxed
camera in the ﬁxed camera conﬁguration in order
to increase the available workspace for the robot
hand. Moreover, we derived that the dynamic
visual feedback system preserved the passivity ofthe visual feedback system by the same strategy