Visual Motion Observer-based Pose Control
via Image Space Navigation Function
Toshiyuki Murao

Hiroyuki Kawai

Masayuki Fujita

Master Program of Innovation
for Design and Engineering
Advanced Institute ofIndustrial Technology
Tokyo 140-0011, Japanmurao-toshiyuki@aiit.ac.jp

Department of Robotics
Kanazawa Institute of Technology
Ishikawa 921-8501, Japanhiroyuki@neptune.kanazawa-it.ac.jp

Department of Mechanicaland Control Engineering
Tokyo Institute of Technology
Tokyo 152-8550, Japanfujita@ctrl.titech.ac.jp

Abstract—This paper investigates passivity-based visual feedback control via an image space navigation function. Firstly, abrief summary of a visual motion observer is given. Next, a visualmotion error system is reconstructed in order to apply to timevarying desired motion. Then, visual motion observer-based posecontrol for a 3-D visual feedback system is proposed. Moreover,a path planner to be appropriate for the visual motion error
system is designed through the image space navigation function
to keep all features into a camera field of view. Finally, simulation
results are shown in order to confirm the proposed method.

Camera5c
gcoTargetObject

gwc

5o

gwo

5w
World Frame

I. I NTRODUCTION
Fig. 1.

Visual feedback control is now a very useful method in robot
control [1]. Recently, Gans and Hutchinson [2] proposed hybrid switched-system control which utilizes image-based andposition-based visual feedback control. In [3], an occlusion
problem for a hybrid eye-in-hand/eye-to-hand multicamera
system was tackled by using an extended Kalman filter anda multiarm robotic cell. In order to avoid joint limits andocclusions, Mansard and Chaumette [4] considered directional
redundancy that only imposes to a secondary control law not
to increase an error of a main task.
Especially, there has been an increase of interest in problems
that are all feature points remain within a camera field of view
since the work of Chaumette [5], while many researchers havetackled various problems for visual feedback control. Cowan
et al. [6] proposed a visual feedback controller to bring a robot
to rest at a desired configuration for the field of view problem
by using navigation functions, similar to artificial potentialfunctions. Chen et al. [7] developed an off-line path plannerbased on an image space navigation function with an adaptive
2 1/2-D visual servoing controller.
In [8], the authors proposed stabilizing receding horizon
control for an eye-in/to-hand visual feedback system which
includes both an eye-in-hand system and an eye-to-hand one as
the special cases. However, this position-based visual feedback
control method through passivity approach can allow featurepoints to leave the field of view.
In this paper, we propose 3-D passivity-based visual feedback control via an image space navigation function. Firstly, abrief summary of a visual motion observer is given. Secondly,

Eye-in-hand visual feedback systems.

a visual motion error system is reconstructed in order to handle
time-varying desired motion. Next, a passivity-based control
law for the visual motion error system is proposed. Then,a path planner to be appropriate for the visual motion error
system is designed through an image space navigation functionin order to keep all features into the camera field of view. The
path planning based on an image space could be of significant
benefit when used in conjunction with the proposed positionbased visual feedback control by using an error definedon a Cartesian space. Finally, the control performance withvisibility maintenance of the proposed control scheme and theprevious one [8] is evaluated through simulation results.
II. V ISUAL M OTION O BSERVER
This section mainly reviews our previous work [8] via
passivity-based visual feedback control with the eye-in-handconfiguration.
A. Body Velocity of Relative Rigid Body Motion
Visual feedback systems with the eye-in-hand configuration
use three coordinate frames which consist of a world frameΣw , a camera frame Σc , and an object frame Σo as in Fig. 1.
Let pco ∈ R3 and eξ̂θco ∈ SO(3) be a position vector anda rotation matrix from the camera frame Σc to the object
frame Σo . Then, a relative rigid body motion from Σc to Σocan be represented by gco = (pco , eξ̂θco ) ∈ SE(3). Similarly,
gwc = (pwc , eξ̂θwc ) and gwo = (pwo , eξ̂θwo ) denote rigid body

xi

fyi
zo

fi

pci

zw

Image Plane
xw
yw

5w

World Frame

Target Object
Fig. 2.

5c
yc

fxi

yo

5o

of the coordinates, we have pci = gco poi , where pci and poi
should be regarded, with a slight abuse of notation, as [pTci 1]Tand [pToi 1]T via the well-known homogeneous coordinate
representation in robotics, respectively (see, e.g., [9]).
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
fi =
,
(3)
zci yci

yi

P
xo

zc

l

gco
poi

Camera Framexc

Pinhole camera model.

motions from the world frame Σw to the camera frame Σc andfrom the world frame Σw to the object frame Σo , respectively.
An objective of position-based visual feedback control is,in general, to bring the actual relative rigid body motion gco
to a reference one gcd . Firstly, we consider the relative rigid
body motion gco in order to achieve the control objective.
The relative rigid body motion from Σc to Σo can be led byusing the composition rule for rigid body transformations ([9],Chap. 2, pp. 37, Eq. (2.24)) as follows:−1gwo .
gco = gwc

(1)

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 [9]. We define the body velocity of theb
T
T T= [vwcωwc] ,camera relative to the world frame Σw as Vwc
where vwc and ωwc represent the velocity of the origin andthe angular velocity from Σw to Σc , respectively ([9] Chap. 2,
Eq. (2.55)).
Differentiating Eq. (1) with respect to time, the body
velocity of the relative rigid body motion gco can be written
as follows (See [8]):
b
b
b
−1 V
= −Ad(gco
Vco) wc + Vwo ,(2)bwhere Vwois the body velocity of the target object relative toΣw and Ad(gco ) is the adjoint transformation associated withgco [9].

B. Image Features of Pinhole Camera Model
The relative rigid body motion gco = (pco , eξ̂θco ) cannot be
immediately obtained in the visual feedback system, becauseb
the target object velocity Vwo
is unknown and furthermore
cannot be measured directly. To control the relative rigid body
motion using visual information provided by computer visionsystem, we use the pinhole camera model with a perspective
projection as shown in Fig. 2. Here, we consider m(≥ 4)
feature points on the rigid target object in this paper.
Let λ be a focal length, poi ∈ R3 and pci ∈ R3 be
the position vectors of the target object’s i-th feature point
relative to Σo and Σc , respectively. Using a transformation

where pci = [xci yci zci ]T . It is straightforward to extend thismodel to m image points by simply stacking the vectors of the
T T] ∈ R2m
image plane coordinate, i.e., f (gco ) := [f1T · · · fm
TT T
3m
and pc := [pc1 · · · pcm ] ∈ R . Hereafter, fab meansf (gab ) for simplicity. We assume that multiple point features
on a known object are given. Under this assumption, the image
feature vector fco depends only on the relative rigid body
motion gco .
C. Estimation Error System
The visual feedback control task requires information of
the relative rigid body motion gco . Since the measurable
information is only the image feature fco in the visual feedback
system, we consider a nonlinear observer in order to estimate
the relative rigid body motion gco from the image feature fco .
bFirstly, using Eq. (2), we choose estimates ḡco and V̄coof
the relative rigid body motion and velocity, respectively as
b
b
−1 V
V̄co
= −Ad(ḡco) wc + ue .

(4)

The new input ue is to be determined in order to drive thebestimated values ḡco and V̄coto their actual values.
In order to establish the estimation error system, we define
the estimation error between the estimated value ḡco and theactual relative rigid body motion gco as−1
gee = ḡco
gco .

(5)
ξ̄ˆθ̄coif and only if
Note that pco = p̄co and eξ̂θco = e
gee = I4 , i.e., pee = 0 and eξ̂θee = I3 . We next define
the error vector of the rotation matrix eξ̂θab as eR (eξ̂θab ) :=sk(eξ̂θab )∨ where sk(eξ̂θab ) denotes 12 (eξ̂θab − e−ξ̂θab ). Using
this notation, the vector of the estimation error is given by
ee := [pTee eTR (eξ̂θee )]T . Note that ee = 0 iff pee = 0 andeξ̂θee = I3 .
Suppose the attitude estimation error θee is small enoughthat we can let eξ̂θee  I + sk(eξ̂θee ). Therefore, using a firstorder Taylor expansion approximation, the estimation error
vector ee can be obtained from image information fco andthe estimated value of the relative rigid body motion ḡco as
follows [8]:(6)ee = Je† (ḡco )(fco − f¯co ),where f¯co is the estimated value of image feature and Je (ḡco ) :SE(3) → R2m×6 is defined as
 T
T
T
T
(ḡco ) Je2(ḡco ) · · · Jem(ḡco )Je (ḡco ) := Je1
(7)
 λ

λx̄ci
0 − z̄2


ˆ
z̄ci
ci
Jei (ḡco ) :=eξ̄θ̄co I −p̂oi ,λȳci
λ0 z̄ci − z̄2ci(i = 1, · · · , m),(8)

and † denotes the pseudo-inverse defined as A† :=(AT A)−1 AT . In the same way as Eq. (2), the estimation error
system can be represented bybb−1 ue + V
Vee
= −Ad(gee
wo .
)

0

where βe is a positive scalar.
D. Visual Motion Observer
Based on the above passivity property of the estimation error
system, we consider the following control law.
(11)

where Ke := diag{ke1 , · · · , ke6 } is the positive gain matrixof x, y and z axes of the translation and the rotation for theestimation error.
bTheorem 1 ([8]): If Vwo= 0, then the equilibrium pointee = 0 for the closed-loop system (9) and (11) is asymptotic
stable.
It should be noted that if the vector of the estimation error
is equal to zero, then the estimated relative rigid body motionḡco equals the actual one gco . The estimation error vector
is configured by available information (i.e.,the measurement
and the estimate) though it is defined by unavailable one.
By the proposed visual motion observer, the unmeasurable
motion gco will be exploited as the part of control law.
Our proposed visual motion observer is composed just as
Luenberger observer for linear systems.
III. V ISUAL M OTION O BSERVER - BASED P OSE C ONTROL
Although the time-varying desired motion is needed to thepath planner, the desired motion in our previous work [8]is assumed to be constant. Thus, we have to reconstruct thevisual motion error system in order to handle the time-varying
desired motion.
A. Pose Control Error System
Let us consider the dual of the estimation error system,
which we call the pose control error system in order to achieve
the control objective. First, we define the pose control error
as follows:−1
gco ,gec = gcd

gco = ḡco gee−1ξθee =(9)Then, we have the following lemma relating the input ue
to the vector form of the estimation error ee .
bLemma 1 ([8]): If Vwo= 0, then the following inequality
holds for the estimation error system (9).
 T
uTe (−ee )dt ≥ −βe(10)ue = −Ke (−ee ),motion ḡco and the estimation error vector ee = [pTee eTR (eξ̂θee ) ]equivalently as follows:(12)

which represents the error between the relative rigid body motion gco and the reference one gcd . It should be remarked thatgco can be calculated by using the estimated relative rigid body

sin

(13)
eR (eeRξ̂θee(eξ̂θee )

)eR (eξ̂θee ),(14)although gco can’t be measured directly. Using the notationeR (eξ̂θ ), the vector of the pose control error is defined as
ec := [pTec eTR (eξ̂θec )]T .
Differentiating Eq. (12) with respect to time, the posecontrol error system can be represented as
b
b
b
b
−1
= −Ad(gecVec) Ad(g−1 ) Vwc + Vcd + Vwo ,
cd

(15)

b
is the body velocity of the reference of the relative
where Vcd
rigid body motion gcd . This is dual to the estimation error
system. Similar to the estimation error system, the pose control
error system also preserves the passivity property.
b= 0 in Eq. (15), thenRemark 1: If gcd is constant, i.e., Vcd
the control error system (15) equals to that of proposed in [8].
Thus, our previous work [8] can be regarded as the special
cases of this study.

B. Passivity of Visual Motion Error System
Combining the estimation error system (9) and the posecontrol one (15), we construct the visual motion observerbased pose control error system (we call the visual motion
error system) as follows:  b  −1
−Ad(gec0IVec)b, (16)
=
u+
Vwo
b
−1
I
0
−Ad(geeVee)bbwhere u := [uTc uTe ]T , uc := Ad(g−1 ) Vwc+ Vcd. Let us define
cd
the error vector of the visual motion error system as x :=[eTc eTe ]T , which consists of the pose control error vector ec
and the estimation error vector ee . It should be noted that if
the vectors of the pose control error and the estimation one
are equal to zero, then the actual relative rigid body motion
gco tends to the reference one gcd when x → 0.
Next, we show an important relation between the input andthe output of the visual motion error system.
b
Lemma 2: If Vwo= 0, then the visual motion error system(16) satisfies
 T
uT (−x)dt ≥ −β, ∀T > 0(17)0

where β is a positive scalar.
Proof: Consider the following positive definite function11
V = pec 2 + φ(eξ̂θec ) + pee 2 + φ(eξ̂θee ). (18)22The positive definiteness of the function V results from theproperty of the error function φ. Differentiating Eq. (18)with respect to time and using the skew-symmetry of thematrices p̂ec and p̂ee , i.e., pTec p̂ec ωuc = −pTec ω̂uc pec = 0 and
pTee p̂ee ωue = −pTee ω̂ue pee = 0 yields0
T −Ad(−pec )V̇ = xu = uT (−x). (19)0−Ad(−pee )

b
Vcd

f

gcd

Visual Motion
Pose
b
gco
ObserverController
Vwc
e
u
e
u
e
e
e
c
c
...
... e ...
...
Ke
Kc

b
Vwc

b
Vwo

Vision f
CameraVisual Motion Observer-based
Pose Controller

Fig. 3.

Block diagram of visual motion observer-based pose control.

Integrating (19) from 0 to T , we obtain
 T
uT (−x)dt = V (T ) − V (0) ≥ −V (0) := −β, (20)0where β is the positive scalar which only depends on the initialstates of gec = (pec , eξ̂θec ) and gee = (pee , eξ̂θee ).
Remark 2: Let us take u as the input and −x as its output.
Thus, Lemma 2 implies that the visual motion error system(16) is passive from the input u to the output −x.
C. Visual Motion Observer-based Pose Control and Stability
AnalysisBased on the above passivity property of the visual motion
error system, we consider the following control law.

Kc 0
u = −K(−x), K :=
,
(21)
0 Ke
where Kc := diag{kc1 , · · · , kc6 } is the positive gain matrixof x, y and z axes of the translation and the rotation for thepose control error.
b= 0, then the equilibrium point x = 0
Theorem 2: If Vwo
for the closed-loop system (16) and (21) is asymptotic stable.
Proof: In the proof of Lemma 2, we have already shownthat the time derivative of V along the trajectory of the system(16) is formulated as Eq. (19). Using the control input (21),
Eq. (19) can be transformed intoV̇ = −xT Kx.

(22)This completes the proof.
Theorem 2 shows Lyapunov stability for the closed-loopbsystem. If the camera velocity Vwc
is decided directly, thecontrol objective is achieved by using the proposed control
law (21). Fig. 3 shows the block diagram of the visual motion
observer-based pose control. It should be noted that the desired
image is not needed in the proposed controller which only
entails the given desired relative rigid body motion gcd .
IV. I MAGE S PACE NAVIGATION F UNCTION - BASED PATHP LANNING
In [5], the inherent problem of the position-based visualfeedback control by using only the error defined on the
Cartesian space, is that the target object cannot always remainin the camera field of view during the servoing. Because theproposed visual feedback control in the previous section is
the position-based method, it may leave the camera field ofview. In this section, a path planner to be appropriate for thevisual motion error system is designed through an image space
navigation function to guarantee to keep all features into thecamera field of view. The control objective in this paper isstated as follows:
Control Objective: The vision camera follows the target
object, i.e., the relative rigid body motion gco (t) is coincidedwith the time-varying desired one gcd (t), which is generated tokeep all features into the camera field of view and converges
the final desired one gcdf .
From the proposed control law for the visual motion error
system, the input to vision camera is designed as follows:
b
b
Vwc
.
= Ad(gcd ) uc − Vcd(23)

Hence, the vision camera input is only needed the bodybvelocity Vcdof the reference of the relative rigid body motion
1
gcd .
Here, we introduce the navigation function-based method
as a technique for constructing artificial potential fields inborder to design Vcd
which can achieve the control objective.
Firstly, we define the desired image feature vector and the final
one as fcd := f (gcd ) and fcdf := f (gcdf ), respectively. The
navigation functions used in this paper are defined as follows:
Definition 1 ([7],[10]): Let D be a space where all featurepoints of the target remain visible, and let fcdf be in theinterior of D. A map ϕ : D → [0, 1] is a navigation function
if it is1) smooth on D (at least a C (2) function);
2) a unique minimum exists at fcdf ;3) admissible on D, i.e., uniformly maximal on the boundary of D;4) a Morse function.
A. Path Planning of Desired Body Velocityb, we derive a
To develop the desired body velocity Vcdbrelationship between fcd defined on the image space and Vcd
defined on the Cartesian space. Differentiating Eq. (3), we
have the following relation between the desired feature pointb
pcdi and the body velocity Vcd: bṗcdi = eξ̂θcd I −p̂oi Vcd.
(24)
Moreover, the relation between the desired image feature fcdi
and the desired feature point pcdi can be expressed as
 λ

cdi
0
− λx
2
zcdi
zcdi
˙
fcdi =(25)ṗcdi .
λ
cdi
0
− λy
zcdi
z2
cdi

Hence, the desired image feature vector and the desired body
velocity can be related asb,f˙cd = JL (gcd )Vcd

(26)

2m×6

where JL (gcd ) : SE(3) → R
is defined as
T
 T
T
T
(gcd ) · · · JLm(gcd )JL (gcd ) := JL1 (gcd ) JL2
(27)
 λ

λxcdi
0
− z2


zcdi
cdi
JLi (gcd ) :=eξ̂θcd I −p̂oi ,
λycdi
λ
0
− z2
zcdi
cdi

1 The relative rigid body motion gb
cd can be obtained solving ġcd = gcd Vcd .

(i = 1, · · · , m).

(28)

Inspired by Eq. (26) and the definition of the navigationbfunction, the desired body velocity Vcd
is designed as follows:b= −kcd JL† (gcd )∇ϕ,
Vcd

gcd

Image Space
Navigation
Function-based
Path Planner

b
Vcd
f

Visual Motion
Observer-based
Pose Controller

b
Vwo
b
Vwc

Vision f
Camera

(29)

T

cd )
denotes the gradient vector
where ∇ϕ(fcd ) := ∂ϕ(f∂fcd
of ϕ(fcd ) and kcd ∈ R is the positive gain. The design ofthe image navigation function ϕ(fcd ) is considered in thenext subsection. Substituting (29) into (26), the velocity ofthe desired image feature vector yields

f˙cd = −kcd JL (gcd )JL† (gcd )∇ϕ.

gcdf

(30)

Here, it is assumed that ∇ϕ(fcd ) ∈ N S(JLT (fcd )) where
N S(·) denotes the null space operator, similar to [7]. Since
fcd is chosen a priori via the off-line path planning routine in(30), this assumption can be satisfied.

Fig. 4. Block diagram of visual motion observer-based pose control withimage space navigation function-based path planner.

„

«T
∂ϕ
∂fcd(
)
2κ−1
sT Ks
1 − η12κ−1 ηf 1
1 − η2m
ηf 2m
=
×diag
2κ+1 , · · · ,2κ+12κ
(1 + ϕ̄)2(1 − η12κ ) 2κ
(1 − η2m
) ﬀ2κ
j
2
2
2
×diag
,
,··· ,. (36)
fxM − fxm fyM − fym
fyM − fym∇ϕ(fcd ) :=It should be noted that fcd → fcdf from (31)–(36) when∇ϕ(fcd ) → 0.
C. Convergence Analysis of Path Planner

B. Image Space Navigation Function
In this subsection, we develop the image space navigation
function ϕ(fcd ) [6], [7]. Here, the image space navigation
function is designed that all image features keep into visible
set.
Firstly, we define two auxiliary functions η(fcd ) : R2m →[−1, 1]2m and s(η) : [−1, 1]2m → R2m as follows:
ﬀ
j
2
2
2
fcd
η(fcd ) = diag,,···fyM − fy–m
» fxM − fxm fyM − fymTfyM + fym
fxM + fxm fyM + fym
−
···
(31)
fxM − fxm fyM − fym
fyM − fym"#T
η2m − ηf 2m
η1 − ηf 1
···
,
(32)
s(η) =
1
1
2κ 2κ
2κ 2κ
(1 − η1 )(1 − η2m)

where η(fcd ) = [η1 (fcd ) η2 (fcd ) · · · η2m (fcd ) ]T and κ >0 ∈ R is an additional parameter to change the potential field.
η(fcdf ) = [ηf 1 (fcd ) ηf 2 (fcd ) · · · ηf 2m (fcd ) ]T : R2m →[−1, 1]2m is defined as same as (31). fxM , fxm , fyM andfym ∈ R denote the maximum and minimum pixel values
along the x- and y-axis, respectively. η and s are the functions
in order to normalize the current pixel value for the maximum
and minimum pixel values, and to define the error between
the current image feature and the final one, respectively. Then,the model space navigation function ϕ̃(η) ∈ R2m → [0, 1] isdefined as
ϕ̄
.
(33)
ϕ̃(η) :=
1 + ϕ̄
Moreover, the objective function ϕ̄(η) ∈ R2m → R is definedas1ϕ̄(η) := sT (η)Ks s(η),(34)
2where Ks ∈ R2m×2m is a positive definite symmetric matrix.
The image space navigation function denoted by ϕ(fcd ) ∈D → R, can be developed as follows:ϕ(fcd ) := ϕ̃ ◦ ϕ̄ ◦ s ◦ η(fcd ),(35)

where ◦ denotes the composition operator. The gradient vector∇ϕ(fcd ) can be represented as

Suppose that ∇ϕ(fcd ) is not a member of the null space
JLT (fcd ), the following theorem concerning the convergenceof the path planner holds.
Theorem 3: Suppose that ∇ϕ(fcd ) ∈ N S(JLT (fcd )) andthe initial desired image feature vector fcd (0) satisfies
fcd (0) ∈ D. Then, the desired image feature vector (30)
ensures that fcd (t) ∈ D and has the asymptotically stableequilibrium point fcdf .
Proof: Consider the following positive definite function:
Vn (fcd (t)) = ϕ(fcd (t)).

(37)

Evaluating the time derivative of Vn (fcd ) along the trajectories
of Eq. (30) gives us
V̇n (fcd (t)) = (∇ϕ)T f˙cd
= −kcd (∇ϕ)T JL (gcd )JL† (gcd )∇ϕ= −kcd (JLT ∇ϕ)T (JLT JL )−1 JLT ∇ϕ≤ −kJLT ∇ϕ2 ,(38)
where we use the property ka2 ≤kcd aT (JLT JL )−1 a, ∀a ∈R6 and k denotes a positive constant. It is clear from Eq. (38)that Vn (fcd ) is a non-increasing function in the sense thatVn (fcd (t)) ≤ Vn (fcd (0)).

(39)

From Eqs. (37) and (39), the condition fcd (t) ∈ D, ∀t > 0
is satisfied for any initial condition fcd (0) ∈ D. By following
LaSall’s Theorem [11], it can be proved that the only invariant
set that satisfies JLT (fcd )∇ϕ(fcd ) = 0 is the origin. Considering the assumption ∇ϕ(fcd ) ∈ N S(JLT (fcd )), we haveshown that ∇ϕ(fcd ) = 0. Therefore, it can be concluded
that fcd → fcdf from Sec IV-B.
Theorem 3 guarantees the convergence of the time-varying
desired image feature vector fcd (t) to the final one fcdf . The
path planner can be designed to keep all features into thecamera field of view based on the image space navigation
function. The block diagram of the visual motion observerbased pose control with the image space navigation functionbased path planner is shown in Fig. 4.

200

f4(6)

f3(6)

f2(6)

f1(6)

0.8
perx [m]

300

0
0
0

100

2

4

6

2

4

6

4

6

pery [m]

0

-0.4

-100

f2(0)f4(0)

-200

-0.8
0
0.5

f1(0)f3(0)perz [m]fy [pixel]

0.4

Camera Field of View

-300
-300

-200

-100
100
0
fx [pixel]

200

0

-0.5

300

-1
0

2

time [s]Pose control translational
Fig. 5. Trajectory of image feature Fig. 6.
points fco (Solid: proposed method, error per .
Dashed: previous one[8]).

Although position-based control can allow feature points toleave the field of view, the principle advantage of it is that
it is possible to describe tasks in terms Cartesian pose as iscommon in many applications, e.g., robotics [1], and that it
does not need a desired image a priori. Thus, the proposed
method which is connected the position-based visual feedback
control and the image-based path planner allows us to extend
technological application area. The main contribution of thispaper is to show that the path planner which always remains
in the camera field of view during the servoing is designed
for the position-based visual feedback control.
Remark 3: It is also interesting to note that the Jacobian
JL (·) between the image feature vector and the body velocity
is exactly the same form as the Jacobian Je (·) for the estimation error which is derived using a first Taylor expansion
approximation.
V. S IMULATION R ESULTS
In this section, we present simulation results for the visualfeedback control with the path planner via the image space
navigation function, compared with the simple constant desired motion proposed in [8]. The control objective is thatthe vision camera tracks the static target object to keep alltarget feature points (four points) inside the camera field ofview. In other words, it is bring the actual relative rigid body
motion gco (t) to a given reference one gcdf using a timevarying reference one gcd (t), and it can be achieved to makeboth the estimation and the pose control errors zero.
The simulation is carried out with the initial condition pco =[0.2 0.2 − 1.35]T [m], ξθco = [0 0 0]T [rad]. The final desired
relative rigid body motion is pcdf = [−0.23 − 0.3 − 1.35]T[m], ξθcdf = [0 0 π/2]T [rad]. This condition means that thevision camera moves from end to end of the field of view
diagonally with optical axis rotation.
The simulation results are presented in Figs. 5 and 6.
Fig. 5 shows the trajectory of image feature points fco . In
Fig. 5, the solid lines denote the trajectory applying the
proposed visual feedback control with the path planner, andthe dashed lines denote those for the simple constant desired
value, respectively. fco (0) := [f1 (0) f2 (0) f3 (0) f4 (0)] andfco (6) := [f1 (6) f2 (6) f3 (6) f4 (6)] shows the values of theimage feature vector in the case of the initial condition andthose of in the case of t = 6 [s], respectively. The control

method have to be designed that the feature points don’t get outof the camera field of view which is denoted as the rectangle inFig. 5. From Fig. 5, it is concluded that the proposed method
can make the vision camera keep all feature points in the fieldof view. Although the convergence to the desired values isalso achieved in the case of the previous method [8] in thesimulation, it corresponds to fail in the actual experiment as
the vision camera miss the target object.
Fig. 6 shows the actual translational control error per , whichis the position error vector between the current relative rigid
body motion gco (t) and the final desired one gcdf , instead of
the time-varying desired one gcd (t). It should be noted thatthe position error with z-axis increases, while the ones with
x-axis and y-axis are monotonically decreasing. This meansthat the vision camera moves away once in order to keep thetarget object. This validates one of the expected advantages ofthe position-based control with the path planner for the visualfeedback system.
VI. C ONCLUSIONSThis paper proposes passivity-based control via an image
space navigation function for 3D eye-in-hand visual feedback
systems. The main contribution of this paper is to show that thepath planner which always remains in the camera field of view
during the servoing is designed for the position-based visualfeedback control. Especially, our proposed method does notneed a desired image a priori. Simulation results are presented
to verify the control performance with visibility maintenanceof the proposed control scheme and the previous one [8].
R EFERENCES[1] F. Chaumette and S. A. Hutchinson, “Visual Servoing and Visual
Tracking,” In:B. Siciliano and O. Khatib (Eds), Springer Handbook of
Robotics, Springer–Verlag, pp. 563–583, 2008.
[2] N. R. Gans and S. A. Hutchinson, “Stable Visual Servoing throughHybrid Switched-System Control,” IEEE Trans. on Robotics, Vol. 23,
No. 3, pp. 530–540, 2007.
[3] V. Lippiello, B. Siciliano and L. Villani, “Position-based Visual Servoing
in Industrial Multirobot Cells using a Hybrid Camera Configuration,”
IEEE Trans. on Robotics, Vol. 23, No. 1, pp. 73–86, 2007.
[4] N. Mansard and F. Chaumette, “Directional Redundancy for Robot
Control,” IEEE Trans. on Automatic Control, Vol. 54, No. 6, pp. 1179–
1192, 2009.
[5] F. Chaumette, “Potential Problems of Stability and Convergence inImage-based and Position-based Visual Servoing,” D. J. Kriegman, G.
D. Hager and A. S. Morse (Eds.), The Confluence of Vision and Control,
Lecture Notes in Control and Information Sciences 237, Springer–
Verlag, pp. 66–78, 1998.
[6] N. J. Cowan, J. D. Weingarten and D. E. Koditschek, “Visual Servoing
via Navigation Functions,” IEEE Trans. Robotics and Automation,
Vol. 18, No. 4, pp. 521–533, 2002.
[7] J. Chen, D. M. Dawson, W. E. Dixon and V. K. Chitrakaran, “Navigation
Function-based Visual Servo Control,” Automatica, Vol. 43, No. 7,
pp. 1165–1177, 2007.
[8] M. Fujita, H. Kawai and M. W. Spong, “Passivity-based Dynamic VisualFeedback Control for Three Dimensional Target Tracking: Stability
and L2 -gain Performance Analysis,” IEEE Trans. on Control Systems
Technology, Vol. 15, No. 1, pp. 40–52, 2007.
[9] R. Murray, Z. Li and S. S. Sastry, A Mathematical Introduction to
Robotic Manipulation, CRC Press, 1994.
[10] E. Rimon and D. E. Koditschek, “Exact Robot Navigation using Artificial Potential Functions,” IEEE Trans. Robotics and Automation, Vol. 8,
No. 5, pp. 501–518, 1992.
[11] H. K. Khalil, Nonlinear Systems (3rd ed.), Prentice Hall, 2002.