Visual Motion Observer-based Pose Control
via Obstacle Avoidance Navigation Function
for Eye-in-Hand Systems
Yusei Tsuruo

Toshiyuki Murao

Hiroyuki Kawai

Masayuki Fujita

Graduate Program in
Master Program of Innovation
Department of Robotics
Department of Mechanical
Mechanical Engineeringfor Design and Engineering
Kanazawa Instituteand Control Engineering
Kanazawa Institute
Advanced Institute ofof Technology
Tokyo Instituteof Technology
Industrial Technology
Ishikawa 921-8501, Japanof Technology
Ishikawa 921-8501, JapanTokyo 140-0011, Japan
Email: hiroyuki@neptune.
Tokyo 152-8550, Japan
Email: integrity@venus. Email: murao-toshiyuki@aiit.ac.jp
kanazawa-it.ac.jp
Email: fujita@ctrl.titech.ac.jp
kanazawa-it.ac.jp

Target Object

Abstract—This paper investigates passivity-based pose control via an obstacle avoidance navigation function for threedimensional (3-D) eye-in-hand visual feedback systems. Firstly,visual motion observer-based pose control for 3-D eye-in-handvisual feedback systems is presented. Next, a path planner tobe appropriate for the visual motion error system is designed
through an obstacle avoidance navigation function to keep
collision-free during servoing. Finally, the effectiveness of the
proposed method is verified through computer simulations.

I. I NTRODUCTION
Visual feedback control is now a very flexible and useful
method in robot control [1]. Recently, Allibert et al. [2]reported comparison results between two image prediction
models for an image-based visual servoing scheme based on
nonlinear model predictive control. Becerra et al. [3] presented
a robust sliding mode control law that exploits epipolargeometries among three views. Zhang et al. [4] proposed a
new 2-1/2-D visual servoing method for nonholonomic mobile
robots based on a novel motion-estimation technique, which
does not need homography/fundamental matrix estimation or
decomposition. The authors have been proposed passivitybased visual feedback control for three-dimensional (3-D)
target tracking in a series of papers [5]–[7].
Obstacle avoidance coupled with accurate path following
control which is to move a vehicle towards a target location
free of collisions with the obstacles has attracted much attention of a great amount of robotics researchers [8]. Huang
et al. [9] proposed a local navigation method with a single
camera for obstacle avoidance using headings to obstacles andtheir angular widths. Cherubini and Chaumette [10] presented
appearance-based visual navigation with obstacle avoidance
for real outdoor environments. This appearance-based visualnavigation, which is one of the few methods for obstacle
avoidance through the camera model, avoids new obstacles byusing a range scanner. Although good vision-based navigation
approaches for the obstacle avoidance problems are presented

5w

5c
Camera-mounted
VehicleFig. 1.

5o

5b1

Obstacle 1

5bn

Obstacle n

Eye-in-hand visual feedback system with obstacles.

in [9] and [10], they are restricted to a ground vehicle.
On the other hand, in robot control, one of the representative
works for obstacle avoidance problems is a method usinga navigation function that is globally convergent potentialfunction proposed by Rimon and Koditschek [11]. The featureof this efficient approach is that automatically merge pathfinding and trajectory generation in a closed-loop fashion.
Some navigation function approaches for visual feedback
systems are reported in [12]–[14]. Cowan et al. [12] proposeda visual feedback controller to bring a robot to rest at a desiredconfiguration for an important problem that are all featurepoints remain within a camera field of view by using navigation functions. Chen et al. [13] and the authors [14] developed
an off-line path planner based on an image space navigation
function with an adaptive 2 1/2-D visual servoing controller
and a stabilizing receding horizon one, respectively. However,
all these approaches [12]–[14] take advantage of navigation
functions to solve the camera field of view problems but not
to deal with the obstacle avoidance problems.
In this paper, passivity-based pose control via a navigation
function for obstacle avoidance is applied to 3-D visual feedback systems with an eye-in-hand configuration as shown inFig. 1. Firstly, passivity-based pose control for an eye-in-handvisual motion error system is proposed. Next, a path planner tobe appropriate for the visual motion error system is designed
through an obstacle avoidance navigation function in orderto move the camera towards a desired pose and away fromobstacles without additional sensors such as a range scanner.
Convergence analysis of the proposed path planner is provided.
Finally, we present simulation results that demonstrate theeffectiveness of the proposed control scheme.
II. V ISUAL M OTION O BSERVER - BASED P OSE C ONTROLThis section mainly reviews our previous work [5] via
passivity-based visual feedback control with an eye-in-handconfiguration.
A. Vision Camera Model
Visual feedback systems with an eye-in-hand configuration
use three and n coordinate frames which consist of a world
frame Σw , a camera frame Σc , a target object frame Σo andn obstacle frames Σb1 , · · · , Σbn as depicted in Fig. 1. Let
pco ∈ R3 and eξ̂θco ∈ SO(3) be a position vector and arotation matrix from the camera frame Σc to the object frame
Σo . Then, a relative pose from Σc to Σo can be represented
by gco = (pco , eξ̂θco ) ∈ SE(3).
The objective of position-based visual feedback control is,in general, to bring the actual relative pose gco to a reference
one gcd . Firstly, we consider the relative pose gco in order toachieve the control objective. The relative pose from Σc toΣo can be led by using the composition rule for rigid bodytransformations as follows:−1gwo .
gco = gwc

(1)

The relative pose involves the velocity of each rigid body.
To this aid, let us consider the velocity of a rigid body as
described in [15]. We define the body velocity of the camera
b
b T
b T T
relative to the world frame Σw as Vwc= [(vwc) (ωwc) ] ,bbwhere vwc and ωwc represent the velocity of the origin andthe angular velocity from Σw to Σc , respectively [15].
Differentiating Eq. (1) with respect to time, the body
velocity of the relative pose gco can be written as follows(See [5]):bbb= −Ad(gco−1 V
Vco) wc + Vwo ,(2)bis the body velocity of the target object relative towhere VwoΣw and Ad(gco ) is the adjoint transformation associated withgco [15].
The relative pose gco = (pco , eξ̂θco ) cannot be immediately
obtained in the visual feedback system, because the targetbobject velocity Vwo
is unknown and furthermore cannot bemeasured directly. To control the relative pose using visual
information provided by computer vision system, we use thepinhole camera model with a perspective projection. Here, we
consider m(≥ 4) feature points on the rigid target object inthis 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 transformationof 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., [15]).
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
,
(3)
fi =
zci yci
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 and observed. Under this assumption, the image feature vector fco depends only on the relativepose gco .
B. Estimation Error System
The visual feedback control task requires information of therelative pose gco . Since the measurable information is only theimage feature fco in the visual feedback system, we consider anonlinear observer (we call a visual motion observer) in orderto estimate the relative pose gco from the image feature fco .
bFirstly, using Eq. (2), we choose estimates ḡco and V̄coof
the relative pose 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 thebto their actual values.
estimated values ḡco and V̄co
In order to establish the estimation error system, we define
the estimation error between the estimated value ḡco and theactual relative pose gco as−1
gee = ḡco
gco .

(5)

We next define the error vector of the rotation matrix eξ̂θei as
rei := sk(eξ̂θei )∨ where sk(eξ̂θei ) denotes 12 (eξ̂θei − e−ξ̂θei ).
Using this notation, the vector of the estimation error is given
T T] .
by ee := [pTee ree
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 pose ḡco as follows:
ee = Je† (ḡco )(fco − f¯co ),(6)

where f¯co is the estimated value of image feature and Je (ḡco )is an image Jacobian-like matrix [5]. In the same way as Eq.
(2), the estimation error system can be represented bybb−1 ue + V
Vee
= −Ad(gee
wo .
)

(7)

C. 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. Firstly, we define the pose control error
as follows:−1
gec = gcd
gco ,

(8)

which represents the error between the relative pose gco andthe reference one gcd . It should be remarked that gco canbe calculated by using the estimated relative pose ḡco and
T T
the estimation error vector ee = [pTee ree] equivalently as
follows:
gco = ḡco gee
sin−1 ree 
ree ,ξθee =
ree 

(9)(10)

although gco cannot be measured directly. Similar to theestimation error vector, the vector of the pose control error
T T] .
is defined as ec := [pTec rec
Differentiating Eq. (8) with respect to time, the pose control
error system can be represented as


b
b
b
b
−1
Vec
= −Ad(gec(11)) Ad(g−1 ) Vwc + Vcd + Vwo ,cdbis the body velocity of the reference of the relative
where Vcd

pose gcd .

D. Visual Motion Observer-based Pose Control
Combining the estimation error system (7) and the posecontrol one (11), we construct the visual motion observerbased pose control error system (we call the visual motion
error system) as follows: b   
−1
−Ad(gec0
VecI)b, (12)
=
u
+
Vwo
b
−1
I
0
−Ad(gee
Vee
)
where
b
b
u := [uTc uTe ]T , uc := Ad(g−1 ) Vwc
+ Vcd
.
cd

(13)

Let us define 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 notedthat if the vectors of the pose control error and the estimation
one are equal to zero, then the actual relative pose gco tends
to the reference one gcd when x → 0.
It can be proved that the visual motion error system (12)is passive from the input u to the output −x by using thefollowing positive definite function:
V = E(gec ) + E(gee ),(14)

where E(gei ) := 12 pei 2 + φ(eξ̂θei ) and φ(eξ̂θei ) := 12 tr(I −eξ̂θei ) is an error function of the rotation matrix.
Based on the passivity property of the visual motion error
system, we consider the following control law.
u = −K(−x), K := diag{Kc , Ke },

(15)

:=
diag{kc1 , · · · , kc6 } and Ke:=
where Kc
diag{ke1 , · · · , ke6 } are the positive gain matrices of x,y and z axes of the translation and the rotation for the posecontrol error and the estimation one, respectively.
b
Theorem 1 ([5]): If Vwo= 0, then the equilibrium pointx = 0 for the closed-loop system (12) and (15) is asymptotic
stable.
Theorem 1 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 (15).
III. O BSTACLE AVOIDANCE NAVIGATIONF UNCTION - BASED PATH P LANNING FOR E YE - IN -H AND
S YSTEMS
Vision-based navigation with obstacle avoidance should
offer great perspectives in many applications, such as surveillance, patrolling, search and rescue or high risk missions. In
this section, as a first step for a vision-based navigation, we
design a path planer through an obstacle avoidance navigation
function for 3-D eye-in-hand visual feedback systems. The
control objective in this paper is stated as follows:
Control Objective: The camera follows the target object,i.e., the relative pose gco (t) is coincided with the time-varying
desired one gcd (t) which is generated to avoid unexpectedobstacles, and which converges the final desired one gcdf .
From the proposed visual feedback control law, the input tothe camera is designed as follows:


b
b
.
(16)
Vwc
= Ad(gcd ) uc − VcdbHence, the camera input is needed the body velocity Vcd
=
b T
b T T
[(vcd ) (ωcd ) ] of the reference of the relative pose gcd 1 .

A. Rotation Error for Path Planning
In rotation control, we consider only convergence to thefinal desired rotation. We define the rotation error between
the time-varying relative desired rotation eξ̂θcd and the final
one eξ̂θcdf as follows:eξ̂θed = e−ξ̂θcdf eξ̂θcd .

(17)

The vector form is defined as red := sk(eξ̂θed ). Differentiating
Eq. (17) with respect to time, the rotation error system for path
planning can be written as
b
b
= ωcd
.
ωed

(18)

B. Visual Motion Observer for Path Planning
Obstacle avoidance navigation function-based path planningneeds the relative position of the obstacles from the camera
frame pcbi . In this section, we design the visual motion
observer in order to estimate the relative position pcbi using
the image feature fcbi := f (gcbi ). Now, we assume that themultiple points on the obstacles can be observed and the
1 The relative pose gb
cd can be obtained solving ġcd = gcd V̂cd .

position of them pbi i is known. The body velocity of therelative pose gcbi can be written as follows:bbVcbb i = −Ad(g−1 ) Vwc
+ Vwb
,
i
cbi

(19)

b
is the i-th obstacle body velocity. Using the bodywhere Vwbivelocity Vcbb i (19), we establish a following estimate model forthe obstacles:b+ ubi ,
V̄cbb i = −Ad(ḡ−1 ) Vwc
cbi

(20)

where ḡcbi and V̄cbb i are the estimated values of gcbi and Vcbb i ,respectively. We define the estimation error gebi between theactual relative pose gcbi and the estimated one ḡcbi as follows:−1
gcbi .
gebi = ḡcb
i

(21)

T T
The vector of the estimation error ebi = [pTebi reb] can bei
obtained by exploiting the image feature fcbi and the estimate
values ḡcbi and f¯cbi as

ebi = Je† (ḡcbi )(fcbi − f¯cbi ).

A smooth vector field on any sphere world with a unique
attractor, must have at least as many saddles as obstacles. The
property of a Morse function whose Hessian at all criticalpoints is non-degenerate establishes that the initial conditions
that bring the system to saddle points are sets of measure zero.
In view of this property, all initial conditions away from setsof measure zero are brought to the unique minimum [16].
In order to design the obstacle avoidance navigation function, we utilize the following function which represents theerror between the desired relative position pcd and the final
one pcdf :s(pcd ) = pcd − pcdf 2κ ,where κ > 0 ∈ R is an additional parameter to change thepotential field. Next, let define the obstacle function which
includes the function η0 for keeping in the space F asM

η(pcd ) =

(22)

ˆ

(23)

through gcbi = ḡcbi gebi . The estimation error system for the
obstacles can be obtained asb
Vebb i = −Ad(g−1 ) ubi + Vwb
.
i
ebi

(24)

ηi (pcd ) =Bi ,ρ20 − pcd2
for i = 0
pcbi 2 − ρ2i . for i = 1 · · · M

Then, the model space navigation function ϕ̃(x) ∈ R2m →[0, 1] and a κ-th root function are defined asxϕ̃(x) =
(28)
μ+x
1

In this subsection, we develop the obstacle avoidance navigation function ϕ(pcd ) [11], [16]. In this paper, we assume thatthe workspace and the obstacles are spherical. This assumption
does not constrain the generality of this work since it has
been proven that navigation properties are invariant under
diffeomorphisms in [16].
Firstly, we define a space D where the camera can move
avoiding obstacles as follows:
D=F −

(27)

and

C. Obstacle Avoidance Navigation FunctionMηi (pcd )i=0Hence, the relative position pcbi can be calculated as follows:
pcbi = eξ̄θ̄cbi pebi + p̄cbi

(26)

(25)

i=1

where F := {pcd : pcd2 ≤ ρ20 } is the camera movablespace which represents a Euclidean 3-dimensional disk withthe radius ρ0 > 0, and Bi := {pcd : pcbi (pcd )2 <ρ2i }, i = 1 . . . M denotes a M -th obstacle space in F withthe radius ρi > 0. We impose the additional constraint that
all obstacles closures are contained in the interior of theworkspace; i.e. pcbi  + ρi < ρ0 , 1 ≤ i ≤ M , and that noneof them intersect; i.e. pcbi − pcbj  > ρi + ρj , 1 ≤ i, j ≤ M .
If pcd ∈ D, the camera keeps collision-free.
The navigation functions used in this paper are defined asfollows:
Definition 1 ([11],[16]): A smooth Morse function ϕ(pcd ): D → [0, 1] is a navigation function if it is1) a unique minimum exists at pcdf ;2) uniformly maximal on the boundary of D.

ρ(x) = x κ ,(29)respectively, where μ > 0 ∈ R is a parameter. The function(29) is important in order to change pcdf to a non-degeneratecritical point. From Eqs. (26)–(29), the obstacle avoidancenavigation function denoted by ϕ(pcd ) ∈ D → R, can be
developed as follows:sϕ(pcd ) = ρ ◦ ϕ̃ ◦ (pcd ) =ηs(pcd )μη(pcd ) + s(pcd )

1
κ

(30)

where ◦ denotes the composition operator. Using a similar
way in [16], it can be verified that the function (30) is thenavigation function if the parameter κ is selected adequately.
D. Path Planning of Desired Body Velocityband the input for
We design the desired body velocity Vcd
the estimation ubi as follows:−e−ξ̂θcd Kdp ∇ϕ(pcd )
b
(31)
Vcd =
−Kdr e−ξ̂θed red
ubi = Kb ebi ,(32):=diag{kdp1 , kdp2 , kdp3 }, Kdr:=
where Kdp
diag{kdr1 , kdr2 , kdr3 } and Kb := diag{kb1 , · · · , kb6 }are the positive gain matrices for the translation and therotation of the desired body velocity
 and for the estimation
input, respectively. ∇ϕ(pcd ) :=∂ϕ(pcd )∂pcd

Twhich denotesthe gradient vector of ϕ(pcd ) can be calculated as Eq. (33) atthe bottom of this page.
We state the main result of this paper concerning theconvergence of the path planner.
bb= 0 and Vwb= 0, and theTheorem 2: Suppose that Vwoiinitial desired relative position pcd (0) satisfies pcd (0) ∈ D.
Then, the desired relative position pcd (t) ensures that pcd (t) ∈D and gcd (t) has the asymptotically stable equilibrium point
gcdf .
Proof: Consider the following positive definite function:
Vn = ϕ(pcd ) + φ(e

ξ̂θed

)+

M


E(gebi ).

(34)

i=1

Evaluating the time derivative of Vn along the trajectories ofEqs. (18), (24), (31), (32) gives us
T ξ̂θed bV̇n = (∇ϕ)T ṗcd + red
e
ωed +

M


eTbi Ad ξ̂θebi Vebb i

(e
)
i=1
T ξ̂θcd b
T ξ̂θed b= (∇ϕ) e
vcd + red e
ωcd
M

+
eTbi Ad ξ̂θebi (−Ad−1(gebi ) ubi )(e)
i=1
T ξ̂θed= −(∇ϕ)T eξ̂θcd e−ξ̂θcd Kdp ∇ϕ − rede
Kdr e−ξ̂θed redM+−eTbi Kb ebi
i=1= −(∇ϕ)T Kdp ∇ϕ − (e−ξ̂θed red )T Kdr e−ξ̂θed redM−eTbi Kb ebi .
(35)
i=1

It is clear from Eq. (35) that Vn is a non-increasing functionin the sense thatVn ≤ Vn (0).

(36)

From Eqs. (34) and (36), the condition pcd (t) ∈ D, ∀t > 0
is satisfied for any initial condition pcd (0) ∈ D. Since theestimation error vector ebi → 0, the position pcbi which
is utilized in the obstacle functions converges actual value.
Thanks to the property of navigation functions [16], it canbe shown pcd (t) → pcdf through ∇ϕ(pcd ) → 0. On theother hand, the rotation eξ̂θcd (t) → eξ̂θedf can be verified
because of e−ξ̂θed red → 0. Therefore, it can be concluded
that gcd (t) → gcdf .
Theorem 2 guarantees the convergence of the time-varying
desired pose gcd (t) to the final one gcdf . The path planner
can be designed to keep collision-free based on the obstacle
avoidance navigation function. The block diagram of thevisual motion observer-based eye-in-hand pose control withthe obstacle avoidance navigation function-based path planneris shown in Fig. 2.
⎛∇ϕ(pcd ) =2μ1+κκ(μη + s) κ

gcdf
fcbi
b
Vwc

gcd

Obstacle
Avoidance
Navigation
Function-based
Path Planner

b
Vcd

fco

Visual Motion
Observer-based
Pose Controller

b
Vwc

b Vb
Vwo
wbi

VisionCameraFig. 2. Block diagram of visual motion observer-based eye-in-hand posecontrol with obstacle avoidance navigation function-based path planner.

Our proposed approach, which deals with the estimation
problem from the visual features explicitly, can be applied to
3-D eye-in-hand visual feedback systems without restriction
to the ground vehicle. Thus, the proposed method which isconnected the visual motion observer-based pose control andthe obstacle avoidance navigation function-based path plannerwithout additional sensors allows us to extend technologicalapplication area. The main contribution of this paper is toprovide that the path planner which can achieve obstacle
avoidance during the servoing is designed for the eye-in-handvisual feedback control.
In our passivity-based visual feedback control approach, theimage Jacobian-like matrix J(·) of the pinhole camera modelis exactly the same form as that of the panoramic camera
model [17]. Therefore, the proposed approach can be applied
to 3-D eye-in-hand visual feedback systems with a panoramic
camera using the image Jacobian-like matrix J(·) in [17].
IV. V ERIFICATIONIn this section, we present simulation results for the visualfeedback control with the path planner via the obstacle avoidance navigation function, compared with the constant desired
relative pose proposed in [5]. The control objective is that thecamera tracks the static target object to avoid two obstacles.
In other words, it is bring the actual relative pose gco (t) to agiven reference one gcdf using a time-varying reference one
gcd (t), and it can be achieved to make both the estimation andthe pose control errors zero.
The simulation is carried out with the initial conditionpco = [−1.25 4.994 0.5]T m, ξθco = [0 0 − π/12]T rad.
The final desired relative pose is pcdf = [0 1.5 0]T m,ξθcdf = [0 0 0]T rad. The other conditions are set as pcb1 =[−0.224 1.095 0]T m, ξθcb1 = [0 0 − π/12]T rad, ρ1 = 0.5m, pcb2 = [−1.414 2.45 0.75]T m, ξθcb2 = [0 0 − π/12]Trad, ρ2 = 0.5 m.
The simulation results are presented in Figs. 3 and 4. Fig. 3
shows the actual pose control error er , which is the error vector
between the current relative pose gco (t) and the final desired
one gcdf . The asymptotic stability can be confirmed by steadystate performance in Fig. 3.
⎛⎝κη(pcd − pcdf ) + pcd − pcdf 2 ⎝M

ηi pcd −
i=1

M


M

j=1 i=0,i=j⎞⎞ηi pcbj ⎠⎠

(33)

perx

1

V. C ONCLUSIONS

0
-1
-2

0

2

4

6

8

0

2

4

6

8

0

2

4

6

8

pery

4
2
0
-2

perz

0.5
0

-0.5

rerz

0.5
0

-0.5

0

2

Fig. 3.

4time [s]68Pose control error er (perx , pery , perz , rerz ).

1.2

zw [m]

0.8
0.4
0

-0.4
5
y

w [

0
m]

-5 -3

-2

-1
xw [m]

0Fig. 4. Trajectory of position pwc with two obstacles: Solid (Red); applying
proposed method: Dashed (Blue); applying previous one[5].

Fig. 4 presents the trajectory of the position of the camera
pwc . In Fig. 4, the spheres and the rectangle show the static
obstacles and target object, respectively. The solid (red) lines
denote the trajectory applying the proposed passivity-basedpose control with the path planner, and the dashed (blue) lines
denote those with the constant desired value. The trajectoryshould be designed that the camera moves toward the rectangle
avoiding the spheres. From Fig. 4, it is concluded that the
proposed method can make the camera avoid all obstacles. Although the convergence to the desired values is also achievedin the case of the previous method [5] in the simulation, it
corresponds to fail in the actual experiment since the camera
hits the obstacles.

This paper proposes passivity-based 3-D eye-in-hand visualfeedback control via an obstacle avoidance navigation function. The main contribution of this paper is to show that thepath planner which can avoid obstacles during servoing isdesigned for 3-D eye-in-hand visual feedback systems. Simulation results are presented to verify the control performanceof the proposed control scheme with obstacle avoidance. Although only simulation results are shown in this submitted version, experimental movies using a Parrot AR.Drone which is a
wifi quadrotor aerial vehicle with two cameras are available at
http://wwwr.kanazawa-it.ac.jp/kawai/resear
ch/ARDrone/ARDronemovies.html.
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