ESMF based multiple UAVs active cooperative observation method in relative velocity coordinates

Abstract

Based on extended set-membership filter (ESMF) and the path planning method in relative velocity coordinates (RVCs), a new 3D multiple Unmanned Aerial Vehicle (UAV) systems active cooperative observation method, high precision cooperative observing a moving target by proper planning the behavior of each member vehicle, is proposed. The new method combines the ESMF based cooperative observation method and the LP-based trajectory generation method in RVCs, where the ESMF based cooperative method is used to obtain the moving target's motion state which is further used as part of the cost function to plan the UAV's behavior by using the planning method in RVCs. The contribution of this paper is: (1) the computational burden of the proposed cooperative algorithm is comparable to single ESMF algorithm; (2) the high precision observation of moving target is always obtainable by considering the optimal observation condition during trajectory planning; (3) the planning algorithm in RVCs, where the trajectory planning can be modeled as a LP problem, is used to optimize the UAV's behavior, thus, in company with (Gu, et al; 2006), the fast application of the new proposed active cooperative observation method is desirable. Finally, the simulations in 3D environments are conducted to verify the feasibility and validity of the method.

Full text

Page 1

28th Chinese Control Conference
Shanghai, P.R. China, December 16-18, 2009


Abstract—based on extended set-membership filter (ESMF)
and the path planning method in relative velocity coordinates
(RVCs), a new 3D multiple Unmanned Aerial Vehicle (UAV)
systems active cooperative observation method, high precision
cooperative observing a moving target by proper planning the
behavior of each member vehicle, is proposed. The new method
combines the ESMF based cooperative observation method and
the LP-based trajectory generation method in RVCs, where the
ESMF based cooperative method is used to obtain the moving
target’s motion state which is further used as part of the cost
function to plan the UAV’s behavior by using the planning
method in RVCs. The contribution of this paper is: 1) the
computational burden of the proposed cooperative algorithm is
comparable to single ESMF algorithm; 2) the high precision
observation of moving target is always obtainable by
considering the optimal observation condition during trajectory
planning; 3) the planning algorithm in RVCs, where the
trajectory planning can be modeled as a LP problem, is used to
optimize the UAV’s behavior, thus, in company with 1), the fast
application of the new proposed active cooperative observation
method is desirable. Finally, the simulations in 3D environments
are conducted to verify the feasibility and validity of the
method.
I. INTRODUCTION
Localization/identification is one of the basic enabling
techniques of mobile robotics [1]. Thus, it is also one of the top
important research tasks in the field of robotics. However,
single mobile robot system may present powerlessness due to
the limited visibility and observation accuracy of equipped
sensors, especially when facing complicated surroundings
and terrible observation conditions. An effective method,
which has been extensively researched in some applications,
to overcome these drawbacks is to make observation by
fusing the observations of a common moving target from
multiple mobile robot systems [1-5].
Cooperative observation is usually on the basis of
estimation technique. Many estimation methods, including
traditional stochastic theory based methods, and ESMF
strategy, have all been considered to construct the
cooperative observation algorithm. For example, work [5]
presents a method that uses a simple re-parameterization of
two dimensional Gaussian distributions to obtain more

Manuscript received March 1, 2009.
F.Gu is with the Graduate School of the Chinese Academy of Sciences and
State Key Laboratory of Robotics, Shenyang Institute of Automation,
Chinese Academy of Sciences, Shenyang 110016, China (e-mail:
fenggu@sia.cn).
Y.Q.He, J.D.Han, and Y.C.Wang are all with the State Key Laboratory of
Robotics, Shenyang Institute of Automation, Chinese Academy of Sciences,
Shenyang 110016, China (e-mail: heyuqing, jdhan, ycwang@sia.cn).
accurate target position estimation from two or more robot
agents. In [1], an optimal cooperative position and velocity
estimation, based on the Cramer-Rao bound, of the ground
moving target (GMT) is investigated aiming at obtaining the
optimal cooperative observation. All of these methods
present the same disadvantages, i.e., the process and
measurement noise should both be stochastic variables with a
prior known mean and covariance, which often mismatches
the reality. ESMF is another estimation method that can be
used in cooperative observation. Different from the
stochastical estimation methods, ESMF assumes that the
noise is Unknown But Bounded (UBB)[6], and it can obtain an
uncertain estimation set which the real system state are
ensured in[7]. With these characteristics, ESMF are very
useful for the observation problem and the use of ESMF in
cooperative observation is discussed in [9] where the
cooperative observation method is a simple combination of
two ESMF algorithm. However, the computational burden of
this method is heavy due to the process of computing the
intersection and its outer bounding ellipsoid, especially in 3D
surroundings.
In all of the preceding researches, the influence of robot
motion on the observation is not considered, while this will
absolutely deteriorate the optimal observation results. Thus,
in this paper, a new on-line cooperative observation method is
introduced, where a new ESMF based cooperative
observation method with reduced computational burden is
first proposed, and then, this method is considered into the
trajectory planning method in RVCs to obtain better
observations by relative motion of robot systems, i.e., the
active cooperative observation method.
II. PROBLEM FORMULATION
The problem of two UAVs cooperatively detecting and
tracking a moving target is researched as shown in Fig. 1, T is
a moving target to be tracked; A1 and A2 denote two UAV
systems, where A1, named main UAV, is responsible for
accurately detecting, planning path of two UAVs and pursuit
the states of the target T by fusing the observation data from
both the main UAV A1 and the accessorial UAV A2. The angle
between the tie line between UAV systems A1, A2 and target T,
i.e. φ is defined as cooperative observation angle.
In general, the dynamic model is necessary in order to
estimate/predict the state of a dynamic system. However, the
kinematics or dynamics of the moving target vehicle is
usually unknown or too complicated to be used. Thus, in this
paper, the following Newton’s laws of motion based
kinematic equation is used to denote the action of the moving
ESMF Based Multiple UAVs Active Cooperative Observation
Method in Relative Velocity Coordinates
Feng Gu, Yuqing He, Jianda Han and Yuechao Wang
Joint 48th IEEE Conference on Decision and Control and
WeCIn5.4
978-1-4244-3872-3/09/$25.00 ©2009 IEEE3008

Page 2

target,
X



where
vectors, ΔT is the sampling time, wT,k is the process noise
bounded by
,,,
[][]
T kT k T k
wQw

positive definite matrix.
3,,
,1,
010
11
T k T k
T k T k
IvTw
X






X









is the position of the target,









v


(1)
, T k

, T k
is the velocity
1
1
T

, QT,k is a symmetric and
x
y
z
T
(x
,,,
,,)
T kT k
y
T k
z
1,k
y
A
1, 1,1,
(,,)
kkk
xz
2, 2, 2,
(,,)
kkk
xyz
2,k
A
O


Fig. 1 Cooperative tracking with two UAVs
The observation of the moving target by UAV systems is
realized by 3D radar sensor. The theory of the observation
can be shown in Fig. 2 and the following equations,
222
,,,,,,,,
1,,
1,,
,,
,,
,,
22
,,,,
()()()
tan
(
t
)
a
)
n
(
i k
r
T ki k T ki kT ki kr i
T k i k
i ki
T k i kT k
T ki k
i k i
T ki
i k
k
xxyyzzn
zz
n
x
yy
n
x
yy
x
x















where ri,k, θi,k, αi,k are the observations of the moving target
denoted as polar coordination. nT,k=(nr,i, nθ,i, nα,i)T is the
observation noises and satisfies
symmetric and positive definite matrix.
1
,,,
[][]1
T
T kT kT k
nRn

 .RT,k is a
x
y
z
T
(x
, i k

, i k

,,,
,,)
T k T k
y
T k
z
, i k
r
, i k
)
A
,,,
(,,
i ki ki k
xyz

Fig. 2 Description of the parameters of radar
III. COOPERATIVE OBSERVATION BASED ON ESMF
A. Extended Set-Membership Filter
ESMF delivers an ellipsoid set given by the following
equation,

ˆˆ
( , )| ()(
E x PxRxxP


1
ˆ
x
)1
nT
x


(2)
where ˆ x is the center of the ellipsoid; P is an envelope matrix
satisfying symmetric and positive definite conditions.
Considering a discrete non-linear system written as:
()
kkk
xf x w

(3)

(4)
wk Rn and nk+1 Rm are respectively process and
measurement noise which satisfy the following inequalities,
 ,
[n
Qk and Rk+1 are both symmetric and positive definite matrix.
Linearizing Eq. (3) near current state ˆkx yields,
1

111
()
kkk
yh x n

1
[][]1
T
kkk
wQw

1

111
][]1
T
kkk
Rn



2
1
ˆˆ
x
ˆ
x
( ) |( ) /|()()
kkk
kkkkkkk
x x

x
xf xf xxxO xw

The ESMF method considers the higher order terms
(H.O.T.) as a part of the process noise and computes its
envelope matrix with interval analysis method [7]. The
linearized Eq. (3) can be rewritten as,
( ) |( ) /
kk
kkk
xx
x f xf xx

Similarly, the output equation can be rewritten as,
() |
() /
kk
kk
kxx
yh x
h x


The new noise bounds can be denoted as,
ˆ
ˆˆ
[][]1
kkk
wQw
 ,
[
n
With the linearized system, ESMF algorithm can be used to
estimate the state of system (3) to (4) by the following steps[9],

Step I: ESMF Prediction Step
ˆˆ (
)
kk k k
x f x


(8)
ˆ
/ (1)/
kkkk kkkk
P A P AQ



Step II: ESMF Update Step
ˆˆˆ
((
kkkkkkk
xxKyh x


(1) ( / (1
kkkkkk
PP




 
(5)
1
ˆˆ
x
ˆ
x
ˆ
|()
kk
kkk
x
xw



(6)
1111
11111
ˆ
x
1
ˆ
ˆ
x
ˆ
n|()
kk
kkk
x
x
x




(7)
1T

1
111
ˆ
ˆˆ
n
][]1
T
kkk
R



1,,
1,,
T
k
(9)
1,11,11,
))
k
(10)
1,1 1,1
1,11,1
))
(1) (

/ (1 ))( / (1))
T
kkkkkkkkkk
PC W CP



where,


(11)
1,1,
1
1,111
1,
1 1,1 1,
ˆ
x
ˆ
x
() / |() /|
ˆ
[(/ (1)) /]
ˆˆ
[()][()]
,
kkkkkk
kkkk
T
kkkkkkkk
T
kkkkk
T
kkkkkkkk
xx
Af xxC h xx
WCPCR
KP C W
yh xW y h x








  




βk (0,1) and ρk+1 (0,1) are used in the process of
determining outer bounding of summation of ellipsoids. They
can be calculated using the methods in [6]. Then estimation
result is E(
1,1
ˆkk
x
1,1kk
P
The main idea of ESMF is to obtain the final estimation by
intersecting prediction uncertain set and the measurement
set[8]. Thus, ESMF is inherent an algorithm to compute the
intersection of two special sets, which is a very useful idea for
cooperative observation.
,
).
B. Cooperative Observation Based on ESMF
The observation result obtained by using ESMF is an
WeCIn5.4
3009

Page 3

uncertainty ellipsoid set denoted as Eq. (2), which the target is
ensured to lie within. Thus, it is obvious that better
observation result can be obtained by intersecting two
observation uncertainty sets. The main idea of the new
proposed method is to embed the process of the intersecting
into the ESMF algorithm itself, i.e. to displace the process of
computing the intersection of the prediction uncertainty set
and the measurement one with the process of computing the
intersection of the following three sets: prediction uncertainty
set and two measurement uncertainty sets, as shown in Fig. 3.

Fig. 3 Main idea of cooperative observation method
The detailed steps of the cooperative observation algorithm,
which is executed in the computer system of main UAV
system, are as follows. The superscript i (i=1, 2) on the left of
the variable denotes the variable is computed using the ith
UAV’s observation data at each sub-step.
Step 1: Single prediction:
ˆ ˆ (
)
kkk k
x f x


(12)
ˆ
/ (1)/
kkk k kkkk
PA P AQ



 
1,,
1,,
T
k

(13)
where,
1,
ˆ
x
() /|
kk
kkk
x
A f xx



Step 2: Cooperative Update:
Sub-step 1 – Fusing the Accessorial UAV Measurement:
ˆˆ
(
kkkkkk
xxKyh x


222
1,11,11,
ˆ
( ))
kk

(14)
222
1,1 1,
22212
1,111,
/ (1)
( / (1)) / (1)
kkkkkk
T
kkkkkkkkkk
PP
PCW CP







(15)
where
C

1,
2222
1 1,1,1
221
1,1
22212
1 1,11
ˆ
x
,
1
|
ˆ
/ (1)/
(/ (1))
ˆˆ
x
1 [

( )][( )]
() /
kkk
T
kkkkkkT kk
T
kkkkkk
T
kkkkkk
k
kk
kx
WCPCR
KPCW
y h xW

yh
h xx















 


222
,1,11
2
ˆ
/ 1/
T kT kRkR
RRR




Sub-step 2 – Fusing the Main UAV Measurement:
2111
1, 1 1, +1
k
11, +1
k
ˆ
x
ˆ
x
ˆ
(())
kkkkkk
Kyh x


(16)
1

21
1,11,1
1

2121 1 2

21
1,111 1,1
( / (1))
(/ (1 )) / (1)
kkkkkk
T
kkkkkkkkkk
PP
PCWCP






(17)
where,

1221211
1 1,11,1
121211
1,11
1

21
2
1
1
12
1 1,11
1
1,1
,1,1,1
2
1, 1
11
11
ˆ
x
ˆ
( / (1))/
( / (1))
ˆ
x
ˆ
x
1 [
 
()][()]
() /|
ˆ
/ 1/
kk
T kT kR
T
kkkkkk T kk
T
kkkkkk
T
kkk
T kR
kkkkk
kkk
x
WCPCR
KPCW
yhW
C h xx
RRR
yh
















 


It is well known that ESMF update step, i.e., Eq. (10) and
(11), is itself an algorithm to find the intersection of the
prediction ellipsoid set and the observation set. Then, we
make full use of this theory to intersect the three sets as shown
in Fig. 4. This process can be divided as two steps:

Fig. 4 Details of intersecting the three sets
1) The first sub-step is to intersect the prediction ellipsoid
set
1,1,
ˆ (
,)
kkkk
E xP

and the accessorial UAV’s observation
set
y
S
.Based on the ESMF algorithm, the result
will be an ellipsoid set that satisfies Eq. (2),
denoted as Etemp;
2) The second sub-step is to intersect the Etemp and the main
UAV’s observation set
2
22
1,1 1,1
ˆ
x
(,)
kkkk
PE

1
yS . This sub-step takes Etemp as the
prediction ellipsoid set and updates it using the main UAV’s
observation, i.e. calculates the intersection of Etemp and the
main UAV’s observation.
However, in some special situations, the two UAVs’
observation sets may not intersect, i.e. Etemp
in order to continue the algorithm recursively, we can directly
take the Etemp=
1,1
ˆ
(,
kk
E x

continue the algorithm recursively.
1
yS = , thus,
11
1,1
)
kkP

as the final result to
IV. ACTIVE COOPERATIVE OBSERVATION METHOD
In this section, based on the cooperative observation
method formulated above, a new active cooperative
observation method is proposed. The main idea of this
method is to embed path planning between the single
WeCIn5.4
3010

Page 4

prediction step and cooperative update step so as to achieve
optimal observation at each time step.

Fig. 5 Target tracking in RVCs
Here the LP-based path planning method in RVCs [10] is
used for optimal observation. With this method the path
planning problem in the dynamic environment can be
described as minimizing an objective function subject to a set
of linear inequalities that are easily embedded into LP planner
[10]. The main idea is as shown in Fig. 5. The arbitrary contour
with a point T denotes the target with velocity
iv.
from Ai to T which is the center of the target.
Tv. Ai is a
moving UAV with velocity
ATi
L

is the ray that originates

DTi
L
is the ray
that originates from Ai in the direction of
where,
ATiTi
vvv

is the UAV-target relative velocity. γATi is the angle between
v
and
AT
L
.
ATi
v


AT

The pursuit theory in RVCs is to make
cone area denoted by AMTNT in Fig. 5 and towards to the
target by adjusting the



(18)
It should be noted that the target is supposed to keep
constant speed during sampling time ΔT, therefore the
equation
ATiAi
vv
 
In our method, the UAV-target relative states can be
calculated according to the target states predicted in single
prediction step. Then the optimization criterion based on path
planning method in RVCs is designed and analyzed in the
following.
ATi
v
lie within the
iv
, i.e., [10, 11]
{|}
ATiiATi DTi
vvvLT
  
is always satisfied.
1) Optimization with Respect to Pursuit Velocity
From the discussion above we can see clearly that if the
UAV wants to track the target with shortest time, the
v
in
ATi
L
direction, i.e.
tiv can be denoted as,


(19)
Therefore we propose to design the pursuit velocity
criterion as,
(,) (
ATk ATk
JLw Lw L
  

where,
1,2, 1,
(, ) (
ATkATkk
LLLv


weight values.
2) Optimization with Respect to Cooperative Observation
When the cooperative observation angle φ is 90 degree, the
two UAVs can achieve optimal observation [1],[9]. The
relationship between φ and the position of the UAVs can be
denoted as,
() (
cos
XXX



component of
ATi

tiv
is maximal [10] as
shown in Fig. 5.

tiATi ATi
vvL

111,2 2,1,

2,
T
,

)
kk
T
vv



(20)
2,
,)
k
v


, w1≥0 and w2 ≥0 are
1,

1, 1,2,1,1,
1,1, 1,2,1, 1,
)
kT k

kk T kk
k T kkk T kk
XXX

X

X








(21)
The position state of UAV
as following:
,1,T kk
X

can be denoted by the
, i k
v


33
3
,1,,
3
3
2

0
i k i ki k
T
ITI
I
XXv
I
TI



















 (22)
Substituting Eq. (22) into Eq. (21) and linearizing it yields,
cos( , ) (,
k
cc cvv
 

where,

01212
/
c X XXX



0121,2,
)
k
T
(23)
2121

12
1 1,1,

, 1,
1
22,
3
2112
12
2121
212
3
2112
1
, 1,
2
2,
()
()
2
2
()
()
2

2
kkT k

k
kkT k
TT
k
TT
XXX

X
T
cXX
XX

XX
X

X

X

XX

X
T
X

XT
cXX
vX
X

XXX

X
XXT v

X
X



  





















Thus, the optimal observation criterion can be designed as,
( ,) (,)
kk
c c cvv


(24)
Then the optimization of cooperative observation angle is to
minimize J2 by adjusting the
, i k
v
By defining a positive variable z1 as [10],
( , ) (,)
kk
zcc cvv


Then, minimizing z1 subjected to the inequality of (25) is
equivalent to minimizing J2. The objective function of (24)
can be rewritten as a linear one
J2=w3z1 (26)
where w3≥0 is the weight value of this objective.
2012 1,2,
T
J

.
10121, 2,1
T
z

(25)
WeCIn5.4
3011

Page 5

3) Optimization with Respect to Collision Avoidance
For multi-UAV cooperation, one of the most important
problems is collision avoidance. In our method, keeping the
UAVs apart from the target can avoid two collisions: the
collision between each UAV and the target; the collision
between the two UAVs. The distance between each UAV and
the target can be denoted as,
dXX


(27)
1,1 1,1, 1,kk T kk
2,12,1,1,kk T kk
dXX


(28)
Substituting Eq. (22) into Eq. (27) and Eq. (28) and
linearizing them, then this criterion can be designed as
000
w d
Jd
w d

where,


12
,DcDc Dc

is the required distance between each UAV
and the target. w4≥0 and w5 ≥0 are the weight values.
Using the same method as the Eq. (24), minimizing J3 is
equivalent to minimizing z2 subjected to the following
inequality,





The objective function (29) can be written as,
J3=z2 (31)
The constraints due to UAV kinematics and dynamics also
have to be considered:
vvv
  
(32)
where,
( )min(
uplimiimax
vjv
vjv


and
imin
v
are the maximum and minimum velocity
change;
maxi
and
mini
are maximum and minimum of
velocities; j=1, 2, 3 denotes the corresponding component of
the vector.
,
4
0
1
30
,
52
00
M k

A k
v
v
Dc












 




 
(29)

012111222
,,/,/
T
TT
dXXdTXXdTXX
   



T
 
1,
4
0
1
202
2,
52
000
00
k
k
v
v
w d
zdDcz
w d

 












 


(30)
,lowlimi i kuplimi

,
,
( ),j v ( )j( ))j
( )max(( ),j v ( )j( ))j
imax

i k
v
lowlimi

imin imini k
v





imax
v
v
v
V. SIMULAITON RESUTLS
The target system model is as Eq. (1) The envelop matrix
of system noise and observation noise are Q=diag(0.0001,
0.0001, 0.0001) and R=diag(0.001, 0.001, 0.001) respectively.
The initial envelope matrix of uncertain area of target is
P0=diag (0.1, 0.1, 0.1).The weight values are w1=0.01,
w2=0.0005, w3=2.7, w4=1, w5=1.1.
Fig. 6 shows the process of cooperative tracking. The solid
line in the middle is the trajectory of target and other two solid
lines are trajectory of UAVs. The dashed lines are the
observation lines connected with each UAV and the target.
Each ellipsoid is the cooperative observation result that gets
smaller and smaller with the tracking and observing going on.
For verifying the excellence performance our method we
supposed that there is a sudden change in the target’s
trajectory. But from Fig. 6 we can see that the two UAVs can
still track the target cooperatively with the proper trajectory
and the convergence of the cooperative observation method
can be ensured, i.e., the observation result can converge to a
steady uncertainty set quickly.

Fig. 6 Whole process of cooperative tracking
The size of an ellipsoid can be indicated by its length of
three axes i.e. the trace of envelope matrix Pk,k in Eq. (2).
Fig.7 shows changing trend of the trace of matrix Pk,k, which
illustrates the uncertain ellipsoid quantitatively. The dashed
line denotes the envelope matrix trace estimated by the ESMF
algorithm based on only a group of measurement data; the
solid line denotes the case when two groups of measurement
data are available. It is obvious from Fig. 7 that the uncertain
ellipsoid set cooperatively estimated by two UAVs can
converge to a smaller steady value (0.0034 vs. 0.0091), which
indicates much higher observation precision. Besides that
there is an increase near Step 5 because of the sudden change
of target trajectory. But after that the uncertain ellipsoid can
still converge, indicating that the proposed method has better
steady performance.

Fig. 7 Trace of envelope matrix of target position
Fig. 8 shows the change of cooperative observation angle
during the tracking process. The cooperative observation
angle can always keep around 90 degree during the tracking
process except near step 5 where the target trajectory has a
WeCIn5.4
3012

Page 6

sudden change. From Fig. 8 the conclusion is that our method
can achieve near-optimal cooperative observation actively
with the moving of the target by path planning the path
properly.

Fig. 8 Change of cooperative observation angle
For discussing the speediness of our method more
specifically, the cooperative observation method proposed in
[9] is compared with our method. Table I is the computational
cost of different algorithms [12]. From the table we can see
clearly the new proposed algorithm in this paper takes
averagely only 0.313s to complete one iteration computation,
which is almost equivalent to that of the single ESMF
algorithm. While the algorithm proposed in [9] takes
averagely 0.391s, which is about 20% more than that of the
new proposed method in this paper. Thus, we can conclude
that the new cooperative observation algorithm is more
suitable to be used in fast applications. More difference
between these two methods can be found in [12].
Table I The comparisons of computational cost
Algorithms
Single ESMF
New Proposed Algorithm
Algorithm in [9]
Time cost (s)
0.311
0.313
0.391
VI. CONCLUSION
An active cooperative observation method, which
combines the LP based path planning method in RVCs with
cooperative observation based on ESMF, was presented in
this paper. With this method the UAVs first plan the path
according to the prediction of target and then updated the
predicted states of the target cooperatively for obtain the more
accurate result. The following advantages can be obtained
with the new proposed algorithm: 1) this method can observe
the target optimally by planning the path of the UAVs
dynamically; 2) fusing the observation result without extra
computation burden apart from ESMF and planning the path
with LP method can ensure the speediness of algorithm; 3)
Besides that, this method avoids introducing the more
approximations which can ensure the accurate observation
result. Finally, the simulation results verify the feasibility and
steady performances of the new proposed method.
REFERENCES
[1] G. Gu, P. R. Chandler, C. J. Schumacher, A. Sparks, and M. Pachter,
“Optimum cooperative UAV sensing based on Cramer-Rao bound,”
IEEE Transactions on Aerospace and Electronic Systems, vol. 42, no.4,
2006.
[2] A. Sanfeliu, J. M. M. Tur and A. C. Murtra;, “Efficient active global
localization for mobile robots operating in large and cooperative
environments,” Proceedings of IEEE International Conference on
Robotics and Automation, pp. 2758-2763 2008.
[3] U. Zengin and A. Dogan. “Cooperative target tracking for autonomous
UAVs in an adversarial environment,” Proceedings of AIAA Guidance,
Navigation, and Control Conference and Exhibit, 2006
[4] D. H. Shim, H. J. Kim and S. Sastry. “Decntralized nonlinear model
predictive control of multiple flying robots, ”Proceedings of IEEE
Conference on Decision and Control, pp. 3621-3626, 2003
[5] A. W. Stroupe, M. C. Martin and T. Balch, “Distributed Sensor Fusion
for Object Position Estimation by Multi-Robot Systems,” Proceedings
of IEEE International Conference on Robot and Automation,
pp.1092-1098, 2001.
[6] B. Zhou, J. Han and G. Liu. “A UD factorization-based nonlinear
adaptive set-membership filter for ellipsoidal estimation,” International
Journal of Robust and Nonlinear Control, 2007.
[7] E. Scholte and M. E. Campbell. “A nonlinear set-membership filter for
on-line applications,” International Journal of Robust and Nonlinear
Control, 13, pp. 1339-13589, 2003.
[8] E. Scholte and M. E. Campbell. “On-line nonlinear guaranteed
estimation with application to a high performance aircraft,”
Proceedings of American Control Conference, pp. 184–190. 2002.
[9] M. E. Campbell and J. Ousingsawat. “On-line estimation and path
planning for multiple vehicles in an uncertain environment,”
International Journal of Robust and Nonlinear Control, 14, pp.
741-766, 2004.
[10] D. Zu, J. Han and D. Tan. “LP-based optimal path planning in
acceleration space, ” Proceedings of IEEE International Conference on
Robotics and Biomimetics, pp. 1340-1345, 2006
[11] D. Zu, J. Han and D. Tan. “MILP-based trajectory generation in relative
velocity coordinates,” Proceedings of IEEE Conference on Decision
and Control, pp. 1975-1980, 2007
[12] F. Gu, Y. He, J. Han and Y. Wang. “On-line cooperative observation
based on ESMF in three dimensional environments,” Proceedings of
AIAA Guidance, Navigation, and Control Conference and Exhibit.
WeCIn5.4
3013