Dynamic equations for particle s ik is de?ned by
d u iketT=d t?W1etTtW2etT
W1etT?àu iketTtc v iketT
W2etT?k1tk2o JetT
ik
àk3o PetT
ik
àk4o QetT
ik
h i
o u iketT
ik
h i2
to u iketT
ik
h i2
&'
8
>>>
>>>
<
>>>
>>>
:
e9T
where c>1.And v iketTis a piecewise linear function of u iketTde?ned by
v iketT?
0if u iketT<0
u iketTif06u iketT61
1if u iketT>1
8
><
>:e10T
In order to dynamically optimize sensor allocation,the particle s ik may alternately modify r ik and c ik,respectively, as follows:
d c iketT=d t?k1
o u iketT
o c iketT
tk2
o JetT
o c iketT
àk3
o PetT
o c iketT
àk4
o QetT
o c iketT
e11Td r iketT=d t?k1
o u iketT
o r iketT
tk2
o JetT
o r iketT
àk3
o PetT
o r iketT
àk4
o QetT
o r iketT
e12T
where o QetT
ik
?àf?1texpeàf iketTu iketTT à1à0:5g is a sig-moid function of the aggregate intention strength f iketT. The graphical presentation of o Q
ik
is shown in Fig.3.
Note that f iketTis related to social coordinations among
the sensors at time t.o Q
o u ik
is a monotone decreasing function. By Eq.(9)the greater the value of f iketT,the more there
are Fig.3.Graphical presentation of o Q
o u ik
.
454X.Feng et al./Information Fusion9(2008)450–464
values of W 2et Tand hence u ik et Twill increase,which implies that the social coordinations will strengthen the current allocation r ik et T.Because o Q o u ik
is a monotone decreasing function,the greater the value of f ik et T,the smaller o Q ik ,the greater ào Q ik
,the greater W 2et Tby Eq.(9)and the greater D r ik et t1Tby Eq.(12).Since r ik et t1T?r ik et TtD r ik et t1T,the greater the value of f ik et T,the greater r ik et t1T.Since u ik is a monotone increasing function,the greater r ik et t1T,the greater u ik f ik et T")
o Q o u ik #)ào Q o u ik ")e9TW 2"ào Q o u ik
")e11T
D r ik et t1T")a
r ik et t1T")e3Tb
u ik "
(a)r ik et t1T?r ik et TtD r ik et t1T;
(b)u ik is monotone increasing function.
In the following,we derive some formal properties of the mathematical model presented above.
Proposition 1.Updating the weights c ik and allotted resource r ik by Eqs.(11)and (12)respectively amounts to changing the speed of particle s ik by W 2et Tof Eq.(9).
Denote the j th terms of Eqs.(11)and (12)by d c ik et T
D E j and d r ik et T
D E j
;respectively.When allotted resource r ik is updated according to (12),the ?rst and second terms of (12)will cause the following speed increments of the parti-cle s ik ,respectively:
h d u ik et T=d t i r
1?
o u ik et To r ik et Td r ik et Td t ()1
?k 1o u ik et To r ik et T !2
e13Th d u ik et T=d t i r
2?
o u ik et To r ik et Td r ik et Td t ()2
?k 2o u ik et To r ik et To J et To r ik et T?k 2
o u ik et To r ik et To J et To u ik et To u ik et To r ik et T
?k 2
o J et To u ik et To u ik et T
o r ik et T !2
e14TSimilarly,the third and the fourth term of Eq.(12)will cause the following speed increments of the particle s ik :
h d u ik et T=d t i r 3?àk 3o P et Tik o u ik et Tik !2
h d u ik et T=d t i r
4
?àk 4
o Q et To u ik et To u ik et T
o r ik et T
!2
Similarly,for Eq.(11),we have h d u ik et T=d t i c
j ;j ?1;2;3;4.We thus obtain
X 4j ?1
?h d u ik et T=d t i c j th d u ik et T=d t i r j
?k 1tk 2
o J et To u ik et Tàk 3o P et To u ik et Tàk 4o Q et T
o u ik et T
!o u ik et To r ik et T
!2
(to u ik et To c ik et T
!2
)?W 2et T:Therefore,updating c ej T
ik and r ej T
ik by (11)and (12),respec-tively,gives rise to the speed increment of particle s ik that is exactly equal to W 2et Tof Eq.(9).
Proposition 2.The ?rst and second terms of Eqs.(11)and (12)will enable the particle s ik to move upwards,that is,the personal utility of sensor A i from object T k increases,in direct proportion to the value of ek 1tk 2T.
According to Eqs.(13)and (14),the sum of the ?rst and second terms of Eqs.(11)and (12)will be
h d u ik et T=d t i r 1th d u ik et T=d t i r 2th d u ik et T=d t i c 1th d u ik et T=d t i c
2
?k 1tk 2o J et To u ik et T !o u ik et To r ik et T !2t
o u ik et T
o c ik et T !2()
?ek 1tk 2Tx 2ik et T?r 2ik et Ttc 2
ik et T ?àu ik et T 2P 0:
Therefore,the ?rst and second terms of (11)and (12)will
cause u ik et Tto monotonically increase.
Proposition 3.For C-GPM,if is very small,then decreas-ing the potential energy P et Tof Eq.(4)amounts to increasing the minimal utility of an sensor with respect to an object,min-imized over S et T.
Supposing that H et T?max i ;k fàu 2ik et Tg ,we have
?exp eH et T=2 2T 2 2
6X n i ?1
X m k ?1
exp eàu 2ik et T=2 2
T
"#2 2
6?mn exp eH et T=2 2T 2 2
:
Taking the logarithm of both sides of the above inequalities gives
H et T62 2ln X n i ?1
X m k ?1
exp eàu 2ik et T=2 2T6H et Tt2 2
ln mn :
Since mn is constant and is very small,we have
H et T%2 2ln X n i ?1
X m k ?1
exp eàu 2ik et T=2 2Tà2 2
ln mn ?2P et T:
It turns out that the potential energy P et Tat the time t rep-resents the maximum of àu 2ik et Tamong all the particles s ik ,which is the minimal personal utility of a sensor with respect to an object at time t .Hence the decrease of potential energy P et Twill result in the increase of the minimum of u ik et T.Proposition 4.Updating c ik and r ik according to Eqs.(11)and (12)amounts to increasing the minimal utility of a sensor with respect to an object in direct proportion to the value of k 3.
The speed increment of particle s ik ,which is related to potential energy P et T,is given by d u ik et Td t ()3
?h d u ik et T=d t i r 3th d u ik et T=d t i c
3
?àk 3o P et To u ik et To u ik et To r ik et T !2to u ik et T
o c ik et T
!2():
X.Feng et al./Information Fusion 9(2008)450–464455
Denote by
d P et Td t
D
E
the di?erentiation of the potential
energy function P et Twith respect to time t arising from using (11)and (12).We have d P et Td t ()?o P et To u ik et Td u ik et T
d t ()3
?àk 3o P et To u ik et T !2o u ik et To r ik et T !2t
o u ik et T
o c ik et T
!2()?àk 3x 2ik et Tu 2ik et Tx 2ik et T?r 2ik et Ttc 2
ik et T ?u ik et T 2
60:
where,x ik et T?
exp ?àu 2ik et T=2 2
X n i ?1
X m k ?1
exp ?àu 2ik et T=2 2
:
,
It can be seen that using Eqs.(11)and (12)gives rise to
monotonic increase of P et T.Then by Proposition 3,the de-crease of P et Twill result in the increase of the minimal util-ity,in direct proportion to the value of k 3.
Proposition 5.Updating c ik and r ik by Eqs.(11)and (12)gives rise to monotonic increase of the whole utility of all the sensors,in direct proportion to the value of k 2.
Similar to Proposition 2,it follows that when a particle s ik modi?es its c ik and r ik by Eqs.(11)and (12),di?eren-tiation of J et Twith respect to time t will not be negative—i.e.,d J R et T
d t D E
P 0,and it is directly proportional to the value of k 2.
Proposition 6.Updating c ik and r ik by Eqs.(11)and (12)gives rise to monotonic decrease of the potential energy Q et T,in direct proportion to the value of k 4.As in the above,we have
d u ik et Td t ()4?àk 4o Q et To u ik et To u ik et To r ik et T !2to u ik et T
o c ik et T !2()
;and
d Q et Td t ()?
o Q et To u ik et Td u ik et Td t ()4
?àk 4o Q et T
o u ik et T !2?o u ik et To r ik et T !2to u ik et To c ik et T !2()60:
Proposition 7.C-GPM can dynamically optimize in parallel
sensor resource allocation for a collection of sensors exhibit-ing different autonomous strengths of pursuing personal util-ity and executing social coordinations.
In summary,by Propositions 1–6,ek 1tk 2Trepresents the autonomous strength for A i to pursue its own personal utility;k 2represents the autonomous strength for A i to take into account the collective utility of all the sensors;k 3rep-resents the autonomous strength for A i to increase the min-imal personal utility among all the sensors;and k 4represents the autonomous strength for constraint satisfac-tion and social coordinations.
Propositions 1,2,3,47show that di?erent sensors may have di?erent degrees of autonomy and their own rational-
ity in pursuing individual and system-wide bene?ts.More-over,di?erent sensors may engage in di?erent social
coordinations with other sensors for a speci?c object.A large variety of social coordinations may be taken into account,including possibly unilateral and unaware social coordinations.
3.3.Convergence analysis
In this subsection,we show that all the particles can con-verge to their stable equilibrium states through algorithm C-GPM.
Lemma 1.If c à1>àW 2et T>0,o W 2et T
o u ik et T<1for u ik et T>1,
and o W 2et T
o u ik et T>1àc for 01;v ik et T?1.
Proof.For c >1,W 1et Tof particle s ik is a piecewise linear function of the stimulus u ik et T,as shown in Fig.4:Segment I,Segment II,and Segment III.By Eq.(9),a point is an equilibrium point,i.e.,d u ik et T=d t ?0,i?àW 2et T?W 1et Tat the point.We see that for the case of c à1>àW 2et T>0,an equilibrium point may be on Segment I,II or III.Note from Eq.(3),u ik et TP 0.Thus we need not con-sider the equilibrium point on Segment I.
Suppose that the particle s ik is at an equilibrium point on Segment III at time t 0,and an arbitrarily small perturbation D u ik to the equilibrium point occurs at time
t 1.Since o W 2et To u ik et T<1,and o W 1et T
o u ik et T?à1for u ik et T>1,we have
c ?
o W 1et Tik to W 2et T
ik h i <0,and D
d u ik d t ?d u ik d t t 1àd u ik d t t 0?d u ik d t
t 1
?D ?W 1et TtW 2et T %o W 1et To u ik et Tto W 2et To u ik et T
!D u ik ?àj c j D u ik :
That means d u ik d t t 1
is always against D u ik ,or in other words,the perturbation will be suppressed and hence the particle s ik will return to the original equilibrium
point.
456X.Feng et al./Information Fusion 9(2008)450–464
Whereas,for an equilibrium point on Segment II,
because o W2etT
o u iketT>1àc and o W1etT
o u iketT
?cà1>0,we have
c?
o W2etT
ik t
o W1etT
ik
!
>0;and
d u ik
t1
%o W1etT
o u iketT
t
o W2etT
o u iketT
!
D u ik?j c j D u ik
such that the perturbation is intensi?ed and the particle s ik departs from the original equilibrium point on Segment II. Therefore,an equilibrium point on Segment II is unstable, and only an equilibrium point on Segment III,e.g.,p4in Fig.4,is stable with u iketT>1;v iketT?1.h
Lemma 2.If c>1;àW2etT<0,and o W2etT
o u iketT<1for
u iketT>1,then the particle s ik will converge to a stable equi-librium point with u iketT>1;v iketT?1.
Proof.Due to c>1andàW2etT<0,an equilibrium point must be on Segment III,e.g.,p6in Fig.4.Moreover,as sta-
ted in the proof of Lemma1,o W2etT
o u iketT<1for u iketT>1guar-
antees any equilibrium point on Segment III to be stable, with u iketT>1;v iketT?1.h
Lemma 3.If c>1,o W2etT
ik <1for u iketT?1t;and
o W2etTo u iketT>1àc for u iketT?0tand u iketT?1à,then the equilib-
rium points s1and s3in Fig.4are unstable and a saddle point, respectively.
Proof.It is straightforward from the proof of Lemma 1.h
Theorem 1.If c>1,o W2etT
ik
<1for u iketTP1t,and o W2etT
o u iketT
>1àc for0t6u iketT61à,the dynamical equation (9)has a stable equilibrium point on Segment III iff the right side of the equation is larger than0for u iketT?1and v iketT?1:
Proof.By Lemmas1–3,the equilibrium points on Segment II and Segment III,except for the saddle point s3in Fig.4, are unstable and stable,respectively.We denote the right side of Eq.(9)by RHS.
Suf?ciency.Assume that RHS?d u ijetT?W1etTtW2etT>0holds for u iketT?1;v iketT?1.It follows that W1etT?àW2etTfor u iketT?1;v iketT?1,namely,it is impossible that the equilibrium point is the intersection
point s3where W1etT?àW2etT.Since RHS?d u ik
d t >0at
point u iketT?1,v iketT?1,the increase of u iketTfrom value 1leads to that the particle s ik converges to a stable equilibrium point on Segment III.
Necessity.Suppose that Eq.(9)has a stable equilibrium point.We need to prove that RHS>0holds for u iketT?1 and v iketT?1.By contrary,if there is RHS60for u iketT?1and v iketT?1,then the equilibrium point must be either at the point s3where RHS?0,or on Segment II
because RHS?d u ik
d t <0would giv
e rise to the decrease of
u iketTfrom value1.Since the point s3and any equilibrium
point on Segment II are all non-stable,we have a
contradiction.h
Theorem 2.If c>1,o W2etT
ik
<1for u iketTP1t;and
o W2etT
ik
>1àc for0t6u iketT61à,then the dynamical equa-
tion(8)has a stable equilibrium point iff
c>1t2?k3t0:25k4 :e15T
Proof.By Eqs.(3),(4),(7),(8)we have
W2etT?f k1tk2àk3x2
ik
etTu2
ik
etT
àk4?e1texpeàf iketTu iketTTTà1à0:5 2g?r2
ik
etT
tc2
ik
etT ?u iketT 2e16T
Note that x2
i
etT;r2
ik
etT;c2
ik
etT61.By Theorem1,for
u iketT?1;v iketT?1,we have
c>1àW2etT
?1tfàk1àk2tk3x2
ik
etTu2
ik
etT
tk4?e1texpeàf iketTu iketTTTà1à0:5 2g?r2
ik
etTtc2
ik
etT ?u iketT 2
61t2?k
3
t0:25k4 :?
Lemma4.If c>1;f iketTP0,and the following conditions
are valid:
k1tk2c>max f1t4?k1tk2t0:25k3tf iketTk4 ;1t2?k3t0:25k4 g:
e18T
then the dynamical equation(9)has a stable equilibrium point
with u ik?1;v ik?1.
https://www.360docs.net/doc/0410065210.html,ing Eq.(16)and noting d x iketT
ik
?2x iketT
u iketTex iketTà1T,we obtain
o W2etT
o u iketT
?f2?àk1àk2tk3x2
ik
etTu2
ik
etT
tk4?e1texpeàf iketTu iketTTTà1à0:5 2
tà2k3x iketT
d x iketT
d u iketT
u2
ik
etTà2k3x2
ik
etTu iketT
à2k4f iketTexpeàf iketTu iketTT
??e1texpeàf iketTu iketTTTà1à0:5
??1texpeàf iketTu iketTT à2
i
eàu iketTTg x2
ik
etT?r2
ik
etT
tc2
ik
etT ?àu iketT ?f2?àk1àk2tk3x2
ik
etTu2
ik
etT
tk4?e1texpeàf iketTu iketTTTà1à0:5 2
t?à4k3x2
ik
etTu3
ik
etTex iketTà1Tà2k3x2
ik
etTu iketT
à2k4f iketTexpeàf iketTu iketTT
??e1texpeàf iketTu iketTTTà1à0:5
??1texpeàf iketTu iketTT à2 eàu iketTTg x2
ik
etT
??r2
ik
etTtc2
ik
etT ?àu iketT
X.Feng et al./Information Fusion9(2008)450–464457
Then from o W2etT
o u iketT
<1for u iketTP1t;and u iketT6x iketT; r iketT;c iketT61,we derive
o W2etT
o u iketT
<2?2eàk1àk2tk3t0:25k4T
t4k3e2=3T2e3=4T3=12 <1;
which leads to Eq.(17).
Similarly,from o W2etT
o u iketT>1àc for0t6u iketT61à,we
have
o W2etT
o u iketT
>4eàk1àk2àk3=4àf iketTk4T>1àc;
which leads to Eq.(18).By Theorem2,therefore the con-clusion is valid.h
Lemma5.If c>1;f iketT60,and the following conditions are valid:
k1tk2max f1t4?k1tk2t0:25k3 ;1t2k3t0:25k4ge20Tthen the dynamical equation(8)has a stable equilibrium point with u ik?1;v ik?1:
Proof.Similar to the proof of Lemma4,from o W2etT
o u iketT<1
for u iketTP1t,and u iketT6;x iketT;r iketT;c iketT61,we derive
o W2etT
o u iketT
<2?2eàk1àk2tk3t0:25k4T
t4k3e2=3T2e3=4T31=12tj f iketTj k4 <1;
which leads to Eq.(19).
And from o W2etT
o u iketT
>1àc for0t6u iketT61à,we have
o W2etT
o u iketT
>4eàk1àk2àk3=4T>1àc;
which leads to Eq.(20).By Theorem2,therefore the con-clusion is valid.h
Theorem3.If c>1,and the conditions(18),(19)are valid, then the dynamical equation(9)has a stable equilibrium point with u ik?1;v ik?1.
Proof.Straightforward,by Lemmas4and5.h
Theorem4.If the conditions(17),(18)remain valid,then C-GPM will converge to a stable equilibrium state.
https://www.360docs.net/doc/0410065210.html,ing Eq.(16)and noting d x iketT
ik ?2x iketTu iketT
ex iketTà1T,we obtain o W2etT
o u iketT
?2?àk1àk2tk3x2
ik
etTu2
ik
etT
è
tk4?e1texpeàf iketTu iketTTTà1à0:5 2
t?à2k3x iketT
d x iketT
d u iketT
u2
ik
etTà2k3x2
ik
etTu iketT
à2k4f iketTexpeàf iketTu iketTT
??e1texpeàf iketTu iketTTTà1à0:5
?1texpeàf iketTu iketTT à2
h i
eàu iketTT
o
x2
ik
etT?r2
ik
etT
tc2
ik
etT ?àu iketT ?f2?àk1àk2tk3x2
ik
etTu2
ik
etTtk4?e1texpeàf iketTu iketTTTà1à0:5 2
t?à4k3x2
ik
etTu3
ik
etTex iketTà1Tà2k3x2
ik
etTu iketTà2k4f iketTexpeàf iketTu iketTT
??e1texpeàf iketTu iketTTTà1à0:5
??1texpeàf iketTu iketTT à2 eàu iketTTg x2
ik
etT?r2
ik
etT
tc2
ik
etT ?àu iketT
For the force?eld F,we de?ne a Lyapunov function LetTby
LetT?à
1
2
X
i;k
ecà1Tv iketT2t
X
i;k
Z t
d v ikexT
d x
fàk1àk2
tk3x2
ik
exTu2
ik
exTtk4?e1texpeàf ikexTu ikexTTTà1
à0:5 2g?r2
ik
exTtc2
ik
exT ?àu ikexT 2d x:
We hence have
j LetTj6
X
i;k
ecà1Tj v iketT2jt
X
i;k
Z t
d v ikexT
d x
ájfàk1àk2
tk3x2
ik
exTu2
ik
exTtk4?e1texpeàf ikexTu ikexTTTà1
à0:5 2gj?r2
ik
exTtc2
ik
exT ?àu ikexT 2d x:
Since condition(18)is valid,v iketT61and u ietT61,it fol-lows that
j LetTj6
X
i;k
ecà1Tt
X
i;k
cwhich implies that LetTis bounded.
In addition,we have
d LetT
d t
?à
X
i;k
ecà1Tv iketT
d v iketT
d t
t
X
i;k
d v iketT
d t
fàk1àk2
tk3x2
ik
etTu2
ik
etTtk4?e1texpeàf iketTu iketTTTà1
à0:5 2g?r2
ik
etTtc2
ik
etT ?àu iketT 2
?à
X
i;k
d v iketT
d u ietT
d u iketT
d t
fàu iketTtc v iketTtf k1tk2
àk3x2
ik
etTu2
ik
etTàk4?e1texpeàf iketTu iketTTTà1
à0:5 2g?r2
ik
etTtc2
ik
etT ?àu iketT 2g
?à
X
i;k
d v iketT
d u iketT
d u iketT
d t
2
60:
458X.Feng et al./Information Fusion9(2008)450–464
Thus LetTwill monotonically decrease with the elapsing time.h
3.4.The parallel C-GPM algorithm
Algorithm C-GPM
As indicated by Theorems1–4,as long as we properly select the parameters k1;k2;k3;k4for the dynamical equa-tions of C-GPM according to Eqs.(17),(19),the conver-gence and stability of the particle dynamics can be guaranteed.
Given the results above,we can construct an algorithm to solve the sensor fusion(resource allocation)problem,as given below.
The algorithm C-GPM has in general a complexity of X(nm),where(nm)is the number of particles(the sum of the number of elements of matrix S).The more realistic time complexity of the algorithm is OeIT,where I is the number of iterations for Step2(the while loop),as much of the inner steps can be carried out in parallel.
4.Simulations
Here we give an example of a robot which represents a multi-sensor fusion problem,and then use our C-GPM method to?nd the optimum solution.
The robot has seven sensors:eye,nose,ear,arm,hand, leg and foot.The robot has a reportoire of three actions: open a door,take dictation,?nd food and take back. Now we describe in detail how to solve the multi-sensor fusion problem,namely,how to allocate sensors to objects.
Let sensors A i;i?1;2;...;7represent the eye,nose, ear,arm,hand,leg and foot of the robot,respectively. Let objects T k;k?1;2;3represent the robot’s actions of opening a door,taking dictation,and?nding food and tak-ing them back,respectively.
Step1.
1.x ik:
First,we establish the relations between sensors and objects(actions).If sensor A i is related to object T k, x ik?1;otherwise,x ik?0.For example,the eye,arm,hand,leg and foot are related to the object of opening a door;the eye,ear,arm and hand are related to the object of taking dictation.We can form the matrix X,as follows:
2.c ik:
Based on the di?erent e?ects of the sensors A1–A7on the objects T1–T3,we can choose di?erent weights c ik.If sensor A i is more important to object T k than to T l(k;l21;2;3), then c ik will be larger than c il.The following must be observed when choosing the weights:
?If x ik?0,then c ik?0,to ensure that object T k which is not related to sensor A i is not allocated any sensor resource.
?06c ik61.
?In order that the sensor resources are fully allocated, P3
k?1
c ik?1;i?1;2; (7)
The weight matrix C we decide on for this particular problem is as follows:
3.f ik:
(1)b ijk:
C-GPM has a clear advantage over other approaches in being able to deal with a variety of coordinations occurring in multi-sensor systems.In Section3,we divide typical coordinations into four categories(I,II,III,IV)of12 types.
Since the nose A2is not related with the object T1of opening a door,the coordination of the eye A1with respect to the nose A2is unilateral coordination,whereby A2 would modify its intention,and A1would not.Therefore,
b
121
?II and b211?I.Since the eye A1and the arm A4
Input:c ik;x ik;f ik
Output:
1.Initialization:
t0
r iketT—Initialize in parallel
2.Whileed u ik=d t?0Tdo
t tt1
u iketT—Compute in parallel according to Eq.(3) d u ik=d t—Compute in parallel according to Eq.(8)
d r iketT=d t—Comput
e in parallel according to Eq.
(11)
r iketTr iketà1Ttd r iketT=d t
d c iketT=d t—Comput
e in parallelaccording to Eq.(10) c iketTc iketà1Ttd c iketT=d t x ik T1T2T3
open a door take dictation?nd food and
take back
A1:eye111
A2:nose001
A3:ear010
A4:arm111
A5:hand111
A6:leg101
A7:foot101
c ik T1T2T3
open a door take dictation?nd food and
take back
A1:eye0.50.250.25
A2:nose001
A3:ear010
A4:arm0.40.30.4
A5:hand0.330.340.33
A6:leg0.3500.65
A7:foot0.3500.65
X.Feng et al./Information Fusion9(2008)450–464459
are both related to the object T1of opening a door,but they themselves are unrelated,the coordination between the eye A1and the arm A4is bilateral coordination,and neither A1nor A4would modify their intention.So b141?III and b411?III.Since the arm A4and the hand A5are both related to the object T1of opening a door and they in?uence each other,the coordination between A4and A5is bilateral coordination,and A4and A5must both modify their intention.So b451?IV and
b 541?IV.
As such,we can obtain all the b ijk values,as shown in the following,where for object T k,if there is no coordina-tion between sensor A i and A j,b ijk is set to‘‘–’’.
(2)f ijk:
According to Eq.(6),
f ijketT?
1if b ijk2LeIIT[LeIIITà1if b ijk2LeIT[LeIVT(
f jiketT?
1if b jik2LeIT[LeIIITà1if b jik2LeIIT[LeIVT(
and so we obtain f ijk as follows:
(3)f ik:
According to Eq.(5),f iketT?
P n
j?1
f ijketTt
P n
j?1
f jiketT, we get f ik as follows:
f ik is equal to the sum of the i th row and the i th column
of the matrixef ijkT
7?7
.
4.r ik:
Initialization:et?0T
b ij1A1A2A3A4A5A6A7 A1–II II III III III III A2I––I I I I A3I––I I I I A4III II II–IV III III A5III II II IV–III III A6III II II III III–IV A7III II II III III IV–b ij2A1A2A3A4A5A6A7 A1–II III III III II II A2I–I I I––A3III II–III III II II A4III II III–IV II II A5III II III IV–II II A6I–I I I––A7I–I I I––
b ij3A1A2A3A4A5A6A7 A1–III II III III III III A2III–II III III III III A3I I–I I I I A4III III II–IV III III A5III III II IV–III III A6III III II III III–IV A7III III II III III IV–f ij1A1A2A3A4A5A6A7 A10111111 A2à100à1à1à1à1 A3à100à1à1à1à1 A41110à111 A5111à1011 A6111110à1 A711111à10 f ij2A1A2A3A4A5A6A7 A10111111 A2à10à1à1à100 A31101111 A41110à111 A5111à1011 A6à10à1à1à100 A7à10à1à1à100
f ij3A1A2A3A4A5A6A7 A10111111 A21011111 A3à1à10à1à1à1à1 A41110à111 A5111à1011 A6111110à1 A711111à10 f ik T1T2T3 A18610 A20010 A3060
A4426
A5426
A6406
A7406
r ik T1T2T3 A11/31/31/3 A2001
A3010
A41/31/31/3 A51/31/31/3 A61/201/2 A71/201/2
460X.Feng et al./Information Fusion9(2008)450–464
Here r ik is initialized by the average values.Alterna-tively,r ik can also be initialized as random numbers between0and1.In fact,based on the experiments we have done,we found that the results are not a?ected by the ini-tialization of R.Whatever r ik is initialized to be,the matrix R is dealt with using the following three steps:
?r ik?0j8x ik?0.If x ik?0,then r ik?0,to ensure that an object T k which is not related to the sensor A i is not allo-cated any sensor resource.
?Non-negativity:06r ik61.If min i;k r ik<0,then let
r ik?r ikàmin i;k r ik.?Normalization.Let r ik?r ik=P3
k?1
r ik,in order that the
sensor resources are fully allocated;that is, P3
k?1
r ik?1;i?1;2; (7)
5.zeRT:
According to Eq.(1),zeRT?P n
i?1
P m
k?1
c ik r ik x ik?4:023.
https://www.360docs.net/doc/0410065210.html,pute in parallel(showing only the?rst evo-lutionary iteration;t=1).
1.D r ik:
According to Eq.(11),we have
D r ik%d r ik=d t?k1o u ik
o r ik
tk2
o J
o r ik
àk3
o P
o r ik
àk4
o Q
o r ik
where,
o u ik
o r ik
?c ik x ik expeàc ik r ik x ikTo J
o r ik
?c ik x ik expeàc ik r ik x ikT
o P o r ik ?
o P
o u ik
á
o u ik
o r ik
?àu iká
expeàu2
ik
=2 2T
P n
i?1
P m
k?1
expeàu
ik
=2 2T
á
o u ik
o r ik
o Q o r ik ?2
X m
k?1
x iká
X n
i?1
X m
k?1
r ik x ikà1
!à
1
1texpeàf ik u ikT
à
1
2
!
á
o u ik
o r ik
2.D c ik:
According to Eq.(10),we have
D c ik%d c ik=d t?k1o u ik
o c ik
tk2
o J
o c ik
àk3
o P
o c ik
àk4
o Q
o c ik
where,
o u ik o c ik ?r ik x ik expeàc ik r ik x ikT
o J o c ik ?r ik x ik expeàc ik r ik x ikT
o P o c ik ?
o P
o u ik
á
o u ik
o c ik
?àu iká
expeàu2
ik
=2 2T
P n
i?1
P m
k?1
expeàu2
ik
=2 2T
á
o u ik
o c ik
o Q o c ik ?à
1
1texpeàf ik u ikT
à
1
2
!
á
o u ik
o c ik
3.r iket?1Tand c iket?1T:
r iket?1T?r iket?0TtD r iket?1T
c iket?1T?c iket?0TtD c iket?1T
k1?0:05k2?0:05k3?0:9k4?0:9 ?0:8
After we get r iket?1T,matrix R should be dealt with
using the three steps just mentioned again.
In addition,after we get c iket?1T,matrix C should be
dealt with similarly,i.e.,
?c ik?0j8x ik?0.
?Non-negativity:Let c ik?c ikàmin i;k c ik.
?Normalization.Let c ik?c ik=
P3
k?1
c ik.
Regarding the coe?cients k1,k2,k3,k4and ,we can
draw the following conclusions from the theoretical proofs
and the experiments we have done:
?When is larger,the convergence speed of C-GPM algo-
rithm is faster.
?If the values of k1,k2,k3,and k4change in direct propor-
tion,the experimental results will hardly be in?uenced.
?In order for the C-GPM algorithm to converge,k3and
k4should be much larger than k1and k2,based on
Eqs.(17)and(19).
At the end of the?rst iteration we get
Ret?1T?
0:36920:31110:3197
001:0000
01:00000
0:33680:31580:3474
0:33310:32600:3409
0:458200:5418
0:458200:5418
B B
B B
B B
B B
B B
B B
B B
@
1
C C
C C
C C
C C
C C
C C
C C
A
Cet?1T?
0:48690:24830:2648
001:0000
01:00000
0:36030:26610:3735
0:33040:32690:3427
0:355400:6446
0:355400:6446
B B
B B
B B
B B
B B
B B
B B
@
1
C C
C C
C C
C C
C C
C C
C C
A
zet?1T?4:0690
Obviously,zet?1T>zet?0T,the robot multi-sensor
fusion problem is being optimized.The evolutionary exper-
imental results for z from t=1to t=120are as follows,
and depicted in Fig.5.
As shown in Fig.5,the convergence speed is faster at
the beginning(from t=0to t=20)of the evolution.The
X.Feng et al./Information Fusion9(2008)450–464461
optimization trend of z re?ects exactly the optimization of the problem.At about t =60,z approaches the maximum of 4.53.At t =96,z reaches its maximum,and stays unchanged in the remainder of the iterations.From t =104to 120,R is always equal to
1:00000:00000:0000
01:000001:000000:00000:00001:00000:00000:00001:00000:000001:00000:0000
1:0000
B
B B
B B
B
B B
B B
B
@1C C C C C C C
C C
C C A Since r ik (t =104)=r ik (t =105)and c ik (t =104)=
c ik (t =105),
d r ik /d t (t =105)=0and d c ik /d t (t =105)=0.
Thus,d u ik d t (t =105)=o u ik o r ik (t =105)d r ik d t (t =105)+o u ik
o c ik
(t =105)d c ik
d t (t =105)=0.Therefore,th
e C-GPM algorithm ends appropriately.This shows that sensor resources are allocated to the objects whose weights are the maximum.Undoubtedly this solution o
f R obtained by C-GPM is the optimum solution as it maximizes z ,which veri?es the approach’s convergence and its ability to arrive at the optimum for a complex problem.The solution at t =120however is not a practical solution because in order to achieve the objective,it does not give the ‘‘less important’’sensors any resource.A more practical solution is probably at around t =20where z is very near its maximum value and every sensor is allocated some resource:
0:88520:02770:087100
1:000001:000000:10410:04260:85330:10370:05650:83980:072300:92770:0723
00:9277
B B B B B B B
B B
B B B
B @
1
C C C
C
C
C C
C
C
C C
C
C A It should be possible to re-formulate the problem model,
perhaps with the addition of more constraints,such that the iterative algorithm will settle at an optimum yet practi-cal solution.
Besides the C-GPM algorithm’s good performance in the mathematical sense,the particles’evolution and motion during the process of resolving the problem are also very interesting.If x ik ?1,then s ik is a particle.For our robot problem,there are 15
particles.
Fig.5.Optimization from t =0to t =120.
s ik T 1T 2T 3A 1?
??A 2?A 3?A 4???A 5??
?A 6??A 7
?
?
462X.Feng et al./Information Fusion 9(2008)450–464
According to Eq.(3),u ik ?1àexp eàc ik r ik x ik T,we get u ik et ?0T,as follows:
u ik is the y-coordinate of particle s ik .The x-coordinate of particle s ik represents the ordinal number of the particle.The initial states of the 15particles in the force ?eld are shown in Fig.6.
When t =5,10,15,120,the states of the 15particles in the force ?eld are shown in Figs.7–10,respectively.
5.Conclusion
In this paper,we describe a new evolutionary approach to multi-sensor fusion based on the coordination general-ized particle model (C-GPM).We study some theoretical foundations of the approach including its convergence.The validity of C-GPM is veri?ed via several formal proofs and by simulations.
u ik et ?0TT 1T 2T 3A 10.15210.07920.0792A 200
0.6321A 30
0.63210
A 40.12370.09430.1237A 50.10320.10610.1032A 60.160500.2775A 7
0.1605
0.2775
Fig.6.The initial state of 15particles in the force
?eld.
Fig.7.When t =
5.
Fig.8.When t =
10.
Fig.9.When t =
15.
Fig.10.When t =120.
X.Feng et al./Information Fusion 9(2008)450–464463
We present the C-GPM approach as a new branch of evolutionary algorithms,which can overcome the limita-tions of other existing evolutionary algorithms in capturing the entire dynamics inherent in the problem,especially those that are high-dimensional,highly nonlinear,and random.
Hence,the C-GPM approach can describe the complex behaviors and dynamics of multiple sensors.
The model treats each possible allocation of resources as a con?guration of particles with‘‘physical’’attributes and then?nds the equilibrium of the system iteratively.A novel aspect of the approach is that it takes into account‘‘uncon-scious’’social coordinations between the sensors.
The C-GPM algorithm can work out the theoretical optimum solution,which is important and exciting.To summarize,the C-GPM approach has the following attrac-tive features:
?Very high parallelism and real-time computational performance.
?A sensor in a multi-sensor system is regarded as being neither fully sel?sh nor fully unsel?sh,and so di?erent sensors can exhibit di?erent degrees of autonomy.
?A variety of complicated social coordinations among the sensors can be taken into account in the process of prob-lem-solving.
?The C-GPM algorithm can work out the optimum solu-tion of any multi-sensor fusion problem.
?C-GPM is a physics-based evolutionary algorithm that can overcome the limitations of other traditional evolu-tionary algorithms in describing the high-dimensional, highly nonlinear,and random behaviors and dynamics of objects.
?There are only?ve coe?cients in the C-GPM algorithm, whose values are all easy to choose. Acknowledgements
We thank the editor and referees for their very clear and useful advice and comments.This work is supported by a Hong Kong University Small Project Funding (200607176155).
References
[1]S.Forrest,Genetic algorithms—principles of natural-selection applied
to computation,Science261(5123)(1993)872–878.
[2]S.Kirkpatrick,C.D.Gelatt,M.P.Vecchi,Optimization by simulated
annealing,Science220(4598)(1983)671–680.
[3]E.Bonabeau,M.Dorigo,G.Theraulaz,Inspiration for optimization
from social insect behaviour,Nature406(6791)(2000)39–42.
[4]J.Kennedy,R.C.Eberhart,Particle swarm optimization,in:Proc.
IEEE Conf.Neural Networks,Piscataway,NJ,vol.IV,1995,pp.
1942–1948.
[5]Z.Ren,C.J.Anumba,O.O.Ugwu,The development of a multi-agent
system for construction claims negotiation,Advances in Engineering Software34(2003)683–696.
[6]Z.Ren,C.J.Anumba,Multi-agent systems in construction state of
the art and prospects,Automation in Construction13(2004) 421–434.
[7]G.Vincent,L.P.Christophe,S.Sami,A multi-agents architecture to
enhance end-user individual based modelling,Ecological Modelling 157(2002)23–41.
[8]C.J.Emmanouilides,S.Kasderidis,J.G.Taylor,A random asymmetric
temporal model of multi-agent interactions:dynamical analysis, Physica D181(2003)102–120.
464X.Feng et al./Information Fusion9(2008)450–464