Show PDF File - robotics - Universidad de Zaragoza

R. Aragüés (1980, Zaragoza) received the M.S. degree in
system engineering and computer science from the
Universidad de Zaragoza, Zaragoza, Spain, in 2008, and
the Ph.D. degree with the Departamento de
Informática e Ingeniería de Sistemas in March 2012.
Her research interests include multirobot perception,
map merging, and distributed consensus in robotic
networks.
Distributed Algorithms on Robotic Networks for Coordination in Perception Tasks
Programa Oficial de Posgrado en Ingeniería Informática
Distributed Algorithms on Robotic Networks
for Coordination in Perception Tasks
PhD Thesis
Rosario Aragüés Muñoz
Departamento de Informática e Ingeniería de Sistemas (DIIS)
Instituto de Investigación en Ingeniería de Aragón
Escuela de Ingeniería y Arquitectura (EINA)
Universidad de Zaragoza

R. Aragüés (1980, Zaragoza) received the M.S. degree in
system engineering and computer science from the
Universidad de Zaragoza, Zaragoza, Spain, in 2008, and
the Ph.D. degree with the Departamento de
Informática e Ingeniería de Sistemas in March 2012.
Her research interests include multirobot perception,
map merging, and distributed consensus in robotic
networks.
Distributed Algorithms on Robotic Networks for Coordination in Perception Tasks
Programa Oficial de Posgrado en Ingeniería Informática
Distributed Algorithms on Robotic Networks
for Coordination in Perception Tasks
PhD Thesis
Rosario Aragüés Muñoz
Departamento de Informática e Ingeniería de Sistemas (DIIS)
Instituto de Investigación en Ingeniería de Aragón
Escuela de Ingeniería y Arquitectura (EINA)
Universidad de Zaragoza

R. Aragüés (1980, Zaragoza) received the M.S. degree in
system engineering and computer science from the
Universidad de Zaragoza, Zaragoza, Spain, in 2008, and
the Ph.D. degree with the Departamento de
Informática e Ingeniería de Sistemas in March 2012.
Her research interests include multirobot perception,
map merging, and distributed consensus in robotic
networks.
Distributed Algorithms on Robotic Networks for Coordination in Perception Tasks
Programa Oficial de Posgrado en Ingeniería Informática
Distributed Algorithms on Robotic Networks
for Coordination in Perception Tasks
PhD Thesis
Rosario Aragüés Muñoz
Departamento de Informática e Ingeniería de Sistemas (DIIS)
Instituto de Investigación en Ingeniería de Aragón
Escuela de Ingeniería y Arquitectura (EINA)
Universidad de Zaragoza

R. Aragüés (1980, Zaragoza) received the M.S. degree in
system engineering and computer science from the
Universidad de Zaragoza, Zaragoza, Spain, in 2008, and
the Ph.D. degree with the Departamento de
Informática e Ingeniería de Sistemas in March 2012.
Her research interests include multirobot perception,
map merging, and distributed consensus in robotic
networks.
Distributed Algorithms on Robotic Networks for Coordination in Perception Tasks
Programa Oficial de Posgrado en Ingeniería Informática
Distributed Algorithms on Robotic Networks
for Coordination in Perception Tasks
PhD Thesis
Rosario Aragüés Muñoz
Departamento de Informática e Ingeniería de Sistemas (DIIS)
Instituto de Investigación en Ingeniería de Aragón
Escuela de Ingeniería y Arquitectura (EINA)
Universidad de Zaragoza

R. Aragüés (1980, Zaragoza) received the M.S. degree in
system engineering and computer science from the
Universidad de Zaragoza, Zaragoza, Spain, in 2008, and
the Ph.D. degree with the Departamento de
Informática e Ingeniería de Sistemas in March 2012.
Her research interests include multirobot perception,
map merging, and distributed consensus in robotic
networks.
Distributed Algorithms on Robotic Networks for Coordination in Perception Tasks
Programa Oficial de Posgrado en Ingeniería Informática
Distributed Algorithms on Robotic Networks
for Coordination in Perception Tasks
PhD Thesis
Rosario Aragüés Muñoz
Departamento de Informática e Ingeniería de Sistemas (DIIS)
Instituto de Investigación en Ingeniería de Aragón
Escuela de Ingeniería y Arquitectura (EINA)
Universidad de Zaragoza

R. Aragüés (1980, Zaragoza) received the M.S. degree in
system engineering and computer science from the
Universidad de Zaragoza, Zaragoza, Spain, in 2008, and
the Ph.D. degree with the Departamento de
Informática e Ingeniería de Sistemas in March 2012.
Her research interests include multirobot perception,
map merging, and distributed consensus in robotic
networks.
Distributed Algorithms on Robotic Networks for Coordination in Perception Tasks
Programa Oficial de Posgrado en Ingeniería Informática
Distributed Algorithms on Robotic Networks
for Coordination in Perception Tasks
PhD Thesis
Rosario Aragüés Muñoz
Departamento de Informática e Ingeniería de Sistemas (DIIS)
Instituto de Investigación en Ingeniería de Aragón
Escuela de Ingeniería y Arquitectura (EINA)
Universidad de Zaragoza

R. Aragüés (1980, Zaragoza) received the M.S. degree in
system engineering and computer science from the
Universidad de Zaragoza, Zaragoza, Spain, in 2008, and
the Ph.D. degree with the Departamento de
Informática e Ingeniería de Sistemas in March 2012.
Her research interests include multirobot perception,
map merging, and distributed consensus in robotic
networks.
Distributed Algorithms on Robotic Networks for Coordination in Perception Tasks
Programa Oficial de Posgrado en Ingeniería Informática
Distributed Algorithms on Robotic Networks
for Coordination in Perception Tasks
PhD Thesis
Rosario Aragüés Muñoz
Departamento de Informática e Ingeniería de Sistemas (DIIS)
Instituto de Investigación en Ingeniería de Aragón
Escuela de Ingeniería y Arquitectura (EINA)
Universidad de Zaragoza

R. Aragüés (1980, Zaragoza) received the M.S. degree in
system engineering and computer science from the
Universidad de Zaragoza, Zaragoza, Spain, in 2008, and
the Ph.D. degree with the Departamento de
Informática e Ingeniería de Sistemas in March 2012.
Her research interests include multirobot perception,
map merging, and distributed consensus in robotic
networks.
Distributed Algorithms on Robotic Networks for Coordination in Perception Tasks
Programa Oficial de Posgrado en Ingeniería Informática
Distributed Algorithms on Robotic Networks
for Coordination in Perception Tasks
PhD Thesis
Rosario Aragüés Muñoz
Departamento de Informática e Ingeniería de Sistemas (DIIS)
Instituto de Investigación en Ingeniería de Aragón
Escuela de Ingeniería y Arquitectura (EINA)
Universidad de Zaragoza

R. Aragüés (1980, Zaragoza) received the M.S. degree in
system engineering and computer science from the
Universidad de Zaragoza, Zaragoza, Spain, in 2008, and
the Ph.D. degree with the Departamento de
Informática e Ingeniería de Sistemas in March 2012.
Her research interests include multirobot perception,
map merging, and distributed consensus in robotic
networks.
Distributed Algorithms on Robotic Networks for Coordination in Perception Tasks
Programa Oficial de Posgrado en Ingeniería Informática
Distributed Algorithms on Robotic Networks
for Coordination in Perception Tasks
PhD Thesis
Rosario Aragüés Muñoz
Departamento de Informática e Ingeniería de Sistemas (DIIS)
Instituto de Investigación en Ingeniería de Aragón
Escuela de Ingeniería y Arquitectura (EINA)
Universidad de Zaragoza

R. Aragüés (1980, Zaragoza) received the M.S. degree in
system engineering and computer science from the
Universidad de Zaragoza, Zaragoza, Spain, in 2008, and
the Ph.D. degree with the Departamento de
Informática e Ingeniería de Sistemas in March 2012.
Her research interests include multirobot perception,
map merging, and distributed consensus in robotic
networks.
Distributed Algorithms on Robotic Networks for Coordination in Perception Tasks
Programa Oficial de Posgrado en Ingeniería Informática
Distributed Algorithms on Robotic Networks
for Coordination in Perception Tasks
PhD Thesis
Rosario Aragüés Muñoz
Departamento de Informática e Ingeniería de Sistemas (DIIS)
Instituto de Investigación en Ingeniería de Aragón
Escuela de Ingeniería y Arquitectura (EINA)
Universidad de Zaragoza

R. Aragüés (1980, Zaragoza) received the M.S. degree in
system engineering and computer science from the
Universidad de Zaragoza, Zaragoza, Spain, in 2008, and
the Ph.D. degree with the Departamento de
Informática e Ingeniería de Sistemas in March 2012.
Her research interests include multirobot perception,
map merging, and distributed consensus in robotic
networks.
Distributed Algorithms on Robotic Networks for Coordination in Perception Tasks
Programa Oficial de Posgrado en Ingeniería Informática
Distributed Algorithms on Robotic Networks
for Coordination in Perception Tasks
PhD Thesis
Rosario Aragüés Muñoz
Departamento de Informática e Ingeniería de Sistemas (DIIS)
Instituto de Investigación en Ingeniería de Aragón
Escuela de Ingeniería y Arquitectura (EINA)
Universidad de Zaragoza

n
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Z p n M p n

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Z p n M p n

y
1000
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0
z 0
−1000 −500 0 500 1000 1500
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y
1000
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0 1000
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z

•
•
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•

•
•
•
•

•
•
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•

•
•
•
•

k−
k
k

k−
k
k

k−
k
k

•
•
•
•

•
•

•

•

(x, y)
360◦
i ∈ N
j ∈ N xri = [xri , yri , θri ]T
i xri , yri ∈ R θri ∈ [−π, π]
i j
xj = [xj, yj] T xj = [xj, yj, θj, ρj] T
xj, yj ∈ R θj ∈ [−π, π] ρj ∈ R≥0 x
xr n
x = [xr, x1...xn] T ,

P i
j zij zj
x
(x, y)
xj = [xj, yj] T xr = [xr, yr, θr] T
yr − yj
zj = h(xr, xj) = arctan
− θr.
xr − xj
xj
(xj, yj) θj
1/ρj mj Rr
cos θj
mj = ,
sin θj
cos θr
Rr =
− sin θr
sin θr
cos θr
j
xj = [xj, yj, θj, ρj] T
[xj, yj] T + 1/ρj mj,
hj = atan2(h y
j , hx j ),
h xy
j = [hx j , h y
j ]T j xr =
[xr, yr, θr] T
h xy
j =
x hj h y
j
= Rr
xj
yj
+ 1/ρj mj −
xr
yr
.
.
ρi > 0
h xy
j =
x hj h y
xi xr
= Rr ρi − + mi .
j
yi yr

ρi = 0
xr1 xr2
xr1 z1j, z2j j xr1
xr2 α α = z1j − z2j dp dt xr1 xr2
z1j
dt dj xr1 j
dj = dp/ tan α.
(0/0) dp = 0 α = 0 + kπ
k ∈ Z
dt
α = 0 + kπ k ∈ Z dp
xr1 xr2
xr1 z1j, z2j j xr1
xr2 θr2
θr2 = z1i − z2i.

z1i − z2i dp < ∞
dmin

j
xr zj
ρ0 xj
dmin
ρmin = 1
dmin
, ρ0 = ρmin
2 , σρ = ρmin
4 ,

ρmin ρ0
[0, ρmin] σρ ρ0
95% ρ [ρ0 − 2σρ, ρ0 + 2σρ] = [0, ρmin]
xj = gj(xr, zj, ρ0) = [xr, yr, θr + zj, ρ0] T .
z1j z2j j
xr1 xr2 xj
z2j ρ0
xj = g(xr1, xr2, z1j, z2j) = [xr2, yr2, θr2 + z2i, ρ0] T , ρ0 =
ci si i ∈ {1, . . . , 2}
s2 c1 − c2 s1
c1 (yr1 − yr2) − s1 (xr1 − xr2) ,
ci = cos(θi + zij), si = sin(θi + zij).
z1j, z2j j xr1, xr2
xj
xj = g x j (xr1, xr2, z1j, z2j) = xr1 s1 c2 − xr2 s2 c1 − (yr1 − yr2) c1 c2
,
yj = g y
s1 c2 − s2 c1
j (xr1, xr2, z1j, z2j) = yr2 s1 c2 − yr1 s2 c1 + (xr1 − xr2) s1 s2
,
s1 c2 − s2 c1
ci si i ∈ {1, . . . , 2}

xri i = 1, 2 zij
j
αij
αij = θri + zij,
i = 1, 2 Sαij αij
(α1j − α2j) 2
Sα1j + Sα2j
> χ 2 0.99,1d.o.f.
xri i = 1, 2 j
αij Sαij
θt
xr1 xr2
yr2 − yr1
θt = arctan
,
xr2 − xr1
Sθt
(α1j − α2j) 2
Sα1j + Sα2j
≤ χ 2 0.99,1d.o.f,
(θt − αij) 2
Sθt + Sαij
≤ χ 2 0.99,1d.o.f,
j = 1, 2

90 ◦
st
st
st
xr 0.01 st 0.03 st 0.03 st
yr 0.01 st 0.03 st 0.03 st
θr 2 ◦ 2.5 ◦ 2.5 ◦
0.125 0.250 0.5 1

•
(xr, yr, θr)
(a − a) 2
Pa
≤ χ 2 0.99,1,
a xr yr θr a Pa
xr yr θr
•
j
xj yj
|a − a|
+
Pa χ 2 0.99,1
≤ 1.5
a a Pa
xj yj
•
xr yr θr
•
•
•
dmin = 0.5m

% final divergence
number of steps to initialize features
100
80
60
40
20
0
4
3.5
3
2.5
2
1.5
1
Final divergence
xy−d
xy−f
xy−l
0.1250.25 0.5
step size
1
% consistent features
100
80
60
40
20
0
Map consistency
xy−d
xy−f
xy−l
0.1250.25 0.5
step size
1
Feature initialization step
xy−d
xy−f
xy−l
0.1250.25 0.5
step size
1
% features used per step
100
80
60
40
20
0
Feature usage
xy−d
xy−f
xy−l
0.1250.25 0.5
step size
1
% steps divergence
% consistent features per step
100
80
60
40
20
0
100
80
60
40
20
0
Trajectory divergence
xy−d
xy−f
xy−l
0.1250.25 0.5
step size
1
Map consistency per step
xy−d
xy−f
xy−l
0.1250.25 0.5
step size
1
x

% final divergence
number of steps to initialize features
100
80
60
40
20
0
4
3.5
3
2.5
2
1.5
1
Final divergence
id−d
id−f
id−l
0.1250.25 0.5
step size
1
% consistent features
100
80
60
40
20
0
Map consistency
id−d
id−f
id−l
0.1250.25 0.5
step size
1
Feature initialization step
id−d
id−f
id−l
0.1250.25 0.5
step size
1
% features used per step
100
80
60
40
20
0
Feature usage
0.1250.25 0.5
step size
1
id−d
id−f
id−l
% steps divergence
% consistent features per step
100
80
60
40
20
0
100
80
60
40
20
0
Trajectory divergence
id−d
id−f
id−l
0.1250.25 0.5
step size
1
Map consistency per step
id−d
id−f
id−l
0.1250.25 0.5
step size
1
x
dmin = 0.5m
0.125

14
12
10
8
6
4
2
0
−2
−4
FINAL MAP: INVERSE−DEPTH Delayed (two observations)
F36
F37
F28
F42
F43
F27
F38
F13
F44
F7
F8 F9
F18
F10
F11
F39
F12
F1 F2
F26
F34
F33
F32 F31
F40
F35
F41
F3 F4
F29
F30
F22
F20 F23
F14
F5 F6
−5 0 5 10 15
F25
F24
F21
F19
F17
F16
F15
14
12
10
8
6
4
2
0
−2
−4
FINAL MAP: INVERSE−DEPTH Delayed Feature not aligned
F28 F27 F26 F25 F24 F23
F34
F17
F33
F40
F32 F31 F30 F29
F22
F35
F36
F37
F38
F41
F42
F43
F44
F7 F8 F9 F10
F18
F11
F39
F12
F1 F2 F3 F4 F5 F6
−5 0 5 10 15
F21
F20
F19
F16
F15
F14
F13
14
12
10
8
6
4
2
0
−2
−4
FINAL MAP: INVERSE−DEPTH Delayed (Finite depth)
F36
F37
F42
F43
F38
F44
F8
F7
F39
F1 F2
F3
−5 0 5 10 15
F34
F35
F28
F27
F33
F40
F41
F32
F26
F31
F25
F4
F24
F17
F29
F30
F22
F16
F5
F21
F20
F19
F23
F15
F14
F13
F18
F9 F10
F11
F12
% final divergence
number of steps to initialize features
100
4
3
2
1
0
80
60
40
20
0
Final divergence
id−l
id−u
xy−f
0.1250.25 0.5
step size
1
% consistent features
100
80
60
40
20
0
Map consistency
id−l
id−u
xy−f
0.1250.25 0.5
step size
1
Feature initialization step
id−l
id−u
xy−f
0.1250.25 0.5
step size
1
% features used per step
100
80
60
40
20
0
Feature usage
id−l
id−u
xy−f
0.1250.25 0.5
step size
1
% steps divergence
% consistent features per step
100
80
60
40
20
0
100
80
60
40
20
0
0.1250.25 0.5
step size
1
F6
Trajectory divergence
id−l
id−u
xy−f
Map consistency per step
id−l
id−u
xy−f
0.1250.25 0.5
step size
1
x
dmin = 0.5m

14
12
10
8
6
4
2
0
−2
−4
FINAL MAP: INVERSE−DEPTH Delayed (two observations)
F36
F37
F28
F42
F43
F27
F38
F13
F44
F7
F8 F9
F18
F10
F11
F39
F12
F1 F2
F26
F34
F33
F32 F31
F40
F35
F41
F3 F4
F29
F30
F22
F20 F23
F14
F5 F6
−5 0 5 10 15
F25
F24
F21
F19
F17
F16
F15
14
12
10
8
6
4
2
0
−2
−4
FINAL MAP: INVERSE−DEPTH Delayed Feature not aligned
F28 F27 F26 F25 F24 F23
F34
F17
F33
F40
F32 F31 F30 F29
F22
F35
F36
F37
F38
F41
F42
F43
F44
F7 F8 F9 F10
F18
F11
F39
F12
F1 F2 F3 F4 F5 F6
−5 0 5 10 15
F21
F20
F19
F16
F15
F14
F13
14
12
10
8
6
4
2
0
−2
−4
FINAL MAP: INVERSE−DEPTH Delayed (Finite depth)
F36
F37
F42
F43
F38
F44
F8
F7
F39
F1 F2
F3
−5 0 5 10 15
F34
F35
F28
F27
F33
F40
F41
F32
F26
F31
F25
F4
F24
F17
F29
F30
F22
F16
F5
F21
F20
F19
F23
F15
F14
F13
F18
F9 F10
F11
F12
% final divergence
number of steps to initialize features
100
4
3
2
1
0
80
60
40
20
0
Final divergence
id−l
id−u
xy−f
0.1250.25 0.5
step size
1
% consistent features
100
80
60
40
20
0
Map consistency
id−l
id−u
xy−f
0.1250.25 0.5
step size
1
Feature initialization step
id−l
id−u
xy−f
0.1250.25 0.5
step size
1
% features used per step
100
80
60
40
20
0
Feature usage
id−l
id−u
xy−f
0.1250.25 0.5
step size
1
% steps divergence
% consistent features per step
100
80
60
40
20
0
100
80
60
40
20
0
0.1250.25 0.5
step size
1
F6
Trajectory divergence
id−l
id−u
xy−f
Map consistency per step
id−l
id−u
xy−f
0.1250.25 0.5
step size
1
x
dmin = 0.5m

14
12
10
8
6
4
2
0
−2
−4
FINAL MAP: INVERSE−DEPTH Delayed (two observations)
F36
F37
F28
F42
F43
F27
F38
F13
F44
F7
F8 F9
F18
F10
F11
F39
F12
F1 F2
F26
F34
F33
F32 F31
F40
F35
F41
F3 F4
F29
F30
F22
F20 F23
F14
F5 F6
−5 0 5 10 15
F25
F24
F21
F19
F17
F16
F15
14
12
10
8
6
4
2
0
−2
−4
FINAL MAP: INVERSE−DEPTH Delayed Feature not aligned
F28 F27 F26 F25 F24 F23
F34
F17
F33
F40
F32 F31 F30 F29
F22
F35
F36
F37
F38
F41
F42
F43
F44
F7 F8 F9 F10
F18
F11
F39
F12
F1 F2 F3 F4 F5 F6
−5 0 5 10 15
F21
F20
F19
F16
F15
F14
F13
14
12
10
8
6
4
2
0
−2
−4
FINAL MAP: INVERSE−DEPTH Delayed (Finite depth)
F36
F37
F42
F43
F38
F44
F8
F7
F39
F1 F2
F3
−5 0 5 10 15
F34
F35
F28
F27
F33
F40
F41
F32
F26
F31
F25
F4
F24
F17
F29
F30
F22
F16
F5
F21
F20
F19
F23
F15
F14
F13
F18
F9 F10
F11
F12
% final divergence
number of steps to initialize features
100
4
3
2
1
0
80
60
40
20
0
Final divergence
id−l
id−u
xy−f
0.1250.25 0.5
step size
1
% consistent features
100
80
60
40
20
0
Map consistency
id−l
id−u
xy−f
0.1250.25 0.5
step size
1
Feature initialization step
id−l
id−u
xy−f
0.1250.25 0.5
step size
1
% features used per step
100
80
60
40
20
0
Feature usage
id−l
id−u
xy−f
0.1250.25 0.5
step size
1
% steps divergence
% consistent features per step
100
80
60
40
20
0
100
80
60
40
20
0
0.1250.25 0.5
step size
1
F6
Trajectory divergence
id−l
id−u
xy−f
Map consistency per step
id−l
id−u
xy−f
0.1250.25 0.5
step size
1
x
dmin = 0.5m

14
12
10
8
6
4
2
0
−2
−4
FINAL MAP: INVERSE−DEPTH Delayed (two observations)
F36
F37
F28
F42
F43
F27
F38
F13
F44
F7
F8 F9
F18
F10
F11
F39
F12
F1 F2
F26
F34
F33
F32 F31
F40
F35
F41
F3 F4
F29
F30
F22
F20 F23
F14
F5 F6
−5 0 5 10 15
F25
F24
F21
F19
F17
F16
F15
14
12
10
8
6
4
2
0
−2
−4
FINAL MAP: INVERSE−DEPTH Delayed Feature not aligned
F28 F27 F26 F25 F24 F23
F34
F17
F33
F40
F32 F31 F30 F29
F22
F35
F36
F37
F38
F41
F42
F43
F44
F7 F8 F9 F10
F18
F11
F39
F12
F1 F2 F3 F4 F5 F6
−5 0 5 10 15
F21
F20
F19
F16
F15
F14
F13
14
12
10
8
6
4
2
0
−2
−4
FINAL MAP: INVERSE−DEPTH Delayed (Finite depth)
F36
F37
F42
F43
F38
F44
F8
F7
F39
F1 F2
F3
−5 0 5 10 15
F34
F35
F28
F27
F33
F40
F41
F32
F26
F31
F25
F4
F24
F17
F29
F30
F22
F16
F5
F21
F20
F19
F23
F15
F14
F13
F18
F9 F10
F11
F12
% final divergence
number of steps to initialize features
100
4
3
2
1
0
80
60
40
20
0
Final divergence
id−l
id−u
xy−f
0.1250.25 0.5
step size
1
% consistent features
100
80
60
40
20
0
Map consistency
id−l
id−u
xy−f
0.1250.25 0.5
step size
1
Feature initialization step
id−l
id−u
xy−f
0.1250.25 0.5
step size
1
% features used per step
100
80
60
40
20
0
Feature usage
id−l
id−u
xy−f
0.1250.25 0.5
step size
1
% steps divergence
% consistent features per step
100
80
60
40
20
0
100
80
60
40
20
0
0.1250.25 0.5
step size
1
F6
Trajectory divergence
id−l
id−u
xy−f
Map consistency per step
id−l
id−u
xy−f
0.1250.25 0.5
step size
1
x
dmin = 0.5m

i, i ′ , j, j ′
r, r ′ , s, s ′
G
t t ∈ N
f i r
Ar,s
Aij
[Aij]r,s
Mi
MG
r i
(r, s) A
(i, j) A
(r, s) Aij
i
n ∈ N G =
(V, E) V = {1, . . . , n}
i j (i, j) ∈ E
Ni i
Ni = {j | (i, j) ∈ E, j = i}.

i mi ∈ N
i
Mi = + mi
ˆxi ∈ R Mi Σi ∈ R Mi×Mi
= 3
(x, y) θ = 2 = 3
xi ∈ R Mi
mi
ˆxi = xi + vi,
vi Σi
m ∈ N
i ∈ {1, . . . , n} mi ≤ m x ∈ R MG
n m MG = n +m
Hi ∈ {0, 1} Mi×MG x
i 0 1
1 xi = Hix
ˆxi = Hix + vi,
vi Σi
T ˆx 1 . . . ˆx T T T
n = H1 . . . H T T T
n x + v1 . . . v T T n .
vi
E[ vT 1 . . . vT T
T
n v1 . . . vT
n ] = diag (Σ1, . . . , Σn)
i Ii ∈
RMG×MG ii ∈ RMG Ii = H T i Σ −1
i Hi, ii = H T i Σ −1
i ˆxi.
n
IG =
n
Ii, iG =
i=1
n
ii,
i=1
ˆxG = (IG) −1 iG, ΣG = (IG) −1 .

i Îi G (t) ∈ RMG×MG
îi G (t) ∈ RMG
Î i G(0) = H T i Σ −1
i Hi, î i G(0) = H T i Σ −1
i ˆxi,
t ≥ 0
Î i G(t + 1) =
n
j=1
Wi,j Îj
G (t), îi G(t + 1) =
n
j=1
Wi,jîj G (t),
ˆxi Σi i Wi,j G
G
lim Î
t→∞
i G(t) = IG/n, lim î
t→∞
i G(t) = iG/n,
IG iG
ˆx i G (t) ∈ RMG
ˆ Σi G (t) ∈ RMG×MG i ∈ V t ≥ 0
ˆx i −1 −1 i i
G(t) = ÎG(t) îG(t), Σˆ i i
G(t) = ÎG(t) ,
lim
t→∞ ˆxi G(t) = ˆxG, lim ˆΣ
t→∞
i G(t) = nΣG,
ˆxG ΣG ˆx i G (t)
ˆx i G (t) i ∈ {1, . . . , n}
x E [ˆx i G (t)] = x t ≥ 0
t ≥ 0
Q i G (t) Qi G (t) ˆ Σ i G (t)
Wi,j = 0 j /∈ Ni
G(t) = (V, E(t))
i t → ∞

i Hi
Hi
i
i Li
i ∈ V
Li = {L i 1, . . . , L i mi } mi f i r f j s
i j L i r = (i⋆, r⋆) L j s = (i ′ ⋆, r ′ ⋆)
i⋆ = i ′ ⋆ and r⋆ = r ′ ⋆.
Lj Hi
Li
Li Li L i r = (i⋆, r⋆)
i⋆ r⋆ i
Li Li
|Li| Li
Li Lj Li ⊆ Lj Li Lj Γ
Li Lj Lij
Li Lj
Γ
(i) Γ(Li, Lj) = Γ(Lj, Li);
(ii) Γ(Li, Li) = Li;
(iii) Γ(Li, Γ(Lj, Lk)) = Γ(Γ(Li, Lj), Lk);
(iv) Li, Lj ⊆ Γ(Li, Lj);

Li, Lj, Lj ′
H
Li Lj Li ⊆ Lj H(Li, Lj) Hi j ∈ {0, 1} |Li| ×|Lj|
Li
Lj H Li ⊆ Lj ⊆ Lj ′ H(Li, Lj) = Hi j
H(Lj, Lj ′) = Hj
j ′ Hi jH j
j ′ = Hi j ′ = H(Li, Lj ′) H
H(Li, Li) = I
Lj j = 1, . . . , n LG
m L1, . . . , Ln
LG = Γ(Γ(. . . Γ(L1, L2), . . . ), Ln).
LG Γ
Li Li i ∈ {1, . . . , n}
Hi H(Li, LG)
LG Hi
LG i
Li
LG i
t
Hi

Lchild i Li
Lchild j j i
Lchild i = Li
i Lchild j1 , . . . , Lchild j
|childi |
childi Li
L child
i
= Γ(Γ(Γ(. . . Γ(L child
j1 , L child
j2 ), . . . ), L child
j |childi | ), Li) i

Lchild i
LG = Lchild root
LG
LG
LG i Hi Li
LG Hi = H(Li, LG) Hi
Îi G (t), îiG (t)
i Li
LG i ∈ {1, . . . , n}
Li(t) t
i i G (t) ∈ RMi(t) I i G (t) ∈ R Mi(t)×Mi(t)
Mi(t) t Mi(t) = |Li(t)| t = 0

i
Li(0) = Li, I i G(0) = Σ −1
i , i i G(0) = Σ −1
i ˆxi.
t Li(t + 1)
Lj(t) j ∈ Ni ∪ {i}
Li(t + 1) = Γ(Γ(. . . Γ(Lj1(t), Lj2(t)), . . . ), LjN (t)),
i +1
{j1, . . . , jNi+1} = Ni ∪ {i} Ii G (t) iiG (t)
I i G(t + 1) =
n
j=1
H j,t
i,t+1 = H(Lj(t), Li(t + 1))
Wi,j(H j,t
i,t+1 )T I j
G (t)Hj,t
i,t+1 , ii G(t + 1) =
n
j=1
Wi,j(H j,t
i,t+1 )T i j
G (t),
t ≥ 0 i ∈ {1, . . . , n}
Î i G(t) = (H i,t
G )T I i G(t)H i,t
G , îi G(t) = (H i,t
G )T i i G(t),
H i,t
G = H(Li(t), LG) diam(G)
t ≥ 0 i ∈ {1, . . . , n}
Î i G(diag(G) + t) = I i G(diag(G) + t), î i G(diag(G) + t) = i i G(diag(G) + t).
Ii G (t)
ii G (t) t = 0
Î i G(0) = H T i Σ −1
i Hi = (H i,0
G )T I i G(0)H i,0
,
i ∈ {1, . . . , n} t i ∈ {1, . . . , n}
Î i G(t + 1) =
n
j=1
= (H i,t+1
G ) T
Wi,jÎj G (t) =
n
j=1
n
j=1
Wi,j(H j,t
G )T I j
G (t)Hj,t G
(H j,t
i,t+1 )T I j
G (t)Hj,t
i,t+1
H i,t+1
G
G
= (Hi,t+1
G
t ≥ 0 i, j ∈ {1, . . . , n} H j,t
G
) T I i G(t + 1)H i,t+1
G ,
= Hj,t
i,t+1 Hi,t+1
G
n
j=1 Wi,j = 1
diam(G) i ∈ {1, . . . , n}
Lj
j = 1, . . . , n t ≥ 0 Li(diam(G)+t) = LG LG
= H(LG, LG) = I
H i,diam(G)+t
G

Hi
Ii G (t) t ≥ 0 i ∈ {1, . . . , n}
Îi G (t)
i
i
t = diam(G)
Ii G (t) t ≥ 0 i ∈ {1, . . . , n}
xi G (t) ∈ RMi(t) i ΣG (t) ∈ RMi(t)×Mi(t) i
x i G(t) = I i G(t) −1 i i G(t), Σ i G(t) = I i G(t) −1 .
i
t → ∞
t
ˆx i G (t)
E ˆx i G(t) = x,
t ≥ 0 i ∈ V Îi G (t)
Qi G (t) ∈ RMG×MG ˆx i G (t)
ˆ Σi G (t)
ˆx i
G(t) − x ˆx i G(t) − x T
Q i G(t) = E
=
−1 i
ÎG(t) n
j=1
([W t ]i,j) 2 H T j Σ −1
j Hj
Î −1 i
G(t) ,
t ≥ 0 i ∈ V Îi G (t) 0 ≤ [W t ]i,j ≤ 1
i, j ∈ V t ≥ 0 ([W t ]i,j) 2 ≤ [W t ]i,j
Q i G(t)
−1 i
ÎG(t) = ˆ Σ i G(t),

ˆ Σi G (t)−Qi G (t)
Qi G (t)
ˆ Σi G (t) ˆ Σi G (t) − QiG (t)
[ ˆ Σi G (t) − QiG (t)]r,r ≥ 0
[ ˆ Σi G (t)]r,r ≥ [Qi G (t)]r,r
ˆΣ i G (t)
Mmax
Mmax = max
i∈{1,...,n} Mi,
dmax
dmax = max
i∈{1,...,n} |Ni|.
G(t) = (V, E(t)) dmax
dmax = max |Ni(t)|.
i∈{1,...,n},t≥0
O(M3 max)
O(dmaxM2 G )
i Ii G (t)
O(dmaxM2 G ) Ii G (t)
(t)
Ii G
O(nM2 max) O(n + m)
O(dmaxnM2 max) O(dmax(n + m))
O(dmaxm)
I i G (t) O(nM2 max) O(n + m)

|λ2(W )|
γ = |λ2(W )|
[Ii G (t)]r,s [ii G (t)]r
|[I i G(t)]r,s − [IG]r,s| ≤ (γ) t√
n max [Ij ,
|[i i G(t)]r − [iG]r| ≤ (γ) t√ n max
j
j
G (0)]r,s − [IG]r,s
[ij G (0)]r
− [iG]r
,
i ∈ {1, . . . , n} r, s ∈ {1, . . . , Mi(t)} t ≥ 0
G(t) = (V, E(t))
τ
{G(t0 + 1), . . . G(t0 + τ)}
τ t0 τ
δ < 1
δ = max {|λ2(W)| | W }
W∈Wτ
Wτ τ W (t)
[Ii G (t)]r,s
[ii G (t)]r
τ
|[I i t
G(t)]r,s − [IG]r,s| ≤ (δ)
|[i i t
G(t)]r − [iG]r| ≤ (δ)
τ √
n max [Ij G
j
(0)]r,s
− [IG]r,s
,
[ij G (0)]r
− [iG]r
,
τ √ n max
j
t
t
τ τ
W = (1/n)11T |λ2(W )| = 0
|[Ii G (t)]r,s − [IG]r,s| ≤ 0 |[ii G (t)]r − [iG]r| ≤ 0 t ≥ 1 i ∈ {1, . . . , n}
r, s ∈ {1, . . . , Mi(t)}

10
8
6
4
2
0
−2
−4
−6
−8
−10
10
8
6
4
2
0
−2
−4
−6
−8
−10
10
8
6
4
2
0
−2
−4
−6
−8
−10
−10 −5 0 5 10
10
8
6
4
2
0
−2
−4
−6
−8
−10
−10 −5 0 5 10
10
8
6
4
2
0
−2
−4
−6
−8
−10
−10 −5 0 5 10
−10 −5 0 5 10
10
8
6
4
2
0
−2
−4
−6
−8
−10
−10 −5 0 5 10
10
8
6
4
2
0
−2
−4
−6
−8
−10
−10 −5 0 5 10
−10 −5 0 5 10
10
8
6
4
2
0
−2
−4
−6
−8
−10
−10 −5 0 5 10
10
8
6
4
2
0
−2
−4
−6
−8
−10
−10 −5 0 5 10

9
t = 1 t = 10
t → ∞
10
8
6
4
2
0
−2
−4
−6
−8
−10
Robot 7. t= 1
−10 −5 0 5 10
10
8
6
4
2
0
−2
−4
−6
−8
−10
Robot 7. t= 10
−10 −5 0 5 10
10
8
6
4
2
0
−2
−4
−6
−8
−10
−10 −5 0 5 10
t = 1 t = 10 t → ∞
t = 1
t = 10
t → ∞
7
20 × 20 m
70 30 m
σx, σy = 0.4 cm σθ = 1
6 m
0.5
xi G (t), ΣiG (t)

10
8
6
4
2
0
−2
−4
−6
−8
−10
−10 −8 −6 −4 −2 0 2 4 6 8
20 × 20 m
1.5
1
0.5
0
−0.5
−1
−1.5
−2
R5
R1
R4
R3
R2
R7
R6
5 5.5 6 6.5 7 7.5 8 8.5 9
G
3 m
10
8
6
4
2
0
−2
−4
−6
−8
−10
−10 −8 −6 −4 −2 0 2 4 6 8

10
8
6
4
2
0
−2
−4
−6
−8
−10
−10 −8 −6 −4 −2 0 2 4 6 8
xi G (t), ΣiG (t) t = 5

10
8
6
4
2
0
−2
−4
−6
−8
−10
−10 −8 −6 −4 −2 0 2 4 6 8
xi G (t), ΣiG (t) t = 5

10
8
6
4
2
0
−2
−4
−6
−8
−10
−10 −8 −6 −4 −2 0 2 4 6 8
xi G (t), ΣiG (t) t = 5

n i, j r, s
G k ∈ N
t ∈ N I n × n
0 n × n
n1 × n2 1 ∈ Rn
W [W ]i,j (i, j) λi(W )
i− vi(W ) λ(W )
n ∈ N
k i mk i ∈ N
i ∈ {1, . . . , n} k
ˆx k i ∈ RMk i Σk i ∈ RMk i ×Mk i Mk i = + mk i
xk i ∈ RMk i mk i
ˆx k i = x k i + v k i ,
v k i Σ k i
k n
n
k m ∈ N
x ∈ R MG n
m MG = n + m i ∈ {1, . . . , n} k
m k i ≤ m H k i ∈ {0, 1} Mk i ×MG
x x k i x k i = H k i x i
x
ˆx k i = H k i x + v k i ,
= 3 (x, y) θ = 2
= 3

obot
feat feat.
robot feat. feat
n
I k G ik G Ik i i k i i ∈
{1, . . . , n}
i
j = i
v k i , v k′
j i = j
k, k ′ ∈ N
i k
k ′ < k v k i , v k′
i I k i ∈ R MG×MG
i k i ∈ R MG i k
I k i = (H k i ) T (Σ k i ) −1 H k i , i k i = (H k i ) T (Σ k i ) −1 ˆx k i ,
i ∈ {1, . . . , n} Ik G ikG k
I k G =
n
I k i , i k G =
i=1
n
i k i .
k
ˆx k G = (I k G) −1 i k G, Σ k G = (I k G) −1 .
(r, s) r
I k i i k i i
R ⊆ {1, . . . , n}
(r, s) r [I k G ]r,s
[i k G ]r [I k i ]r,s [i k i ]r i ∈ R
i=1
[I k G]r,s =
[I k i ]r,s, [i k G]r =
[i k i ]r.
i∈R
i∈R

i [I k j ]r,s [i k j ]r
j ∈ R i i
j = i (r, s) r [I k j ]r,s = 0
[i k j ]r = 0 i
[I k G]r,s = [I k i ]r,s, [i k G]r = [i k i ]r,
I k i i k i j i = j
n ∈ N
k = 1, 2, . . .
Gk = (V, E k )
k V = {1, . . . , n}
i j (i, j) ∈ E k
N k
i i k
N k
i = {j | (i, j) ∈ E k , j = i}.
i ∈ V
N k
i
i ∈ {1, . . . , n} ui ∈ R
xi(t), wi(t) ∈ R
˙xi(t) = −γ xi(t) −
[WP ]i,j [xi(t) − xj(t)] +
[WI]j,i [wi(t) − wj(t)] + γ ui,
j=i
˙wi(t) = −
[WI]i,j [xi(t) − xj(t)] ,
j=i
WP WI
[WP ]i,j = [WP ]j,i = [WI]i,j =
[WI]j,i = 0 i j j /∈ Ni γ > 0
j=i

n u ∈ R n =
(u1, . . . , un) T x ∈ R n = (x1, . . . , xn) T w ∈ R n = (w1, . . . , wn) T
˙x(t)
=
˙w(t) −LI
−γI − LP L T I
0
x(t)
w(t)
γI
+
0
u,
LP LI WP
WI
LP = diag(WP 1) − WP , LI = diag(WI1) − WI.
ex(t) ∈ R n
ex(t) = x(t) − 11T
n u,
Π Π = I − 11T i ∈ {1, . . . , n}
n
γ LP LI
rank(LI) = n − 1,
ε ∈ R is such that Π(LP + L T P )Π 2εΠ,
γ > 0 is chosen such that γ + ε > 0,
u x(0), w(0) ex(t)
0 t → ∞
xi(t) i ∈ V
i xi(t), wi(t)
[WP ]i, j = [WI]i, j = 0 j /∈ Ni
WM ∈ R n×n
WM [WM]i,j = [WM]j,i = 0 i j
WM
WM = W T M WM1 = 1 1 T WM = 1 T 1
λ(WM) ∈ (−1, 1)
LM = diag(WM1) − WM = I − WM,
0
λ(LM) ∈ (0, 2) LM
ε = 0 Π(LM + L T M )Π = 2LM LM
LM γ > 0
WP = WI = WM
LW ∈ R n LP = LI = LW L T W = LW W

LW = diag(W 1) − W
W = WM
λi(A) i A vi(A)
λi LW r = 1 √ 1 LW
n
0 [r S2 . . . Sn] = [r S] LW
k i ∈ V
ui ∈ R xG ∈ R ui
xavg ∈ R
xG =
n
i=1
ui, xavg = 1
n
n
i=1
ui = 1
n xG.
i ∈ V xi(t) ∈ R xavg
Ni
LW ∈ R n LP = LI = LW
L T W = LW W LW = diag(W 1) − W
h
γ
W = WM
LW = LM
r ∈ R n LW λ1(LW ) = 0
r = 1/ √ n.
S2, . . . , Sn n − 1 LW [r S2 . . . Sn] = [r S]
LW
[r S] T LW [r S] = diag (λ1(LW ), . . . , λn(LW )) ,
λ1(LW ) ≤, . . . , ≤ λn(LW )
LW
λ2(LW ), . . . , λn(LW ) L (−1)
W
L (−1)
W = (I − rrT ) LW + rr T −1 (I − rr T ).

i ∈ V bi = b(λi(LW ))
bi = b(λi(LW )) = (γ + λi(LW )) 2 − (2 λi(LW )) 2 .
LW h > 0
x(t + 1) x(t) hγI
= A + u, with
w(t + 1) w(t) 0
−γI − µLW LW
A = I2n×2n + h
−LW 0
.
µ > 0
µ = 1
h γ
h γ
xi(t) → xavg t → ∞
x(t) → 1xavg x∗ w∗ ∈ R n
r L (−1)
W
x∗ = rr T u = 1xavg, w∗ = rr T w(0) − γL (−1)
W u,
A
LW
λi(LW ) LW
λi(A) λn+i(A) A
λi(A) = 1 − h (γ + λi(LW ) + bi) /2, λn+i(A) = 1 − h (γ + λi(LW ) − bi) /2,
i ∈ V bi λ1(LW ) = 0 λ1(A) =
1 − hγ λn+1(A) = 1
−γI − LW LW
Z =
, A = I2n×2n + hZ
−LW 0
A Z i ∈ {1, . . . , 2n}
λi(A) = 1 + hλi(Z), vi(A) = vi(Z).

Y = P T ZP
r S2 . . . Sn 0
P =
0 r S2 . . . Sn
,
[r S2 . . . Sn] LW
Z Y
λi(Z) = λi(Y ), vi(Z) = P vi(Y ).
Y
T −γI − [r S] LW [r S] [r S]
Y =
T LW [r S]
−[r S] T LW [r S] 0
diag(−γ − λ1, . . . , −γ − λn) diag(λ1, . . . , λn)
Y =
−diag(λ1, . . . , λn) 0
,
λi = λi(LW ) (LW ) Y vi(Y ) =
λi(Y )vi(Y ) Y i ∈ V
λi(Y ) = − γ + λi + bi
2
, λn+i(Y ) = − γ + λi − bi
.
2
vi(Y ) vn+i(Y ) i−
(n + i)−
[vi(Y )]i = 1, [vi(Y )]n+i = −λi(LW )/λi(Y ),
[vn+i(Y )]i = 1, [vn+i(Y )]n+i = −λi(LW )/λn+i(Y ).
A
h γ
,
γ ≥ 3/2 λn(LW ),
hγ < 3/2,
λn(LW ) LW A
λn+1(A) = 1
A LW
λi(LW ) LW
γ h λi(A) ±Im [bi]
0 A

λi > 0 i ∈
{2, . . . , n} bi γ ≤ bi ≤ 2/ √ 3γ
λi(Y ) λn+i(Y ) −4/3 γ ≤ λi(Y ) < −γ
−1/3 γ ≤ λn+i(Y ) < 0 i ∈ {2, . . . , n}
λ1(A) = 1 − hγ 1 h > 0
γ > 0 −1/2
1 i ∈ {2, . . . n} λi(A) = 1 + hλi(Y ) < 1 λn+i(A) = 1 + hλn+i(Y ) <
1 h > 0 λi(Y ) < 0, λn+i(Y ) < 0 λi(A) ≥ 1 − 4/3hγ > −1
λn+i(A) ≥ 1 − 1/3hγ > 1/2 λi(A) λn+i(A)
1 λn+1(A) = 1
γ ≥ 3 h < 3/(2γ)
LM
λ1(LM) = 0 0 < λi(LM) < 2 i ∈ {2, . . . , n}
λeff(A) A
λn+i(A) ≥ 1/2
λn+i(A) i ∈ {2, . . . , n} λn+2(A) λi(A)
i λi(A) ≥ 0 λn+i(A)
i λi(A) < 0
λn(A) λeff(A) = max{λn+2(A), −λn(A)}
LW
G
h > 0 γ > 0
u ∈ R n x(0) ∈ R n w(0) ∈ R n
x(t) ∈ R n w(t) ∈ R n
lim
t→∞ x(t) = x∗, lim w(t) = w∗,
t→∞
t → ∞ x∗ w∗ β = 2 √ 10/3
exw(t) = x(t) T , w(t) T T
T − x∗ , wT T
∗ t
exw(t) 2 ≤ βλ t eff(A) exw(0) 2 .
h γ
|λn+1(A)| = 1 |λi(A)| < 1
λeff(A) λt eff (A) 0 t → ∞
exw(t)2
t

exw(t)
exw(t + 1) = A exw(t), exw(t) = A t exw(0).
C = P T AP = I + hY,
P Y ezy
ezy(t) = P T exw(t), exw(t) = P ezy(t),
exw(t) 2 = ezy(t) 2
ezy(t + 1) = C ezy(t), ezy(t) = C t ezy(0),
ezy(0) = (z(0) T , y(0) T ) T − (z T ∗ , y T ∗ ) T
T r u
z∗ =
0n−1
, y∗ =
r T w(0)
−γ(S T LW S) −1 S T u
,
r T w(0) = [y(0)]1 [ezy(0)]n+1
C
ezy(0) =
2n
i=1
i ∈ {1, . . . , 2n}
ai vi(C),
λi(C) = λi(A), and vi(C) = vi(Y ),
λi(A) vi(C)
ai an+i [ezy(0)]i [ezy(0)]n+i
ezy(0) ai an+i i ∈ V
[ezy(0)]i = ai + an+i,
[ezy(0)]n+i = −aiλi(LW )/λi(Y ) − an+iλi(LW )/λn+i(Y ),
i = 1 [ezy(0)]1 = a1 + an+1 [ezy(0)]n+1 = 0
a1 a1 = [ezy(0)]1

an+1 = 0
ai an+i i ∈ {2, . . . , n}
ai = − λi(LW
)
[ezy(0)]n+i +
bi
λi(Y )
λi(LW ) [ezy(0)]i
,
an+i = λi(LW
)
[ezy(0)]n+i + λn+i(Y )
λi(LW ) [ezy(0)]i
,
bi
bi
t
ezy(t) = C t ezy(0) =
i ∈ {2, . . . , n}
2n
ai λi(C) t vi(C),
[ezy(t)]1 = (λ1(A)) t [ezy(0)]1, [ezy(t)]n+1 = 0,
i=1
[ezy(t)]i = ci,n+i [ezy(0)]n+i + ci,i [ezy(0)]i,
[ezy(t)]n+i = −ci,n+i [ezy(0)]i + cn+i,n+i [ezy(0)]n+i,
ci,n+i = λi(LW ) (λn+i(A)) t − (λi(A)) t /bi,
ci,i = λn+i(Y )(λn+i(A)) t − λi(Y )(λi(A)) t /bi,
cn+i,n+i = −λi(Y )(λn+i(A)) t + λn+i(Y )(λi(A)) t /bi.
ezy(t) 2
2 t
ezy(t) 2
2 =
+
+
n
i=2
n
i=1
([ezy(t)]i) 2 + ([ezy(t)]n+i) 2 = (λ1(A)) 2t ([ezy(0)]1) 2 +
(c 2 i,n+i + c 2 i,i)([ezy(0)]i) 2 +
n
i=2
(c 2 i,n+i + c 2 n+i,n+i)([ezy(0)]n+i) 2 +
n
2ci,n+i(ci,i − cn+i,n+i)[ezy(0)]i[ezy(0)]n+i,
i=2
2ci,n+i(ci,i − cn+i,n+i) = − 2λi(LW )(γ + λi(LW ))
b 2 i
(λn+i(A)) t − (λi(A)) t 2 .

k1a k2b |k1a−k2b| ≤ max{k1, k2} max{a, b}
(λn+i(A)) t (λi(A)) t ≥ 0 t 1/bi ≤
1/γ λi ≤ 2/3γ max{−λi(Y ), −λn+i(Y )} ≤ 4/3γ
n
|[ezy(0)]i[ezy(0)]n+i| ≤
i=2
c 2 i,n+i ≤ (2/3) 2 λ 2t
eff(A),
max{c 2 n+i,n+i, c 2 i,i} ≤ (4/3) 2 λ 2t
eff(A),
|2ci,n+i(ci,i − cn+i,n+i)| ≤ (20/3 2 )λ 2t
eff(A).
n
i=2
exw(t) 2 = ezy(t) 2
(max{|[ezy(0)]i|, |[ezy(0)]n+i|}) 2 ≤ ezy(0) 2
2 .
ezy(t) 2
2 ≤ 40/32 λ 2t
eff(A) ezy(0) 2
.
exw(t) 2 ≤ 2 √ 10/3 λ t eff(A) exw(0) 2 ,
λeff(A) = max{λn+2(A), −λn(A)}
λ2(LW ), λn(LW ) LW
λn(LW ) λ2(LW )
λn(A) λn+2(A)
h ∗ −λn(A) = λn+2(A)
h ∗ = 4/(2γ + λn(LW ) + λ2(LW ) + bn − b2).
i u k i
k = 1, . . . , K
1
n
n
i=1 uk i k
k = 1, . . . , K
k
L l
L − l(K − 1)
2

Consensus iterations
...
t=1
t=2
t=l
Input update steps
k=1 k=2 ... k=K
i ∈ {1, . . . , n} u k i
k = 1, . . . , K
l
L
L
l
l k ∗
ɛ k
k exw(lk ∗) / 2 k exw(0) ≤ ɛ l 2 ∗ k
l k ∗ ≥ (log (ɛ) − log (β)) / log (λeff(A)) .
k k + 1
k L k W
Gk = V, E k k [r Sk 2 . . . Sk n] = [r Sk ]
Lk W r
Lk W Ak Lk W
k uk = (uk 1, . . . , uk n)
xk (t) = (xk 1(t), . . . , xk n(t)) wk (t) = (wk 1(t), . . . , wk n(t))
k xk ∗ wk ∗ ∈ Rn
...

k
x k ∗ = rr T u k = 1x k avg, w k ∗ = rr T w k (0) − γ(L k W ) (−1) u k ,
r (L k W )(−1) λ⋆
λ⋆ = max
k∈{1,...,K} λeff(A k ).
h > 0 γ > 0
k
i
k = 1
x k i (0) = 0w k i (0) = 0 u k i ←
k = 1, . . . , K − 1
t = l
[x k i (t), w k i (t)] = consensusalg u k i , x k i (0), w k i (0)
x k i (0) = x k i (t) w k i (0) = w k i (t)
u k i ←
k = K
t = L − (K − 1)l
[x k i (t), w k i (t)] = consensusalg u k i , x k i (0), w k i (0)
(i) k = 1 (ii)
α σ
α = αk for k = 1, σ = max
k∈{1,...,K−1} σk,
with αk = r T u k 2 2 + γ 2 (L k W ) (−1) u k 2 2
1/2 ,
and σk = (r T (u k − u k+1 ) 2 2 + γ 2 (L k W ) (−1) u k − (L k+1
W )(−1) u k+1 2 2) 1/2 .
G k x k i (t) i x k avg
k

V
G k
k ∈ {1, . . . , K} x k (t) ∈ R n w k (t) ∈ R n
lim
t→∞ xk (t) = x k ∗, lim w
t→∞ k (t) = w k ∗,
t → ∞ x k ∗ w k ∗ e k xw(t) =
[(x k (t)) T , (w k (t)) T ] T −[(x k ∗) T , (w k ∗) T ] T k ∈ {1, . . . , K} t
t
e k xw(t) ∞ ≤ e k xw(t) 2 ≤ αfk(t) + σgk(t),
fk(t) = β k λ t+(k−1)l
⋆ , gk(t) = βλ t k−2
⋆ (βλ l ⋆) p ,
l
β = 2 √ 10/3 λ⋆ α σ
x k (t) ∈ R n w k (t) ∈ R n x k ∗ w k ∗
l
p=0
k
exw(l) ≤ βλ 2 l
k
⋆ exw(0) .
2
ek xw(l) k ek+1 xw (0)
k + 1
e k+1
xw (0) = e k xw(l) +
x k ∗
w k ∗
−
x k+1
∗
w k+1
∗
k
exw(t) ≤ β 2 k λ t+(k−1)l
1
⋆ exw(0)
2
+ βλ t k−2
⋆ (βλ l ⋆) p
k−p−1
x∗
p=0
w k−p−1
∗
− x k−p
∗
− w k−p
∗
.
2
,
k = 1
k = 1 t = 0 e1 xw(0) = [(−x1 ∗) T , (−w1 ∗) T ] T
e1 xw(0)2 α
T T T (a , b ) 2 = a2
2 2 + b2 2
k = 1, . . . , K − 1 k ((x∗ − xk+1 ∗ ) T , (wk ∗ − wk+1 ∗ ) T ) T σk
2
σ

k − 1 k
β k λ t+(k−1)l
⋆
α1 + βλ t ⋆
p=0
k−2
(βλ l ⋆) p σk−p−1,
βkλ t+(k−1)l
⋆ α1 βλt ⋆(βλl ⋆) k−p−1σp p = 1, . . . , k−2
βλt ⋆σk−1
k − 1 k
k
k
βλ t ⋆αk.
αk
σk
αk
k
l uk Gk t < l
k +1 l
Gk+1 uk = uk+1
σk+1 γ(L k+1
W )(−1) uk2
σk+1
t l
MG×MG
MG
(r, s)
r r, s ∈ {1, . . . , MG}
{I, r, s} {i, r} u k i , x k i (t), w k i (t)
M 2 G + MG
1
2 MG(MG + 1)
MG + 1
2 MG(MG +1) MG +M 2 G

k [I k i ]r,s [i k i ]r
i ∈ V
u k i {I,r,s} = [I k i ]r,s, u k i {i,r} = [i k i ]r,
r, s ∈ {1, . . . , MG} x k i (t)
t i ∈ V I k i (t) ∈ R MG×MG
i k i (t) ∈ R MG
[I k i (t)]r,s = x k i (t) {I,r,s}, [i k i (t)]r = x k i (t) {i,r},
r, s ∈ {1, . . . , MG} i ∈ V k ∈ {1, . . . , K} t = 0, 1, . . .
ˆx k i (t) ∈ R MG Σ k i (t) ∈ R MG×MG
t k
ˆx k i (t) = I k i (t) −1 i k i (t), Σ k i (t) = I k i (t) −1 /n.
(i) I k i i k i j (ii)
i k i (t) I k i (t)
I k i (t)
i ∈ V
Gk k ∈ {1, . . . , K}
K ˆx K i (t) ΣK i (t) i ∈ V
ˆx K G ΣKG
lim
t→∞ ˆxK i (t) = ˆx K G , lim
t→∞ Σ K i (t) = Σ K G .

i ∈ V
G k k ∈ {1, . . . , K}
I k avg i k avg k
I k avg = 1
n
n
j=1
I k j , i k avg = 1
n
n
i k j .
α{I,r,s} σ{Ir,s} α σ u k i {I,r,s} = [Ik i ]r,s
α{i,r} σ{i,r} u k i {i,r} = [ik i ]r αI σI αi
σi I αi
αI = max
r,s∈{1,...,MG} α{I,r,s}, σI = max
r,s∈{1,...,MG} σ{I,r,s},
αi = max α{i,r}, σi = max σ{i,r}.
r∈{1,...,MG}
r∈{1,...,MG}
i ∈ V k ∈ {1, . . . , K} r, s ∈ {1, . . . , MG} t ≥ 0 [I k i (t)]r,s
[i k i (t)]r
i t
|[I k i (t)]r,s − [I k avg]r,s| ≤ αIfk(t) + σIgk(t),
|[i k i (t)]r − [i k avg]r| ≤ αifk(t) + σigk(t),
fk(t) gk(t)
ˆθ k i (t) i x
ˆx k i (t) i ∈ V
k ∈ {1, . . . , K} t
x
E ˆx k i (t) Ik = E i (t)
−1 k
ii (t) = x.
I k i (t) i k i (t)
I 1 j , . . . , I K j i 1 j, . . . , i K j
j=1

k = 1 t = 0
I k i (t) i k i (t)
I k i (t) =
i k i (t) =
n
[Φ(k, t)]i,j I k j +
j=1
n
[Φ(k, t)]i,j i k j +
j=1
n
k−1
[Ω(k, t, p)]i,j I
p=1 j=1
p
j ,
k−1
n
p=1 j=1
Φ(k, t), Ω(k, t, p), Ψ(t1, t2) ∈ R2n×2n
t
hγI
Φ(k, t) = Ψ (τ + (k − 1)l, t − 1 + (k − 1)l)
0
τ=1
l
hγI
Ω(k, t, p) = Ψ (τ + (p − 1)l, t − 1 + (k − 1)l)
0
τ=1
Ψ(t1, t2) = A(t2) . . . A(t1 + 1)A(t1),
[Ω(k, t, p)]i,j i p
j ,
A(t + kl) = Ak (t) t k
ˆx k j j x
ˆx k j = H k i x + v k j , with E v k j = 0,
i k j = (H k j ) T (Σ k j ) −1 ˆx k j
i k j = (H k j ) T (Σ k j ) −1 v k j + I k j x.
ik i (t)
i k n
i (t) =
j=1
n
[Φ(k, t)]i,j(H k j ) T (Σ k j ) −1 v k j +
k−1
[Ω(k, t, p)]i,j(H
p=1 j=1
k−p
j
n
[Φ(k, t)]i,j I k j +
j=1
) T (Σ k−p
n
j
) −1 v k−p
j
+
k−1
[Ω(k, t, p)]i,j I
p=1 j=1
p
j
Ik i (t)x Ik i (t) ˆx k i (t) = (Ik i (t)) −1ik i (t)
ˆx k i (t) = x + (I k i (t)) −1
n
+
(I k i (t)) −1
k−1
n
j=1
[Ω(k, t, p)]i,j(H
p=1 j=1
k−p
j
[Φ(k, t)]i,j(H k j ) T (Σ k j ) −1 v k j
) T (Σ k−p
j
x,
) −1 v k−p
j
v k j k ∈ {1, . . . , K} j ∈ V
ˆx k i (t) x
,
,
.

A(t + kl)
t k Ψ (t1 + kl, t2 + kl)
(A k ) t2−t1+1
K = 5
20
10
0
−10
−20
−30
−40
−20 −10 0 10 20
9
k k + 1 20
9
k = 1 k = 2 L = 1000
l = 0
l = 1(L/K)
4
l = (L/K)
l = 0
k = K l = 1(L/K)
4
50 k = 1 50
k = K
200
l = (L/K) l = 1
4 (L/K)
k = 1, 2, . . . , 4
k = K 800
L ˆ θK i (L) ΣK(L)
θi

15
10
5
0
−5
−10
−15
Robot 1
Robot 2
Robot 3
Robot 4
Robot 5
Robot 6
Robot 7
Robot 8
Robot 9
−5 0 5 10 15 20 25
15
10
5
0
−5
−10
−15
−5 0 5 10 15 20 25
k = 1 k = 2
9 k = 1 k = 2
20 40
k = 2
5
4
3
2
1
0
x 10 4
l=0
l=1/4(L/K)
l=L/K
map updates, l=1/4(L/K)
map updates, l=L/K
0 200 400 600 800 1000
160
140
120
100
80
60
40
20
0
0 200 400 600 800 1000
|| ˆ θk i (t) − ˆ θk c || L ||Tr(Σk θi ˆ (t)) − Tr Σk c (t) || L
L || ˆ θk i (t)− ˆ θk c ||
||Tr(Σk θi
ˆ (t)) − Tr Σk c(t) || l = 0
k = 5 l = 1
4 (L/K)
50
200
l = (L/K) 200
800

15
10
5
0
−5
−10
−15
−5 0 5 10 15 20 25
15
10
5
0
−5
−10
−15
−5 0 5 10 15 20 25
1
i = 1 ˆ θK i (L)
(L/K)
ΣK θi
ˆ (L) l = 1
4
ˆ θK G ΣKG
ˆ θ K G ΣK G
l = 1
4 (L/K)
µ

15
10
5
0
−5
−10
−15
−5 0 5 10 15 20 25
15
10
5
0
−5
−10
−15
−5 0 5 10 15 20 25
1
i = 1 ˆ θK i (L)
(L/K)
ΣK θi
ˆ (L) l = 1
4
ˆ θK G ΣKG
ˆ θ K G ΣK G
l = 1
4 (L/K)
µ

15
10
5
0
−5
−10
−15
−5 0 5 10 15 20 25
15
10
5
0
−5
−10
−15
−5 0 5 10 15 20 25
1
i = 1 ˆ θK i (L)
(L/K)
ΣK θi
ˆ (L) l = 1
4
ˆ θK G ΣKG
ˆ θ K G ΣK G
l = 1
4 (L/K)
µ

n ∈ N n
Gcom = (Vcom, Ecom)
Vcom = {1, . . . , n} i j
(i, j) ∈ Ecom Ni i
Ni = {j | (i, j) ∈ Ecom}.
i Si mi
Si = {f i 1, . . . , f i mi }.
Si
Sj j ∈ Ni
Si
Robot C
f C
1
f C
2
f B
2
Robot B
f B
1
f D
1
X
f D
2
Robot D
f A
2
f A
1
Robot A
A B C D
A f A 1 f D 2 f B 1 B
f B 1 f C 1 C f C 1 f D 1
f D 1 f D 2
i, j k
r, r ′ , s, s ′ , r th

i th f i r A, Ar,s [A]r,s
(r, s) Aij (i, j)
Ik k × k 0k1×k2 k1 × k2
F
Si Sj, ij ∈ Nmi×mj
i 1 if fr and f j s are associated,
[ij]r,s =
0 otherwise,
r = 1, . . . , mi s = 1, . . . , mj F
F Si
F (Si, Si) = ii = I
ij
mi
r=1
[ij]r,s ≤ 1
r = 1, . . . , mi s = 1, . . . , mj
mj
s=1
[ij]r,s ≤ 1,
i j
Si Sj F (Si, Sj) = ij = T
(F (Sj, Si))
ji =
T
n
n
n F
Si Sj i, j ∈ {1, . . . , n}
Gcen = (Fcen, Ecen) Fcen f i r i = 1, . . . , n
r = 1, . . . , mi f i r f j s [ij]r,s = 1

Camera A Camera B
Camera D Camera C
Camera A Camera B
Camera D Camera C
Camera A Camera B
Camera D Camera C
(f A 1 , f B 1 ) (f A 2 , f B 2 ), (f A 2 , f B 1 ).
A C B D
Gcom.
Gcen
Si
Gcen
Gcen
n
n
Gcom = (Vcom, Ecom)
Gdis = (Fdis, Edis) ⊆ Gcen,
Fdis = Fcen, f i r f j s
Gcen i j
Edis = {(f i r, f j s ) | (f i r, f j s ) ∈ Ecen ∧ (i, j) ∈ Ecom}.

msum |Fdis| = n i=1 mi = msum
df Gdis
Gdis, dv Gcom
df ≤ msum dv ≤ n. ∈ Nmsum×msum Gdis
⎡
⎤
ij =
=
⎢
⎣
11 . . . 1n
n1 . . . nn
F (Si, Sj) if j ∈ {Ni ∪ i},
0 otherwise.
⎥
⎦ ,
i
ij, j = 1, . . . , n.
Gcom,
Gdis
Gdis
Gdis
Gdis |V|
∈ {0, 1} |V|×|V| , i, j ∈
{1, . . . , |V|} t ∈ N (i, j) t th t ,
t i j.
i ∈ Vcom t
Xij(t) ∈ N mi×mj , j = 1, . . . , n, t ≥ 0,
Xij(0) =
I, j = i,
0, j = i,
Xij(t + 1) =
ikXkj(t),
k∈{Ni∪i}
ik

[ t ]ij ∈ N mi×mj t
i j Xij(t) i
[ t ]ij
i, j ∈ {1, . . . , n} t ∈ N
Xij(t) = [ t ]ij,
t = 0 0 = I i, j ∈ {1, . . . , n} [ 0 ]ii = I
[ 0 ]ij = 0 Xij
t > 0
[ t ]ij =
n
ik[t−1 ]kj =
k=1
k∈{Ni∪i}
ik[ t−1 ]kj,
ik = 0 k /∈ {Ni ∪ i} i, j ∈ {1, . . . , n}
t > 0, Xij(t − 1) = [t−1 ]ij
Xij(t) =
ikXkj(t − 1) =
ik[t−1 ]kj = [t ]ij.
k∈{Ni∪i}
k∈{Ni∪i}
Xij(t) = [ t ]ij t > 0.
Xij(t)
t i j.
.
Gdis
[Xij(t)]r,s > 0
f i r f j s
df
df msum
¯
∼ ¯ , r s
[]r,s > 0 ⇔ [ ¯ ]r,s > 0 []r,s = 0 ⇔ [ ¯ ]r,s = 0.
Xij(t)

i, ti t, Xij(t) ∼
Xij(t − 1) j = 1, . . . , n. i ti
¯ Xij(t) Xij(t), [Xij(t−1)]r,s = 0 [Xij(t)]r,s >
0. | ¯ Xij(t)|, f j s ∈ Sj
Gdis f i r ∈ Si t ti, Xij(ti) ∼
Xij(ti − 1) ∀j n
j=1 | ¯ Xij(ti)| = 0
ti f i r Gdis.
ti,
ti + 1
i
df + 1
Gdis
df, n
j=1 | ¯ Xij(df + 1)| = 0 i = 1, . . . , n.
j t Xij(t) i
i
Xij(t) ∼ Xij(t − 1).
Xij(ti).
f i r
f i r ′ r = r′
f j s f j
s ′, s = s ′ ,
[Xii(ti)]r,r ′ > 0;
[Xij(ti)]r,s > 0 [Xij(ti)]r,s ′ > 0.
i Xij(ti)

t
Gdis.
i r(0) = {[i1]r,1, . . . , [in]r,mn} ∈ {0, 1} msum
f i r [ i r(0)]u, u = 1, . . . , msum, u th
i
i r(t), r ∈ Si j ∈ Ni
j s(t), j ∈ Ni, s ∈ Sj
r ∈ Si
j ∈ Ni, s ∈ Sj | [ij]r,s = 1
i r(t + 1) = i r(t) ∨ j s(t)
r ′ ∈ Si ∃ u ∈ {1, . . . , msum} [ i r(t)]u =
[ i r ′(t)]u = 1
i r(t + 1) = i r(t) ∨ i r ′(t)
i r(t + 1) = i r(t) ∀r ∈ Si
2m 2 sum.
i r

i ∈ Vcom, r ∈ Si.
i r
m 2 sum .
2m 2 sum.
y i r(t) i r(t), [ i r(t − 1)]u = 0
[ i r(t)]u = 1, u = 1, . . . , msum.
Gdis t Si.
ti i r(ti) = i r(ti − 1) ∀r y i r(ti) = 0
Si,
ti, ti +1.
ti +1
i
j t i r(t), r = 1, . . . , mi
i i
i r(t) = i r(t − 1).
min(df, 2n)
df
df ≤ 2n
Gcom, ℓ.
n − ℓ
3ℓ
+ 1
4
ℓ/2 ℓ + 1
ℓ/2 + 1
(ℓ/2)

u, ℓ/4
ℓ + 1
n−ℓ. 2n− 5
4 ℓ+1.
3 2n.
n − ℓ n − ℓ
Cam. A
Cam. B
Cam. C
Cam. D
1 1
1 1 1
1 1
1 1 1
1 1
1 1 1
1 1 1
1 1 1
1 1
1 1 1
1 1 1
Cam. A
Cam. B
Cam. C
Cam. D
1 1 1
1 1 1 1 1
1 1 1
1 1 1 1 1
1 1 1
1 1 1
1 1 1 1 1
1 1 1 1
1 1 1
1 1 1 1
1 1 1 1 1
Cam. A
Cam. B
Cam. C
Cam. D
1 1 1 1 1 1 1
1 1 1 1 1 1 1
1 1 1
1 1 1 1 1 1 1
1 1 1 1 1 1 1
1 1 1
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 1
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
Cam. A
Cam. B
Cam. C
Cam. D
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 1
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 1
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
1 1 1
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
f A 1 ,
A 1 (2) = A 1 (1) ∨ D 1 (1).
f B 1 f D 2 A 2 (2) = A 2 (1)∨ B 1 (1)∨ D 2 (1). A 3 (2) = A 3 (1)∨ B 3 (1).
f A 1 f A 2 f C 1

3 < df = 7
f A 1 f A 2 f B 1 , f B 2 , f C 1 , f C 2 , f D 1
f D 2 .
Gdis
C Gdis
C i r ∈ {1, . . . , mi} f i r ∈ C.
C R ⊆ Vcom
i ∈ R C ˜mi Gdis C = 0
Gdis
C Gdis
C
C Cq
∪ Cq = C, and Cq ∩ Cq ′ = ∅,
q
q, q ′ = 1, 2, . . .
f i r f i r ′
˜mi ≥ 2 C i ∈ R
˜mj ≥ 2 j ∈ R
R C i⋆
C
i⋆ = arg max
i∈R
˜mi.
C
˜mi⋆
i ∈ R
maxi∈R ˜mi = ˜mi⋆

i
C i i = i⋆
˜mi⋆
f i⋆
r ∈ C Cq
f j s f i r
[ij]r,s = 0
j
f i r
f j s f i r
[ij]r,s = 0
˜mi⋆
i
Xi1(ti), . . . , Xin(ti)
˜mi⋆ Cq
f i⋆ ∈ C Cq
f i⋆ ∈ Cq f j s Cq f i r
[ij]r,s = 1 j i i
f i r Cq
f i r
f i r ′ Cq
f i r i Cq
(a) f i r Cq i
(b) (c) f i r Cq i [ij]r,s
j j

[ji]s,r (d) i f i r Cq
i ∈ Vcom
Gdis G ′ dis
t = n Cq
Cq G ′ dis
Cq
Cq
q ∈ {1, . . . , ˜mi⋆}
(i) n
n
Cq
(ii) Gdis
C Gdis
Cq Gdis
Cq G ′ dis
Cq
G ′ dis (iii)
Cq
(iv)
Cq
C
2 ˜mi⋆ ˜mi⋆ ≥ 2
C
˜mi⋆
G ′ dis
i

C
2 ˜mi⋆ ≥ 4
|C| < 4 3 f i r, f i r ′, f j s
(f i r, f j s ) (f i r ′, f j s )
2
Xij(ti)
ij
ij j ∈ {1, . . . , n}
f i r i Li r = (i⋆, r⋆) ∈ N2 i⋆ r⋆ f i r
f j s , f j′
s ′ , . . . Gdis
i, j, j ′ , . . .
(i, r), (j, s), (j ′ , s ′ ), . . . (i⋆, r⋆)
i f i r Li r = (i⋆, r⋆)
j, j ′ , . . .
L i r = L j s = L j′
s ′ = · · · = (i⋆, r⋆) .
f i r Gdis
i L i r (i, r)
f i r
Gdis
f i r f j s L i r = L j s
Gdis
˜ Si ⊆ Si
i ∈ {1, . . . , n} | ˜ Si|
i Li L i r
f i r ∈ Si \ ˜ Si Xij | ˜ Si| × | ˜ Sj| ¯r = row (f i r)

f i r ∈ ˜ Si Xij
¯r ∈ {1, . . . , | ˜ Si|} ¯s = col(f j s ) ˜ Sj
Ãij ∈ N | ˜ Si|×| ˜ Sj| Aij
i j
i
˜ Si ← {f i 1, . . . , f i mi } Li ← ∅
L i r = (i, r) f i r f i r
| ˜ Si| > 0
f i r f i⋆
r⋆
L i r = (i⋆, r⋆) f i r
f i r [i⋆, r⋆]
L i r = (i⋆, r⋆) f i r
f i r
L i r = (i, r) f i r
L i r f i r
Li ← Li ∪ {L i r} ˜ Si ← ˜ Si \ {f i r}
i
˜Si = {f i 1, . . . , f i mi }, Li = ∅.
i j ∈ Ni
ij ∈ N mi×mj
f i r
[ij]r,s = 0 for all j ∈ Ni, j = i, and all s ∈ {1, . . . , mj}.
f i r i L i r
L i r = (i, r), Li = Li ∪ L i r, ˜ Si = ˜ Si \ {f i r}.
˜ Si
Gdis
˜Si i ∈ {1, . . . , n} i Xij ∈ N | ˜ Si|×| ˜ Sj|
j = 1, . . . , n (Gdis) ˜ Si
˜ Sj f i r f j s
[Xij]¯r,¯s > 0,

¯r = row(f i r) ¯s = col(f j s ) i
i f i r
i i
(i⋆, r⋆)
i⋆
f i⋆ i⋆
r⋆ , fr ′ , . . .
⋆
Xi1, . . . , Xin i ˜mj
j f i r ¯r = row(f i r)
˜mj = f j s | [Xij]¯r,¯s > 0, with ¯s = col(f j s ) .
˜m⋆ ˜mj j ∈ {1, . . . , n} i⋆
f i⋆ i⋆
r⋆ , fr ′ ⋆ , . . . f i r ¯r = row(f i r)
i⋆ = min {j | ˜mj = ˜m⋆} , {r⋆, r ′ ⋆, . . . } = s | [Xii⋆]¯r,¯s > 0 with ¯s = col(f i⋆
s ) .
f i r i⋆
f i⋆
r⋆
i⋆ = min j | [Xij]¯r,¯s > 0 for some ¯s ∈ {1, . . . , | ˜
Sj|} ,
r⋆
= s | [Xii⋆]¯r,¯s > 0 with ¯s = col(f i⋆
s ) ,
¯r = row(f i r) i f i r L i r = (i⋆, r⋆)
L i r = (i⋆, r⋆), Li = Li ∪ L i r, ˜ Si = ˜ Si \ {f i r}.
Cq (i⋆, r⋆)
i⋆ r⋆
f i r
(i⋆, r⋆)
i f i r
L i r = (i⋆, r⋆), Li = Li ∪ L i r, ˜ Si = ˜ Si \ {f i r}.

f i r ∈ ˜ Si
i f i r
[ Ãij]¯r,¯s = 0, j ∈ Ni, j = i, ¯s ∈ {1, . . . , | ˜ Sj|},
¯r = row(f i r)
i
˜ Si
i Sij Xij
C.
[]r,s =
ers if []r,s = 1,
−1 otherwise,
ers fr fs.
• err = 0, ∀r
• ers ≥ 0, ∀r, s
• ers = esr, ∀r, s
• ers = er ′ s ′ ⇔ [r = r′ ∧s = s ′ ]∨[r = s ′ ∧s =
r ′ ]
C. C

•
•
i
C
Gdis
C
r(0) = {[C]r,1, . . . , [C]r,c}, r = 1, . . . , ˜mi
r(t + 1) = maxs∈C, [C]r,s≥0(r(t), s(t)rs)
r(t + 1) = r(t) ∀r ∈ ˜mi
i r r ′
(s, s ′ ) :
[r]s = [r ′]s ′, s = s′
s ′′ = s, [r]s = [r]s ′′,
s ′′ = s ′ , [r ′]s ′ = [r ′]s ′′
r(0) = {[C]r,1, . . . , [C]r,c}, r ∈ {1, . . . , ˜mi}.
˜mi
r(t + 1) = max
s∈C, [C]r,s≥0 (r(t), s(t)rs),
rs
r s.
r, s ∈ C [C]r,s ≥ 0 r
srs.

, s ∈ C [C]r,s ≥ 0,
C, r s
r ≥ srs s ≥ rsr
rs = sr = −1
sr , srs ≥
r, r = srs.
fr u th
[r(t + 1)]u,
[r(t + 1)]u =
⎧
⎨max([r(t)]u,
[s(t)]s) [C]r,s ≥ 0 ∧ u = r
max([r(t)]u, [s(t)]r) [C]r,s ≥ 0 ∧ u = s
⎩
max([r(t)]u, [s(t)]u) [C]r,s ≥ 0 ∧ r = u = s
,
C
r ∈ C, [r(t)]r = 0, ∀t ≥ 0.
[r(t)]s ′ → [C]r ′ ,s ′ = er ′ s ′
r ′ = arg min d(r, r
[] r ′′ ,s ′=1
′′ ),
d(r, r ′′ ) r ′′ r
fr ′ fr fs ′.
fr, r th
r, [r(t + 1)]r,
[r(t + 1)]r = max
s∈C, [C]r,s≥0 ([r(t)]r, [s(t)]s).
[r(0)]r = err = 0,
r, [r(t)]r = 0, ∀t ≥ 0.
(fr ′, fs ′).
C
Cr ′ = {r | d(r, r′ ) < d(r, s ′ )},
Cs ′ = {s | d(s, s′ ) < d(s, r ′ )}.

′ ∈ Cr ′ s′ ∈ Cs ′
s ′th
Cr ′ r′th Cs ′
[r(t)]s ′, r ∈ Cr ′, [s(t)]r ′, s ∈ Cs ′.
r ∈ Cr \ r ′ ,
[r(t + 1)]s ′ = max
r ′′ ([r(t)]s ′, [r ′′(t)]s ′).
∈Cr ′, [C] r,r ′′≥0
r = s ′ = r ′′ . Cs ′
C fr ′
[r ′(t + 1)]s ′ = max ([r ′(t)]s ′, [r(t)]s ′, [s ′(t)]r ′).
r∈Cr ′,[C] r ′ ,r≥0 Cs
[s(t + 1)]r ′ = max
s ′′ ([s(t)]r ′, [s ′′(t)]r ′)
∈Cs ′, [C] s,s ′′≥0
[s ′(t + 1)]r ′ = max ([s ′(t)]r ′, [s(t)]r ′, [r ′(t)]s ′).
s∈Cs ′,[C] s ′ ,s≥0 Cr ′ Cs ′
max ([r(0)]s ′, [s(0)]r ′),
r∈Cr ′, s∈Cs ′
r ∈ Cr ′ \ r′ fs ′, [r(0)]s ′ = −1.
s ∈ Cs ′ \ s′ , [s(0)]r ′ = −1. r′ s ′ ,
[r ′(0)]s ′ = er ′ s ′ = es ′ r ′ = [s ′(0)]r ′ ≥ 0 > −1.
c
(fr ′, fs ′), er ′ s ′. r ∈ Cr ′, [r]s, s ∈ Cs ′ \s′ ,
[s ′]s.
ℓ
Cℓ
∀r ′ , s ′ ∈ Cℓ, s ′ = r ′
∀r ′ ∈ Cℓ, s ′ ∈ Cℓ, s ′ = arg min s∈Cℓ d(r′ , s),
[r ′]s ′ → max ers.
r,s∈Cℓ
[r ′]s ′ → max ers.
r,s∈Cℓ

1, . . . , rℓ, Cℓ.
(fr1, fr2), (fr2, fr3) . . . , (frℓ , fr1).
r2 fr1,
[r1]r2 = max([r1]r2, [r2]r1, [rℓ ]r2),
fr1
fr2 Cℓ
[r1]r2 [r2]r1.
[r2]r1 = max([r2]r1, [r1]r2, [r3]r1),
[r3]r1.
[rℓ−1 ]r1 [rℓ ]r1,
[rℓ ]r1 = max([rℓ ]r1, [rℓ−1 ]r1, [r1]rℓ ).
[r1]r2 = [r2]r1, [r2]r1 = [r3]r1, . . . ,
[rℓ−1 ]r1 = [rℓ ]r1 [rℓ ]r1 = [r1]rℓ
[r1]r2 = [r1]rℓ = [r]r1, ∀r ∈ Cℓ \ r1.
Cℓ
r ∈ Cℓ, [r]r ′ = [r]r ′′, ∀r′ , r ′′ ∈ Cℓ \ r.
ℓ − 1
¯ Cℓ = C \ Cℓ.
s ∈ ¯ Cℓ
• ∃ r ∈ Cℓ fr fs
• s Cℓ
∈ ¯ Cℓ r ∈ Cℓ
r r ′ ∈ Cℓ,
s r, s
r s
Cℓ.
r
Cℓ r ′ ∈ Cℓ \ r, [s]r ′ [r]r ′
[s]r
r fr fs Cℓ,

Camera B
1
7
Camera A Camera F
9
Errors
8
Camera C Camera D
6
3
4
2
Camera E
Camera B
Camera A Camera F
Camera C Camera D
Camera E
Camera B
1
7
Camera A Camera F
9
8
Camera C Camera D
0 1
0 9 6
1 0 7
9 0
7 0 8 2
8 0 4
2 4 0 3
6 3 0
0 1 7
0 9 3 6
1 0 7 8 2
9 0 6
1 7 0 8 4 3
7 8 0 4 3
6 7 8 8 0 3
6 9 2 4 3 0
0 1 7 8 2
0 9 2 4 3 6
1 0 7 8 4 3
9 0 3 6
1 6 7 0 8 8 3
1 6 7 8 0 8 3
1 6 7 9 8 8 0 3
6 7 9 8 8 3 0
0 1 7 8 4 3
0 7 9 8 8 3 6
1 6 0 7 8 8 3
9 0 2 4 3 6
1 6 7 9 0 8 8 3
1 6 7 9 8 0 8 3
1 6 7 9 8 8 0 3
1 6 7 9 8 8 3 0
0 6 1 7 8 8 3
1 0 7 9 8 8 3 6
1 6 0 9 7 8 8 3
9 7 0 8 8 3 6
1 6 7 9 0 8 8 3
1 6 7 9 8 0 8 3
1 6 7 9 8 8 0 3
1 6 7 9 8 8 3 0
6
3
4
2
Camera E
0 6 1 9 7 8 8 3
1 0 7 9 8 8 3 6
1 6 0 9 7 8 8 3
1 9 7 0 8 8 3 6
1 6 7 9 0 8 8 3
1 6 7 9 8 0 8 3
1 6 7 9 8 8 0 3
1 6 7 9 8 8 3 0
r
(fs, fs ′),
fr fr ′
[r]s = [r ′]s ′, s = s′
s ′′ = s, [r]s = [r]s ′′,
s ′′ = s ′ , [r ′]s ′ = [r ′]s ′′,
[r]s, [r]s = [r ′]s ′ = ess ′.
˜mi
˜mi

0 6 1 9 7 8 8 3
1 0 7 9 8 8 3 6
1 6 0 9 7 8 8 3
1 9 7 0 8 8 3 6
(f A 2 , f B 2 ).
f B 1 f C 1
f D 1
f C 1
r
f B 1 ,
f A 1 f C 1 .
f D 1 f E 1
f A 2 f B 2 .

f B 1 f C 1
7 20 × 20 m
70
30 m
σx, σy = 0.4 cm
σθ = 1
6 m 0.5
10
8
6
4
2
0
−2
−4
−6
−8
−10
−10 −8 −6 −4 −2 0 2 4 6 8
20 × 20 m
3 m
F (Si, Sj)
(i, j) ∈ Ecom

10
8
6
4
2
0
−2
−4
−6
−8
−10
−10 −8 −6 −4 −2 0 2 4 6 8
1.5
1
0.5
0
−0.5
−1
−1.5
−2
R5
R1
R4
R3
R2
R7
R6
5 5.5 6 6.5 7 7.5 8 8.5 9
3 m
Gdis

10
8
6
4
2
0
−2
−4
−6
−8
−10
−10 −8 −6 −4 −2 0 2 4 6 8
1.5
1
0.5
0
−0.5
−1
−1.5
−2
R5
R1
R4
R3
R2
R7
R6
5 5.5 6 6.5 7 7.5 8 8.5 9
3 m
Gdis

Gdis
10
8
6
4
2
0
−2
−4
−6
−8
−10
−10 −8 −6 −4 −2 0 2 4 6 8
xi G (t), ΣiG (t) t = 5

Gdis
10
8
6
4
2
0
−2
−4
−6
−8
−10
−10 −8 −6 −4 −2 0 2 4 6 8
xi G (t), ΣiG (t) t = 5

Gdis
10
8
6
4
2
0
−2
−4
−6
−8
−10
−10 −8 −6 −4 −2 0 2 4 6 8
xi G (t), ΣiG (t) t = 5

Gdis
10
8
6
4
2
0
−2
−4
−6
−8
−10
−10 −8 −6 −4 −2 0 2 4 6 8
xi G (t), ΣiG (t) t = 5

i, j
e, e ′ e i
j e = (i, j) A, Ar,s [A]r,s
(r, s) ⊗ Ir
r × r 0r×s r × s
A Aij A = [Aij] A = (B1, . . . , Br)
A Aii = Bi Aij = 0 i = j
n ∈ N
n ∈ N
{p G 1 , . . . , p G n } G p G i = x G i , y G i , θ G i
∈ SE(3)
i ∈ {1, . . . , n} m ∈ N
p i j ∈ SE(3)
j i
G = (V, E) V = {1, . . . , n} E m
|E| = m e = (i, j) ∈ E i j i

j G
i
Ni
Ni = {j | (i, j) ∈ E or (j, i) ∈ E}.
A ∈ {0, 1, −1} n×m
Ai,e =
⎧
⎨
⎩
−1 e = (i, j)
1 e = (j, i)
0
, for i ∈ {1, . . . , n}, e ∈ {1, . . . , m},
Wi,j
G n
G
−π/2 < θi < π/2 i ∈ V
a ∈ V a = 1
p a a = 03×1
a
V a = V \ {a}
A a ∈ {0, 1, −1} n−1×m a
A
e = (i, j) ∈ E G = (V, E)
z θ e z xy
e j i
Σ z θ e Σ z xy
e
ˆp a i i ∈ V a a
−π/2 < θi < π/2 i ∈ V
i ∈ V p− xi ∈ R p ze ∈ R p
e = (i, j) ∈ E i j
ze = xj − xi + ve,
ve ∼ N (0p×p, Σze)
ˆx cen
i i ∈ V
ˆx a i

0 < θi < π i ∈ V
p G i = (R G i , T G
i ) ∈ SE(3) i
G R G i T G
i
⎡
R G i = ⎣
cos θ G i − sin θ G i 0
sin θ G i cos θ G i 0
0 0 1
⎤
⎦ , T G
i =
⎡
⎣
x G i
y G i
0
⎤
⎦ ,
z = 0 z
p i j = (R i j, T i j ) ∈ SE(3) j i
p i j
p
i = (R
i , T
i ) ∈ SE(3) i
p i j
G
p
i = (R
i , T
i )
T
G
T
G = 1
n
n
j=1
T
j .
R G
(RG , T G ) (R
G , TG )
RG = (R G )T T G = − (R G )T T
G i ∈ V
pG i = (RG i , T G
i )
R G i = R G R
i , T G
i = R G T
i + T G .
G
p G i
p G i = (R G i , T G
i )

G i
pi G = (Ri G , T i G ) pGi pi G
R G i = R i T G
G , Ti = − R i T i
G TG, R i G = R GT i
i , TG = − R GT G
i Ti .
pi j
(Ri j, T i j ) ∈ SE(3)
G
i ∈ V
T i G (t) ∈ R3 T i G (0) = 0 t ∈ N
T i G(t + 1) =
j∈Ni∪{i}
Wi,j(R i jT j
G (t) + T i j ),
Wi,j G
G i ∈ V
t → ∞ T i G (t) i ∈ V
i T i G
lim
t→∞ T i G(t) = T i G.
Ti(t) = R
i T i G(t) + T
i , T i G(t) = (R
i ) T Ti(t) − (R
i ) T T
i ,
T i G (t) (R i , T
i )
Ti(t + 1) =
Wi,jR
i R i jR j Tj(t) +
Wi,j(R
i R i jT j + R
i T i j + T
i )
j∈Ni∪{i}
=
j∈Ni∪{i}
j∈Ni∪{i}
Wi,jTj(t),
j∈Ni∪{i} Wi,j = 1 i ∈ V R i Ri jRj = R = I R i Ri jT j + R i T i j + T
i =
T
= 0 i, j ∈ V
lim
t→∞ Ti(t) = 1
n
n
j=1
Tj(0) = 1
n
n
R
j=1
jT j
G
(0) + Tj = 1
n
n
j=1
T
j = T
G,

lim
t→∞ T i G(t) = (R
i ) T T
G − (R
i ) T T
i = − (R
i ) T R G
= − R G i
T R G T
i + T G
= − R G i
T T G − (R
i ) T R G
T R G T
i
T T G
i = T i G,
R i R G
R
G = arg min
R n
d 2 (R , R
i ),
d 2 (R
i , R) R R
i
i=1
d 2 (R , R
i ) = − 1
2 Tr [log((R ) T R
i )] 2 .
log log : SO(3) → so(3)
log(R) =
03×3
β = 0,
β
2 sin β (R − RT ) β = 0,
β = arccos Tr{R}−1
n 2
i=1 d2 (R i , R)
θi
π/2
Ri G (t) i
SO(3)
i Ri G (0) = I
t ∈ N
R i G(t + 1) = R i G(t) exp(ui(t)), ui(t) =
Wi,j log(R i G(t) T R i jR j
G (t)),
j∈Ni∪{i}
Wi,j G
exp exp : so(3) → SO(3)
I α = 0,
exp(ui(t)) = sin α I + α ui(t) 1−cos α + α2 u2 i (t) α = 0,
1
α = 2Tr{uTi (t)ui(t)}
± π
G
2
lim
t→∞ R
i R i G(t) = R
G,

i ∈ V
n 2 log(ɛ −1 )
ɛ
n ∈ N
{p a 1, . . . , p a n} p a i = [x a i , y a i , θ a i ] i ∈ {1, . . . , n}
a m ∈ N
a ∈ V p a a = 03×1
a = 1 V a = V \{a}
e = (i, j) ∈ E G = (V, E)
z θ e z xy
e j
i Σ z θ e Σ z xy
e zθ ∈ R m zxy ∈ R 2m
Σzθ ∈ Rm×m Σzxy ∈ R 2m×2m m
zθ = (z θ 1, . . . , z θ m) T , zxy = ((z xy
1 ) T , . . . , (z xy
m ) T ) T ,
Σzθ = Diag(Σ z θ 1 , . . . Σ z θ m ), Σzxy = blkDiag(Σ z xy
1 , . . . Σ z xy
m ).
i ∈ V ˆp a i
˜ θ a V ∈ Rn
a V
zxy
ˆp a V = ((ˆxa V )T , ( ˆ θ a V )T ) T
˜ θV a ∈ Rn−1
a

zθ ∈ Rm Σzθ ∈ Rm×m
± π
2
˜θ a V a = Σ˜ θ a V a Aa Σ −1
zθ zθ, Σ˜ θ a V a = A a Σ −1
zθ (Aa ) T −1 ,
Aa ∈ {0, 1, −1} n−1×m
a A
˜ θa i = 0 i = a ˜ θa V ∈ Rn Σ˜ θa R
V
n×n
V a
˜θ a V = (0, ( ˜ θ a V a)T ) T , Σ˜ θ a V
= Diag(0, Σ˜ θa ).
Va z θ e e = (i, j)
i
˜ θ a V
e = (i, j) ∈ E ˜ Re ∈ R 2×2 ˜ Se ∈ R 2×2
˜ θi i
˜Re = R( ˜ θ a
cos ˜a θi i ) =
sin ˜ θa i
− sin ˜ θa i
cos ˜ θa i
,
Se
˜ = S( ˜ θ a
− sin ˜a θi i ) =
− cos ˜ θa i
cos ˜ θa i
− sin ˜ θa i
,
˜ R ∈ R 2m×2m m
˜R = R( ˜ θ a V) = blkDiag( ˜ R1, . . . , ˜ Rm).
w ∈ R2m+(n−1)
Σw
˜zxy
w = ˜θV a
˜R 0 zxy
=
0 In−1
˜θV a
,
T K J Σzxy 0 K 0
Σw =
,
0 In−1
0 Σ˜ θV a
J T In−1
K ∈ R 2m×2m J ∈ R 2m×(n−1)
zxy ˆ θ
K = ˜ R, and Je,i = ˜ Se z xy
e if e = (i, j) for some j, and Je,i = 02×1 otherwise.
ˆx a V a ∈ R2(n−1)
a ˆ θ a V a ∈ Rn−1

˜θ a V a ˆpa V a ∈ R3(n−1)
ˆp a Va =
ˆx a V a
ˆθ a V a
= Σˆp a VaBΣ−1 w w, Σˆp a Va = BΣ −1
w B T −1 ,
B = blkDiag ((A a ⊗ I2), In−1) Σw w
ˆp a V ∈ R3n V a
ˆp a V = (0 T 3×1, (ˆp a Va)T ) T , Σˆp a V = blkDiag(03×3, Σˆp a ).
V
ˆx a V a ˆ θ a V a
ˆx a V a = L−1 (A a ⊗ I2)Υ˜zxy
I2m + JΣˆ θ a V a J T Υ˜zxyE
˜R zxy,
ˆθ a V a = (Aa Σ −1
zθ (Aa ) T ) −1 A a Σ −1
zθ zθ + Σˆ θ a V a J T Υ˜zxyE ˜ R zxy, where
Υ˜zxy = ( ˜ RΣzxy ˜ R T ) −1 , E = (A a ⊗ I2) T L −1 (A a ⊗ I2)Υ˜zxy − I2m,
Σˆ θ a V a = ((Σ˜ θ a V a )−1 − J T Υ˜zxyEJ) −1 , L = (A a ⊗ I2)Υ˜zxy(A a ⊗ I2) T ,
ˆp a V
ˆx a V ˆ θa V
˜ θ a V a Σ˜ θ a V a
˜θ a V a = Σ˜ θ a V a Aa Σ −1
zθ zθ, Σ˜ θ a V a = (Aa Σ −1
zθ (Aa ) T ) −1 .
w
Σw
w =
˜zxy
˜θ a V a
˜Rzxy
=
˜θ a V a
˜RΣzxy
, Σw =
˜ RT + JΣ˜ θa Va J T JΣ˜ θa Va Σ˜ θa Va J T Σ˜ θa Va
ˆp a Va =
ˆx a V a
ˆθ a V a
T
.
= (BΣ −1
w B T ) −1 BΣ −1
w w.

ˆx a Va ˆ θa Va Υw = Σ−1 w
Υw =
Υ˜zxy −Υ˜zxyJ
−J T Υ˜zxy Σ −1
˜θ a Va + J T Υ˜zxyJ
,
Υ˜zxy Υ˜zxy = ( ˜ RΣzxy ˜ R T ) −1
A B
C D
−1 =
E F
G H
E = A −1 + A −1 B D − CA −1 B −1 CA −1 = A − BD −1 C −1 ,
F = −A −1 B D − CA −1 B −1 = − A − BD −1 C −1 BD −1 ,
G = − D − CA −1 B −1 CA −1 = −D −1 C A − BD −1 C −1 ,
H = D − CA −1 B −1 = D −1 + D −1 C A − BD −1 C −1 BD −1 .
Υˆp a = (BΣ−1
Va w BT ) Σˆp a
Va Υˆp a = Va Σˆp a = Va a (A ⊗ I2)Υ˜zxy(Aa ⊗ I2) T −(Aa ⊗ I2)Υ˜zxyJ
−J T Υ˜zxy(Aa ⊗ I2) T Σ −1
˜θ Va + J T Υ˜zxyJ
Σˆx Σ ˆx, ˆ θ
Σ T
ˆx, ˆ θ
Σˆ θ
Σˆ θ = ((Σ˜ θ a V a )−1 − J T Υ˜zxyEJ) −1 ,
,
, with
Σˆx = L −1 + L −1 (A a ⊗ I2)Υ˜zxyJΣˆ θ J T Υ˜zxy(A a ⊗ I2) T L −1 ,
Σ ˆx, ˆ θ = L −1 (A a ⊗ I2)Υ˜zxyJΣˆ θ ,
E = (A a ⊗ I2) T L −1 (A a ⊗ I2)Υ˜zxy − I,
L = (A a ⊗ I2)Υ˜zxy(A a ⊗ I2) T .
˜ θa V
i ∈ V
˜ θa i (t) ∈ R i = a
t ∈ N
˜θ a i (0) = 0, ˜ θ a i (t + 1) = ˜ θ a i (t), for i = a.
i ∈ Va t = 0 ˜ θa i (0)
t ∈ N
(Σzθ) e −1˜a θj (t) + C −1
i (Σzθ) e −1˜a θj (t),
˜θ a i (t + 1) = C −1
i ci + C −1
i
e=(i,j)∈E
e=(j,i)∈E

ci = −
Ci =
e=(i,j)∈E
e=(i,j)∈E
(Σ z θ e ) −1 z θ e +
(Σ z θ e ) −1 +
e=(j,i)∈E
e=(j,i)∈E
(Σ z θ e ) −1 z θ e,
(Σ z θ e ) −1 .
Υ˜ θ a V a
η˜ θ a V a ˜ θ a V a
Υ˜ θ a V a = (Σ˜ θ a V a )−1 = A a Σ −1
zθ (Aa ) T , η˜ θ a V a = Aa Σ −1
zθ zθ.
C Υ˜ θ a V a
C = Diag([Υ˜ θ a V a ]2,2, . . . , [Υ˜ θ a V a ]n,n),
D D = C − Υ˜ θa
Va
Υ˜ ˜θ θa Va a Va = η˜ θa Va , θ˜ a
Va = C −1 D˜ θ a Va + C−1η˜ θa .
Va ˜θ a Va(t + 1) = C−1D ˜ θ a Va(t) + C−1η˜ θa ,
Va t = 0 ˜ θ a V a(0) Aa Σ −1
zθ zθ A a Σ −1
zθ (Aa ) T
i − th ˜ θa Va
˜ θa i (t) ˜ θa i i ∈ Va
C−1D
ρ(C −1 D) < 1,
˜ θ a i (t) i = a t
ρ(C −1 D) ρ(C −1 D)
Σzθ ˜ θ a i (t)
i ˜ θ a i ˜ θ a V a
Ci ci i ∈ V a
z θ e Σ z θ e e = (j, i)
e = (i, j) ˜ θ a j (t) ˜ θ a i (t + 1)
j ∈ Ni

tmax
¯ θ a i tmax ¯ θ a i = ˜ θ a i (tmax)
zxy
˜zxy
¯ θ a i ¯ R =
R( ¯ θ a V a) ¯ θ a i ˜ θ a i
¯R i ∈ V
¯z xy
e = ¯ Rez xy
e , for all e = (i, j) ∈ E.
¯ θ ˜ θ
¯zxy ˜zxy
ˆp a Va
Σw
Σ˜ θa
Va ˆp a V i ∈ V ˆp a i (t) ∈ R3
ˆx a i (t) ∈ R2 ˆ θa i (t) ∈ R ˆp a V (t)
pa i (t) i ∈ V
ˆp a i (0) = 03×1, ˆp a i (t + 1) = ˆp a i (t), for i = a.
i ∈ V a t = 0 ˆp a i (0)
ˆpi(t) t ∈ N
ˆp a i (t + 1) =
M1 M2
Mi =
M3 M4
ˆx a i (t + 1)
ˆθ a i (t + 1)
, fi(p a
f1
V(t)) =
f2
= M −1
i (fi(ˆp a V(t)) + mi) ,
m1
, mi =
m2
.
Υ ˜z xy
e Υ˜zxy e = (i, j) ∈ E
Υ ˜z xy
e
= ˜ Re(Σ z xy
e )−1 ( ˜ Re) T .

Mi
M1 =
Υ xy
˜z + e
e=(i,j)∈E
M2 =
e=(i,j)∈E
M3 =
e=(i,j)∈E
M4 =
e=(i,j)∈E
e=(i,j)∈E
Υ ˜z xy
e
e=(j,i)∈E
˜Se z xy
e ,
(z xy
e ) T ( ˜ Se) T Υ ˜z xy
e ,
Υ ˜z xy
e ,
(z xy
e ) T ( ˜ Se) T Υ xy ˜Se ˜z z e
xy
e +
(Σ xy
ze e=(i,j)∈E
)−1 +
(Σ xy
ze e=(j,i)∈E
)−1 .
fi(ˆp a V (t))
ˆp a V (t) = (ˆxa V (t)T , ˆ θ(t) a
V )T
f1 =
Υ xy ˜Se ˜z z e
xy
e ˆ θ a j (t),
f2 =
e=(i,j)∈E
Υ ˜z xy
e ˆxa j (t) +
e=(j,i)∈E
(z xy
e ) T ( ˜ Se) T Υ ˜z xy
e ˆxa j (t) −
mi
m1 = −
e=(i,j)∈E
m2 = −
−
e=(i,j)∈E
e=(i,j)∈E
Υ xy
˜z e ˜zxy e +
e=(j,i)∈E
Υ ˜z xy
e ˆxa j (t) +
e=(i,j)∈E
Υ xy
˜z e ˜zxy e +
(z xy
e ) T ( ˜ Se) T Υ xy
˜z e ˜zxy e +
(Σ z θ e ) −1˜ θ a j −
e=(j,i)∈E
e=(i,j)∈E
e=(j,i)∈E
(Σ z xy
e )−1ˆ θ a j (t) −
e=(i,j)∈E
(Σ z θ e ) −1˜ θ a j +
Υ ˜z xy
e
e=(j,i)∈E
˜Sez xy
e ˜ θ a i −
(z xy
e ) T ( ˜ Se) T Υ xy ˜Sez ˜z e
xy
e ˜ θ a i
e=(i,j)∈E
(Σ z θ e ) −1˜ θ a i +
(Σ z xy
e )−1ˆ θ a j (t).
e=(j,i)∈E
e=(j,i)∈E
Υ ˜z xy
e
(Σ z θ e ) −1˜ θ a i .
˜Sez xy
e ˜ θ a j ,
ˆpi(t) i ∈ V
ˆp a i = [(ˆx a i ) T ˆ θ a i ] T G
i = a ˆp a i (t) = 0
Va ˆp a i
i ∈ Va Υˆp a
Va ˆp a Va Υˆp a Va = (Σˆp a Va)−1
Υˆp a = Va
L −A a Υ˜zxyJ
−J T Υ˜zxy(A a ⊗ I2) T A a Σ −1
zθ (Aa ) T + J T Υ˜zxyJ
,
L Υ˜zxy BΣ −1
w
BΣ −1
a (A ⊗ I2)Υ˜zxy
w =
−J T Y˜zxy
−(A a ⊗ I2)Υ˜zxyJ
A a Υ −1
zθ (Aa ⊗ I2) T + J T Υ˜zxyJ
.

i
ˆp a i =
Υˆp a Va ˆpaV a = B Σ−1 w w,
ˆx a i
ˆθ a i
= M −1
i (fi(ˆp a V) + mi) ,
Mi fi(pa V (t)) mi
M = blkDiag(M2, . . . , Mn)
ˆq a Va ˆpaV a ˆqa Va =
(ˆx a 2) T θˆ a
2, . . . , (ˆx a n) T T θˆ a
n
Υˆp a V a Υˆq a V a ˆpa i (t) i ∈ V a
ˆp a i = [(ˆx a i ) T θˆ a
i ] T ρ(M −1 (M − Υˆq a Va)) < 1
ρ(I − M −1 Υˆq a Va) < 1.
λ(I − M −1Υˆq a Va) = 1 − λ(M −1Υˆq a Va) 0 < λ(M −1Υˆq a V
0 < λ(M −1Υˆq a V
M −1 Υˆq a Va λ(M −1Υˆq a V
λmin(M −1 )
λmax(Υˆq a
Va ) 0 < λmin(M −1 )
λmax(Υˆq a
Va ) 0 < λ(M −1Υˆq a Va) a) < 2.
a)
a) ≥
λ(M −1Υˆq a Va) < 2
Υˆq a Va
adj(G) ⊗ I3 adj(G)
I3 3 × 3
s = 1
adj(G)ij = 0 for |i − j| > s.
Υˆq a V a s′ = 1
λmax(M −1 Υˆq a Va) < 2s′ = 2.
¯ θa V ˜ θa V

pi
i
xi
i ∈ {1, . . . , n} p− xi ∈ R p
xj −xi
e = (i, j) ∈ E G = (V, E)
i ze ∈ R p j
ze = xj − xi + ve,
ve ∼ N (0p×p, Σze) z ∈ R mp Σz ∈ R mp×mp
m
z = (z T 1 , . . . , z T m) T , Σz = blkDiag(Σz1, . . . , Σzm),
G
i Ni
n z
a ∈ V a = 1
ˆx a a = 0p a
ˆx a V a ∈ R(n−1)p
ˆx a V a = ((ˆxa 2) T , . . . , (ˆx a n) T ) T V a = V\{a} a
ˆx a V a = Σˆx a V a (Aa ⊗ Ip) Σ −1
z z, Σˆx a V a = (A a ⊗ Ip)Σ −1
z (A a ⊗ Ip) T −1 ,
A a ∈ R (n−1)×m G
a ˆx a V = (0Tp , (ˆx a Va)T ) T Σˆx a V =
blkDiag 0p×p, Σˆx a Va
a
i
Ni

ˆx a V
ˆx a Va
Υˆx a Va = η η Υ
ˆx a Va Σˆx a Va η = (A a ⊗ Ip) Σ −1
z z, Υ = (A a ⊗ Ip) Σ −1
z (A a ⊗ Ip) T .
ˆx a Va(t) ∈
R (n−1)p ˆx a Va(0) t
D, N Υ = [Υij]
ˆx a V a(t + 1) = D−1 N ˆx a V a(t) + D−1 η,
D = blkDiag(Υ22, . . . , Υnn), N = D − Υ.
ˆx a V a(t) ˆxa V a J = D−1 N
ρ(J) = ρ(D −1 N) < 1
Υ
j /∈ Ni Υij = Υji = 0p×p
Υ η i ∈ V a
i
i ∈ V ˆx a i
ˆx a V a = ((ˆxa 2) T , . . . , (ˆx a n) T ) T
i ∈ V ˆx a i (t) ∈ R p t = 0
ˆx a i (0) = 0p t i ∈ V ˆx a i (t)
ˆx a i (t + 1) =
MiBij ˆx a j (t) +
j∈Ni
e=(j,i)∈E
MiΣ −1
ze ze −
e=(i,j)∈E
MiΣ −1
ze ze,
Mi Bij p×p Mi = 0 i = a Mi = (
j∈Ni Bij) −1 i = a
⎧
⎨ Σ
Bij =
⎩
−1
ze + Σ−1 ze ′ if e = (i, j), e ′ Σ
= (j, i) ∈ E
−1
ze
Σ
if e = (i, j) ∈ E, (j, i) /∈ E
−1
ze if e = (j, i) ∈ E, (i, j) /∈ E
.

G Σz1, . . . , Σzm
m
p− Σz
D −1 N D N
ρ(D −1 N) < 1.
Υ Υ Z p
n−1
Υ
M p
n−1 Υ+ΥT ∈ M p
n−1
n − 1 n
Υ = [Υij] i, j ∈ Va |Va | = n − 1
Υ
Υij Υ
Υij =
−Bij if j ∈ Ni, j = i
0 if j /∈ Ni, j = i
, and Υii =
Bij,
i, j ∈ V a Bij Bij
Bij ≻ 0 −Bij ≺ 0 Υ Z p
n−1
J ⊂ V a i ∈ J
j∈J Υij ≻ 0
J = V a Υ
a Υ = [Υij] i, j ∈ V a i a
a ∈ Ni Υii Bia
j∈V a Υij 0 i ∈ V a
Υij = 0 if a /∈ Ni, and
j∈V a
j∈Ni
Υij = Bia ≻ 0, when a ∈ Ni.
j∈V a
A ≻ B A B A−B
≺

G a ∈ Ni i ∈ Va
J Va i ∈ J Va
Υij = 0 if Ni ⊆ J , and Υij =
Bij ≻ 0, otherwise.
j∈J
j∈J
j∈Ni\J
G J Va
i ∈ J J a
j∈J Υij ≻ 0
u2, . . . , un = 1 Υ ∈ M p
n−1 Υ
Υ + ΥT ∈ M p
n−1 ρ(D−1N) < 1
G Σz1, . . . , Σzm
m p− Σz
i ∈ V
ˆx a i (t) i ∈ V
lim
t→∞ ˆxa i (t) = ˆx a i ,
ˆx a i
ˆx a V Σˆx a
V
a ˆx cen
V Σˆx cen
V
ˆx cen
V = (I − Hcen) ˆx a V, Σˆx cen
V = (I − Hcen) Σˆx a V (I − Hcen) T ,
where Hcen = (1n ⊗ Ip) (1n ⊗ Ip) T /n.
ˆx cen
V
Σˆx cen
V
blkTr
Σˆx cen
V
Σˆx a V
blkTr Σˆx cen
V
a ∈ V
blkTr Σˆx a
, V
Tr Σˆx cen
≤ Tr Σˆx V
a
. V
Pij Qij p × p
Σˆx a V = [Pij] Σˆx cen
V = [Qij] i, j ∈ V
blkTr Σˆx a V
=
n
Pii.
P = [Pij] i, j ∈ {1, . . . , n}
blkTr(P ) = n
i=1 Pii
i=1

Σˆx cen
V
Qii = Pii − 1
n
n
j=1
(Pij + Pji) + 1
n 2
n
n
j=1 j ′ =1
Pjj ′,
i ∈ V
blkTr Σˆx cen
n n
= Qii = Pii − V
i=1 i=1
1
n n
Pij
n
i=1 j=1
− (1n ⊗ Ip) T Σˆx a V (1n ⊗ Ip)/n.
= blkTr Σˆx a V
Σˆx a V (1n ⊗ Ip) T Σˆx a V (1n ⊗ Ip) 0
blkTr Σˆx cen
− blkTr Σˆx V
a
0
V
A Tr(blkTr(A)) = Tr(A)
blkTr Σˆx cen
− blkTr Σˆx V
a
0 blkTr Σˆx V
cen
V
blkTr Σˆx a
V
Tr(Σˆx cen)
= Tr(blkTr(Σˆx cen
V V )) ≤ Tr(blkTr(Σˆx a V )) = Tr(Σˆx a V ).
Tr(Σˆx a V
) − Tr(Σˆx cen
V
) = 1
n
n n i=1 j=1 Tr(Pij).
ˆx a a
0
k Σˆx a V = [Pij] Pij = 0
i, j ∈ {1, . . . , k} blkTr(Σˆx cen
V ) = blkTr(Σˆx a V ) − n i=k+1
n
i=k+1
n
j=k+1 Pij/n 0
n
j=k+1 Pij/n,
i ∈ V ˆx cen
i
ˆx cen
V = ((ˆxcen 1 ) T , . . . , (ˆx cen
n ) T ) T
ˆx cen
1 + · · · + ˆx cen
n = 0 (1n ⊗
Ip)(I − Hcen) = 0 i j ˆx cen
i = ˆx cen
j − ˆx a j + ˆx a i
ˆx a V = ((ˆxa 1) T , . . . , (ˆx a n) T ) T
ˆx a V
ˆx cen
i (0) = 0 i ∈ V
t
ˆx cen
i (t + 1) =
Wi,j(ˆx cen
j (t) − ˆx a j + ˆx a i )
j∈Ni∪{i}
i ∈ V W = [Wi,j] Wi,j > 0
(i, j) ∈ E Wi,j = 0 j /∈ Ni Wi,i ∈ [α, 1] Wi,j ∈ {0} ∪ [α, 1]
i, j ∈ V α ∈ (0, 1]
−ˆx a j + ˆx a i ze
e = (j, i)
ˆze ˆz = (A ⊗ Ip) T ˆx a V , ˆz = ((ˆz1) T , . . . , (ˆzm) T ) T
t
ˆx a i , ˆx a j ˆx a i (t), ˆx a j (t)

i ∈ V
ˆx cen
i (t) ∈ R p t = 0 ˆx cen
i (0) = 0 t
i ∈ V ˆx cen
i (t)
ˆx cen
i (t + 1) =
j∈Ni∪{i}
Wi,j(ˆx cen
j (t) + ˆx a i (t) − ˆx a j (t)),
ˆx a i (t), ˆx a j (t) i j t
Wi,j
i ∈ V G
ˆx cen
i (t) i ∈ V
i ˆx cen
i
lim
t→∞ ˆxcen i (t) = ˆx cen
i .
ecen(t) = (ˆx cen
1 (t) − ˆx cen
1 ) T , . . . , (ˆx cen
n (t) − ˆx cen
n ) T T
n t G
t
||ecen(t)||2 ≤ λ t eff(W)||ecen(0)||2 + 2p(n − 1)σJλ t t
k ρ(J)
eff(W)
,
λeff(W)
J J = D −1 N D N
σJ J W
ecen(0)
t = 0
ecen(t)
ea(t) ∈ R (n−1)p
ecen(t) = (ˆx cen
1 (t)) T , . . . , (ˆx cen
n (t)) T T cen − ˆx ˆxcen 1 ) T , . . . , (ˆx cen
n ) T T
V
k=1
V = (ˆx cen
˜ea(t) = (ˆx a 2(t) T , . . . , ˆx a n(t) T T − ˆx a V a ˆx a V a = (ˆx a 2) T , . . . , (ˆx a n) T T
i = 1 a ea(t)
(0 T p , ˜ea(t) T ) T
j∈Ni∪{i} ˆxa i (t) = ˆx a i (t)
ˆx cen
V ˆx cen
V = (I − Hcen)ˆx a V
ecen(t) = (W ⊗ Ip)ecen(t − 1) + ((In − W) ⊗ Ip)ea(t − 1) + P ˆx a V,
P ˆx a V
P = I − (W ⊗ Ip) − (I − (W ⊗ Ip))(I − Hcen) = (I − (W ⊗ Ip))Hcen.

(W ⊗ Ip)Hcen = Hcen P = 0
ecen(t) = (W ⊗ Ip)ecen(t − 1) + ((In − W) ⊗ Ip)ea(t − 1) =
= (W ⊗ Ip) t t−1
ecen(0) + (W ⊗ Ip) t−k−1 ((I − W) ⊗ Ip) ea(k).
k=0
ecen(t)
ecen(t)2 ≤ λ t t−1
eff(W)ecen(0)2 + 2 λ t−k−1
eff (W)ea(k)2,
((W − I) ⊗ Ip) 2 ≤ 2 W
ea(t)2
˜ea(t) ∈ R (n−1)p ea(t) = (0, ˜e T a (t)) T J
VJ = [vp+1(J), . . . , vnp(J)] λJ = Diag (λp+1(J), . . . , λnp(J))
J = VJ λJ V −1
J ||vi(J)||2 = 1
˜ea(t)
k=0
˜ea(t) = J˜ea(t − 1) = J t ˜ea(0).
˜ea(0) σp+1, . . . , σnp
˜ea(0) =
np
i=p+1
σivi(J),
˜ea(t) t
˜ea(t) = VJλ t JV −1
J VJ [σp+1, . . . , σnp] T =
np
i=p+1
σivi(J)λ t i(J).
σJ = max np
i=p+1 |σi| ρ(J) = max np
i=p+1 |λi(J)| t ≥ 0
||˜ea(t)||2
||ea(t)||2 = ||˜ea(t)||2 ≤ p(n − 1)σJρ t (J).
ecen(t)2 ≤ λ t t−1
eff(W)ecen(0)2 + 2p(n − 1)σJ λ t−k−1
eff (W)ρ k (J),
k=0

ρ(J) = λeff(W)
||ecen(t)||2 ≤ λ t eff(W)||ecen(0)||2 + 2p(n − 1)σJλ t−1
eff (W)t,
ρ(J) = λeff(W)
||ecen(t)||2 ≤ λ t eff(W)||ecen(0)||2 +
2p(n − 1)σJ
ρ(J) − λeff(W) (ρt (J) − λ t eff(W)).
λeff(W) < 1 G λt eff (W)||ecen(0)||2
t → ∞
ecen(0) ρ(J) = λeff(W) λt eff (W)t
t ≥ λeff(W)
1−λeff(W) t → ∞ ρ(J) = λeff(W)
(ρ t (J) − λ t eff (W)) λeff(W)
ρ(J) < 1 limt→∞ ||ecen(t)||2 = 0
||ecen(t)||2 = 0 ecen(t) = 0
Z p n M p n
A = [Aij] A ∈ R np×np Aij p × p
i, j ∈ {1, . . . , n}
A Zp n Aij i, j ∈ {1, . . . , n}
Aij 0 i, j ∈ {1, . . . , n}, j = i. ˆ Zp n A ∈ Zp n
Aii ≻ 0 i ∈ {1, . . . , n} A M p n A ∈ ˆ Zp n
u1, . . . , un > 0
n
ujAij ≻ 0 for all i ∈ {1, . . . , n}.
j=1
A ∈ Zp n ∀J ⊂ {1, . . . , n}
i ∈ J
j∈J Aij
≻ 0 π
j≥i Aπ(i),π(j) ≻ 0 i ∈ {1, . . . , n}
A ∈ Zp n u1, . . . , un > 0
n
Aijuj 0, for all i ∈ {1, . . . , n}.
j=1
π {1, . . . , n}
Aπ(i),π(j)uπj ≻ 0, for all i ∈ {1, . . . , n}.
A ∈ M p n
j≥i

A + A T ∈ M p n, D = blkDiag (A11, . . . , Ann) , and A = D − N.
ρ (D −1 N) < 1
7
20 × 20 m
10
8
6
4
2
0
−2
−4
−6
−8
−10
−10 −8 −6 −4 −2 0 2 4 6 8
20 × 20 m
1.5
1
0.5
0
−0.5
−1
−1.5
−2
−2.5
R1
R3
R4
R2
R5
R7
R6
−9.5 −9 −8.5 −8 −7.5 −7 −6.5 −6 −5.5 −5
3 m

3 m
G
i
(T i G (t), Ri G (t))
(T
G , R G ) (T i G (t), Ri G (t))
i pG i G
x− y−
0.11 cm 2.12 cm
0.43
2
1.5
1
0.5
0
−0.5
−1
−1.5
−2
−2.5
−3
t=0
t=1
t=3
t=7
−9 −8.5 −8 −7.5 −7 −6.5 −6 −5.5 −5
t = 0 i
(T i G (t), Ri G (t))
(T i G (t), Ri G (t))
(T i j (t), Ri j(t))
t = 1, 3, 7
, R )
(T
G
G
n = 20
4 m ± π
2
x y 6 cm 1

−6
−6.5
−7.5
−8.5
0 4 8 10
1
0
−1
−2
0 4 8 10
1
0.5
0
−0.5
0 4 8 10
x− y−
R1 100
8
7
6
5
4
3
2
1
0
R 15
R 20
R 1
R 10
−5 −4 −3 −2 −1 0 1 2 3 4 5
4m
25 cm x− y−
13 cm 4
1.7
ρ(C −1 D)
ρ(I − M −1Υˆq a Va)
ρ(C −1 D) ρ(I−M −1 Υˆq a V a)
t = 50 ¯ θ a V
˜ θ a V 0.16◦
0.01 t = 100 8.5e−05
R 5

3.38 ◦ 1.87 ◦
x cm cm
y cm cm
4.03 ◦ 1.66 ◦
ρ(C−1D)
ρ(I − M −1Υˆq a Va)
t = 50 t = 100 t = 200
0.16 ◦ 0.01 ◦ 8.5e − 05 ◦
x cm cm cm
y cm cm cm
0.29 ◦ 0.12 ◦ 0.11 ◦
t = 200
200
¯ θa V ˜ θa V t = 200
1.64 cm 0.48 cm x− y− 0.11
t = 100 t = 50
1.74 cm 0.84 cm x− y− 0.29
˜θ a V i
j
i j
d 5% d x 0.7% d
y
2.5
t = 0 0

5
4
3
2
1
0
−1
−2
−3
−4
R 2
R 1
R 7
R 3
R 4
R 8
R 5
R 6
R 9
R 10
0 2 4 6 8 10
6
4
2
0
−2
−4
R 2
R 1
R 7
R 3
R 4
R 8
R 9
R 6
R 5
R 10
−6
−2 0 2 4 6 8 10 12
6
4
2
0
−2
−4
R 2
R 1
R 7
R 3
R 4
R 8
R 9
R 5
R 6
R 10
−6
−2 0 2 4 6 8 10 12
6
4
2
0
−2
−4
R 1
R 2
R 7
R 3
R 4
R 8
R 5
R 9
R 6
R 10
−6
−2 0 2 4 6 8 10 12
n = 20
4
i ∈ V Σ ˆx i V
blkTr(Σ ˆx i V )

−2
−2.5
−3
−3.5
−4
Centr.
t=0
t=1
t=3
t=7
t=15
t=31
t=63
t=100
−4.5
8 8.5 9 9.5 10 10.5 11 11.5
1
0
−1
−2
−3
−4
Centr.
t=0
t=200
t=400
t=600
t=800
t=1000
t=1200
t=1400
−5
−2 0 2 4 6 8 10
t = 0
10
8
6
4
2
0
R 6
R 14
R 20
R 7
R 8
R 2
R 13
R 19
R 11
R 15
R 3
R 16
R 18
R 17
R 4
R 9
R 5
R 12
R 10
R 1
0 2 4 6 8 10
11
10
9
8
7
6
R 8
R 14
R 19
R 13
R 18
R 3
5
1 2 3 4 5 6
R19
10
×10 e = (i, j) ∈ E
4 i
ze j e = (i, j)

i ∈ V
blkTr(Σˆx i )
V
blkTr(Σˆx cen)
V
R3 R12
R1
ˆx cen
V Σˆx cen
V
a = R1 ˆx a V Σˆx a
V
R1
λ 2
2.5
2
1.5
1
0.5
cen
R 3
R 12
R 1
0
0 0.5 1 1.5 2
λ
1
10
8
6
4
2
0
cen
R 1
0 2 4 6 8 10
blkTr(Σ ˆx i V )
blkTr(Σ ˆx i V ) i ∈ V
blkTr(Σˆx cen
V )
R1
R3 R12
R1
n 1000
ˆx a i (t) ˆx cen
i (t)
100
n
t i ∈ V
R1 R3 R12
100

35
30
25
20
15
10
5
0
0 200 400 600 800 1000
18
16
14
12
10
8
6
4
2
2.6
2.4
2.2
2
1.8
1.6
1.4
1.2
1
0.8
900 920 940 960 980 1000
100
0
0 200 400 600 800 1000
1.1
1.08
1.06
1.04
1.02
1
0.98
0.96
0.94
900 920 940 960 980 1000
100
100
100
i ∈ V
R1 R3 R12
9001000
6.5.5
9001000
6.5.5

35
30
25
20
15
10
5
0
0 200 400 600 800 1000
18
16
14
12
10
8
6
4
2
2.6
2.4
2.2
2
1.8
1.6
1.4
1.2
1
0.8
900 920 940 960 980 1000
100
0
0 200 400 600 800 1000
1.1
1.08
1.06
1.04
1.02
1
0.98
0.96
0.94
900 920 940 960 980 1000
100
100
100
i ∈ V
R1 R3 R12
9001000
6.5.5
9001000
6.5.5

35
30
25
20
15
10
5
0
0 200 400 600 800 1000
18
16
14
12
10
8
6
4
2
2.6
2.4
2.2
2
1.8
1.6
1.4
1.2
1
0.8
900 920 940 960 980 1000
100
0
0 200 400 600 800 1000
1.1
1.08
1.06
1.04
1.02
1
0.98
0.96
0.94
900 920 940 960 980 1000
100
100
100
i ∈ V
R1 R3 R12
9001000
6.5.5
9001000
6.5.5

k−

k
k
k k−
max − max −
k + 1
n

n
n
P
S
S
i Si
Si = HiP(k + 1|k)H T i + σ 2 z.

P(k + 1|k) = I Si
Si = 1 + σ 2 z + 2/r 2 ,
r = (xk+1 − xi) 2 + (yk+1 − yi) 2
Si
r
Pxx 0
P(k + 1|k) = blkDiag I,
0 Pyy
, I ,
Pyy = kPxx k > 1
y r xk+1 = r cos α
yk+1 = r sin α Si
Si = 1 + σ 2 z +
1 + Pxx
r 2
+ (k − 1)Pxx
r2 cos2 (α),
Si Si α =
0 + nπ, n ∈ Z α = π + nπ, n ∈ Z
2
∂2Si = 2(k − 1)Pxx
∂α2 r2 (1 − 2 cos2 (α)).
k > 1 Si α = 0 + nπ, n ∈ Z α =
π
2 + nπ, n ∈ Z Si
Si
r
α
π
2
90º
70º
50º
90º
α = 90, 70 50
α < π/2
A = (B1, . . . , Br) A Aii = Bi Aij = 0 i = j
70º
50º

ˆxr(k) = (ˆxk, ˆyk, ˆ θk) T
k ˆxi(k) = (ˆxi, ˆyi) T
k i ∈ {1, . . . , m} ˆx(k) = (ˆxr(k) T , ˆx1(k) T . . . , ˆxm(k) T ) T
k P(k) xg = (xg, yg, θg) T
xk k+1 ˆxr(k) xg
x k k+1 = (⊖ˆxr(k)) ⊕ xg,
⊖
⎡
−ˆxk cos
⊖ˆxr(k) = ⎣
ˆ θk − ˆyk sin ˆ θk
ˆxk sin ˆ θk − ˆyk cos ˆ θk
−ˆ ⎤
⎦ .
θk
⊕
ˆxr(k) xg
xk k+1 k
k + 1
x k ⎡
(xg − ˆxk) cos
k+1 = ⎣
ˆ θk + (yg − ˆyk) sin ˆ θk
−(xg − ˆxk) sin ˆ θk + (yg − ˆyk) cos ˆ ⎤
θk
⎦ .
θg − ˆ θk
¯x(k +
1) = x(k+1|k) = (¯xr(k+1) T , ¯x1(k+1) T . . . ¯xm(k+1) T ) T P(k+
1|k) ˆx(k), P(k) xk k+1 =
(xodom, yodom, θodom) Podom ¯xr(k + 1) = (¯xk+1, ¯yk+1, ¯ θk+1) T
¯xi(k+1) = (¯xi, ¯yi) T i ∈ {1, . . . , m} xk k+1
Podom = Diag σ2 x, σ2 y, σ2
θ σx = Kxd
σy = Kyd d = (xg − ˆxk) 2 + (yg − ˆyk) 2

¯x(k + 1) = ((¯xr(k + 1)) T , ¯x1, ¯y1, . . . , ¯xm, ˆym) T = ((ˆxr(k) ⊕ x k k+1) T , ˆx1, ˆy1, . . . , ˆxm, ˆym) T ,
P(k + 1|k) = J1P(k)J T 1 + J2PodomJ T 2 ,
⊕ ˆxr(k) x k k+1
ˆxr(k) ⊕ x k ⎡
ˆxk + xodom cos
k+1 = ⎣
ˆ θk − yodom sin ˆ θk
ˆyk + xodom sin ˆ θk + yodom cos ˆ ⎤
θk
⎦ ,
ˆθk + θodom
J1 J2
J1 = blkDiag(j1, I), J2 = ((j2) T , 0) T ⎡
1 0 −xodom sin
j1 = ⎣
,
ˆ θk − yodom cos ˆ θk
0 1 xodom cos ˆ θk − yodom sin ˆ ⎤
θk
⎦ ,
⎡
cos
j2 = ⎣
0 0 1
ˆ θk − sin ˆ θk
sin
0
ˆ θk cos ˆ θk 0
0 0 1
⎤
⎦ .
¯xi(k + 1) = (¯xi, ¯yi) T
¯xr(k + 1) = (¯xk+1, ¯yk+1, ¯ θk+1) T
−(¯xi − ¯xk+1) sin
hi(¯xr(k + 1), ¯xi(k + 1)) = atan2
¯ θk+1 + (¯yi − ¯yk+1) cos ¯ θk+1
(¯xi − ¯xk+1) cos ¯ θk+1 + (¯yi − ¯yk+1) sin ¯
.
θk+1
∂hi
Hi =
0 · · · 0
∂¯xr(k+1)
∂hi
∂¯xr(k + 1) =
∂hi ∂hi
∂¯xk+1 ∂ ¯yk+1
∂hi/∂ ¯ θk+1 = −1
∂hi
∂¯xk+1
∂hi
∂¯xi
= −
=
¯yk+1 − ¯yi
∂hi
∂ ¯ θk+1
(¯xk+1 − ¯xi) 2 2 ,
+ (¯yk+1 − ¯yi)
¯yk+1 − ¯yi
(¯xk+1 − ¯xi) 2 2 ,
+ (¯yk+1 − ¯yi)
∂hi
∂¯xi(k+1)
,
∂hi
∂¯yk+1
∂hi
∂¯yi
0 · · · 0
,
∂hi
∂¯xi(k + 1) =
=
= −
∂hi
∂¯xi
∂hi
∂ ¯yi
¯xk+1 − xi
(¯xk+1 − ¯xi) 2 2 ,
+ (¯yk+1 − ¯yi)
¯xk+1 − ¯xi
h H
,
(¯xk+1 − ¯xi) 2 2 ,
+ (¯yk+1 − ¯yi)
h = (h T 1 , . . . , h T n) T , H = (H T 1 , . . . , H T n) T .

ˆx(k + 1) = x(k + 1|k + 1) P(k + 1|k + 1)
ˆx(k + 1) = ¯x(k + 1) + K(z − h), P(k + 1|k + 1) = (I − KH)P(k + 1|k),
K = P(k + 1|k)H T S −1 , S = HP(k + 1|k)H T + R,
ν = z − h S K
z
z = h ν
ˆx(k + 1) = ¯x(k + 1) R
σ 2 zI σz
x = (xk+1, yk+1) f(x)
f(x) = Tr(P(k + 1|k + 1)),
P(k + 1|k + 1)
x(0) = (xk, yk), ˙x = −∇f(x),
x(0) = (xk, yk), x(t + 1) = x(t) − h∇f(x(t)),
(xk, yk) x(t) x t ∇f(x(t)
f x(t) h
˙x = 0 x(t + 1) − x(t)

8
6
4
2
0
−2
F1
Initial map
−4
−6 −4 −2 0 2
MAP at Step 2, features: 1
4 6
−4
−6 −4 −2 0 2 4 6
8
6
4
2
0
−2
Cost function for a map with a single feature
F1
0.25
0.2
0.15
0.1
0.05
5.5
5
4.5
4
3.5
3
Cost function for a map with a single feature
−3.5 −3 −2.5 −2 −1.5 −1
(x, y)
n
ˆxi Σi = Pi(k + 1|k + 1)
i ∈ {1, . . . , n}
ˆxG =
n
i=1
H T i Σ −1
i Hi
−1 n
i=1
H T i Σ −1
i ˆxi, ΣG = (
n
i=1
H T i Σ −1
i Hi)
−1
,
Hi
F1
0.25
0.2
0.15
0.1
0.05

8
6
4
2
0
−2
F1
Initial map
−4
−6 −4 −2 0 2
MAP at Step 2, features: 1
4 6
−4
−6 −4 −2 0 2 4 6
8
6
4
2
0
−2
F1
Cost function
x 10
10
−4
9
8
7
6
5
4
3
5.5
5
4.5
4
3.5
3
Cost function
−3.5 −3 −2.5 −2 −1.5 −1
(x, y)
F
F (x1, · · · , xn) =
m
j=1
min
i fij(xi)),
xi i k + 1 i ∈ {1, . . . , n} fij
j i
i xi
fij(xi) Pi(xi)
i xi
[Pi(xi)]jj Pi(xi) j
fij
fij(xi)) = ([Pi(xi)]jj).
F1
x 10
10
−4
9
8
7
6
5
4
3

i
1 · · · m
x 1 i fi1(x 1 i ) · · · fim(x 1 i )
xK i fi1(xK i ) · · · fim(xK i )
x l1
1
x ln
n
1 · · · m
f11(x l1
1 ) · · · f1m(x l1
1 )
fn1(xln n ) · · · fnm(xln n )
min mini fi1(x li
i ) · · · mini fim(x li
i )
i
K
K ≤ mi i ∈ {1, . . . , n}
i ∈ {1, . . . , n} K
x 1 i , · · · , x K i
x 1 i , · · · , x K i i
fij(x l i)
l ∈ {1, . . . , K} j ∈ {1, . . . , m} i ∈ {1, . . . , n} j
i fij(x l i) = ∞ l ∈ {1, . . . , K}
i fij
x l1
1 , · · · , xln n m j=1 mini fij(x li
i ))
F

W
λ2(W ) W
λ2(W )
λ2(LW )
λn(LW ) LW
λ2(LW ) W
λ2(LW ) LW = I − W
λ2(W ) = 1−λ2(LW )
n ∈ N i ∈ {1, . . . , n}
G = (V, E) V = {1, . . . , n} E
(i, j) ∈ E i j n × n
C G Cij = 0 (i, j) /∈ E j = i
Cii A ∈ {0, 1} n×n

G
n k k ∈ N
i, j t t ∈ Rt≥0
Aij [A]ij
(i, j) A
(b1, . . . , br) A Aii = bi Aij = 0
λi(A) vi(A) ith A
λA
(λ1(A), . . . , λr(A))
VA
A∞
[v1(A), . . . , vr(A)]
n ∞− A maxi j=1 |Aij|
A2
A maxi λi(AT A)
ρ(A)
A maxi |λi(A)|
Ir
r × r
0r 1r r
A
L L = diag(A1) − A
λ⋆(L) λi(L)
Aij =
1 (i, j) ∈ E
0
, for i, j ∈ V.
G Ni
i i Ni = {j | (i, j) ∈ E}
di i Ni dmax = maxi∈V di
L ∈ R n×n G
L = diag(A1) − A.
A L
L
λ1(L) ≤ λ2(L) ≤ · · · ≤ λn(L).
L
v1(L) = 1/ √ n λ1(L) = 0 L1/ √ n =
0
G
0 = λ1(L) < λ2(L) ≤ · · · ≤ λn(L); and

λn(L) ≤ 2dmax
G λ⋆(L) λ2(L)
L
i ∈ V
˙xi(t) = ui(t), with xi(t), ui(t) ∈ R,
ui i
ui(t) = −
(xi(t) − xj(t)).
j∈Ni
x = [x1, . . . , xn] T u = [u1, . . . , un]
˙x(t) = −Lx(t), x(0) = x0.
G
lim
t→∞ xi(t) =
n
xi(0)/n.
xi(t)
i
ei(t) i
xi(t) t ˆxi(t)
i=1
ei(t) = ˆxi(t) − xi(t), for i ∈ V,
e(t) = [e1(t), . . . , en(t)] T c1, α
c1 > 0 and 0 < α < λ⋆(L).
fi(t, ei(t)) = |ei(t)| − c1e −αt ,
i fi(t, ei(t)) > 0 i
ˆxi

i xi(t) xj(t)
(i) (ii)
x0
λ⋆(L)
λ⋆(L)
α
λ⋆(L)
λ⋆(L)
(i)
C∞
max−
(ii) C k 1
k
∞ ρ(C) C
C
n
i ∈ V
Ĉij(k) (i, j) k− C [C k ]ij j ∈ V
k = 0 i Ĉij(k)
Ĉii(0) = 1, and Ĉij(0) = 0 for j ∈ V \ {i}.
k ≥ 1 i
Ĉij(k + 1) =
Cij ′Ĉj ′ j(k), for j ∈ V.
j ′ ∈Ni∪{i}

i
j ∈ Ni
i n n k
C
k ≥ 0 k− C C k
i i− Ck k = 0
I C0 k ≥ 1
Ck+1 = CC k (i, j)
[C k+1 n
]ij = Cij ′[Ck ]j ′ j.
j ′ =1
C Cij ′ = 0 j′ /∈ Ni ∪ {i}
[C k+1 ]ij =
Cij ′[Ck ]j ′ j,
j ′ ∈Ni∪{i}
i
[ Ĉ(k)]ij = [C(k)C(k − 1) . . . C(0)]ij, for j ∈ V.
n
k j, j ′
Ĉj ′ j(k)
i
ID(i)
i ∈ V
li(k) ID(j) j ˜ Cij(k)
(i, j) k− C [C k ]ij j ID(j) ∈ li(k)
k = 0 i ∈ V ˜ Cii(k)
˜Cii(0) = 1, li(0) = {ID(i)},
Ni
k ≥ 1 i lj(k)
li(k)
li(k + 1) =
lj(k).
j∈Ni∪{i}

i ˜ Cij(k) ˜ Cij(k) = 0
j
ID(j) ∈ li(k + 1) and ID(j) /∈ li(k).
i ˜ Cij(k) ID(j) ∈ i(k + 1)
˜Cij(k + 1) =
j ′ ∈Ni∪{i},ID(j)∈ j ′(k)
Cij ′ ˜ Cj ′ j(k),
˜ Cij(k) ID(j) ∈ li(k)
li(k)
i ∈ V k ≥ 0
˜Cij(k) = 0 ID(j) /∈ li(k) C
k− C C k
li(0) = {ID(i)}, Ĉii(0) = 1, Ĉij(0) = 0, for j = i,
li(k + 1) =
j∈Ni∪{i}
{lj(k)}, and Ĉij(k + 1) =
j ′ ∈Ni∪{i}
Cij ′Ĉj ′ j, for all j ∈ V.
j /∈ li(k) Ĉij
k = 0 k i ∈ V
j /∈ li(k) Ĉij = 0 j k + 1 j /∈ li(k + 1)
j /∈
j ′ ∈Ni∪{i} {lj ′(k)} j′ ∈ Ni ∪ {i} Ĉj ′ j = 0
Ĉij(k + 1) =
j ′ ∈Ni∪{i}
Cij ′0 = 0.
˜ Cij(k) Ĉij(k)
k ≥ 0 i ∈ V j ∈ li(k) k ≥ 0 i ∈ V j /∈ li(k)
˜ Cij(k) Ĉij(k) = 0
ID(j) ∈ i(k + 1)
˜Cij(k + 1) =
j ′ ∈Ni∪{i},ID(j)∈ j ′(k)
Cij ′ ˜ Cj ′ j(k).
j /∈ lj ′(k) Ĉj ′ j(k) = 0 Ĉj ′ j(k) = ˜ Cj ′ j(k) k k + 1
˜Cij(k + 1) =
Cij ′Ĉj ′
j(k) +
Cij ′0,
j ′ ∈Ni∪{i},ID(j)∈ j ′(k)
Ĉij(k + 1)
j ′ ∈Ni∪{i},ID(j)/∈ j ′(k)

i [C k ]ij j ∈ V
i − th C k
i ID(i)
n
i ∈ V ki li(k) =
li(k − 1)
n = |li(ki)|
ki = min{k | li(k) = li(k − 1)}.
li(k − 1) (k − 1)− i
k
k li(k −1)
i
n = |li(ki)|
C
.
ρ(C) ≤ C, and ρ(C) = lim C
k→∞ k 1
k .
∞− .∞
C k ∞ = max
i∈V
n
|[C k ]ij|,
max−
i i − th C k
C k
ci(k) ˜ Cij(k)
ci(k) =
| ˜ Cij(k)|.
j=1
ID(j)∈li(k)
max − ci(k)
βi(k) = ci(k), βi(k + τ + 1) = max
j∈Ni∪{i} βj(k + τ),

T = diam(G) βi(T )
i ∈ V ci(k)
β1(k + T ) = · · · = βn(k + T ) = max
i∈V ci(k).
β ∗ i (k) i k ≥ 1
β ∗ i (k) = (βi(k + T )) 1
k = (max
j∈V
cj(k)) 1
k .
ci(k) i
i − th C k β ∗ i (k)
k T k + T
max −
T
max −
C
i
C k → ∞ β ∗ i (k)
ρ(C) C
k ≥ 1
lim
k→∞ β∗ i (k) = ρ(C), for all i ∈ V,
( √ n) −1
k β ∗ i (k) ≤ ρ(C) ≤ β ∗ i (k).
k−
C k ˜ Cij(k) = 0 ID(j) /∈ li(k) ci(k)
i − th C k max −
C k β ∗ i (k)
β ∗ i (k) = (C k ∞) 1
k
ρ(C) = lim
C
k→∞ k 1
k
ρ(C) = (ρ(C k )) 1
∞ = lim β
k→∞ ∗ i (k),
k ≤ C k 1
k
∞ = β ∗ i (k),
C
C2 = maxi λi(C 2 ) ρ(C) = maxi |λi(C)|
ρ(C) = C2 = C k 1
k
2 .

C k 2 C k ∞ C k
( √ n) −1 C k ∞ ≤ C k 2
( √ 1
−
n) k β ∗ i (k) = ( √ 1
−
n) k C k 1
k
∞ ≤ C k 1
k
2 = ρ(C),
n
λ⋆(L)
L
G C
λ⋆(L) ρ(C)
λ⋆(L)
ρ(C)
C L
C = I − βL − 11 T /n.
λ1(C) = 0, λi(C) = 1 − βλi(L), for i ∈ {2, . . . , n}.
ρ(C) λ⋆(L)
λ⋆(L) = (1 − ρ(C))/β, if 0 < β < 1/λn(L).
L + r11 T /n L i ∈
{2, . . . , n} r
λ1(L + r11 T /n) = r, and λi(L + r11 T /n) = λi(L) for i ∈ {2, . . . , n}.
VL L
VL = 1/ √ n, v2(L), . . . , vn(L)
= 1/ √ n, ˜
VL ,

L
V T L LVL = λL = diag(0, λ2(L), . . . , λn(L)),
L + r11 T /n
V T L
T 1T 1/n 1T VL/ ˜
L + r11 /n VL = λL + r
√ n
˜V T L 1/√n ˜ V T L 11T VL
˜
1 0
= λL + r
0 0
= diag(r, λ2(L), . . . , λn(L)),
1T 1 = n 1T VL
˜ = 0
L C
T
λi(C) = 1 − βλi L + (1/β)11 /n , for i ∈ V,
r = 1/β
β
C
β = ε/λn(L), for some ε ∈ (0, 1),
λ1(C) = 0, and for i ∈ {2, . . . , n},
λi(C) = 1 − ελi(L)/λn(L).
λn(L) ≥ λi(L) > 0 i ∈ {2, . . . , n}
1 > λ2(C) ≥ · · · ≥ λn(C) > λ1(C) = 0,
C λ2(C)
C = I − βL − 11 T /n
1/ √ n v1(C) C
λ1(C) = 0 k ≥ 1
C k = (I − βL − 11 T /n) k = (I − βL) k − 11 T /n,
I − βL
n n β
n β = ε/(2dmax)
β < 1/λn(L) max −
β
k
Ĉ = I − βL
ˆ λi(k) n k
n
n

ε ∈ (0, 1)
β = ε/(2n)
Ĉ = I − βL k ≥ 1 i ∈ V
Ĉij(k) (i, j) k− Ĉ [Ĉk ]ij
ID(j) ∈ li(k)
[ Ĉk ]ij = [C k ]ij + 1/n,
C
k i
ĉi(k) =
| Ĉij(k) − 1/n| + (n − |li(k)|)/n,
ID(j)∈li(k)
max − maxj∈V ĉj(k) ˆ β ∗ i (k)
ˆβ ∗ 1
i (k) = (max ĉj(k)) k .
j∈V
i ∈ V k ≥ 1
ˆλi(k) = 1 − ˆ
⋆
βi (k) /β.
i
k → ∞ ˆ λi(k)
λ∗(L)
lim ˆλi(k) = λ∗(L), for i ∈ V,
k→∞
k ≥ 1 λ⋆(L)
ˆλi(k) ≤ λ⋆(L) ≤ ( √ n) −1
k ˆ λi(k) + (1 − ( √ n) −1
k )/β.
β = ε/(2n) 0 < β < 1/λn(L) ε ∈ (0, 1) λn(L) ≤
2dmax < 2n dmax
λ∗(L) = (1 − ρ(C))/β C
C = I − βL − 11 T /n = Ĉ − 11T /n
i ∈ V Ĉij(k) [ Ĉk ]ij ID(j) ∈ li(k) [ Ĉk ]ij = 0
ID(j) /∈ li(k)
[C k ]ij = [ Ĉk ]ij − 1/n, for ID(j) ∈ li(k),
[C k ]ij = −1/n, for ID(j) /∈ li(k),
i ∈ V k ≥ 1 ĉi(k) i−
C k ˆ β ∗ i (k) ˆ β ∗ i (k) = C k 1
k
∞
( √ n) −1
k C k 1
k
∞ ≤ ρ(C) ≤ C k 1
k
∞,
C

˙x(t) = −Lˆx(t) = u(t), ei(t) = ˆxi(t) − xi(t),
|ei(t)| ≤ c1e −α(t)t ,
α
ˆα(k) = ˆ λ(k), for k ∈ N, and α(t) = γ ˆα(k), for t ∈ [k, k + 1),
0 < γ < 1 ˆ λ(k)
ˆα(k) = ˆ λ(k) ≤ λ⋆(L).
σz = 1 σx = 0.01d
σy = 0.01d σθ = 2.5
d

15
10
5
0
−5
−10
r1
r2
r3
gt
F6
−15
−20 −10 0 10
F16
15
10
5
0
−5
−10
gt
Global map
F20
F13 F15
F1
F6
F19
F14
F5
F4
F18
F7
F16
F10
F12 F8 F11
F9
F2
F17
F3
−15
−20 −10 0 10
15
10
5
0
−5
−10
r1
r2
r3
gt
F6
−15
−20 −10 0 10
F16
15
10
5
0
−5
−10
gt
Global map
F9
F13
F1
F5
F6 F7 F8
F12
F2
F17
F19
F3
F20
F15
F14
F18
F4 F16
F10
F11
−15
−20 −10 0 10
15
10
5
0
−5
−10
r1
r2
r3
gt
F6
−15
−20 −10 0 10
F16
15
10
5
0
−5
−10
gt
Global map
F9
F13
F1
F5
F6 F7 F8
F12
F2
F17
F19
F3
F14
F18
F4 F16
F10
F11
F20
F15
−15
−20 −10 0 10
ˆxG, ΣG

F 16 r1
F 16
F 16
F 16
F 6, F 7, F 8, F 3
n = 20
ˆ λi(k) i ∈ V k
ˆ λi(k)
k λ∗(L)
λ∗(L) ( √ n) −1
k λi(k) ˆ +
(1 − ( √ n) −1
k )/β λ∗(L) k
λ∗(L)
I − βL − 11T /n
y(k) = w(k)/normalization cons.(w(k)),
w(k + 1) = (I − βL)y(k) − deflation cons.(y(k)),
Tcons = 10, 25, 50, 100
I − βL w(k)
z(t + 1) = Wz(t)
W
ˆ λi(k)
( √ n) −1
k ˆ λi(k) + (1 − ( √ n) −1
k )/β
i ∈ V w(k +
1) T y(k)/y T (k)y(k)

10
8
6
4
2
0
0 2 4 6 8 10
10 × 10
e = (i, j) ∈ E
4
30
20
10
0
−10
−20
−30
λ * (L)
Estimated λ(k)
Upper−bound
−40
0 20 40 60 80 100
ˆ λi(k) i ∈ V
k
λ∗(L)
λ∗(L) ( √ n) −1
k ˆ λi(k) + (1 − ( √ n) −1
k )/β
λ∗(L) k λ∗(L)
λ∗(L)
Tcons
n

14
12
10
8
6
4
2
0
−2
λ * (L)
Estimated λ(k)
Upper−bound
PowerIt 10
PowerIt 25
PowerIt 50
PowerIt 100
0 200 400 600 800 1000
Iterations
λ∗(L)
Tcons = 10, 20, 50 100
λ∗(L) λ∗(L)
Tcons
14
12
10
8
6
4
2
0
−2
λ * (L)
Estimated λ(k)
Upper−bound
PowerIt 10
PowerIt 25
PowerIt 50
PowerIt 100
0 200 400 600 800 1000
Messages per agent
i n

14
12
10
8
6
4
2
0
−2
λ * (L)
Estimated λ(k)
Upper−bound
PowerIt 10
PowerIt 25
PowerIt 50
PowerIt 100
0 200 400 600 800 1000
Iterations
λ∗(L)
Tcons = 10, 20, 50 100
λ∗(L) λ∗(L)
Tcons
14
12
10
8
6
4
2
0
−2
λ * (L)
Estimated λ(k)
Upper−bound
PowerIt 10
PowerIt 25
PowerIt 50
PowerIt 100
0 200 400 600 800 1000
Messages per agent
i n

60m × 45m
30 m 30 m

20
10
0
−10
−20
−30
−40
−20 −10 0 10 20
9

15
10
5
0
−5
−10
−15
F23
−5 0 5 10 15 20 25
R8
R7
R9
R6
R1
R5
R2
R3
R4
R8
R9
R7
R6
R1
t ((t − 1) mod 9) + 1 (t
mod 9) + 1
I i G (t) ii G (t)
40 IG iG
x
4
R5
R2
R3
R4
R8
R7
R9
R6
R1
X
R5
R2
R3
R4

20000
10000
0
2000
1000
−1000
0 20 40
20000
10000
0
0 20 40
20000
10000
0
0 20 40
Ii G (t) Ii G (t) Ii G (t)
0
0 20 40
2000
1000
0
−1000
0 20 40
2000
1000
0
−1000
0 20 40
ii G (t) ii G (t) iiG (t)
x− 40
Ii G (t)
ii G (t)
xi G (t) ΣiG (t)
i xG
ΣG Qi G (t)
xi G (t) xG Σi G (t) nΣG
Qi G (t) ΣiG (t)

4
2
0
3000
2000
1000
x 10 4
0 20 40
4
2
0
x 10 4
0 20 40
4
2
0
x 10 4
0 20 40
Ii G (t) Ii G (t) Ii G (t)
0
0 20 40
3000
2000
1000
0
0 20 40
3000
2000
1000
0
0 20 40
ii G (t) iiG (t) ii G (t)
Ii G (t) iiG (t)
Ii G (t) x−
IG
γ = |λ2(W )|
W = 1
⎡
2
⎢ 1
⎢
3 ⎢ 0
⎢
⎣
1
1
0
1
. . .
0
0
0
0
1
1
0
1
0 0 0 1 2
γ = 0.96 1
t ((t − 1)
mod 9) + 1 (t mod 9) + 1 τ = 8
⎤
⎥ .
⎥
⎦

5.8
5.6
5.4
0.2
−0.2
0 20 40
5.8
5.6
5.4
0 20 40
5.8
5.6
5.4
0 20 40
xi G (t) xiG (t) xiG (t)
0
0 20 40
0.2
0
−0.2
0 20 40
0.2
0
−0.2
0 20 40
Σi G (t) ΣiG (t) ΣiG (t)
x− 40
xi G (t)
Σi G (t) i
xi G (t)
Qi G (t)
ΣG
Qi G (t)
Σi G (t)
nΣG
W (t) t
δ = 0.89
τ = 1
δ = 0.80
0.80 t ≤ 0.96 t ≤ 0.89 t/8 t = 0, 1, . . .
1 5 20

23
22.5
22
0.5
−0.5
0 20 40
23
22.5
22
0 20 40
23
22.5
22
0 20 40
xi G (t) xiG (t) xiG (t)
0
0 20 40
0.5
0
−0.5
0 20 40
0.5
0
−0.5
0 20 40
Σi G (t) ΣiG (t) ΣiG (t)
x− 40
xi G (t)
Σi G (t) i
xi G (t)
Qi G (t)
ΣG
Qi G (t)
Σi G (t)
nΣG
9
1
5 1
1 6 7
9
20
Ii G (t)
ii G (t)
5 5

15
10
5
0
−5
−10
−15
−5 0 5 10 15 20 25
15
10
5
0
−5
−10
−15
−5 0 5 10 15 20 25
xG ΣG
Time (secs.)
1.4
1.2
1
0.8
0.6
0.4
0.2
Fixed graph
Switching graph
Link failure graph
0
0 5 10 15 20 25 30 35 40
Iterations
Ii G (5)
5 5 9
I i G
(t) 9
9
8 9 8 9 15

1 7
16
γ = 1.8 h = 0.8 λ⋆ = 0.97
L = 500 K = 5
k k + 1 10
l = 25 k =
1, . . . , 4 L − (K − 1)l = 400
l = 50 300 k = K
l = (L/K) = 100
|[I k i (t)]r,s − [I k avg]r,s|/σI
|[i k i (t)]r − [i k avg]r|/σi L
k = 1, . . . , K − 1 l = 100
l = 50
l = 25

6
4
2
0
−2
−4
ST
R1
R2
R3
R4
R5
R6
R7
R8
−5 0 5 10 15 20
5
4
3
2
1
0
−1
−2
−3
−4
−5
R 1
R 8
R 7
R 2
R 3
R 5
R 4
R 6
0 2 4 6 8 10 12
7.5 m
k = K l = 25
l = 50 l = 100
l = 50
l = 100
l = 25 L ˆx k i (t) Σk i (t)
ˆx k G Σk G
k = K
l = 100 L
t l = 100
l = 100 k = 1

6
4
2
0
−2
k=1
k=2
k=3
k=4
k=5
−4
−8 −6 −4 −2 0 2 4 6 8
k = 1, . . . , 5
10 k = 4 k = 5
k = 4 k = 5
k = 4
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
l=25
l=50
l=100
0
k=1 k=2 k=3 k=4 k=5 end
3
2.5
2
1.5
1
0.5
l=25
l=50
l=100
0
k=1 k=2 k=3 k=4 k=5 end
maxi,r,s |[I k i (t)]r,s − [Ik avg ]r,s|/σI maxi,r |[i k i (t)]r − [ik avg ]r|/σi
L maxi,r,s |[I k i (t)]r,s −
[I k avg]r,s|/σI maxi,r |[i k i (t)]r −[i k avg]r|/σi l = 25
k = 1, . . . , K − 1
l = 100
k = 1, . . . , K − 1
l = 50
k = 2, . . . , K

6
4
2
0
−2
−4
ST
GM
R1
R2
R3
R4
R5
R6
R7
R8
−5 0 5 10 15 20
ˆx k i (t) Σk i (t) i = 1
L l = 100
ˆx k G ΣkG k = K
3
2.5
2
1.5
1
0.5
Bound
Algorithm 1
Zero−init
0
k=1 k=2 k=3 k=4 k=5 end
3
2.5
2
1.5
1
0.5
Fixed graph
Switching graph
0
k=1 k=2 k=3 k=4 k=5 end
maxi,r |[i k i (t)]r − [ik avg]r|/σi fk(t)αi/σi + gk(t) maxi,r |[i k i (t)]r − [ik avg]r|/σi
L l = 100
t l = 100
G
t k G
l = 100
G t = k
G G
l = 100

9 9
9
t
C
10%

10
0
−10
−20
−30
−40
−50
−20 −10 0 10 20
8
6
4
2
0
−2
−4
−5 0 5 10 15
Robot 1
Robot 2
Robot 3
Robot 4
Robot 5
Robot 6
Robot 7
Robot 8
Robot 9
9
9
1

(C)
msum
msum Gcom
Gcom
n
m

(C)
msum
msum Gcom
Gcom
n
m

(c)
msum

y
1000
500
−500
2000
1000
0
z 0
−1000 −500 0 500 1000 1500
x
y
1000
500
2000
0
1000
z 0
−500
0
x
500
1000
y
1000
0
−1000
2000
1500
1000
500
z
0
−500
x
0
500
y
1000
0
−1000
3000
2000
1000
z 0
−1000
0
x
1000
1000
500
0
−500
−1000
−1000 −500 0 500 1000
x
y
0 1000
2000
z

y
−1000 500 2000
x
y
z
−2000 0 2000
x
z

y
1000
0
−1000
3000
2000
1000
z
0
−1000
−2000
−2000
−1000
8000
6000
4000
2000
0
−2000
R3
R2
−3000 −2000 −1000 0 1000 2000 3000 4000 5000 6000
R2, R3, R6,
R9
R9
x
0
R6
1000
2000
3000

−200
200 400
0
R9
R7
R5
R2
R6
−600 −400−200 0 200 400
R8
R4
500
R3
0
R1
−500
−1000
R8
R8
R8
R2
R2
R2
R5
R4
R5
R4
R4
R5
500
500
500
R9
R9
R9
R3
R3
R3
0
0
0
R1
−200
200 400
R1
0 R7
R6
−500−200
200 400
0 R7
R6
−500−200
200 400
R1
0 R7
R6
−500
−600
−400−200 −600 −600
−400−200 −400−200
0 −1000
200 0 −1000
0 −1000
400
200 200
400
400
R9
R5
R2
R8
R4
500
R3 0
R9
R5
R2
R8
R4
500
R3
0
R9
R5
R2
R8
R4
500
R3
0
−200
200 400
0 R7
−600
−400−200
R1
R6
−500−200
200 400
0 R7
−600
−400−200
0 −1000
200
400
R1
R6
0
200
400
−500−200
200 400
0 R7
−600
−400−200
−1000
R1
R6
0
200
400
−500
−1000
R1 R1

R1
R1
2000
1000
0
−1000
−1000
2000
1000
0
−1000
−1000
0
0
1000
1000
2000
3000
2000
1000
−1000
−1000
Common
Exclusive (R1)
4000
3000
2000
1000
0
−1000
0
0
1000
2000
3000
0
−1000
Common
Exclusive (R1)
Exclusive (R3)
Exclusive (R6)
2000
1000
0
−1000
−1000
4000 0
3000
2000
1000
1000
2000
2000
3000
0
−1000
Common
Exclusive (R1)
Exclusive (R2)
Exclusive (R3)
Exclusive (R5)
Exclusive (R6)
Exclusive (R7)
Exclusive (R9)
4000
3000
2000
1000
2000
1000
0
−1000
−1000
0
1000
2000
3000
0
−1000
Common
Exclusive (R1)
Exclusive (R2)
Exclusive (R3)
Exclusive (R4)
Exclusive (R5)
Exclusive (R6)
Exclusive (R7)
Exclusive (R8)
Exclusive (R9)
4000
3000
2000
1000
2000
1000
0
−1000
−1000
0
1000
2000
3000
3000
0
−1000
0
−1000
Common
Exclusive (R1)
Exclusive (R3)
Exclusive (R5)
Exclusive (R6)
Exclusive (R7)
4000
3000
2000
1000
Common
Exclusive (R1)
Exclusive (R2)
Exclusive (R3)
Exclusive (R4)
Exclusive (R5)
Exclusive (R6)
Exclusive (R7)
Exclusive 4000 (R8)
Exclusive 3000 (R9)
2000
1000
R1 0 − 5

2000
1000
0
−1000
−1000
0
1000
2000
3000
−1000
0
1000
Common
Exclusive (R1)
Exclusive (R2)
Exclusive (R3)
Exclusive (R4)
Exclusive (R5)
Exclusive (R6)
Exclusive (R7)
Exclusive (R8) 4000
Exclusive 3000 (R9)
2000
R1

2000
1000
0
−1000
−1000
0
1000
2000
3000
−1000
0
1000
Common
Exclusive (R1)
Exclusive (R2)
Exclusive (R3)
Exclusive (R4)
Exclusive (R5)
Exclusive (R6)
Exclusive (R7)
Exclusive (R8) 4000
Exclusive 3000 (R9)
2000
R1

2000
1000
0
−1000
−1000
0
1000
2000
3000
−1000
0
1000
Common
Exclusive (R1)
Exclusive (R2)
Exclusive (R3)
Exclusive (R4)
Exclusive (R5)
Exclusive (R6)
Exclusive (R7)
Exclusive (R8) 4000
Exclusive 3000 (R9)
2000
R1

2000
1000
0
−1000
−1000
0
1000
2000
3000
−1000
0
1000
Common
Exclusive (R1)
Exclusive (R2)
Exclusive (R3)
Exclusive (R4)
Exclusive (R5)
Exclusive (R6)
Exclusive (R7)
Exclusive (R8) 4000
Exclusive 3000 (R9)
2000
R1

2000
1000
0
−1000
−1000
0
1000
2000
3000
−1000
0
1000
Common
Exclusive (R1)
Exclusive (R2)
Exclusive (R3)
Exclusive (R4)
Exclusive (R5)
Exclusive (R6)
Exclusive (R7)
Exclusive (R8) 4000
Exclusive 3000 (R9)
2000
R1

2000
1000
0
−1000
−1000
0
1000
2000
3000
−1000
0
1000
Common
Exclusive (R1)
Exclusive (R2)
Exclusive (R3)
Exclusive (R4)
Exclusive (R5)
Exclusive (R6)
Exclusive (R7)
Exclusive (R8) 4000
Exclusive 3000 (R9)
2000
R1

2000
1000
0
−1000
−1000
0
1000
2000
3000
−1000
0
1000
Common
Exclusive (R1)
Exclusive (R2)
Exclusive (R3)
Exclusive (R4)
Exclusive (R5)
Exclusive (R6)
Exclusive (R7)
Exclusive (R8) 4000
Exclusive 3000 (R9)
2000
R1

2000
1000
0
−1000
−1000
0
1000
2000
3000
−1000
0
1000
Common
Exclusive (R1)
Exclusive (R2)
Exclusive (R3)
Exclusive (R4)
Exclusive (R5)
Exclusive (R6)
Exclusive (R7)
Exclusive (R8) 4000
Exclusive 3000 (R9)
2000
R1

2000
1000
0
−1000
−1000
0
1000
2000
3000
−1000
0
1000
Common
Exclusive (R1)
Exclusive (R2)
Exclusive (R3)
Exclusive (R4)
Exclusive (R5)
Exclusive (R6)
Exclusive (R7)
Exclusive (R8) 4000
Exclusive 3000 (R9)
2000
R1

2000
1000
0
−1000
−1000
0
1000
2000
3000
−1000
0
1000
Common
Exclusive (R1)
Exclusive (R2)
Exclusive (R3)
Exclusive (R4)
Exclusive (R5)
Exclusive (R6)
Exclusive (R7)
Exclusive (R8) 4000
Exclusive 3000 (R9)
2000
R1

2000
1000
0
−1000
−1000
0
1000
2000
3000
−1000
0
1000
Common
Exclusive (R1)
Exclusive (R2)
Exclusive (R3)
Exclusive (R4)
Exclusive (R5)
Exclusive (R6)
Exclusive (R7)
Exclusive (R8) 4000
Exclusive 3000 (R9)
2000
R1

2000
1000
0
−1000
−1000
0
1000
2000
3000
−1000
0
1000
Common
Exclusive (R1)
Exclusive (R2)
Exclusive (R3)
Exclusive (R4)
Exclusive (R5)
Exclusive (R6)
Exclusive (R7)
Exclusive (R8) 4000
Exclusive 3000 (R9)
2000
R1

O(n)

O(n)

p

p

p

i ∈ V zi(0) ∈ R
W ∈ R n×n
≥0 Wi,j > 0 (i, j) ∈ E Wi,j = 0
j /∈ Ni Wi,i ∈ [α, 1] Wi,j ∈ {0} ∪ [α, 1] i, j ∈ V
α ∈ (0, 1] G i ∈ V
zi(t) t ≥ 0
n
zi(t + 1) = Wi,j zj(t),
j=1
t → ∞ zi(t) i ∈ V
lim
t→∞ zi(t) = z⋆ = 1
n
zj(0),
n
i ∈ V i zi(t)
Wi,j = 0
j /∈ Ni e(t) = (z1(t), . . . , zn(t)) T − (z⋆, . . . , z⋆) T
t t ||e(t)||2/||e(0)||2 < ɛ
n 2 log(ɛ −1 )
W ∈ R n×n
Wi,j =
⎧
⎨
⎩
1
1+max{|Ni|,|Nj|}
j=1
if j ∈ Ni, j = i,
0 if j /∈ Ni, j = i,
1 −
j∈Ni Wi,j, if j = i,
i, j ∈ V j = i |Ni| |Nj| i j
W

60×45m

20
10
0
−10
−20
−30
−40
−20 −10 0 10 20
9