RSS-Based Sensor Localization in Underwater Acoustic Sensor Networks 1 2 3 2 Tao Xu , Yongchang Hu , Bingbing Zhang and Geert Leus
Tsinghua University, China, & Tianjin 712 Communication Broadcasting Co. Ltd, 2 Circuits and Systems (CAS) group, Delft University of Technology, the Netherlands 3 Fac. EE, National University of Defence Technology, 410073, Changsha, China 1
Objectives
Optimization Problems
CRLB
Simulation Results
Underwater acoustic (UWA) sensor localization approaches 2 received signal strength (RSS) measurement
Non-convex but Maximum likelihood (ML) problem:
In R2, 4 anchors located at (59, 26), (35, 5),(10, 40) and (26, 1).
10 anchors is randomly deployed; β = 2; use CVX to solve SDPs.
argmin
N X
x∈RM
+ 10βlog10di + α di)
2
55 RSS−based, α(f)=0.001
i
(f)
RSS−based, α =0.1
x∈R (f ) γi
= 10
2.0
and
1.5
(f )
+ γ di −
(f ) λi
1
1.0
0.5
=
(f ) (f ) γi α ln 10
FDRSS−based, [0.001, 0.1] 40
0.85 0.8
35
0.75
0.65 100 80
100 60
80
10β
80
100 60
60
40 0
60
y−axis
x−axis
20 0
0
30 25
40
20
20 0
y−asix
80 40
40
20
x−axis
• TOA and TDOA: Synchronization is expensive,
20
unpredictable velocity; • AOA: no line-of-sight (LOS) ; • XRSS: practical simplicity of implementation;
15
complicated underwater channels and hence very few attention: • For certain water depths, the Urick propagation model shall be considered; • The transmission loss: ψTL(f, d) = 10βlog10d + α(f )d + ξ (f,d) • β is the path-loss exponent (PLE); • α(f ) is the absorption coefficient (dB/km) and obtained
by Thorp’s empirical formula f2 f2 0.11 1+f 2 + 44 4100+f 2 + 2.75 × 10−4f 2 + 0.003; • ξ (f,d) contains residual factors; • The
RSS measurement: (f ) (f ) (f ) di (f ) Pi = P0 − 10βlog10 d0 − α (di − d0) + ni ,
• di = kx − sik2 > 0 and d0 is the reference distance,
normally d0 = 1 m (f ) • P0 is measured power at d0, i.e., the transmit power. (f ) • ni = ξ (f,0) − ξ (f,d) is the Gaussian distributed measurement noise;
Important Result
10 5 −8
We introduce a new smooth optimization model for RSS-based acoustic localization. 2 We propose a semi-definite programming (SDP) approach using the RSS measurement. 3 Another SDP approach is also developed using the FDRSS measurement. 1
−6
−2
0 2 Variance of n(f) (dB) i
4
6
8
10
4
6
8
10
35
N=8, Good Placement N=4, Good Placement N=8, Bad Placement
30
RSS-based Approach
−4
FDRSS-based Approach 25
• First introduce two " # " new variables: # ddT d . d h T i •D = d 1 = ; T 1 d 1# " # " xxT x . x h T i •X = x 1 = ; T 1 x 1 2 • d collects all di, di = [D]i,i and di = [D]N +1,i; T h i I −s x i T • [D]i,i = x 1 T T = Trace [XSi] −si si si 1
• Then
the semi-definite relaxation (SDR) dumps the rank constrains Rank(X) = Rank(D) = 1. • Finally, introduce the slack variables ti and solve the semi-definite programming (SDP): min
N X
D,X,ti i
ti, (f ) λi [D]i,i
(f )
s.t. − ti < + γ [D]N +1,i − 1 < ti [D]i,i = Trace [XSi] X 0, D 0 [X]M +1,M +1 = [D]N +1,N +1 = 1
RMSE(m)
• However,
i
0.9
0.7 0 100
RSS−based, discard λ (f)
45
0.95
N X (fk ) 2 λ i di i
(f ) (f ) Pi −P0 −α(f ) 10β
1
RMSE (m)
minM
RSS−based, α(f)=0.01
50
Convex problem:
• Location-awareness
is important task for underwater acoustic sensor networks. • Measurements for localization:
(f )
Cost Function
Introduction
(f ) (Pi
Cost Function
1
• The
frequency differential RSS (FDRSS) measurement: (∆) (∆) (∆) (∆) Pi = P0 − α (di − d0) + ni , k 6= p
(∆) • Pi • (∆)
(fk ) Pi (fk )
20
15
(fp) Pi (fp)
(fp) (fk ) (∆) and P0 = P0 − P0 ; (fp) (∆) (fk ) and ni = ni − ni
= − α =α −α • Apply least squares (LS) criterion on (∆) (∆) (∆) α di − ηi = ni with (∆) (∆) (∆) (∆) ηi = P0 + α − Pi • Similarly, solve the SDP: min
N X (∆)2
D,X i
α
[D]i,i −
(∆) (∆) 2α ηi [D]N +1,i
s.t. [D]i,i = Trace [XSi] X 0, D 0 [X]M +1,M +1 = [D]N +1,N +1 = 1
10 −8
−6
−4
−2
0 2 (f) Variance of ni (dB)
Contact Emails: •
[email protected]
+
(∆) ηi
2
,
(Tao Xu) •
[email protected] (Yongchang Hu)
