Efficient Semidefinite Relaxation for Robust ... - Semantic Scholar

EFFICIENT SEMIDEFINITE RELAXATION FOR ROBUST GEOLOCATION OF UNKNOWN EMITTER BY A SATELLITE CLUSTER USING TDOA AND FDOA MEASUREMENTS Kehu Yang† , Lizhong Jiang† and Zhi-Quan Luo‡ †

ISN Lab, Xidian University South Taibai Road #2, 710071, Xi’an, China ‡ Dept of ECE, University of Minnesota 200 Union Street SE. Minneapolis, MN 55455 Email: † [email protected], † [email protected], ‡ [email protected] ABSTRACT

2. PROBLEM FORMULATION

In this paper, we consider the problem of geolocating an unknown emitter by a satellite cluster. We formulate the problem as the maximum likelihood location estimation by using TDOA and FDOA measurements and provide efficient convex relaxations for this nonconvex optimization problem. We also propose a formulation for robust geolocation in the presence of satellite orbit perturbations. Simulation results confirm the efficiency and superior performance of the convex relaxation approach as compared to the existing least squares based approach when large measurement noise and orbit perturbations are present. Index Terms— Emitter geolocation, maximum likelihood estimation, semidefinite programming, TDOA, FDOA. 1. INTRODUCTION

where

Recently, the problem of localizing a radiating emitter using time difference of arrival (TDOA) and frequency difference of arrival (FDOA) measurements, which are collected by a group of synchronized passive receivers, has received significant attention due to its importance to many applications such as surveillance, navigation, maritime search and rescue, and wireless sensor networks. There have been a number of existing approaches for this localization problem by using either TDOA measurements [2] or TDOA and FDOA measurements [1, 3, 4]. However, these approaches have their own inherent limitations in practical applications. For example, the SDP methods [2] by using only TDOA measurements require at least four receivers to guarantee a unique solution. The weighted least squares (WLS) [1] is quite sensitive to sensor location errors and large measurement noise. The GMM-ITS [3] by using two moving sensors is only for localizing an emitter on the flat earth. The multidimensional scaling analysis (MDS) [4] requires at least five sensors, or there is no noise subspace. In practice, the number of formation-flying satellites for geolocation of unknown emitter is often limited to few, typically 3, due to the cost and the feasibility. Therefore, for a cluster of 3 formationflying satellites, TDOA and FDOA information at one measurement time will provide 4 independent measurement equations to solve the location of the unknown emitter. It is obvious that the solution is unique. In this paper, we formulate this geolocation problem as the maximum likelihood location estimation using TDOA and FDOA measurements, and provide efficient convex relaxations for this nonconvex optimization problem. We also propose a formulation for robust geolocation in the presence of satellite orbit perturbations.

978-1-4577-0539-7/11/$26.00 ©2011 IEEE

We consider a scenario whereby a cluster of passive satellite receivers collaborate to localize an unknown emitter on the earth, with the satellite orbits assumed to be known (possibly with perturbations). In the ensuing mathematical formulation, we use the notation M to denote the number of satellites in the cluster, x ∈ R3 the location of the unknown emitter, s˜j and sj respectively the true and estimated location of the j-th satellite at given measurement time, ˜s˙ j and s˙ j respectively the corresponding true and estimated velocity, Δsj the location error corresponding to the j-th satellite, ε (> 0) the bound of the location errors, c the speed of light, and I  {1, 2, · · · , M } the index set. Define (1) s˜i  si + Δsi ∈ Ai (ε)

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˘ ¯ Ai (ε)  si + ei , ei  ≤ ε , i ∈ I.

(2)

Also, we define τio 

‚ 1‚ ‚x − s˜i ‚, c

i ∈ I.

(3)

According to the above definition, for static emitters the derivative of τio with respect to time t can be denoted by 1 ‚ (s˜i − x)T s˜˙i . νio = dτio /dt = ‚ c‚x − s˜i ‚

(4)

The unknown emitter localization using TDOA and FDOA measurements can be straightforwardly formulated as the following maximum likelihood (ML) estimation problem min

x,t,f

M M M M 1 XX 1 XX 2 (t − t − τ ) + (fi − fj − νij )2 i j ij σT2 i=1 j>i σF2 i=1 j>i

subject to

‚ 1‚ ‚x − s˜i ‚ = ti , c 1 ‚ ‚ (s˜i − x)T s˜˙i = fi , c‚x − s˜i ‚

i ∈ I.

(5)

where τij = τio − τjo + nij and νij = νio − νjo + vij are respectively the TDOA and FDOA measurements, and the measurement noise nij , vij (i, j ∈ I, i > j) are assumed to be independent and Gaussian distributed with zero mean and variances of σT2 and σF2 , respectively.

ICASSP 2011

Notice that (5) is a nonlinear and non-convex problem. Denote the objective function of (5) as θ and notice that j» – ff j» – ff T t F f θ = tr E + tr V , (6) T T 1 1 t f where t  [t1 , · · · , tM ]T , f  [f1 , · · · , fM ]T , T  ttT , F  ffT, » – » – GT G −GT τ GT G −GT ν E , V  , T T T T τ τ ν ν −τ G −ν G ˜T ˆ τ = τ12 , · · · , τ1M , τ23 , · · · , τ2M , · · · , τM −1,M , ˆ ˜T ν = ν12 , · · · , ν1M , ν23 , · · · , ν2M , · · · , νM −1,M , (7) and G is a constant matrix defined similar to that in [2]. It is obvious that E  0 and V  0. ‚ ‚ ‚ In (6), T ii and T ij can be represented by the constraint x − s˜i ‚/c = ti in the same way performed in [2]. The constraint (s˜i − ‚´ ` ‚ x)T s˜˙i / c‚x − s˜i ‚ = fi in (5) can be equivalently written as fi ti =

1 (s˜i − x)T s˜˙i , c2

for all J ⊆ I, X X 2 X“ 1 ‚ ‚ ‚ ‚ ” ‚x − s˜i ∓ s˜˙i ‚2 − 1 ‚s˜˙i ‚2 (ti ± fi )2 ≤ fi + c2 c2 i∈J i∈J i∈J ` ´` ´T =⇒ t ± f t ± f  F + R± » ` ´T – 1 t±f ⇐⇒  0, t ± f F + R± the ML estimation problem (5) can be cast into the following SDP: – ff j» – ff j» F f T t E + tr V tr min T T 1 f 1 t x,z,T ,t,F ,f + δ1 subject to T ii =

(9)

Substituting t2i =

‚

1 ‚ x c2

‚ ‚ ‚ 1‚ ‚x − s˜i ∓ s˜˙i ‚2 − 1 ‚s˜˙i ‚2 . c2 c2

(10)

‚2 − s˜i ‚ into (9), we have

(ti ± fi )2 = fi2 + For ∀J ⊆ I, we have X

(ti ± fi )2 =

i∈J

X i∈J

fi2 +

X“ 1 ‚ ‚ ‚ ‚ ” ‚x − s˜i ∓ s˜˙i ‚2 − 1 ‚s˜˙i ‚2 , 2 2 c c i∈J

and X X 2 X“ 1 ‚ ‚ ‚ ‚ ” ‚x − s˜i ∓ s˜˙i ‚2 − 1 ‚s˜˙i ‚2 (ti ± fi )2 = fi + 2 2 c c i∈I i∈I i∈I ` ±´ 2 2 (11) ⇐⇒  t ± f  = f  +tr R “` ` ´ ´` ´T ” = tr F + R± , (12) ⇐⇒ tr t ± f t ± f

R± ij

–T » 1 I s˜i ± s˜˙i = 2 xT −1 c 1 ` ´T ` ˙ ´ − 2 s˜˙i s˜j , c

x z

–»

s˜j ± s˜˙j −1

»

I xT

x z

–»

s˜i −1

–

By using Nt TDOA and FDOA measurements, the robust geolocation of an unknown emitter in the presence of satellite orbit perturbations can be performed by maximizing the worst-case likelihood function and formulated by min v1 + v2 (»

–˛˛ ˛ ˛ ˛ Δsik  ≤ ε, ff ≤ v1 ik = mod(i − 1, M ) + 1, i ∈ L – » –T » f f V ≤ v2 1 1

subject to sup (13)

where z = xT x. By using the following relaxations: – » I x  0, xT x ≤ z ⇐⇒ T z x – » T t ttT  T ⇐⇒  0, T 1 t – » F f f f T  F ⇐⇒  0, T 1 f

–T »

3. ROBUST GEOLOCATION IN THE PRESENCE OF SATELLITE ORBIT PERTURBATIONS

x,t,f

–

s˜i −1

F ii

i=1

where δ1 and δ2 are positive constants for penalization. Notice that suitable δ1 and δ2 are often needed to have a good solution. The formulation (14) can be easily extended to the case where Nt independent TDOA and FDOA measurements are available.

where the entries of R± in (11) and (12) are denoted by »

1 c2

M X

, ˛» –» –T » –˛˛ ˛ 1 ˛ s˜i I x s˜j ˛ T ij ≥ 2 ˛ ˛, −1 ˛ xT z c ˛ −1 –» –T » – » 1 I x s˜j ± s˜˙j s˜i ± s˜˙i R± = ij xT z −1 −1 c2 1 ` ˙ ´T ` ˙ ´ − 2 s˜i s˜j , c – » » ` ´T – T t 1 t±f  0,  0, tT 1 t ± f F + R± – » – » I x F f  0,  0, i, j ∈ I, (14) T T 1 x z f

(8)

1 (s˜i − x)T s˜˙i . c2

T ij + δ2

i=1 j=1

which is also equivalent to fi2 + t2i ± 2fi ti = fi2 + t2i ± 2

M X M X

ti = where

t 1

–T

»

E

t 1

(15)

(˜ si − x)T ˜s˙ i 1 ˜ i  , fi = x − s , i∈L ˜i  c c x − s

h iT (1) (1) (N ) (N ) t = [t1 , . . . , tM Nt ]T = t1 , . . . , tM , . . . , t1 t , . . . , tM t iT h (1) (1) (N ) (N ) f = [f1 , . . . , fM Nt ]T = f1 , . . . , fM , . . . , f1 t , . . . , fM t

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i h t) t) ˜ = [˜ ˜M Nt ] = s ˜(1) ˜(1) ˜(N ˜(N S s1 , . . . , s ,...,s 1 ,...,s 1 M ,...,s M i h ˜˙ = [˜s˙ 1 , . . . , ˜s˙ M Nt ] = ˜s˙ (1) , . . . , ˜s˙ (1) , . . . , ˜s˙ (Nt ) , . . . , ˜s˙ (Nt ) S 1 1 M

where λ ≥ 0,

Here we assume that s˙ i = ˜s˙ i for a short period of time (this is reasonable in practice). According to (1), the satellite location due to orbit perturbation can be denoted by

ΔsTik (x − si ) 1 x − si  − + o (Δsik ) , c c x − si  „ (x − si )T s˙ i s˙ i fi = + ΔsTik c x − si  c x − si  « (si − x) (x − si )T s˙ i − + o (Δsik ) . c x − si 3

ˆ ,ˆ x,z,T t,F ,f ,v1 ,v2 ,λ

(17)

(18)

(19)

(20)

Under the first-order approximation, (19) can be denoted by fi tˆi =

–˛ ˆt −  ˆt −  ˛˛ subject to sup E ˛ 1 1 ˛ Δsik  ≤ ε, ff ≤ v1 , ik = mod(i − 1, M ) + 1, i ∈ L – » –T » f f V ≤ v2 , 1 1 1 tˆi = x − si  , c ˛ ˛ ˛ ˛ 2ρ ˛fi tˆi − 1 (si − x)T s˙ i ˛ ≤ √ s˙ i  , ˛ ˛ c2 c ζ M Nt i ∈ L, (21) √ where  ≤ ρ = ε M Nt /c is used instead of the box constraint |i | ≤ ε/c (i ∈ L) and ζ ≥ 1 is a penalty factor to tighten the last inequality constraint. Similar to the manipulation in [2] by using S-Procedure, the first constraint in (21) is equivalent to # " » – A −Aˆt − b I 0 ` ´T λ , (22) 2 0 −ρ μ − v1 − Aˆt + b –T

#

) E

,

i∈L

(23)

where D = diag(s˙ 1  , · · · , s˙ M ), a diagonal matrix under matlab notation, δ1 and δ2 are positive constants for penalization, and ρ should be chosen according to the satellite orbit perturbations.

and the problem (15) can be reformulated as min v1 + v2 (»

i,j∈L

λ ≥ 0, ζ ≥ 1, i, j ∈ L, j > i,

ΔsTik s˙ i ΔsTik (si − x) (si − x)T s˙ i (si − x)T s˙ i + − , 2 2 c c c2 si − x2

x,t,f

ˆt 1

subject to " # λI − A Aˆt + b ` ´T  0, Aˆt + b v1 − μ − λρ2 # ) (" – ff j» Tˆ ˆt F f V ≤ v2 , E , tr μ = tr T T ˆt f 1 1 –» –T » – » 1 I x si si Tˆ ii = 2 , −1 −1 xT z c ˛» –» –T » –˛˛ ˛ 1 I x s s ˛ ˛ i j Tˆ ij ≥ 2 ˛ ˛, −1 ˛ xT z c ˛ −1 –» –T » – » 1 1 I x sj ± s˙ j si ± s˙ i R± − 2 s˙ Ti s˙ j , T ij = 2 −1 −1 z x c c " # ` ´ ˆt ± f T ` 1 ´  0, ˆt ± f F + R± + c ζ √4ρM N D t # " – » – » Tˆ ˆt I x F f  0,  0,  0, T ˆt xT z fT 1 1

Let ΔsTik (x − si ) 1 , tˆi = x − si  , i = c c x − si  ˜T ˆ ˆt = tˆ1 , . . . , tˆM Nt ,  = [1 , . . . , M Nt ]T .

Tˆ ˆtT

T A, b and C are entries of E as denoted by (16), and Tˆ  ˆtˆt . T Define F  f f . By following the formulation steps for (14), (21) can be cast into the following SDP: X X Tˆ ij + δ2 v1 + v2 + δ1 F ii min

based on which ti and fi can be represented in the context of the first-order Taylor expansion: ti =

Tˆ

μ  ˆt Aˆt + 2b t + C = tr

M

L  {1, 2, · · · , M Nt }, mod(x, y) denotes modulus after division, E and V are defined similar to (7). And E can be represented by – » A b . (16) E= T C b

˜i = si + Δsik , ik = mod(i − 1, M ) + 1, i ∈ L, s

(" T

4. SIMULATION RESULTS

»

In the simulation, geolocation is performed by 3 formation-flying satellites, whose true locations (×106 m) and velocities (×103 m/s) are obtained by STK (Satellite Tool Kit) software : 2 3 0.36342996 0.11357633 0.23854764 ˜ = 4 3.48075996 3.48503735 3.52664895 5 , S 6.02885310 6.03626176 6.00838440 2 3 −7.551450 −7.560729 −7.557304 ˜˙ = 4 0.197114 0.061600 0.130985 5 . S 0.341411 0.106695 0.223161 The emitter location on the earth surface is generated as a sample of random variable x0 = [0.37780202, 3.15597773, 5.52147726]T × 106 m, which is within the coverage of a satellite formation [here we use S-type formation]. We here consider two geolocation scenarios, one is with the orbit perturbations while the other is without. There are two procedures to solve (14) and (23), respectively. Procedure 1: Step 1: Choose a pair of (δ1 , δ2 ) (δi ∈ [10−6 , 10−1 ]). Use solver SDPT3 in the matlab package CVX to solve the SDP (14), ˆ of the unknown emitter; and obtain the location estimate x

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ˆ as a starting point for a further local optimizaStep 2: Use x tion by applying any standard nonlinear optimization routine to the objective function of (5) [we use a Newton type method]. 3

10

Step 2: Same as that in Procedure 1. Using these two procedures, we plot the RMSE of the geolocation error of the unknown emitter versus the TDOA and FDOA measurement noise and satellite orbit perturbations in Fig.1 and Fig.2 for the two scenarios, respectively. In Fig.1, σt and σf denotes the TDOA and FDOA measurement noise standard deviation respectively. σf = nσt means that the FDOA measurement noise standard deviation is assumed n times that of TDOA,“TF-1-CRLB” denotes corresponding CRB, and so on. It is seen that the RMSE can approach the CRB for all the situations and the FDOA measurements play more important role than that of the TDOA when its measurement noise is smaller than that of the TDOA. In Fig.2, we set orbit perturbations as truncated Gaussian√ distribution with σs = 25m, i.e., the orbit standard deviation is 3σs2 . We set ζ = 10. It is seen that the RMSE also approach the CRB for almost all the situations. In order to compare the performance of our proposed SDP robust approach with the WLS method [1], we plotted in Fig. 3 the RMSE curves (versus the measurement noise and orbit perturbations) of the proposed robust geolocation method (abbreviated as Robust-SDP) and the weighted least squares method (abbreviated as WLS), respectively. Due to limited space, we do not list the five satellite locations and velocities, or the location of the unknown emitter. It is seen from Fig. 3 that our proposed robust approach still achieve the CRB when larger measurement noise and orbit perturbations are present. For each scenario, a total of 3000 Monte Carlo runs are performed. Our methods are implemented using Matlab v.7.1 on a HP personal computer with a 2.8-GHz pentium dual core CPU. Each simulation run including local search requires less than 0.5s.

TFí1(σ =σ )

2

10

f

t

TFí1íCRLB(σ =σ ) f

t

TFí2(σ =0.5σ ) f

t

TFí2íCRLB(σ =0.5σ ) f

t

TFí3(σ =0.1σ ) f

t

TFí3íCRLB(σf=0.1σt) TFí4(σ =0.05σ ) f

t

TFí4íCRLB(σ =0.05σ ) f

1

10

5

10

15

20

σt ( ns )

t

25

30

Fig. 1. RMSE of geolocation without orbit perturbations

3

10

N =1,σ =σ t

RMSE (m)

−6 −1 Step √ 1: Choose a pair of (δ1 , δ2 ) (δi ∈ [10 , 10 ]) and set ρ = 3M Nt σs and ζ ≥ 1. Use solver SDPT3 in the matlab packˆ; age CVX to solve the SDP (23), and obtain the location estimate x

RMSE ( m )

Procedure 2:

f

t

N =1íCRLB,σ =σ t

f

t

N =1,σ =0.5σ t

f

t

N =1íCRLB,σ =0.5σ t

2

10

f

t

N =3,σ =σ t

f

t

N =3íCRLB,σ =σ t

f

t

N =3,σ =0.5σ t

f

t

N =3íCRLB,σ =0.5σ t

f

t

N =5,σ =σ t

f

t

N =5íCRLB,σ =σ t

f

t

N =5,σ =0.5σ t

t

N =5íCRLB,σ =0.5σ t

1

10

f

5

10

15

f

20

t

t

25

σ (ns) (σ = 25m)

30

s

Fig. 2. RMSE of robust geolocation using Nt measurements with orbit perturbations

5. REFERENCES [1] K.C.Ho, Xiaoning L and L.Kovavisaruch, “Source Localization Using TDOA and FDOA Measurements in the presence of Receiver Location Errors: Analysis and Solution”, IEEE Trans. Signal Process., vol. 55, No.2, Feb. 2007.

[3] Darko Musicki, Regina Kaune and Wolfgang Koch, “Mobile Emitter Geolocation and Tracking Using TDOA and FDOA Measurements”, IEEE Trans. Signal Process., vol. 58, No.3, pp. 1863-1874, Mar. 2010. [4] He-Wen Wei, Rong Peng, Qun Wan, Zhang-Xin Chen and Shang-Fu Ye, “Multidimensional Scaling Analysis for Passive Moving Target Localization With TDOA and FDOA Measurements”, IEEE Trans. Signal Process., vol. 58, No.3, pp. 16771688, Mar. 2010.

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WLS RobustíSDP CRLB(σ = 0.5σ )

5

10

f

t

4

10

RMSE ( m )

[2] Kehu Yang, Gang Wang, and Zhi-Quan Luo, “Efficient Convex Relaxation Methods for Robust Target Localization by a Sensor Network Using Time Differences of Arrials”, IEEE Trans. Signal Process., vol. 57, No.7, July 2009.

6

10

3

10

2

10

1

10

0

10

0

1

2

3

4

5

6

σ ( ns ) (σ = cσ ) t

s

7

8

9

10

t

Fig. 3. RMSE of robust SDP and WLS: σs = cσt and σf = 0.5σt (5 satellites with orbit perturbations)