Robust Convex Approximation Methods for TDOA-Based Localization - PowerPoint PPT Presentation

Robust Convex Approximation Methods for TDOA-Based Localization under NLOS Conditions Anthony Man-Cho So Department of Systems Engineering & Engineering Management The Chinese University of Hong Kong (CUHK) (Joint Work with Gang Wang) DIMACS Workshop on Distance Geometry Theory and Applications 29 July 2016

Source Localization in a Sensor Network • Basic problem: Localize a signal-emitting source using a number of sensors with a priori known locations • Well-studied problem in signal processing with many applications [Patwari et al.’05, Sayed-Tarighat-Khajehnouri’05] : – acoustics – emergency response – target tracking – ... • Typical types of measurements used to perform the positioning: – time of arrival (TOA) – time-difference of arrival (TDOA) – angle of arrival (AOA) – received signal strength (RSS) • Challenge: Measurements are noisy A. M.-C. So, SEEM, CUHK 29 July 2016 1

TDOA-Based Localization in NLOS Environment • Focus of this talk: TDOA measurements – widely applicable – better accuracy (over AOA and RSS) – less stringent synchronization requirement (over TOA) • Assuming there are N +1 sensors in the network, the TDOA measurements take the form t i = 1 c ( � x − s i � 2 − � x − s 0 � 2 + E i ) for i = 1 , . . . , N, where – x ∈ R d is the source location to be estimated, – s i ∈ R d is the i -th sensor’s given location ( i = 0 , 1 , . . . , N ) with s 0 being the reference sensor, – d ≥ 1 is the dimension of the ambient space, – c is the signal propagation speed (e.g., speed of light), – 1 c E i is the measurement error at the i -th sensor. A. M.-C. So, SEEM, CUHK 29 July 2016 2

TDOA-Based Localization in NLOS Environment • In this talk, we assume that the measurement error E i consists of two parts: – measurement noise n i – non-line-of-sight (NLOS) error e i : variable propagation delay of the source signal due to blockage of the direct (or line-of-sight (LOS)) path between the source and the i -th sensor • Putting E i = n i + e i into the TDOA measurement model, we obtain the following range-difference measurements: d i = � x − s i � 2 − � x − s 0 � 2 + n i + e i for i = 1 , . . . , N. A. M.-C. So, SEEM, CUHK 29 July 2016 3

Assumptions on the Measurement Error • Localization accuracy generally depends on the nature of the measurement error. • The measurement noise n i is typically modeled as a random variable that is tightly concentrated around zero. • However, the NLOS error e i can be environment and time dependent. It is the difference of the NLOS errors incurred at sensors 0 and i . As such, it needs not centered around zero and can be positive or negative/of variable magnitude. • We shall make the following assumptions: – | n i | ≪ � x − s 0 � 2 (measurement noise is almost negligible) – | e i | ≤ ρ i for some given constant ρ i ≥ 0 (estimate on the support of the NLOS error is available) A. M.-C. So, SEEM, CUHK 29 July 2016 4

Robust Least Squares Formulation • Rewrite the range-difference measurements as d i − � x − s i � 2 − e i = � x − s 0 � 2 + n i . Squaring both sides and using the assumption on n i , we have ( d i − e i ) 2 − 2( d i − e i ) � x − s i � 2 + � s i � 2 2 − 2 s T − 2 � x − s 0 � 2 n i ≈ i x − � s 0 � 2 2 + 2 s T 0 x � � � s 0 � 2 2 − � s i � 2 2 − d 2 2( s 0 − s i ) T x − 2 d i � x − s i � 2 − = i + e 2 i + 2 e i ( � x − s i � 2 − d i ) . • In view of the LHS, we would like the RHS to be small, regardless of what e i is (provided that | e i | ≤ ρ i ). A. M.-C. So, SEEM, CUHK 29 July 2016 5

Robust Least Squares Formulation • This motivates the following robust least squares (RLS) formulation: N � � � � 2 � 2( s 0 − s i ) T x − 2 d i r i − b i + e 2 min max i + 2 e i ( r i − d i ) x ∈ R d , r ∈ R N − ρ ≤ e ≤ ρ i =1 subject to � x − s i � 2 = r i , i = 1 , . . . , N. Here, b i = � s 0 � 2 2 − � s i � 2 2 − d 2 i is a known quantity. • Note that the inner maximization with respect to e is separable. Hence, we can rewrite the objective function as   2 N   � � � � 2( s 0 − s i ) T x − 2 d i r i − b i + e 2   S ( x , r ) = max i + 2 e i ( r i − d i ) . �   − ρ i ≤ e i ≤ ρ i   i =1 � �� � Γ i ( x , r ) • Note that both objective function and the constraints are non-convex. Moreover, the S -lemma does not apply. A. M.-C. So, SEEM, CUHK 29 July 2016 6

Convex Approximation of the RLS Problem • By the triangle inequality, � � � 2( s 0 − s i ) T x − 2 d i r i − b i + e 2 i + 2 e i ( r i − d i ) � � � � � � + � 2( s 0 − s i ) T x − 2 d i r i − b i � e 2 ≤ i + 2 e i ( r i − d i ) � . • It follows that � � � 2( s 0 − s i ) T x − 2 d i r i − b i + e 2 Γ i ( x , r ) = max i + 2 e i ( r i − d i ) � − ρ i ≤ e i ≤ ρ i � � � � � + � e 2 � 2( s 0 − s i ) T x − 2 d i r i − b i ≤ max i + 2 e i ( r i − d i ) � . − ρ i ≤ e i ≤ ρ i • Key Observation: � � � = ρ 2 � e 2 max i + 2 e i ( r i − d i ) i + 2 ρ i | r i − d i | . − ρ i ≤ e i ≤ ρ i A. M.-C. So, SEEM, CUHK 29 July 2016 7

Convex Approximation of the RLS Problem • Hence, � � � + ρ 2 � 2( s 0 − s i ) T x − 2 d i r i − b i Γ i ( x , r ) ≤ i + 2 ρ i | r i − d i | . • Observation: The function Γ + i given by � � � + ρ 2 Γ + � 2( s 0 − s i ) T x − 2 d i r i − b i i ( x , r ) = i + 2 ρ i | r i − d i | is non-negative and convex. • Thus, N � � � 2 S + ( x , r ) = Γ + i ( x , r ) i =1 is a convex majorant of the non-convex objective function of the RLS problem. A. M.-C. So, SEEM, CUHK 29 July 2016 8

Convex Approximation of the RLS Problem • Using the convex majorant, we have the following approximation of the RLS problem: N � �� � � 2 � + ρ 2 � 2( s 0 − s i ) T x − 2 d i r i − b i min i + 2 ρ i | r i − d i | (ARLS) x ∈ R d , r ∈ R N i =1 subject to � x − s i � 2 = r i , i = 1 , . . . , N. • This can be relaxed to an SOCP via standard techniques: min η 0 x ∈ R d, r ∈ R N η ∈ R N, η 0 ∈ R � � � + ρ 2 � 2( s 0 − s i ) T x − 2 d i r i − b i subject to i + 2 ρ i | r i − d i | ≤ η i , i = 1 , . . . , N, � x − s i � 2 ≤ r i , i = 1 , . . . , N, � η � 2 2 ≤ η 0 . (SOCP) A. M.-C. So, SEEM, CUHK 29 July 2016 9

Convex Approximation of the RLS Problem • Alternatively, observe that �� � � � + ρ 2 � 2( s 0 − s i ) T x − 2 d i r i − b i i + 2 ρ i | r i − d i | � � � � 2( s 0 − s i ) T x − 2 d i r i − b i + ρ 2 = max ± i ± 2 ρ i ( r i − d i ) . • Hence, Problem (ARLS) can be written as N � min τ i x ∈ R d , r ∈ R N i =1 � 2 ≤ τ i , � � � + ρ 2 2( s 0 − s i ) T x − 2 d i r i − b i subject to ± i ± 2 ρ i ( r i − d i ) i = 1 , . . . , N, � x − s i � 2 2 = r 2 i = 1 , . . . , N. i , • The above problem is linear in τ and Y = yy T , where y = ( x , r ) . A. M.-C. So, SEEM, CUHK 29 July 2016 10

Convex Approximation of the RLS Problem • Hence, we also have the following SDP relaxation of (ARLS): N � min τ i Y ∈ S d + N i =1 y ∈ R d + N, τ ∈ R N (SDP) subject to some linear constraints in Y , y , and τ , � Y � y � 0 . y T 1 A. M.-C. So, SEEM, CUHK 29 July 2016 11

Theoretical Issues • When is (ARLS) equivalent to the original RLS problem? In particular, when does the convex majorant Γ + i ( x , r ) equal the original function Γ i ( x , r ) ? • Does (SDP) always yield a tighter relaxation of (ARLS) than (SOCP)? • Do the relaxations yield a unique solution? A. M.-C. So, SEEM, CUHK 29 July 2016 12

Exactness of Problem (ARLS) • Consider a fixed i ∈ { 1 , . . . , N } . Recall � � � 2( s 0 − s i ) T x − 2 d i r i − b i + e 2 Γ i ( x , r ) = max i + 2 e i ( r i − d i ) � − ρ i ≤ e i ≤ ρ i � � � + ρ 2 Γ + � 2( s 0 − s i ) T x − 2 d i r i − b i i ( x , r ) = i + 2 ρ i | r i − d i | If ρ i = 0 , then Γ i ( x , r ) = Γ + • Proposition: i ( x , r ) . Otherwise, Γ i ( x , r ) = Γ + i ( x , r ) iff 2( s 0 − s i ) T x − 2 d i r i − b i ≥ 0 ; i.e. (using the definition of b i ), ( n i + e i ) 2 − 2 � x − s 0 � 2 ( n i + e i ) ≥ 0 . (1) • Interpretation: Recall that n i + e i is the measurement error associated with � x ∗ − s i � 2 − � x ∗ − s 0 � 2 , where x ∗ is the true location of the source. – Scenario 1: n i + e i ≤ 0 or n i + e i ≥ 2 � x − s 0 � 2 (so that (1) holds) e.g., x ∗ ↔ s 0 highly NLOS but x ∗ ↔ s i almost LOS – Scenario 2: 0 < n i + e i < 2 � x − s 0 � 2 (so that (1) fails) e.g., x ∗ ↔ s 0 almost LOS but x ∗ ↔ s i mildly NLOS A. M.-C. So, SEEM, CUHK 29 July 2016 13