Closed-form Solution for Joint Localization and Synchronization in Wireless Sensor Networks With and Without Beacon Uncertainties

Abstract

In Wireless Sensor Networks (WSNs), the use of the same set of measurement data for simultaneous localization and synchronization is potentially useful for achieving higher estimation accuracy, and lower communication overhead and power consumption. In this paper, we first analyze the impact of asynchronous sensor nodes (SNs) on the accuracy of time-based localization schemes, and the impact of inaccurate SN location information on the accuracy of synchronization based on packet delay measurement, to illustrate the necessity and significance of simultaneous localization and synchronization of SNs. We then consider the joint localization and synchronization problem for two cases. In the first case, we assume that the beacon information is perfectly known. The Maximum Likelihood (ML) estimator is first formulated, which is computationally expensive. A new closed-form Joint Localization and Synchronization I (JLS-I) estimator is then proposed to provide a computationally efficient solution. In the second case, we assume that the beacon locations and timings are known inaccurately, and develop the ML and JLS-II estimators accordingly. JLS-II is based on Weighted Least Square and Generalized Total Least Square, and is of low complexity. The Cramer-Rao Lower Bounds (CRLBs) and the analytical Mean Square Errors of the proposed estimators are derived, and we also analytically show that JLS-I can achieve the corresponding CRLB. Simulation results demonstrate the effectiveness of the proposed estimators compared to other approaches. With only a three-way message exchange, JLS-I can attain the CRLB and JLS-II can provide close to optimal performance in their respective scenarios. They are also robust against the Geometric Dilution of Precision problem, and outperform existing algorithms in NLOS scenarios. Our results demonstrate the advantages of JLS-I and JLS-II in reduced computational complexity with lower power consumption and communication cost while achieving high estimation accuracy. They are therefore attractive solutions to the simultaneous localization and synchronization problem in WSNs where energy and network resources are the most important considerations.

Appendices

Appendix 1: Derivation of $\text{cov} ({\mathbf{e}}_{1}^{\prime} )$ and C _U

For notation simplicity, we define the following symbols. The observed timing information of the beacons are grouped as ${\mathbf{o}} = [\theta_{1} ,\Upomega_{1} , \ldots ,\theta_{N} ,\Upomega_{N} ]^{T}$, and their error vectors are denoted as $\Updelta {\mathbf{o}} = [\Updelta \theta_{1} ,\Updelta \Upomega_{1} , \ldots ,\Updelta \theta_{N} ,\Updelta \Upomega_{N} ]^{T}$ with covariance matrix C _o . The covariance matrix of each observed beacon location vector is defined as ${\mathbf{C}}_{{{\mathbf{x}}_{j} }} = E[{\mathbf{x}}_{j} {\mathbf{x}}_{j}^{T} ],j = 1, \ldots ,N$. Note that C _o and ${\mathbf{C}}_{{{\mathbf{x}}_{j} }}$can be determined from C _β . Since $\theta_{j} = {1 \mathord{\left/ {\vphantom {1 {\left( {1 + \varepsilon_{j} } \right)}}} \right. \kern-0pt} {\left( {1 + \varepsilon_{j} } \right)}} = 1 - \varepsilon_{j} + \sum\limits_{k = 1}^{\infty } {\left( {\varepsilon_{j} } \right)^{k} }$ and $\varepsilon_{j}$ is usually much smaller than 10⁻³, we can only maintain the linear order terms, i.e., $\theta_{j} \approx 1 - \varepsilon_{j}$. Therefore, we can just approximate the variance of θ _j using that of ε _j .

We first derive ${\text{cov(}}{\mathbf{e}}_{ 1}^{\prime} )$. Recalling the definition of ${\mathbf{e}}_{ 1}^{\prime}$ in (49), we can rewrite ${\mathbf{e}}_{ 1}^{\prime}$ as

$$ {\mathbf{e}}_{1}^{\prime} = {\mathbf{e}}_{1} + {\mathbf{Q}}\Updelta {\mathbf{o}}, $$

(79)

where e ₁ is defined in (14), and

$$ {\mathbf{Q}} = diag\left[ {\begin{array}{*{20}c} {{\mathbf{Q}}_{1} ,} \hfill & { \cdots ,} \hfill & {{\mathbf{Q}}_{N} } \hfill \\ \end{array} } \right],\,{\mathbf{Q}}_{j} = \left[ {\begin{array}{*{20}c} {cR_{j}^{1} - c\Upomega_{j}^{0} } & { - c\theta_{j}^{0} } \\ { - cT_{j}^{2} + c\Upomega_{j}^{0} } & {c\theta_{j}^{0} } \\ \vdots & \vdots \\ \end{array} } \right]_{M \times 2} . $$

(80)

Since the range measurement noise e ₁ is assumed to be uncorrelated with the beacon timing error Δo, the covariance matrix of ${\mathbf{e}}_{ 1}^{\prime}$, ${\text{cov(}}{\mathbf{e}}_{ 1}^{\prime} )$, is then given by

$$ {\text{cov}}\left( {{\mathbf{e}}_{1}^{'} } \right) = {\mathbf{C}}_{{\mathbf{n}}} + {\mathbf{QC}}_{{\mathbf{o}}} {\mathbf{Q}}^{T} . $$

(81)

We then derive C _U .

$$ {\mathbf{C}}_{{\mathbf{U}}} = \left[ {\begin{array}{*{20}c} {E\left[ {\Updelta {\mathbf{A}}_{2}^{T} \Updelta {\mathbf{A}}_{2} } \right]} \hfill & {E\left[ {\Updelta {\mathbf{A}}_{2}^{T} {\varvec{\delta}}_{2} } \right]} \hfill \\ {E\left[ {{\varvec{\delta}}_{2}^{T} \Updelta {\mathbf{A}}_{2} } \right]} \hfill & {E\left[ {{\varvec{\delta}}_{2}^{T} {\varvec{\delta}}_{2} } \right]} \hfill \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {{\mathbf{C}}_{{\mathbf{A}}} } \hfill & {{\mathbf{C}}_{{{\mathbf{A\delta }}}} } \hfill \\ {{\mathbf{C}}_{{{\mathbf{A\delta }}}}^{T} } \hfill & {{\mathbf{C}}_{{\varvec{\delta}}} } \hfill \\ \end{array} } \right]. $$

(82)

Recalling the definition of ΔA ₂ and δ = Δh ₂ − e ₂ in (54) and (30), we have

$$ {\mathbf{C}}_{{\mathbf{A}}} = 4\sum\limits_{j = 1}^{N} {{\mathbf{C}}_{{{\mathbf{x}}_{j} }} } ,\,{\mathbf{C}}_{{{\mathbf{A\delta }}}} = 4\sum\limits_{j = 1}^{N} {{\mathbf{C}}_{{{\mathbf{x}}_{j} }} {\mathbf{x}}_{j}^{0} } , $$

$$ {\mathbf{C}}_{{\varvec{\delta}}} = \left( {1 + \Upomega^{02} } \right)E\left[ {\Updelta \theta^{2} } \right] + E\left[ {\Updelta \left( {\theta \Upomega } \right)^{2} } \right] - 2\Upomega^{0} E\left[ {\Updelta \theta \Updelta \left( {\theta \Upomega } \right)} \right] + 4\sum\limits_{j = 1}^{N} {r_{j}^{02} E\left[ {\Updelta r_{j}^{2} } \right]} + 4\sum\limits_{j = 1}^{N} {{\mathbf{x}}_{j}^{0T} {\mathbf{C}}_{{{\mathbf{x}}_{j} }} {\mathbf{x}}_{j}^{0} } . $$

(83)

where E[Δθ ²], E[Δ(θΩ) ²], E[ΔθΔ(θΩ] and E[Δr ² _j ] can be easily determined from cov(φ ₁) in (51).

Appendix 2

Defining Θ = [θ, θΩ]^T, ${\mathbf{r}} = [r_{ 1} , \ldots ,r_{N} ]^{T}$ and applying the chain rule, we can express (62) in the product form as

$$ {\text{FIM}}\left( {\varvec{\rho}} \right) = \left[ {\begin{array}{*{20}c} {\frac{{\partial {\mathbf{f}}}}{{\partial {\varvec{\Uptheta}}^{0} }}\frac{{\partial {\varvec{\Uptheta}}^{0} }}{{\partial {\varvec{\Upphi}}^{0} }},} & {\frac{{\partial {\mathbf{f}}}}{{\partial {\mathbf{r}}^{0} }}\frac{{\partial {\mathbf{r}}^{0} }}{{\partial {\mathbf{x}}^{0} }}} \\ \end{array} } \right]^{T} {\mathbf{C}}_{{\mathbf{n}}}^{ - 1} \left[ {\begin{array}{*{20}c} {\frac{{\partial {\mathbf{f}}}}{{\partial {\varvec{\Uptheta}}^{0} }}\frac{{\partial {\varvec{\Uptheta}}^{0} }}{{\partial {\varvec{\Upphi}}^{0} }},} & {\frac{{\partial {\mathbf{f}}}}{{\partial {\mathbf{r}}^{0} }}\frac{{\partial {\mathbf{r}}^{0} }}{{\partial {\mathbf{x}}^{0} }}} \\ \end{array} } \right] = {\mathbf{D}}^{T} \times {\text{FIM}}^{\prime} \times {\mathbf{D}}, $$

(84)

where

$$ {\mathbf{D}} = \left[ {\begin{array}{*{20}c} {{\mathbf{0}}_{N \times 2} } & {\frac{{\partial {\mathbf{r}}^{0} }}{{\partial {\mathbf{x}}^{0} }}} \\ {\frac{{\partial {\varvec{\Uptheta}}^{0} }}{{\partial {\varvec{\Upphi}}^{0} }}} & {{\mathbf{0}}_{2 \times 2} } \\ \end{array} } \right]_{{\left( {N + 2} \right) \times 4}} ,\quad {\text{FIM}}^{\prime} = \left[ {\begin{array}{*{20}c} {\frac{{\partial {\mathbf{f}}}}{{\partial {\mathbf{r}}^{0} }},} & {\frac{{\partial {\mathbf{f}}}}{{\partial {\varvec{\Uptheta}}^{0} }}} \\ \end{array} } \right]^{T} {\mathbf{C}}_{{\mathbf{n}}}^{ - 1} \left[ {\begin{array}{*{20}c} {\frac{{\partial {\mathbf{f}}}}{{\partial {\mathbf{r}}^{0} }},} & {\frac{{\partial {\mathbf{f}}}}{{\partial {\varvec{\Uptheta}}^{0} }}} \\ \end{array} } \right]. $$

(85)

It is trivial to verify that ${\mathbf{G}}_{1} = [{{\partial {\mathbf{f}}} \mathord{\left/ {\vphantom {{\partial {\mathbf{f}}} {\partial {\mathbf{r}}^{0} }}} \right. \kern-0pt} {\partial {\mathbf{r}}^{0} }},{{\partial {\mathbf{f}}} \mathord{\left/ {\vphantom {{\partial {\mathbf{f}}} {\partial {\varvec{\Uptheta}}^{0} }}} \right. \kern-0pt} {\partial {\varvec{\Uptheta}}^{0} }}]$. With the relationship W ₁ = C ⁻¹ _n , the equivalence of cov(φ ₁)⁻¹ and FIM′ is obvious by comparing (24) with (85).

Substituting (32) into (33), and applying the above conclusion, i.e., cov(φ ₁)⁻¹ = FIM′, we have

$$ \text{cov} (\varphi_{2} ) = \left( {{\mathbf{G}}_{2}^{T} \left( {{\mathbf{B}}_{2} {\text{FIM}}^{\prime - 1} {\mathbf{B}}_{2}^{T} } \right)^{ - 1} {\mathbf{G}}_{2} } \right)^{ - 1} . $$

(86)

Since B ₂ is a (N + 2) × (N + 2) matrix with full rank, (86) can be rewritten as

$$ \text{cov} (\varphi_{2} ) = \left( {{\mathbf{G}}_{2}^{T} {\mathbf{B}}_{2}^{ - T} {\text{FIM}}^{\prime}{\mathbf{B}}_{2}^{ - 1} {\mathbf{G}}_{2} } \right)^{ - 1} . $$

(87)

Similarly, since B ₃ is a 5 × 5 matrix with full rank if the SN is neither on the x axis nor on the y axis, or can be made full rank with proper coordinate transformation if the SN is on either the x or y axis, cov(φ ₃) can be expressed as (88) by substituting (87) into (41).

$$ \text{cov} (\varphi_{3} ) = \left( {{\mathbf{G}}_{3}^{T} {\mathbf{B}}_{3}^{ - T} {\mathbf{G}}_{2}^{T} {\mathbf{B}}_{2}^{ - T} {\text{FIM'}}{\mathbf{B}}_{2}^{ - 1} {\mathbf{G}}_{2} {\mathbf{B}}_{3}^{ - 1} {\mathbf{G}}_{3} } \right)^{ - 1} . $$

(88)

Consequently, the error covariance matrix of JLS-I is derived from (77) and (88) as

$$ \text{cov} ({\varvec{\rho}}) = \left( {{\mathbf{B}}_{4}^{T} {\mathbf{G}}_{3}^{T} {\mathbf{B}}_{3}^{ - T} {\mathbf{G}}_{2}^{T} {\mathbf{B}}_{2}^{ - T} {\text{FIM'}}{\mathbf{B}}_{2}^{ - 1} {\mathbf{G}}_{2} {\mathbf{B}}_{3}^{ - 1} {\mathbf{G}}_{3} {\mathbf{B}}_{4} } \right)^{ - 1} . $$

(89)

It can be easily identified that CRLB(ρ) and cov(ρ) are identical if

$$ {\mathbf{D}} \cong {\mathbf{B}}_{2}^{ - 1} {\mathbf{G}}_{2} {\mathbf{B}}_{3}^{ - 1} {\mathbf{G}}_{3} {\mathbf{B}}_{4} . $$

(90)

Premultiplying B ₂ at both sides of (90), we can easily observe that proving (90) is equivalent to showing (91).

$$ {\mathbf{B}}_{2} {\mathbf{D}} \cong {\mathbf{G}}_{2} {\mathbf{B}}_{3}^{ - 1} {\mathbf{G}}_{3} {\mathbf{B}}_{4} . $$

(91)

Substituting B ₂, G ₂, B ₃, G ₃, B ₄ and D with the expressions in (30), (38), (76) and (85) respectively, using simple algebra manipulations, and ignoring the noise in G ₂, the correctness of (91) can be easily verified.

Consequently, we establish that

$$ \text{cov}_{{{\text{JLS}} - {\text{I}}}} ({\varvec{\rho}}) = {\text{CRLB}}\left( {\varvec{\rho}} \right). $$

(92)

It is worthy to note that the above proof does not depend on the specific form of C _n , and hence the analysis also applies to the scenario when measurement errors are correlated as long as they are Gaussian.