Circular Uncertainty method for range-only localization with imprecise sensor positions

Abstract

This study provides an effective new method to solve the range-only localization in the presence of sensor position errors. In practice, the sensors can stay only within a limited region whereas the target can be far from there. To increase the estimation capability, some peripheral measurements with moving sensors can be obtained, which results in the issue of imprecise sensor positions. In these situations, sensor positions also become unknown parameters which need to be jointly estimated together with the target location. Because of the large number of unknown parameters, reaching the global minimum becomes a significant challenge. Our study is dedicated to build a robust localization scheme for these scenarios. We propose a new search strategy, namely Circular Uncertainty which allows the localization system to safely find the global minimum of maximum likelihood cost function in case of imprecise sensor positions. Circular Uncertainty not only makes it possible to reach maximum likelihood estimation, but also significantly simplifies this task. Our solution is based on the observation that when the initial estimation is disturbed with new measurements, the disturbed estimation moves along the Circular Uncertainty which can be viewed as a circular valley along the cost surface. The new method is compared to nonlinear least squares as well as the squared range weighted least-squares algorithm which was previously designed in the literature specifically for localization with imprecise sensor positions. Since the proposed solution obtains maximum likelihood estimation, it attains Cramer Rao lower bound, where other competing methods partly fail.

Appendix

In this section, the MLE cost function of range-only localization in the presence of uncertainties of sensor positions is explained in detail. The multivariate likelihood function of all parameters in ${\vec {q}}$ based on the distance-to-target and sensor position measurements can be written as the following,

$$\begin{aligned} p\left( {\vec {D},\vec {X},~\vec {Y};\vec {q}} \right)= & {} \frac{1}{{\sqrt{\left( {2\pi } \right) ^{N} ~~\left| {C_{D} } \right| } ~}}e^{{ - ~\left( {\vec {D} - ~\vec {\mu }_{d} } \right) ~~C_{D} ^{{ - 1}} \left( {\vec {D} - ~\vec {\mu }_{d} } \right) ^{T} ~}} ~~\nonumber \\&\cdot \frac{1}{{\sqrt{\left( {2\pi } \right) ^{N} ~~\left| {C_{X} } \right| } ~}}e^{{ - ~\left( {\vec {X} - ~\vec {\mu }_{x} } \right) ~~C_{X} ^{{ - 1}} ~\left( {\vec {X} - ~\vec {\mu }_{x} } \right) ^{T} }} \nonumber \\&\cdot ~~\frac{1}{{\sqrt{\left( {2\pi } \right) ^{N} ~~\left| {C_{Y} } \right| } ~}}~~e^{{-~\left( {\vec {Y} - ~\vec {\mu }_{y} } \right) ~~C_{Y} ^{{ - 1}} ~\left( {\vec {Y} - ~\vec {\mu }_{y} } \right) ^{T} }} \end{aligned}$$

(51)

where $C_D$, $C_X$ and $C_Y$ are the covariance matrices of the distance-to-target measurements as well as the measurements of x and y position of the sensors respectively. ${\vec {\mu }}_d$ is the vector of the distance-to-target values as a vector valued function of the parameters in ${\vec {q}}$,

$$\begin{aligned} \vec {\mu }_{d} = \left[ {\begin{array}{c} {\sqrt{\left( {x - x_{1} } \right) ^{2} + ~\left( {y - y_{1} } \right) ^{2} } }\\ \vdots \\ {\sqrt{\left( {x - x_{i} } \right) ^{2} + ~\left( {y - y_{i} } \right) ^{2} } } \\ \vdots \\ {\sqrt{\left( {x - x_{N} } \right) ^{2} + ~\left( {y - y_{N} } \right) ^{2} } } \\ \end{array} } \right] ^{T} \end{aligned}$$

(52)

Furthermore, ${\vec {\mu }}_x$ and ${\vec {\mu }}_y$ are the vectors of the x positions and the y positions of the sensors respectively based on the parameters in ${\vec {q}}$. Because of the measurement model that is defined in (2), (3) and (4), all measurements are independent, so the covariance matrices are diagonal ones as the followings,

$$\begin{aligned} C_D= & {} \left[ {{\begin{array}{l@{\quad }l@{\quad }l@{\quad }l} {\sigma _D ^{2}}&{} &{} &{}\\ &{} {\sigma _D ^{2}}&{} &{}\\ &{} &{} \ddots &{}\\ &{}&{}&{} {\sigma _D ^{2}} \\ \end{array} }} \right] _{NxN} \end{aligned}$$

(53)

$$\begin{aligned} C_X= & {} C_Y =\left[ {{\begin{array}{l@{\quad }l@{\quad }l@{\quad }l} {\sigma _S ^{2}}&{} &{} &{}\\ &{} {\sigma _S ^{2}}&{} &{}\\ &{} &{} \ddots &{}\\ &{}&{}&{} {\sigma _S ^{2}} \\ \end{array} }} \right] _{NxN} \end{aligned}$$

(54)

Therefore, (51) can be rewritten as the following expression,

$$\begin{aligned} p\left( {\vec {D},\vec {X},~\vec {Y};\vec {q}} \right)= & {} {} \left( \frac{1}{{\sqrt{{2\pi ~\sigma _{D} ^{2} ~}}}}\right) ^{N}e^{{ - ~\frac{1}{{\left( {\sigma _{D} } \right) ^{2} }}\mathop \sum \nolimits _{{i = 1}}^{N} \left( {\sqrt{\left( {x - x_{i} } \right) ^{2} + ~\left( {y - y_{i} } \right) ^{2} } - ~D_{i} } \right) ^{2} ~}} ~\nonumber \\&\cdot \left( \frac{1}{\sqrt{{2\pi \sigma _s ^{2} } }}\right) ^{N}e^{- \frac{1}{\left( {\sigma _S } \right) ^{2}} \mathop \sum \nolimits _{i=1}^N \left( {X_i -x_i } \right) ^{2}} . \left( \frac{1}{\sqrt{{2\pi \sigma _s ^{2} }}}\right) ^{N} e^{ - \frac{1}{\left( {\sigma _S } \right) ^{2}} \mathop \sum \nolimits _{i=1}^N \left( {Y_i -y_i } \right) ^{2}}\nonumber \\ \end{aligned}$$

(55)

Finally, the log-likelihood of the all the parameters in ${\vec {q}}$ can be written as the following,

$$\begin{aligned} \ln p\left( {\vec {D},\vec {X},~\vec {Y};\vec {q}} \right)= & {} - ~\frac{1}{{\left( {\sigma _{D} } \right) ^{2} }}\mathop \sum \limits _{{i = 1}}^{N} \left( {\sqrt{\left( {x - x_{i} } \right) ^{2} + ~\left( {y - y_{i} } \right) ^{2} } - ~D_{i} } \right) ^{2}\nonumber \\&- ~\frac{1}{{\left( {\sigma _{S} } \right) ^{2} }}~\mathop \sum \limits _{{i = 1}}^{N} \left[ {\left( {X_{i} - x_{i} } \right) ^{2} + ~\left( {Y_{i} - y_{i} } \right) ^{2} } \right] + K \end{aligned}$$

(56)

where is K is a constant such that,

$$\begin{aligned} K=- N \ln \left( {\sqrt{2 \pi \left( {\sigma _D } \right) ^{2}}} \right) -2 N \ln \left( {\sqrt{2 \pi \left( {\sigma _S } \right) ^{2}}} \right) \end{aligned}$$

(57)