Abstract
Triangulation uncertainty is the uncertainty associated when we try to locate an unknown target with the help of three anchor nodes resulting in formation of a triangulated region. Efficient triangulation leads to superior accuracy and lower rate of errors in sensor networks. Although sufficient work has been done to compute localization uncertainties, there is dearth of work pertaining to triangulation uncertainty. The existing problems are: first, localization incurs large computation cost, necessitating some hierarchy or clustering techniques. Second, linear, non-linear and optimization-based solvers invariably simplify the occurrence of errors during estimation of localization. To solve these problems, the present work proposes a range free assistive approach in detecting symmetric triangulations. This approach combined with semidefinite programming of the cost function is shown to exhibit improved localization performance. Numerical results show that the RMS errors is reduced by using triangulation assisted node deployment. The results are compared with the standard weighted least square method for different number of anchor nodes.
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Abbreviations
- APIT:
-
Approximate point in triangulation
- DV:
-
Distance vector
- SLAM:
-
Simultaneous Localization and mapping
- NLOS:
-
Non-Line of Sight
- GMM:
-
Gaussian Mixture Model
- SDP:
-
Semidefinite programming
- T-LOC:
-
Triangulation Localization
- WLS:
-
Weighted least squares
- MLE:
-
Maximum Likelihood Estimator
- R-O:
-
Radio obstructed link
- nR-O:
-
Unobstructed link
- U:
-
Triangulation Uncertainty
- GMD:
-
Gaussian Mixture distribution
- RMSE:
-
Root mean square error
- RSSr:
-
Received Signal Strength range
- CDF:
-
Cumulative Distribution function
- CRLB:
-
Cramer Rao Lower Bound
- \(N_{a} ,\;N_{na} ,\;N\) :
-
Number of anchor nodes, non-anchor nodes, total number of nodes
- \({\mathbf{x}}_{i} ,\;{\mathbf{x}}_{j}\) :
-
Position of anchor nodes and non-anchor nodes, respectively
- \(\alpha ,\;\beta\) :
-
Pool of anchor nodes, pool of non-anchor nodes
- \(\gamma_{{ii^{\prime}}}\) :
-
Range measurement between \(i^{th}\) and \(i^{\prime}\) pair of nodes
- wii :
-
Additive white Gaussian noise
- \(\sigma_{{ii^{\prime}}}^{2}\) :
-
Variance of \(i^{th}\) and \(i^{\prime}\) pair of nodes
- \(\delta_{{ii^{\prime}}}\) :
-
Range skew between node \(i\) and \(i^{\prime}\)
- \(\angle s_{i} t_{g} s_{{i^{\prime}}}\) :
-
Internal angle of the target vertex node \(t_{g}\)
- \(RSSr\left( {s,t_{g} } \right)\) :
-
Received signal strength range between sensor node \(s\) and target node \(t_{g}\)
- \(U\left( {s_{i} ,{\kern 1pt} s_{{i^{\prime}}} ,{\kern 1pt} {\kern 1pt} t_{g} } \right)\) :
-
Node triangulation uncertainty
- \(\nu_{n}\) :
-
Spatial uncertainty Cluster for \(n^{th}\) target node
- \(n^{th}\) :
-
Number of target node considered
- \(i,i^{\prime},k \in \alpha\) :
-
Anchor nodes
- \(\varepsilon_{U}\) :
-
Positive value
- \(N_{i,k,n}\) :
-
Number (\(N\)) of anchor nodes (subscript \(i\) and \(k\)) responsible for the triangulation of the \(n^{th}\) target node (subscript \(n\))
- Η:
-
Number of links
- \({\mathbf{r}} = \left[ {x^{r} {\kern 1pt} y^{r} } \right]^{T}\) :
-
Common reflection point in case of range obstruction
- \(\left( {t - 1} \right)\) :
-
Time instance during which measurements are taken
- \(\psi_{A}\) :
-
Maximum Likelihood Estimator
- \({\mathbf{\varphi }}\) :
-
Constraint on uncertainty matrix
- \({{\varvec{\upzeta}}}\) :
-
Matrix containing two sets \({{\varvec{\upzeta}}}_{{\mathbf{a}}}\) for anchor nodes and \({{\varvec{\upzeta}}}_{{\mathbf{b}}}\) for non-anchor nodes
- \(\overline{D}_{jm}\) :
-
Upper limit of the uncertainty skew
- \(\Delta \tau_{{ii^{\prime}m}}\) :
-
Upper limit on the absolute difference of uncertainty skew
- \(\rho_{jm} \;and\;g_{jm}\) :
-
Slack variables
- \(M_{jj}^{\left( t \right)}\) :
-
Mobility constraint for \(j^{th}\) node between time instances \(\left( {t - 1} \right)\) and \(t\)
- \(d_{jm}\) :
-
l2-Norm between node \(j\) and node \(m\)
- \(f_{\eta = 0} \left( {\gamma_{jm} ;\hat{d}_{jm} ,H_{1} } \right)\) :
-
Likelihood ratio of no obstruction
- \(f_{\eta > 0} \left( {\gamma_{jm} ;\hat{g}_{jm} ,H_{0} } \right)\) :
-
Likelihood ratio of single or multiple obstructions
- \(eval_{\max }\) :
-
Maximum number of evaluations
- \(Th_{jm}\) :
-
Detection threshold
- \(a\) :
-
Kurtosis factor for skew estimates
- \({{\varvec{\uptheta}}} = \left[ {{\mathbf{l}}^{T} \, {\mathbf{q}}^{T} } \right]^{T}\) :
-
Unknown parameter to be estimated
- \(\Delta {\tilde{\mathbf{\tau }}}\) :
-
Measured location vector of target by non-anchor nodes
- \({{\varvec{\upgamma}}}\) :
-
Measured location vector of target by anchor nodes
- \({\mathbf{J}}_{\theta }\) :
-
Fisher Information matrix (FIM) of unknown parameter \({{\varvec{\uptheta}}}\)
- \({\mathbf{L}}_{1} ,\;{\mathbf{L}}_{2} ,\;{\mathbf{L}}_{3}\) :
-
FIM component for anchor-anchor node link, sensor-anchor node link, and sensor-sensor node link
- \({\mathbf{l}} = \left[ {x_{1}^{T} {\kern 1pt} {\kern 1pt} x_{2}^{T} {\kern 1pt} {\kern 1pt} \ldots x_{{N_{a} }}^{T} } \right]\) :
-
Location vector of anchor nodes
- \({\mathbf{K}}\) :
-
Transformation matrix for triangulation uncertainty skew constraint
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Prateek, Arya, R. T-LOC: RSSI-based, range-free, triangulation assisted localization for convex relaxation with limited node range under uncertainty skew constraint. J Ambient Intell Human Comput 14, 7063–7077 (2023). https://doi.org/10.1007/s12652-021-03559-1
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DOI: https://doi.org/10.1007/s12652-021-03559-1