Table 1 Summary of the localization algorithms

LS (median)

(A ^T A)⁻¹ A ^T{med(b _1:P )}

ANLOS

\(\mathbf {x}^{({k}+1)}=\mathbf {x}^{({k})}+ \left (\sum _{{i}\in \Lambda } \frac {1}{\sigma _{{i},\text {SE}}^{2}}\left \{{\mathbf {g}_{{i}}^{({k})}}{\mathbf {g}_{{i}}^{({k})}}^{{T}}\right \}\right.\)

\( \qquad \qquad \left. +\sum _{{q}\in \Lambda ^{c}} \frac {1}{\sigma _{{q},\text {SA}}^{2}}\left \{{\mathbf {g}_{{q}}^{({k})}}{\mathbf {g}_{{q}}^{({k})}}^{{T}}\right \}\right)^{-1}\)

\(\,\qquad \qquad \times \left (\sum _{{i}\in \Lambda } {\mathbf {g}_{{i}}^{({k})}}\frac {1}{\sigma _{{i},\text {SE}}^{2}}({r}_{{i},\text {SE}} -{f}_{{i}}(\mathbf {x}^{({k})})) \right. \)

\(\quad \qquad \quad \left. +\sum _{{q}\in \Lambda ^{c}} {\mathbf {g}_{{q}}^{({k})}}\frac {1}{\sigma _{{q},\text {SA}}^{2}}(\text {med}(\mathbf {r}_{{q},1:{P}}) -{f}_{{q}}(\mathbf {x}^{({k})}))\right)\)

ATLS

\((\mathbf {A}^{{T}}\mathbf {A})^{-1}\mathbf {A}^{{T}}\widehat {\mathbf {b}}\)

Gauss-Newton (median)

\(\mathbf {x}^{({k}+1)}=\mathbf {x}^{({k})}+ \left (\sum _{{q}=1}^{{M}} \frac {1}{\sigma _{{q},\text {SA}}^{2}}\left \{{\mathbf {g}_{{q}}^{({k})}}{\mathbf {g}_{{q}}^{({k})}}^{{T}}\right \}\right)^{-1}\)

\(\times \left (\sum _{{q}=1}^{{M}} {\mathbf {g}_{{q}}^{({k})}}\frac {1}{\sigma _{{q},\text {SA}}^{2}}(\text {med}(\mathbf {r}_{{q},1:{P}}) -{f}_{{q}}(\mathbf {x}^{({k})}))\right)\)

M-estimator

\(\text {min}_{\mathbf {x}} \bigg \{\sum _{{i}=1}^{{M}} \rho ({r}_{{i}}(\mathbf {x}))\bigg \}\)

ρ(t)=\(\frac {1}{2}{t}^{2}\) if |t|≤γ, \(\rho ({t})=\gamma |{t}|-\frac {1}{2}\gamma ^{2}\) elseif |t|>γ,

r _i (x)=[med(b _1:P )] _i −[A x] _i , [·] _i denotes the ith component of [·],

γ is the tuning constant (γ>0).

ATWLS

\((\mathbf {A}^{{T}}{\mathbf {C}_{\widehat {\mathbf {b}}}}^{-1}\mathbf {A})^{-1} \mathbf {A}^{{T}}{\mathbf {C}_{\widehat {\mathbf {b}}}}^{-1}\widehat {\mathbf {b}}\)

LMedS

(1) Calculate the m subsets of three measurements.

(2) For each subset S, compute a location by trilateration

in closed-form LS solution (\([\widehat {{x}}_{\text {LS}}, \widehat {{y}}_{\text {LS}}]^{{T}}\)).

(3) For each solution, the residues R _s are

obtained as \(\mathbf {R}_{s}=[({r}_{1}-\widehat {r}_{1})^{2}, ({r}_{2}-\widehat {r}_{2})^{2},\cdots, ({r}_{{M}}-\widehat {r}_{{M}})^{2}]^{{T}}\)

where \(\widehat {{r}}_{{i}}=\sqrt {(\widehat {{x}}_{\text {LS}}-{x}_{{i}})^{2}+(\widehat {{y}}_{\text {LS}}-{y}_{{i}})^{2}}\;\; ({i}=1,\cdots,{M})\)

and the median of the residues is obtained.

(4) Find the minimum median of the residues.