# Table 1 Summary of the localization algorithms

Algorithm Solution
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.