Algorithm 1.

Joint magnetic calibration and localization

Input: The number of iterations L, the stop threshold δ of EM algorithm, the recorded magnetic field data Y = {y}t=1:T, initial sensor parameters ${\mathbf{\Omega }}_s^0=\{{\mathbf{G}}^0,{\mathbf{b}}^0,{\mathbf{H}}^0,\mathbf{s}\}$, initial prior distribution ${\mathbf{\Omega }}_p^0=\left\{{\mathbf{m}}_0^0,{\mathbf{P}}_0^0\right\}$ and initial noise covariances ${\mathbf{\Omega }}_c^0=\left\{{\mathbf{Q}}^0,{\mathbf{R}}^0\right\}$. Hf = 0.
1:	for n = 1 : L do
2:	Run URTSS with Y and $\left\{{\mathbf{\Omega }}_s^{n-1},{\mathbf{\Omega }}_p^{n-1},{\mathbf{\Omega }}_c^{n-1}\right\}$ to estimate ${\mathbf{X}}^{n-1}={\left\{{\mathbf{m}}_t^s,{\mathbf{P}}_t^s\right\}}_{t=1:T}^{n-1}$;
3:	Update $\left\{{\mathbf{\Omega }}_p^{n-1},{\mathbf{\Omega }}_c^{n-1}\right\}$ to $\left\{{\mathbf{\Omega }}_p^n,{\mathbf{\Omega }}_c^n\right\}$ with respect to ${\mathbf{\Omega }}_s^{n-1}$, Y and Xn−1 according to (9)–(13);
4:	if Hf = 0 then
5:	Update ${\mathbf{\Omega }}_s^{n-1}$ to ${\mathbf{\Omega }}_s^n$ by computing {Gn, bn} with respect to Rn, Hn−1, Y and Xn−1 according to (15);
6:	else
7:	Update ${\mathbf{\Omega }}_s^{n-1}$ to ${\mathbf{\Omega }}_s^n$ by alternately optimizing {G, b} and H based on curvilinear search according to (15), (17), (18) and (20).
8:	end if
9:	Compute the J(n) = [E(log(L)]n with respect to $\left\{{\mathbf{\Omega }}_s^n,{\mathbf{\Omega }}_p^n,{\mathbf{\Omega }}_c^n\right\}$, Y and Xn−1 according to (8);
10:	if n > 1, Hf = 1 and $\left|\frac{J(n)-J(n-1)}{J(n-1)}\right|<\delta$ then
11:	break, end;
12:	end if
13:	if n > 1, Hf = 0 and $\left|\frac{J(n)-J(n-1)}{J(n-1)}\right|<\delta$ then
14:	break or Hf = 1 (optional), end;
15:	end if
16:	end for
Output: The optimized {Ωs, Ωp, Ωc} and X.