Erratum to “DOA-Based Cooperative Positioning in Wireless Sensor Network With Factor Graph Technique”

In the above article [1], several typos were uncorrected prior to publication. 1)Square operations are missing after (39). The correction is as follows: \begin{align*} \frac{\partial \theta _{i}}{\partial x_{1}} &= \frac{Y_{i}-y_{1}}{r_{i}^{2}} \tag{36}\\ \frac{\partial \theta _{i}}{\partial y_{1}} &= \frac{-X_{i}+x_{1}}{r_{i}^{2}} \tag{37}\\ \frac{\partial \theta _{c}}{\partial x_{1}} &= \frac{y_{2}-y_{1}}{r_{c}^{2}} \tag{38}\\ \frac{\partial \theta _{c}}{\partial y_{1}} &= \frac{-x_{2}+x_{1}}{r_{c}^{2}} \tag{39} \end{align*} where $r_{i}=\sqrt{(X_{i}-x_{1})^{2}+(Y_{i}-y_{1})^{2}}$ with $i=\lbrace 1,2\rbrace$, and $r_{c}=\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}$. 2)The numerators in the two brackets in the last line of (41) were written incorrectly. The equation should be as follows: \begin{align*} \sigma _{\hat{\theta }_{c}}^{2} &= \left[\frac{\partial g(\hat{x}_{2},\hat{y}_{2})}{\partial \hat{x}_{2}}\right]^{2}\sigma _{\hat{x}_{2}}^{2}+\left[\frac{\partial g(\hat{x}_{2},\hat{y}_{2})}{\partial \hat{y}_{2}}\right]^{2}\sigma _{\hat{y}_{2}}^{2}+\sigma _{\hat{\theta }_{c}}^{2} \\ &= \left(\frac{\Delta y_{c}}{\Delta x_{c}^{2}+\Delta y_{c}^{2}}\right)^{2}\sigma _{\tilde{x}_{2}}^{2} + \left(\frac{-\Delta x_{c}}{\Delta x_{c}^{2}+\Delta y_{c}^{2}}\right)^{2}\sigma _{y_{2}}^{2} + \sigma _{\tilde{\theta }_{c}}^{2} \tag{41} \end{align*}