## 3 Error Analysis
The error can be expressed as the difference between the actual distances and the estimated distances, i.e.,  
\[\mathbf{E}=\mathbf{D}(\mathbf{p}(t_{N}))-\mathbf{R}=[e_{1},e_{2},\ldots,e_{N}] \tag{7}\]  
To show the impact of the AUV's velocity and MAC delay on the measurement errors, the velocity vector of the AUV is defined by \(V(t)=[v_{x}(t),v_{y}(t),v_{z}(t)]\) where each component indicates velocity with respect to the \(x\), \(y\), \(z\)-axis respectively. Let \(\tau_{i}\) denote MAC delay between the \(i\)th and (\(i+1\))th transmission, and then the position receiving the \(i\)th packet is \[x(t_{i}) =x(t_{i})+\int_{t_{1}}^{t_{N}}v_{x}(t)dt,\] \[y(t_{i}) =y(t_{i})+\int_{t_{1}}^{t_{N}}v_{y}(t)dt, \tag{8}\] \[z(t_{i}) =z(t_{i})+\int_{t_{1}}^{t_{N}}v_{z}(t)dt\]  
where,  
\[t_{i}=t_{1}+\sum_{j=2}^{N-1}\tau_{i}. \tag{9}\]  
If the minimum and maximum speed of the AUV moving in a direction is \([V_{min},V_{max}]\), the lower and upper bounds of the distance measurement error are derived when the AUV's velocity is \(V_{min}\) and \(V_{max}\) respectively, and the ranging errors are distributed within the bounds. If the AUV's constant velocity is \(v\), and then the error vector is denoted by \(E_{v}=[e_{v,1},e_{v,1},...,e_{v,N}]\). The mean of \(e_{i}\) presented in (7) is  
\[\bar{e}_{i}=\frac{1}{(v_{max}-v_{min})}\int_{v_{min}}^{v_{max}}e_{v,i}dv, \tag{10}\]  
and the variance of the error vector can be expressed as below:  
\[\sigma_{i}^{2}=\frac{1}{(v_{max}-v_{min})}\int_{v_{min}}^{v_{max}}(e_{v,i}- \bar{e}_{i})^{2}dv, \tag{11}\]  
By calculating \(\sigma_{i}^{2}\) with (11), the FIM can be derived. As mentioned above, CRLB is equal to the inverse of FIM; the CRLB of the ToA-based localization is  
\[\mathbf{I}(\mathbf{p})=\left[\frac{\partial\mathbf{R}(\mathbf{p})}{\partial \mathbf{p}}\right]^{T}\mathbf{C}^{-1}\left[\frac{\partial\mathbf{R}(\mathbf{p })}{\partial\mathbf{p}}\right] \tag{5}\]0  
where \([\mathbf{I}]_{i,i}^{-1}\) means the position estimation error with respect to the \(i\)th coordinate, i.e., \(i\) = 1, 2, and 3 indicates \(x\), \(y\), and \(z\)-coordinate respectively. We suppose that the AUV is moving in a direction and its velocity changes from 0 to 3 m/s, i.e., \([v_{min},v_{max}]\) is [0, 3] m/s. The ANs are put on the corners and the MAC delay \(\tau\) increases as the communication range increase. Figure 3 plots the CRLB with respect to the AUV's velocity. The figure reveals that the localization performance decreases as increase the AUV's velocity. Furthermore, the longer the communication distance due to the enlarged network size is, the greater the CRLB is as shown in \[\mathbf{I}(\mathbf{p})=\left[\frac{\partial\mathbf{R}(\mathbf{p})}{\partial \mathbf{p}}\right]^{T}\mathbf{C}^{-1}\left[\frac{\partial\mathbf{R}(\mathbf{p })}{\partial\mathbf{p}}\right] \tag{5}\]1 The results suggest that the MAC delay is significant error source in AUV localization. In addition, in contrast to the general localization theory that the more the number of ANs participating on localization is, the higher the localization accuracy can be achieved, the CRLB grows as the number ANs increase because the required time to receive the all packets is proportional to the number of ANs. The results indicate that the MAC delay should be eliminated in the AUV localization.