## 3 Traditional ToF-Based Method  
Consider the actuator position as denoted by \(A\) and the 12 receiving positions of the sensor denoted by \(R_{i}\) (\(i=1,2,\ldots,12\)) in the virtual calculation domain. For any target in the calculation domain with the coordinate \((x,y,z)\), \(T_{i}\) was the ToF along an actuator-target-sensor path \(A\)-target-\(R_{i}\)  
\[T_{i}=\frac{1}{V}(\sqrt{\left(x^{A}-x\right)^{2}+\left(y^{A}-y\right)^{2}+ \left(z^{A}-z\right)^{2}}+\sqrt{\left(x^{R_{i}}-x\right)^{2}+\left(y^{R_{i}}-y \right)^{2}+\left(z^{R_{i}}-z\right)^{2}}), \tag{1}\]  
where \((x^{A},y^{A},z^{A})\) and \((x^{R_{i}},y^{R_{i}},z^{R_{i}})\) were coordinates of the actuator position and of the \(i\)-th receiving position, respectively. \(V\) was the speed of ultrasound in water. There were three unknown target parameters, \((x,y,z)\), in Equation (1). The solution to Equation (1) was a root locus, as shown in Figure 2. For any point on the ellipsoid, the sum of the distance to the receiver and the transmitter was one distance measurement (\(T_{i}\cdot V\)), which was a constant. Therefore, the locus was an ellipsoid whose focuses were actuator position \(A\) and receiving position \(R_{i}\). Since calculation was conducted in the calculation domain, only part of the ellipsoid in the calculation domain was shown. The ellipsoid implied the possible locations of the target for a certain ToF. In the traditional ToF-based method, there are two steps involved in target localization. First, the curves--the intersection of any two ellipsoids--are obtained. Then, the target location is given by seeking the intersection of all curves. As shown in Figure 3, in the case of using four receiving positions, there were four ellipsoids. Curve1 is the intersection of the blue and green ellipsoids. Curve2 is the \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Object** & **x/mm** & **y/mm** & **z/mm** & **Object** & **x/mm** & **y/mm** & **z/mm** \\ \hline Target center & 65 & 108.5 & 45 & Actuator position & 60 & 0 & 40 \\ Receiving position 1 & 20 & 40 & 150 & Receiving position 7 & 20 & 0 & 150 \\ Receiving position 2 & 20 & 40 & 110 & Receiving position 8 & 20 & 0 & 110 \\ Receiving position 3 & 60 & 40 & 150 & Receiving position 9 & 60 & 0 & 150 \\ Receiving position 4 & 60 & 40 & 110 & Receiving position 10 & 60 & 0 & 110 \\ Receiving position 5 & 100 & 0 & 150 & Receiving position 11 & 100 & 40 & 150 \\ Receiving position 6 & 100 & 0 & 110 & Receiving position 12 & 100 & 40 & 110 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The coordinates of the actuator, the 12 receiving positions, and the target. Figure 2: The relation between a single time-of-flight (ToF) measurement and the ellipsoid. intersection of the blue and yellow ellipsoids. Then, the point at which all curves (Curve1, Curve2, etc.) passed through was considered the target location. However, the measured ToF values were not absolutely accurate due to noises and many other factors. There was no point at which all curves passed through when the ellipsoids were drawn based on real-life ToF measurements. The points at which some, but not all, curves passed through were considered as pseudo-targets. As shown in Figure 4, we marked down Curve1-Curve6. Curve1 is the intersection of the blue and green ellipsoids. Curve2 is the intersection of the blue and pink ellipsoids. Curve4 is the intersection of the blue and yellow ellipsoids. Curve5 is the intersection of the pink and yellow ellipsoids. Curve6 is the intersection of the green and yellow ellipsoids. Pseudo-target 1 is the point at which Curve1-Curve3 passed through but Curve4-Curve6 did not. Pseudo-target 2 and pseudo-target 3 were obtained in the same way. Figure 3: Target localization result using the traditional ToF-based method. Four ellipsoids are given by accurate ToF. In addition, ToF measurement errors also gave rise to disjoint ellipsoids and disjoint curves.