## 4 Calibration  
Calibration of a LPS is essential before the system can be used to localize static nodes or mobile devices. Contrary to localization the _LOSNUS_ calibration uses ToF and requires at least four receivers, six transmitters and a known reference distance in the currently used form. Using ToF was required because otherwise convergence of the calibration algorithm was not satisfying. Calibration is performed by using six transmitters and four receivers where six out of the 30 coordinates are used for defining the coordinate system. Using the ToF and the speed of sound 24 equations can be found. Using proper minimization algorithms, the 24 unknown coordinates can be obtained. Let \(\mathrm{T_{i}}=(\mathrm{x_{T_{i}}},\mathrm{y_{T_{i}}},\mathrm{z_{T_{i}}})^{ \mathrm{T}},1\leq i\leq 6\) be the transmitters, \(\mathrm{R_{j}}=(\mathrm{x_{R_{j}}},\mathrm{y_{R_{j}}},\mathrm{z_{R_{j}}})^{ \mathrm{T}},1\leq j\leq 4\), c the speed of sound and \(\mathrm{t_{ij}}\) be the ToF between transmitter i and receiver j. The defining equations are then given by  
\[\mathrm{t_{ij}c}=\left\|\mathrm{T_{i}}-\mathrm{R_{j}}\right\| \tag{4}\]  
Due to the application of the cones for broadening the main lobe, the acoustic wave no longer directly propagates between the transmitter and receiver but first propagates from the transmitter to the opening of the cone. From there, it propagates to the receiver. Therefore, the previous equation has to be modified as shown in Equation (5) where the length of the acoustical cone is l.  
\[\mathrm{t_{ij}c}=\left\|\mathrm{T_{i}}-\mathrm{R_{j}}\right\|+1 \tag{5}\]  
Figure 7: (**a**) Calculation of the receiver positions by all fifteen combinations in case of no error. All positions are close and spread by the DOP. (**b**) In case of a single outlier only five combinations are still valid and all others deliver different results. The assumed ToA error for this example was 1 cm. It can be shown that if a minimization algorithm solves equation system Equation (5) it also solves a scaled version of the system by a factor \(\mathrm{a}\), _i.e._, a geometrical expansion or compression. \[\mathrm{t_{ij}}\mathrm{ca}=\left\|\mathrm{aT_{i}}-\mathrm{aR_{j}}\right\|+ \mathrm{a}\cdot\mathrm{l} \tag{6}\]  
This is an important result because no matter which speed of sound has been used the result of a calibration process is a scaled version of the physical system with the following consequence: Let \(\mathrm{r}\) be a reference distance between to arbitrary receiver position. Then a post scaling factor can be calculated as  
\[\mathrm{a}=\frac{\mathrm{r}}{\left\|\mathrm{R_{i}}-\mathrm{R_{j}}\right\|} \tag{7}\]  
We therefore have exchanged the problem of knowing the accurate temperature to the problem of knowing a reference distance. Note that there has been placed no restriction on the type of reference distance which makes realization very easy. In this system, the reference path has been realized by using a mounted receiver on a linear belt and fixing two positions. The uncertainty of the belt was assumed to be 50 \(\mathrm{\SIUnitSymbolMicro m}\) as manufacture data was not complete. The calibration algorithm can be summarized as following: Based on guesses of the speed of sound, receiver and transmitters positions, a minimization algorithm is executed delivering positions \(\mathrm{T_{i}}^{\prime}\), \(\mathrm{R_{i}}^{\prime}\) and a cone length \(\mathrm{l}^{\prime}\). As the speed of sound was only relatively correct but not absolutely the output is a scaled version of the system. By calculation of the factor \(\mathrm{a}\) according to Equation (7) the final calibrated transmitter and receiver positions as well as the cone length are given by  
\[\mathrm{T_{i}}=\mathrm{T_{i}^{\prime}a\;R_{i}}=\mathrm{R_{i}^{\prime}a\;l=al^ {\prime}} \tag{8}\]  
Another important result is that a calibration performed with this approach is dependent on the accuracy of the reference path.