# Improving IMES Localization Accuracy by Integrating Dead Reckoning Information
## 4 Trilateration Positioning Method  
Trilateration is a geometric method that is used to estimate the location of an object by measuring its distances from multiple reference points. It is a well-known technique used in indoor positioning [7; 23], and it requires three or more distance measurements to estimate a location in a two-dimensional space. Based on the IMES radio signal propagation model that was described in Equation (2), the distance from an IMES transmitter can be estimated using a given CNR of the received IMES radio signal. Therefore, the location of an IMES receiver can be estimated by applying trilateration with distance measurements from multiple IMES transmitters. Figure 2 shows the configuration of our test environment for IMES trilateration. Figure 2: Experimental configuration of IMES-radio-based indoor positioning. Six IMES radio transmitters are mounted on the ceiling of an indoor environment. These transmitters broadcast the IMES radio modulated with its location and identity information. An IMES receiver is mounted on a mobile robot that moves freely within the indoor environment. The IMES receiver can receive IMES radio signals from multiple transmitters simultaneously and decode the location and identity information of each transmitter. Then, the CNR of the received IMES radio signal is used to estimate its location. We use the \(X_{i}\), \(Y_{i}\) coordinate system to denote the location of the \(i\)-th IMES transmitter, which is known beforehand. \(x,y\) denotes the location of a mobile robot or IMES receiver, which is unknown and should be estimated. The CNR of the received signal from the \(i\)-th IMES transmitter is represented as \(cnt_{i}\). \(N\) denotes the identity number of the IMES transmitter from which the IMES radio signal is received. Given a set of CNR measurements received, denoted as \([cnt_{1},cnt_{2},\ldots,cnt_{N}]\), the corresponding distance \([d_{1},d_{2},\ldots,d_{N}]\) can be calculated according to Equation (1) as follows:  
\[d_{i}=\sqrt{10^{\frac{x-cnt_{i}}{5_{Y}}}-h^{2}} \tag{3}\]  
Thus, we can obtain the following observation equations:  
\[d_{1} =\sqrt{(x-X_{1})^{2}+(y-Y_{1})^{2}}\] \[d_{2} =\sqrt{(x-X_{2})^{2}+(y-Y_{2})^{2}}\] \[\vdots\] \[d_{N} =\sqrt{(x-X_{N})^{2}+(y-Y_{N})^{2}} \tag{4}\]  
According to Equation (4), the trilateration problem can be formulated as a nonlinear equation solving problem: given the measurement \([d_{1},d_{2},\ldots,d_{N}]\) and the known parameters \([X_{1},Y_{1},X_{2},Y_{2},\ldots,X_{N},Y_{N}]\), try to find the solution of \([x,y]\) that best fits the nonlinear Equation (4). Figure 3 shows the trilateration problem. Figure 3: Trilateration problem. The empirical Gauss-Newton method can be used to solve the nonlinear equations iteratively. Given an initial estimate of \([x,y]\) as \([x_{0},y_{0}]\), the nonlinear equations of Equation (4) can be linearized around \([x_{0},y_{0}]\), as follows:  
\[d_{1} =d_{1}(x_{0},y_{0})+\frac{\partial d_{1}}{\partial x}|_{(x_{0},y_{ 0})}\Delta x+\frac{\partial d_{1}}{\partial y}|_{(x_{0},y_{0})}\Delta y\] \[d_{2} =d_{2}(x_{0},y_{0})+\frac{\partial d_{2}}{\partial x}|_{(x_{0},y_{ 0})}\Delta x+\frac{\partial d_{2}}{\partial y}|_{(x_{0},y_{0})}\Delta y\] \[\vdots\] \[d_{i}=\sqrt{10^{\frac{x-cnt_{i}}{5_{Y}}}-h^{2}} \tag{3}\]0  
where \(\Delta x\), \(\Delta y\) is the increase in \(x,y\) that will be added to the initial estimate of \(x_{0},y_{0}\) to generate the estimate of \(x,y\) for the nextiteration step. According to Equation (5), we can obtain a set of linear equations about \(\Delta x\), \(\Delta y\) as follows:  
\[\mathbf{A} \begin{bmatrix}\Delta x\\ \Delta y\end{bmatrix}