# Improving IMES Localization Accuracy by Integrating Dead Reckoning Information
## 5 Integration of IMES Radio and PDR Information  
To realize a practical indoor positioning system, dead reckoning technology is introduced because it can provide comparatively accurate measurements of an object's relative movements if it is in a short period. Other positioning technology, such as radio positioning, will be used to estimate the initial position or to modify the error accumulation during dead reckoning over an extended duration. This combination is meaningful because it is cost efficient and reliable. Thus, we would like to develop an indoor positioning system that combines IMES radio positioning and dead reckoning. The extended Kalman filter (EKF) is used for integrating IMES radio and dead reckoning, because it is a classic solution for dynamic tracking with different sensor measurements (note that a particle filter also can be used for the multi-sensor integration, but considering the use for embedded systems, we chose the EKF, because its computational load is low). In our system, the IMES radio receiver is mounted on a mobile robot platform that moves randomly within the indoor environment, as shown in Figure 2. The relative displacement of a mobile robot can be measured through its wheel encoder, and the received IMES radio signal is considered a sensor measurement that can be used to modify the dynamic tracking of the robot location. Let \(x,y\) denote the coordinate of the mobile robot and \(\theta\) its orientation angle. Thus, the state variable of the system can be written as follows:  
\[\mathbf{X}=[x,y,\theta]^{T} \tag{11}\]  
According to the wheel encoders of the mobile robot, we can obtain the relative distance and orientation angle that the robot moved in a time step, denoted as \(\Delta d\) and \(\Delta\theta\), respectively. It can also be considered an input variable \(\mathbf{U}\) of the system. \[\mathbf{U}=[\Delta d,\Delta\theta]^{T} \tag{12}\]  
Thus, we could easily obtain the dynamical model of the system as follows:  
\[x^{k+1} =x^{k}+\Delta d^{k}cos(\theta^{k})+\omega_{1}\] \[y^{k+1} =y^{k}+\Delta d^{k}sin(\theta^{k})+\omega_{2}\] \[\theta^{k+1} =\theta^{k}+\Delta\theta^{k}+\omega_{3} \tag{13}\]  
Figure 6: Histogram of estimation error of IMES radio-based trilateration. where superfix \(k\) and \(k+1\) denote time step \(k\) and time step \(k+1\), respectively. \(\mathbf{\omega}=\left[\omega_{1},\omega_{2},\omega_{3}\right]^{T}\) denotes the system noise, which is assumed to be drawn from a zero mean multivariate normal distribution with covariance \(\mathbf{\Omega}\). The observation equations of the system can be obtained as follows:  
\[d_{1}^{k} =\sqrt{(x^{k}-X_{1})^{2}+(y^{k}-Y_{1})^{2}}+\psi_{1}\] \[d_{2}^{k} =\sqrt{(x^{k}-X_{2})^{2}+(y^{k}-Y_{2})^{2}}+\psi_{2}\] \[\vdots\] \[\mathbf{X}=[x,y,\theta]^{T} \tag{11}\]0  
where \(d_{1}^{k},d_{2}^{k},\ldots,d_{N}^{k}\) is the distance between the robot and the IMES transmitters, which is calculated using the received radio CNR. It is also considered an observation variable \(\mathbf{\mathrm{Z}}\) of the system. \[\mathbf{X}=[x,y,\theta]^{T} \tag{11}\]1  
where \(N\) is the number of IMES radio transmitters whose radio signal is successfully received. \((X_{1},Y_{1}),(X_{2},Y_{2}),\ldots,(X_{N},Y_{N})\) are coordinates of the corresponding IMES radio receivers. \(\mathbf{\psi}=[\psi_{1},\psi_{2},\ldots,\psi_{N}]^{T}\) denotes the observation noise, which is assumed to be drawn from a zero mean multivariate normal distribution with covariance \(\mathbf{\mathrm{Y}}\). The dynamical equation and observation equation of the system can also be organized as follows:  
\[\mathbf{X}=[x,y,\theta]^{T} \tag{11}\]2 \[\mathbf{X}=[x,y,\theta]^{T} \tag{11}\]3  
This is a typical nonlinear system; thus, we need to use the EKF for state estimation. We could linearize the equations about the current estimated state and then apply [PERSON]'s equations.