# Improving IMES Localization Accuracy by Integrating Dead Reckoning Information
## 6 Positioning Experiments
### Robot Localization
#### 6.1.2 Experimental Results  
Figure 9a,b shows comparisons between the actual path and the estimated path in the case of using the initial values of correct position and orientation and the case of incorrect values, respectively. In these figures, the purple dot, the blue dot with a line, the black line, the green line and the red line correspond to the transmitter position, position estimated by trilateration, actual path, path estimated by dead reckoning and path estimated by Kalman filtering, respectively.  
These results show that the blue dots representing IMES trilateration are widespread (which means low accuracy), but by using the Kalman filtering, the estimation paths are significantly improved. As seen in Figure 9a, the performance of the dead reckoning is higher than the Kalman filtering, even if the convergence period of the EKF in the initial stage of the estimation process is not considered. However, Figure 9b shows that the dead reckoning cannot recover the incorrect initial position and orientation, but the Kalman filtering corrects the wrong initial condition and converges to a nearly correct path.  
The accuracy of position estimation during path following is also analyzed. Figures 10 and 11 respectively show the time series data of the position estimation errors for both the trilateration and the EKF and their histograms in the case of using incorrect initial values.  
As can be seen in these figures, the estimation error of the Kalman filtering case is far smaller than that of the trilateration case. Trilateration has an average estimation error of 766 mm, and an error at the 95 cumulative percentage is 1474 mm, whereas the estimation errors of EKF are respectively 217 mm and 458 mm.  
Figure 12 shows the time series of the parameters of the EKF. Figure 12a shows the sequence of the innovation of transmitter #175 (_i.e._, measurement residual expressed as \(\mathbf{Z}^{k+1}-h(\mathbf{X}^{k+1|k},\psi)\) in Equation (19), which means the difference between the actual and predicted observation values of the distance between the receiver and transmitter antennas). If the system is properly modeled in the EKF and the system and observation noise are zero-mean Gaussian, the innovation sequence becomes a white noise. However, as seen in the figure, since the sequence has a sort of periodic pattern, such conditions are not completely met. Figure 12b shows the time series of the error covariance, \(\mathbf{P}^{k+1|k+1}\), represented in Equation (19) and the actual position estimation error. Note that Cov. \(X\) and Cov. \(Y\) in the figure actually mean the square root of the diagonal elements of \(\mathbf{P}^{k+1|k+1}\), which also represent the standard deviation of the uncertainty for the x- and y-components of the estimated position. As seen in the figure, the estimation errors are far larger than the covariance values. This also means that the position estimation algorithm does not completely meet the Kalman filtering conditions mentioned above.  
Figure 9: Experimental result of robot position estimation. (**a**) Correct initial position and orientation; (**b**) Incorrect initial position and orientation.  
Figure 11: Histogram of position estimation error.  
Figure 12: Time series of EKF parameters of robot localization. **(a)** Innovation sequence; **(b)** Estimation error and error covariance.  
Figure 10: Position estimation error (experiment with robot).