## 5 Simulation Experiments
### Analysis for Improved UWB/IMU Tight Combination Positioning Model  
To verify the effectiveness of the algorithm, this study compared the pure UWB (i.e., only ranging UWB) positioning system with the improved model based on the UWB/IMU tight combination and EKF filter positioning system. The pure UWB positioning system only uses the UWB ranging information for position calculation. The position information output is unstable and has a large position offset, with the maximum error reaching 2 m.  
The improved model could control the error within 0.3 m during 75% of the experiment and converge the positioning trajectory to the real value faster, which shows that the improvement based on the combined positioning model of EKF could effectively track the position of the target. The IMU could remove the data that cannot be identified by the filter in the traditional model in advance, and the remaining data optimize the UWB positioning results. This greatly eliminates the impact of NLOS interference on pure UWB systems [27; 28].  
The comparison chart of trajectory and positioning results is shown in Figure 7. The chart shows that along the Y-axis direction, the absolute value of the maximum deviation of the pure UWB reached 1.444 m, the maximum deviation of the conventional UWB/IMU reached 0.408 m, and the absolute value of the maximum deviation of the UWB/IMU was only 0.232 m.  
In Table 1, error reduced by \(1-(0.232/1.444)=83.93\%\), \(1-(0.232/0.408)=43.14\%\), and standard deviation reduced by \(1-(0.0998/0.8567)=88.35\%\), \(1-(0.0998/0.1283)=22.21\%\), so the improved UWB/IMU tight combination positioning results (as the greed line) were far better than the pure UWB (as the red color line) shown in Figure 7.  
The I-UWB/IMU was the improved UWB/IMU tight combination positioning model proposed in this paper, and UWB/IMU is the conventional UWB/IMU tight combination positioning model.  
In this experiment, the time interval was \(1\), \(i=1\), \(j=0\), \(k=0.02\); the system matrix was \(F_{i}\) and the control matrix was \(\Gamma\).  
\[F=\begin{bmatrix}I_{2}&I_{2}&1/2I_{2}\\ O_{2}&I_{2}&I_{2}\\ O_{2}&O_{2}&I_{2}\\ \end{bmatrix},\Gamma=\begin{bmatrix}1/6I_{2}\\ 1/2I_{2}\\ I_{2}\\ \end{bmatrix} \tag{36}\]  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Method/Model** & **Maximum Deviation (m)** & **Standard Deviation (m)** \\ \hline Pure UWB & 1.444 & 0.8567 \\ \hline UWB/IMU & 0.408 & 0.1283 \\ \hline I-UWB/IMU & 0.232 & 0.0998 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Deviations in I-UWB/IMU, UWB/IMU, and pure UWB positioning.  
According to Equations (34) and (35), the rank of the matrix \(F\) was 3, and the system did not satisfy the complete controllability. In the case of the double-loop accumulated error in this study, the \(F\) and \(\Gamma\), as shown in Equation (37), are as follows:  
\[F=\begin{bmatrix}I&I&1/2I\\ O&I&I\\ O&O&I\end{bmatrix},\Gamma=\begin{bmatrix}1/6I\\ 1/2I\\ I\end{bmatrix} \tag{37}\]  
According to Equation (35), the controllable matrix of the system could be calculated as  
\[P=\begin{bmatrix}0.1667&1.1667&3.1667\\ 0.5000&1.5000&2.5000\\ 1.0000&1.0000&1.0000\end{bmatrix} \tag{38}\]  
The rank of the matrix \(P\) was 3, and the system was completely controllable, proving that the error of the system was controllable with the elimination approach of the double-loop accumulated error.