## 3 Experiment and Analyses
### Analyses of the Dynamic Experiment
#### 3.2.2 The Linear Constraint to Recognize the Real Peak  
The linear constraint used in this section can be divided into two parts. In one of the experiments, there is a straight track. The rover station can be controlled to move along the rail for a short distance. Only the coordinate of the real peak will change along a straight line, which is mentioned in Section 2.4. In the other one, a dolly is taken as a carrier instead of the fixed rail. However, the method is the same as the former.  
After static initialization, the rover station is controlled to move along the straight rail for a short distance. AFM is applied during the dynamic perturbation dataset and a series of coordinates corresponding to the large AFVs can be obtained (including the wrong peaks). The correct trajectory should be a beeline so that the real peak can be found through this constraint strategy. The coordinates of the large AFVs are shown in Figure 9.  
Figure 9 shows the results of AFM from static to dynamic linear constraint stage. The scatter is drawn every five epochs (interval: 0.1 s) in order to make the graph more clear and intuitive. According to Figure 9, only one trajectory is always in line with the characteristics of the linear constraint and the other coordinates are all discrete distribution. It indicates that the real peak is uniquely identified. In addition, the right picture of Figure 9 is the actual photograph of the fixed rail and motion carrier (rover station).  
The first coordinate in correct trajectory is the coordinate corresponding to real peak at static initialization stage. In the program, the coordinates of static epoch and the first epoch of dynamic linear constraint should be close. The candidate lines could be found and the slopes of lines would be obtained. The coordinates of the first and second epochs can also form a straight line, whose slopes should be similar to the former (the first epoch and static epoch). Therefore, the right trajectory can be found by setting the threshold of the slopes difference. Once the real peak is determined, the corresponding coordinate can be used as the initial value of LAMBDA, so as to achieve high accuracy dynamic indoor positioning.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**X Coordinate** & **Y Coordinate** & **AFV** & **remark** & **X Coordinate** & **Y Coordinate** & **AFV** \\ \hline \(-0.68\) & \(-0.52\) & \(0.9364\) & Real Peak & \(-0.52\) & \(-0.56\) & \(0.8957\) \\ \(-0.50\) & \(-0.16\) & \(0.9147\) & Wrong Peak & \(-0.86\) & \(-0.88\) & \(0.8916\) \\ \(-0.50\) & \(-0.14\) & \(0.9138\) & Wrong Peak & \(-1.04\) & \(-0.16\) & \(0.8831\) \\ \(-1.04\) & \(-0.86\) & \(0.9069\) & Wrong Peak & \(-0.34\) & \(-0.18\) & \(0.8599\) \\ \hline \hline \end{tabular}
\end{table}
Table 4: The large AFVs of the dynamic experiment (static initialization stage).  
In order to verify the reliability of the linear constraint to remove the wrong peaks another test is conducted. In this test, the dolly is not fixed on the track; instead, it can be controlled to move in any directions. We put the antenna on the top of the dolly and control it to move about 30 cm in a certain direction. AFM is also used to obtain the coordinates of each epoch and the result is shown in Figure 10.  
Figure 10: The trajectory obtained by AFM during the linear constraint (random direction), the right image shows the dolly for linear constraint test  
Figure 9: The trajectory obtained by AFM during the linear constraint (along rail), the right image shows the fixed rail for linear constraint test  
According to Figure 10, only one trajectory is always in line with actual linear motion. The real peak can be identified by the method described in the preceding paragraph. It reveals that, instead of using the fixed rail, the wrong peaks can be also removed with a similar approach in practical applications and only a short distance linear constraint is needed. The right side of Figure 10 is the actual photograph of the base and rover station whose antenna is placed on a radio-controlled dolly.