**6. Experimental Case Study**

To validate the simulation results, a parrot AR drone has been used for the experimental case study. The drone has the following technical specifications as given by the manufacturer:


The parrot AR drone has been developed as a low cost drone by parrot company and is quite customizable. The code is open source and can be modified according to the necessity. It has a variety of sensors and the data can be obtained from them and processed as needed. For the present case study, the velocity and acceleration measurements have been considered. For information about the algorithm used by the drone to calculate its speed, the readers are suggested to refer to [25].

The employed drone has been observed to have a negative systematic error in the velocity measurements obtained from the sensors present in the drone itself. So, the velocity is being underestimated by the sensors of the drone. It has also been observed that the systematic error is not constant for all runs. Each individual run had a systematic error that may be different from the other runs. So, only an interval of values can be estimated and the error can not just be compensated.

By performing a large number of runs of the drone, the interval for the systematic error has been estimated and this was used to construct the internal membership function of the RFV for the measured velocity. The constructed RFV assumed to be centered at zero velocity can be seen in Figure 10.

**Figure 10.** RFV of the velocity constructed from the data. The blue line represents the external membership function and the red line represents the internal membership function.

The measured acceleration, on the other hand, does not have any systematic contributions to uncertainty. Hence, the RFV can be constructed by simply using a probabilitypossibility transformation on the probability distribution of the acceleration.

The drone was made to fly for a few seconds to cover a distance of approximately 4 m. The velocity and acceleration data from the sensors is obtained from the drone every 5 ms using a software program that links the computer with the drone using the Wi-Fi network. The alternative possibilistic KF described in Section 4 was used to provide the filtered velocity and acceleration predictions with their respective uncertainties as well as compensate partially for the systematic error in the velocity measurements provided by the drone.

The velocity estimates provided by the KF were integrated to get the estimated distance traveled by the drone. Similarly, the velocity measurements directly obtained by the drone were integrated as well, to get the distance that the drone traveled according to the sensors present in the drone.

At the end of every run, the actual distance from the starting point was been measured. Measuring tape was used to do this since the error in the distance calculated using the velocity data from the sensors is quite high and the precision of the measuring tape is enough to be deemed negligible. Several runs were made and the distances estimated by the KF and those estimated according to the sensor data were compared with the actual distance traveled by the drone. To facilitate a comparison between the alternative KF defined in this paper and the possibilistic KF defined in [23], the sensor data was processed using both the KFs seperately.

The results using the possibilistic KF defined in [23] can be seen in Figure 11. The green line represents the distances estimated according to the velocity measurements obtained directly from the sensors in the drone. The blue line represents the distance obtained from the velocity estimates of the defined possibilistic KF. The black line represents the actual distance traveled by the drone. Finally, the red lines represent the upper and lower bounds for the uncertainty.

**Figure 11.** Distances obtained from the velocity estimates of the possibilistic KF (blue line). The predicted uncertainty intervals (red lines). Actual distance traveled by the drone (black line) and distances estimated according to the velocity measurements obtained directly from the sensors in the drone (green line). Green line and blue line are almost the same.

It can be seen that the distances estimated by the possibilistic KF are quite close to the distances from the sensors. The blue line and the green line in Figure 11 are almost the same and that is why only the green dots and the blue line can be seen in the figure. However, the real measurements lie inside the uncertainty limits of the distances provided by the KF.

**Figure 12.** Distances obtained from the velocity estimates of the defined alternative possibilistic KF (blue line). The predicted uncertainty intervals (red lines). Actual distance traveled by the drone (black line) and distances estimated according to the velocity measurements obtained directly from the sensors in the drone (green line).

The results using the alternative KF defined in this paper can be seen in Figure 12. Again, the green line represents the distances estimated according to the velocity measurements obtained directly from the sensors in the drone. The blue line represents the distance obtained from the velocity estimates of the defined possibilistic KF. The black line represents the actual distance traveled by the drone. Finally, the red lines represents the upper and lower bounds for the uncertainty.

For an easier comparison, Figure 13 shows again the distances obtained using the modified possibilistic KF (green line) and those obtained using the alternative possibilistic KF (blue line) along with the actual distance traveled by the drone (red line).

**Figure 13.** Distances obtained from the velocity estimates of the defined alternative possibilistic KF (blue line). The distances obtained from the velocity estimates of the modified alternative possibilistic KF (green lines). Actual distance traveled by the drone (red line).

A comparison of the results obtained from the two KFs has also been given in Table 2. From Table 2, it can be clearly seen that the distance obtained using the alternative KF defined in this paper is much more accurate and closer to the real measurements than the distances obtained from the sensor measurements or those obtained from the possibilistic KF defined in [23].

**Table 2.** Comparison of the distance estimates of the drone obtained from the two KFs.


Additionally, it can be easily seen that the width of the uncertainty limits associated with the distance (red lines) are also smaller in Figure 12 compared to that in Figure 11. The same can be verified from Table 2.

This confirms that the systematic error in the velocity is being compensated quite efficiently using the defined alternative possibilistic KF and the overall uncertainty associated to the predictions is being decreased as well.

#### **7. Conclusions**

The modified possibilistic KF defined in [23] is capable of propagating the systematic contributions to uncertainty effectively. This paper defines an alternative possibilistic KF which also decreases the effects of the systematic uncertainty contributions on the final measurement and therefore can be considered an improved version of the KF defined in [23].

The same simulated case study as in [23] has been considered to facilitate an easy comparison and the results obtained using the KF defined in this paper have been shown along with the results obtained by using the KF defined in [23]. The obtained results show that the proposed KF provides a compensation of the systematic uncertainty and decreases the overall uncertainty associated to the predictions.

The only requirement to use this method is that the direction of the residual systematic error should be known. This requirement is not so difficult to be satisfied in the era of big data. In any case, if not satisfied, the modified possibilistic KF defined in [23] is still valid and can be successfully applied. A possible area of application of the alternative possibilistic KF proposed in this paper could be in PTP networks where the network traffic is being monitored and thereby it can be evaluated if the transmission delay is higher from master to slave or from slave to master, thus identifying the direction of the systematic error in the calculation of the offset. Therefore, this method could be used to further decrease the uncertainty associated with the time predictions provided by the KF.

**Author Contributions:** Conceptualization, H.V. J. and S.S.; methodology, H.V. J. and S.S.; software, H.V. J.; validation, H.V. J.; formal analysis, H.V. J. and S.S.; investigation, H.V. J. and S.S.; resources, H.V. J. and S.S.; data curation, H.V. J. and S.S.; writing—original draft preparation, H.V. J.; writing review and editing, S.S.; visualization, H.V. J. and S.S.; supervision, S.S. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.
