The performance comparisons between the KT scheme, the CLT scheme, and the proposed PLT and GPLT algorithms are conducted under the rural environment. Fig. 6.4 illustrates the scenarios with various numbers of BSs (i.e. the Nk values) that are available at different time instants. It can be seen that the number of BSs becomes insufficient (i.e. Nk< 3) from the time interval of t = 102 to 150 sec. The total simulation interval is set as 150 seconds.
Figs. 6.5 to 6.7 illustrate the performance comparisons of the trajectory, the velocity, and the acceleration tracking using the four algorithms. The estimated trajectories obtained from these schemes are illustrated via the solid lines; while the true trajectories are denoted by the dashed lines. The locations of the BSs are represented by the red empty circles as in Fig. 6.5.
The acceleration is designed to vary at time t = 40, 55, and 120 sec from ak = (ax,k, ay,k)
= (0.5, 0), (-1, 1), (0, 0) to (0.2, -0.5) m/sec2 (as shown in Fig. 6.7). It is noted that the number of BSs becomes insufficient during the second acceleration change (i.e. at t = 102 sec). By observing the starting time interval between t = 0 and 101 sec (where the number of BSs is sufficient), the four algorithms provide similar performance on location tracking as shown in the x-y plots in Fig. 6.5. As illustrated in Figs. 6.6 and 6.7, it can be seen that the KT scheme can provide better performance on the velocity and acceleration tracking during the transient phase (i.e from t = 0 to 10 sec). The reason is attributed to its compromise
−10000 −500 0 500 1000 1500 2000 2500 3000 2000
4000
Kalman Tracking (KT) Scheme
y (m)
−500 0 500 1000 1500 2000 2500
1000 2000 3000
Cascade Location Tracking (CLT) Scheme
y (m)
−10000 −500 0 500 1000 1500 2000 2500 3000
2000 4000
Predictive Location Tracking (PLT) Scheme
y (m)
−10000 −500 0 500 1000 1500 2000 2500 3000
2000 4000
Geometric−assisted Predictive Location Tracking (GPLT) Scheme
x (m)
y (m)
Figure 6.5: Trajectory Tracking of the MS Using the KT (Top Plot), the CLT (2nd Plot), the PLT (3rd Plot), and the GPLT (Bottom Plot) Schemes (Solid Lines: Estimated Trajectories;
Dashed Lines: True Trajectories; Red Empty Circles: the Position of the BSs)
0 50 100 150
Figure 6.6: Velocity Tracking of the MS Using the KT (Left Plots), the CLT (Middle-Left Plots), the PLT (Middle-Right Plots), and the GPLT (Right Plots) Schemes (Solid Lines:
Estimated Trajectories; Dashed Lines: True Trajectories)
0 50 100 150
Figure 6.7: Acceleration Tracking of the MS Using the KT (Left Plots), the CLT (Middle-Left Plots), the PLT (Middle-Right Plots), and the GPLT (Right Plots) Schemes (Solid Lines:
Estimated Trajectories; Dashed Lines: True Trajectories)
between the estimated state variables, ˆxk, ˆvk, and ˆak. However, the KT scheme results in the worst performance between the four schemes after the transient phase (as shown in Figs. 6.6 and 6.7). Owing to the utilization of an external location estimator within the KT scheme, the estimation errors are increasingly accumulated caused by the potential inaccuracy of the estimator.
During the time interval between t = 102 and 150 sec with inadequate signal sources, it can be observed that only the proposed GPLT scheme can achieve satisfactory performance in the trajectory, the velocity, and the acceleration tracking. The estimated trajectories obtained from both the KT and the CLT schemes diverge from the true trajectories due to the inadequate number of measurement inputs. It is noticed that the inaccuracy within the PLT scheme is primarily resulted from the implicitly worse geometric layout at certain time instants, which will further be explained by the GDOP plot as in Fig. 6.10.
Moreover, Figs. 6.8 and 6.9 illustrate the average position error and the RMSE (i.e.
characterizing the signal variances) for location estimation and tracking of the MS. The four
0 30 60 90 120 150
Figure 6.8: Performance Comparison: Average Position Error vs. Simulation Time (sec)
0 30 60 90 120 150
Figure 6.9: Performance Comparison: RMSE vs. Simulation Time (sec)
100 105 110 115 120 125 130 135 140 145 150 0
1 2 3 4 5 6 7 8 9 10
Time (sec)
GDOP (m)
PLT GPLT
Figure 6.10: Comparison of the Mean GDOP Values (Associated with Their Confident In-tervals) Between the PLT and the GPLT schemes During the Time Interval with Deficient Signal Sources
location tracking schemes are compared based on the same simulation scenario as shown in Fig. 6.4. It can be observed from both plots that the proposed GPLT and PLT algorithms outperform the conventional KT and CLT schemes. During the time interval of 40 to 55 sec while the acceleration changes, the RMSEs obtained from these four schemes slightly deviate for the acceleration adjustment. The main differences between these algorithms occur while the signal sources become insufficient after the time instant of t = 102 sec. The proposed GPLT scheme can still provide consistent location estimation and tracking; while the other three algorithms result in augmented estimation errors. The major reason is attributed to the assisted information that is fed back into the location estimator while the signal sources are deficient. Furthermore, the GPLT algorithm outperforms the PLT scheme (especially under the situations with the number of BSs equal to 1) primarily due to its exploitation of the GDOP criterion.
The comparison of the mean GDOP values (associated with their confident intervals) between the PLT and the GPLT schemes is illustrated in Fig. 6.10. It is noted that the averaged GDOP values are computed based on 25 simulation runs. The mean GDOP values
are compared only during the time interval with deficient signal sources, i.e. while the virtual BSs and the virtual measurements are exploited in both schemes. It can be observed that the GDOP values obtained from the GPLT algorithm are consistent during the simulation period with reasonable variations. On the other hand, the GDOP values acquired from the PLT scheme result in larger variations, especially during the time interval of t = 129 to 141 sec. The results are consistent with those estimation errors as acquired from Figs. 6.8 and 6.9 that worse GDOP value will result in incorrect location estimation of the MS. During the time interval of t = 102 to 128 sec, the GDOP values obtained from both schemes are considered similar, which represent that comparable geometric topology are formed by their individual virtual BSs. The geometric effect will not be an influential factor to the estimation error for the MS. On the other hand, during the time interval of t = 129 to 141 sec, sudden deviates in the GDOP values are observed by using the PLT scheme. The larger average position error and the RMSE within the PLT algorithm (as seen from Figs. 6.8 and 6.9 at around t = 135 sec) can therefore be attributed to the corresponding increased GDOP values and variations.
Nevertheless, with the adoption of the minimal GDOP criterion, the proposed GPLT scheme can still maintain consistent GDOP values under different numbers of available signal inputs.
The resulting estimation error and RMSE can consequently be controlled within a reliable interval. The effectiveness of the GPLT algorithm is therefore perceived, especially under insufficient signal sources (i.e. Nk = 1 and 2).
Chapter 7
Conclusion
In this paper, the Predictive Location Tracking (PLT) and the Geometric-assisted Predictive Location Tracking (GPLT) schemes are proposed. The predictive information obtained from the Kalman filtering formulation is exploited as the additional measurement inputs for the location estimator. With the feedback information, sufficient signal sources become available for location estimation and tracking of a MS. Moreover, the GPLT algorithm adjusts the locations of its virtual Base Stations based on the GDOP criterion. It is shown in the simula-tion results that the proposed GPLT algorithm can provide consistent accuracy for locasimula-tion estimation and tracking even under the environments with insufficient signal sources.
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