Chapter 4 Fourth-order FMP Filter to Track High Mobility Warship
4.4 Speed-dependent Fourth-order FMP Filter to Track High Mobility
Target in Motion
Section 4.3 discussed the speed-dependent fourth-order FMP filter to track high mobility warship with the condition that own ship is motionless.
But it is much more common that own ship is also in motion. This Section explores the speed-dependent fourth-order FMP filter to track high mobility target under the condition of both own ship and target are in motion
4.4.1 Input Model of Own Ship and Target
The input model of own ship and target is the same model of Section 4.2. As parameters and are used to control target's speed in Equations (4.3) - (4.6), in order to find the performance of filter influenced by speed, and are looped from 1 to 150 with step of 15. After 1000 times of simulation, 10000 sets of data of and speed will be obtained.
4.4.2 Noise Addition
The observed position is the output obtained from the radar measurements and therefore includes an error. In this study, the noisy observation was obtained by corrupting the true state with zero mean random white Gaussian noise of which standard deviation, , is 10 .
Figure (4.22) and Figure (4.23) show the error distribution in the observation.
-40 -30 -20 -10 0 10 20 30 40
1 51 101 151 201 251 301 351 401 451 501 551 601 651 701 751 801 851 901 951 1001
E as t- W es t er ro r,
mSample number
Figure 4.22 East-West error in the observation of speed-dependent 4th order FMP filter to track high mobility target under the condition of both
own ship and target in motion
-40 -30 -20 -10 0 10 20 30 40
1 51 101 151 201 251 301 351 401 451 501 551 601 651 701 751 801 851 901 951 1001
N o rt h -S o u th e rr o r,
mSample number
Figure 4.23 North-South error in the observation of speed-dependent 4th
4.4.3 Determination of Optimal
Figure (4.24) shows the enlarged view of true, observed, predicted and smoothed position trajectories lies in the interval [17800, 18600] in the X-axis. Figure (4.25) shows the prediction tracking error which stands for the difference between true position and predicted position while Figure (4.26) shows the smoothing tracking error which stands for the difference between true and smoothed position. Afterwards, set varies for 0 to 1 with step of 0.01 and calculated the total prediction tracking error and total smoothing tracking error of each step.
5000 5100 5200 5300 5400 5500 5600
17900 18000 18100 18200 18300 18400 18500
N o rt h -S o u th p o si ti o n ,
mEast-West position, m
Predicted Position True Position Smoothed Position Observed Position
Figure 4.24 Enlarged view of tracking result of speed-dependent 4th order FMP filter to track high mobility target under the condition of both own
ship and target in motion
0 10 20 30 40 50 60 70
1 51 101 151 201 251 301 351 401 451 501 551 601 651 701 751 801 851 901 951 1001
T ra c k in g e rr o r,
mSample number
Figure 4.25 Prediction tracking error of speed-dependent 4th order FMP filter to track high mobility target under the condition of both own ship
and target in motion
0 10 20 30 40 50 60 70
1 51 101 151 201 251 301 351 401 451 501 551 601 651 701 751 801 851 901 951 1001
T ra c k in g e rr o r,
mSample number
Figure 4.26 Smoothing tracking error of speed-dependent 4th order FMP filter to track high mobility target under the condition of both own ship
and target in motion
0 10000 20000 30000 40000 50000 60000 70000 80000
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Sum of TrackingError, m
ξ value
Sum of Prediction Tracking Error Sum of Smoothing Tracking Error
Figure 4.27 Sum of prediction tracking error and smoothing tracking error of speed-dependent 4th order FMP filter to track high mobility target under
the condition of both own ship and target in motion
Figure (4.27) shows the sum of total prediction tracking error and total smoothing tracking error when value varies from 0 to 1. In this case the damping parameter, was selected as 0.68 for prediction purpose.
4.4.4 Relationship between Speed and Optimal
As mentioned above, curve fitting is used to find the relationship between the speed and optimal . Setting coefficients in Equations (4.1) - (4.4) looped from 1 to 150 with step of 15, after plotting 10000 sets of points, it is clear to know that it exists a connection between and . Since the data between ξ and v in Figure (4.28) are linearly related, it is reasonable to assume an approximation of a linear relation as
. (4.10)
Using the obtained 10000 sets of average speed and optimal , the curve fitting result can be obtained as Figure (4.28).
= 4E-052- 0.006+ 0.7674
0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9
0 10 20 30 40 50 60
, m/s
Figure 4.28 Curve fitting result of speed-dependent 4th order FMP filter to track high mobility target under the condition of both own ship and target
in motion
With the result of 10000 sets of data, the Equation (4.10) can be obtained as follows:
× ∙ ∙ (4.11)
4.4.5 Advantages of Speed-dependent FMP Filter
the speed-dependent fourth-order FMP filter has a higher performance than the classic fourth-order FMP filter with constant parameter.
Quantitatively, the total prediction tracking error and total smoothing tracking error are used to evaluate the performance of two filters. As shown in Table below 4.6, the classic fourth-order FMP filter demonstrates the best performance when =0.50 with the prediction tracking error amounting to 18,762.83 . On the contrary, speed-dependent FMP filter depicts its optimal performance with total prediction tracking error resulting to 16,593.72 . The accuracy in tracking is therefore improved by 2170.11 equivalent to 11.57% higher. Similarly, the smoothing accuracy is increased by 409.28 by employing the speed-dependent FMP filter. Meanwhile, the standard deviation of prediction tracking error and smoothing tracking error of speed-dependent FMP filter are a little smaller than them of classic fourth-order FMP filter, it indicates that the stability of speed-dependent FMP filter is relatively higher than that of classic fourth-order FMP filter.
Filter type
Prediction tracking error ()
Smoothing tracking error ()
Std. of prediction tracking error
Std. of smoothing tracking error Classic fourth order FMP
filter 18,762.83 10,274.56 10.16 5.31 Speed-dependent fourth
order FMP filter 16,593.72 9,865.28 8.89 5.17 Table 4.6 Performance comparison of classic fourth-order FMP filter and
speed-dependent fourth-order FMP filter to track high mobility target under the condition of own ship and target in motion
Table 4.7 shows the tracking performance comparison result of speed-dependent third-order FMP filter and speed-dependent fourth-order
FMP filter to track high mobility target under the condition of own ship and target in motion. The prediction tracking error of speed-dependent fourth-order FMP filter amounts to 16,593.72 while the speed-dependent third-order FMP filter depicts its optimal performance with total prediction tracking error resulting to 21,059.89 . The accuracy in prediction is therefore improved by 4466.17 equivalent to 21.2%
higher. Similarly, the smoothing accuracy is increased by 944.68 by employing the speed-dependent fourth-order FMP filter. Meanwhile, the standard deviation of prediction tracking error and smoothing tracking error of speed-dependent fourth-order FMP filter are smaller than them of speed-dependent third-order FMP filter which indicates that the stability of speed-dependent fourth-order FMP filter is relatively higher.
Filter type
Prediction tracking error ()
Smoothing tracking error ()
Std. of prediction tracking error
Std. of smoothing tracking error Speed-dependent third
order FMP filter 21,059.89 10,809.96 10.95 5.56 Speed-dependent fourth
order FMP filter 16,593.72 9,865.28 8.89 5.17 Table 4.7 Performance comparison of speed-dependent third-order FMP filter and speed-dependent fourth-order FMP filter to track high mobility
target under the condition of own ship and target in motion
4.5 Performance of Fourth-order FMP Filter to Track Random