# The Delusion of Maneuver Detection

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THE DELUSION OF MANEUVER OR ACCELERATION

DETECTION

J. W. Bell,bellco@infowest.com

October 27, 2012

ABSTRACT

The purpose of this note is to demonstrate by example that a smooth transition occurs in

the MSE (or RMSE) of a tracking estimator when a target accelerates; and that major spikes

actually occur only when the target stops accelerating. As a result, it appears to be a delusion to

believe that a maneuver or increase in acceleration can be effectively detected and used.

KEY WORDS

Tracking Filter, Tracking, Kalman Filter, Optimal Filter, Optimal Estimation, Mean Square

Estimation, Least Squares Method, Minimum Mean Square Error

mailto:bellco@infowest.commailto:bellco@infowest.commailto:bellco@infowest.commailto:bellco@infowest.com
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EXAMPLE SCENARIO

Consider the following scenario: Target has constant velocity for 10 scans, accelerates at

3g for 10 scans, travels at constant velocity for 10 scans, decelerates at 1.5g for 10 scans, and

ends up traveling at a constant velocity for the final 10 scans.

Assume 5-point one step prediction. As we saw in Example 5 of [1], the sample period at

which the MSE of a 5-point 2nd

order predictor applied to a target with a maximum acceleration

of 3g's crosses over the a 5-point 3rd

order predictor variance (MSE) is c = 0.235 sec.

Consider the optimal 5-point FOE matching that sample period of c = 0.235 for

comparison with the standard 5-point 2nd

and 3rd

order predictors.

The optimal FOE 5-point one step prediction weights are wopt= [0.1 -0.35 -0.3 0.25, 1.3].

The 3rd

order 5-point one step predictor weights are w3 = [0.6 -0.6 -0.8 0.0 1.8], and the 2nd

order 5-point one step predictor weights are w2 = [-0.4 -0.1 0.2 0.5 0.8].

Figure 1 shows the plots of the one step predicted RMSEs (used rather than the MSE to

reduce the scale of the plot) comprised of the theoretical variance (based on a measurement noise

variance of unity) plus the actual bias squared. Since the variance remains constant, the only

variation in the plots is due to changes in the bias. Track initiation is ignored for simplicity.

Therefore, the first sample in the plots actually start on the 5th

scan; and the acceleration begins

on the 6th sample (10th scan in the plot).

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ANALYSIS AND DISCUSSION

The most striking feature of these plots is that all of the estimators make a smooth RMSE

transition when the target begins to accelerate at the 6th

sample (or decelerate at the 26th

sample).

In fact, major transition spikes occur only at the end of acceleration at the 15th

sample (or

deceleration at the 35th

sample).

Thus, in this case a maneuver detector or acceleration estimator would be expected to

kick in only after the target has stopped accelerating (or decelerating). Therefore, not only would

adaptivity increase the error as suggested in [2], it would cause more havoc by calling for an

estimator matched to an acceleration which no longer existsa major double whammy.

Figure 1

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Other things to note are as follows:

(1) The 2nd

order RMSE does equal the 3rd

order RMSE for max acceleration of 3g as designed.

(2) The 3rd

Order RMSE has double spikes for both acceleration and deceleration.

(3) Both 2nd

and 3rd

order RMSEs have spike widths of 4 samples for both acceleration and

deceleration.

(4) Optimal predictor RMSE has both lower spikes than 2nd

and 3rd

order predictors and spike

widths of only 2 samples in both caseshalf that of both the 2

nd

and 3

rd

order predictors.

FORMULAS AND NUMBERS

Normalized 3rd

order predictor variance:)2)(1(

699||

22

3

NNN

NN = 4.6, for N = 5.

15.2RMSE . This we see pretty much everywhere in Figure 1 except for the large spikes as

well as the small blip at N=7 during acceleration and the almost imperceptible blip at N=27

during deceleration.

Normalized 2nd

order predictor variance:)1(

24|| 22

NN

N = 1.1, for N = 5.

05.1RMSE . This occurs everywhere there is no acceleration or deceleration: between N = 1 to

5, N = 20 to 25, and N = 40 to 45.

Normalized 2nd order predictor bias squared at worst case acceleration of 3gs:

2

3

22

2T

a

= 3.5 where

2

2

36

)2)(1(

NNT =49 and

)4)(1(

180

2 22

22

NNN

a= 1/14

for N = 5. This occurs between N = 10 to 15.

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Normalized 2nd

order predictor mean square error (MSE) at worst case:

2

3

22

2

22

|| Ta

MSE

= 4.6 for N=5. This matches the 3rd order variance at maximum

acceleration of 3gs as designed. Again 15.2RMSE . This occurs between N = 10 to 15.

Normalized FOE predictor MSE:

2

33

22

2

3

2

3

2

2 )1(2

)|||(|

fT

afMSE ,

wheref3 is the interpolation factor (or fraction) between the 2nd

and 3rd

order predictors. For the

optimum at worst case of 3gs, this reduces to )|||(| 23,32

2 optopt fMSE , where

2

2

2

3

2

3 |||||| .

By design the optimal FOE in this case is based on the optimal fraction, or interpolation

factor off3opt = 0.5. Therefore, 85.2)6.41.1(5.0)|||(|5.02

3

2

2 MSE . 69.1RMSE .

This occurs also at maximum acceleration between N = 10 to 15.

On the other hand, if there is no bias, the FOE predictor MSE reduces to the normalized

variance: )|||(|var 232322 fiancenomalizedMSE , which can be written as

975.1)1.136.4(4

1)||3|(|

4

1)|||(|

4

1|| 22

2

3

2

2

2

3

2

2

2 xMSE p .

40.1RMSE . This occurs during no acceleration or deceleration: N = 1 to 5, N = 20 to 25, and

N=40 to 45.

CONCLUSION

Detecting the onset of a target maneuver or acceleration is shown in this example to be

problematic. Spikes in the RMSE at the end of a maneuver or acceleration suggest that a change

in the tracking estimator based on maneuver or acceleration detection is likely to cause more

havoc by introducing the change after the fact. Therefore, not only would adaptivity increase the

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error as suggested in [2], it would call for an estimator matched to an acceleration which no

longer existsa major double whammy.

REFERENCES

[1] Bell, J. W., The Fall acy Of Using A 2ndOrder Estimator On Acceleration,

http://site.infowest.com/personal/j/jeffbell/The2ndOrderEstimatorFallacy.pdf

[2] Bell, J. W., Tracking Estimator Adaptivity: What Pri ce?,