Explicit Expressions for Debiased Statistics of 3D. Converted Measurements. The method for tracking based on Kalman Elter with debiased consistent converted ...
[3]
Harris, E J. (1978) On the use of windows for harmonic analysis with the discrete Fourier transform. Proceedings of the ZEEE, 66, 1 (Jan. 1978).
Cartesian position (x,y,z) [2, 31
x = p ' coscp . cos6, y = p.sinip.cos6,
Explicit Expressions for Debiased Statistics of 3D Converted Measurements
z
= p . sin6.
The measured position (p,, pm,6,) can be expressed as
pm=p+j, The method for tracking based on Kalman Elter with debiased consistent converted 2D measurements was given and discussed in [l,41. In this work explicit expressions for debiasing compensation terms and dehiased covariance statistics related to the 3D case are presented. The proposed procedure can be employed in active sonar systems or long range radar systems especially when the cross-range errors are significantly large relative to the range errors.
INTRODUCTION
Linear (linearized) tracking in Cartesian coordinates making use of nonlinear (polar or spherical) measurements can be generally performed in two ways. The first approach, in its generic form, is based on measurements converted to a Cartesian domain and results in correlated error statistics. The second technique based on an extended Kalman filter paradigm results in a mixed coordinate filter. The explicit solutions for the mean and covariance of the converted 2D measurements were derived in [I]. It was shown that for certain practically important levels of the cross-range measurement error the mean of the errors is significant and requires debiasing compensation. A simple test (in terms of range, range accuracy, and angular accuracy) to check whether bias is a problem can be found in [4]. In this work explicit formulas for the debiased consistent characteristics related to the 3D spherical measurements are derived. The debiasing terms as well as the covariance characteristics are given for both the true location of the target and for the measured values of the target position.
cp,=cp+'p,
-
6,=6+8
where the errors ( p ,'p, 6) are assumed to be Gaussian mutually independent with zero mean and diagonal covariance cov(j, Cp,6) = diag(oz, o$,0;).
-
A.
True Bias and Covariance
The errors defined as
(X,?,?) in Cartesian coordinates
X=xm-xx, I.
(1)
j=ym-y,
are
Z=z,-z
where (x,, y,, z,) denotes the converted measurements (i.e., the Cartesian coordinate measurements obtained by application of the transformation (1) to the measured position (p,, (p,,6,)).Assuming zero-mean Gaussian statistics of the spherical measurement errors one can derive the true bias pt(p,p,9) and covariance R,(p, cp, 9) of the converted measurements:
and
II. ANALYSIS OF CONVERTED MEASUREMENT ERRORS
The spherical coordinate system is defined by the range p , azimuth cp, and elevation 6 with respect to the Manuscript received November 8, 1996; revised August 17, 1998 IEEE Log No. T-AES/35/1/01516. 0018-9251/99/$10.00 @ 1999 IEEE 368
IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 35, NO. 1 JANUARY 1999
where
where CY,
= sin2cp . sinh u i
G,x= sin2cpm . sinh o; + cos2cp, . cosh a;
+ cos2'p. cosh u i
ai, = sin29., cosha,2
a, = sin2cp .coshui + cos2 cp . sinh o;
+ cos2 cp,.
sinha;
tiz = sin26, . cosh a$ + cos219, . sinh a :
+ cos219 . sinh a: a,), = sin26. sinh ai + cos26 . cosh ai. CY, = sin219 . cosho:
Gxy = sin2B, . sinh
ai + cos26, . cosh
0:
6,= sin2pm . sinh 20; + cos2 ym. cosh2a; j Y= sin2pm . cosh 20; + cos2 cp, . sinh 2a;
6. Average True Bias and Covariance
Since expressions (2) and (3) are conditioned on the true position (p, cp, 6)they cannot be used directly. To make them more practicable the expected value of the true moments &I, cp, 6) and covariance R,(p, cp, 19) should be evaluated taking into account the measured position (p,,cp,,d,). The expected bias p a z T
PO
= E [ P ~ ( P , P1 ,~m,(~m,6ml ~) = [P:,P:,P~I
and covariance R, RF
R, = E[R,(p,cp,fl) 1 P,,(P,,~,)
RiY
RF
C]
R,", R,Yy R,Y'
=
[ R Z R,Y"
conditioned on (p,, cp,, 6,) are called the average true bias and the average true covariance, respectively [l, 41. After some straightforward algebra one obtains
. cos 6, . (e-0~e-0Tj - e-0%/2e-d/21 7
P:
=Pm.COSpm
1
P: =Pm'SlnPm . cosfl,, . ( e - o ~ e - ~-i e-c:/2e-4/2) 7
7
6,= sin26, . cosh 20: + cos26, . sinh 2 4 bXy= sin26, . sinh 20: + cos26, .cosh 2 ~ : . C.
Consistency Test
The static situation is examined when only the single scan converted measurement errors are taken into account. The analysis is performed by applying the standard statistical consistency check based on the chi-square distribution [2, 41. The test for when debiasing is needed is entirely the same as in [4] for the polar to Cartesian conversion, except that it has to be done for both angles. The two-dimensional dynamic situation in which the converted measurements are utilized in tracking is examined in [ 1, 41. In order to perform the consistency test, the following sample average (sample mean) of the normalized error squared (NES [ 1, 2, 41) associated with converted measurement errors is considered .
p , ~= p . sin6 a
m
.(e-4 m
and
N
- e-+)
(4)
tL
where denotes the 3-dimensional vector of converted measurement errors in realization i, compensated for the hypothesized bias, is the hypothesized covariance of the errors, and N denotes the assumed number of test samples. The mean value (ensamble average) of the statistic (4) is equal to 3 when there is no bias and the assumed hypothesized covariance is matched to the actual error covariance. If the errors are jointly Gaussian then the distribution of N G is chi-square with 3N degrees of freedom. Using N = 1000 samples of converted measurements for each accuracy one obtains the following acceptance region for the 99.8% probability bounds [2]
e,:,
[~~~~~(0.001),x~~,= ~ (12.76' 0 . 9 9 1000,3.24. 9)] 10001. The assumed true object position is at p = lo5 m with azimuth cp = 45" and elevation 6 = 45". The standard deviations of measurement errors are as follows: ap = 50 m, aiDE [0.1", 1001 and a4 = {0.1", 10"). CORRESPONDENCE
369
6,
,
REFERENCES
0
Lerro, D., and Bar-Shalom, Y. (1993) Tracking with debiased consistent converted measurements versus EKE IEEE Transactions on Aerospace and Electronic Systems, 29, 3 (July 1993), 1015-1022. [2] Bar-Shalom, Y., and Li, X.-R. (1993) Estimation and Tracking: Principles, Techniques, and Software. Boston: Artech House, 1993. [3] Blackman, S. (1986) Multiple Target Tracking with Radar Applications. Dedham, MA: Artech House, 1986. [4] Bar-Shalom, Y., and Li, X.-R. (1995) Multitnrget-Multisensor Tracking: Principles and Techniques. Storrs, CT: YBS Publishing, 1995. [l]
0
2
4
6
8
1
Fig. 1. Average NES for debiased conversion (pa,Ra)and standard conversion ( p L ,RL) evaluated at measured position.
The average NES for debiased conversion with the average true bias pa and the average true covariance R, evaluated at each measured position is plotted in Fig. I as a function of ov parametrized by L T ~(the chi-square bounds are also indicated). The same experiment has been performed for the standard coordinate conversion based on linear approximation (p,,RL) of error bias and covariance [2]. This conversion, which is obtained by taking the first-order terms of the Taylor series expansion of (1) evaluated at the measured position (,om,cp,,6,), results in ILL
=0
Comments on "Bearings-Only and Doppler-Bearing Tracking Using Instrumental Variables"
The work carried out by Prof. Y. T. Chan and W. Rudnicki [l] is very much useful for studies in underwater target motion analysis. They presented a simple procedure so that the dimension of the target state vector need not be increased even if the number of frequency tonals are more than one available. This is appreciable.
sin cpm . cos 6,
The results obtained for ( p L , R L )are also shown in Fig. 1 (for o8 = 10" the linearized approximation ( p L ,R,) becomes completely inconsistent and the corresponding plot is out of the figure scale). It is easily seen that the covariance R, is consistent (at least approximately since the errors are not Gaussian) even for relatively large measurement errors, while the conversion based on the standard linear approximation results in consistent characteristics only for a significantly narrower range of measurement errors. PIOTR SUCHOMSKI Dept. of Automatic Control Faculty of Electronics, Telecommunication and Computer Science Technical University of Gdansk Narutowicza 11/12, 80-952 Gdansk Poland
370
-pm . cos cp, . sin 6,
1
While implementing their work, I tried to rederive the equations. There is a confusion in [l, eq. (4b)l. The initial covariance matrix, according to me, should have been
p0 = diag
(60 sin
402 - (60cos 3 12
31 9
instead of Po = diag
(60si11/3~)~ 402 __ 3
(~OCOS/~~,)~ 3
Manuscript received March 8, 1998. IEEE Log NO. T-AES/35/1/01517. 0018-9251/99/$10.00 @ 1999 IEEE
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