Underwater Passive Target Tracking from a Stationary Observer Using Modified Gain Extended Kalman Filter

Underwater Passive Target Tracking from a Stationary Observer Using Modified Gain Extended Kalman Filter

ABSTRACT: Target tracking in underwater ,for a stationary observer, observability is less compared to moving observer. Modified Gain Extended Kalman Filter (MGEKF) developed by Song and Speyer [2] was proven to be suitable algorithm for angles only passive target tracking applications in air. In this paper, this improved MGEKF algorithm is explored for underwater applications with some modifications. In underwater, the noise in the measurements is very high, turning rate of the platforms is low and speed of the platforms is also low when compared with the missiles in air. These characteristics of the platform are studied in detail and the algorithm is modified suitably for tracking applications in underwater. Monte- Carlo simulated results for one typical scenario is presented for the purpose of explanation. From the results it is observed that this algorithm is suitable for stationary observer in underwater passive target tracking using angles only measurements.

1. INTRODUCTION: In the ocean

   environment, an observer monitors noisy sonar bearings and elevations from a radiating target.

   Song and Speyer [2], was the successful contribution for angles only passive target tracking applications in air. This MGEKF algorithm was further improved by P.J. Galkowiski and M.A. Eslam [4]. In this paper, this improved MGEKF algorithm is explored for underwater applications with some modifications. In underwater, the noise in the measurements is very high, turning rate of the platforms is low and speed of the platforms is also low when compared with the missiles in air. These characteristics of the platform are studied in detail and the algorithm is modified suitably for tracking applications in underwater.

   Section 2 deals with mathematical modelling of bearing and elevation measurements. Section 3 describes the implementation of the filter and section 4 is about the the results obtained in simulation.

2. MATHEMATICAL MODELING: Let a

   target be at a point P and the observer be at the origin, as shown in Fig-1. The measurement vector ,Z, is written as

   B

   B

   tan-1 rx

   B r

   m

   m

   The measurements are extracted from a single

   Z= y

   (1)

   stationary observer and the observer processes

   m

   tan-1 rxy

   these measurements to find out target motion parameters-Viz., range, course, bearing,

   rz

   elevation and speed of the target. Here the

   Where

   B and

   are zero mean, uncorrelated

   measurements are nonlinear; making the whole process nonlinear. However, the modified gain extended kalman filter (MGEKF) developed by

   normally distributed errors in the bearing (Bm) and elevation ( m ) measurements respectively.

   Let the state vector be

   Xs x y

   T

   z rx ry rz

   (2)

   simplified and the above equation is rewritten

   The measurement matrix H is given by as

   cosB

   sinB

   2cosB(r r ) 2sinB(r r )

   0 0 0 r

   r 0

   2 rxy = x x – y y

   (9)

   H =

   0 0 0

   xy

   cos sinB

   xy

   cos cosB

   sin

   (3)

   sin(B- B )

   sin(B- B )

   r

   r r

   Again eqn. (9 ) is rewritten as

   sin( B – B ) = cosB( rx rx ) sinB( ry ry )

   (10)

   1. HORIZONTAL PLANE AND BEARING MEASUREMENTS :-

      If the range in horizontal plane is

      rx2 ry2 , then the estimated range be

      rxy

   2. ELEVATION ANGLE MEASUREMENT:

   In the previous section , it is seen that

   2 2

   2 2

   rxy rx ry (4)

   tan1 rx

   B generates

   By adding

   rxy

   and

   rxy

   , eqn. (5) is obtained . ry

   r + r = r sinB + r cosB + r sinB + r cosB

   (5)

   sin(B – B ) = cosB(rx rx ) sinB(ry ry )

   . (11)

   xy xy x y x y

   rxy

   adding both sides r

   – r sinB – r sinB – r cosB + r cosB

   to the above

   Similarly

   tan1

   xy generates

   x x y y

   equation and after straight forward simplification, the following equations are obtained.

   rz

   sin( – ) = cos (rxy rxy ) sin (rz rz )

   r

   (12)

   rxy

   + rxy

   ( rx – rx )(sinB – sinB ) + (ry – ry )(cosB – cosB )

   =

   1 cos(B – B )

   After simple trigonometric manipulations, eqn.

   rxy

   • rxy

     (6)

     ( rx – rx )(sinB + sinB ) + (ry – ry )(cosB + cosB )

     =

     1 cos(B – B )

     1. is written as

        rxy – rxy =

        (B + B )

        (B + B )

        using (6) and (7)

        (7)

        ( rx – rx )sin

        2

        cos

        + (ry – ry )cos 2 (B – B )

        2

        (13)

        2r

        = ( r

        – r )

        sinB- sinB

        • sinB + sinB

        Substituting (13) in (12)

        xy x

        x 1 cos(B – B )

        1 cos(B – B )

        (8)

        ( r – r

        )sin (B + B ) + (r

        – r

        )cos (B + B )

        cosB – cosB

        cosB + cosB

        cos x x

        2 y y 2

        (ry – ry )

        sin( – ) =

        r

        1 cos(B – B)

        1 cos(B – B)

        cos (B – B)

        2

        Eqn.(8) can be simplified as

        sin (r

        r )

        sinB- sinB sinB + sinB

        r z z

        1 cos(B – B ) 1 cos(B – B ) =

        (1 cos(B – B )(sinB – sinB ) – (1 cos(B – B )(sinB + sinB ) 1 cos2 (B – B )

        (14)

        Eqn. (10) and eqn.(14) are rewritten in matrix

        the coefficients of ( rx – rx ) and ( ry – ry )

        are

        form as

        (B – B )

        cos B

        rxy

        B + B

        • sinB 0

        rxy

        B + B

        day, say 15000 meters, is utilized in the calculation of initial position estimate of the target is as

        X(0|0)=[10 10 10 15000sin Bm (0)sin m(0)

        sin cos

        cos cos T

        2

        2

        – sin

        15000sin m(0)cos Bm(0) 15000cos m(0) ]

        B – B

        B – B

        ( – )

        B – B

        B – B

        cos r

        cos r

        r

        where Bm (0) and m(0) are initial bearing and

        2

        2

        2

        2

        rx – rx

        elevation measurements.

        4. SIMULATION RESULTS :

        r – r

        (15)

        All raw bearings and elevation

        y y

        rz – rz

        As true bearing is ot available, it is replaced by

        measured bearing in eqn.(15) and obtained eqn.(16) as follows.

        measurements are corrupted by additive zero mean Gaussian noise with a r.m.s level of 0.3 degree. Corresponding to a tactical scenario in which the target is at the initial range of 20000 meters at initial bearing and elevation of 0.5 and 45 degrees respectively. The target is assumed to be moving at a constant course of 130 degrees. Observer is assumed to be stationary. The results have been ensemble

        cosBm

        rxy

        • sinBm rxy

   averaged over several Monte Carlo runs. The errors

   0 in estimates are plotted in Fig.2. It is observed that

   this required accuracy is obtained from 800 seconds

   B + B

   B + B

   (B – B )

   sin m cos

   cos Bm + B cos

   .

   2

   2

   onwards and so this algorithm seems to be very

   ( – )

   – sin

   much useful for underwater passive target tracking

   B

   – B

   B – B

   r

   cos m r

   cos m r

   when observer is stationary, for a moving target.

   2

   2

   2

   2

   REFERENCES:

   rx – rx

   rx – rx

   [1]. S. Farahmand , G. Giannakis , G. Leus and Z.

   r – r = g

   r – r

   (16)

   Tian "Sparsity-aware Kalman tracking of target

   y y

   rz – rz

   y y

   rz – rz

   signal strengthson a grid", Proc. 14th Int. Conf.

   Inform. Fusion, pp.1 -6 2011

   [2].T.L.Song & J.L. Speyer," A stochastic Analysis

   Considering x, y and z also, g is

   0

   0

   0

   0

   0

   0

   cosBm

   m

   m

   rxy rs. sin

   Bm B

   given by

   m

   m

   -sinBm rxy rs.sin

   Bm B

   of a modified gain extended kalman filter with applications to estimation with bearing only measurements", IEEE Trans. Automatic Control

   0 Vol. Ac -30, No.10, October 1985, PP 940-949.

   [3]. E. Masazade, M. Fardad and P. K. Varshney,

   Sparsity-Promoting Extended Kalman Filtering

   g

   0 0 0

   cos sin

   2

   cos cos

   2

   – sin

   for Target Tracking in Wireless Sensor

   Bm B

   Bm B

   r Networks, IEEE Signal Processing Letters, vol.

   2

   2

   r cos

   r cos

   2

   2

   19, no.12, December 2012.

   [4]. P.J. Galkowiski and M.A. Islam, An

3. IMPLEMENTATION OF THE ALGORTIHM :

The above mentioned improved algorithm is implemented using MGEKF (MGEKF equations are not given in this paper due to space constraint. These equations are available in [4].)for underwater passive target tracking as follows. As only bearing and elevation measurements are available, the velocity components of the target are assumed to be each 10 m/sec which is very close to the realistic speed of the vehicles in underwater. The range of the

alternate derivation of the Modified Gain Function of Song and Speyer ", IEEE Trans. Automatic Control Vol. 36, No.11, November 1991, PP 1323-

1326.

Y

Z

rxyz

P(x,y,z)

rz

90

B

rxy

rx

P1(x,y)

ry

X

Fig.2(a). Error in range estimate versus time

Fig.2(c). Error in Course estimate verses time

Fig.1. A typical target observer geometry

Fig.2(b). Error in Speed estimate versus time