8
Multisensor, multisite tracking filter F.R. Castella Indexing rerms: Multisite frackingfilfer, Airborne rargets, Marine platform Abstract: A fusion filter has been designed for the tracking of airborne targets over a spherical Earth which can accommodate different sensor types at different geographical sites. It is assumed that sensor contacts are exchanged between sites in a timely fashion and all sensor contacts from the same object update the track filter. Registration and correlation mechanisms are assumed to be in place so that only sensor contacts derived from target A update the master track associated with target A. The filter is updated by one scalar meas- urement quantity at a time: either slant range, bearing angle, elevation angle or Doppler (i.e. range rate) from a local or remote sensor. This feature considerably simplifies the filter equations and allows the easy mixture of different sensor types. Performance of this filter is demonstrated with 2D, 3D and 4D sensors at either one or two sites and manoeuvring or nonmanoeuvring target trajectories. The lilter is general enough to allow the localisation of a target in three dimensions with only contacts from a single unstabilised 2D sensor on a rolling and pitching marine platform. 1 Introduction The fusion of contact data from multiple sensors distrib- uted at multiple sites can enhance track continuity and accuracy. To achieve these improvements, the following requirements are necessary: (i) a communication link must exist between sites so that contact data can be exchanged ; (ii) registration and correlation mechanisms must be in place such that site A can update a target track in its own coordinate system with a contact meas- urement from site B on the same target without the intro- duction of any appreciable bias; (iii) a tracking or fusion filter must exist to integrate multiple reports from multi- ple sites in some optimal fashion. This paper addresses the tracking or fusion filter design. References 2 and 3 discuss the data fusion problem and the various system architecture options and their implications. The least complex approach involves sensor level tracks from various sensors being combined at a fusion centre into master track files. No associated contact data is exchanged in this approach and thus this approach entails the greatest loss of information in fusing the data. The most complex approach involves sensor contacts being exchanged between sensors and sites and ~ IF& 1994 Paper lOlOF (E5, E15), first received 15th March and in revised form 22nd November 1YY3 The author is at the Johns Hopkins Applied Physics Laboratory, Johns Hopkins Road, Laurel, Maryland 20723-6099, USA IEB Proc.-Radar, Sonor Nuuig., Vol. 141, No. 2, April 1994 the master track files being created from these sensor contacts. We have the latter approach in mind for the fusion filter design proposed here. Reference 4 considers a general approach to the development of a master track file based on track data from colocated passive and active sensors. Reference 5 presents approaches for quantifying the performance of the functions of data association (labelling measurements from different sensors, at differ- ent times, that correspond to the same object) and data fusion (combining measurements from different times and/or different sensors). The performance of these func- tions is analysed for such cases as closely spaced objects with minimal clutter and isolated objects, with less than 100% detection probability, in a heavy clutter or false alarm background. Generic extended Kalman filters (EKF) are recommended for the fusion operation in Ref- erences 3 and 5 but the specifics of the design are not given. In this paper we develop the equations for a fusion filter intended for tracking airborne targets over a spher- ical Earth. Various sensor types and various geographical sites can be accommodated with this filter design. Finally, we demonstrate the performance of this filter with 2D, 3D and 4D radar sensors at either one or two sites. A six-state extended Kalman filter (EKF) is recom- mended for this multisensor, multisite fusion application. Slant range, bearing angle, elevation angle and Doppler measurements from the local sensors or remote sensors can be used to update this tracking filter. (Some or all of these measurements may be available from local and remote sensors.) The state vector for a target is (xkyjzi) where x corresponds to the east component, y corres- ponds to the north component and z to the zenith com- ponent at the local site on a spherical Earth. iji are the corresponding target rates in these three directions. Target accelerations are assumed to be of short enough duration that the estimate of L, y and z are not war- ranted. Sustained accelerations and manoeuvres are handled by sensing these manoeuvres and enlarging the filter bandwidth, if necessary. The Kalman filter gain vector K allows an update of these coordinates with non- linear measurements via the measurement vector H, which involves the partial derivatives of the scalar mea- surement M with respect to each element of the state vector. In the case of a three-dimensional measurement, we iterate the filter equations three times, the first for the slant range measurement, the second for bearing angle and the third for elevation angle. Therefore, each mea- surement is a scalar quantity. The partial derivatives are derived for both local and remote measurements. In the latter case, the geodetic coordinates of local and remote of J.R. Moore who developed a MATLAB simula- tion program of this filter and generated all graphical and numerical results. 75

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Multisensor, multisite tracking filter

F.R. Castella

Indexing rerms: Multisite frackingfilfer, Airborne rargets, Marine platform

Abstract: A fusion filter has been designed for the tracking of airborne targets over a spherical Earth which can accommodate different sensor types at different geographical sites. It is assumed that sensor contacts are exchanged between sites in a timely fashion and all sensor contacts from the same object update the track filter. Registration and correlation mechanisms are assumed to be in place so that only sensor contacts derived from target A update the master track associated with target A. The filter is updated by one scalar meas- urement quantity at a time: either slant range, bearing angle, elevation angle or Doppler (i.e. range rate) from a local or remote sensor. This feature considerably simplifies the filter equations and allows the easy mixture of different sensor types. Performance of this filter is demonstrated with 2D, 3D and 4D sensors at either one or two sites and manoeuvring or nonmanoeuvring target trajectories. The lilter is general enough to allow the localisation of a target in three dimensions with only contacts from a single unstabilised 2D sensor on a rolling and pitching marine platform.

1 Introduct ion

The fusion of contact data from multiple sensors distrib- uted at multiple sites can enhance track continuity and accuracy. To achieve these improvements, the following requirements are necessary: (i) a communication link must exist between sites so that contact data can be exchanged ; (ii) registration and correlation mechanisms must be in place such that site A can update a target track in its own coordinate system with a contact meas- urement from site B on the same target without the intro- duction of any appreciable bias; (iii) a tracking or fusion filter must exist to integrate multiple reports from multi- ple sites in some optimal fashion. This paper addresses the tracking or fusion filter design.

References 2 and 3 discuss the data fusion problem and the various system architecture options and their implications. The least complex approach involves sensor level tracks from various sensors being combined at a fusion centre into master track files. No associated contact data is exchanged in this approach and thus this approach entails the greatest loss of information in fusing the data. The most complex approach involves sensor contacts being exchanged between sensors and sites and

~

IF& 1994 Paper lOlOF (E5, E15), first received 15th March and in revised form 22nd November 1YY3 The author is at the Johns Hopkins Applied Physics Laboratory, Johns Hopkins Road, Laurel, Maryland 20723-6099, USA

IEB Proc.-Radar, Sonor Nuuig., Vol. 141, No. 2, April 1994

the master track files being created from these sensor contacts. We have the latter approach in mind for the fusion filter design proposed here. Reference 4 considers a general approach to the development of a master track file based on track data from colocated passive and active sensors. Reference 5 presents approaches for quantifying the performance of the functions of data association (labelling measurements from different sensors, at differ- ent times, that correspond to the same object) and data fusion (combining measurements from different times and/or different sensors). The performance of these func- tions is analysed for such cases as closely spaced objects with minimal clutter and isolated objects, with less than 100% detection probability, in a heavy clutter or false alarm background. Generic extended Kalman filters (EKF) are recommended for the fusion operation in Ref- erences 3 and 5 but the specifics of the design are not given. In this paper we develop the equations for a fusion filter intended for tracking airborne targets over a spher- ical Earth. Various sensor types and various geographical sites can be accommodated with this filter design. Finally, we demonstrate the performance of this filter with 2D, 3D and 4D radar sensors at either one or two sites.

A six-state extended Kalman filter (EKF) is recom- mended for this multisensor, multisite fusion application. Slant range, bearing angle, elevation angle and Doppler measurements from the local sensors or remote sensors can be used to update this tracking filter. (Some or all of these measurements may be available from local and remote sensors.) The state vector for a target is ( x k y j z i ) where x corresponds to the east component, y corres- ponds to the north component and z to the zenith com- ponent at the local site on a spherical Earth. iji are the corresponding target rates in these three directions. Target accelerations are assumed to be of short enough duration that the estimate of L, y and z are not war- ranted. Sustained accelerations and manoeuvres are handled by sensing these manoeuvres and enlarging the filter bandwidth, if necessary. The Kalman filter gain vector K allows an update of these coordinates with non- linear measurements via the measurement vector H , which involves the partial derivatives of the scalar mea- surement M with respect to each element of the state vector. In the case of a three-dimensional measurement, we iterate the filter equations three times, the first for the slant range measurement, the second for bearing angle and the third for elevation angle. Therefore, each mea- surement is a scalar quantity. The partial derivatives are derived for both local and remote measurements. In the latter case, the geodetic coordinates of local and remote

of J.R. Moore who developed a MATLAB simula- tion program of this filter and generated all graphical and numerical results.

75

sites are required. This is necessary in order to express a remotely measured quantity in terms of coordinates in the local system..The filter is general enough to allow measurements from an unstabilised platform, such as a rolling and pitching ship and an unstabilised 2D sensor measuring only slant range and bearing. For this case, roll and pitch measurements at the time of the contact are required in addition to ship heading. With appre- ciable roll and pitch the filter converges to the correct target location in three coordinates even with a single 2D sensor. With two noncolocated 2D sensors the con- vergence to the correct target location occurs with smaller errors during the convergence process. The con- vergence is most rapid when a 3D measurement exists from either the local or remote platform.

2 Analysis

The extended Kalman filter (EKF) equations required for this application can be derived [1] and are as follows:

x k l k - 1 = @ ( ( T ) x k - I l k - I (1)

p k l k - l = @ ( T ) p k - l l k - i @ ( T ) T -t Qk (2) K k = P k I k - l H ? ( H k P k I k - l H ? + (3)

xkIk = x k l k - l + K k [ M k - M d X k I k - i ) l (4)

‘klk = ( ‘ - K k H k ) p k l k - l (5) where X, the state vector, is composed of north, east and zenith components at the local site and is given by

x = C X i ~ i Y i Y i z l ~ l l T (6) The state transition matrix is

1 r 0 0 0 0 r o 1 0 0 0 0 1

(7)

where T is the time increment between updates and will generally vary between updates. M p ( X k I k - ,) implies a predicted value of the measurement evaluated with the predicted state vector. The subscript k I k - 1 indicates a prediction up to the time of measurements k based upon k - 1 measurements and the superscript T indicates transpose vector or matrix, whichever is appropriate. P is the 6 x 6 covariance matrix of the errors and the meas- urement matrix H , is the following row vector of partial derivatives evaluated at X k l k -

(8)

Note that H k varies as the scalar measurement type varies (i.e. local slant range, remote Doppler, etc.).

The plant noise Qk is given [6] by

02 =[: 3 Each scalar qi is a plant noise spectral density with units of lengthz/time3 = acceleration’/hertz. The selection of qi parametric values will be addressed later in this paper. Since the EKF will be updated by one scalar measured quantity at a time, the quantity Rk is simply the measure- ment variance of that scalar quantity with units of length’ for slant range measurements, radian’ for angular measurements of bearing or elevation, or (length/time)’ for Doppler measurements. Also, due to this update pro- cedure, the matrix inversion indicated in eqn. 3 becomes simply a scalar division operation, a significant computa- tional simplification. We now derive expressions for the various measurements, from either local or remote plat- forms, in terms of components of the local state vector x 1 . . . i, and the elements of the measurement vector H k of eqn. 8. The former quantities are needed to evaluate M p ( X k , k - l ) in the innovations residual of eqn. 4, while the latter quantity is utilised in eqns. 3 and 5.

2.1 Derivation of expressions for predicted measurements for stabilised platforms

The geometry is illustrated in Figs. 1 and 2.

, I

remote \ /

system\ / v Earth’s

centre

Fig. 1 Earth centre, local and remote coordinate systems

Fig. 2 Spherical Earth geometry with platform unit vectors

IEE Proc.-Radar, Sonar Navig., Vol. 141, No . 2, April 1994 76

Vectors R I and R , are vectors from the centre of the Earth to local platform 1 and remote platform 2. TI and T, are vectors from platforms 1 and 2 to the target of interest. Local and remote measurements, indicated by subscripts 1 and 2, can be expressed as follows in terms of platform 1 coordinates and rates.

Slant range:

s1 = I T, I = (x: + y: + Zf)”, SZ = I TZI = IR, - R , + Ti I = IAR + Ti I

where

AR = R , - R ,

Bearing as measured clockwise from north:

T , , P l xI tan B, = - - - - TI ‘91 Y l

Elevation as measured above the x, y plane:

T, . i, [(T, . P J 2 + ( T 2 . 92)2]”’

tan E , =

Doppler :

d - dt

s --IT,\

d dt

= - (x: + yf + zy/,

For convenience, we have defined Doppler as the rate of change of slant range. In reality, the Doppler frequency in hertz in equal to 2S/I where 1 is the wavelength of the radiation. x i , y,, zi are the x, y, z coordinates of the target as measured at site i where i = 1 is the local site and i = 2 is the remote site. Similarly ii, ji, ii, are the unit vectors at site i where ii points in the east direction, ji in the north direction and ii to the zenith. The expressions for local sensor measurements given by eqns. 12, 15, 17 and 19 are obtained in a straightforward manner. The expressions for remote sensor measurements require some more work.

In calculation of s2 , the target slant range from site 2 we first express the difference vector R,-R, in an Earth centered coordinate system and then transform to a site 1 I E E hoc.-Radar, Sonar Navig., Vol. 141, No. 2, April 1994

coordinate system. This is accomplished as follows:

AR = RI - R ,

= i [ ( R n + h,) COS pi COS L ,

- (Ro + h2) COS ~2 COS L,]

+ j [ ( R o + h,) cos pl sin L,

- ( R , + h,) cos p, sin L,]

+ i [ ( R o + h , ) sin p1 - (R, + h,) sin p21

2 ARx + j AR, + i ARz (21)

where P, j , i are Earth centred coordinate system unit vectors as shown in Fig. 2 and

R, = Earth radius hi = altitude of platform i above spherical Earth Li = longitude of platform i (+ indicates east of

pi = latitude of platform i (+ indicates north of Greenwich)

equator)

The transformation to a site 1 coordinate system is accomplished via the matrix

cos L, -sin pl cos L , -sin p1 sin L, cos p, cos p, cos L , cos p, sin L , sin p,

Thus S, expressed in terms of site 1 coordinates is

Sz = [(XI + AR,,)’ + oil + AR,JZ + (21 + AR,,)21”2

(23) since

T, = T, + AR

= (xi + A R d k i + ( ~ i + AR,i)i i

(24) + (ZI + A U f l

when expressed in terms of site 1 unit vectors.

measurement can be derived to be Using eqn. 23 the expression for a remote Doppler

s , = (xi + AR&1 + (YI + ARyJ31 + (ZI + ARzAi1 S ,

(25)

To evaluate the expressions for tan E, and tan E, in eqns. 16 and 18 the indicated vector dot products have to be performed. This can be accomplished by first express- ing site 2 unit vectors in terms of site 1 unit vectors as follows:

where the 3 x 3 transformation matrix N, which applies

I1

for a spherical Earth, has the following elements:

N,, = COS AL

N, , = sin pl . sin AL

N,, = -cos p1 . sin AL

N, , = -sin p, . sin AL

N,, = sin pl sin p, cos AL + cos pi cos p,

N,, = sin pi cos p2 - sin p, cos p, cos AL

N3, = cos p, sin AL

N3, = sin p, cos pl - cos p, sin p, cos AL

N,, = sin p z sin p, + cos p, cos p, cos AL (27) and

AL = L, - Ll

pi and L, are the latitude and longitude of the ith plat- form, respectively. N was obtained in a two-step process by transforming from site 2 to the centre of the Earth and then out to site 1. When eqns. 26 and 27 are applied to eqns. 16 and 18 we get

tan E , = ' 2 cx: + y;] (29)

where

' 2 = + ARxl)N31 f (Y1 + ARy1)N32

+ ('1 + ARzl)N33

x2 = (XI + ARXl)NlI + ( Y , + ARJNI2

+ ('1 + ARz1)N13

Y , = (xi + ARxiINzi + I v i + ARyi)N22

+ (zl + ARz1)N23 (30) The expressions for local measurements S,, E , , E, and S, are tabulated in Table 1 along with the appropriate mea- surement vector H evaluated via eqn. 8 for each case. The corresponding quantities for remote S,, E , , E , and S, are tabulated in Table 2.

2.2 Derivation of expressions for predicted measurements for unstabilised platforms

In some cases, naval platforms possess a 2D sensor which measures slant range and bearing only with the latter quantity measured with respect to the ship centre line. This bearing measurement is called deck bearing and can be shown to be given by the following expressions when both local and remote platforms possess such sensors:

+ + ARz1)v23

E', is the local platform deck bearing and E; is the remote platform deck bearing.

The matrices U and V are defined by

(33) U = R(l'P(1)"')

78

(34) V = R(Z)p(2"(Z)N

where N is the same matrix of eqns. 26 and 27. H") is not related to the measurement matrix of eqn. 8. Matrices H('), PcO and R"' are

(35)

(36)

(37)

where

H"' = heading with respect to north of ith platform (i = 1, 2 for local, remote)

P(') = pitch angle of ith platform R'" = roll angle of ith platform

It has been assumed that pitch occurs first and its posi- tive sense is bow down, while roll occurs second and its

Table 1 : Formulae for local stabilised coordinates

a Slant range

d Doppler

positive sense is port down. The corresponding meas- urement vector H for these deck bearing measurements can be derived from the bearing quantities of Table 2 with the following substitutions:

Replace N with V for remote deck bearing measure- ments. Replace N with U and set ARxl = AR,, = ARzl = 0 for local deck bearing measurements.

1EE Proc.-Radar, Sonar Navig., Vol. 141, No. 2, April 1994

Slant range quantities of Tables 1 and 2 remain the same for this type sensor. It will be shown via an example under numerical results that sensors of this type can localize a target in 3D if sufficient roll and pitch of the platform are present.

2.3 Selection of plant noise parameters The plant noise spectral densities of eqns. 9 and 10 require the specification of parameters q l , q2 and q3 cor- responding to x, y and z spectral densities. For our appli- cation, 4 , = y2 = q3 = 4 where 4 has dual values dependent upon the output of single site manoeuvre detectors. The lower value for 4 is qL and is specified to achieve a desired steady-state rate accuracy when only a single sensor is operating. The upper value for 4 is 4” and is specified to bound the bias due to a Gg’s manoeuvre when a single sensor is operating. When at least one of the single site manoeuvre detectors declares a manoeuvre the q value for the multisensor track is increased from the lower value qL to the upper value 4”. The single site manoeuvre detector is composed of several optimised a, p, y filters which operate on single site contacts for a track in several coordinates. Parameters have been selec- ted such that good acceleration estimates are obtained with the minimum of contacts. When an acceleration exceeding the threshold is no longer sensed by the single site manoeuvre detectors, the 4 value is reduced to the lower limit value q L .

3 Simulation results

Four scenarios are presented which illustrate the per- formance of the multisensor multisite filter. These scen- arios are as follows:

Scenario I : single platform with an unstabilised 2D radar sensor;

Scenario 2: two platforms, each with an unstabilised 2D radar sensor;

Scenario 3: single platform with an electronically sta- bilised 3D radar sensor ;

Scenario 4: two platforms, one with an electronically stabilised 3D radar sensor the the other with an elec- tronically stabilised 4D radar sensor.

For scenario 1, the parameters of platform 1 of Table 3 apply while for scenario 2 the parameters of platforms 1 and 2 apply. For scenario 3 the parameters of platform 3 apply while, for scenario 4, the parameters of platforms 3 and 4 apply. In the upper part of Table 3, from left to right, we list the platform number, latitude and longitude location in degrees of the platform, range, bearing, ele- vation (if applicable) and Doppler (if applicable) one sigma accuracies in the stated units. The last column of the upper part of Table 3 tabulates the update period or revisit time of the radar in seconds. In the lower part of Table 3 are tabulated the platform number, platform heading, plant noise values qL and qu , maximum roll and pitch angles and roll and pitch periods. Initialisation

phases for the assumed sinusoidal roll and pitch angles are also specified but they are not tabulated. The present- ed results are typical of one Monte Carlo trial.

3.1 Results for scenario 1 The actual target trajectory is shown in Fig. 3 in the ground plane coordinates of platform 1 (i.e. x1 vs. yl in nmi). The target has a bearing of 45" and a heading of 225", so it is radially inbound. Trajectories are flown in the local xl, y l , z1 coordinate system. Target speed is 850 fps at a constant z1 value of 5 nmi. Fig. 4 plots the

1OOr

901

70-

L 40

30

20

l o t 01

20 40 60 80 100 x -posltlon. nmi

Fig. 3 Case I single 2D sensor, actual target trajectory

70 -

30

20

'0 20 40 60 80 103 x-position.nmi

Fig. 4 Case I single 2D sensor,filtmed target trajectory

filtered values of xI and y,, corresponding to the first and third elements of the state vector X of eqn. 6. Thus, noise has been added to the sensor measurements and filtering according to eqns, 1-5 has taken place. Close exam- ination of Figs. 3 and 4 reveal close agreement after an initialisation phase. The z, position estimate, correspond- ing to element 5 of the state vector, is shown in Fig. 5.

15r

Fig. 5 Case I : single 2D sen~or,filtered z-position

We see that the filter converges to the actual target z1 value of 5 nmi after a lengthy period of time and with a large negative estimate during the transient phase. However, we must keep in mind that this is a severe case of a single 2D sensor, ordinarily without any indication of the z1 component.

A few additional comments are now related about the tracking for 2 0 sensors. Single point initialisation is used with an assumed target altitude, that is height above the Earth of 15 kft. Position and velocity covariances in the P covariance matrix (off diagonal terms) are initially set equal to zero while rate variances are initially set at (500 kts') in all three dimensions. At each filter update time, after a range and bearing update, a pseudoelevation update is also performed. This pseudoelevation update assumes the target is still at an altitude of 15 kft above the Earth, but with a sigma value of the pseudoelevation measurement equal to 0.3 radians or 17 degrees. Since this corresponds approximately to the elevation coverage of the 2D sensor, this feature guarantees that the z1 estimate does not converge to a large negative z1 value.

3.2 Results for scenario 2 The target trajectory of Fig. 3 was also used for this scen- ario which employs two platforms separated by 50 nmi at

Table 3: Platform and sensor pnrameters for cases 1 4

Platform No. Latitude (") Longitude (") uR (ft) us (") uz (") uR (fps) T (s)

1 0 0 100 0.17 NA* NA 5 2 0 100 0.17 NA NA 5 0.83 3 0 0 500 0.5 0.5 NA 2 4 0 0.83 500 0.5 0.5 0.003 2

Platform No. H ("1 qL (ttz/s3) q,, (h2/s3) R,., ("1 P,,, (") zrO,, (s) r.,tch ( 5 )

1 0 1000 4000 10 3 7.8 10.0 2 0 1000 4000 10 3 9.2 15.9 3 0 1000 5000 NA NA NA NA 4 0 1000 5000 NA NA NA NA

80

* Not applicable.

I E E Proc.-Radar, Sonar Navig., Vol. 141, No. 2, April I994

the equator. Fig. 6 plots the filtered target trajectory for this scenario, while Fig. 7 plots the filtered z1 position. Upon comparing Figs. 4 and 6 we see less noise for the latter results which should be the case since twice as

100

901

801

70 -

50

30

20

x- posti ion, nmi

Fig. 6 Case 2 two 2D sensors,Jiltered target tralectory

--_--I

100

- 3 0 ~ ' " " . . " 1 150 200 250 300 350 400 450 500 50 100 1irne.s

Fig. 7 Case 2: two 2D sensors,filtered z-position

much data is being used (i.e. two sensors instead of one). Comparing Figs. 5 and 7 indicates that the two platform scenario also converges to the correct z , value in a some- what shorter period of time and with lesser errors during the transient phase.

It was attempted to limit z , values to numbers greater than zero but, when that was done, convergence took approximately twice as long. Thus, it was better to operate the filter in a 'hands off mode for this 2D appli- cation.

3.3 Results for scenario 3 The trajectory of interest for scenarios 3 and 4 is shown by the solid line in Fig. 8. z1 has a constant value of 5 nmi with an initial bearing and heading of 0" and 18W, respectively. Initial range is about 100 nmi. Target speed is the same as previously, or 850 fps. A 3 g manoeuvre occurs between 30 and 58 seconds so that, after this time period, the target heading is 0" or outbound. The track lilter estimates for x, and y, are indicated by the aster- isks. Lags are noticeable at the beginning and end of the manoeuvre. Sensor accuracies from Table 3 are 500 ft in

IEE Proc.-Radar, Sonar Navig., Val. 141, No. 2, April I994

range, 0.5" in bearing and 0.5" in elevation. The single sensor manoeuvre estimator output obtained using slant range measurements is shown in Fig. 9. This is used to vary the filter plant noise parameter q. Note the accurate indication of target acceleration whlch is obtained. The manner in which the filter gains are adjusted upon sensing of a manoeuvre is illustrated in Fig. 10. In this

104

I .* I t

-2 -6 -4 0 2 4 x-position. nmi

941

Fig. 8 Case 3' single 30 sensor, actual and$ltered trajectory

I

-3L 20 i o 60 eo I d 0 llo time.s

Fig. 9 3 g manoeuvre from 30-58 s

Case 3: single 3D sensor, estimated target acceleration

O b 20 40 60 80 l b o 120 time,s

Fig. 10 updating with range - position gain (unitless)

Case 3: single 3D sensor, y - position and rate gains when

~~- rate gain (in uniis of lis)

81

plot are shown, as a function of time, the y coordinate position and rate gains of the filter used with a slant range measurement update. It can be seen that the gains have achieved steady-state, and then they get ‘bumped’ upward twice when the acceleration estimate exceeds the threshold of one gee or 32.2 ft/s2. Close examination of Figs. 9 and 10 reveal a slight delay between the time at which the manoeuvre threshold is exceeded, at which time the 4 value is increased from qL to 4u, and the filter gains respond. This is to be expected, owing to the finite bandwidth of the tracking filter.

Filter errors were examined and found to be in the expected normalised range of f 3. This examination was accomplished using the following measures of normalised error :

(38) (39)

(40) xf, y,, z , are the filtered x , y, z values, actual refers to trajectory values without noise and F‘L refers to the ii element of the filtered covariance matrix P. If the filter is performing correctly, these normalised errors should be in the range ofk3. The actual values of these errors, with the normalisation factor removed, are tabulated in Table 4 in the form of an average error p, standard deviation of this error c and RMS error equal to ( p 2 + cZ)li2. For scenario 3 the RMS error in x is 0.4785 nmi, in y it is 0.0827 nmi and in z it is 0.4007 nmi.

( x , - x ~ ~ ~ ~ ~ ~ ) / ( P ( , ) ( ~ ~ ~ ) = normalised x error (yf - yacrual)/(P{3)(1i2) = normalised y error

(zf - z . , , ~ ~ ~ ) / ( P { ~ ) ( ~ ~ ~ ) = normalised z error

Table 4: Filter errors in nmi for scenarios 3 and 4

Scenario 3 0.01 59 0.4782 0.4785 Scenario 4 -0.0584 0.1434 0.1548

4 U” e;+u;)”’ Scenario 3 -0.031 1 0.0766 0.0827 Scenario 4 -0.0233 0.0779 0.081 3

U. U. (U! + ~

Scenario 3 -0.0303 0.3996 0.4007 Scenario 4 0.0039 0.41 80 0.4180

3.4 Results for scenario 4 This scenario varies from the previous one by the addi- tion of platform 4 whose characteristics are listed in Table 3. This is a 4D sensor which measures range, bearing and elevation with the same accuracies of plat- form 3, but it also has a range rate measurement accu- racy of 0.003 fps or 10.8 knots. This platform has been displaced by 50 nmi in the easterly direction along the equator. The solid line plot of Fig. 11 shows the idealised trajectory in the local platform coordinate system .(i.e. platform 3 coordinate system). This is the same trajectory as used in scenario 3 as can be seen by comparing the solid line plots of Figs. 8 and 11. Also indicated in Fig. 11 are the filtered trajectory estimates indicted by asterisks. Notice twice as many data points and smaller lags when comparing Fig. 11 with Fig. 8.

Single sensor manoeuvre estimator outputs, now available from each platform, are not significantly differ- ent from that of Fig. 9. Filter errors were again examined and found to be in the expected normalised range of 3. However, when we examine the actual unnormalised errors, we get the results listed in Table 4. We see that the addition of the 4D second sensor has reduced the RMS x

a2

error by a factor of three while the y and z RMS errors remain essentially unchanged.

4 Conclusions

A tracking filter has been designed for the tracking of airborne targets over a spherical Earth which can accom-

-2 -6 -L 0 2 b X-position. nmi

Case 4 .(D + 4 0 sfmars, actual undfiltered trajectory

9 d

Fig. 11

modate different type sensors at different sites. New sensors can be added at any time with the only require- ment that the appropriate measurement matrix of Table 1 or Table 2 be utilised in the filter equations and the sensor measurement variance to be specified. It is also possible to expand the sensor types of Tables 1 and 2 to more varied sensors such as phased array radars measur- ing direction cosines, bistatic radars and IFF with mode C , where target altitude is available via a sensor on the airborne target. Angle-only measurement sensors can be handled by using the relevant portions of Table 1 and 2. This is possible since a single scalar sensor measurement at a time updates the filter. In the case of an angle-only sensor, the filter can be updated with the bearing meas- urement first and second with the elevation measurement or vice versa. The performance of this filter has been demonstrated with 2D, 3D and 4D sensors at either one or two sites and manoeuvring and non-manoeuvring target trajectories.

5 References

1 GELB, A.: ‘Applied optimal estimation’ (MIT Press, 1974) 2 BLACKMAN, S.S.: ‘Multiple-target tracking with radar applica-

tions’(Artech House, Inc., 1986) 3 ALSPACH, D.L.: ‘An approach to the multisensor integration

problem’. IEEE proceedings of EASCON, 19x3, pp. 149-157 4 CHAUDHURI, S.P.: ‘A general approach to the development of

passive/active sensor data fusion’. IEEE proceedings of 1985 Amer- ican control conference, Vol. 2, pp. 823-828

5 BROIDA. T.J.: ‘Performance prediction for multisensor tracking systems: kinematic accuracy and data association performance’, Proc. SPIE-Int. Soc. Opt. Eng., 1989,1198, pp. 256-271

6 CASTELLA, F.R.: ‘An adaptive two-dimensional Kalman tracking filter’, IEEE Trans., 1980, AES-16, (6), pp. 822-829

IEE Proc.-Radar, Sonar Navig., Vol. 141, No. 2, April 1994