A New Adaptive Model for Radar and Optronic Trackers

Copyright © IFAC Identification and System Parameter Estimation 1982 . Washington D.C . . USA 1982

A NEW ADAPTIVE MODEL FOR RADAR AND OPTRONIC TRACKERS C. Bozzo*, C. Blondel**, B. Dellery***, A. Guilbert* and E. Siffredit ·DTCN, Centre d'analyse de Systemes, CAPCA, DCAN Toulon, France ··Societe DIGITONEIDIGILOG, Aix-en-Provence, France ···Societe STUDEC, Toulon, France t SOCI-ete CSEE Toulon, France

Abstract. Model designing methods are an essential step towards the synthesis of filters associated with the problems of target motion analysis_ The model selected by most authors (SINGER model) only gives a very incomplete account of both the path characteristics and the laws of motion. A more systematic approach has been introduced and developed since 1977 by C. Bozzo and A. Guilbert. Such an approach consists in taking the fundamental choices relating both to the system of co-ordinates (Cartesian, polar, cylindrical/ co-ordinates) and to the reference system (connected with the platform, the antenna or the trajectory) on the basis of the knowledge of the status vector structure required by the variables to be calculed (position, speed, acceleration, etc ... ), by the closing variables (first and second order stress equations, et·c ... ) and by the control variables (filter Q-controllability). This approach is fully justified by the satisfactory results obtained on real data (Radar, Infra-Red, TV Trackers) with models which had been intuitively del~rmined in the past

(i,

log D, tan A models, etc •.. ) and which

adapt their gains and noise covariances from the knowledge of the pseudoinnovation process. SCOPE

vis-a-vis the conventional estimation and filtering procedures (KALMAN's filter, extended KALMAN's filter, etc ... ). Let us mention : - The continuous discrete aspect of the envisaged proc~ss~s~--------- The ~2~:li~~~!_~~E~£! of the dynamic and measurement equations. - The selection of the envisaged parametric space, hence the !YE~_2~_!~E!~~~~!~!i2~ (relative or absolute), or the nature of the datum, and of the system of coordI~;t~s ~dopt~d-to represent the dynamics of the moving object tracked, of the constitution of the state vector etc ... - The noise own characteristics (gaussian or non gaussian, non stationariness, non additivity, relationship, etc .•. ). He will call "measuring variables" the variables that permit correcting the a-priori knowledge of the system status to which is associated .the concept of ~:~bs~rv~~i~i!Y of the model. - The selection of the statistic parameters associated with the variable (dynamic noise) permitting to characterize the maneuvers of the tracked moving object. - Such maneuvers being "a priori" unknown of the watcher and having to be :llOdelled within a probabilistic approach, we will call "monitoring variables" the random variables which thus affect the g:£~~!~~~~~~i!i!Y of the model.

The problem is about monitoring a process supporting a number of sensors for either 3-D tracking or 2-D tracking (the plane problem). We will hereafter only discuss the general problem of 3-D tracking since 2-D tracking and angular tracking without knowledge of the distance (passive direction finding) are particular cases. The problem considered therefore cons is ts in es tab 1 ishing ,,,i th accuracy the position of a (~oving) object assumed to be non co-operative (a maneuvering object) as well as the parameters that control its trajectory and the law of motion over that trajectory. It is assumed that deviation elements (side and site) are available, as defined by two angles (between the watcherto-moving object axis and the instantaneous position of the axis of the angular tracking device). The measurements are clearly dependent upon the type of sensors (RADAR, OPTRONIC ... ). PROBL~MS

OF

CF~CTERISATION

0=

AND

MODELLING Characteri7.ation of the processes envisaged. Such characterization refers to a number of basic concepts which should be analysed 785

C. Bozzo et al,.

786

Continuous-discrete aspect of the processes. The state equation of the system is, by essence, continuous. However, the observation equation is discrete, since the signal processing devices and the implementation of the algorithms for estimation and control require digital processing facilities. The overall process considered is therefore of the "continuous discrete" type. Non linear aspect of dynamic and measurement equations. lfuatever the system of coordinates retained and the state vector adopted, the dynamic and measurement equations are always non-linear. In a few very specific cases, and for given trajectories, it is possible to make either equation linear. But since the trajectories are not determined a priori, such very specific cases must be considered exceptional. The two equations defining the process in the continuous field have, therefore, the following form, which is not the most general form insofar as it is implicity assumed that the noises are additive (while there is a priori no reason for them to be so) :

~ dX t dZ

f(xt,t) dt + G(xt,t) df\ ( I)

h(xt,t) dt + dn t

t

EM[d S m[dn

t

T dS ] t

Q( t) o(t)

T dn ] = R(t) o(t) t t

(2)

wand v being noises with covariance matrix Q and R, x the State and Z the measurement. Choice of the reference and coordinate systems. Analysis of the basic choices. We can a priori envisage any combination of th e three major types of choices which follow Choice A (State) : choice of a reference point attached to the observer (a ship, for instance), to the sensor (its antenna, for instance), or to the trajectory. Choice B (State) : choice of cartesian, cylindrical or polar coordinate systems. Choice C (measurement) : choice of an abs o lute or relative reference point. It should be noted, however, that: - measurements are always polar : azimuth and elevation angles, ranges, Doppler, gyrometric velocities, deviations, etc ... - measurements are, as a rule, effected as relative measurements, and when they are given as absolute measurements, the relative-toabsolute conversion is done at the sensor level. We will hereafter only consider the more conventional combinations (A,B,C). We will discuss, in particular: - The case of the Cartesian coordinates (which is discussed in more detail, for the so called SINGER model in references [13] to [15] and the case of polar coordinates, as the cylindrical coordinates are of limited interest.

- The case of the observer-bound reference point and that of the trajectory-bound reference point. The problem raised when changing ever from relative coordinates of measurements to absolute coordinates of measurements can be dealt with according to three procedures. - Adding a complementary measurement noise representing the disturbances associated with platform displacement (see ref. [18]). - Linearizing the measurement equation z ~ h (x) taking into account the non-linearities associated with the absolute-to-relative conversions. This is a complex procedure which is delicate to apply. - Linearizing the converse (if any) of measurement equation x

=h

-I

(z).

Choice of the composition of the state vector. The composition of the state vector results from the following choices : a) Choice of the PEi~~E~ variables (Cartesian or polar coordinates ; variables or variable functions to be estimated). S) Choice of the ~~£~~~~El variables : (expression of equations at constraint or closing conditions) and choice of the monitoring variables. Y) Choice of the £~~Pl~~~~!~El states for adapting the filter (see para. adaptive filtering) : innovation, gain, etc •.. Equations at constraint and closing conditions. Once the primary state vector is selected and the primary variables are fixed, it is necessary to write the relationships that link the derivatives of those variables in order to set up the dynamic equation. The relationships between the primary variables, their derivatives and, the components of the state vector cannot be established without taking into account "the equations at constraint" or "closing conditions" of the first or second order which reflect the linear and angular velocities and accelerations and result from the basic equations of the spot kinematics. Taking for example, the 3-D problem illustrated in figure I, it can easily be shown that, in the antenna trihedron for example, the following relationships exist

rV

- D ~L - 2D ~L + D S1 S ~ 2 D - D(S1 L + ~2) S

rS

D Q + 2D Q + D Q Q L L S S

r

L

(3;

Equations (3) appears as conditions at constraint for the various components of the state vector, and provide the closing condition of the second order, i.e the possibility to write a matricial relationship connecting all the second order derivatives of a Dolar primary state {A.S.D.} or {a (A), S (S), o (D)} and to work out a ~2~~B~~~~~~_££~pl~!~ dynamic equation.

787

A New Adaptive Model for Radar and Optronic Trackers

It can easily be seen that a model utilizin~ the closing conditions is of the form:

f(xt,t) dt + L dB t h(x ,t) dt + dn t

(~'i th

(4)

t

the notations of formulae (1».

L being a stationary matrix and h(xt,t) being linear when the state vector selected has primary components that include the polar variables actually measured: azimuth and range, or azimuth and converse of the range, in the 2-D problem, for instance. Monitoring vCl.riables The variables selected as monitoring variables are assumed to be ~arkovian and represented by a Markovian process whose order (degree of the associated dynamic equation) and parameters make it possible to characterize the Qcontrollability of the process considered, hence the maneuvering possibilities of the tracked object around the identified oean !~~~!~E~' Considerine, f~r-Instance-(;~e SINGER model) the first order case (but this is not compulsory), the monitoring variable 6 will be characterized by an exponential correlation equation in the form: 2

II' SS (')= OQ e

-0,

The converse of parameter a characterizes the average duration of a maneuver over the monitoring variable considered. The variance of the white noise U affecting the B formatting filter characterizes the amplitude of the maneuvers. Thus, the choice of monitoring variables is essential. It is important that the Markovian process associated with the physical variable be representative of the actual maneuvering "capabilities" of the tracked object. Usually the choice con~i~t~ in taking the Cartesian accelerations X,Y,Z (SINGER model [14]) or the polar accelerations r , rS (PERRIOT-MATHONA model [1], BOZZO, L BLOND EL and SIFFREDI model [5]) as monitoring variables, other choices are possible, howeve~ with equally good results : - velocity of moving object V , altitude aB R and heading ~ (GUILBERT model [18] and [19]). - tangential acceleration y , curvature f and algebric twist J (PASSERON model [4]).

Adaptive filtering. Parameters related to measurement noise. Measurement noises, or disturbances in this measurement, can b.e divided into three major categories :

- noises with a nil average and a known or unknown distribution to be identified (covariance matrix etc.). Such noises are usually considered white and can be introduced into the model. - random biases occuring at random times (e.g. distributed according to a POISSON's law) and with a random amplitude. - biases with a permanent deterministic character whose characteristic can be known (measured) : dissymetry at the antennas, control curves, etc ••. Parameters ·r elating to the dynamic noise. The dynamic noise reflects the controllability of the tracked moving object. In large maneuvers, it is therefore necessary to identify the new characteristics of that dynamic noise in order to prevent divergence of the associated filter (the model being no longer representative of the actual maneuvering conditions of the object). We can try to identify the co-variance matrix of the dynamic noise and/or its mean :value provided it is non nil. This is the assumption made by MOOSE, Mc CABE and GHOLSON, [15] and [16] an assumption which has been studied in detail by SIFFREDI [17]. STUDY OF 3-D TRACKING - EQUATIONS AT CONSTRAINT AND CLOSING CONDITIONS Observer - bound trihedron In the cartesian trihedron

(8x , 8y ' 8z ),

we

can ~rrite the classic relationships that link the Cartesian cooedinates (position, velocity, acceleration) according to the polar coordinates defining the measurements. It can be seen that all the equations are strongly paired. In order to split them apart it is necessary to : - select the antenna trihedron - introduce assumptions as to the trajectory of the moving abject tracked - process the three coordinates individually, purposedly ignoring the pairings. This is SINGER's classic approach, which leads to a linear filter. SINGER's approach [14] • For each coordinate (X for example) a status vector is written which includes the successive derivatives. The n-derivative is the monitoring variable. In the case n = 2, for instance, acceleration is considered as the output of a formatting filter whose input is a white noise. This type of model was introduced as far back as 1966 by DEFFONTAINES [13]. Its properties were studied by LINARD and De LARMINAT [20], among others, who demonstrated that : - only the triple integrator filter gives rise to filters with a nil final error (velocity or acceleration step) - it is possible to introduce, with those models, a concept of majorating model in the spectrum sense. This majorating model permits

C. Bozzo et al.

7SS

evaluation of the estimation errors resulting from insufficient knowledge of the statistical parameters (see also [11]). Considering. for instance. the model of SINGER. it can be observed that the characterization selected amounts to writing :

= 0 Q2

e

-a(T)

(S)

XX

X 4

2 = rl + rl 2 + rv _ D S D S D X2 + X2 + X9 X - X2 4 S 6 3

(~)

.,'"

-2rlD rlL + rlS Wv + rlS EL Ws fL + rlS ES WL - D

0

- 2 X4 Xs + X6 ~ + X6 XIO Ws

Xs c::QJ c::

~

The SINGER model was utilized with success in solving the RADAR tracking problem. It has a serious drawback. homever. in that it brings about estimates that "streak" over a maneuver. For more detailed studies on the subject. see BOZZO and LEGRAND [111.MICHEL [12].FITZGERALD [ S] and [10]. FRIEDLAND [ 9] etc ...

. 2

D

S

+X

0

()

6

X

ll

w - X3 X L 7

()()

....c::

X6

.....'"0 u

- 2 rlD rlS - rlL Wv - rlL EL Ws rs rlL ES WL + - D - 2 X4 X6 - Xs Wv - Xs XIO Ws

Antenna-bound trihedron

- Xs XII wL + X3 Xs

~2~~1_~!_£12~i~g_£2~~i!i2~~·

In addition to the advantages it offers as regards the representation of the closing conditions. this trihedron is basically interesting in that is leads (under its "relative coordinates" version) to de-paired measurement errors.

~T

~2~:~i!_~~~ia!i2~_~2~~!·

L

- W + ES W ) L V

-Drl L - 2DrlL + DrlS (WV + EL Ws + ES WL) ..

D - D(rl

2

L

2

+ rlS)

The W being the gyro velocities measured at the antenna and the rl being the same terms. but with respect ,t o the object under tracking (see fig. 3). g~!~_~~£!2!

XI - A sin S

Xs

rlL

X9 = rvr

X2

X6

rlS

X IO

EL

X3 a~ D

X 7

r

XII

ES

X = rlD 4

Xs

rS

A cos S

L

. .

tI)

>'''

El

c::

c::

'" QJ

.... 0 ~

p,.

~

El

·0

()

X = A sin S + AS cos S I

X2 X6 - Wv

X = A cos S - AS sin S 2

-

D X= 3 - D2

~

~

3

with

Xs = - as Xs + Vs

rlS

s

•

rlv =-AsinS

x9

a v x9 + Vv

D

rlD =0

RESULTS OBTAINED UNDER ACTUAL CONDITIONS Figures 2 to 6 show the results obtained with BSS model for combined Radar/Optronic tracking. The filter used is an Extended KALMAN Filter (EKF). the characteristics of the measurement noise being evaluated in an adaptive way. In the same example. it would be observed that the results obtained with SINGER's model are unsatisfactory (too much streacking during maneuvers) while the filters worked out on the BBS model yield consistent results. Utilization of the closing conditions provides sturdy estimators, with satisfactory maneuvering performances . ACKNOWLEDGEMENTS

g~!!:_!:g!!~Ei2!!~

~

....Ec:: §S

- (XL X7 + VL

A cos S

7

•

It can be seen that the rotation speed of the sighting line rlV or Wv must be known (e.g. using a gyro).

rlS - Ws + EL Wv - (rl

QJ

X6 - W +X IO ~ S rlL

.~ ~ ~

The equations defining this model are as follows :

.

XII =:lS - Ws + EL Wv

-

X3 X4

Xs - XI X6

We wish to thank Mr. G. SALUT (LAAS. Toulouse) and Professors Ph. de LARMINAT and C. DONCARLI for their kind advice and useful criticisms. BIBLIOGRAPHY [ I] Perriot-Mathonna D. (Mars 1979). Le filtrage de Kalman adaptatif. Application a la poursuite de cibles manoeuvrantes. Revue technigue Thomson-CSF Vol. 12 nOI.

A New Adaptive Model for Radar and Optronic Trackers

[ 2] Snlut G., Aguilar J., Favier G., Alengrin G. (Sept. 1979). Optimal Joint Adaptive Estimation of Parameter and State of a Linear Stochastic System with Application to Tracking. Questiio V. 3 n03. [3] Bozzo C., Legrand W. (Juin 1973). Model Error Sensitivity via Kalman Filtering in the Identification of Unforced Dynamical Linear Systems. Application to Radar Tracking Problems. Third IFAC Symposium on Identification and System Parameter Estimation. La Haye. [ 4] Passeron L. (Juin 1981). Modelisation et traitement de trajectoires. Huitieme Collogue GRETSI sur le traitement du signal et ses applications. Nice [ 5] Blondel C. (Oct. 1980). Analyse de differentes modelisations en vue de l'estimation des elements de cinematique de la trajectoire d'un mobile. Diplome d'Etudes Approfondies, Universite d'Aix-Marseille Ill. [ 6] Doncarli C., De Larminat Ph. et Linard & (Jan. 1979). Etude et application d'algorithmes d'identification recursive multivariable a l'estimation de la cinematique d'un but. Rapport final de Convention DCAN de Toulon CAPCA-ESNM Universite de Nantes. [ 7] Ramachandra K. V. (Nov. 1979). SteadyState Covariance Matrix Determination for a Three Dimensional Kalman Tracking Filter. IEEE Transactions on Aerospace and Electronic Systems Vol. AES-15 n06 . [ 8] Fitzgerald R. G. Dimensionless Design Data for Three State Tracking Filters. Raytheon Company Bedford, Massachussets. [ 9] Friedland B. (Nov. 1973). Optimum Steady State Position and Velocity Estimation Using Noisy Sampled Position Data. IEEE Transactions on Aeros~ace and Electronic Systems Vol. AES n 6. [10] Fitzge ;-ald R. J. (1975) Target Tracking Filters. Electronic Prog (U.S.A.) Tome 17 nO I. [11] Bozzo C. A. et Legrand H. (Juin 1973). Differents aspects des problemes de modelisation et de sensibilite en filtrage lineaire. Journees LAAS GRETSI. Le Filtrage Numerique et ses applications. Toulon. [12] Michel C. (1974). Deternination des parametres caracteristiques du mouvement de deux mobiles par une methode de filtrage optimal. Diplome d'Etudes Approfondies Faculte de St-Jerome Marseille. [13] Deffontaines E. (1966). Prediction d'une trajectoire a partir de mesures anterieures bruitees et echantillonnees. Memorial de l'Artillerie Fran~aise Tome 40 2eme fascicule Paris.

789

[ 14] Singer R. A. (Juil. 1970). Estimating Optimal Tracking Filter Performance for Manned Maneuvering Targets. IEEE Transactions on Aerospace and Electronic Systems. Vol. AES-6 n04. [ 15] Moose L., Mc Cabe H. (Fev. 1980). Adaptive Target Tracking for Underwater Maneuvering Targets. Department of Electrical Engeneering, Virginia Polytechnic Institute and State University Blacksburg. [ 16] Gholson N. H., Moose R. L. (Avril 1976). A Comparison of two Approaches to the three Dimensional Target Tracking Problem Involving Non-Linear State Measurements. Proceedings of the Eighth Annual South Eastern Symposium on System Theory Knoxville (USA) Tennessee. [ III Siffredi E. (Oct. 1979). Poursuite d'un mobile par filtrage adaptatif : determination de la position et du vecteur vitesse. Diplome d'Etudes Approfondies Universite d'Aix-Marseille Ill. [ 18] Guilbert A. (Jan. 1976). Azimetrie et Poursuite 2D en goniometrie passive. Rapport d'Etude S 03 083 GESTA DCAN de Toulon. [19] Guilbert A., Bozzo C. (Mai 1979). Differents aspects des problemes de modelisation et d'estimation des mouvements relatifs d'un mobile dans le plan, a partir de mesures angulaires bruitees et echantillonnees. Septieme Colloque sur le traitement du signal et ses applications. GRETSI Nice. [20] Linard A. (Sept. 1980). Analyse et Identification de modeles de Trajectoire en vue du filtrage autoadaptatif. These de Doctorat d'Ingenieur Universite de Nantes.

Figure 1

790

C. Bozzo et al,.

0 . tII141ni. (rd/s)

Distance(m)

6062.0

Estimati.on

0 .14767

Filtered

8032.6 8017.11

Distance

0 .06020

EKF

". omputed

Time -0.027286.=::::=-_ _ _ __ __ _ _ _ _ _ _ _ _ _ __ _ 61.440

BSS Model

511511. I

BBS Model

0 .011146

51188.5 51173.8

0 .10383

113.143

114.8411

611.5411

611.251

l1li.1154

71.557

Computed nS

0 .13184

EKF

73.360

0 .011082

51144.4

Estimation of !:"i. ..,"

0.041181

69211.7

= ___=---:-::-_ _Time __

511t5.0'J,f.,,,,---:-.,-:,::-::---:-.,.--:-_ _,..,..,._ _ 13.780 14.320 14.880 15.440 111.000

111.1160

17.120

17.880 -0.03223 _0.07324l..-_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _.--:::..Time

Figure 2 3 .22441

T

61.440

Azimuth (rd)

113.143

64.846

I

BBS Model

611.5411

611.251

111.1154

71 .557

73.3110

Figure 5

EKF 3.22277

Filtered Azimuth

3.22236 3.22195

0 .373011

3.22154

~ (rd/s)

0 .211l1li7

3.22113

0 .22011&

3.2207:i

Time 3.22031 L _ _ _ _ _ _ _ _ _...lL...lL-=iL-:>L_ _ _ _ _ _ _ _ 40.960

41 .634

42.3011

42.1183

43.l1li7

44.331

45.006

45.880

~ (Command) (Line of sight)

0.14483 0.0118711

-0 . 00732~==~====~====~====~==~__~__--~_

Figure 3

2 .080

13.3411

24.1117

36.886

47. 1114

118.423

111.1111

80.1180

0.10028 0 .16040

Site (rd)

0.0111141

0 .15975

Estimation of

~

= DD

0.018114

BBS Model

-0.02232

0.15911

EKF

0 . 111847

- 0 .0113111 . 0.104011 L._ _~--~--_--~--_-~£...,~--~ 2.080 13.3411 24.1117 36.1186 47. 1114 118.423 1111.l1li1 80.11110

0 .15783

Filtered Site

0.15718 0 .15654

Figure 6

0 . 155110 0 .1 5525 0.15461 0 . 153117 11.780

10.1111

12.023

13.154

14.286

Figure 4

l11j117

111.54Il

17.880