Handover analyis for tactical guided weaponsusing the adjoint method

M. Weiss ∗

TNO Organization, The Hague 2509 JG, The Netherlands

D. Bucco †

DSTO Organization, Edinburgh SA 5111, Australia

The adjoint method for guided loop analysis has been traditionally ap-plied for miss distance estimation. This paper presents an application ofthe adjoint method for estimating the pointing error of a homing seeker atthe end of an inertial guidance flight phase. This enables to use the adjointmethod to analyse the handover from midcourse guidance to homing guid-ance, by putting in perspective the contribution of each error source to thesucces/failure of the handover.

I. Introduction

Requirements for increased intercept envelopes and higher accuracy for modern guidedweapons impose frequently that the guided weapon be launched before locking on the target.Typically, after launch and flight stabilization, the missile enters a midcourse guidance phaseusing inertial, or aided-inertial navigation. At a predefined moment, the weapon tries toacquire the target with its own sensors. After target acquisition, the weapon enters thehoming guided phase in which acceleration commands are generated based essentially on theown target sensor information. Naturally, the ultimate performance of the guided weaponis determined by its effect on the target. Nevertheless, without a proper handover from themidcourse guidance phase to the homing guidance phase the missile is simply lost in mostpractical cases. Consequently, the design of the midcourse guidance phase of the flight has toensure that the missile will be able to properly detect the target at the end of the midcoursephase.

There are basically two reasons that may cause the handover to fail. Firstly, the sensormay not be sensitive enough to detect the target from the noisy signal at the handovermoment. Secondly, the target may not be in the field of view of the sensor at the handovermoment. The first effect can be mitigated by choosing the handover moment to occur as the

∗Scientific Researcher, TNO Defence, Security and Safety, P.O.Box 96864, The Hague, Netherlands.†Head Weapon Robotics, Weapons Systems Division, DSTO Edinburgh, P.O. Box 1500, Edinburgh

SA5111, Australia.

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missile is sufficiently close to the target. This will generally improve the signal-to-noise ratioin the seeker, and consequently the probability of detection. However, the chance that thetarget will fall outside of the field of view will increase as the relative distance between themissile and the target decreases. First of all, this is a natural consequence of the geometricconfiguration: as the target is closer to the missile, the error in estimating the target positionwill translate in larger angle positioning error of the line-of-sight. Moreover, navigation errorsof the missile and prediction errors of the target position are likely to increase with the timeof flight, and both these errors will determine an incorrect positioning of the boresight lineof the seeker with respect to the true line of sight.

In this work we will describe an approach to handover analysis and design using theadjoint method. This method has been extensively used in the past for miss distance analysis(see e.g.1), but its application to the handover situation is novel.

The basic idea of the adjoint method as it is applied to miss distance analysis of guidanceloops is to express the error budget of the miss distance in terms of the various noise anddisturbance sources using the impulse response of the so-called adjoint model. The adjointmodel is obtained through simple manipulation rules of the original model. In most of theliterature the procedure to obtain the adjoint is explained in terms of the block schemerepresentation of the original system. Lately a state-space approach was presented in.2

Since the adjoint method strives for error budgets, its application is intrinsically limited tolinear(ized) models for which the superposition principle is applicable. Therefore a first stepin applying the adjoint method to the handover problem is to obtain a linearized model thatgenerates good approximation of the handover errors. Subsequently, the adjoint response ofthe linearized model will provide the error budget for the analysis.

In this work, we consider a very simple formulation of the handover model, limited to2D intercepts in which the missile and the target are modelled as kinematic point models.However, in future work, the method will be extended to more realistic 3D models, that takeinto account the 6 degrees of freedom of the flying objects.

II. Modeling and linearization of a 2D intercept

The planar kinematic model of the missile is described by the following differential equa-tions

x = V cos γ, x(to) = xo , y(to) = yo,

y = V sin γ,

V γ = aN(t), γ(to) = γo, (1)

V = aL(t), V (to) = Vo,

where (x, y) are the position coordinates of the missile, V is its velocity and γ its course, aN (t)represents the lateral acceleration profile, and aL(t) represents the longitudinal accelerationprofile.

For this model we will assume that Vo > 0 and that aL has the typical profile of aboost/coast missile. In this case, system (1) is well-posed and will describe the planartrajectory of the missile for reasonable times of flight.

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Target

Missile

Line-of-sightθ

λγ

VT

V

(xTd, yT d)

(xd, yd)

y

x

Figure 1. Intercept geometry at target detection moment.

We assume that the missile is able to measure both the acceleration components, and usethese measurements to estimate its own position, velocity and course. For simplicity, we willassume that the navigation system of the missile simply integrates the kinematic equations(1) to obtain these estimates. Thus the equations of the navigation system are

˙x = V cos γ, x(to) = xo , y(to) = yo,˙y = V sin γ,

V ˙γ = aN,m(t), γ(to) = γo, (2)

˙V = aL,m(t), V (to) = Vo,

whereˆdenote estimated quantities and aN,m and aL,m represent the measured componentsof the acceleration.

Let us denote the handover time by td. At time td, the missile will direct its seeker inthe direction of the estimated line of sight. We assume that at this time the target will bewithin the effective range of the sensor. In this case, the success of the handover is measuredin terms of the probability that the actual position of the target will fall within the field ofview of the seeker. Neglecting the angle of attack of the missile, and denoting by xT and yT

the position coordinates of the target, the pointing angle of the seeker at time td is given by

θd = γ(td) − λ(td) = γ(td) − arctanyT (td) − y(td)

xT (td) − x(td). (3)

The “true” or correct pointing angle is defined as the angle between the geometric line ofsight and the true missile course and can be written as

θd = γ(td) − arctanyT (td) − y(td)

xT (td) − x(td). (4)

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The diference between these two angles will be denoted by

εθd = θd − θd

and represents the pointing error. As we explained before, we shall use this quantity tocharacterize the probability that the handover will succeed.

If we denote estimation errors at time td by εXd, εY d, εXT d, εYT d, εγd defined as

εXd = x(td) − x(td),

and the analogues, and assuming that these errors are relatively small, the pointing errorcan be approximated with the following first-order formula

εθd = εγd − xT (td) − x(td)

R2d

(εYT d − εY d) +yT (td) − y(td)

R2d

(εXT d − εXd), (5)

where Rd denotes the relative distance between missile and target at time td.The navigation errors εXd, εY d, and εγd can be approximated as outputs of the lineariza-

tion of system (1) along the nominal missile trajectory:

εx = εV cos γ − εγV sin γ,

εy = εV sin γ + εγV cos γ,

εγ =εaN

V− aN

V 2εV , (6)

εV = εaL,

where εaLand εaN

stand for the accelerometer measurement errors, whereas εx, εy, εV andεγ are the estimation errors for the position coordinates, velocity and course.

The errors in the target position estimation are the result of two distinct factors: thetracking errors that reflect the performance of the sensors on the launching platform and theprediction errors that reflect target maneuvering after the last track update. The trackingerrors can be easily modelled as an additive random error to the geometric position of thetarget at the track update time. To describe the prediction errors, we need to choose a modelfor the target maneuvers. For simplicity, we use here a white noise acceleration model (seee.g. [3, Section 6.2]), but the method can be applied to any other linear maneuver model.This means that the target position is described by

xT = vxT ,

vxT = axT , (7)

yT = vyT ,

vyT = ayT ,

where axT and ayT are assumed to be delta-correlated noise of zero mean.We assume that linear prediction based on the last target position and velocity track is

used to estimate the current target position, i.e.

xT (td) = xT (tu) + vxT (tu)(td − tu), (8)

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where tu is the time of the last target position and velocity update. For simplicity, wewill take here tu = to that means that the guided weapon uses target data available atinitialization. However, if the weapon has an uplink capability, it is possible that tu > to.It follows that the prediction errors on the target prediction satisfy the following differentialsystem

εxT = εvxT , εxT (to) = xT (to) − xT (to),

(9)

εvxT = −axT , εvxT (to) = vxT (to) − vxT (to),

(10)

and similarly for the y-coordinate.Combining (6) and (9), we can write the pointing error from (5) as the value at time td

of the output of the linear system

x = A(t)x + B(t)u , x(to) = xo (11)

εθ = C(t)x,

where

x =[

εx εy εγ εV εxT εvxT εyT εvyT

]T

u =[

εaN εaL axT ayT

]T

xo =[

0 0 0 0 εxTo εvxTo εyTo εvyTo

]T

A(t) =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 0 −V (t) sin γ(t) cos γ(t)

0 0 V (t) cos γ(t) sin γ(t)

0 0 0 −aN (t)V 2(t)

0 0 0 0

0

0

0 1 0 0

0 0 0 0

0 0 0 1

0 0 0 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(12)

B(t) =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 0 0 0

0 0 0 01

V (t)0 0 0

0 1 0 0

0 0 0 0

0 0 −1 0

0 0 0 0

0 0 0 −1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(13)

C(t) =[−yT (t)−y(t)

R2(t)xT (t)−x(t)

R2(t)1 0 yT (t)−y(t)

R2(t)0 −xT (t)−x(t)

R2(t)0

](14)

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The application of the adjoint method to this problem is only justified if the linear system(11) is a good approximation of the nonlinear intercept model. This needs to be verified foreach specific application separately, and we will do it for the numerical case study consideredin Section IV. For the moment we will assume that this condition is satisfied and we deducein the next section an analytic expression for the variance of the pointing angle error.

III. Analytic expressions for the variance of the pointing angleerror

According to,2 we introduce the adjoint state and output response at time td satisfying

xadj(t) = AT (td − t)xadj(t), xadj(0) = CT (td), (15)

yadj = BT (td − t)xadj(t),

Following the theory in2 (Proposition 3.2), the variance of the pointing error can be writtenas a function of the adjoint state and output response according to the formula

σ2εθ

= [xadj(td − to)]T Qox

adj(td − to) +

∫ td

to

[yadj(td − τ)]T V (τ)yadj(td − τ)dτ, (16)

where Qo is the variance matrix of the initial value of xo and V (τ) is the intensity of theinput that is assumed to be a delta-correlated signal.

In our particular case,

V (τ) = diag{σ2aN , σ2

aL, σ2axT , σ2

ayT} = const (17)

Qo = diag{0, 0, 0, 0, σ2xTo, σ

2vxTo, σ

2yTo, σ

2vyTo}. (18)

If we denote by xadji , i = 1, . . . , 8, the components of the adjoint state vector and by yadj

j ,j = 1, . . . , 4, the components of the adjoint output vector, it is easy to solve the system(15) for analytical expressions of these components. Straightforward calculations lead to the

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following expressions:

xadj1 (t) = −yTd − yd

R2d

, (19)

xadj2 (t) =

xTd − xd

R2d

, (20)

xadj3 (t) = 1 +

yTd − yd

R2d

∫ t

0

V (td − τ) sin γ(td − τ)dτ

+xTd − xd

R2d

∫ t

0

V (td − τ) cos γ(td − τ)dτ, (21)

xadj4 (t) = −yTd − yd

R2d

∫ t

0

cos γ(td − τ)dτ +xTd − xd

R2d

∫ t

0

sin γ(td − τ)dτ

−∫ t

0

aN(td − τ)

V 2(td − τ)xadj

3 (τ)dτ, (22)

xadj5 (t) =

yTd − yd

R2d

, (23)

xadj6 (t) =

yTd − yd

R2d

t, (24)

xadj7 (t) =

xTd − xd

R2d

, (25)

xadj8 (t) =

xTd − xd

R2d

t, (26)

yadj1 (t) =

1

V (td − t)xadj

3 (t), (27)

yadj2 (t) = xadj

4 (t), (28)

yadj3 (t) = −xadj

6 (t), (29)

yadj4 (t) = −xadj

8 (t). (30)

Introducing these expressions in (16) and using (17) and (18) we obtain the following ex-pression for the error budget of the pointing angle

σ2εθ

= (31)

σ2xTo(

yTd − yd

R2d

)2 + σ2vxTo(

yTd − yd

R2d

)2(td − to)2

+σ2yTo(

xTd − xd

R2d

)2 + σ2vyTo(

xTd − xd

R2d

)2(td − to)2

+σ2aN

∫ td−to

0

[yadj1 (τ)]2dτ + σ2

aL

∫ td−to

0

[yadj2 (τ)]2dτ

+

[σ2

axT (yTd − yd

R2d

)2 + σ2ayT (

xTd − xd

R2d

)2

](td − to)

3

3

This expression can readily be applied in trade-off studies for a quick estimate of theconsequences of a design decision on the pointing error.

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IV. Numerical case study

To illustrate the theoretical developments presented in the previous section we considerthe following case study. A ground-to-air missile is launched against a target. The missileflies initially towards a predicted intercept point (PIP) that is calculated in such a way thatboth missile and target would reach it in the same time provided that the target does notchange its initial velocity and course.

It is assumed that the longitudinal acceleration of the missile is essentially determinedby the thrust profile, so that aL(t) is a given function of time. The normal acceleration aN isdetermined by the requirement that the missile approaches the PIP. We choose an approachstrategy based on the Proportional Navigation Guidance Law

aN,c = NVcλ, (32)

where Vc is the closing velocity of the missile to the PIP, and λ is the angular rotation withrespect to the inertial reference frame of the line between the current missile position andthe PIP. If the coordinates of the PIP are denoted by xP , yP , then λ is computed accordingto the formula

λ =vyM(xM − xP ) − vxM(yM − yP )

(xM − xP )2 + (yM − yP )2, (33)

and Vc is computed according to the formula

Vc =vxM(xM − xP ) + vyM(yM − yP )√

(xM − xP )2 + (yM − yP )2. (34)

The response of the autopilot of the missile is approximated by a first-order element, that isthe lateral acceleration of the missile satisfies

τ aN + aN = aN,c, (35)

Formulas (32),(33), (34) and (35) define the midcourse guidance law used by the missile toapproach the PIP.

Two types of studies were run with this model in order to illustrate the application ofthe adjoint method for the problem at hand.

In the first type of study the intercept point was kept fixed and the handover time td wasvaried. The sigma value of the pointing error of the antenna was computed firstly using thenonlinear model, secondly using the linearized model, and finally using the adjoint method.The results are shown in Figure 2. It is clear that the three methods provide very closeresults. This can be verified for several different numerical values, validating the assumptionthat the linear model is a good approximation of the original intercept model and can beused for performance analysis. In turn this provides the justification for using the adjointmethod for estimating the pointing error variance.

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1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Detection time−to−go [s]

σ θ [deg

]

Monte Carlo nonlinear

Monte Carlo linear

Adjoint method

Figure 2. Sigma value of the antenna angle positioning error as a function of the time-to-goat handover obtained by Monte Carlo simulations of the nonlinear and of the linear modelsand by the adjoint method.

Having established the validity of the adjoint method, we can use it in performancestudies that would otherwise require an impressive number of Monte Carlo simulations.For example, we have varied both the time-to-go at handover and the horizontal interceptrange and calculated in each case the variance of the pointing error. By comparing it toan appropriate threshold representing the field of view of the seeker, it is easy to obtaina feasible range estimate as a function of the time-to-go at target detection. The result ispresented in Figure 3 and is obtained in just a fraction of the time necessary to obtain itthrough Monte Carlo simulation.

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2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 70

1

2

3

4

5

6

7

8

9

10x 10

4

Detection time−to−go [s]

Dow

nran

ge a

t int

erce

pt [m

]

Feasible intercept ranges as a function of detection time

Figure 3. Feasible range estimate as a function of time-to-go at handover

Variable Value Observations

aL(t)0 5 10 15 20 25

−40

−20

0

20

40

60

80

100

120

140

Time [s]

Long

. acc

. [m

/s2 ]

An arbitrary profile emulating arocket-boosted missile.

Np 3

τ 0.1 [s]

(xP , yP ) (50000, 300) [m] For the second study, xP variesbetween 5000 and 100000.

(vxT , vyT ) (−300, 0) [m/s]

σaxT = σayT 3 [m/s2]

σaN = σaL 0.1 [m/s2]

Table 1. Values of the parameters used for the numerical examples.

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V. Conclusions

This work presents a novel application of the adjoint method to the performance evalua-tion of guidance loops. Whereas most of the studies concentrate on miss distance evaluation,we applied the adjoint method to evaluate the pointing angle error of the target seeker atthe end of a midcourse guidance flight phase. This is an important performance indicatorfor the probability of detection and, therefore, for the probability of a successful intercept.

The work reported here can be extended to deal with the three dimensional case. Thiswill be reported elsewhere.

References

1Zarchan, P., Tactical and Strategic Missile Guidance, Vol. 176 of Progress in Austronautics and Aero-nautics , AIAA, 3rd ed., 1997.

2Weiss, M., “The adjoint method for missile performance analysis on state-space models,” AIAA J.Navigation, Guidance and Control , Vol. 58, No. 2, 2005, pp. 236–248.

3Bar-Shalom, Y. and Li, X.-R., Estimation and Tracking Principles, Techniques and Software, ArtechHouse, 1993.

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