[American Institute of Aeronautics and Astronautics AIAA Guidance, Navigation and Control Conference...

Linear Quadratic Differential Games Guidance Law

for Imposing a Terminal Intercept Angle

Vitaly Shaferman∗ and Tal Shima∗∗

Technion - Israel Institute of Technology, Haifa, 32000,Israel

A linear quadratic differential games guidance law that enables imposing a predeter-mined terminal intercept angle is presented. The guidance law is derived for arbitraryorder linear missile dynamics using a quadratic cost function. The obtained guidance lawis dependent on the well known zero effort miss distance and on a new variable denotedzero effort angle. Theoretic conditions for the existence of a saddle point solution arealso derived. It is shown that imposing the terminal angle constraint raises considerablythe gains of the guidance law and requires a higher maneuverability advantage from themissile. The performance of the proposed guidance law is investigated using a non-lineartwo dimensional simulation of the missile’s lateral dynamics and relative kinematics, whileassuming first order dynamics for the target’s evasive maneuvers. Using a Monte Carlostudy it is shown that for the investigated problem a target can be intercepted with anegligible miss distance and terminal intercept error even when the target maneuvers andthere are large initial heading errors.

I. Introduction

The terminal angle in which a target is intercepted is a vital parameter in a missile guidance problemas it effects warhead lethality and the target’s capability to effectively employ countermeasures. Usingconventional guidance laws, such as proportional navigation (PN), the interception angle can not be imposed.It is mainly a function of the initial collision triangle and the target maneuver.

In Ref. 1 a simple rendezvous of an interceptor and a non-maneuvering target was solved, using an optimalcontrol framework. The formulation included quadratic penalties on the terminal relative displacement andrelative terminal velocity to the required rendezvous course. This formulation can be used to impose aterminal intercept angle in such a scenario by selecting the ratio between the relative terminal velocity andthe closing speed. In Ref. 2 a similar formulation was used to derive linear quadratic optimal guidance laws(OGL) and linear quadratic differential game (LQDG) based laws for maneuvering target scenarios. Theguidance laws were derived using a linearization around the collision course. This formulation does not enabledirect intercept angle control. However, for the OGL case a required intercept angle can be reformulated toa required relative velocity, resulting in an indirect intercept angle control.

Previous works on direct intercept angle control mainly include OGLs and modified PN type laws. InRef. 3 an optimal control law for impact angle error and miss distance minimization was proposed for areentry vehicle pursuing a target, moving at a constant velocity on the ground. The guidance law wasformulated in a missile fixed Cartesian coordinate system and minimized a quadratic cost function. InRefs. 4, 5 optimal control laws, for a missile with arbitrary order dynamics intercepting a stationary target,were proposed with a similar cost function and a line of sight (LOS) fixed coordinate system. The proposedlaw was implemented for a lag-free and a first-order lag missile systems. The same formulation was usedin Ref. 6 with a time-to-go weighted energy cost function, to shape the missile’s trajectory. One of theunderlying assumptions in Refs. 3–6 is that the target is stationary or moving at a constant velocity on theground, therefore these guidance laws can not be applied against a moving and maneuvering target.

In Ref. 7 an optimal guidance law for impact angle control of a maneuvering ship and a missile withvarying velocity was proposed. The guidance law was formulated in a Cartesian coordinate system with

∗Graduate Student, Department of Aerospace Engineering; [email protected]∗∗Senior Lecturer, Department of Aerospace Engineering; This work was supported in part by a Horev Fellowship through

the Taub Foundation; [email protected]

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American Institute of Aeronautics and Astronautics

AIAA Guidance, Navigation and Control Conference and Exhibit18 - 21 August 2008, Honolulu, Hawaii

AIAA 2008-7302

Copyright © 2008 by the authors. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

one axis pointing towards the required missile terminal heading. The authors assumed a constant, andrelatively low, target maneuver and a small initial heading error. The resulting guidance law requiredextensive information including the perpendicular position and speed relative to the terminal heading, andthe target’s path angle and angular rate. To fulfill these requirements a suboptimal estimation filter wasintegrated in the proposed solution. Ref. 8 presented a biased PN law for an impact with a predefinedmissile attitude angle against a moving non-maneuvering target. A time varying component was added tothe LOS rate of the PN guidance law. For the derivation linearization and time-to-go estimation were notrequired. It was shown through simulation that the guidance law provides acceptable performance againstslowly maneuvering targets. In Ref. 9 optimal control planar interception laws against maneuvering targetswith known trajectories were presented. The authors derived open loop guidance laws for a lag-free missilewith constraints on the initial and final flight-path angles of the interceptor and a cost on the flight timeand control effort of the missile. The solution requires numerically finding four constants and the number ofinterceptor acceleration sign switches along the trajectory.

In this paper we propose a closed loop LQDG type guidance law, for arbitrary order linear missiledynamics, which enables imposing a predetermined intercept angle on a moving and maneuvering target.Due to the complexity of the derivation, analytic terms are derived only for the ideal missile dynamics case.The remainder of this paper is organized as follows: In the next section the mathematical models used forthe guidance law’s derivation and simulation are presented. The guidance law with constrained terminalimpact angle (LQDG-CTIA) is derived in Sec. III and then analyzed in Sec. IV and Sec. V. A simulationstudy is presented in Sec. VI, followed by concluding remarks.

II. Models Derivation

A skid to turn roll stabilized missile is considered. The 3D motion of such a missile can be separated to twoperpendicular planes, which can be solved independently. The solution to one such plane will be presentedin this paper. The 3D motion is derived by combining the solutions in both planes. We first present the fullnon-linear kinematics equations of the interception problem, which will be used in the non-linear simulationin Sec. VI. Then, the linearized equations which will be used to derive the proposed guidance law will bepresented.

A. Non-Linear Kinematics

In Fig. 1 a schematic view of the planar endgame geometry is shown, where XI − OI − ZI is a Cartesianinertial reference frame. We denote the missile and target by the subscripts M and T , respectively. Thespeed, normal acceleration, and flight path angles are denoted by V , a, and γ, respectively; the range betweenthe adversaries is r, and λ is the angle between the LOS and the XI axis. The remaining terms will bedefined in the next subsection.

Neglecting the gravitational force, the engagement kinematics, expressed in a polar coordinate system(r, λ) attached to the missile, is

r = Vr (1a)λ = Vλ/r (1b)

where the speed Vr isVr = − [VM cos (γM − λ) + VT cos (γT + λ)] (2)

and the speed perpendicular to the LOS is

Vλ = −VM sin (γM − λ) + VT sin (γT + λ) (3)

During the endgame, the target and missile are assumed to move at a constant speed. In addition we assumefirst order lateral maneuver dynamics for the target

aT = (acT − aT ) /τT (4a)

γT = aT /VT (4b)

where τT is the time constant of the target dynamics and acT is the maneuver command.

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ZI

XIOI

aTN

aMN

λ0

M

T

zr

Z

VT

VM

λ0

LOS0

γT + γM

X

aM

γT

γM

λ

aT

Figure 1. Planar engagement geometry.

In endo-atmospheric interception the majority of the interceptor’s lift is usually obtained by generatingan angle of attack, a process that has dynamics. The interceptor has steering devices such as canard or tailthat can generate angle of attack, and, neglecting servo dynamics, possibly instantaneous maneuvers. Weassume that the dynamics during the endgame can be represented by arbitrary order linear equations

xM = AMxM + BMuM (5a)γM = aM/VM (5b)

whereaM = CMxM + dMuM (6)

and xM is the state vector of the interceptor’s internal state variables with dim(xM ) = n. We denote thepart of the acceleration without dynamics, if it exists, as the direct lift, while the part with dynamics asthe specific force. These equations can represent the closed loop missile dynamics. They can also representthe linearized open loop dynamics in which case the obtained guidance law will actually be an integratedguidance-autopilot controller.

B. Linearized Kinematics for Guidance Law Derivation

The endgame kinematics used for the guidance law derivation was presented in Fig. 1. The X-axis, alignedwith the LOS used for linearization, is denoted as LOS0. z is the relative displacement between the targetand the missile normal to this direction. The target and missile accelerations normal to LOS0 are denoted byaTN and aMN , respectively; and the intercept angle is (γT +γM ). We denote the required value of (γT +γM )as xc

4.The derivation of the guidance law in this paper will be performed based on a linearized model. If during

the endgame the missile and target deviations from the collision triangle are small, i.e. the endgame isinitiated with a collision triangle satisfying closely the requirement on xc

4 and the target’s maneuver relativeto its speed is small, then the linearization is justified. This initialization can be performed for exampleby a non-linear mid-course guidance law. If on the other hand the scenario is not initiated on the requiredcollision triangle (as in some of the cases simulated in section Sec. VI) and/or the target has a large maneuvercapability relative to its speed, then we resort to extended linearization or state dependent Riccati equation(SDRE) like techniques.10 We will linearize the equations of motion at each step of time and then solve theassociated SDRE. Therefore, the guidance law will be calculated at each time step with LOS0 reset to the

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current LOS between the adversaries. To obtain an implementable scheme we will seek an analytic solutionfor this equation, thus not requiring the online solution of the Riccati equation at each time step.

The state vector of the linearized problem is

x =[

z z aTN (γT + γM ) xTM

]T

(7)

The equations of motion are

x =

x1 = x2

x2 = x3 − aMN

x3 = (v − aTN )/τT

x4 = aTN/(VT cos(γT0 + λ0)) + aMN/(VM cos(γM0 − λ0))xM = AMxM + BMuM

(8)

where v is the target’s acceleration command normal to LOS0.Let us define u = uMcos (γM0 − λ0) and the missile and target velocity components on LOS0 as v′T and

v′M

v′T = VT cos(γT0 + λ0) (9)v′M = VM cos(γM0 − λ0) (10)

The matrix form of the equation set is therefore

x = Ax + Bu + Cv (11)

where

A =

[Ak A12

[0]n×4 AM

], B =

0−dM

0dM/v′M

BM/ cos(γM0 − λ0)

, C =

00

1/τT

0[0]n×1

(12)

and

Ak =

0 1 0 00 0 1 00 0 −1/τT 00 0 1/v′T 0

, A12 =

[0]1×n

−CM cos(γM0 − λ0)[0]1×n

CM cos(γM0 − λ0)/v′M

(13)

with [0] denoting a matrix of zeros with appropriate dimensions.Once a collision triangle is reached the speed Vr is constant and the interception time, given by tf =

−r0/Vr, can be assumed fixed. For the guidance law implementation we approximate time-to-go by

tgo = −r/Vr (14)

III. Guidance Law Derivation

We seek optimal strategies for both adversaries that satisfy a saddle point, i.e. if one of the adversariesdeviates from its optimal strategy then it can not gain.

The quadratic cost function chosen for our zero-sum game is

J =a

2x2

1(tf ) +b

2(x4(tf )− xc

4)2 +

12

∫ tf

0

(u2 − γ2v2)dt (15)

where the weights a, b are non-negative and γ2 is a measure of the target’s maneuvering capability relativeto that of the missile. The missile, using u as its control, wishes to minimize this cost function; while thetarget, using v as its control, wishes to maximize it. Note that γ2 should be selected based on intelligenceinformation on the expected targets and should be tuned based on simulation results performed using thisintelligence information, similarly, to other LQDG2 guidance laws.

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A. Order Reduction

Ref. 1 introduced a transformation enabling reducing a problem’s order. This transformation is sometimesdenoted as terminal projection. We will use a similar transformation.

For this two sided optimization problem the transformation is

Z(t) = DΦ(tf , t)x(t) (16)

where Φ(tf , t) is the transition matrix associated with Eq. (11), and D is a constant matrix

D =

[1 0 0 0 [0]1×n

0 0 0 1 [0]1×n

](17)

The time derivative of the new state vector Z(t) is

Z = DΦ(tf , t)Bu + DΦ(tf , t)Cv (18)

which is independent of the states of the problem.Using this new state vector Z(t) the cost function from Eq. (15) can also be expressed as

J =a

2Z2

1 (tf ) +b

2(Z2(tf )− xc

4)2 +

12

∫ tf

0

(u2 − γ2v2)dt (19)

Note that besides reducing the order of the problem, the two variables of the new state vector Z(t) have animportant physical meaning. Z1(t) is known as the zero effort miss (ZEM) which is the miss distance if fromthe current time onwards both the missile and the target will not apply any control. We will refer to thenew term Z2(t) as zero effort angle (ZEA), which, in an analogy to the ZEM, is the impact angle if from thecurrent time onwards both the missile and the target will not apply any control. We denote (Z2(t)− xc

4) asthe zero effort angle error (ZEAE).

B. Optimal Controllers

The Hamiltonian of the problem is

H =12u2 − 1

2γ2v2 + λ1Z1 + λ2Z2 (20)

The adjoint equations are{

λ1 = − ∂H∂Z1

= 0 ; λ1(tf ) = aZ1(tf )λ2 = − ∂H

∂Z2= 0 ; λ2(tf ) = b(Z2(tf )− xc

4)(21)

yielding the solutions {λ1(t) = aZ1(tf )λ2(t) = b(Z2(tf )− xc

4)(22)

The optimal controllers for the missile and the target satisfy u∗ = argu

min H and v∗ = argv

maxH.

For obtaining an analytic solution for the optimal control laws of the two adversaries (u∗ and v∗) we willassume dealing with adversaries having ideal dynamics, i.e. zero-lag. This simplifies the time derivatives ofZ to

Z(t) =

[−(tf − t)

1/v′M

]u +

[(tf − t)1/v′T

]v (23)

and the guidance laws satisfy

∂H

∂u= 0 ⇒ u∗(t) = λ1(tf − t)− λ2/v′M (24a)

∂H

∂v= 0 ⇒ v∗(t) =

1γ2

[λ1(tf − t) + λ2/v′T ] (24b)

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Substituting Eqs. (24) into Eq. (23) and integrating from t to tf yields two coupled algebraic equations.Solving for Z1(tf ) and Z2(tf ) and substituting the solution into Eqs. (24) yields the optimal strategies

u∗(t) =Nu

ZEM

t2go

Z1(t) + NuZEAE

v′Mtgo

(Z2(t)− xc4) (25a)

v∗(t) =Nv

ZEM

t2go

Z1(t) + NvZEAE

v′Mtgo

(Z2(t)− xc4) (25b)

where

Z1(t) = z + ztgo (26a)Z2(t) = γT + γM (26b)

NuZEM = at3goK1(tgo) +

bK3(tgo)t2go

K2(tgo)

(aV1K1(tgo)t3go

2− 1

v′M

)(27a)

NuZEAE =

btgo

v′MK2(tgo)

(aV1K1(tgo)t3go

2− 1

v′M

)(27b)

NvZEM =

1γ2

[at3goK1(tgo) +

bK3(tgo)t2go

K2(tgo)

(aV1K1(tgo)t3go

2+

1v′T

)](27c)

NvZEAE =

btgo

γ2v′MK2(tgo)

(aV1K1(tgo)t3go

2+

1v′T

)(27d)

and

V1 =v′T γ2 + v′Mv′Mv′T γ2

, V2 =v′2M − v

′2T γ2

v′2Mv

′2T γ2

(28a)

K1(tgo) =3γ2

3γ2 − (1− γ2) at3go

(28b)

K2(tgo) = 1− bV2tgo −3abV 2

1 γ2t4go

4(3γ2 − (1− γ2) at3go

) (28c)

K3(tgo) =3aV1γ

2t2go

2(3γ2 − (1− γ2)at3go

) (28d)

Note that the closed form solutions of the optimal guidance laws for the two players presented in Eqs. (25)are linear with respect to both the ZEM and the ZEAE.

C. Guidance Law Implementation

The required information for the implementation of the guidance laws according to Eqs. (25) is: Z1(t), Z2(t),tgo, and v′M , v′T . By using the small deviation from collision triangle assumption the displacement z normalto the initial LOS can be approximated by

z ≈ (λ− λ0) r (29)

Differentiating Eq. (29) with respect to time yields

z + ztgo = −Vrt2goλ (30)

Using this expression, the Z1(t) of Eqs. (26) can be expressed as

Z1(t) = −Vrt2goλ (31)

By using a radar seeker and an inertial navigation system (INS): v′T , v′M , Vr, and γM can be directlymeasured, while λ, γT and the tgo must be estimated. Note that the tgo can be estimated using Eq. (14),but a more accurate tgo estimation might be needed, such as the one presented in Ref. 5.

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IV. Navigation Gains

Let us now examine the values of the navigation gains derived in Eqs. (27). By choosing b → 0 we nolonger impose a terminal impact angle, resulting with

NuZEM (b → 0) =

3aγ2t3go

3γ2 − (1− γ2) at3go

(32a)

NvZEM (b → 0) =

3at3go

3γ2 − (1− γ2) at3go

(32b)

NuZEAE(b → 0) = Nv

ZEAE(b → 0) = 0 (32c)

and the degeneration of the guidance laws in Eqs. (25) to the LQDG presented in Ref. 2. If we also choosea →∞, which dictates zero miss, the navigation gains further degenerate to

NuZEM (b → 0, a →∞) =

3γ2

(γ2 − 1)(33a)

NvZEM (b → 0, a →∞) =

3(γ2 − 1)

(33b)

NuZEAE(b → 0, a →∞) = Nv

ZEAE(b → 0, a →∞) = 0 (33c)

also presented in Ref. 2. And by considering a non-maneuvering target scenario (γ → ∞), we get the wellknown optimal PN guidance law with NZEM = 3.

For a perfect intercept with some account for terminal intercept angle, we would require a → ∞ and afinite b, resulting with the following navigation gains

NuZEM (a →∞) =

3γ2

(γ2 − 1)+

6bγ2V1tgo

∆b

(3γ2V1

2(1− γ2)+

1v′M

)(34a)

NuZEAE(a →∞) =

4b(γ2 − 1)tgo

v′M∆b

(3γ2V1

2(1− γ2)+

1v′M

)(34b)

NvZEM (a →∞) =

3(γ2 − 1)

+6bV1tgo

∆b

(3γ2V1

2(1− γ2)− 1

v′T

)(34c)

NvZEAE(a →∞) =

4b(γ2 − 1)tgo

v′Mγ2∆b

(3γ2V1

2(1− γ2)− 1

v′T

)(34d)

where∆b = 4(1− γ2) + btgo

(3γ2V 2

1 − 4(1− γ2)V2

)(35)

Fig. 2 presents the time history of the navigation gains for a perfect intercept, γ = 10, and a fewrepresentative values of b. Note that the navigation gains are constant with respect to tgo for b → 0 andb →∞ and increase with tgo for values of b in between.

The weight γ is a design parameter associated with the expected maneuvering capability of the targetcompared to the interceptor. Thus, for smaller values of γ (more maneuverable target) larger gains areobtained. If the target is not expected to maneuver, a large value of γ should be chosen and vise versa.Imposing a perfect intercept and intercept angle (or large values of a and b to improve performance, for thatmatter) might result with a conjugate point. In such a case, in order to obtain a solution in the entire gamespace, performance must be compromised by using smaller values of a and b.

A conjugate point does not exist if and only if the gains are finite2 for all 0 < tgo ≤ tf . For examplein Eqs. (33) the gains diverge for γ = 1, therefore a conjugate point exists for γ ≤ 1 and the solution mayno longer be optimal. The critical value of γ is denoted γcr, and to avoid a conjugate point we must selectγ > γcr.

For a perfect intercept and perfect intercept angle we would require a, b →∞ resulting in the degeneration

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0 2 4 6 8 100

2

4

6

8

NZ

EM

ub = 1

b = 105

b = 106

b = ∞

0 2 4 6 8 100

1

2

3

4

NZ

EA

Eu

tgo

(sec)

b = 1

b = 105

b = 106

b = ∞

0 2 4 6 8 100

0.05

0.1

0.15

0.2

0.25

0.3

NZ

EM

v

b = 1b = 105

b = 106

b = ∞

0 2 4 6 8 100

0.05

0.1

0.15

0.2

NZ

EA

Ev

tgo

(sec)

b = 1

b = 105

b = 106

vM

’ = 500 m/s, vT’ = 300 m/s, γ = 10

b = ∞

Figure 2. LQDG-CTIA navigation gains for perfect intercept (a →∞) and γ = 10

of the navigation gains of Eqs. (27) to

NuZEM (a, b →∞) =

6γ2

∆∞(2V2 + V1/v′M ) (36a)

NuZEAE(a, b →∞) = − 1

v′2M∆∞

(6γ2V1v

′M + 4(1− γ2)

)(36b)

NvZEM (a, b →∞) =

6∆∞

(2V2 − V1/v′T ) (36c)

NvZEAE(a, b →∞) = − 1

v′Mv′T γ2∆∞

(6γ2V1v

′T − 4(1− γ2)

)(36d)

where∆∞ = 3V 2

1 γ2 − 4V2(1− γ2) (37)

Note that the navigation gains for the a, b →∞ case are not a function of the time-to-go.Fig. 3 presents the navigation gains for perfect intercept and perfect intercept angle (a, b → ∞), for

v′M = 500(m/s), and v′T = 300(m/s). Clearly, as γ → 5 the gains diverge. The navigation gains for this caseare not bounded when ∆∞ → 0, therefore if there exists a γcr for which ∆∞(γcr) → 0, then a conjugatepoint exists for γ ≤ γcr and the solution may not be optimal. For this sample case γcr = 5, and consequentlya conjugate point exists for γ ≤ 5. We will derive the conditions for the existence of an optimal solution forthe perfect intercept and perfect intercept angle case in the next section.

By assuming a non-maneuvering target scenario (γ →∞) the navigation gains further degenerate to

NuZEM (a, b, γ →∞) = 6 (38a)

NuZEAE(a, b, γ →∞) = 2 (38b)

NvZEM (a, b, γ →∞) = Nv

ZEAE(a, b, γ →∞) = 0 (38c)

Which are the navigation gains of the simple rendezvous presented in Ref. 1, and which also appear inRefs. 2, 11.

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5 6 7 8 9 100

5

10

15

20

25

30

35

40

45

50

N

γ

NZEMu

NZEAEu

NZEMv

NZEAEv

Figure 3. LQDG-CTIA navigation constants for perfect intercept and intercept Angle

V. Perfect Intercept and Perfect Intercept Angle

Using the LQDG-CTIA guidance law we can guarantee perfect intercept and perfect intercept angle,provided a solution exists. As we have seen in the previous section, a conjugate point may exist in somescenarios. In such scenarios an optimal solution to the game does not exist for some tgo ≥ 0. In this sectionwe will present the optimal trajectories obtained for the perfect intercept and intercept angle case, and studythe conditions for the existence of a solution to the differential game.

Let us perform the following transformation

Z(t) =

[Z1(t)

Z2(t)− xc4

](39)

The new state Z(t) uses the ZEAE as the second state instead of the ZEA. The dynamic equation in the newspace is the same as in Eq. (23) for the original state vector Z(t). Substituting Eqs. (25) into this equationyields the following optimal trajectory dynamics

˙Z∗1 (t) = (NvZEM −Nu

ZEM )Z∗1 (t)tgo

+ (NvZEAE −Nu

ZEAE)v′M Z∗2 (t) (40a)

˙Z∗2 (t) =(

NuZEM

v′M+

NvZEM

v′T

)Z∗1 (t)t2go

+(

NuZEAE +

v′Mv′T

NvZEAE

)Z∗2 (t)tgo

(40b)

The resulting equation set is a second order linear time varying ordinary deferential equation (ODE) set. Forthe perfect intercept and perfect intercept angle case Nu

ZEM , NuZEAE , Nv

ZEM , and NvZEAE have been shown

to be constants with respect to time (see Eqs. (36)), and the ODE set can be analytically solved yielding

Z∗1 (t) = C3tαgo + C4t

βgo (41a)

Z∗2 (t) =(α− a11)

a12C3t

α−1go +

(β − a11)a12

C4tβ−1go (41b)

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where

a11 = (NuZEM −Nv

ZEM ) (42a)a12 = (Nu

ZEAE −NvZEAE) v′M (42b)

a21 = −(

NuZEM

v′M+

NvZEM

v′T

)(42c)

a22 = −(

NuZEAE +

v′Mv′T

NvZEAE

)(42d)

and

δ =√

(a11 − a22)2 − 2 (a11 − a22) + 4a12a21 + 1 (43a)

α =12

(a11 + a22 + 1 + δ) (43b)

β =12

(a11 + a22 + 1− δ) (43c)

C3 =1tαf

[(1− α− a11

δ

)Z10 +

(a12tf

δ

)Z20

](43d)

C4 =1

tβf

[(α− a11

δ

)Z10 −

(a12tf

δ

)Z20

](43e)

for which Z10 and Z20 are the initial conditions of Z1 and Z2 respectively.Fig. 4 presents a 3D plot (tgo, ZEM, ZEAE) of optimal trajectories from various initial conditions. It can

be seen that all the optimal trajectories terminate at the (0,0,0) coordinate as expected. For presentationsimplicity Fig. 5 presents similar trajectories on two intertwined 2D plots. The optimal trajectories startwith various non-zero initial ZEM values and with nulled ZEAE initial conditions.

00.5

11.5

2

−400

−200

0

200

400−150

−100

−50

0

50

100

150

tgo (sec)ZEM (m)

vM

"=500 m/s, vT"=300 m/s, γ= 10

ZE

AE

(d

eg)

Figure 4. LQDG-CTIA perfect intercept game space, without a conjugate point, γ = 10

Thus far we have shown trajectories emerging from a saddle point solution of the game. As we have seenin the previous section a conjugate point may exist, in which case the solution might not be optimal anymore. The non existence of a conjugate point has been shown to be a sufficient condition for the existence

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0 0.5 1 1.5 2−50

0

50

100

ZE

M (

m)

0 0.5 1 1.5 2−10

0

10

20

30

tgo (sec)

ZE

AE

(d

eg)

vM

"=500 m/s, vT"=300 m/s, γ= 10

Figure 5. LQDG-CTIA perfect intercept game space, without a conjugate point, γ = 10

of a saddle point solution to the game.2 A conjugate point exists if and only if P(t)

P = PAAT P + P(γ−2CCT −BBT )P (44a)P(tf ) = Qf (44b)

of the Riccati equation associated with the original problem is finite1 (the same is true of course for thereduced order problem). In turn P(t) is finite if and only if the solution navigation gains are bounded.2

Therefore, a saddle point exists if the navigation gains are bounded.Let us now consider the conditions for the existence of a conjugate point in the perfect intercept and

intercept angle scenario. As we have seen in the previous section, the navigation gains are unbounded when∆∞ → 0, where ∆∞ is defined in Eq. (37). By substituting the definitions of V1 and V2 from Eqs. (28a) intoEq. (37) we obtain

∆∞ =1

v′2Mv

′2T γ2

[3

(v′T γ2 + v′M

)2 − 4(v′2M − v

′2T γ2

) (1− γ2

)](45)

and by equating the result to zero and reordering we can find the values of γ that satisfy ∆∞(γ) = 0

−v′2T γ4 +

(4

(v′2M + v

′2T

)+ 6 (v′T v′M )

)γ2 − v

′2M = 0 (46)

This quadratic equation in γ2 obviously has a maximum. By substituting γ2 = 0 and γ2 = 1 in Eq. (46) weget −v

′2M and 3(v

′2M + v

′2T ) respectively. We can therefore conclude that the equation has two roots: the first

γ2 < 1 and the second γ2 > 1. We define γ∞ as the larger root between the two and no conjugate pointexists in this case for γ > γ∞.

Fig. 6 presents the values of γ∞ as a function of v′M and v′T . Note that the maneuver advantage requiredfor the missile over the target is smaller when v′M , the missile’s velocity component on LOS0, is smaller.The intercept angle is controlled by changing the path angle γM according to γM = aMN/(v′M ). Therefore,it is clear that for a slower missile the needed acceleration to control the path angle is smaller. This analysisdoes not consider the ability of the missile to generate the required acceleration, which, in aerodynamicallycontrolled missiles, is obviously a function of the missile’s lift, which in turn is a function of the missile’svelocity.

Let us now consider the perfect intercept case without requiring a perfect intercept angle. In that casethe navigation gains are unbounded when ∆b → 0, where ∆b is defined in Eq. (35). By equating ∆b to zero

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vM

’ (m/s)

v T’ (

m/s

)

8

6

5

4

3

2.5

2.5

200 300 400 500 600 700 800200

300

400

500

600

700

800

Figure 6. LQDG-CTIA conditions on γ for perfect intercept and intercept Angle (a, b →∞)

and substituting Eq. (37), we can derive the conditions for which the gains are unbounded and therefore aconjugate point exists

∆b = 4(1− γ2) + btgo∆∞ = 0 (47)

The critical tgo denoted tcrgo from Eq. (47) is

tcrgo = arg

tgo

[∆b(tgo) = 0] = −4(1− γ2)b∆∞

(48)

It is apparent from Eq. (48) that for every 1 < γ ≤ γ∞, and for every positive b, there exists a positive tcrgo

for which ∆b → 0, and therefore the navigation gains of Eqs. (34) are unbounded, at that tgo. This meansthat for a perfect intercept without a perfect intercept angle constraint (b < ∞) a conjugate point does notexist for scenarios lasting 0 < tf < tcr

go. On the other hand a conjugate point exists for scenarios in whichtf ≥ tcr

go. Note that for the perfect intercept and intercept angle case (b →∞), we can see from Eq. (48) thatfor every 1 < γ ≤ γ∞, a conjugate point exists at tcr

go → 0, and therefore the game does not have a solution.Fig. 7 presents the maximal scenario duration for a perfect intercept game for v′M = 500m/s, and

v′T = 300m/s. The figure presents tcrgo as a function of selected values of the design parameters γ and

b. As expected we can obtain a saddle point solution to the game for practical engagement scenarios bycompromising intercept angle performance. We can also see that for small values of b we converge to thewell known LQDG solution for which γcr = 1. Another interesting observation that arises from Fig. 7 isthat adding a requirement on the intercept angle, by increasing the value of b, dramatically increases themaneuvering advantage needed by the missile over the target. This is not surprising, because the interceptorneeds to reshape its trajectory to generate the required angle for every change in the target’s path angle.This result limits the usefulness of the proposed guidance law to missiles with high maneuvering capabilitywith respect to the intercepted targets.

VI. Simulation Study

The performance of the guidance law developed above is investigated in this section via numerical sim-ulations, using the non-linear kinematics and missile and target dynamics presented in Sec. II A. We will

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1 2 3 4 5 60

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

γ

t go

cr (

sec)

vM

’ = 500 m/s, vT’ = 300 m/s

b = 104

b = 105

b = 106

b = ∞ , γ∞

Figure 7. LQDG-CTIA maximal scenario duration for perfect intercept (a →∞)

first outline the non-linear test scenarios, then demonstrate and investigate the performance of the guidancelaw using a non-linear simulation, and finally present the performance using a Monte Carlo study.

A. Scenarios

Two interception scenarios are used for the performance analysis. The first scenario, in which the targetperforms a constant maneuver, will be used to demonstrate and evaluate the guidance law. This scenariowas chosen due to its simplicity and ease of presentation. The second scenario, in which the target performsa random evasive maneuver, will be used in our Monte Carlo study to evaluate the guidance law in a morerealistic setting.

The engagements are initiated in head-on. The vehicles are not necessarily initially flying along therequired collision triangle. Thus, we continuously update γM0, γT0, and λ0 used for the guidance lawimplementation.

The initial range is 2000m. The target speed is VT = 300m/s and its maneuver capability is aT = 5g. Wewill use a missile and target first order time constants of τM = 0.1sec and τT = 0.1sec, respectively. In thefirst scenario VM ∈ {300, 500, 700}m/s and the target is performing a constant maximum maneuver. In thesecond scenario the missile’s speed is 500m/s and the target performs a square wave evasive maneuver (usingits maximum acceleration capability) with a period of ∆T and a phase of ∆φ relative to the beginning of thesimulation. Due to the soft limit nature of the control law derivation, we will not impose a saturation valuein the simulation, but we will comment on the missile’s maximal acceleration values during the analysis.

B. Guidance law demonstration and investigation

We first present the ability of the proposed guidance law to intercept the target in the required angle. Fig. 8presents the trajectories obtained using the LQDG-CTIA guidance law (note the different scaling in theX and Y axis). By choosing xc

4 = γT + γM = 0 the commanded impact angle is head-on . When b → 0the trajectories coincide with those obtained using the classic LQDG. When the weight b is increased themissile’s trajectory is shaped so as to intercept the target in the required impact angle.

Fig. 9 presents the time history of the missile’s acceleration aM and the intercept angle for the sameengagements. Note that as b is increased the final intercept angle error is decreased from about 45 (deg),when b = 0 (which is the classic LQDG) to less than 1 (deg) for b = 108. This is expected from the

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formulation of the guidance law, but this does not come without a price. The missile’s maximal accelerationis increased from amax

M ≈ 13.5g to amaxM ≈ 85g, with the maximal acceleration being at the end of the

engagement similarly to the b = 0 case, which is usually unwanted.The large acceleration ratio between the missile and the target is mainly due to the large angle correction

required. One important conclusion that can be deducted is that for the implementation of this guidancelaw a substantial acceleration advantage is needed.

0 500 1000 1500 20000

50

100

150

200

250

X (m)

− Z

(m

)

v

M=500 m/s, v

T=300 m/s, a

T=50 m/s2,

λ0=0 deg,γ

M0 = 0 deg,

γT0

= 0 deg, r0=2000 (m)

Missile, b=0

Missile, b=106

Missile, b=108

Target

Figure 8. LQDG-CTIA with a head-on impact angle command, a = 105,γ = 7

0 0.5 1 1.5 2 2.5 3−1000

−500

0

500

a M (

m/s

2 )

0 0.5 1 1.5 2 2.5 30

10

20

30

40

50

Time (sec)

γ M+γ

T (

deg

)

b=0

b=106

b=108

Command

Figure 9. LQDG-CTIA with a head-on impact angle command, a = 105,γ = 7

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We have an additional design parameter γ, that describes the target’s maneuvering capability comparedto the missile. As we have seen in the theoretical analysis of this guidance law, for v′M = 500m/s andv′T = 300m/s we obtain γ∞ = 5 and a saddle point exists for γ ≥ γ∞. Fig. 10 presents the same engagementscenario but for different values of γ > 5. For the different values of γ, interception is obtained with therequired terminal angle.

0 500 1000 1500 20000

50

100

150

200

250

X (m)

− Z

(m

)

v

M=500 m/s, v

T=300 m/s, a

T=50 m/s2,

λ0=0 deg, γ

M0 = 0 deg,

γT0

= 0 deg, r0=2000 (m)

Missile, γ=6Missile,γ=10Missile, γ=20Target

Figure 10. LQDG-CTIA trajectories with an head-on impact angle command,with a = 105,b = 108,γ = 7

The weights a and b, are design parameters reflecting a tradeoff between the allowable miss distance,intercept angle error, and lateral acceleration. We have chosen a = 105 and b = 108 for most simulationruns. These values were selected to give equal weight to approximately 1m in miss distance, 2deg in interceptangle, and 25g in sustained acceleration for a scenario duration of 2 sec.

Fig. 11 presents trajectories for different requirements on the final intercept angle. The scenario is thesame as before with a = 105, b = 108 and γ = 7. The miss distances and intercept angle errors were verysmall in these runs. The worst miss distance in the tested range was approximately 0.6m, and the worstintercept angle error was approximately 3.5deg. Similar results have also been obtained for different initialheading errors.

Fig. 12 presents the sensitively of the guidance law to the missile’s speed VM . In these runs the peakaccelerations increased as the missile’s speed increased, being approximately 62g, 75g, and 79g for speedsof 300m/s, 500m/s, and 700m/s, respectively. Note that the maneuver advantage required from the missileover the target is smaller when VM , the missile’s speed is smaller, as expected from the analytic derivationin Sec. V.

C. Monte Carlo study

Thus far we have evaluated the guidance law in a scenario where the target performs a constant maneuver. Inactual engagements the target might perform unpredictable evasive maneuvers. Therefore, we will evaluatethe guidance law in a different scenario, presented in Sec. A, in which the target performs a square waveevasive maneuver, with a period of ∆T and a phase of ∆φ.

To evaluate the guidance law, a Monte Carlo study consisting of 200 simulation runs for each test pointwas performed. In these simulations, for each sample run a random value was chosen for ∆φ from a uniformdistribution, in the range ∆φ = [0, 1]. The period was chosen to be ∆T = 2sec. In order to properly evaluatethe guidance law, the statistics (mean and standard deviation) of all the components of the cost functionwere evaluated.

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0 500 1000 1500 2000−300

−200

−100

0

100

200

300 −30

−15

0

15

30

45

60

75

X (m)

− Z

(m

)

vM

=500 m/s, vT=300 m/s, a

T=50 m/s2,

λ0=0 deg, γ

M0 = 0 deg,

γT0

= 0 deg, r0=2000 (m)

MissileTarget

Figure 11. LQDG-CTIA planar trajectories for various impact angle commands,with a = 105,b = 108,γ = 7

0 500 1000 1500 20000

50

100

150

200

250

300

350

400

X (m)

− Z

(m

)

vT=300 m/s, a

T=50 m/s2, λ

0=0 deg,

γM0

= 0 deg, γT0

= 0 deg, r0=2000 (m)

Missile, vM

=300 m/s

Missile, vM

=500 m/s

Missile, vM

=700 m/s

Target

Figure 12. LQDG-CTIA planar trajectories for various missile speeds, with a = 105,b = 108,γ = 7

Figs. 13, 14, and 15 present the statistics of the miss distance, intercept angle error, and the control effortrespectively, as a function of the commanded intercept angle. It is clear that the LQDG-CTIA guidance law,yields good performance with an expected miss of less than 0.7m and an expected angle error of less than2deg. It is also clear that the miss and intercept angle error are only moderately effected by the commandedintercept angle, while the control effort is larger for larger intercept angle requirements. The increase in

16 of 19


control effort for large intercept angles is expected, due to the larger curvature and longer trajectoriesrequired in these cases.

−40 −30 −20 −10 0 10 20 30 400

0.5

1

1.5

E(m

iss)

[m

]

−40 −30 −20 −10 0 10 20 30 400

0.2

0.4

0.6

0.8

σ (m

iss)

[m

]

Commanded intercept angle [deg]

Figure 13. Miss distance for various intercept angles against an evading target, with : a = 105, b = 108, γ = 7,∆T = 2sec, ∆φ = [0, 1]

−40 −30 −20 −10 0 10 20 30 400

1

2

3

4

E(A

ng

le E

rr)

[deg

]

−40 −30 −20 −10 0 10 20 30 400

1

2

3


σ (A

ng

le E

rr)

[deg

]

Figure 14. Intercept angle error for various intercept angles against an evading target, with : a = 105, b = 108,γ = 7, ∆T = 2sec, ∆φ = [0, 1]

17 of 19


−40 −30 −20 −10 0 10 20 30 400

2

4

6x 10

5

E(∫u

2 dt)

[m

2 /s3 ]

−40 −30 −20 −10 0 10 20 30 400

5

10x 10

4


σ (∫u

2 dt)

[m

2 /s3 ]

Figure 15. Control effort for various intercept angles against an evading target, with : a = 105, b = 108, γ = 7,∆T = 2sec, ∆φ = [0, 1]

VII. Conclusions

In this paper a closed form solution of a differential games based guidance law, which enables imposinga predetermined terminal intercept angle, was developed. The navigation gains of the guidance law werestudied, and their behavior was analyzed for the perfect intercept and perfect intercept angle case. Theequations for the optimal trajectories in this case were derived, analytically solved, and plotted. A conjugatepoint analysis was also performed.

Even when the scenario was initiated with large deviations from the collision triangle and the targetperformed a hard maneuver, causing large deviations from an initial collision triangle, the guidance lawexhibited excellent performance by providing near zero miss distance and terminal intercept angle error. Toobtain this performance the missile must have a large maneuver advantage over the target.

The capability demonstrated by this guidance law to impose a terminal intercept angle can greatlyimprove warhead lethality, resulting in possible warhead size reduction. It may also decrease the target’sability to effectively employ countermeasures.

References

1Bryson, E. A. and Ho, C. Y., Applied Optimal Control , Blaisdell Publishing Company, Waltham, Massachusetts, 1969,pp. 154–155, 282–289.

2Ben-Asher, J. Z. and Yaesh, I., Advances in Missile Guidance Theory, Vol. 180, Progress in Astronautics and Aeronautics,AIAA, Reston, VA, 1998, pp. 25–90.

3Kim, M. and Grider, K. V., “Terminal Guidance for Impact Attitude Angle Constrained Flight Trajectories,” IEEETransactions on Aerospace and Electronic Systems, Vol. AES-9, No. 6, 1973, pp. 852–859.

4Ryoo, C. K., Cho, H., and Tahk, M. J., “Closed-Form Solutions of Optimal Guidance with Terminal Impact AngleConstraint,” Proc. IEEE Conf. on Control Applications, Vol. 1, Istanbul, Turkey, 2003, pp. 504–509.

5Ryoo, C. K., Cho, H., and Tahk, M. J., “Optimal Guidance Laws with Terminal Impact Angle Constraint,” Journal ofGuidance, Control, and Dynamics, Vol. 28, No. 4, 2005, pp. 724–732.

6Ryoo, C. K., Cho, H., and Tahk, M. J., “Time-to-Go Weighted Optimal Guidance With Impact Angle Constraints,”IEEE Transactions on Control System Technology, Vol. 14, No. 3, 2006, pp. 483–492.

7Song, T. L., Shin, S. J., and Cho, H., “Impact Angle Control for Planer Engagements,” IEEE Transactions on Aerospaceand Electronic Systems, Vol. 35, No. 4, 1999, pp. 1439–1444.

8Kim, B. S., Lee, J. G., and Han, H. S., “Biased PNG Law for Impact with Angular Constraint,” IEEE Transactions onAerospace and Electronic Systems, Vol. 34, No. 1, 1998, pp. 277–288.

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9Idan, M., Golan, O., and Guelman, M., “Optimal Planar Interception with Terminal Constraints,” Journal of Guidance,Control, and Dynamics, Vol. 18, No. 6, 1995, pp. 1273–1279.

10Friedland, B., Advanced Control System Design, Prentice-Hall, Englewood Clis, NJ, 1996, pp. 110–112.11Ohlmeyer, E. J. and Phillips, C. A., “Generalized Vector Explicit Guidance,” Journal of Guidance, Control, and Dy-

namics, Vol. 29, No. 2, 2006, pp. 261–268.

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