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Head Pursuit Guidance for Hypervelocity Interception

Oded M. Golan * and Tal Shima † RAFAEL, Haifa 31021, Israel

A new guidance law for intercepting targets in a novel head-pursuit engagement is presented. The guidance law positions the interceptor missile ahead of the target, on its flight trajectory, so that both fly in the same direction. The missile speed is planned to be lower than that of the target, and therefore the target closes in on the interceptor missile. Using this approach the closing speed is significantly reduced relative to a head-on engagement; compared to a tail-chase engagement, the low closing speed is achieved with reduced energy requirements. The guidance law is similar to deviated pure pursuit but with a time varying lead angle. Analytic solutions of the relative trajectories and interception envelopes are given for the case of a non-maneuvering target. The performance of the new guidance law is studied through simulation for the case of a maneuvering target and interceptor employing a continuous or bang-bang maneuver device. The implementation of the new guidance scheme may dramatically reduce the requirements from missile subsystems for the interception of high speed targets, such as ballistic missiles.

I. Introduction NTERCEPTION of ballistic missiles (BMs) is a formidable challenge. Several anti-ballistic missile systems (e.g. Arrow, THAAD and PAC-3) were developed in recent years for this task using intercepting missiles that are

launched from the protected territory against the incoming BMs. The interception in these scenarios is typically head-on, with a very high closing speed. This imposes severe requirements on the interceptor systems such as precise detection of the target from large distances by the onboard seekers, and very fast response time of the missile subsystems.

Recently a system for destroying BMs, flying predictable trajectories, was proposed1. In the suggested method, the interceptor is positioned ahead of the target, on its flight trajectory, so that both fly in the same direction. The missile speed is planned to be lower than that of the target, and therefore the target closes in on the precursor interceptor missile. Using this approach the closing speed is significantly reduced relative to a head-on engagement; compared to a tail-chase engagement, the low closing speed is achieved with reduced energy requirements.

Various classic guidance methods and their derivatives have been examined for implementation in the different stages of an exo-atmospheric interception of BMs. Some of these methods are described next. In Ref. 2 a modified version of proportional navigation (PN) guidance law3 was proposed for implementation in the coast phase. A variable bias was applied to the actual line-of-sight (LOS) to account for engine burn. The terminal guidance in a hypervelocity exo-atmospheric orbital interception was studied in Ref. 4. The control energy expenditure is reduced by constraining the expected final state to a function of the estimation error. An optimal guidance algorithm was proposed in Ref. 5 for the interception of a non-maneuvering target decelerated by atmospheric drag. Its implementation requires knowledge on many scenario states, obtained from a nonlinear state estimator. In a recent paper6 a differential game guidance law was proposed against targets having known speed and lateral acceleration limit profiles. It was shown that in a BM interception scenario such a guidance law provides a significant improvement in the homing accuracy compared to a guidance law derived based on a model with constant velocities and lateral acceleration limits.

To perform the final trajectory shaping and to ensure the required unique interception geometry outlined in Ref. 1 a guidance law that can impose a final interception geometry is sought. In Ref. 7 an optimal guidance law was developed for intercepting targets while imposing initial and final flight path angles. The algorithm requires the solution of a two point boundary value problem and thus can be implemented against targets with known trajectories. The simple guidance law of pure pursuit8 (PP) and its derivative deviated PP9 (DPP) can also impose a

* Chief Systems Engineer; odedgol@rafael.co.il; Senior Member AIAA. † System Engineer; shima_tal@yahoo.com; Senior Member AIAA.

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AIAA Guidance, Navigation, and Control Conference and Exhibit16 - 19 August 2004, Providence, Rhode Island

AIAA 2004-4885

Copyright © 2004 by O. M. Golan and T. Shima. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

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final interception geometry. In PP the interceptor is aimed at the target and it is intercepted from its rear10. The DPP guidance law is an extension of PP in that the missile is aimed at a constant lead angle to the target and the target is intercepted from a constant angle, dependent on the lead angle and velocity ratio9. Such guidance laws require minimal knowledge of the interception state variables and future target maneuvers. However, they impose severe maneuver requirements from the interceptor (especially PP) and require a velocity advantage of the interceptor. The guidance law proposed in this paper is similar to DPP but with a time varying lead angle. It enables performing the trajectory proposed in Ref. 1, also against maneuvering targets, while the interceptor has a velocity disadvantage.

The remainder of this paper is organized as follows: In the next section, the engagement is outlined and the new guidance law is derived. Then, the performance of the guidance law and the requirements on its parameters are analyzed analytically. This is followed by a simulation study of a scenario between a target performing constant maneuvers and an interceptor employing continuous or bang-bang maneuver devices. Concluding remarks are offered in the last section.

II. Head pursuit guidance law A schematic view of the proposed engagement outline is shown in Figure 1. The interceptor trajectory is shown

to have three stages: first stage for approach, second for trajectory bending and third, endgame, in which the kill vehicle (KV) conducts final corrective maneuvers. The missile is launched towards the BM in a head-on trajectory, and at a predetermined time is steered to bend its flight trajectory until reaching a so-called trajectory matching flight mode. The drawing illustrates exo-atmospheric interception in which thrust is used in order to achieve trajectory bending and end-game maneuvers.

In the beginning of the endgame phase the interceptor is flying close to the target predicted trajectory at a lower speed. The objective of the endgame guidance is to correct the position and flight direction errors. Note the unconventional final geometry in which the target approaches the interceptor from its rear end. Such geometry also relaxes the requirements from the interceptor seeker dome since it is not exposed to the high aerodynamic heating. However, it requires special adaptation of the kill mechanism.

Figure 1: Schematic view of a BM interception engagement.

A roll controlled interceptor is considered. For the relatively short time interval of the endgame (with small changes in the flight direction) the motion of such an interceptor can be separated into two perpendicular channels and the guidance problem can be treated as planar in each of those channels. The planar endgame geometry is shown in Figure 2. The target T is located behind the slower interceptor I . The velocity, the maneuvering acceleration,

APPROACH PHASE

TRAJECTORY BENDING PHASE

ENDGAME PHASE

stage I separation

stage II separation

KV

target

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and the flight path angles are denoted by V , a and γ , respectively; the range between the target and interceptor is r , and λ is the LOS angle relative to a fixed reference. The angle θ is the instantaneous target direction of flight relative to the LOS. This angle is varying even when the target is non-maneuvering due to the interceptor motion. The angle δ , the interceptor flight direction relative to the LOS, is the guidance control variable.

Figure 2: Planar engagement geometry.

The engagement kinematics is expressed in a polar coordinate system ( ),r λ attached to the target

cos cos

( sin sin ) /I T

I T

r V V

V V r

δ θλ δ θ

= − = −. (1)

We assume constant speeds and define the non-dimensional parameter K as the speed ratio

/ 1I TK V V < . (2) Note that we design the guidance law specifically for a scenario in which the target has a speed advantage.

It is required that at interception ( 0r = ) both interceptor and target will fly in the same direction, hence

lim0

0r

θ→= , (3)

and

lim0

0r

δ→= . (4)

The objective of the head pursuit (HP) guidance law is to bring the precursor interceptor to the interception point such that Eqs. (3) and (4) hold. Since the interceptor is slower than the target, this final geometry is achieved when the target head approaches the interceptor tail, hence the name of the suggested law.

The proposed HP guidance law is similar to DPP but with the lead angle δ being time dependent. Specifically, the interceptor lead angle is required to be proportional to the target flight direction relative to the LOS:

nδ θ= . (5) Note that this relation guarantees that δ vanishes with θ .

Implementation of the HP guidance law in a missile with realistic maneuver dynamics and limits requires more consideration. It is necessary to find the relation between the angular condition of Eq. (5) and the interceptor acceleration.

For aerodynamically controlled vehicles, the maneuvering acceleration is roughly normal to the velocity vector. Hence, /a Vγ = , and differentiating Eq. (5) it is straight forward to show that

/ ( )1I T Tna V nγ λ= − − . (6)

•

•

r

λT

θ

λ

Iγ λ δ= +

I

δ

Tγ λ θ= +

IV

TV

Ta

Ia

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The first term accounts for the target maneuver and the second for the line of sight dynamics. In order to account for initial heading errors, an additional term is included following the suggestions in Ref. 9. The new term is an odd function of the heading error, weighted by a monotonic decreasing function of the range. Thus, the implementable HP guidance law is:

( ) sin( ) /21I T I Ia nKa n V V n rλ κ δ θ= − − − − . (7) It requires estimation of the target acceleration; target and interceptor velocity vectors; and relative position. The

required information can be obtained using standard onboard sensors. For an exo-atmospheric interceptor the maneuvering acceleration must be obtained by thrust. The thrust can be

exerted in any desired direction, for example normal to the interceptor, normal to the LOS, or normal to the instantaneous predicted target trajectory. We examine a case where the thrust is perpendicular to the missile velocity vector. Since rocket engines usually perform best in specified nominal thrust levels, a bang-bang implementation of the HP guidance law is required. One form of such an implementation is

0 sign( ) if0 otherwiseI

a n n Da

δ θ δ θ− − − >=

(8)

where 0a is the nominal thrust acceleration, and D is a dead zone introduced in order to limit the fuel expenditure. Note that this form of the HP guidance law requires less information than the continuous version, but still needs the estimation of the target and interceptor velocity vectors and LOS direction.

III. Ideal interception scenario In this section we address an ideal scenario with no target maneuvers and interceptor maneuver dynamics and

bounds. Hence, the performance of the HP guidance law given in Eq. (5) is analyzed. The relationship between the guidance constant n , the interceptor and target speed ratio K , and the initial conditions, is studied.

Lemma 1: Against a non-maneuvering target, a necessary condition for performing the HP interception is

/1n K> . (9)

Proof: The non-maneuvering target path angle Tγ is constant and hence λ θ=− . For HP interception the following relation must hold at 0r →

sgn( ) sgn( )θ θ=− . (10) Close to interception the angles δ and θ are small and therefore from Eq. (1) we obtain

( )1TV Knr

λ θ≈ − , (11)

and using the relationship λ θ=−

( )1TV Knr

θ θ≈− − . (12)

Using Eq. (12), for Eq. (10) to hold, the relationship given in Eq. (9) must hold and the Lemma is proved. □

Lemma 2: Against a non-maneuvering target, and given /1n K> , the sufficient condition for performing the HP interception is

3161

KnKn

θ −<−

. (13)

Proof: By the relationship λ θ=− and Eqs. (1), (5)

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( )sin sin /TV K n rθ θ θ=− − . (14)

Using a 4th order approximation of the sine function we obtain

( ) ( ) / /2 31 1 6TV Kn Kn rθ θ θ =− − − − +∆ , (15)

where ∆ is a nonnegative term, dependent on the error in the sine series expansion and ,K n . Since 1n > and using the condition given in Eq. (13), θ is a monotonic function. For 0r → Eq. (3), and consequently Eq. (4),

hold. Since 0∆≥ , the condition given in Eq. (13) is sufficient and the Lemma is proved. □

Based on Eq. (13) the solution of

arg ( ) /( )36 1 1 0mx Kn Knθ

θ θ = − − − = , (16)

is plotted in Figure 3 for different values of K . The minimum value of n agrees with the condition given in Eq. (9). Note that for every speed ratio K there is an optimal value of n that guarantees a maximum angle mxθ .

Figure 3: Conditions for HP interception; 4th order approximation.

Since, as stated in Lemma 2, the condition given in Eq. (13) is sufficient, the true value of mxθ is larger. For comparison, a solution of the 6th order series is also examined, and the results are plotted in Figure 4 . It can be observed that for the same values of n and K a higher value of mxθ is achieved.

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Figure 4: Conditions for HP interception; 6th order approximation.

For a non-maneuvering target, the equations of motion can be solved in the polar ( ),r λ coordinates. From Eq. (1) we obtain:

cos cossin sin

I T

I T

dr V V rd V V

δ θλ δ θ

−=−

. (17)

Substituting the HP guidance law of Eq. (5) and rearranging

( )dr f dr

θ λ= , (18)

where

cos cos( )sin sin

K nfK n

θ θθθ θ−=− . (19)

Using the relationship Tγ θ λ= + and assuming a non-maneuvering target, i.e. T constγ = , we can write

( )dr f d

rθ θ=−

. (20)

By direct integration of Eq. (20) we obtain

( ) ( ){ }exp 0

0

r g gr

θ θ= −, (21)

where

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g f dθ∫ . (22)

For some values of n an analytical solution of Eq. (22) can be obtained. For example, for 2n = the solution is:

( ) ( )

( ) ( )

ln ln

ln

2

22

2

112 1

2 2 1 2 14 1

KgK

K K KK

θ ψ ψ

ψ

−= + −−

− + − − − , (23)

and for 3n = it is:

( ) ( )

( ) ( )

ln ln

ln

2

4 2

113 1

3 1 2 5 1 33 1

KgK

K K K KK

θ ψ ψ

ψ ψ

−= + −−

− − − + + − , (24)

where

( )tan /2ψ θ . (25)

Solutions of Eq. (21) are shown in Figure 5 for 2n = and different values of K . It can be observed that at the limiting case when 1nK → interception is achieved but the required directional conditions are obtained only close to interception. As the speed ratio is increased, the easier it becomes for the interceptor to position itself ahead of the target and reach the desired interception conditions.

Figure 5: Trajectories in target coordinate system; 2n = .

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IV. Non-ideal interception scenario

In this section a scenario with a maneuvering target is studied using simulations. Two different interceptors are examined: one with a continuous unbounded controller and the other employing a bang-bang maneuver device. In both cases perfect information is assumed.

The performance of the implementable HP guidance law of Eq. (7) is investigated for the continuous case using the simulation parameters summarized in Table 1.

Table 1: Simulation parameters – continuous controller

Kinematics Guidance Initial conditions //

[ , , ]

1600190020 0 20

I

T

T

V m sV m sa g

=== −

23

nκ==

0

0

0

3

20

20

r km

θδ

=

=−=−

Three different scenarios are investigated: a non-maneuvering target, and targets performing constant maneuvers of 20g or 20g− . Figure 6 presents the interceptor and target trajectories in an Inertial coordinate system. Despite the large initial heading error of 20 degrees, the interceptor gradually approaches the target future trajectory and tracks it until it is caught up by the target. In Figure 7 the interceptor relative trajectories in non-rotating polar coordinates attached to the target are shown. It can be seen that even in the presence of target maneuvers and initial heading errors interception is achieved with 0θ = , as required. The missile acceleration profiles are plotted in Figure 8. It can be observed that the acceleration approaches that of the target as the two vehicles come nearer. In the case of a maneuvering target, the acceleration difference at interception is due to the different speeds and turning radius of the two vehicles. The difference in the scenario duration, due to the different interception trajectories, is also evident.

Figure 6: Inertial Trajectories; continuous unbounded interceptor controller.

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Figure 7: Relative trajectories in target non-rotating polar coordinates; continuous unbounded interceptor

controller.

Figure 8: Missile acceleration profile; continuous unbounded interceptor controller.

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Next, the bang-bang controller of Eq. (8) is applied. We examine the same scenario as in the continuous case. The simulation parameters are summarized in Table 2.

Table 2: Simulation parameters – bang-bang controller

Kinematics Guidance Initial conditions //

160019000

I

T

T

V m sV m sa g

===

0

220505

na gD mrad

msτ

====

0

0

0

3

20

20

r km

θδ

=

=−=−

0a is the thrust acceleration level, D is the dead zone and τ is a first order time constant representing delays in turning the thrust on and off. We compare the performance of the bang-bang and the continuous controllers.

Figure 9 presents the interceptor and target trajectories in an inertial coordinate system. As for the case with continuous unbounded controller, despite the large initial heading error of 20 degrees relative to the direction required by the guidance law, the interceptor performs the HP interception. The radius of turn for the bang-bang controlled interceptor is larger due to the saturated acceleration. Yet, as shown in Figure 10 interception is achieved with a very small heading error. The error in θ , due to the dead zone in the control logic and the time delay, represents the design trade-offs between system cost and performance. The acceleration profiles are plotted in Figure 11. The bang-bang nature of the maneuvers is immediately evident. The maneuver starts with initial thrusting in the negative direction. Note the longer time required in the saturated case to turn the interceptor to the correct flight direction towards the target's future trajectory. This is followed by free flight, and finally positive thrusting in order to turn the flight direction back and align it with the target's path. When the desired flight direction is reached, the controller maintains the flight course by short corrections, each time the directional error exceeds the imposed limit. This results in high frequency chattering that is unavoidable with this kind of on-off control.

Figure 9: Inertial Trajectories; continuous and bang-bang interceptor controllers.

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Figure 10: Relative trajectories in target polar coordinates; continuous and bang-bang interceptor

controllers.

Figure 11: Missile acceleration profile; continuous and bang-bang interceptor controllers.

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The bang-bang controller behavior is similar for the other scenarios that were investigated for the continuous controller, where the target is maneuvering. It should be noted however, that the interceptor should have higher maneuvering capability in order to compensate for the target's maneuver.

V. Conclusions A new guidance law for intercepting targets in a novel head-pursuit engagement was presented. The guidance

law has a similar form to the well known deviated pure pursuit but with a time varying lead angle. Analytic solutions for the case of a non-maneuvering target were presented and bounds on the guidance constant and the intercept envelope were determined. Performance was demonstrated via simulation for the interception of a maneuvering target and initial heading errors. The results indicate the feasibility of the proposed engagement using the precursor interceptor.

The implementation of the new guidance scheme against high speed targets reduces considerably the closing speed relative to a head-on engagement. Thus, the stringent requirements imposed on the onboard seekers can be relaxed. Compared to a tail-chase engagement, the same low closing speed can be achieved with reduced energy requirements because of the planned lower speed of the interceptor.

Further study of the robustness of the HP guidance law and analysis of its efficiency compared to other possible guidance laws, will be the subject of future work.

VI. References 1Golan O.M., Rom H. and Yehezkely O., “System for Destroying Ballistic Missiles” US Patent No. 6,209,820 B1, April 3,

2001. 2Zes, D., "Exo-Atmospheric Intercept Using Modified Proportional Guidance with Gravity Correction for Coast Phase",

AIAA 32nd Aerospace Sciences Meeting, Reno NV, Jan. 1994, paper 94-0209. 3Zarchan, P., Tactical and Strategic Missile Guidance, Progress in Astronautics and Aeronautics, Vol. 176, AIAA Inc.,

Washington D.C., 1997. 4Alfano, S. and Fosha Jr., C. E., "Hypervelocity Orbital Intercept Guidance Using Certainty Control", AIAA Journal of

Guidance, Control, and Dynamics, Vol. 14, No. 3, 1991, pp. 574-580. 5Hough, M. E., "Optimal Guidance and Nonlinear Estimation for Interception of Decelerating Targets", AIAA Journal of

Guidance Control and Dynamics, Vol. 18, No. 2, 1995, pp. 316-324. 6Shima, T. and Shinar, J., “Time Varying Pursuit Evasion Game Models with bounded Controls”, AIAA Journal of Guidance,

Control, and Dynamics, Vol. 25, No. 3, 2002, pp. 425-432. 7Idan M., Golan O.M. and Guelman M., “Optimal Planar Guidance Laws with Terminal Constraints”, AIAA Journal of

Guidance, Control and Dynamics, Vol. 18, No. 6, 1995, pp. 1273-1279. 8Yuan, L. C.-L., "Homing and Navigational Courses of Automatic Target Seeking Devices", Journal of Applied Physics, Vol.

19, 1948, pp. 1122-1128. 9Shneydor N. A., Missile Guidance and Pursuit – Kinematics, Dynamics and Control, Series in Engineering Science,

Horwood Publishing, England, 1998. 10Bruckstein, A. M., "Why the Ants Trails Look so Straight and Nice", The Mathematical Intelligencer, Vol. 15, No. 2, 1993,

pp. 59-62.