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1 American Institute of Aeronautics and Astronautics

Trajectory-Shape-Varying Missile Guidance for Interception of Ballistic Missiles during the Boost Phase

John A. Lukacs, IV* Defense Technical Information Center, Ft. Belvoir, VA 22060-6218

Oleg A. Yakimenko† Naval Postgraduate School, Monterey, CA 93943-5107

The paper presents a near-optimal guidance law that has been developed using the direct method of calculus of variations. By the direct access to controlling the shape of the trajectory, this guidance law seeks to maximize the kinetic energy transfer upon interception from a surface launched medium range interceptor missile to a ballistic missile target during the boost phase of flight. Mathematical models of a two-stage liquid-fueled medium-range ballistic missile and a velocity-limited endo-atmospheric interceptor missile with onboard active radar guidance are used to demonstrate the guidance law’s performance. This law will utilize the interceptor's onboard computer and active radar sensors to independently predict an intercept point, solve the two-point boundary-value problem, and determine the best feasible flight path to that point. While determining a truly optimal flight path would require significant computing power and therefore cannot be realized online, the proposed near-optimal flight path can be calculated onboard the interceptor in fractions of a second and updated in multiple times during the intercept without significant technological advancements in the interceptor’s onboard hardware. That near-optimal guidance path is then converted into a set of command functions and fed back into the control system of the interceptor. By modifying just a few parameters affecting the higher-order derivatives at the intercept point, the optimization algorithm varies the shape of a three-dimensional trajectory over a wide range without compromising the constraints on controls or jeopardizing satisfaction of the final conditions. An example features the guidance solution for probably the most difficult scenario of maximizing kinetic energy upon impact by forcing it to occur at a right angle.

Abbreviations BM = Ballistic Missile BPI = Boost Phase Intercept DoF = Degree-of-Freedom ESG = Engagement Sequence Groups ICBM = Intercontinental Ballistic Missile LoS = Line of Sight LQ = Liner Quadratic PF = Penalty Function PI = Performance Index PN = Proportional Navigation UAV = Unmanned Air Vehicle

I. Introduction OR many years now, the United States Department of Defense has expended great effort to develop an integrated intercontinental ballistic missile (ICBM) defense system through a layered, defense in depth strategy.

An ICBM’s speed and altitude leave little room for error by the defender, and any strategy must include systems capable of defeating a ballistic missile (BM) at each of its three distinct phases – boost, midcourse, and terminal - which the Missile Defense Agency labels “Engagement Sequence Groups” (ESG) as shown in Fig.1. * Lieutenant, United States Navy, [email protected]. AIAA Student Member. † Research Associate Professor, Department of Mechanical and Astronautical Engineering, Code MAE/Yk, [email protected], AIAA Associate Fellow.

F

AIAA Guidance, Navigation and Control Conference and Exhibit20 - 23 August 2007, Hilton Head, South Carolina

AIAA 2007-6538

Copyright © 2007 by Oleg Yakimenko and John Lukacs. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.


It is the portion of the effort directed toward the boost phase of the Ballistic Missile Defense Programs that is the focus this paper. The boost phase ESG is concerned with developing methods and technologies to conduct Boost Phase Intercept (BPI). Intercepting a missile in its boost phase is the ideal solution for a ballistic missile defense, since the missile is very vulnerable during this phase of its flight. The missile is relatively slow while struggling to overcome gravity, has a very visible exhaust plume, and cannot deploy countermeasures.1,2 Yet the challenges needing to be overcome are immense: countering the large and changing acceleration rates, reliable scanning and tracking, and very short reaction time being among the most daunting.1,2 A variety of weapon systems are under development for conducting BPI, including airborne lasers, space-based intercept missiles, and ground-based intercept missiles. None of these systems is totally operational, though several look promising.

A missile’s guidance law is one of the largest single factors affecting its ability to intercept a target. Yet, when discussing mission success in the BPI ESG, intercepting the target is only one factor; another major consideration is the ability to kill the target, i.e., using the available kinetic energy as effectively as possible.3-5 Early ballistic missile defense concepts recognized that simple warhead effects are not sufficient to destroy an ICBM and initiated development of hit-to-kill technologies.6,7 The relative sizes of a nominal ICBM and an interceptor means that the interception must maximize the kinetic energy transferred to the ICBM in order to be effective, which suggests the need to control the geometry of the interception. As it will be shown in the next section, current guidance laws do not address this aspect, leaving the actual intercept geometry to be the result of the guidance law and the relative capabilities of the missiles instead of an input into the guidance law. This is a reasonable course of action when all that is necessary to kill the target is to get the missile within the limits of the proximity fuse the case with most surface-to-air engagements. It breaks down, however, when dealing with an ICBM. The desire for hit-to-kill end game conditions, coupled with the need to maximize the kinetic energy transfer, means the intercept geometry cannot be left to chance and must be controlled as an input to the guidance law.

Obviously, the actual missile guidance laws are never disclosed. However, the basic ideas behind the guidance laws of the vast majority of interceptors are well known.8-15 The objective of this paper is to design a guidance law from scratch, casting the problem as an optimal control problem with the objective of generating the interceptor’s entire flight path rather than synthesizing the instantaneous optimal control based on the estimate of the intercept time. This near-optimal flight path to interception should minimize the distance traveled and the time to intercept, simultaneously maximizing kinetic energy transfer by controlling the intercept geometry. This will be done by utilizing the direct method of calculus of variations combined with inverse dynamics technique (rather than relying on feedback loop closed with the “optimal” control synthesized by solving the linear quadratic problem) to produce a near-optimal flight path using the missile’s onboard sensors and computers in real time.16

The paper is organized as follows. Section II reviews the existing guidance laws, emphasizing the progress from the classical proportional-navigation type laws to modern trajectory-shaping guidance laws, and evaluating their potential effectiveness at a BPI (although they were specifically designed for the midcourse and terminal intercept). Before addressing the development of a new guidance law, Section III deals with the simulation models for typical ICBM target and interceptor missile, concentrating on kinematic three Degree-of-Freedom (DoF) models (Ref.17 contains more sophisticated 6DoF for future research to capitalize upon). Section IV develops and describes the essence of a new direct-method-based guidance law continuously calculated onboard the missile as a complete solution of a two-point boundary-value problem, followed by Section V presenting some simulation results and discussing the feasibility of employing such methods in a real-world scenario. The paper ends with conclusions.

II. Review of the Existing Guidance Laws There are a wide variety of guidance laws available for missile guidance processing. Many of them, such as Beam Rider

and Pure Pursuit, are not acceptable for this particular application at all, while others, Proportional Navigation and its variants, might be acceptable, but they do not assure unconditional satisfaction of the final conditions and do not address trajectory shaping at all. So-called explicit and kappa guidance laws do address the geometry at impact though they still do not solve two-point boundary-value problem but rather exploit the optimal control law synthesized from maximization of the Hamiltonian for the simplified (linearized) system. The following provides more details on these classical and modern control laws.

Figure 1. Engagement Sequence Groups.1


A. Classical Guidance Laws Beam Rider (also called Command Line-of-Sight) is among the simplest forms of command guidance. The missile simply

flies along a tracker beam that is continuously pointed at the target. In other words, the inertial velocity vector is continuously pointed at the target. The beam and the missile generally originate from the same position, but not always. The guidance commands are proportional to the angular error between the missile and the beam, and if the missile remains exactly on the beam it will hit the target. There is no consideration for the capabilities of the target, and the missile is not directed to orient itself to lead the target in order to anticipate its movements. The missile therefore requires much greater maneuvering capabilities in the moments just prior to interception. This form of guidance law also assumes that the shooter will have a direct line of sight (LoS) for the entire engagement, and that the diffraction of the beam will be negligible, which is obviously not true over the distances required for a BPI.6

Pure Pursuit overcomes this limitation by having the missile supply the targeting data instead of a third-party observer. This is, in effect, Beam Rider Guidance, where the beam is generated onboard the missile. As before, the missile only follows the beam and the guidance commands are proportional to the angular error between the missile and the LoS to the target. If x and V denote the vectors of interceptor’s current position and velocity, and Tx denotes the vector of current position of a target, then the three-dimensional acceleration command can be expressed as (to make the text flow, hereinafter we might slightly change the notations used in the original source)

( )1~ cos ( , )V LOS V LOS V− × ×a u u u u u . (1)

In Eq.(1) 1V V −=u V (V = V ) and 1( )LOS T T

−= − −u x x x x are the corresponding unit vectors. The expression inside the module on the right-hand side of Eq.(1) is proportional to the angular error between the vector V and LoS, and the double cross product defines the direction of acceleration to be perpendicular to the vector V and lying in the plane including this vector and LoS. This overcomes the diffraction of the beam, but still does not command the missile to orient itself to lead the target. It thus requires similarly large maneuvering capabilities to complete the interception.11,12

Both of aforementioned guidance laws are severely limited in their effectiveness, are used only at short range against non-maneuvering or relatively slow targets, and require significant interceptor capabilities relative to the target. Obviously, this is not an accurate description of the requirements for a BPI.6

Another family of guidance laws involves Proportional Navigation (PN) and its derivatives. Of the three basic guidance laws addressed so far, proportional navigation is the most versatile, and therefore most frequently implemented, making it the guidance law of choice in nearly all modern guided missiles. Using angular velocity vector of the LoS LOSω and a unit vector Vu , a three-dimensional pure PN guidance law becomes

LOSVNV= ×a ω u . (2)

In Eq.(2) the cross product is again used to assure that the acceleration vector a is normal to the vectors V and LOSω , and the proportionality constant N determines the amount of interceptor lead with respect to the target (usually N=2…5 depending on the interceptor’s maneuvering capabilities). The guidance system always tries to point the differential (relative) velocity vector at the target (maintaining the “engagement triangle”).10,14

Corrected for the gravity g and interceptor’s longitudinal acceleration/deceleration projected into the LoS plane longa , the three-dimensional PN guidance law (3) becomes

LOSlongNV= − −a ω u a g (3)

(such a guidance law is termed compensated PN).14

The acceleration vector a can be further split into two controls, for instance expressed in the interceptors wind frame {w} as ( ) 0 1 01 ( )( ) 0 0 1

y wi

z

n tR t

n t g⎡ ⎤ ⎡ ⎤

=⎢ ⎥ ⎢ ⎥−⎣ ⎦⎣ ⎦a (4)

(here wi R is the rotation matrix that converts a vector from the inertial coordinate frame {i} to {w}). The missile

seeker measures the LoS rate and the PN guidance law converts that into an acceleration command by the guidance computer.

By definition guidance systems are feedback systems and there is no general closed-form solution to PN guidance equations (since the motion of the target is not known). Hence, the big question is in determining proportionality constant N providing the best performance (the smallest miss distance or the largest probability of kill). Obviously, it depends on the geometry of engagement, relative speeds, target maneuvering, and time to intercept (time-to-go). It can be shown11 that N=1 converges the PN guidance equation (2) to that of pure pursuit (1), and N→∞ ( 0LOS →ω ) leads to parallel navigation, which is optimal for nonmaneuvering targets ( T const=V → const=V ). A lot of efforts has been made to make this constant variable and adaptive to the current engagement parameters: relative distance, speed (energy), orientation, maneuverability, etc.


(for example, see Refs.12,15 and references therein). To this end another variation of PN, the augmented PN tries to accommodate an extra term to account for the target’s acceleration aT

10

2LOS

T longNNV= + − −a ω u a a g . (5)

(In our case, although a booster does not execute evasive maneuvers, any longitudinal acceleration, perpendicular to the LoS will appear as a target maneuver.10) Of course, that implies a detailed knowledge of target parameters, specifically the target’s acceleration, which can be derived from an advanced Kalman filter, such as an Extended Kalman Filter or an Alpha-Beta-Gamma Filter.11 These may not always be sufficiently accurate, especially during the boosting phase, which features staging events, where the acceleration values change radically.10,18 In addition, there are several more reasons why the guidance laws based on Eqs.(1)-(5) might fail: variable speed, time lags in system’s dynamics, saturation of the control surfaces’ capabilities (components of the normal acceleration (4)), etc.

Unlike the classical laws (1)-(5), optimal-control-theory-based laws rely on more detailed dynamics of both the interceptor and target. They cast the intercept problem as an optimization problem of minimizing a performance index (PI)

0

1 12 2( ) ( )

ftT T

Tf f Tf ft

J dt= − − + ∫x x S x x a Ra . (6)

subject to dynamic constraint T T

T T

− −⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤= +⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥− − −⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦

x x x x0 I 0a

V V V V0 0 I& &

& & . (7)

The idea is to minimize the terminal miss distance (the first term in Eq.(6) with the weighting matrix S) and limit control efforts or indirectly time-to-go 0go ft t t= − (the second term with the weighting matrix R).

The synthesis of the optimal control for the linear dynamics of the miss distance when the target is not maneuvering, obtained by Bryson,19 yielded

3

3( ) ( , )

3go

gogo

tt t t

t=

+a m

R, (8)

where ( )( , ) ( ) ( ) ( ) ( )go T T got t t t t t t= − + −m x x V V (9)

is an estimate of the zero-effort miss distance. If 0=R , i.e. there is no control-effort limit (unconstrained optimization), then Eq.(8) converges to the proportional navigation law of Eq.(2) with N=3 (with the near-collision course assumptions)

3

3( ) ( , )gogo

t t tt

=a m . (10)

Therefore, the optimal guidance law from the standpoint of minimizing the control effort is simply augmented PN with the proportionality constant N equal to 3 (this value resulted from the fact that it is a quadratic form of the acceleration a appearing under the integral in Eq.(6)). For the maneuvering target, i.e. for the dynamic constraint

T T

T T

T T

− −⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥− = − + −⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦

x x 0 I 0 x x 0V V 0 0 I V V I a

a 0 0 0 a 0

& &

& &

&

(11)

an estimate of the miss distance includes the target acceleration aT 10,12,15

( ) 212( , ) ( ) ( ) ( ) ( ) ( )go T T go T got t t t t t t t t= − + − +m x x V V a . (12)

(that’s how it gets into the augmented PN (5)). For the highly maneuverable targets, further modifications assume a certain evasive maneuver so that even jerk can be accounted for in estimating the zero-effort miss distance

( , )got tm .12

In their problem formulation and derivation Bryson and Ho used implicit and explicit assumptions necessary in order for the solution to result in PN.15 The optimality of PN is therefore dependent on the deviation from the real-world implementation model and those assumptions. The performance of the PN-based guidance laws is obviously sensitive to those deviations. Also it depends on the PI, control constraints (saturation), terminal constraints, assumptions on the availability of the target acceleration information, system dynamics, etc.

Hundreds of papers were developed to address these issues by reformulating the optimal control problem (using another model, PI, etc.). For example, some papers addressed the issue of interceptor dynamics in the form of the


first-, second- and higher-order lag in the guidance law (10),(12).12,13 Specifically, for the time lag represented by a linear first-order system with the time constant T

11c sT

=+

aa

, (13)

(where a is the achieved missile acceleration, and ca is the commanded missile acceleration), or the dynamic constraint

1 1

T T

T Tc

T T

T T

− −⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥− −− −⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥= +⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥−⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦

0 I 0 0 0x x x x0 0 I I IV V V V

a0 0 0 I 0a a0 0 0 I Ia a

& &

& &

&

&

, (14)

the original zero-lag guidance law (10),(12) transfers to the new one by simply adjusting the proportionality constant N and adding an additional term:

( )( )2 2

1( ) ( ) ( ) ( ) ( ) ( ) ( )

2

gotgo

c T T go T gogo go

e tN Nt t t t t t t f tt t

− + −′ ′= − + − + −a x x V V a a (15)

where 4

23 2

6 ( )

2 6 6 (1 2 ) 3(1 )go go

go got t

go go go

t f tN

t t t e e− −′ =

− + − + −, 2

1( )

gotgo

gogo

e tf t

t

− + −= , and 1

go got t T −= ( lim 3got

N→∞

′ = ).10,12,13,20

Along with the optimal control theory the same results can be obtained by using the Schwartz inequality.10,21 A more general approach deals with the system dynamics described by the state vector z and controlled by the vector u in the following linearized form (which would be very difficult to define for the highly nonlinear kinematics and aerodynamics of a BPI)

( ) ( ) ( ) ( ) ( )t t t t t= +z F z G u& . (16) Then for the quadratic PI similar to (6)

( )0

1 12 2

ftT T Tf f

t

J dt= + +∫z Sz z Qz u Ru (17)

the optimal control for this linear quadratic (LQ) problem is found to be ( , ) ( )Tt t= −u x RG Px (18)

where P is the solution of the Riccati nonlinear differential equation10,12,13,22 1 T−− = + − +P PF FP PGR G P Q& (19)

subject to ( )ft =P S . Under the same conditions this approach also yields the PN solution obtained by Bryson, i.e. Eqs.(10),(9)).

Many optimal guidance problems are reducible to special cases of LQ differential zero-sum games.13,23 The result is still of the form (2) with some optimal gain 2 13(1 )N γ − −= − accounting for the relative maneuverability γ.19,13,24

The specifics of the synthesized control (10),(9) and other similar controls obtained via of the classical optimal control theory (employing indirect methods) is that even for the simplified linear model they do not solve the boundary problem (do not assure a zero miss distance), but rather express the current control command via the unknown time-to-go got . A number of papers therefore addressed the issue of estimating got 12,13,10,24-26 or making the solution less sensitive to it.24

B. Trajectory-Shaping Guidance Laws Along with hitting the target and spending less control effort, it might be also desirable to shape the missile

trajectory near impact, which none of the guidance laws considered so far address explicitly. This problem is similar to that of shaping the trajectory for a spacecraft to meet certain final conditions on speed or approach angle. Hence, a significant body of papers has addressed this issue as well.22-23,27-43

Specifically, the optimal control theory (Schwartz inequality) applied to the problem of minimizing the PI

0

12

ftT

t

J dt= ∫ a Ra . (20)


subject to the dynamic constraint (11) and boundary conditions ( ) ( ) 0T f ft t− =x x and ( ) ( )T f f ft t− =V V V , where

fV determines the final geometry, yields10

( )( ) ( )2 2

4 ( ) ( ) ( ) ( ) 2 ( ) ( )( ) ( )T T go T f go

c Tgo go

t t t t t t t tt t

t t

− + − − += + +

x x V V x x Va a (21)

As seen, the trajectory-shaping guidance law appears to be very similar to the augmented PN law (10),(12) (with N=4 and the doubled multiplier for the target acceleration term doubled) plus an additional term, proportional to the difference between the true LoS and the desired LoS at impact.

We may slightly regroup Eq.(21) and, assuming a non-maneuvering target, obtain another form of trajectory-shaping guidance law:

( )( ) ( )( )2

6 ( ) ( ) ( ) ( ) 2 ( ) ( )( ) T T go f T

cgogo

t t t t t t tt

tt

− + − − −= +

x x V V V V Va . (22)

If 0f =V , and ( ) ( ) 0T Tt t= =x V , Eq.(22) simplifies to the guidance law used to land the Apollo on the moon:10

2

6 ( ) 4 ( )( ) go

cgo

t t tt

t+

= −x V

a (23)

The same result was obtained by Cherry27 even without optimal control theory. He simply assumed the following acceleration profile of the vehicle

1 2( ) ( )c got c c t t= + −a (24) and determined the two coefficients that satisfy the terminal conditions on the coordinate and velocity vectors assuming known got . For c2=0, he ended up with Eq.(22).

Considering the LQ problem (17) in the vertical plane with the scalar analog of the dynamic constraint (11) including the impact angle θ explicitly, Kim and Grider28 ended up having a guidance law in the form of

1 2 3( ) ( )( ( ) ( )) ( )( ( ) ( )) ( )( ( ) ( ))c go T go T go Ta t f t x t x t f t V t V t f t t tθ θ= − + − + − (25) where fi, i=1,2,3 are some nonlinear functions of time-to-go. (Attempting to cope with a missile deceleration due to maneuvering, the authors of Ref.29 included even more terms but again premultiplied by some functions of got .) Making some additional assumptions, their followers,30 showed that Eq.(25) indeed converges to Eq.(22). However, they stated that in order to allow more aggressive maneuvering in the initial phase (when the near-collision course assumptions used to derive Eqs.(21) and (22) are not applicable) and avoid excessive maneuvering at the end of the intercept, the time-varying navigation constants should be used instead of firm 6 and 2 in Eq.(22) with a lager value at the initial phase and a smaller value at the final phase. They suggested them to be the functions of the current engagement geometry, i.e. range to impact point R to the target and range closing rate R& .30

Following the Cherry’s approach, Lin and Tsai also claimed that the coefficients of 6 and 2 in Eq.(22) are not optimal and made them dependent on the range R and vector of missile parameters p to accommodate the important constraints on the angle-of-attack and sideslip-angle31,11

( )( ) ( )( )1 2

2

( , ) ( ) ( ) ( ) ( ) ( , ) ( ) ( )( ) T T go f T

cgogo

k R t t t t t k R t tt

tt

− + − − −= +

p x x V V p V V Va (26)

They chose to maximize the terminal speed rather than to minimize the control acceleration as in Eq.(20) and apparently achieved 40% improvement compared to the law of Eq.(22).

Ohlmeyer32,33 also attempted to restrict the control actions at the end of the trajectory. In order to do this he replaced the PI (20) with

0

12 ( )

ft T

nft

J dtt t

=−∫

a Ra . (27)

That allowed him to express the varying coefficients in Eq.(26) as 1 ( 2)( 3)k n n= + + and 2 ( 1)( 2)k n n= + + , that converge to 6 and 2 when n→0.

Following Cherry,27 all guidance laws which express the formulas for the steering commands directly in terms of the current and desired terminal values of the components of the position and velocity vectors (Eqs.(8),(10),(15),(21)-(23),(26)) are referred to as Explicit guidance laws or E-guidance laws. The further development of the E-guidance law (26) led to another approach, kappa-guidance that exploits the curvature and torsion along the intercept trajectory parameterized along the arc length or range to the predicted impact point.34-40


Starting from solving the LQ regulator problem for the linearized kappa-guidance dynamic equations in the vertical plane,34 later extended to the three-dimensional (3D) case,36 it led to the interesting case of a 3D guidance law based on the involute of the target’s trajectory assuring maximum terminal speed and a 90-degree impact angle for the target performing the constant-rate flat turn.40 The motivation behind switching from controlling accelerations to controlling geometric parameters of the 3D curve was to gain more control of the shape of the trajectory directly (meaning that the guidance laws (21)-(23),(26) do not allow one to fully control it).

While PN and E-guidance (kappa-guidance) laws apparently offer the only possible real-time solution to the intercept problem so far, they were all developed for the midcourse or terminal phase of an intercept and therefore have three fatal characteristics which can preclude the interceptor from impacting the target during a BPI: the dynamics of the missile controls system and controls limitations (which are the curse for all aforementioned guidance laws), the lack of knowledge of the end-game environment, and the inability to fully control the intercept geometry while still satisfying all constraints. For a BPI the lack of knowledge of end-game environment is one of the major problems as missiles use fuel for a fraction of the flight to generate a set amount of thrust and speed, after which the missile must glide to the target under a constant drag- and gravity-induced deceleration. The amount of available maneuver capability depends on the missile speed and altitude of engagement. As will be shown in the next section, the higher speeds and lower engagement altitudes of a BPI work to increase the missile capability.43 Therefore, trajectories should be flown to maximize the missile velocity and minimize the intercept altitude so that there is sufficient acceleration left to intercept the target. Yet, the above guidance laws do not consider the end-game prior to achieving it; the effect is to guide the missile along the most direct path and hope that the interception will take place under atmospheric conditions that are acceptable to the interceptor and to try to minimize the acceleration at each moment with the hope that there will be enough capability remaining to complete the end-game. The assumption that target maneuvers will be sufficiently small enough that the missile will have enough capability to counter them is usually accurate; however, the immense accelerations involved in a BPI makes active management of the missile’s acceleration critical to ensuring the end-game is successful. And again, since the near-collision course assumptions were used by many of the discussed guidance laws, the interception geometry of any engagement under the above guidance laws will not be input to the guidance law as a fixed requirement but rather a result of the relative speeds of the missile and target, the launch geometry, and the maneuvers involved. Meanwhile, this geometry is crucial factor in ensuring the interceptor is capable of destroying the BM, i.e., an overtaking intercept (the interceptor approaches the target from the rear) is significantly less desirable than a head-on or a right-angle intercept.

To conclude this section, let us go back to the reason why the PN- and E-guidance type laws were and still remain the only alternative. Cherry in a paper written in the early 60s,27 discussed this topic while dealing with a spacecraft guidance rather than more sophisticated guidance of a missile in the atmosphere. Among other interesting thoughts he stated the following:

“The problem of optimization has given rise in this decade to a great deal of mathematical speculation, research, and discovery. Some effective sophisticated numerical optimization methods have been programmed (often in double precision arithmetic) on high-speed, large-memory digital computers. The programs construct a single optimum guided trajectory in approximately 10 to 20 minutes. These techniques will not be applicable for many years to real-time control of spacecrafts with compact, light-weight guidance and control computers. Attempts have been made to precompute one nominal optimum trajectory (or even hundreds of perturbed-from-nominal optimum trajectories) and fit thrust angle regimes in-flight to the precompiled optimum trajectories. These attempts often encounter serious numerical and control problems. Furthermore, these methods, which are inherently dependent on precomputed nominal trajectories, do not have the flexibility that an explicit, direct, in-flight solution to the equations of motion has.

The numerical optimization methods, such as the method of steepest ascent, for example, are very useful for establishing target performance figures for more practical spacecraft guidance schemes. This does not mean that the sophisticated optimization computer programs consistently produce performance figures which are better than, or even as good as, the performance figures produced by the guidance law in this paper. The theoretical performance of the steepest ascent or gradient method is not realized by the digital computer programs which mechanize the algorithm. The programs are beset with numerical roundoff difficulties in computing the influence functions and gradient. It is probably for this reason that the E-guidance method frequently produces performance figures slightly superior to the theoretically optimum numerical optimization programs.”

And another quote from Lin’s mid 80s paper:31

“Many long- and medium-range missile guidance studies have shown that optimal trajectory shaping promises an extended range with more favorable end-game conditions … However, in a three-dimensional, target-intercept flight, direct application of the optimal control theory will result in a two-point boundary value problem that involves several arbitrary parameters so that analytical solutions cannot be obtained without a lot of approximations. The problem can be further complicated with the lift, thrust and drag, and control constraints forced by structural and angle-of-attack limits. This


increases the computation time so that it is not feasible to implement the resulting solutions of the missile performance within an onboard digital computer. In view of this complexity in the problem setup, either indirect methods or direct methods based on nonlinear programming are used to solve the sensitivity and convergence problems. However, both methods require very fast onboard microprocessor technology for real-time, on-line operation.”

So here is the point: we simply were, and apparently still are, unable to solve the two-point boundary-value optimization problem either using indirect methods or direct methods for more or less realistic system’s model in the real-time! That’s why the problem was simplified to the point where an analytical solution for the “optimal” control accelerations (or curvature and torsion) could be obtained. And in spite of the fact that a two-point boundary-value problem is not solved and some of the real-life constraints can not be satisfied a priori, these explicit guidance laws are all we have.

This paper makes an attempt to show that in fact today it is already possible to solve a two-point boundary-value problem and satisfy all the constraints in real-time utilizing the direct methods of calculus of variations; specificly a method that has already proved its real-time solution capability and robustness more than a decade ago. The method we are going to address next can use any complex nonlinear dynamics and aerodynamics, assures satisfaction of any terminal conditions, i.e. has full control over the intercept geometry, and is capable of employing any performance index, i.e. is not limited to that of (17). But before we proceed let us address the specifics of modeling the ICBM and an interceptor in a BPI.

III. Modeling and Simulation This section specifies the ballistic target and interceptor models used in the simulations. To start with, Fig.2

shows a comparison of the relative sizes of several ICBMs. The interceptor model is a medium range surface-to-air missile, which includes active-homing terminal guidance – the key feature that allows for the accuracy necessary to intercept a BM in the boost phase. The interceptor missile is modeled to be roughly ¼ the size of the ICBM.

Thrust/drag models were rigorously developed for an ICBM (Table 1)45 and interceptor (Table 2)3-5 using the approach laid out in Refs.7,10,39-43 and nominal data about their general characteristics.17 The developed simulations model a two-stage, boosting BM that reaches intercontinental velocities, as well as a two-stage, boosting interceptor. According to Refs.3,4 ballistic missile interceptors might have a maximum fly-out velocity of 4.0 or even nearly 6.0 km/s suggesting that for the current study we can use quite a wide range of interceptor’s thrust in Table 2. It should be pointed out however that the specific characteristics (which are not known for sure anyway) are not so important here. What’s

important though is that the interceptor’s aerodynamic data, depending on Mach number and spatial angle of attack are represented by the look-up tables rather than as analytical dependences to demonstrate the robustness (real-time capabilities) of the developed guidance law.

An example of simulated BM ballistic path to be intercepted, where the asterisks represent the staging events, is shown on Fig.3. The initial (launch) elevation angle 85θ = ° was chosen to maximize the range while still recognizing the restrictions on launching such a large missile.10 As noted in Ref.10, the flat earth equations which were used in simulation are only moderately accurate over the course of the rocket’s entire flight, but since the focus of this paper is only on the boost phase, the accuracy of the termination position is irrelevant. During the boost phase the accuracy of the flat earth equations is very good.

Figure 2. Representative liquid-fueled

ICBMs44

-1 0 1 2 3 4 5

x 106-2-1012

x 106

0

1

2

x 106

X position (m)Y position (m)

Z po

sitio

n (m

)

Figure 3. Ballistic missile flight path.


Table 1. Generic ICBM data.45

Overall Characteristics Stage 1 Stage 2 Length 32 m Diameter 2.2m 1.335m Payload 750-1000kg Length 16m 14m Range 3500–4300km Launch Weight ~60,000 kg 15,200kg Stages 2 Thrust ~1,030,000N 130,350N Thrust Chambers 4,1 Fuel / Oxidizer TM-185 / AK-27I Type LR ICBM Propellant Mass - 12,912kg Burn Time ~125s 110s

Table 2. Generic interceptor data.3-5

Overall Characteristics Stage 1 Stage 2 Length 6.5m Diameter 0.53 0.34m Payload 115kg Length 1.72m 4.78m Range 150km Launch Weight 712kg 686kg Stages 2 Thrust - - Thrust Chambers 1,1 Propellant Mass 468kg 360kg Burn Time 6s 14s

The BM thrust is given by a set of time-based values, shown in Fig.4a. It drops sharply at 130 seconds and 240 seconds to represent the staging events. Figure 5 shows the boost phase acceleration and velocity profiles, which are within the required range for a BM10 (the acceleration is of the order of 3.5g and a velocity at the end of the boost phase reaches nearly 6km/s). The two-dimensional graphs, presented in Fig.6, show the altitude versus time profiles for the entire flight and the boost phase only, which compares favorably with the expected altitude values.10

Based on this profile, determination of the available time for a surface-launched missile to intercept BM, one of the critical values, can now be addressed. The uppermost limit of the atmosphere according to the WGS-84 standard atmospheric model is about 86km. Even at 86km, however, an endoatmospheric missile has a hard time maneuvering due to the low density of the surrounding atmosphere. Thus, the maximum allowable intercept value must be lowered; in this case 50-60km will be considered the upper limit. The BM achieves this altitude between 130s and 140s. This is one of the most significant limitations on the BPI problem, since any station actively monitoring the launch area will still need 45-60s to detect, track, analyze, and engage the target. The simulations will assume a 60s delay in the interceptor launch.

Figure 7 shows the time history of the interceptor’s available thrust, which is also given as a set of time-based values. The thrust drops sharply at 6s and 20s to represent the staging events.

a)0 50 100 150 200 250

0

2

4

6

8

10

12x 10

5

Thru

st (N

)

Time (sec) b)0 50 100 150 200 250

0

1

2

3

4

5

6

7

8x 10

4

Rock

et M

ass (

kg)

Time (sec) 0 50 100 150 200 250

0

5

10

15

Time (sec)

Velocity (km/s)Acceleration (g)

Figure 4. Time histories of available thrust (a) and corresponding change in BM’s mass (b).

Figure 5. The BM acceleration and velocity profiles for the boost phase.

While the speed profile for the target, shown in Fig.5, was calculated based on the assumption of its ballistic trajectory, which is again quite accurate during the boost phase, the speed profile for the interceptor is something


much more uncertain. The axial thrust force is defined as the difference between the thrust T and drag D, and therefore the axial load factor is given by

( ) ( , , , )( )

y zx

T t Dn

m t gα β δ δ−

= , (28)

where α and β are the angles of attack and sideslip, while δy and δz are the deflections of the control surfaces in y and z channels. These latter variables are defined by two controls, side yn and normal zn load factors, which are returned from the guidance law. To be more specific yn and zn are defined as

( , , )( )

yy

Yn

m t gα β δ

= and ( , , )

( )z

zL

nm t gα β δ

= . (29)

where Y and L stand for the side force and lift, respectively. Since the interceptor flies a spatial trajectory and may experience huge lateral (side and normal) accelerations, the second term in the numerator of Eq.(28), the drag, can vary significantly with the guidance law in effect and specifics of the simulation. However, to show the feasibility of the developed model and also have some nominal interceptor’s speed profile, some generalized data produced by the model after running several simulations was obtained and plotted (Fig.8).

0 500 1000 1500 2000

0

500

1000

1500

2000

2500

3000

Time (sec)

Alti

tude

(km

)

0 50 100 150 200 2500

50

100

150

200

250

300

350

400

450

Time (sec) 0 5 10 15 20 25

0

0.5

1

1.5

2

2.5x 10

5

Time (sec)

Thru

st (N

)

0 5 10 15 20 25400

600

800

1000

1200

1400

1600

1800

Time (sec)

Inte

rcep

tor M

ass

(kg)

Figure 6. Developed BM altitude profile: entire flight (left) and boost phase only (right).

Figure 7. Time histories of available thrust (left) and corresponding change in interceptor’s mass (right).

0 10 20 30 40 50 60 70 80 90 100-500

0

500

1000

1500

2000

2500

3000

3500

Time (sec)

Velocity (m/s)

Acceleration (m/s2)

0

10

20

30

0

0

0

0

0

I

I

I

I

I

I

t t

IyIz

xxxV

nn

γ

=

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥Ψ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

Ballistic

Flight Path

Possible intercept angles based on variation of derivatives to maximize kinetic energy

0

10

20

30

1

2

3

BM

BM

BM

BM

BM

BMt t

xxxVVV

=

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

0ˆ ˆ

ft t t∆ = −

0

2

ˆ

1

3

ˆˆˆ

BMf

BMf

BMf t t t

xxx

= +∆

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

0

1

2

3

ˆ

1

2

3

ˆˆˆ

If

If

If

I

t t

BMf

BMf

BMf

IfIf t

f

xxx

V

xxx

γ

= +∆

⎡ ⎤=⎢ ⎥=⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

⎥⎣ Ψ⎢ ⎦

fτ

Z

Y

X

O

Figure 8. Generic velocity and acceleration profile for the interceptor model.

Figure 9. The intercept geometry and formulation of the optimization problem.

IV. Development of a New Guidance Algorithm The guidance law suggested in this section determines the near-optimal flight path from the interceptor current

position to a predicted BM position and then derives the set of control commands necessary to execute that flight. First, following Ref.51, we formulate an optimization problem for the BM interception as follows (Fig.9).

Likely intercept time


A. Problem Formulation and Proposed Control Scheme Among all admissible trajectories,

[ ] { }6 61 2 3 0( ) ( ), ( ), ( ), ( ), ( ), ( ) , ( ) , ,T

ft x t x t x t V t t t S S t Z E t t tγ ⎡ ⎤= Ψ ∈ = ∈ ⊂ ∈ ⎣ ⎦z z (30) that satisfy:

- the system of differential equations (dynamic constraints):

1

2

3

cos coscos sin

sin

x Vx Vx V

γ ψγ ψγ

=== −

&

&

&

( )1

1 1

( sin )cos

cos

x

z

y

V g ngV ngn V

γγ γ

γ

−

− −

= −= −

Ψ =

&

&

& (31)

where 1x , 2x and 3x are the North-East-Down coordinates of the interceptor’s center of gravity; V , γ and ψ are the interceptor’s speed, path angle and heading; xn is the axial load factor; yn and zn – the other two

components of the load factor (see Eq.(29)) – constitute the vector of controls, ( ) ( ), ( )Т

y zt n t n t⎡ ⎤= ⎣ ⎦u , 2 2U E∈ ⊂u (according to Eq.(28) we have no direct control over xn );

- the (current) initial conditions: 1 0 10( )x t x= , 2 0 20( )x t x= , 3 0 30( )x t x= , 0 0( )V t V= , 0 0( )tγ γ= , 0 0( )tΨ = Ψ , 0 0( )y yn t n= , 0 0( )z zn t n= ,

i.e., 0 0( )i ix t x= , 0 0( )i ix t x=& & , 0 0( )i ix t x=&& && , 1, 2,3i = ; (32) - the final conditions:

1 1( ) BMf fx t x= , 2 2ˆ( ) BM

f fx t x= , 3 3ˆ( ) BMf fx t x= , ( )f fV t V= , ( )f ftγ γ= , ( )f ftΨ = Ψ , ( ) 0y fn t = , ( ) cosz f fn t γ= ,

i.e. ( )i f ifx t x= , ( )i f ifx t x=& & , ( )i f ifx t x=&& && , 1,2,3i = , (33) based on the prediction of where BM should be at time ft

210 0 02

ˆ ˆ ˆˆˆBM BM BM BMf go got t= + +x x V a , (34)

where 0ˆ BMx , 0ˆ BMV and 0ˆ BMa , are the estimates of BM’s position, speed and acceleration vectors at time 0t t= ,

and 0go ft t t= − is an estimate of time-to-go; - the constraints, imposed on the intercept altitude

3 3max( )fx t x≤ , (35)

where 3max 60x km= , - controls10

max

( )1( ) ( )

1 ( )y

z

n tn q

n t

⎧ ⎫⎡ ⎤⎡ ⎤⎪ ⎪= − ≥⎢ ⎥⎨ ⎬⎢ ⎥⎢ ⎥⎣ ⎦⎪ ⎪⎣ ⎦⎩ ⎭

ξ u 0 , (36)

which happen to be dependent on the dynamic pressure 20.7q pM= (p is the static density determined by the altitude -x3 and Mach number M),

- and on the controls derivatives

max

( )1( )

1 ( )y

z

n tn

n t

⎧ ⎫⎡ ⎤⎡ ⎤⎪ ⎪= − ≥⎢ ⎥⎨ ⎬⎢ ⎥⎢ ⎥⎣ ⎦⎪ ⎪⎣ ⎦⎩ ⎭

η u 0&

&&

(37)

find the optimal trajectory, ( )t∗z , that minimizes some PI, J, and the corresponding optimal control, ( )t∗u . The choice of PI for this specific application will be addressed later in this section, but first let us make three

important comments. First, while the initial conditions for the interceptor are simply its current conditions and therefore fixed, its final conditions are in fact variable. We can vary a position of the impact point by assigning different tf (see Eq.(34)). Also, the direction of interceptor’s velocity vector at the impact point is not defined ( fΨ and fγ are variable), and we have no direct control over the magnitude of the interceptor’s final velocity, Vf (because the thrust profile is fixed). We can only try to minimize the interceptor’s control actions and got to assure it does not fall below a certain limit.


Secondly, it should be noted that as opposed to the standard formulation, the constraints on the control derivatives (37) are added to account for actuator’s dynamics (deflection of the control surfaces δy and δz). Being accounted for at the stage of reference trajectory generation implicitly assures more accurate tracking of this trajectory later on.

Lastly, it should be noted that all four variables appearing on the right-hand side in the equation (34) are not known for sure. For the BM boost phase it is very difficult to rely on the estimate of acceleration even if it is known with the sufficient accuracy. The BM position and velocity vectors are also only known with certain accuracy too. Time-to-go is not known at all and is only assessed during optimization itself, meaning that future nonlinear trajectory of the boosting target is assumed known as a function of time-to-go. Generally speaking, it means that there is no guaranty that at the final point the equality ( ) ( )BM

f ft t=x x holds. However, while the interceptor approaches BM, the estimates of all four parameters can be made more and more accurate. Hence, to assure the high probability of kill (enforce the equality ( ) ( )BM

f ft t=x x ), we would need to be able to solve the optimization problem during the intercept several times, as often as possible. The block-diagram of the control system that accounts for such periodical updates is shown in Fig.10. The inner loop represents the common feedback control for tracking the reference trajectory and the outer loop provides periodical updates of this reference trajectory.

Neither of indirect methods of calculus of variations stated in Section II can solve the above two-point boundary-value optimization problem with a varied right end in real-time. Moreover, not many direct methods can handle it either; especially in this situation, where this trajectory has to be refined every other second or so to assure a continuity of the controls (meaning the requirement to satisfy up to the second-order derivative of the interceptor coordinates in the initial point for every cycle of the optimization as required by Eq.(32)). The direct method of calculus of variations presented earlier in Ref.51, however, was specifically developed for similar problems and proved to be able to compute feasible trajectories in the fractions of a second. The usage of this method assures the following:

- the boundary conditions including high-order derivatives (Eqs.(32),(33)) are satisfied a priori, - the control commands are smooth and physically realizable (Eqs.(36),(37) hold), - the method is very robust and is not sensitive to small variations in the input parameters; - only a few variable parameters are used, thus ensuring that the iterative process during optimization

converges well and that the continuous update of the solution allows reliable path following even with no standard feedback (Kc=0 on Fig.10 can be zero),

- the near-optimal solutions obtained with this method are very close to the optimal one (despite the inherited feature of all direct methods limiting possible trajectory variations within a certain class of basis functions (to be discussed in the next subsection).

Another important feature of this method is that it allows handling any complex PI. Therefore, we are not limited to simple standard PIs like the one of Eq.(17), or time, or fuel expenditure, etc. (We will take advantage of this feature when introducing the compound PI later on.)

This direct method developed more than a decade ago and applied then for a manned aircraft, has recently been employed in different real-time applications in academia and proved to be a reliable tool, easily tuned to a specific problem.52 One of its latest applications also includes planning and conducting time-critical missions of multiple UAVs.53 The differences between the problem formulated above with that of Ref.51 is that the intercept problem implies fluid rather than soft final boundary conditions and that there is no authority over the thrust profile, leaving only two controls as opposed to three controls for the conventional aircraft.

( )tu

KKcc

Trajectory GeneratorTrajectory Generator

InterceptorInterceptor

InterpolatorInterpolator

Update SwitchUpdate Switch

( )tz

( )ref tu

∗u∗z( )ref tz 11……100 100 HzHz

0.010.01……0.5 0.5 HzHz

TargetTarget

fz0z

SensorsSensorsIMU/GPSIMU/GPS

ˆ ( )BM tz

( )tz

Figure 10. Suggested feedback control.


B. Reference Trajectory Applying the aforementioned direct method to the intercept problem leads to the following computational

routine. It starts from assuming the intercept trajectory ix , 1,2,3i = to be represented as a higher-order polynomial in some abstract parameter, a virtual arc τ :

2 3 43 4 5 6 7

2 3 4 52 3 4 5 6 7

2 3 4 5 61 2 3 4 5 6 7

20 1 2

( ) ( )1 1 1 1 ( ) ( )2 3 4 5

1 1 1 1 1 ( ) ( )2 6 12 20 30

1 1( ) ( )2

i x i i i i i i

i x i i i i i i i

i x i i i i i i i i

i xi i i i

x P a a a a a

x" P a a a a a a

x' P a a a a a a a

x P a a a

τ τ τ τ τ τ

τ τ τ τ τ τ τ

τ τ τ τ τ τ τ τ

τ τ τ τ

′′′

′′

′

′′′ = = + + + +

= = + + + + +

= = + + + + + +

= = + + + 3 4 5 6 73 4 5 6 7

1 1 1 16 24 60 120 210i i i i ia a a a aτ τ τ τ τ+ + + +

(38)

The coefficients ika , 1, 2,3i = , 0,...,7k = can be determined by solving the following system of linear algebraic equations

0

1

2

32 3 4 5 6 71 1 1 1 1 1

42 6 24 60 120 2102 3 4 5 61 1 1 1 1

52 6 12 20 302 3 4 51 1 1 1

2 3 4 5 62 3 4

7

1 0 0 0 0 0 0 00 1 0 0 0 0 0 00 0 1 0 0 0 0 00 0 0 1 0 0 0 010 10 0 10 0 0 1

i

i

i

i

f f f f f f f i

f f f f f f i

f f f f f i

f f f f i

aaaaaaaa

τ τ τ τ τ τ ττ τ τ τ τ τ

τ τ τ τ ττ τ τ τ

⎛ ⎞⎛⎜ ⎟⎜⎜ ⎟⎜⎜ ⎟⎜⎜ ⎟⎜⎜ ⎟⎜⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝⎝ ⎠

0

0

0

i

i

i

if

if

if

if

if

xxxxxxxx

⎛ ⎞⎞⎜ ⎟⎟ ′⎜ ⎟⎟⎜ ⎟⎟ ′′⎜ ⎟⎟ ′′′⎜ ⎟⎟ = ⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟′⎜ ⎟⎜ ⎟

⎜ ⎟⎜ ⎟ ′′⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟′′′⎠ ⎝ ⎠

. (39)

Equations (39) resolved for coefficients ika , 1, 2,3i = , 0,...,7k = yield

0 0 1 0 2 0 3

0 0 0 04 2 3 4

05 2

, , , 4 16 60 120 360 480 840 840

30 60 420 600

i i i i i i i

if i if i if i if ii

f f f f

if i ifi

f

a x a x a x a xx x x x x x x x

a

x x x xa

τ τ τ τ

τ

′ ′′ ′′′= = = =′′′ ′′′ ′′ ′′ ′ ′− + − − − −

= + + +

′′′ ′′′ ′′ ′′+ − += + 0 0 0

3 4 5

0 0 0 06 3 4 5 6

0 0 07 4 5 6

2340 2700 5040 5040

60 80 780 900 4080 4320 8400 8400

35 35 420 420 2100 2100

i if i if i

f f f

if i if i if i if ii

f f f f

if i if i i ifi

f f f

x x x x

x x x x x x x xa

x x x x x xa

τ τ τ

τ τ τ τ

τ τ τ

′ ′− − ++ +

′′′ ′′′ ′′ ′′ ′ ′− − − − − −= + + +

′′′ ′′′ ′′ ′′ ′ ′+ − + += + + 0

7

4200 4200if i

f

x xτ

− ++

(40)

and the final arc fτ becomes the first optimization (varied) parameter. It is worth noting that almost any intercept trajectory obtained as a result of the classical and modern guidance

laws of Section II can be approximated with a lower-order polynomial, meaning that all of them are obviously a subset of the variety of trajectories that can be produced with the reference functions (38),(40). Another point is that by having the Cartesian coordinates parametrised using the reference functions (38) we escape the necessity to integrate the kinematic (the first three) equations of the system (31).

Now, the reason for choosing some abstract parameter τ (not time, not path length) as an argument for the reference functions (38) is to allow optimizing (varying) the spatial trajectory without interfering into the speed profile,51,52 which is determined in our case by the axial load factor (28) (given by the thrust profile T(t)). A connection between the virtual arc and time domains is established using the virtual speed λ as

( ) ddtτλ τ = . (41)

Using a virtual arc is a key alement in the proposed approach.


Now, we must make a determination of the boundary conditions in the virtual arc domain (right-hand side vector in Eq. (39)). The initial conditions (32) are all defined from the current interceptor state up to the second-order derivative in the time domain as follows:

10 0 0 0

0 20 0 0 0

30 0 0

cos coscos sin

sin

x Vx Vx V

γ ψγ ψ

γ

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥= =⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦

x&

& &

&

, 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0

cos cos sin cos cos sincos sin sin sin cos cos

sin cosf

f

V V VV V V

V V

γ γ γ γγ γ γ γ

γ γ γ

⎡ ⎤Ψ − Ψ −Ψ Ψ⎢ ⎥= Ψ − Ψ +Ψ Ψ⎢ ⎥⎢ ⎥− −⎣ ⎦

x

& &&

& &&&&

& &

. (42)

In Eq.(42) the derivatives 0V& , 0γ& , and 0Ψ& are taken from the dynamic equations (31). For more flexibility we will also vary the components of the third-order derivative in the initial point 0x&&& . To be more specific, its magnitude,

0x&&& , and two angles, defining a direction, will be used as the second, third and fourth varied parameters. Regarding the final conditions (33), we can do the same assuming fγ and fΨ to be fifth and sixth optimization

parameters and guessing on the value of fV (to be adjusted during optimization):

1

2

3

cos coscos sin

sin

f f f f

f f f f f

f f f

x Vx Vx V

γ ψγ ψ

γ

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥= =⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦

x&

& &

&

, cos cos sin cos cos sincos sin sin sin cos cos

sin cos

f f f f f f f f f f f

f f f f f f f f f f f f

f f f f f

V V VV V V

V V

γ γ γ γγ γ γ γ

γ γ γ

⎡ ⎤Ψ − Ψ −Ψ Ψ⎢ ⎥= Ψ − Ψ +Ψ Ψ⎢ ⎥⎢ ⎥− −⎣ ⎦

x

& &&

& &&&&

& &

. (43)

In order to ensure a smooth flight path at the end of the trajectory and avoid any flair maneuver, we additionally assume

f =x 0&&& . (44) The usage of the virtual speed (41) allows the recalculation of the initial and final boundary conditions,

transforming them from the time domain to the virtual arc domain using the obvious relations: d dd dt

τ λτ

′= =xx x& , 2( )d d

d dtλ τ λ λ λτ′

′′ ′ ′= = +xx x x&& ,

23 2 2 2( ) 3 ( )d d

d dtλ λ λ τ λ λ λ λ λ λ λ

τ′′ ′ ′+ ′′′ ′′ ′ ′ ′′ ′= = + + +

x xx x x x&&& . (45)

When rearranged, the first two relations define the first- and second-order τ-derivatives of the interceptor coordinates as

1λ−′ =x x& , 2 1λ λ λ− −′′ ′= −x x x&& & . (46) Specifically, in the boundary points

10 0 0λ

−′ =x x& , 1f f fλ

−′ =x x& , 2 10 0 0 0 0 0λ λ λ− −′′ ′= −x x x&& & , 2 1

f f f f f fλ λ λ− −′′ ′= −x x x&& & . (47) The values of 0x , 0x& , 0x&& , 0x&&& , fx , fx& , fx&& , fx&&& (Eq.(42)-(44)) converted to 0x , 0′x , 0′′x , 0′′′x , fx , f′x , f′′x , f′′′x

using (47) are now ready to be used in Eq.(40) to define the coefficients of the reference functions (38). The dependence ( )λ λ τ= is being established during optimization and for the boundary conditions for λ and

λ′ we may choose 0 0Vλ = , 10 0 0V Vλ −′ = & , f fVλ = , and 1

f f fV Vλ −′ = & . It should be noted that scaling the virtual speed ( )λ τ does not really matter - the higher values of λ will simply result in the larger fτ , leaving parameters in the

time domain unaltered. If the virtual arc fτ happens to be of the order of the actual path length, then the virtual speed is of the order of the actual speed. Therefore, the fact that at the boundary points we allow virtual and actual speed to be the same implies that the virtual arc fτ will be of the order of the actual path length (which, by the way, allows for making a qualified initial guess on fτ during optimization).

Equations (38) through (40), explaining how the boundary conditions can be met a priori, constitute the spirit of the proposed approach. Changing fτ does not affect the boundary conditions, but do change the shape of a candidate trajectory allowing to vary time histories of the components of the state ( )tz and control vector ( )tu as well as ft . Therefore we have gained an opportunity to optimize the performance index without jeopardizing the boundary-value problem. Moreover, for the problem at hand we can additionally vary fγ , fΨ and 0′′′x providing the proposed guidance law with extra degrees of freedom.

To end this section, it should be noted that equations (46) hold not just for x but for any other parameter as well. For instance, we will use the first of them to integrate the speed equation (the fourth one in Eq.(31)) in the τ-domain:

1 1( ) ( sin )xV V g nτ λ γ λ− −′ = = −& . (48)


Because of the preset thrust profile which cannot be altered, we have no other choice other than to integrate the speed equation. Yet, this is the only equation to be integrated since the two remaining dynamic equations in the system (31) are determined via inverse dynamics to be addressed next.

C. Discretization and Inverse Dynamics The numerical solution proceeds with computation of all parameters along the reference trajectory ( )xiP τ over a

fixed set of N points (for instance, N=100) equidistantly placed along the virtual arc [0; ]fτ with the interval of

1f

Nτ

τ∆ =−

. (49)

We take the very next value for the virtual arc 1j jτ τ τ−= + ∆ , 2,...,j N= (50)

and compute the corresponding parameters of the trajectory from ( )xiP τ , ( )x iP τ′ , and ( )x iP τ′′ (Eq.(38)). Having these, we proceed with defining flight path angle and heading as follows

3;

2 21; 2;

arctg jj

j j

x

x xγ

′−=

′ ′+, 2;

1;

arctg jj

j

xx′

Ψ =′

. (51)

Their derivatives can be defined via direct differentiation as 2 2

3; 1; 2; 3; 1; 2; 22 2 3 / 2

1; 2;

( ) ( )cos

( )j j j j j j

j jj j

x x x x x xx x

γ γ′ ′′ ′′ ′′ ′ ′+ − +

′ =′ ′+

, 2; 1; 1; 2; 22

1;

cosj j j jj j

j

x x x xx

′′ ′ ′′ ′−′Ψ = Ψ

′′. (52)

Next, we numerically integrate Eq.(48) to obtain 1 1( )j j jV V V τ τ− −′= + ∆ . (53)

(the model for thrust and drag used here was addressed in Section III). The next step is to compute the corresponding time interval, the elapsed time, and virtual speed λ

32

; ; 11

11

( )2

i j i ji

jj j

x xt

V V

−=

−−

−∆ =

+

∑, 1 1j j jt t t− −= + ∆ , and

1j

jtτλ−

∆=∆

. (54)

Finally, the controls yn and zn are found by rearranging terms in the two last equations of (31) 1

; cosy j j j j jn V g λ γ− ′= Ψ , 1; cosz j j j j jn V g λ γ γ− ′= + . (55)

The time lag represented, for example, by a first-order system (13) can also be easily inverted as well ; ; 1

; ;1

y j y jcy j y j

j

n nn n T

t−

−

−= +

∆, ; ; 1

; ;1

z j z jcz j z j

j

n nn n T

t−

−

−= +

∆. (56)

D. Penalty Function and Performance Index When all parameters (states and controls) are computed in each of N points, we can compute PI and the penalty

function (PF). According to Eqs.(35)-(37), the PF can be formed as follows:

( )( )( )( )( )

2

3; 3max

2

; max

2

; max

2

; max

2

; max

max 0;

max 0; ( )

0.01,1,1, , max 0; ( )

max 0;

max 0;

jj

y j jj

c cz j jj

y jj

z jj

x x

n n q

w w n n q

n n

n n

⎡ ⎤−⎢ ⎥⎢ ⎥⎢ ⎥−⎢ ⎥⎢ ⎥⎡ ⎤∆ = −⎢ ⎥⎣ ⎦⎢ ⎥⎢ ⎥−⎢ ⎥⎢ ⎥

−⎢ ⎥⎣ ⎦

& &

& &

, (57)

where the penalties on violation of constraints on controls and their derivatives are scaled with the weighting factor cw ( ~ 10cw ). (If the constraints on the control’s derivatives (37) were accounted for via Eq.(56), then 0cw = .) Now, let us address the issue of choosing the most appropriate PI. First of all, it worth mentioning that one of the

advantages of the proposed direct method is that it can handle any complex (compound) PI. For instance, in one of


the earlier similar applications the probability of target kill (in a stochastic simulation) was considered as a PI explicitly.54 Here we are dealing with the deterministic approach, hence we should choose some physical parameter(s) somehow related to the probability to kill. One commonly used PI is miss distance (as in the first term in Eq.(6)). In our particular case, since the method allows for satisfying any terminal conditions a priori, i.e. assures a zero miss distance, we should look at some other parameter. Another choice of a PI could be a time of intercept,

got , or physical path length sf. We would also aim at maximizing an impact velocity, or minimizing 1fV − , which by

the way neither of the PIs of Section II (except those in Ref.31) accounts for at all. The proposed direct method allows combining all three criteria into a single weighted compound PI:

11 1 2 2 3 3J J J

f go fJ w k w k t w k Vτ −= + + . (58)

where ik , 1, 2,3i = are the scaling factors, and iw , 1, 2,3i = ( 1 2 3; ; 1J J Jw w w⎡ ⎤ =⎣ ⎦ ) are the weighing coefficients to

balance multiple objectives. Note that for the sake of computational simplicity we use fτ instead of sf , as they are proportional to each other.

However, the preliminary analysis revealed that all three mentioned criteria are obviously related. The sooner interceptor impacts the target and the shorter (less curved) path it flies, the larger impact velocity can be achieved (see Fig.8). Moreover, because we have no direct control over the final interceptor’s velocity, this last term in Eq.(58) can be excluded with absolutely no harm:

1 1 2 2J J

f goJ w k w k tτ= + . (59) Not only the remaining terms in Eq.(58) assure the fastest intercept with the maximum impact velocity, but they also ensure that the numerical solution is actually physically realizable. Without these two parameters, the system might continue to optimize the intercept well beyond the capabilities of the missile or even physical reality. One example is that, in a purely mathematical sense, there is no problem with a negative velocity (yet in the physical world that makes no sense) and the algorithm might try a program that would intercept after the missile velocity has gone past zero and into negative numbers (of course, the missile would stop flying long before it even reaches zero). One might be tempted to include a myriad of parameters to cover all possible eventualities; however, including these two parameters sufficiently accounts for nearly every physical limitation and eliminates the need for having a long list of cost variables.

In summary, the guidance algorithm described above with the PI of the form of Eq.(59) assures a zero miss distance by construction of Eq.(38); avoids sharp maneuvers right before the impact by satisfying Eq.(44); takes into account changing constraints on the controls (with the dynamic pressure q as shown in Eq.(36)) by zeroing the PF (57); indirectly maximizes fV and minimizes fτ and tgo via the PF (59); and allows for controlling the three-dimensional shape of the intercept trajectory directly by varying impact angles fγ and fΨ in Eq.(42). None of the guidance laws of Section II assures even a single of the mentioned criteria against maneuvering target, not to mention their combination.

Although the algorithm employing a PI in the form of Eq.(59) works perfectly well, the authors wanted to explore other objectives as well. To this end it was suggested to add one more, trajectory-shaping term as shown below:

( )1 1 2 2 3 3

,BMf fJ J J

f go BMf f

J w k w k t w kτ= + +V V

V V. (60)

The third term was chosen in order to maximize the angle of impact upon interception. This reflects the need to go beyond simply hitting the target and instead to maximize the kinetic energy in order to ensure the interceptor disables the target, as described in Section II. Therefore, the authors propose to pursue three objectives, which are: to minimize the dimensions of the intercept maneuver (first term), to minimize time-to-go (second term), and to maximize the impact angle of the interception (hit the BM at 90-degree angle) (third term).

In Eq.(60), using the factors 1k , 2k , and 3k allows for scaling each term appropriately, so that they all are roughly equal to each other when optimized. Failure to weight them properly will obviously skew the PI (e.g., the anticipated intercept time is counted in tens of seconds while the cosine of the impact angle will vary from zero to one). Additionally, through the weighting functions 1w , 2w , and 3w , a trade-off analysis can be conducted and variables can be included or excluded as desired (for instance, making 3 0w = brings PI back to the form of Eq.(59)).


The computational routine of the proposed algorithm looks as follows. With the estimates 0ˆ BMx , 0ˆ BMV and got

(we will not rely on the estimate of 0ˆ BMa ), and guesses of the values of varied parameters fτ , fγ , fΨ and 0′′′x , we determine the initial and final boundary conditions (42)-(44), and compute the candidate trajectory (38)-(40). Then, with a fixed thrust profile, we inverse the missile dynamics and determine the remaining states (51), and controls (55). Then we estimate the PF (57), and the PI (60). Now we can refine the estimate of got and try to adjust some of the variable parameters to drive the PF to zero (within certain tolerance) and minimize the PI.

All computations are performed in the MATLAB/Simulink development environment, employing the “fminsearch” or “fmincon” functions to carry out the optimization by varying six optimization parameters and evaluating the PI and PF together (fminsearch) or separately (fmincon). Once the minimum value of the PI has been found, the developed algorithm returns the required control time history to the missile guidance system, which can then execute the commands and fly the derived flight path. Since the missile system can be programmed with sufficient data to compensate for its control system time constant, the system lag can be effectively negated, thus eliminating a source of error. Updating the guidance system every several seconds (Fig.10) results in increasingly accurate final intercept positions and further ensures the mission kill.

V. Simulation Results This section presents the result of simulation of optimizing the intercept trajectory as explained in the previous

section. During the simulation, the BM path data is called by the interceptor to mimic the missile’s onboard sensors. The interceptor’s launch point was chosen to be roughly 150nm from the BM launch position. The interceptor is launched with a 60s delay after BM. The first portion of the interceptor flight is the vertical launch, which is assumed to last 6s, followed by a 4s period in which the missile arcs over toward the target, as it would do under guidance from under some assumed ground control station. At 10 seconds, the state of the interceptor and the state of BM is “seen” and input into the system. At this point the missile would not have an independent radar picture of the target, so instead would be fed targeting data, as is done by many surface-to-air missile systems. This can continue indefinitely until the interceptor has its own radar fix on the ballistic missile. Then the interceptor “sees” the location and velocity of BM at the appropriate intervals by coordinating the launch time of the interceptor with the launch time of BM. At each control update cycle (according to the proposed control scheme of Fig.10) the optimization problem is solved.

The following only addresses solving the optimization problem at the very first instance, when the missile’s computer has to find the intercept trajectory for the first time. Once it defines this trajectory and starts to control the missile to follow it, all subsequent updates of the intercept trajectory are based on the very good knowledge about the parameters defining this intercept trajectory and therefore the run-time for the following optimization cycles is only a fraction of that needed for the first time. A first “guess” at time-to-go and flight distance is done by a simple iterative process17 that takes the known velocity profile of the interceptor (Fig.8) and iterates an initial intercept time using a first order approximation of the ballistic path (see example on Fig.3), i.e. the known velocity profile of the target (Fig.5). Also, the program creates an initial guess for the final values of the flight path angle ˆ0.5 BM

f fγ π γ= +

and heading ˆ BMf fπΨ = +Ψ . This serves only as an initial start point and must only be mildly accurate, since the

program will optimize the time and flight path as it operates. The accuracy of the initial guesses only serves to decrease the necessary computation time.

As expected, about 20 different candidate paths were tested before arriving in the neighborhood of the final value, and then about 40 additional different flight paths were looked over before arriving at the final solution (Fig.11). An example of the near-optimal intercept (just a single control update cycle) is shown in Fig.12a in a three-dimensional plot. A star on the BM trajectory designates the point when the interceptor activated the guidance law (the point from which the state of BM was extracted). It should be noted that the computed time-to-intercept, got , corresponds to the point on the ballistic flight path; this is the major difference between the proposed algorithm and most predictive intercept guidance laws, where got is only an educated guess – the proposed algorithm has actually tested the proposed flight path and determined the precise got . A plot of the position and velocity of the ballistic missile overlaid upon the final optimal flight path presented on Fig.12b shows the final intercept geometry, where the lines extending from the position point represent the current velocity vector at that moment (the relative scale shows the relative speeds and the circle at the end denotes “forward” for each). From this angle of view it is clear that the interceptor does indeed intercept the ballistic missile at a near right angle as was intended.


The next question to ask is whether it is feasible and physically possible for the interceptor missile to fly the derived path. Figure 13 addresses this by showing that the required control inputs, c

yn and czn , are within the

limitations of the system, where the dashed line represents the maximum capability of the system at each point based on the dynamic pressure of the missile as prescribed by Eq.(36). (This snapshot only shows the capabilities for the chosen flight path, yet while the algorithm is running those capabilities are individually recalculated each time and vary widely based on the specifics of the flight path geometries being tested). The constraints on the control derivatives (37) are also satisfied. Note that the optimal intercept trajectory uses the full capabilities of the missile, as it supposed to be. Also note that the required forces are near zero at the intercept point, demonstrating no flair maneuver or high-g maneuver was conducted (which complies with Eq.(44)). This suggests that the full kinematic energy of the missile is directed into the target, which was the original intent of the algorithm.

a) b) Figure 11. The PF (a) and PI (b) change during optimization.

a) b) Figure 12. Near-optimal intercept trajectory (b), and the planar (x-z) projection of the near-optimal intercept with the

velocity vectors (b).

As seen in Fig.11, the PF was indeed influencing the choice of trajectory. If the required control effort exceeded the maximum capabilities of the missile, the candidate intercept trajectory was rejected as infeasible. As seen from Fig.11a at 18th iteration one of two PF’s components has reached its minimums already, but because the second component is not zero the optimization goes on. At 28th iteration another component reaches zero but the second constraint is still violated, so the optimization procedure contines. The final path has no penalty assigned to it.

Finally, Figs.14 and 15 show the values of each term compounding the PI. As seen in Fig.14, the impact angle indeed goes to 90 degrees therefore its contribution to the PI becomes negligible. Although the two remaining variables to be optimized, tgo and fτ , have no target value, it is seen that the algorithm has indeed tried numerous values to converge to the best ones (Fig.15).

A “chattering” observed in Figs.14 and 15 is not a bad thing as it might be thought of and is caused by three reasons. First, the final conditions are changing at every iteration because of the updated value of time-to-go (see Eq.(34)). Second, as mentioned the gradient-free variable-step method is used for optimization. Therefore, when the candidate solution violates the constraints the algorithm performs a back step and then proceeds with the smaller step (in the same direction). Although this method might be considered as not robust, in fact compared to the gradient methods it is more robust, finding the solution when it exists reliably and fast enough (even with a smaller number of performance index evaluations); not to mention that it might be quite difficult to assure smooth gradients


for the aerodynamic lookup table data. Third, as mentioned already too, the first two objectives of the performance index (60) are conflicting with the third one because, for example, minimization of the virtual arc is not necessarily consistent with achievement of the 90-deg impact angle objective (the latter tends to maximize arc length particularly if the initial conditions are not “friendly” to such a crossing angle). Blending three objectives together using the weighting coefficients apparently causes a certain flatness of the performance index near the optimal solution. And lastly it should be noted that a chattering occurs near the Pareto frontier (the set of solutions where no further improvements can be made in all objectives simultaneously), meaning that we are in the vicinity of the solution already – the PI itself (Fig.11b) does not demonstrate large chattering compared to those of Figs. 14 and 15. As seen from Figs.14 and 15 we are almost done after 31 iterations and the remaining ones are only needed to tune the final solution to meet specifications on the tolerance of varied parameters.

Figure 13. Time histories of controls.

Figure 14. Iterative values of the impact angle.

a) b) Figure 15. Iterative values of the final arch fτ (a) and intercept time got (b).

VI. Conclusions To address the needs for the development of the advanced guidance law overcoming fatal characteristics of

modern guidance in a specific application, a new guidance algorithm has been developed and tested in simulations. This algorithm relies on the direct method of calculus of variations developed earlier51 and has three key elements: the optimal solution is searched for within the limited variations space, namely high-order polynomials; the algorithm limits it even further by forcing unconditional satisfaction of the higher-order derivatives at the both ends of the trajectory and leaving only a few (six) varied parameters; and finally it employs inverse dynamics allowing to avoid integration of the dynamic equations and therefore drastically speed up the optimization routine. The trajectory optimization problem is then converted into a nonlinear programming problem and solved numerically. The developed algorithm is extremely robust and requires only a moderate level of computational power even in the considered application when the final boundary conditions are not fixed. It therefore can easily be incorporated into existing missile guidance systems. The future extension of this study will include addressing the following issues: performing a trade-off analysis of PI’s weighting coefficients (to determine if the chosen terms of the compound PI are the most influential choices, or if they are still redundant, or if other choices may provide better results);


checking how inaccuracies in the estimation of the BM parameters affect the probability of the interception; perform Monte Carlo simulation on launch-interception scenario to determine possible limitations of the algorithm and reliable interception zones; and finally performing robustness analysis using the high-fidelity 6DoF models17 as well as auto-pilot models in the presence of disturbances.

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