1

Multi-Missile Interception Integrating New Guidance Law and Game Theoretic Resource Management

Mo Weia, Genshe Chena, Jose B. Cruz, Jrb, and Erik Blaschc

a- Intelligent Automation, Inc., 15400 Calhoun Dr, Suite 400, Rockville, MD 20855, {mwei, gchen}@i-a-i.com

b- The Ohio State University, 2015 Neil Ave, Columbus, OH 43202, cruz@ece.osu.edu c- AFRL, WPAFB, OH, USA, Erik.Blasch@wpafb.af.mil

Abstract—Traditional missile interception often focuses on simplified scenarios such as one-to-one or multi-to-one interception. Recently, battlefield situations pose new difficulties for missile defense systems, which make traditional interception systems inefficient. The problems revolve around two aspects: 1) The guidance law insufficiency (traditional forms PN, DGL/1, and DGL/C); and 2) Resource management insufficiency. This paper fuses game theoretic resource management and a noise level related guidance law to existing missile defense system which is called Differential Game Law Type M (DGL/M). Intensive simulations show that this approach demonstrates improvements over existing methods.1 2

TABLE OF CONTENTS

1. INTRODUCTION......................................................1 2. TECHNICAL APPROACH OF MULTI-MISSILE INTERCEPTION SYSTEM.............................................4 3. SIMULATIONS ........................................................8 4. CONCLUSIONS .......................................................9 ACKNOWLEDGEMENTS ...........................................10 REFERENCES ...........................................................10 BIOGRAPHY .............................................................12

1. INTRODUCTION

Traditionally, the missile interception scenario is a one-to-one scenario, including one invading missile (with or without maneuvering capability) and one antimissile. Although missiles fly in 3D space, the analyses are mainly focused on 2D planes. This is because it is proved [1-2] that in missile interception area, 3D analysis can be decomposed into analyses on two perpendicular planes to simplify the analysis. Situations are formulated relative to a reference Line of Sight (LOS), which is the reference direction and will remain fixed during the interception procedure (see Fig. 1). In addition, the speeds (invading missile’s and antimissile’s) along the LOS axis are assumed fixed, thus allowing easy calculation of collision time ft , (the time

when the distance between invading missile and antimissile

1 1 1-4244-1488-1/08/$25.00 ©2008 IEEE. 2 IEEEAC paper#1390, Version 7, Updated 2007:12:04

along the LOS direction is zero). In addition, both missiles are modeled as point masses.

The simplest invading missiles are the “pure” long-distance ballistic missiles, where the word “pure” means the ballistic missile has no maneuverability. For such invading missile, as long as launched, the whole flight path is determined and the antimissile can simply aim at a point on the flight path at appropriate time. Since the possible noise distortions and estimation errors, the guidance laws applied by antimissiles are Proportional Navigation (PN) [1, 3-5], which was widely used in the missile-versus-aircraft research area.

The PN guidance law is as follows

.'

cu N V λ= (1)

where u is the command for the antimissile, 'N is the effective navigation ratio (usually values between 3 and 5 are taken), cV is the closing velocity, and

.λ is the LOS rate

(the change rate of the angle between current direction from antimissile to invading missile and the initial LOS, see Fig. 1). It can be shown that (Eq. 1) is approximately equivalent to the following form, which is derived from linear quadratic optimal control formulation [3-5]

' '.

2 2 ( )go P Ngo go

N Nu y y t Z E Mt t

⎛ ⎞= + =⎜ ⎟⎝ ⎠ (2)

where got is the time-to-go (the time between current time t

and the collision time ft ), and y is the evader’s relative

position on the axis perpendicular to the LOS. The PNZEM in (2) is the calculation approach of Zero Effort Miss (ZEM) distance, which is the predicted final miss distance between the invading missile and the antimissile if no further corrective effort is applied. We will see via the expression in (Eq. 2) and ZEM, that it is easy to develop relationships between traditional PN and newer guidance laws.

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Fig. 1 Missile interception illustration

Since “pure” ballistic missiles are easy to intercept, invading missile designers added maneuverability to invading missiles allowing for varying missile inflight path changes. Some Tactical Ballistic Missiles (TBM) [2] have such capabilities. They can maneuver in terminal period (or the “end-game”, the final period of the flight of the invading missile) in a random mode. They have shorter flight distances and thus, the search/response time of the antimissile systems is squeezed. Two facts should be noted: 1) Invading missile would apply a random maneuver so that the antimissile could not follow a pre-defined route to intercept them; and 2) Invading missile designers have no interest in imposing too frequent maneuvers for the invading missiles during the flight, for this not only wastes fuel but also could not increase the final miss distances on the average [6-7].

The guidance laws applied by antimissiles are still mainly PN. This is because: 1) at this stage the maneuverability (such as lateral acceleration) of antimissile is usually much higher than invading missiles and the maneuvers of invading missiles can be roughly treated in the same way as noises; and 2) advances in estimation and computer technology could partly compensate for the loss of response time.

It should be noted that until recently, the guidance strategies of most existing antimissiles are still heavily based on PN, even the very recent ARROW and PAC-3 antimissile systems [2, 8-9]. Many ongoing researches are still focused on PN- related topics, such as the Augmented Proportional Navigation (APN) laws [10], spiraling related laws (here spiraling means a periodic conical pitching and yawing motion) [12-15], time-varying effective navigation ratio [11], etc. These extensions advance PN along different directions and are suitable for different situations.

Experiences over decades prove that PN (and PN-related guidance laws) is simple, easy to use, and efficient when intercepting invading entities with much worse maneuverability than interceptors. Such entities typically

include manned aircrafts, old style non-maneuvering long-distance missiles, satellites, space debris, etc. The disadvantages of PN mainly lie in two aspects: 1) It will saturate when got is too small since the calculated

command goes to infinity; and 2) it does not consider other information such as inertias and current accelerations of invading missiles and antimissiles. PN-related guidance laws have the same problems.

Since traditional PN does not consider acceleration information (which might be available via estimation during the interception procedure) and kinematics inertia of invading missiles and antimissiles, some researchers fused Pursuit Evasion (PE) differential game theory [45-46] to missile interception. Differential PE game theory can find a variety of applications including criminal pursuing situations, collision avoiding designs in transportation systems, space debris tracking and collection, and other related areas. In typical PE games, pursuer(s) wish to capture evader(s), while evader(s) try to avoid capture. For a pursuer-evader pair, if at some point in time the distance between the pursuer and evader is less than some unit distance, the pursuer captures the evader. If the evader avoids this range forever, the evader wins.

Such PE consideration models the invading missile as the evader (E) and the antimissile as the pursuer (P). The evader and the pursuer’s dynamics are expressed as first order differential equations with time constants Eτ and Pτ , respectively.

The result of this effort is the augmented state space description of the interception system and the corresponding Differential Game Law Type 1 (DGL/1) [16].

.max max

/1( )DGL go E Pu u sign ZEM u sign y y t Z Z⎛ ⎞= = + + Δ −Δ⎜ ⎟⎝ ⎠ (3)

where ( )2 ex p ( ) 1y

E E E E EZ a τ θ θΔ = − + − (4)

( )2 e x p ( ) 1yP P P P PZ a τ θ θΔ = − + − (5)

where /E g o Etθ τ= (6)

/P g o Ptθ τ= (7)

and yEa is the evader’s current acceleration along the y

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direction (the direction perpendicular to the initial LOS). yPa can be explained similarly.

It is natural to note that EZΔ and

PZΔ are the compensative terms corresponding to the influences from evader’s current lateral acceleration and pursuer’s current lateral acceleration, respectively. As a result, theoretically

/ 1D G LZ E M in (3) is more suitable for predicting ZEM than the corresponding

P NZ E M in (2). In addition, since DGL/1 is derived from the fusion of PE game theory and modern optimal control theory, the worst situation is already considered implicitly in this guidance law. As a result, DGL/1 is more robust than PN, which is solely derived from optimal control theory. Under relatively loose constraints [17-18], theoretically, DGL/1 can reduce the final miss to zero without the drawbacks of the saturation in the optimal feedback control solution.

However, DGL/1 requires yEa , which is not easy to obtain

( yPa can be assumed accurately known to the antimissile

itself). The measurement of yEa , which is a higher order

characteristic compared to y and .y , involves much large

errors and much slower convergence rates in estimation. Extensive simulations indicate that under heavily noisy environments DGL/1 actually performs worse than PN or PN-related laws [10, 19-20], although theoretically DGL/1 should be “better than” them. For this reason, DGL/1’s and its guidance laws are not used in existing antimissile products.

To improve DGL/1’s performance, researchers consider compensating for the errors introduced by the slow convergence of y

Ea estimation. This resulted in DGL/C:

( ).

max max/( )DGL C go E Pc

u u sign ZEM u sign y y t Z Z⎛ ⎞= = + + Δ − Δ⎜ ⎟⎝ ⎠ (8)

where

( ) exp( )estE Ec

E

tZ ZτΔΔ = Δ − (9)

and the esttΔ is the assumed delay of y

Ea estimation. The difficulty in applying DGL/C is in determining

e s ttΔ . While a “good”

esttΔ can improve accuracy of the antimissile, a “bad” esttΔ can generate results even worse than what can be obtained via DGL/1. Shima et al [21] once suggested a time-varying

e s ttΔ formula that can provide better performance over relatively wider scope than “wild guesses”. Generally, it is difficult to obtain an

appropriatee s ttΔ , which severely limits the applicability in

realistic antimissiles.

Almost all researches about missile interception are based on the guidance laws introduced above, which we will call “traditional guidance laws”. Over decades such laws and corresponding slightly extended versions have proved their efficiency either in theory and/or in practice.

Problems to Basic Missile Interception Strategies

Recently real battlefield situations pose new difficulties for missile defense systems, which makes traditional guidance laws inefficient. The problems lie mainly in two aspects: 1) The guidance law problems; and 2) Resource management problems.

Recent technological advances have greatly improved the maneuverability of both invading missiles and antimissiles. However, such improvements have benefited invading missiles more due to their lower starting point. The result is that now the antimissiles no longer have great (previously 10 times or more) maneuverability advantages over the invading missiles. Currently ratios are often less than 5:1, sometimes even less than 2:1. This will cause more severe problems with PN terminal guidance saturation and results in much larger miss distance. Similarly, DGL/1 and DGL/C will also have worse performance when interceptor missiles have only a marginal maneuverability advantage.

Invading missiles might have carefully selected time-varying intentional spiraling movements to interfere with antimissile’s guidance law calculation and consume more fuel of the antimissile, compared to earlier cases in which spiraling movements were mainly unintentional, small, and caused by manufacturing inaccuracies. In such cases, such spiraling motion can not be ignored and guidance laws should carefully compensate for them.

Guidance laws such as PN, DGL/1 and DGL/C all assume that the missile speeds along the LOS direction are constant. This is not true any more for modern intelligent invading missiles. Modern intelligent invading missiles can control their flight velocities if necessary. As a result, the got in

laws are no longer fixed and have to be estimated and updated over the whole time horizon, which will increase uncertainty and computation.

The derivation of traditional guidance laws all assume that the separation theorem holds, which states that under noisy situations an optimal controller and optimal estimator designed separately are optimal jointly. However, this theorem is true only for linear quadratic Gaussian control problems with unbounded controls. While the linear

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quadratic assumption does not harm the law derivation too much, modern missile flight estimation and tracking include special complex clutters and the overall effect is not very close to Gaussian. The assumption about unbounded control is obviously not true, too. These facts force researchers to design the system according to partial separation theorem, which says the estimator can still be independently designed, but the controller should be designed based on estimator design. In addition, it should be noted that when combined with the controller, traditional 2D estimation approaches (such as 2D extended Kalman filters, particle filtering, etc.) often produce disappointing performance under new time-varying fast-maneuvering 3D invading missile situations. This cannot be tolerated as in old style invading missile interceptions.

In addition, when multiple antimissiles aim at the same invading missile, cooperative coordination and formation control will be a problem. If every antimissile applies exactly the same guidance law with the same parameters, the interception success probability might not be impressively improved, for a single mode of elusive maneuver from the invading missile would be efficient for escaping from all such antimissiles. How to adapt the guidance of cooperative antimissiles poses great difficulty for missile interception researchers.

Modern and future missile attack scenarios are often assumed to have multiple attack waves and in each wave there might be multiple invading missiles. Modern missile defense systems often require optimal resource management which can balance the tradeoff between success probability, safety zone boundaries (the safety margin), cost, etc.

In addition, the defense system should be able to adapt the missile engagement plan online efficiently. This involves tradeoff between computation, accuracy, response time, etc. The system should reconfigure the resource engagement at appropriate time and under appropriate situations. Due to the tradeoff, usually the engagement should not go to two extremes: 1) Re-plan every timestep (which will make the system consume too much computation resources and cause bad fast-response performance); or 2) Only plan at the start of the game (which might make the system difficult to deal with online new situations).

Moreover, the system should also provide the best guidance law(s) for the antimissiles engaged. The best laws for different antimissiles might be different from one another. This should be done according to the available information such as the invading missile type, invading missile position/velocity, antimissile type, antimissile fuel, what kind(s) of airborne radar(s) antimissile missile carries, etc.

Outline of This Paper

In this paper, we propose a multi-missile interception framework, which can deal with the problems discussed above. It integrates and advances recent studies on pursuit-evasion game based guidance laws and game theoretic resource management.

This paper is organized as follows. In Section 2, we will summarize the technical approach, which includes the solution of the guidance problem and resource management problem. Section 3 describes the experimental results and explanations. Section 4 provides conclusions for the paper.

2. TECHNICAL APPROACH OF MULTI-MISSILE

INTERCEPTION SYSTEM

Framework of Our Multi-Missile Interception Framework

The framework of our multi-missile interception simulation system has the following fully coupled hierarchical major parts: 1) Signal Processing Level (to search, identify, tracking possible invading missiles based on the information provided by various sensors); and 2) Strategy Level (to generate the optimal and efficient engagement strategies and guidance strategies). 3) Execution Level (to generate specific control commands for antimissiles in the whole time horizon); 4) Interception Evaluation Model (to automatically evaluate executions of missile interception and determine whether new missiles need to be launched or the whole engagement plan should be recalculated); 5) Knowledge and Information Module (to store, provide, and automatically update the knowledge and rules about missile interception); 6) Human User Interface Module (to provide an easy-to-use graphic interface so that human commanders can conveniently choose scenarios from typical situations, modify default scenarios to model new situations, or construct completely new scenarios). In this paper, we mainly address the strategy level and execution level.

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Fig. 2: A multi-missile interception approach

It might be illustrative to note that Execution Level and Strategy Level correspond to the three classes (PN, DGL/1, and DGL/C) of new problems under modern missile interception situations-Guidance law problems and Resource management problems, respectively, although they are tightly coupled due to complexity of the system. In this paper, we focus on the strategy level and execution level.

The information flow under this framework is as follows. At this user interface module, human commander(s) determine the scenario that the whole system will simulate. After the scenario is set up, the missile interception system will process signals obtained via various sensors such as Electro-Optical (EO) sensors and Infrared (IR) sensors. The strategy level supervises both the overall resource management (antimissile engagement) and the individual interception strategies (guidance law). A Resource engagement algorithm will determine how many and which antimissiles will be launched for interception. Once the interception strategies are determined, each antimissile (also an agent) will calculate and execute the guidance law at the execution level. At each timestep, evaluation module will determine whether invading missiles are destroyed, whether the overall engagement should be modified, whether current evaluation results have “knowledge value” and should be deposited in knowledge module. Over the whole procedure, the knowledge module performs important aiding roles.

Strategy Level

Input to strategy level is the entities’ agent information (such as types, speeds, accelerations, locations, ranges, heading directions, motion patterns, etc.), which is the output of the signal processing level. Outputs of strategy level include answers to

1) Which antimissile(s) are assigned to which invading missile?

2) Which antimissile applies which guidance laws? 3) Which initial parameters and reference biases (if

necessary) should be fed to guidance laws?

The assignment algorithm will pursue optimality via multiple objective optimization theory. This is because usually there is more than one criterion stating whether an assignment is “good” or “bad”. In addition, some of these criteria are contradictory. For example, success probability and cost are contradictory: to ensure higher interception success probability, an efficient approach is to launch more antimissiles against one invading missiles. However, the cost control requires launching as few as possible antimissiles. Another efficient approach is to launch a more advanced antimissile which has smaller time constant,

higher acceleration bounds, faster maximum velocity, more complex airborne radars and computer chips, etc. However, every complexity adds the cost. Similarly, (1) a larger safety zone (or larger safety margin) and cost are contradictory; and (2) a more advanced guidance law (including the corresponding estimation strategy) and faster response time are contradictory.

One possible approach to coordinate so many contradictory criteria is to apply Pareto game theory [29, 43-44]. A sufficient condition to obtain a Pareto solution is to take a convex linear combination of the different criteria. This is equivalent to the proven weighting average method, which assigns different weights to different criteria and takes the average so that the multi-objective optimization problem becomes a single objective optimization problem. Each set of weights corresponds to a different Pareto solution. If each criterion is linear in the decision variables, the assignment problem is reduced to a linear programming optimization problem [30]. However, it should be noted that a linear programming optimization algorithm might not converge, even for some small scale problems [30] if the criteria are not linear. Nonlinear programming should be used but computation might be prohibitive. This is the reason why for engagement problems people usually pursue a suboptimal solution instead of the optimal solution. We will apply proven suboptimal suggestions [31-32] to simplify the optimization so that computation and optimality can be balanced.

In our system the weighting average method is as follows. It may be modified later and more considerations may be added.

1 1 2 2 3 3 4 41

P P P PeN

j j j j

jPayoff W W W W

=

= × + × + × + ×∑ (10)

P a y o f f : the payoff function on which the scalar optimization problem is based.

, 1 , 2 , 3 , 4iW i = : the weights.

eN : the number of invading missiles involved.

1P j : the probability that the j th invading missile can be successfully intercepted

2P j : the interception distance safety margin, which is the distance between the collision point and the target protected by antimissiles. The bigger the better.

3P j : the interception time safety margin, which is the time between the collision time and the assumed time that the invading missile reaches the target protected by antimissiles. The bigger the better.

4P j : the gain/loss ratio value, which is calculated as

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1(1 P )j− × (the destruction that can be caused by

j th invading missile)/(the sum of the costs of the antimissiles engaged to j th invading missile).

Choosing the weights greatly alters the payoff. Our system provides a combined approach in which both experiences from commanders and automatic learning/feedback adaptation results are applied. The details of machine learning can be found in [33-35]. Our system’s combined approach is as follows. As a preliminary step each criterion should be normalized. For example, divide each criterion by its expected maximum value. The criteria generally have different physical meanings and unless they are normalized, it would be difficult to interpret the meaning of the weights. If they are normalized, then the weights can be interpreted as the percent contribution of the individual criteria to the total pay-off function.

1) Offline: The system first obtains and deposits the weights via automatic learning based on collected training scenarios.

2) Pre-online: If the default weights do not satisfy human commander, the commander can modify them. The modifications will overwrite the suggested default values and serves as the initial weights of the system engagement criteria.

3) Pre-online: The commander decides whether “automatic adaptation” should be chosen. If yes, in the later missile interception procedure the initial weights might be automatically adjusted by the system according to feedbacks. Otherwise, the initial weights will be fixed all over the time horizon.

4) At each timestep, the system will evaluate the situation and decide whether a reconfiguration is necessary. If “automatic adaptation” is chosen, the system will automatically update the weights.

When there are imperfect information links between the antimissiles, each antimissile group (the antimissiles among which the communication links are not completely broken so that each group member knows the information about all other members in the same group) will apply the same engagement and reconfiguration strategy.

In addition, the strategy level will also provide guidance law suggestion served as the initial guidance laws for antimissile agents. Note that antimissile agents might modify the law parameters by themselves according to real time information. The details of guidance law in our system will be explained in the next subsection (the execution level).

Execution Level

Inputs to execution level are engagement plan and guidance law suggestions. The execution level will translate them

according to the specific guidance law to calculate specific commands for the next timestep. The commands will be applied to the antimissiles so that the whole battlefield evolves under noisy environments. If necessary, the execution level will modify the parameters of the guidance law.

The objective of the guidance law in our system is to try our best to solve the new problems in modern battlefield discussed in subsection 1.2.2 in an integrated mode. The guidance law integrates and advances recent researches including multi-player PE game [45-46], state space modeling with time-varying parameters [2], acceleration estimation delay [36], reachable sets [2, 36], 3D/multi-model estimator [37], bi-control (concord-tail control) [38], airborne imaging information processing [39], optimal cooperative biasing [40], intentional spiraling [14], etc. to a closed form guidance law. For simplification, we label the guidance law in our system as “DGL/M”, which means it is a modification and integration of existing differential game based guidance laws. In addition, simulations prove the advantages of our integrated law in both perfect information situations and imperfect information situations.

Assumptions:

1) The terminal (end-game) trajectory can be linearized around LOS.

2) The time-varying speeds of the invading missile and antimissile are measured accurately enough so that they can be assumed to be known.

3) The minimum turn radius of invading missile and antimissile are known.

4) Invading missiles and antimissiles can be approximated by linear control dynamics.

5) The maneuvering dynamics of the antimissile can be approximated by canard and tail equations.

6) The invading missile has no information about the antimissiles, except the fact that there might be intercepting antimissiles thus it should have some maneuvers to elude the interceptors.

7) The antimissile has noisy measurements of the invading missile’s relative position.

8) Invading missile might have intentional spiraling. 9) Antimissile might have airborne imaging radars so that

the heading direction of invading missile might be available during the terminal period.

Note that 8) and 9) are optional. That is to say, although the guidance law will have terms corresponding to 8) and 9), such terms will be automatically zero or not be affected if 8) and/or 9) do not hold. Thus whether 8) or 9) exist does not affect the guidance law’s expression and application.

The time-varying state space model is as follows. According

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to the fact that 3D space can be perpendicularly decomposed to two identical planes, the model is presented on the xy plane for simplification. For xz plane the model is similar. For physical meanings of elements (except

EJ ) in the state vector, see Fig. 1 and PN, DGL/1 and DGL/C.

EJ is the jerk of the invading missile (the derivative of the invading missile’s acceleration).

.

P EX A X B u C u v= + + + (11)

where

.

1 2 3 4 5 6 7[ , , , , , , ] [ , , , , , , ]T TP E P E EX y y a a J x x x x x x xφ φ= = (12)

0 1 0 0 0 0 00 0 -1 1 - - 0

0 0 -1/ 0 0 0 0

0 0 0 -1/ 0 0 0

0

xP xE

P

E

a a

A

ττ=

2

0 1/ 0 0 0 0

0 0 0 1 / 0 0 0

0 0 0 - 0 0 0

P

E

V

V

ω

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

(13)

2 m i nP[ 0 0 / ( r ) 0 0 0 0 ] T

P PB V τ= (14)

2 m inE[ 0 0 0 / ( r ) 0 0 0 ]T

E EC V τ= (15)

m ax/cP P Pu a a= (16)

m ax/cE E Eu a a= (17)

where PV and

EV are all time-varying parameters. minPr

and minEr are the minimum radius of the antimissile and

invading missile, respectively. xPa and xEa are Pa ’s and

Ea ’s projections on the x axis, respectively. v is the process noise. ω is the invading missile’s spiraling frequency. Note that when the interceptor missile has bi-control (both concord and tail) with direct lifts [38], the Pu

is the equivalent value calculated so that the same Pa is achieved as

11

11

Pc cc

c P

Pt tt

t P

P Pc Pt

a ddu s

a ddu s

a a a

τ

τ

⎧ −= +⎪ +⎪⎪ −⎪ = +⎨ +⎪⎪ = +⎪⎪⎩

(18)

where cd and

td are non-negative direct lifts from canard control and tail control, respectively.

The guidance law for the antimissile (canard-tail bi-control) is

( )

( ) ( ( , / ) )c n e w

t n e w t g o P

u s ig n Z E M

u s ig n Z E M s ig n f d t

αβ τ

=

= (19)

where

1 2 ( )new go P E aP aEZEM x x t Z Z Z Z Z Biasφ φ ωλ= + + −Δ + Δ + Δ −Δ + Δ + (20)

( , ) (1 )( 1)f e ζδ ζ δζ δ ζ−= + − + − (21)

1, 0, 0α β α β+ = ≥ ≥ (22)

( 2 ( / 2 ) 1 ) BB i a s m N R= − − (22)

where

N : the total number of the antimissiles engaged to the same invading missile.

m : when there are multiple antimissiles engaged to the same invading missile, m is the index of the antimissile among. [1, ]m N∈ .

λ : Noise adjustment factor, calculated as

1SN R

SN Rλ =

+ (28)

where S N R is the signal-to-noise ratio. Simulations show that this adjustment can efficiently improve performance of DGL/M even in highly noisy environments, which often cause big problems to existing complex guidance laws.

BR is the kill radius of the antimissile. When the miss distance is within this range, the invading missile will be killed by the antimissile with probability 1.

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5 ( ( , ) )P P f P g oZ x t t V tφΔ = Λ − (23)

6 ( ( , ) )E E f E g oZ x t t V tφΔ = Λ − (24)

4 72 3

1 cos( ) sin( )go go got t tZ x xω

ω ω ωω ω

− −Δ = + (25)

23 ( ( / ) ( , ) )a P P g o P P fZ x t t tτ ψ τΔ = + Ξ (26)

/ 24 ( ( / ) ( , ))Etcenter

aE E go E E fZ x e t t tτ τ ψ τ−Δ = + Ξ (27)

where

Δ t: the time delay between the time when the of invading missile changes its maneuver command and the time when such change is detected by the antimissile.

( ) 1e ζψ ζ ζ−= + − (29)

/

2

1( , ) ( ) , ,( )

if f ft t t

i f xiti i

et t a d d d i P EV

ς τ

ζ ξζ ς ξ ζ

τ ς

−

Ξ = =∫ ∫ ∫ (30)

( , ) ( ) , ,ft

i f itt t V d i P Eζ ζΛ = =∫ (31)

/ min max4

/4 4

/2 min min max4

( ) , if ( ) ( )

0.5 ( )

0.5 ( )( / )(1 ), if ( ) ( )

E

E

E

t d dE E

tcenter

t d dE E E E

x t t e sign a sign a

x x t t e

sign x V r e sign a sign a

τ

τ

τ

−

−

−

⎧ − =⎪⎪= −⎨⎪+ − =⎪⎩

(32)

where

/ /min min4

/ /max min4

( ) ( / )(1 )

( ) ( / )(1 )

E E

E E

t tdE E E

t tdE E E

a x t t e V r e

a x t t e V r e

τ τ

τ τ

− −

− −

= − − −

= − + − (33)

For simplification and clarity, we do not repeat the extensive mathematical derivations of this law (For details, see [2, 14, 36-40]). Instead, we choose to briefly clarify physical meanings of the main items in the law so that it is more illustrative. If necessary, we will compare them with the corresponding earlier guidance laws.

a) newZ E M : the predicted zero effort miss distance in this law. As we can see, it is constructed by augmenting/compensating DGL/C.

b) ,α β : the weight factors for antimissile’s concord and tail controls, respectively. According to researches in missile flight control [38], most of the time canard control is better than tail control. However, for antimissiles when the distance between the antimissile

and the invading missile is short, concord control is very easy to saturate. At this time to add a tail control can improve the performance.

c) PZ φΔ : the ZEM caused by

Pφ .

d) EZ φΔ : the ZEM caused by

Eφ .

e) a PZΔ : the ZEM caused by current acceleration of the

antimissile. f)

a EZΔ : the ZEM caused by current acceleration of the invading missile (not considering the spiraling).

g) Z ωΔ : the ZEM caused by the spiraling of the invading missile. If no such compensating term (such as PN, DGL/1, DGL/C), the guidance law will force the antimissile to weave thus causing a great deal of trajectory dispersion, causing both fuel wasting and final large miss distance. This is because such guidance laws think the maneuver is non-rotating and will continue in current direction thus lead the antimissile in the wrong way. By compensating the guidance law with Z ωΔ , such problems can be avoided.

Under noisy environments, 1 2 3 4 5 6 7, , , , , ,x x x x x x x in the law should be replaced by the corresponding estimates. Note that when the antimissile has airborne imaging equipment, the estimation of 4 ( )x t can be modified by imaging fusion (by identifying the heading direction of the invading missile). The estimation of the state variables will be performed by the 3D estimator introduced in [37] and the Δ t will be estimated by the multi-model adaptive estimator investigated in [42]. Note the complex coordinate transformation (between rotation coordinate system and xyz perpendicular coordinate system) in the 3D estimator.

The simulations in the next subsection show the efficiency of the modified guidance law (DGL/M) via comparison with the popular PN, DGL/1 and DGL/C under different situations.

3. SIMULATIONS

To illustrate the advantage of the new guidance law DGL/M, over existing guidance laws, we implemented the three most popular guidance laws (PN, DGL/1, and DGL/C) in practical applications. We have compared them under different situations (perfect/imperfect information). For each kind of comparison we ran 100 times and took the average. Each time there are three invading missiles at the same time, and we might have 3, 5, or 10 antimissiles available. The results show that in most cases the new guidance law has smaller final miss distances, higher interception success probability, and smoother flight path

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(so that less control energy is necessary). Since the complexity of the new guidance law (plus the complex 3D/multi-model estimation), the computation time of the new guidance law is larger. However, as we have simulated, such computation time is still largely acceptable under certain conditions. We used 3G CPU/1G Memory computer operated by Windows XP.

The comparison of the four guidance laws in perfect information hit-to-kill scenario is as follows. There are three invading missiles and three antimissiles. In this scenario we set the bombing range of the antimissiles as 0.2 feet so that we can simulate situations in which “hit-to-kill” is required. Another reason why we set the bombing range so small is that in perfect information situation the miss distances of all four guidance laws are very small and if we set the bombing range large the interception success probability will not have any difference (all will be 1 or nearly 1).

Table 1 Comparison with perfect information

Guidance Law PN DGL/1 DGL/C DGL/M Average final

miss distance (in feet)

0.3445 0.1186 0.1119 0.1015

Interception success

probability ( BR =0.2 feet)

0.8167

0.8967

0.9000

0.9200

The comparison of the four guidance laws in imperfect information scenario is as follows. There are three invading missiles and five antimissiles thus there exist cooperative interception against the same invading missile.

Table 2 Comparison with imperfect information

Guidance Law

PN DGL/1 DGL/C DGL/M

Average final miss distance

(in feet)

117.4268 92.0071 92.0754 42.1482

Interception success

probability ( BR =20 feet)

0.7167

0.3400

0.2633

0.7433

A comparison of the interception procedures of different interception strategies is as follows (Fig. 3). Plotted curves are the time varying interception miss distances. We can see that our DGL/M interception strategy is smoother than other strategies. This will help to avoid wasting fuels and unnecessary formation/reconfiguration calculations. The

simplest and oldest PN strategy has the second smoothest path. Compared to PN, DGL/1 and DGL/C have larger derivations during procedure, although theoretically they are considered as superior to PN. This is because the estimation errors in the additional compensating terms in DGL/1 and DGL/C overwhelmed the benefits from the compensating terms. Among DGL/1 and DGL/C, DGL/C is slightly better than DGL/C because it provides the estimation delay factor to partly compensate for the estimation errors. Conceptually, DGL/M is more successful because it has more carefully constructed factors for estimation error compensating.

0 5 10 15 20 25 30 35-20

0

20

40

60

80

100

120

time in seconds

Mis

s in

met

ers

comparison of miss distances of different approaches

BMD Bicontrol Imagingfusion

PN

DGL/1

DGL/C

Fig. 3 Comparison of the interception miss distances

The computation time comparison is as follows. DGL/M requires more computation but it is still at around 0.1 second level. We will continue to refine DGL/M algorithm so that the computation is more efficient (in seconds, including estimation).

Table 3 Comparison with imperfect information

Guidance Law PN DGL/1 DGL/C DGL/M Computation time for each antimissile at each timestep

0.0313

0.0512

0.0532

0.1227

4. CONCLUSIONS

In this paper, we reviewed major research efforts about missile interception, investigated new problems emerged in modern multi-missile interception situations, and developed a general framework for advanced multi-missile

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interception simulation, which can accommodate a large class of realistic missile defense scenarios and simulate different degrees of optimal strategies. We introduced our Differential Game Law Type M (DGL/M) that combines advantages of game-theoretic missile interception and resource management. We have illustrated the advantages of our approach via simulations.

ACKNOWLEDGEMENTS

Dr. Mo Wei’s research was partly supported by the Army under contract number W911NF-06-C-0016. Dr. Genshe Chen, Dr. Jose B. Cruz, and Dr. Erik Blasch’s research was partly supported by the MDA under contract number HQ0006-07-C-7733. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the Army or MDA.

The authors thank Drs. Khanh Pham from AFRL/VSSA and Mou-Hsiung Chang from ARO for their high degree of technical involvement in the project and support throughout the effort.

REFERENCES

[1] Adler, F. P., “Missile Guidance by Three Dimensional Proportional Navigation,” Journal of Applied Physics, Vol. 27, No. 5, 1956, pp. 500-5-7.

[2] Shima, T., Shinar, J., and Weiss, H., “New Interceptor Guidance Law Integrating Time-Varying and Estimation-delay Models,” Journal of Guidance, Control, and Dynamics, Vol. 26, No. 2, March-April 2003.

[3] Willems, G., “Optimal Controllers for Homing Missiles,” U.S. Army Missile Command, Redstone Arsenal, Als., Rept. RE-TR-68-15, Sept. 1968.

[4] Willems, G., “Optimal Controllers for Homing Missiles with Two Time Constants,” U.S. Army Missile Command, Redstone Arsenal, Als., Rept. RE-TR-69-20, Oct. 1969.

[5] Price, C. F., “Optimal Stochastic Guidance Laws for Tactical Missiles,” The Analytical Sciences Corporation, Reading, Mass., Rept. TR-170-2, Sept. 1971.

[6] Dionne, D. and Michalska, H., “Decision-Directed Adaptive Estimation and Guidance for an Interception Endgame,” Journal of Guidance, Control, and

Dynamics, Vol. 29, No. 4, pp. 970-980, July-August 2006.

[7] Shinar, J. and Steinberg, D., “Analysis of Optimal Evasive Maneuvers Based on a Linearized Two-Dimensional Model,” Journal of Aircraft, Vol. 14, No. 8, 1977, pp. 795-802.

[8] Hughes, D., “Next Arrow Test This Summer After Scoring Direct Hit,” Aviation Week and Space technology, Vol. 146, No. 12, 1997, p. 34.

[9] Philips, H. E., “PAC-3 Missiles Seeker Tests Succeed,” Aviation Week and Space technology, Vol. 150, No. 12, 1999, P. 30.

[10] Anderson, G. M., “Comparison of Optimal Control and Differential Game Intercept Missile Guidance Laws,” Journal of Guidance and Control, Vol. 4, No. 2, March-April 1981.

[11] Zarchan, P., Tactical and Strategic Missile Guidance, Fourth Edition, Progress in Astronautics and Aeronautics, AIAA, Reston, VA 2002.

[12] Zarchan, P., “Proportional Navigation and Weaving Targets,” Journal of Guidance, Control, and Dynamics, Vol. 18, Sept.-Oct. 1995, pp. 969-974.

[13] Marks, G. M., “Multiple Models Adaptive Estimation (MMAE) for Improving Guidance Performance against Weaving Targets,” AIAA Guidance, Navigation, and Control Conference and Exhibit, August 2006, Keystone, Colorado.

[14] Zarchan, P., “Tracking and Intercepting Spiraling ballistic Missiles,” Proceedings of IEEE PLANS Conference, San Diego, CA, March 2000.

[15] Shima, T. and Shinar, J., “Time-Varying Linear Pursuit-Evasion Game Models with Bounded Controls,” Journal of Guidance, Control, and Dynamics, 2002, 25(3), pp. 425-432.

[16] Shinar, J., “Solution Techniques for Realistic Pursuit-Evasion Games,” Advanced Control Dynamics Systems, 1981, 17, pp. 63-124.

[17] Shinar, J., and Shima, T., “A Game Theoretical Interceptor Guidance Law for Ballistic Missile Defense,” Proceedings of the 35th Conference on Decision and Control, Kobe, Japan, Dec. 1996.

[18] Lipman, Y., Shinar, J., and Oshman, Y., “Stochastic Analysis of the Interception of Maneuvering

11

Antisurface Missiles,” Journal of Guidance, Control, and Dynamics, Vol. 20, No. 4, July-August, 1997.

[19] Nesline, F. W., “A New Look at Classical vs Modern Homing Missile Guidance,” AIAA Guidance and Control Conference, Colorado, Aug. 6-8, 1979.

[20] Shinar, J. and Turetsky, V., “On Improved Estimation for Interceptor Guidance,” Proceedings of the American Control and Conference, Anchorage, AK May 8-10, 2002.

[21] Shima, T., Oshman, Y., and Shina, J., “Efficient Multiple Model Adaptive Estimation in Ballistic Missile Interception Scenarios,” Journal of Guidance, Control, and Dynamics, Vol. 25, No. 4, July-August 2002.

[22] Tartakovsky, A. G., and Blazek, R., “Effective Adaptive Spatial-Temporal Technique for Clutter rejection in IRST,” SPIE Proceedings: Signal and Data Processing of Small Targets, Vol. 4048, pp. 85-89, 2000.

[23] Baraboski, E. and Guerci, J. R., “Knowledge-aided Adaptive Radar at DARPA,” IEEE Signal Processing Magazine, pp. 41-50, Jan. 2006.

[24] Wicks, M. C., Rangaswamy, M., Adve, R., and Hale, T. B., “Space-Time Adaptive Processing,” IEEE Signal Processing Magazine, pp. 51-65, Jan. 2006.

[25] Shan, T. J., and Kailath, T., “Adaptive Beaming for Coherent Signals and Interference,” IEEE Transaction of Acoust., Speech, Signal Processing, Vol. ASSP-33, pp. 527-536, June 1985.

[26] Blasch, E. P., “Sensor Management Issues for SAR target Tracking and Identification,” Euro Fusion 99, Stratford-upon-Avon, UK, 1999, pp. 279-286.

[27] Rozovskii, B., Tartakovsky, A., and Yaralov, G., “An Adaptive Bayesian Approach to Fusion of Imaging and Kinematic Data,” Proceedings of the 2nd International Conference on Information Fusion (Fusion’ 99), Vol. 2, Sunnyvale, CA, 6-8 July, 1999, pp. 1029-1036.

[28] Balsch, E. P., Cilpepper, E. B., and Johnson, J. J., “Using an Adaptive INS, Monopulse MUSIC, and STAP (AIMS) for Targeting,” IEEE National Radar Conference, Dallas, TX, May 12-13, 1997, pp. 225-230.

[29] Ahmed, E.; Elgazzar, A. S., "On Local Prisoner's Dilemma Game with Pareto Updating Rule,"

International Journal of Modern Physics C, Volume 11, Issue 08, pp. 1539-1544 (2000).

[30] Chivatal, V. Linear Programming, New York: W. H, Freeman and Company, 1983.

[31] Li, D., Cruz, J. B., and Schumacher, C. J., "Stochastic Multi-Player Pursuit Evasion Differential Games," International Journal of Robust and Nonlinear Control, Published online May 16, 2007, to appear.

[32] Schouwenaars, T., Feron, E., Moore, B. D, and How, J., “Mixed Integer Programming for Multi-vehicle Path Planning,” European Control Conference, September 2001.

[33] Kaelbling, L. P., M. L. Littman, and A. W, Moore. 1996, “Reinforcement Learning: A Survey,” Journal of Artificial Intelligence Research 4: 237-285.

[34] Minton, Steven, editor. 1993. Machine Learning Methods for Planning. San Francisco: Morgan Kaufmann.

[35] Mechner, David. A. 1998. All Sytems Go. The Sciences 38 (Jan/Feb 1998): 32-7.

[36] Shinar, J., and Glizer, V. Y., “Solution of a Delayed Information Linear Pursuit-Evasion game with Bounded Controls,” International Game Theory Review, Vol. 1, Nos. 3&4 (1999), pp. 197-217.

[37] Weiss, H., and Hexner, G., “A Simple Structure for a High Performance 3-D Tracking Filter,” Technion-Israel Inst. Of Technology, Dep. Of Aerospace Engineering, TAE Rept. 858, Haifa, Israel, Feb., 2001, pp. 15-23.

[38] Shima, T. and Golan, O. M., “Bounded Differential Games Guidance Law for Dual-Controlled Missiles,” IEEE Transactions on Control Systems Technology, Vol. 14, No. 4, July 2006.

[39] Arad. D, Improving the Interception Performance of Air to Air Missiles Using Target Orientation Observations, PhD thesis, Heshvan: Haifa, Israel, 2001.

[40] Lipman, Y., Analysis of Anti-missile Defense Scenarios by Using a Linear Model, D. Sc. Dissertation, Technion-Israel Inst. Of Technology, Haifa, Israel, may 1995 (in Hebrew).

[41] Thukrai, A. and Innocenti, “A Sliding Mode Missile Pitch Autopilot Synthesis for High Angle of Attack

12

Maneuvering,” IEEE Transaction on Control, System, Technology, Vol. 6, No. 3, pp. 359-371, May 1998.

[42] Shinar, J., and Turetsky, V., “Improved Estimation Is a Prerequisite for Successful Terminal Guidance,” Proceedings of IMechE Vol. 219, Part G: J. Aerospace Engineering, Special Issue, 2005, pp. 145-156.

[43] Zadeh, L. A., "Optimality and Non-Scalar-Valued Performance Criteria," IEEE Trans. Automat. Contr., Vol. AC-8, pp. 59-60, Jan. 1963.

[44] Cunha, N. O. and Polak, E., "Constrained Minimization under Vector-Valued Criteria in Finite Dimensional Spaces," Journal of Mathematical Analysis and Applications, Vol. 19, pp. 103-124, July 1967.

[45] Mo Wei, Genshe Chen, Jose B. Cruz, Jr., Leonard Hayes, and Mou-Hsiung Chang, “A Decentralized Approach to Pursuit-Evasion Games with Multiple Superior Evaders,” Proceedings of IEEE Conference on Intelligent Transportation Systems, Canada, 2006.

[46] Li, D., Cruz, J.B., Jr., Chen, G., Kwan, C., and Chang, M.-H., “A Hierarchical Approach To Multi-Player Pursuit-Evasion Differential Games,” Proceedings of Joint IEEE Conference on CDC-ECC05, Spain, 2005.

BIOGRAPHY

Mo Wei has a BS degree in EE from Northern JiaoTong University and a master’s degree in EE from Tsinghua University. He received Ph. D degree from Department of Electrical and Computer Engineering, Ohio State University in Autumn of 2006. His research interests include game control and corresponding applications, such as

non-ideal games, coupling game theory, sharing creditability game theory and bottom-line game theory, multiplayer games with civilian players, decentralized multiplayer pursuer-evader games, game application in networking, etc.

Genshe Chen received his B. S. and M. S. in electrical engineering, Ph. D in aerospace engineering, in 1989, 1991 and 1994 respectively, all from Northwestern Polytechnical University, Xian, P. R. China. He did postdoctoral work

at the Beijing University of Aeronautics and Astronautics and Wright State University from 1994 to 1997. He worked at the Institute of Flight Guidance and Control of the Technical University of Braunshweig (Germany) as an Alexander von Humboldt research fellow and at the Flight Division of National Aerospace Laboratory of Japan as a STA fellow from 1997 to 2001. He was a Postdoctoral Research Associate in the Department of Electrical and Computer Engineering of The Ohio State University from 2002 to 2004. Since February 2004, Dr. Chen has been with the Intelligent Automation, Inc., Rockville, MD. He has served as the Principal Investigator/Technical lead for more than 15 different projects, including maneuvering target detection and tracking, joint ATR and tracking, cooperative control for teamed unmanned aerial vehicles, a stochastic differential pursuit-evasion game with multiple players, multi-missile interception, asymmetric threat detection and prediction, space situation awareness, and cyber defense, etc. He is currently the program manager in Networks, Systems and Control, leading research and development efforts in target tracking, information fusion and cooperative control. His research interests include guidance and control of aerospace vehicle, GPS/INS/image integrated navigation systems, target tracking and information fusion, cooperative control and optimization for military operations, computational intelligence and data mining, hybrid system theory and Markov chain, signal processing and computer vision, cooperative and non-cooperative game theory, Bayesian networks, Influence Diagram, and GIS.

Jose B. Cruz, Jr. received his B.S. degree in electrical engineering (summa cum laude) from the University of the Philippines (UP) in 1953, the S.M. degree in electrical engineering from the Massachusetts Institute of Technology (MIT), Cambridge in 1956, and the Ph.D. degree in electrical engineering from the University of Illinois,

Urbana-Champaign, in 1959. He is currently a Distinguished Professor of Engineering and Professor of Electrical and Computer Engineering at The Ohio State University (OSU), Columbus. Previously, he served as Dean of the College of Engineering at OSU from 1992 to 1997, Professor of electrical

and computer engineering at the University of California, Irvine (UCI), from 1986 to 1992, and at the University of Illinois from 1965 to 1986. He was a Visiting Professor at MIT and Harvard University, Cambridge, in 1973 and Visiting Associate Professor at the University of California, Berkeley, from 1964 to 1965. He served as Instructor at UP in 1953–1954, and Research Assistant at MIT from 1954 to 1956. He is the author or coauthor of six books, 21 chapters in research books, and numerous articles in research journals and refereed conference proceedings.

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Dr. Cruz was elected as a member of the National Academy of Engineering (NAE) in 1980. In 2003, he was elected a Corresponding Member of the National Academy of Science and Technology (Philippines). He is a Fellow of the Institute of Electrical and Electronics Engineers, Inc. (IEEE) elected in 1968, a Fellow of the American Association for the Advancement of Science (AAAS), elected 1989, a Fellow of the American Society for Engineering Education (ASEE), elected in 2004, and a Fellow of the International Federation on Automatic Control (IFAC) elected in 2007. He received the Curtis W. McGraw Research Award of ASEE in 1972 and the Halliburton Engineering Education Leadership Award in 1981. He is a Distinguished Member of the IEEE Control Systems Society and received the IEEE Centennial Medal in 1984, the IEEE Richard M. Emberson Award in 1989, the ASEE Centennial Medal in 1993, and the Richard E. Bellman Control Heritage Award, American Automatic Control Council (AACC), 1994. In addition to membership in NAE, ASEE, and AAAS, and IEEE, he is a Member of the Philippine American Academy for Science and Engineering (Founding member, 1980, President 1982, and Chairman of the Board, 1998–2000), Philippine Engineers and Scientists Organization (PESO), National Society of Professional Engineers, Sigma Xi, Phi Kappa Phi, and Eta Kappa Nu. He served as a Member of the Board of Examiners for Professional Engineers for the State of Illinois, from 1984 to 1986. He served on various professional society boards and editorial boards, and he served as an officer of professional societies, including IEEE, where he was President of the Control Systems Society in 1979, Editor of the IEEE Transactions on Automatic Control, a Member of the Board of Directors from 1980 to 1985, Vice President for Technical Activities in 1982 and 1983, and Vice President for Publication Activities in 1984 and 1985. He served as Chair (2004–2005) of the Engineering Section of the American Association for the Advancement of Science (AAAS).

Erik Blasch received his B.S. in mechanical engineering from MIT and Masters in mechanical and industrial

engineering from Georgia Tech and MBA, MSEE, from Wright State University and a PhD from WSU in EE. Dr. Blasch also attended Univ of Wisconsin for an MD/PHD in Mech. Eng until being called to Active Duty

in the United States Air Force. Currently, he is a Fusion Evaluation Tech Lead for the Air Force Research Laboratory, Adjunct Professor at WSU, and a reserve Maj with the Air Force Office of Scientific Research. Dr. Blasch was a founding member of the International Society of Information Fusion (ISIF) and the 2007 ISIF President. Dr. Blasch has many military and civilian career

awards; but engineering highlights include team member of the winning ‘91 American Tour del Sol solar car competition, ’94 AIAA mobile robotics contest, and the ’92 AUVs competition where they were first in the world to automatically control a helicopter. Since that time, Dr. Blasch has focused on Automatic Target Recognition, Targeting Tracking, and Information Fusion research compiling 200+ scientific papers and book chapters. He is active in IEEE and SPIE including regional activities, conference boards, journal reviews and scholarship committees.