THE DIFFERENTIAL GAME OF GUARDING A TARGET∗ M. PACHTER, E. GARCIA AND D. CASBEER Abstract. The differential game of guarding a target originally posed by Isaacs is considered. The game is played in the Euclidean plane where a target set T is located. The protagonists are an Attacker (A) and the Defender (D). Both have “simple motion” ` a la Isaacs; that is, their speed VA and VD is constant, VD ≥ V A, and they can turn on a dime. Player A strives to come as close as possible to the target set T before being intercepted by the defender D, which tries to preclude A from reaching T . The optimal state feedback strategies of players A and D are derived whereupon the respective winning regions of A and D are characterized and the game of kind is solved. We also comment on generalizations, extensions, and the case where the target T is a dynamic point target and the D and T team plays cooperatively to defeat A. In this paper, a contribution is made to the, admittedly small, repertoire of differential games which can be solved in closed form.
1. Introduction. Scenarios where dynamic agents are engaged in pursuit and evasion are correctly analyzed in the framework of dynamic games [4], [7], [1], [9]. Approaches based on dynamic Voronoi diagrams have been used in scenarios with several pursuers in order to capture an evader within a bounded environment, as shown in [9] and [2]. In aerospace applications, the Target-Attacker-Defender (TAD) problem has been extensively studied: The work in [13] considered a dynamic target and provided a game theoretical analysis of the TAD problem using classical guidance laws for both the Attacker and the Defender. On the other hand, reference [14] addressed the TAD game where the Target is stationary. It is this stationary target defense differential game first formulated by Isaacs in [10] that is considered in this paper. The game evolves in the Euclidean plane and the target set T is assumed convex, for example, T is a circular disk, a convex polygon, or T is a point target. The protagonists are an Attacker (A) and the Defender (D). Both have “simple motion” `a la Isaacs, that is, their speed is constant and they can turn on a dime. The speed of A is VA and the speed of D is VD where VD ≥ VA . The non-dimensional Defender/Attacker speed ratio is µ≡
VD ≥ 1, VA
and in [10] it is assumed the speed ratio µ = 1. Player A’s objective is to come as close as possible to the target T before being intercepted by the defender D. The defender’s objective is to preclude A from reaching T . The target defense differential game is illustrated in Fig. 1, where the target set T is bounded. We denote the boundary of the target set by ∂T . The case where the target set is of infinite extent, for example a coast line [11], is discussed in Section 6. Since µ ≥ 1 once D comes in contact with A, the interception of A is effected. In general, the Defender could be endowed with a capture circle of radius l and the speed ratio µ could be greater than one. When the speed ratio µ = 1, once D comes in contact with A the latter comes under the control of D, that is, A will be pushed around by D without being able to break off contact with D and when µ > 1, once the A − D separation becomes l (≥ 0), A is captured by D and the game is over. Hence, the game terminates when A reaches T without having been intercepted by D, or A ∗ THE VIEWS EXPRESSED IN THIS ARTICLE ARE THOSE OF THE AUTHORS AND DO NOT REFLECT THE OFFICIAL POLICY OR POSITION OF THE UNITED STATES AIR FORCE, DEPARTMENT OF DEFENSE, OR THE U.S. GOVERNMENT
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Fig. 1. Guarding a Target
comes in contact with D as close as possible to the target set T , at which time A is captured by D. In the target defense differential game the problem parameters are the speed ratio µ ≥ 1 and D’s capture radius l ≥ 0. In this paper we will assume that the capture range l is short, that is, it is l ≈ 0 - we are interested in point capture. Also, we will focus on the case where the speed ratio is µ = 1. The paper is organized as follows. A formal analysis of the differential game of guarding a Target using optimal control theory and the theory of differential games is presented in Section 2. This provides the justification for the geometric method [10] used to characterize the optimal state feedback strategies for players A and D as explained in Section 3. We then turn to the solution of the game of kind: For a specified target set T , the boundary separating the winning regions of players A and D is constructed in Section 4. The optimal placement of the Defender D such that the area of vulnerability of T is minimized is obtained in Section 5. The interesting scenarios where the target set T is not bounded, the Defender/Attacker speed ratio µ > 1, the requirement for point capture of A by D is relaxed, and the dynamic target defense differential game, are briefly discussed in Section 6. Lastly, conclusions are given in Section 7. 2. Analysis. We solve the differential game of guarding a target using the method of optimal control and differential games, and we work in the four dimensional realistic state space ζ T = (xA , yA , xD , yD )T ∈ R4 . We first address the case where the stationary Target is a point target. The normalized dynamics are given by
(2.1)
x˙ A y˙ A x˙ D y˙ D
= cos χ, = sin χ, = cos ψ, = sin ψ,
xA (0) = xA0 yA (0) = yA0 xD (0) = xD0 yD (0) = yD0
where χ and ψ are the respective heading angles of A and D.
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3
Concerning the terminal condition, the cost/payoff function is evaluated when D intercepts A at time tf where xA (tf ) = xD (tf ) yA (tf ) = yD (tf ).
(2.2)
The terminal cost/payoff function is 1 [(xA (tf ) − xT )2 + (yA (tf ) − yT )2 ]. 2 We have a Mayer type differential game. The terminal time tf is free and the terminal manifold T is the hyperplane in R4 defined by xA 1 0 −1 0 yA = 02×1 . (2.4) 0 1 0 −1 xD yD (2.3)
J(xA (tf ), yA (tf ); xT , yT ) =
The co-state λT = (λxA , λyA , λxD , λyD ) ∈ R4 and λ = −Vx , where V is the Value function of the differential game. The Hamiltonian of the differential game is H = λxA cos χ + λyA sin χ + λxD cos ψ + λyD sin ψ. Theorem 2.1. Consider the Differential Game of Guarding a Target (2.1)-(2.3). The optimal headings of the Defender and of the Attacker are constant under optimal play. They are given, respectively, by the state feedback control laws yA −yD ψ(xA , yA , xD , yD ; xT , yT ) = π2 − arcsin √ 2 2 (xA −xD ) +(yA −yD ) (2.5) y − 1 (y +y )(x −x )− x − 1 (x +x )(y −y ) T A D A D T A D A D 2 2 − arctan 2 (xA −xD )2 +(yA −yD )2 and (2.6)
yA −yD χ(xA , yA , xD , yD ; xT , yT ) = π2 − arcsin √ (xA −xD )2 +(yA −yD )2 y − 1 (y +y )(x −x )− x − 1 (x +x )(y −y ) T A D A D T A D A D 2 2 . + arctan 2 (xA −xD )2 +(yA −yD )2
In the region of win of D, the Value function V (x) is C 1 and is explicitly given by xA +xD −2xT (xA −xD )+ yA +yD −2yT (yA −yD ) √ (2.7) . V (xA , yA , xD , yD ; xT , yT ) = 2 2 2
(xA −xD ) +(yA −yD )
Proof. The optimal control inputs (in terms of the co-state variables) are obtained from min max H χ
ψ
and they are given by λxA
(2.8)
cos χ∗ = q
(2.9)
cos ψ ∗ = − q
λ2xA + λ2yA
sin χ∗ = q
,
λxD λ2xD + λ2yD
,
λ yA λ2xA + λ2yA
sin ψ ∗ = − q
λ yD λ2xD + λ2yD
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The co-state dynamics are λ˙ xA = λ˙ yA = λ˙ xD = λ˙ yD = 0; hence, all the co-states are constant and therefore the optimal controls χ∗ ≡ constant and ψ ∗ ≡ constant. In other words, in the realistic (x, y) plane the optimal trajectories are straight lines! Concerning the solution of the attendant TPBVP on 0 ≤ t ≤ tf in R8 , we have 4 initial states specified by (2.1) and we need 4 more conditions for the terminal time tf . In this respect, introduce the augmented Mayer payoff/cost function Φ : R4 → R1 Φ(xA , yA , xD , yD ; xT , yT ) ,
1 [(xA − xT )2 + (yA − yT )2 ] + ν1 (xA − xD ) + ν2 (yA − yD ) 2
where ν1 and ν2 are Lagrange multipliers. The Pontryagin Maximum Principle (PMP) or Dynamic Programming yield the transversality/terminal conditions λ(tf ) = −
∂ Φ(x(tf )) ∂x
that is, (2.10) (2.11) (2.12) (2.13)
λxA = xT − xA (tf ) − ν1 λyA = yT − yA (tf ) − ν2 λxD = ν 1 λ yD = ν 2
At this point, we have equations (2.10)-(2.13) plus equations (2.2) which yield 6 conditions. Since we need only 4 conditions we use the additional two conditions to eliminate the introduced Lagrange multipliers ν1 and ν2 from equations (2.10)-(2.13) and we obtain (2.14) (2.15)
λxA + λxD = xT − xA (tf )
λyA + λyD = yT − yA (tf )
Thus, we have 4 relationships for the terminal time tf : equations (2.2) and (2.14)(2.15). Finally, the time tf is specified by the PMP/Dynamic Programming requirement that the Hamiltonian H(x(t), λ(t), χ, ψ)|tf = 0 which, for this problem, takes the form (2.16)
λxA cos χ∗ + λyA sin χ∗ + λxD cos ψ ∗ + λyD sin ψ ∗ = 0.
So, we have enough relationships to solve the differential game. The existence of a solution is predicated on the initial state: If the initial positions of the Attacker and the Defender are such that the Target is located on the Defender’s side of the orthogonal bisector of AD, then a solution exists and the Defender will be able to interpose itself between the Target and the Attacker to guard/protect the former. In view of the above and without loss of generality, assume that the state in the realistic plane is such that xD = −xA , yA = 0, and yD = 0, as shown Fig. 2. Because the optimal trajectories of A, D, and T are straight lines and VD = VA we have that xA (tf ) = 0 xD (tf ) = 0 yA (tf ) = yD (tf )
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Fig. 2. Optimal Strategies
Let y , yA (tf ) = yD (tf ). Also, let xA = xA (t′ ) be the instantaneous positions at some time t′ < tf . Hence, from equations (2.1) we obtain the following 0 = xA + (tf − t′ ) cos χ, y = (tf − t′ ) sin χ,
0 = −xA + (tf − t′ ) cos ψ, y = (tf − t′ ) sin ψ, In addition, equations (2.14)-(2.15) can be written as follows λxA + λxD = x T λ yA + λ yD = y T − y From the △ADI in Fig. 2 we conclude that tf − t′ = generality, assume that t′ = 0, and then q tf = x2A + y 2 .
p
x2A + y 2 . Without loss of
Thus, having used the theory of differential games, we are now able to reduce the solution of the zero-sum differential game of degree to the optimization of a cost/payoff function of one variable, namely: q J(y; xA , xT , yT ) = (yT − y)2 + x2T . (2.17) In terms of y, the A and D players’ optimal headings are y xA (2.18) , sin χ∗ = p 2 . cos χ∗ = − p 2 2 xA + y xA + y 2 y xA (2.19) , sin ψ ∗ = p 2 . cos ψ ∗ = p 2 xA + y 2 xA + y 2
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Using eqs. (2.9) and (2.19) yields the following relationships (2.20)
−q
λxD λ2xD
+
λ2yD
=p
xA x2A + y 2
−q
,
λ yD λ2xD
+
λ2yD
=p
y x2A + y 2
.
Similarly, from eqs. (2.8) and (2.18) we obtain (2.21)
λxA q
λ2xA + λ2yA
= −p
xA 2 xA +
y2
λ yA
,
q
λ2xA + λ2yA
=p
y x2A
+ y2
.
We have four equations (2.14), (2.15), (2.20), and (2.21) in the four unknowns λxA , λyA , λxD , and λyD . The solution is xA 1 [xT − xA (tf ) − 2 y 1 y λyA = [yT − yA (tf ) − 2 xA 1 xA λxD = [xT − xA (tf ) + 2 y 1 y λyD = [yT − yA (tf ) + 2 xA λxA =
yT − yA (tf ) ] xT − xA (tf ) ] yT − yA (tf ) ] xT − xA (tf ) ]
By substituting yA (tf ) = y and xA (tf ) = 0 we obtain (2.22) (2.23) (2.24) (2.25)
xA 1 [xT − yT − y ] 2 y 1 y λyA = [yT − y − xT ] 2 xA 1 xA λxD = [xT + yT − y ] 2 y 1 y λyD = [yT − y + xT ] 2 xA λxA =
which specify the co-states in terms of the states at time tf . So far we have not used eq. (2.16) which, in view of eqs. (2.8),(2.9) is equivalent to q q λ2xA + λ2yA − λ2xD + λ2yD = 0. It is however more convenient to use eq. (2.16) in conjunction with eqs. (2.18)-(2.19) for the optimal controls and eqs. (2.22)-(2.25) for the co-states. Doing so we obtain the quadratic equation in y (yT − y)(
x2A y
+ y) = 0
which has the real solution y ∗ = yT plus two imaginary roots which are irrelevant to the game under consideration. Inserting y = yT into eqs. (2.18) and (2.19) yields the optimal state feedback strategies of A and D y xA (2.26) , sin χ∗ = p 2 . cos χ∗ = − p 2 2 x A + yT xA + yT2 y xA (2.27) , sin ψ ∗ = p 2 . cos ψ ∗ = p 2 xA + yT2 xA + yT2
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Fig. 3. Optimal Heading Angles in Realistic Space
The Value function is C 1 and is given by (2.28)
V (xA , xT , yT ) = J(y ∗ ; xA , xT , yT ) for all xT < 0
where the function J is given by eq. (2.17). Using the theory of differential games, the original problem was simplified to an optimization problem in one variable, y. The optimal headings χ∗ and ψ ∗ are state feedback control strategies and they are functions of the instantaneous values of the state (xA , yA , xD , yD ) ∈ R4 and of the Target location (xT , yT ) in the realistic space - see Fig. 3. In the realistic space the optimal headings are obtained as follows. Let p (xA − xD )2 + (yA − yD )2 xA − xD cos θ , dA yA − yD sin θ , dA
dA ,
xA + xD yA + yD cos θ − xT − sin θ 2 2 yT − 21 (yA + yD ) (xA − xD ) − xT − 12 (xA + xD ) (yA − yD ) p = . (xA − xD )2 + (yA − yD )2
h , yT −
The strategies of the players are determined by the coordinates (xA , xT , yT ) in a reduced state space and are therefore functions of dA , h, and θ: From Fig. 3, the
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optimal heading of the Defender is 2h π ψ(xA , yA , xD , yD ; xT , yT ) = − θ − arctan 2 dA π yA − yD = − arcsin p 2 (xA − xD )2 + (yA − yD )2 y − 1 (y + y )(x − x ) − x − 1 (x + x )(y − y ) T D A D T A D A D 2 A 2 − arctan 2 (xA − xD )2 + (yA − yD )2 and the optimal heading of the Attacker is 2h 3π − θ − arctan χ(xA , yA , xD , yD ; xT , yT ) = 2 dA π yA − yD = − arcsin p 2 (xA − xD )2 + (yA − yD )2 y − 1 (y + y )(x − x ) − x − 1 (x + x )(y − y ) T D A D T A D A D 2 A 2 + arctan 2 . 2 2 (xA − xD ) + (yA − yD ) Finally, in D’s region of win the Value function of the target defense differential game is given by (2.7). As illustrated in Fig. 4, a polygonal Target is not strictly convex and so there exist configurations in the state space where the geometric solution yields optimal strategies which are not unique: the Attacker and the Defender each choose a separate aimpoint, say IA and ID , on the segment I1 I2 which runs parallel to the side AB of the polygon - see Fig. 4. This however is not a show stopper. After a short time, the orthogonal bisector of the segment AD will cease being parallel to the side AB of the polygon and a unique aimpoint I will reappear - the end result being that the distance V0 is a lower bound for the value of the game and the Defender can make the value arbitrarily close to V0 . This should come as no surprise because in zero-sum games the non-uniqueness of the optimal strategies is not an issue - the optimal strategies are interchangeable and the Value of the game is unique. 3. Geometric Solution. The analysis above is brought to bear on the target defense differential game as it evolves in the Euclidean plane. It is the justification for Isaacs’ geometric solution [10] as discussed herein. The Euclidean plane where the target defense differential game is played is partitioned into two sets. When the speed ratio µ = 1, the set of positions in the plane reachable by A before being possibly reached by D is the half plane whose boundary is the orthogonal bisector of the segment AD and which contains point A; we refer to A’s half plane, HA . Conversely, the set of positions in the plane reachable by D before being possibly reached by A is the half plane whose boundary is the orthogonal bisector of the segment AD and which contains point D; we refer to D’s half plane HD . Player A “wins” if and only if \ HA T 6= ∅
In other words, player A can reach the target set T without being intercepted by D. If however \ HA T = ∅ ,
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Fig. 4. Non-unique A and D Strategies for a Polygonal Target
player A heads toward the point I on the orthogonal bisector of the segment AD which is closest to the convex target set T . Same for player D. Their optimal strategies are straight lines, as proved in Section 2. The optimal strategies, as well as Isaacs’ original geometric optimality proof from [10] are illustrated in Fig. 5. The analysis in Section 2 provides the justification for the geometric solution. When the speed ratio µ > 1, that is, the Defender is faster than the Attacker, the boundary of the set of positions in the plane reachable by A before being possibly reached by D is an Apollonius circle C. The radius of the Apollonius circle is R = µ µ2 −1 dA , where dA is the A-D separation. Its center O is on the straight line through
the points A and D, at a distance 1 µ2 −1 dA . A “wins” iff
µ2 µ2 −1 dA
from D; the length of the segment OA is
C
\
T 6= ∅
C
\
T =∅
If however
A heads toward the point I ∈ C which is closest to the convex target set T . 4. Solution of the Game of Kind. The winning region of the Attacker A is constructed as follows. Given the Target T and Defender D, the boundary of A’s winning region, denoted by B, is constructed for target sets defined by circular discs and polygonal sets. 4.1. Circular Disc. Consider the case where the Target set T is a circular disc. The boundary B of A’s winning region is illustrated in Fig. 6 and is constructed as follows:
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Fig. 5. Optimal and non-optimal play. During optimal play both A and D head toward point I which is the point on the orthogonal bisector of the segment AD closest to the target set T . When one player plays non-optimally (here the Attacker), the opponent (here the Defender) immediately exploits this by heading toward I ′ which is the point of the orthogonal bisector of the segment A′ D ′ closest to the Target set and the miss distance increases.
1. Draw a radial emanating from T . 2. Draw a tangent to ∂T ⊥ the radial from Step 1.1 3. Pick a point A, such that the tangent from Step 2 is the orthogonal bisector of DA. Note that the orthogonal bisector of DA is tangent to the target set T . Thus, the boundary B is the orthotomic of ∂T . The boundary B of A’s winning region is characterized by the optimality principle and the following holds Theorem 4.1. Consider the case where the target set T is a circular disc of radius r. In polar coordinates the Right Hand Side of the (symmetric) boundary B of the winning region of A is (4.1)
R(θ) = 2(r + d cos θ), 0 ≤ θ ≤ θc
where θc = arcsin( dr ) + π2 for d > r, and θc = π otherwise. Thus, the boundary B is a Lima¸con of Pascal. Also, the optimal placement of the Defender with respect to a circular Target of radius r is at the center of the Target set and then the minimal area of vulnerability is S ∗ = 4πr2 . Proof. It is well known that the orthotomic of a circle is a Lima¸con of Pascal. Also, 1. when d < 12 r → A’s winning region is convex, 2. when d > 12 r → the boundary B has an indentation, and 1 This
tangent is labeled as the “Orthogonal bisector” in Fig. 6.
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Fig. 6. Construction of B
3. when d = r → the boundary B is a Cardioid: R(θ) = 2r(1+cos θ), 0 ≤ θ ≤ π. See Fig. 7. Note that when d > r, the Lima¸con has an inner loop; however the inner loop is irrelevant to the differential game under study. Hence, in the corresponding Figures, the inner loop is not drawn - see Fig. 8. Furthermore, the area of A’s winning region is (4.2)
S = 2(2r2 + d2 )π ∀ 0 ≤ d ≤ r,
and when d ≥ r, the boundary B has a cusp and the area of A’s winning region is p r S = 2(2r2 + d2 )[π − arccos( )] + 6r d2 − r2 (4.3) d When d > r the boundary B is not smooth and, as shown in Fig. 9 the indentation angle is given by the relation r + d cos θc = 0. Solving for θc we obtain (4.4)
π r θc = arcsin( ) + d 2
When 0 ≤ d ≤ r the vulnerable area as a function of the parameter d is S(0) = 4πr2 . In addition, S(d) > S(0) for 0 < d ≤ r. Now, when d > r we can show that (4.5)
arg min S(d) = r. d≥r
This can be done by computing 2 dS r 2 2 √ −r √3rd + + 2d(π − arccos( = 2 (2r + d ) )) dd d d d2 −r 2 d d2 −r 2 1 r 2 2 3 √ = 2 d d2 −r2 (3rd − rd − 2r ) + 2d(π − arccos( d )) = 4 d√dr2 −r2 (d2 − r2 ) + d(π − arccos( dr )) √ ≥ 4 dr d2 − r2 + π2 >0
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A's winning region
Target Area
r
T
}
D cusp
d
B
Fig. 7. Circular Target, d = r
A's winning region
Target Area
r
T
} d
cusp
D
B
Fig. 8. Circular Target, d > r
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Fig. 9. Critical Geometry
Thus, for d ≥ r, S(d) is an increasing function of its argument and therefore the minimum value is attained at d = r. This minimum value is S(r) = 6πr2 and we can see that S(r) > S(0). Then d∗ = 0, and the Defender should be placed at the center of the circular Target set. The minimal ensuing vulnerable area is S ∗ = 4πr2 . 4.2. Polygonal Target. Consider the case where the target set T is a polygon. For this case, we can regard each vertex of the polygon as a circle of radius r = 0. Then, the orthotomic curve is given by R(θ) = 2d cos θ which is the equation of a circle. When T is a convex n-gon, the boundary B of the winning region of A consists of n circular arcs: (4.6)
B = ∪ni=1 Ci
where Ci are circular arcs of radius DTi centered at the polygon’s vertices Ti , i = 1, ..., n. 5. Optimization. Consider the case where T is a convex n-gon whose vertices are Ti = (xi , yi ), i = 1, ..., n. From the geometry in Figs. 10 and 11 we deduce that the area enclosed by the boundary B, that is, the area S of A’s winning region, is (5.1)
S = 2 ∗ Area of P olygon +
n X i=1
(π − ∠Ti )d2i
where di ≡k DTi k. From Eq. (5.1) we infer that S is akin to the moment of inertia of the planar figure i formed by the polygon where point masses π−∠T are attached to its vertices; ∠Ti 2π is the internal angle at vertex Ti of the polygon. From mechanics we know that the
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d1
T1
d1
d2 d1
T2
d5
d2
d2
D
d3 d3
d5
T5
d4
d5
T3
d4
T4 d3
d4
Fig. 10. Construction of B, D ∈ T
moment of inertia of a planar rigid body is minimal about an axis through its center of mass and the center of mass of a rigid body is calculated by averaging. Hence, the following holds.
Theorem 5.1. To minimize the polygonal target’s area of vulnerability S, the Defender D positions himself at the center of mass of the n-point configuration T1 , ..., Tn , i and ∠Ti is the internal angle at the vertex where the “mass” of point Ti is wi = π−∠T 2π ∗ Ti of the polygon. The optimal position of D is D∗ = (x∗D , yD ), where
x∗D =
(5.2)
n X π − ∠Ti i=1
2π
∗ x i , yD =
n X π − ∠Ti i=1
2π
yi
and consequently the minimal vulnerable area is n X xi xi+1 ∗ ∗ 2 [det( S = (5.3) ) + (π − ∠Ti )[(xi − x∗D )2 + (yi − yD ) ] yi yi+1 i=1
where in eq. (5.3) xn+1 ≡ x1 , yn+1 ≡ y1 . Remark. Note that
Pn
i=1
wi = 1, as required.
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1
d1
T1
d1
d2 T2
d5 d2
T5
d1
d5
d2 d3
d5
T3
d4
d3
T4 d4
D
Fig. 11. Construction of B, D ∈ /T
5.1. Examples. Example 1. Consider a pentagon where the coordinates of each vertex and the corresponding internal angles in radians are given by x = [0 4 11.056 10.12 5.52]T y = [0 0 2.54 7.45 7.45]T ∠T = [0.933 2.796 1.728 1.7592.208]T The optimal Defender position is D∗ = [5.754 3.318] with a corresponding minimal vulnerable area of S ∗ = 309.99. Fig. 12 shows the polygonal Target set, the optimal position to place the Defender, and the boundary of the minimal area of vulnerability. Example 2. Consider a polygon with 7 vertices. The coordinates of each vertex and the corresponding internal angles are given by x = [0 5.3 7.769 10.581 3.8 1.011 − 0.829]T
y = [0 0 3.398 12.053 13.793 12.689 5.521]T
∠T = [1.72 2.2 2.83 1.51 2.51 2.2 2.74]T The optimal Defender position is D∗ = [4.413 6.938] with a corresponding minimal vulnerable area of S ∗ = 564.44. Fig. 13 shows the polygon Target set, the optimal position to place the Defender, and the boundary of the minimal vulnerable area.
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12
10
8
6
4
D 2
0
-2
-4
-6 -5
0
5
10
15
Fig. 12. Example 1. Target: –. B: –.
20
15
10
D 5
0
-5
-10
-5
0
5
10
15
20
Fig. 13. Example 2. Target: –. B: –.
5.2. Smooth Target Set. When the target’s set boundary C = ∂T is a smooth curve, it is convenient to think of the boundary C of the target set T as specified using intrinsic coordinates, that is, it is parameterized with respect to the arc length s; and R(s) is the radius of curvature of C at s. The winning region of A is obtained as follows. Approximate C by a polygonal curve, as shown in Fig. 14. The following holds (5.4)
π − ∠T = θ(x + dx) − θ(x)
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Fig. 14. Approximating C by a Polygon Line
and consequently let the weight (5.5)
w≡
1 dθ 1 1 1 [θ(x + dx) − θ(x)] = ds = ds 2π 2π ds 2π R(s)
Hence, in view of Theorem 5.1, we have
Theorem 5.2. To minimize the area of vulnerability S, the Defender D positions ∗ himself at D = (x∗D , yD ), where Z Z 1 1 1 1 ∗ ∗ xD = x(s) y(s) (5.6) ds, yD = ds 2π C R(s) 2π C R(s) 6. Extensions. When the target T is not bounded, e.g., an infinite half-plane, and if the speed ratio µ = 1, then HA T6= ∅ and A always wins - see Fig. 15. Hence, in this case, for the target defense differential game to be interesting we need, the speed ratio µ > 1. When the Attacker is endowed with a standoff weapon whose range is R we have to enlarge the defended target set accordingly. A circular disc target of radius r will become a circular disc target of radius r + R. When the target set T is not connected and the Attacker A ∈ / convhull(T ), set T := convhull(T ). One must also consider the case where the capture radius of D, l > 0. To analyze the target defense differential game in the case where the problem parameters µ > 1 and l > 0, we note that the winning regions of the Attacker and the Defender will be constructed along the lines discussed in Sections 4 and 5 on the provision that:
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Pachter et al.
Fig. 15. Coastal Defense
When µ > 1 and l = 0: The Orthogonal Bisector is replaced by an Apollonius circle When µ = 1 and l > 0: The Orthogonal Bisector is replaced by an arc of a hyperbola When µ > 1 and l > 0: The Orthogonal Bisector is replace by a Cartesian oval
In this paper the solution of the Target defense differential game when the Target is static is given. But consider a dynamic point Target: A scenario where a mobile target must be actively defended against a homing missile is described in [3]. The dynamic target defense differential game is illustrated in Fig. 16. Denote the speed of the fleeing Target as VT and the Attacker/Target speed ratio ν = VVA > 1. D and T T form a team that plays against A. This differential game has been addressed in references [12, 8, 6, 5]. The case where the speed ratio µ < 1 has some of the attributes of the obstacle tag differential game where singular surfaces are present and is not considered herein. 7. Conclusion. Isaacs’ Target Defense differential game is revisited and Isaacs’ geometric method is justified. The game of kind is solved: The winning regions of the Attacker A and Defender D are characterized. The optimal positioning of the Defender so as to minimize the area of vulnerability of the Target set T is calculated. It is also indicated how to address the target defense scenarios where the target set is not bounded, the requirement of point capture is relaxed, and the Defender is faster than the Attacker; the interesting case where the target is a point target, but it is a dynamic target, has previously been addressed by the authors. Because the Defender is not slower than the Attacker and point capture is considered, we have a primary flow field only of optimal trajectories and there are no singular surfaces. Thus, the results obtained in this paper are in closed form, which bodes well for their applicability to realistic target defense scenarios with military applications. In this
The Differential Game of Guarding a Target
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Fig. 16. The Dynamic Target Defense Differential Game
paper an additional example to the, admittedly small, repertoire of differential games which can be solved in closed form is provided. REFERENCES [1] Steve Alpern, Robbert Fokkink, Roy Lindelauf, and Geert-Jan Olsder. The “Princess and Monster” game on an interval. SIAM J. Control and Optimization, 47(3):1178–1190, 2008. [2] Efstathios Bakolas and Panagiotis Tsiotras. Optimal pursuit of moving targets using dynamic Voronoi diagrams. In 49th IEEE Conference on Decision and Control, pages 7431–7436, 2010. [3] R. L. Boyell. Defending a moving target against missile or torpedo attack. IEEE Transactions on Aerospace and Electronic Systems, AES-12(4):522–526, 1976. [4] Pierre Cardaliaguet. A differential game with two players and one target. SIAM J. Control and Optimization, 34(4):1441–1460, July 1996. [5] E. Garcia, D. W. Casbeer, and M. Pachter. Cooperative strategies for optimal aircraft defense from an attacking missile. Journal of Guidance, Control, and Dynamics, 38(8):1510–1520, 2015. [6] E. Garcia, D. W. Casbeer, K. Pham, and M. Pachter. Cooperative aircraft defense from an attacking missile using proportional navigation. In 2015 AIAA Guidence, Navigation, and Control Conference, Paper AIAA 2015-0337, 2015. [7] Sergey A Ganebny, Sergey S Kumkov, St´ ephane Le M´ enec, and Valerii S Patsko. Model problem in a line with two pursuers and one evader. Dynamic Games and Applications, 2(2):228–257, 2012. [8] E. Garcia, D. W. Casbeer, K. Pham, and M. Pachter. Cooperative aircraft defense from an attacking missile. In 53rd IEEE Conference on Decision and Control, pages 2926–2931, 2014. [9] Haomiao Huang, Wei Zhang, Jerry Ding, Dusan M Stipanovic, and Claire J Tomlin. Guaranteed decentralized pursuit-evasion in the plane with multiple pursuers. In 50th IEEE Conference on Decision and Control and European Control Conference, pages 4835–4840, 2011. [10] R. Isaacs. Differential Games. New York: Wiley, 1965. [11] S. Lee, G. E. Dullerud, and E. Polak. On the real-time receding horizon control in harbor defense. In American Control Conference, 2015. [12] M. Pachter, E. Garcia, and D. W. Casbeer. Active target defense differential game. In 52nd Annual Allerton Conference on Communication, Control, and Computing, pages 46–53, 2014. [13] Ashwini Ratnoo and Tal Shima. Guidance strategies against defended aerial targets. Journal of Guidance, Control, and Dynamics, 35(4):1059–1068, 2012. [14] R. H. Venkatesan and N. K. Sinha. The target guarding problem revisited: Some interesting revelations. In 19th IFAC World Congress, 2014.
