Many-on-One Pursuit. M. Pachter. AFIT Wright-Patterson A.F.B., OH 45433

Transcription

1 Many-on-One Pursuit M. Pachter AFIT Wright-Patterson A.F.B., OH The views expressed on these slides are those of the author and do not reflect the official policy or position of the United States Air Force, Department of Defense, or the US Gov. October 13, 2018 M. Pachter (2018) Many-on-One Pursuit October 13, / 64

2 AI and Control The views expressed on this slide are those of the author and do not reflect the official policy or position of the US Air Force, DoD, or the US Gov. Current state of AI Enumeration, ML/overfitting, Heuristics/hacking, ES/no AI -It s like alchemy So what is real AI? AI is decision making powered by mathematics and machine computing An attempt at a definition in the Control context AI: Optimal control in the presence of uncertainty and adversarial action. Example: Solution of a DG The adversary is an algorithm and the human player is a loser. M. Pachter (2018) Many-on-One Pursuit October 13, / 64

3 Pursuit and Evasion M. Pachter (2018) Many-on-One Pursuit October 13, / 64

4 N-on-One Pursuit and Evasion Figure: Multi Pursuers M. Pachter (2018) Many-on-One Pursuit October 13, / 64

5 Three Agents M. Pachter (2018) Many-on-One Pursuit October 13, / 64

6 Apollonius Circle Figure: Apollonius Circle M. Pachter (2018) Many-on-One Pursuit October 13, / 64

7 Pincer Movement Pursuit Figure: Solution of Two Cutters and Fugitive Ship Game M. Pachter (2018) Many-on-One Pursuit October 13, / 64

8 No Way Out M. Pachter (2018) Many-on-One Pursuit October 13, / 64

9 Breakout M. Pachter (2018) Many-on-One Pursuit October 13, / 64

10 P 2 Redundant Figure: One Cutter Action M. Pachter (2018) Many-on-One Pursuit October 13, / 64

11 Reduced State Space Figure: Rotating Reference Frame M. Pachter (2018) Many-on-One Pursuit October 13, / 64

12 Reduced State Space The state space of the Two Cutters and Fugitive Ship differential game is reduced to the first quadrant of R 3 : R 3 1 {(x P, x E, y E ) x P 0, y E 0} Symmetry allows us to confine our attention to the case where x E 0, that is, the state will evolve in the positive orthant of R 3 : R 3 + = {(x P, x E, y E ) x P 0, x E 0, y E 0}, The three-state nonlinear dynamics of the differential game are ẋ P = 1 2 (cos χ cos ψ), x P(0) = x P0 (1) ẋ E = µ cos φ 1 2 (cos χ + cos ψ) + 1 y E 2 x P (sin χ sin ψ), x E (0) = x E0 (2) ẏ E = µ sin φ 1 2 (sin χ + sin ψ) 1 x E (sin χ sin ψ), y E (0) = y E0 (3) 2 x P M. Pachter (2018) Many-on-One Pursuit October 13, / 64

13 R 1 /R 1,2 Boundary Figure: Critical Configuration M. Pachter (2018) Many-on-One Pursuit October 13, / 64

14 Soln. of Game of Kind Theorem 1 During optimal play 1. The Evader is singlehandedly captured in PP by P 1 if the state is in the set R 1 ; the set R 1 is the wedge formed by the quarter planes {(x P, x E, y E ) x P = 0, x E 0, y E 0} and {(x P, x E, y E ) x E = µx P, x P 0, y E 0}. 2. The Evader is singlehandedly captured in PP by P 2 if the state is in the set R 2 ; the set R 2 is the mirror image of R 1 about the plane x E = The Evader is cooperatively and isochronously captured by P 1 and P 2 if the state is in the wedge shaped set R 1,2 = {(x P, x E, y E ) µx P x E µx P, x P 0, y E 0} M. Pachter (2018) Many-on-One Pursuit October 13, / 64

15 Soln. of Game of Kind M. Pachter (2018) Many-on-One Pursuit October 13, / 64

16 Soln. of Game of Kind, l > 0 Figure: Positive Orthant, l > 0 M. Pachter (2018) Many-on-One Pursuit October 13, / 64

17 Soln. of Game of Degree in R 1,2 Theorem 2 E is isochronously captured by the P 1 &P 2 team in R 1,2 = {(x P, x E, y E ) µx P x E µx P, x P 0, y E 0} The optimal state feedback strategies are y I sin ψ =, cos ψ = xp 2 + y I 2 y I x P x 2 P + y 2 I sin χ =, cos χ = (5) xp 2 + y I 2 xp 2 + y I 2 sin φ y I y E =, cos φ x = E (6) (y I y E ) 2 + xe 2 (y I y E ) 2 + xe 2 Time-to-capture/Value function: V (x P, x E, y E ) = xp 2 + y I 2 ; the function y I (x P, x E, y E ) = 1 1 µ 2 [y E + sign(y E ) µ 2 ye 2 + (1 µ2 )(µ 2 xp 2 x E 2 )] (7) M. Pachter (2018) Many-on-One Pursuit October 13, / 64 x P (4)

18 Proof by Isaacs Method We begin from the end. The terminal surface in R 1,2, T, is a circular arc in the plane x E = 0. T = {(x P, x E, y E ) x P = l cos ξ, x E = 0, y E = l sin ξ, 0 ξ π 2 } (8) Interested in the inward pointing normals n to the terminal surface T : They set the terminal conditions of the costate vector But although the problem formulation is physically sound (l > 0), the terminal manifold T is rank deficient The normals to the terminal surface at a point of the surface are not unique. From eq. (8) we extract the terminal costates cos ξ λ(tf ) = a b sin ξ M. Pachter (2018) Many-on-One Pursuit October 13, / 64

19 Proof by Isaacs Method The scalars a > 0, b < 0, and 0 ξ π 2 ; in the half of R 1,2 where x E < 0, b > 0 and in the plane x E = 0, b = 0. The size l of the pursuers capture set plays no role in setting the terminal costate exclusively interested in point capture: l 0. The costate λ is related to the partials of the Value function V (x P, x E, y E ): λ xp = V xp, λ xe = V xe and λ ye = V ye. The Hamiltonian H = [(λ y E + y E λ xe x E λ ye x P ) sin ψ + (λ xe + λ xp ) cos ψ + (λ ye y E λ xe x E λ ye x P ) sin χ + (λ xe λ xp ) cos χ] + µ(λ ye sin φ + λ xe cos φ) (9) Maximizing the Hamiltonian in χ and ψ and minimizing the Hamiltonian in φ helps to characterize the optimal controls. M. Pachter (2018) Many-on-One Pursuit October 13, / 64

20 Proof by Isaacs Method Euler-Lagrange Equations ẋ P = 1 2 [ λ xp λ xe (λ λ xe xp )2 + (λ y E λ xe x E λ ye ye x P ) 2 + (λ xe + λ xp ) ], x P (0) = x (λ xe + λ xp ) 2 + (λ ye + y P0 E λ xe x E λ ye x P ) 2 ẋ E = λ xe µ 1 λ 2 + xe λ2 2 [ λ xp λ xe ye (λ λ xe xp )2 + (λ y E λ xe x E λ ye ye x P ) 2 (λ xe + λ xp ) ] (λ xe + λ xp ) 2 + (λ ye + y E λ xe x E λ ye x P ) y E [ 2 x P y E λ xe x E λ ye x P λ ye (λ xe λ xp ) 2 + (λ ye y E λ xe x E λ ye x P ) 2 M. Pachter (2018) Many-on-One Pursuit October 13, / 64

21 Proof by Isaacs Method λ ye ẏ E = µ 1 λ 2 + xe λ2 2 [ ye + x E λ ye y E λ xe x P y E λ xe x E λ ye x P λ ye ] (λ xe + λ xp ) 2 + (λ ye + y E λ xe x E λ ye x P ) 2 1 x E [ 2 x P λ ye (λ xe λ xp ) 2 + (λ ye y E λ xe x E λ ye x P ) 2 y E λ xe x E λ ye x P λ ye (λ xe λ xp ) 2 + (λ ye y E λ xe x E λ ye x P ) 2 x E λ ye y E λ xe x P λ ye ], y E (0) = y (λ xe + λ xp ) 2 + (λ ye + y E0 E λ xe x E λ ye x P ) 2 M. Pachter (2018) Many-on-One Pursuit October 13, / 64

22 Proof by Isaacs Method λ xp = 1 2 y E λ xe x E λ ye x 2 P [ x E λ ye y E λ xe x P y E λ xe x E λ ye x P λ ye (λ xe λ xp ) 2 + (λ ye y E λ xe x E λ ye x P ) 2 λ ye ], λ (λ xe + λ xp ) 2 + (λ ye + y xp (t f ) = a cos ξ E λ xe x E λ ye x P ) 2 λ xe = 1 λ ye [ 2 x P y E λ xe x E λ ye x P λ ye (λ xe λ xp ) 2 + (λ ye y E λ xe x E λ ye x P ) 2 x E λ ye y E λ xe x P λ ye, λ (λ xe + λ xp ) 2 + (λ ye + y xe (t f ) = ab E λ xe x E λ ye x P ) 2 M. Pachter (2018) Many-on-One Pursuit October 13, / 64

23 Proof by Isaacs Method λ ye = 1 2 λ xe x P [ x E λ ye y E λ xe x P λ ye (λ xe + λ xp ) 2 + (λ ye + y E λ xe x E λ ye x P ) 2 y E λ xe x E λ ye x P λ ye, λ (λ xe λ xp ) 2 + (λ ye y xe (t f ) = a sin ξ E λ xe x E λ ye x P ) 2 Inserting the expressions which characterize the optimal controls into the Hamiltonian (9) yields the optimal smooth Hamiltonian H = [ (λ xe λ xp ) 2 + (λ ye y E λ xe x E λ ye ) x 2 P + (λ xe + λ xp ) 2 + (λ ye + y E λ xe x E λ ye x P ) 2 ] µ λ 2 x E + λ 2 y E M. Pachter (2018) Many-on-One Pursuit October 13, / 64

24 Proof by Isaacs Method The Hamiltonian vanishes and evaluating the optimal Hamiltonian at t = t f yields a relationship which encompasses the parameters a, b and ξ: 1 = cos ξ [ (b cos ξ) 2 cos 2 ξ + (b cos ξ) 2 sin 2 ξ + (b + cos ξ) 2 cos 2 ξ + (b + cos ξ) 2 sin 2 ξ]a µ b + sin ξ a The three parameters, 0 ξ π 2, a > 0 and b feature in the solution of the Euler-Lagrange equations. BUT: The vanishing optimal Hamiltonian yields cos ξ a = 1 2 ( b cos ξ + b + cos ξ ) µ b + sin ξ cos ξ So The Euler-Lagrange equations yield a family of optimal trajectories parameterized by two independent parameters b and ξ. Can fill in our three dimensional state space region R 1,2. M. Pachter (2018) Many-on-One Pursuit October 13, / 64

25 Proof by Isaacs Method The Euler-Lagrange equations are integrated in retrograde fashion starting out from the initial condition x P = 0, x E = 0, y E = 0, λ xp = a cos ξ, λ xe = ab, λ ye = a sin ξ The solution of the Euler-Lagrange equations which yield a trajectory which terminates at the initial state (x P0, x E0, y E0 ) is x P (t) = (1 t t f )x P0 x E (t) = (1 t t f )x E0 (10) where t f = 1 1 µ 2 y E (t) = (1 t t f )y E0, 0 t t f (1 µ 2 )(x 2 P 0 x 2 E 0 ) + (1 + µ 2 )y 2 E 0 + 2y E µ 2 y 2 E 0 + (1 µ 2 )(µ 2 x 2 P 0 x 2 E 0 ) M. Pachter (2018) Many-on-One Pursuit October 13, / 64

26 Proof by Isaacs Method To complete the proof it behooves on us to show that (x P0, x E0, y E0 ) R 1,2, b, 0 < ξ < π 2 and t f > 0 s.t. the following holds. x P0 = cos ξ t f (11) x E0 = b µ b 2 + sin 2 ξ t f (12) 1 y E0 = (1 µ b 2 + sin 2 ξ ) sin ξ t f (13) We must be able to solve the three equations (11)-(13) in the three unknowns b, ξ and t f. M. Pachter (2018) Many-on-One Pursuit October 13, / 64

27 Proof by Isaacs Method It all boils down to proving the existence of a solution 0 ξ π 2 to y E0 cos ξ = x P0 sin ξ µ 2 xp 2 0 xe 2 0 cos 2 ξ (14) It is where ξ = Arccos( x) x = (1 µ 2 )(x 2 xp 2 P 0 xe 2 0 ) + (1 + µ 2 )ye y E0 µ 2 (xp ye 2 0 ) (1 µ 2 )xe xp xe ye xE 2 0 ye xP 2 0 ye 2 0 2xP 2 0 xe 2 0 and consequently (1 x)x b = µ 2 ( x P 0 x E0 ) 2 x sign(x E ), t f = x P0 x M. Pachter (2018) Many-on-One Pursuit October 13, / 64

28 The Geometric Method is Correct Theorem 2 In the Two Cutters and Fugitive Ship differential game, in the state space region R 1,2 where both Pursuers P 1 and P 2 actively engage the Evader, the players optimal state feedback strategies are sin ψ = y E + µ 2 ye 2 + (1 µ2 )(µ 2 xp 2 x E 2) (1 µ 2 )(xp 2 x E 2) + (1 + µ2 )ye 2 + 2y E µ 2 ye 2 + (1 µ2 )(µ 2 xp 2 x E 2) cos ψ = (1 µ 2 )x P (1 µ 2 )(xp 2 x E 2) + (1 + µ2 )ye 2 + 2y E µ 2 ye 2 + (1 µ2 )(µ 2 xp 2 x E 2) χ = π ψ M. Pachter (2018) Many-on-One Pursuit October 13, / 64

29 The Geometric Method is Correct sin φ = 1 µ µ 2 y E + µ 2 ye 2 + (1 µ2 )(µ 2 xp 2 x E 2 ) (1 µ 2 )(xp 2 x E 2) + (1 + µ2 )ye 2 + 2y E µ 2 ye 2 + (1 µ2 )(µ 2 xp 2 x E 2) cos φ = 1 µ (1 µ 2 )x E (1 µ 2 )(xp 2 x E 2) + (1 + µ2 )ye 2 + 2y E µ 2 ye 2 + (1 µ2 )(µ 2 xp 2 x E 2) M. Pachter (2018) Many-on-One Pursuit October 13, / 64

30 The Geometric Method is Correct The Value function V (x P, x E, y E ) = 1 1 µ 2 (1 µ 2 )(x 2 P x 2 E ) + (1 + µ2 )y 2 E + 2y E µ 2 y 2 E + (1 µ2 )(µ 2 x 2 P x 2 E ) The field of primary optimal trajectories/regular characteristics covers the entire state space R1 3 No singular surfaces here. The geometric method yields the correct solution of the differential game. M. Pachter (2018) Many-on-One Pursuit October 13, / 64

31 Discussion Addressed the Two Cutters and Fugitive Ship differential game. 1. Provided the solution of the Game of Kind: Partitioned the state space into the R 1, R 2 and R 1,2 regions. 2. Proved the validity of the geometric solution using Isaacs method. -Optimal Hamiltonian is smooth. -Value function is C 1. -Optimal flow field covers state space. -Regular characteristics only no singular surfaces here. Proving the validity of the geometric solution was, surprisingly, difficult: Used the known optimal solution provided by the geometric method. M. Pachter (2018) Many-on-One Pursuit October 13, / 64

32 N-on-One Pursuit and Evasion Figure: Multi Pursuers M. Pachter (2018) Many-on-One Pursuit October 13, / 64

33 N > 2 Pursuers Safe Region (SR) Must solve 1 2N(N 1) quadratic equations. Have M N active Apollonius circles. Must construct the intersection of M Apollonius disks. SR = N 1 i M C i SR: Convex, has M corners, the sides of the M- gon are circular arcs. M. Pachter (2018) Many-on-One Pursuit October 13, / 64

34 Safe Region N = M = 3, µ = 2 Figure: Reuleux Triangle SR M. Pachter (2018) Many-on-One Pursuit October 13, / 64

35 Optimality Principle Solve the OR constrained Obnoxious Facility Location problem: Optimality Principle The N cities are {P 1,..., P N } The constraint set is the SR Aim point is the Obnoxious Facility (P, i 1,..., i l ) = arg max P SR min 1 i 3 dist(p i, P) I = P Active Pursuers: P I 1,...,P I l M. Pachter (2018) Many-on-One Pursuit October 13, / 64

36 Optimal Group Pursuit N pursuers + one evader; simple motion; capture range l 0. Speed ratios: µ i > 1, i = 1,..., N. Sate Feedback Capture Strategy Algorithm Given the pursuers and the evader s instantaneous positions 1. Calculate the centers and radii of the N Apollonius circles/disks. 2. Find BSR vertex candidates by solving 1 2N(N 1) sets of bivariate quadratic equations in the variables x and y requires the solution of 1 2N(N 1) quadratic equations. This designates the set of M N active pursuers via the positivity of the quadratic equations discriminants. 3. Calculate the points antipodal to E on the circumference of each of the M active Apollonius circles. This requires the solution of a quadratic equation in total, M quadratic equations are solved. M. Pachter (2018) Many-on-One Pursuit October 13, / 64

37 Optimal Group Pursuit 4. Check whether the antipodal point is in all the remaining M-1 Apollonius disks requires checking M(M 1) quadratic inequalities. If such an antipodal point exists and say, it resides on the circumference of the ith Apollonius circle, designate it the aim point I of Pursuer i and of the Evader the pursuit degenerates into a tail chase/pp where only Pursuer i is active. 5. If no such antipodal point exists and because a vertex V = (x, y) of the BSR/M-gon must be included in the Apollonius disks of all the M active pursuers, if M > 2, to determine the M vertices of the BSR/M-gon, the M(M 1)(M 2) quadratic inequalities (x i x Oj ) 2 + (y i y Oj ) 2 ρ 2 j, j {1,..., M,, j i}, 1 i M(M 1) must be checked only M inequalities will be satisfied, rendering the BSR s/m-gon s vertices Have constructed the SR. M. Pachter (2018) Many-on-One Pursuit October 13, / 64

38 Optimal Group Pursuit 6. Potential aim point : Vertex V k of BSR farthest from E Formed by the intersection of the Apollonius circles C i(k ) and C j(k ) This identifies two critical pursuers, i(k ) and j(k ), which cooperatively capture the evader by heading to the aim point I = V k BSR Also E heads to the aim point I = V k BSR. Have pincer movement pursuit with P i(k ) and P j(k ) on a CC with E. Pursuer closest to E, but not P i(k ) or P j(k ), employs PP. Remaining N 3 pursuers are redundant. The Value of the game V (P 1,..., P N, E) = dist(e, V k ) (15) M. Pachter (2018) Many-on-One Pursuit October 13, / 64

39 Optimal Group Pursuit 7. Additional candidate aim point P provided by the solution of Constrained Obnoxious Facility Location The role of the cities is assumed by the pursuers, the constraint set is SR. (P, i 1,...i l ) = arg max P SR min 1 i M dist(p i, P) (16) dist(p, P i 1 ) = dist(p, P i 2 ) =...dist(p, P i l ) R, l 1 Optimal location P of the obnoxious facility: Potential aim point I Consider the condition dist(v k, P i(k )) R (17) Let R ɛ d R + ɛ. The optimal aim point is specified as follows P if dist(v k, P i(k )) < R ɛ I = V k if dist(v k, P i(k )) > R + ɛ E if dist(v k, P i(k )) = d M. Pachter (2018) Many-on-One Pursuit October 13, / 64

40 Optimal Group Pursuit If condition (17) does not hold: I = P Int(SR). The l pursuers who are closest to P head toward I = P. The remaining N l pursuers are redundant. E runs toward the aim point I = P in the interior of the SR Upon arriving to P Int(SR), E stops. Awaits isochronous capture by designated pursuers P i 1,...P i l. The Value V (P 1, P 2, P 3, E) = dist(p, P i 1 ) µ If condition (17) holds: I = V k Pursuers P i(k ) and P j(k ) head toward the vertex V k. If Condition (17) holds with equality the pursuers head toward E. M. Pachter (2018) Many-on-One Pursuit October 13, / 64

41 Three Pursuers N = M = 3 (a) If pursuers P 1 P 2 P 3 is obtuse, E heads to the vertex k of the BSR. The vertex V k BSR being generated by C i(k ) C j(k ) Capture of E effected by pursuers P i(k ) and P j(k ) (b) If pursuers P 1 P 2 P 3 is acute, is O SR? i) If O SR eq. (17) E heads toward the vertex V k BSR and as in case (a) capture of E is effected by P i(k ) and P j(k ). ii) If O SR and dist(v k, P i(k )) < R, eq. (17) E heads toward O in the interior of the SR, and so do P 1, P 2, P 3. Since E Interior(SR) E arrives at O before the pursuers. Upon arriving to O, E stays put, awaiting his isochronous capture by the three pursuers. The Value V = R µ. If however dist(v k, P i(k )) > R, eq. (17) E V k and so do P i(k ) and P j(k ), irrespective of whether O is in the SR. The Value V = 1 µ dist(v k, P i(k )) M. Pachter (2018) Many-on-One Pursuit October 13, / 64

42 Three Pursuers Symmetric State O SR dist(v k, P i(k )) = d State symmetric E c O V k s.t. dist(e, E c ) δ, 0 < δ << 1 The pursuers team will incur a slight loss in optimality: iii) If state is symmetric the pursuers head toward E. The Evader has his pick: Head toward O or head toward V k The Value of the game is V = 1 µ R M. Pachter (2018) Many-on-One Pursuit October 13, / 64

43 Three Pursuers N = M = 3, µ = 2 P 1 C 3 P 2 I 1, 3 I 1, 2 C 1 C 2 E O P 3 I 2, 3 I * Figure: P 2, P 3 and E head to I 2,3, Redundant Pursuer 1 Employs PP M. Pachter (2018) Many-on-One Pursuit October 13, / 64

44 Figure: Hexagon in Game with Three Pursuers M. Pachter (2018) Many-on-One Pursuit October 13, / 64 Three Pursuers N = M = 3, µ = 2 Important: P 1 E < P 2 E and P 2 O < P 2 I 2,3

45 Three Pursuers N = M = 3, µ = 2 Figure: Game with Three Pursuers M. Pachter (2018) Many-on-One Pursuit October 13, / 64

46 Three Pursuers N = M = 3, µ = 2 C 3 P 1 C 2 I 1, 3 E I 1, 2 P 2 O * I 2, 3 P 3 C 1 Figure: P 1, P 2, P 3 and E head Toward O M. Pachter (2018) Many-on-One Pursuit October 13, / 64

47 Three Pursuers Symmetry N = M = 3, µ = 5 4 Figure: E is at Circumcenter of Equilateral P 1 P 2 P 3 M. Pachter (2018) Many-on-One Pursuit October 13, / 64

48 Three Pursuers Symmetry N = M = 3, µ = 5 4 (4).png Figure: Symmetric State M. Pachter (2018) Many-on-One Pursuit October 13, / 64

49 Three Pursuers Symmetry N = M = 3, µ = 5 4 Symmetry goes hand in hand with singularity Symmetry, and E heads South A state is reached where O SR and dist(p 2, V k ) = R Without pursuers dead zone parameterized by 0 < δ << 1, E will dither. P 2 & P 3 will zigzag Time-to-Capture > V = 1 µ R Chattering is anathema to optimality Inclusion of pursuers dead zone is crucial step: E won t dither Price paid: Slight reduction in optimality M. Pachter (2018) Many-on-One Pursuit October 13, / 64

50 Three Pursuers Summary N = M = 3 Optimal Solution 1. If E P 1 P 2 P 3, E captured by 2 active pursuers. 2. If E P 1 P 2 P 3 & P 1 P 2 P 3 obtuse, E captured by 2 active pursuers. 3. If E P 1 P 2 P 3 and P 1 P 2 P 3 is acute but vertex of BSR is outside hexagon, E captured by 2 active pursuers. Interesting scenario: 4. E P 1 P 2 P 3, P 1 P 2 P 3 is acute and SR Hexagon Pursuers team might incur a slight loss in optimality, BUT: Evader strategy with no dithering. Capture strategy of team of pursuers with no chatter. When N 3 the pursuit-evasion problem is much harder M. Pachter (2018) Many-on-One Pursuit October 13, / 64

51 Discussion Only when we have a complete set of optimal trajectories filling a capture region separated from the escape region by a closed barrier, or possibly, the region of capturability is the whole state space this, as specified by the solution of the Game of Kind has a differential game been solved. Only a few 3-D pursuit-evasion games solved, none in a higher dimension. Pursuit-evasion differential games in 3-D solved which are an acceptable model of conflict situations of interest so that their relevance is preserved: Games with no singular surfaces, at least for a range of parameters. Examples 2 Cutters & Fugitive Ship, Guarding a Target DG, Active Target Defense. The secret sauce is provided by the following insight. M. Pachter (2018) Many-on-One Pursuit October 13, / 64

52 Target Defense Differential Game Three States, V D V A 1 Figure: Guarding a Target M. Pachter (2018) Many-on-One Pursuit October 13, / 64

53 Defending a Coastline: Infinite Size Target Three States, V D V A 1 M. Pachter (2018) Many-on-One Pursuit October 13, / 64

54 Active Aircraft Defense Three States y T VT n D VD w A VA m x Figure: The Dynamic Target Defense Differential Game V D V A and the speed ratio µ = V A V T > 1. D and T form a team that plays against A. M. Pachter (2018) Many-on-One Pursuit October 13, / 64

55 Discussion Theorem 3 The solution of zero-sum differential games by solving the open-loop minmax optimal control problem using the two sided Pontryagin Maximum Principle and synthesizing the players state feedback optimal strategies in receding horizon optimal control fashion is valid iff the application of Isaacs method results in primary optimal trajectories only, a.k.a., regular characteristics. When this is the case and the players have simple motion the geometric method applies. We have proven that the application of Isaacs method to the Two cutters and Fugitive Ship Differential Game yields an optimal flow field which covers the 3-D state space. This is the reason why in the Two Cutters and Fugitive Ship Differential Game the geometric method is applicable and the correctness of Isaacs solution has been proved. Optimal strategies provide true AI action M. Pachter (2018) Many-on-One Pursuit October 13, / 64

56 Extension: Slow Pursuer P 2 µ 2 = 1 Figure: Slow Pursuer is Redundant M. Pachter (2018) Many-on-One Pursuit October 13, / 64

57 Extension: Slow Pursuer P 2 µ 2 = 1 Figure: Capture Affected by Fast Pursuer Only M. Pachter (2018) Many-on-One Pursuit October 13, / 64

58 Extension: Active Slow Pursuer µ 2 = 1 Figure: Evader is Isochronously Captured by Fast and Slow Pursuer M. Pachter (2018) Many-on-One Pursuit October 13, / 64

59 Two Slow Pursuers Figure: Escape from Two Pursuers whose Speed Ratio µ = 1 M. Pachter (2018) Many-on-One Pursuit October 13, / 64

60 Multiple Agents Network of Slow Pursuers All Agents Have Simple Motion & Equal Speed Point Capture (l = 0) Stroboscopic Information Pattern: Pshenichnyi E Convhull({P 1,..., P N }) E Escapes M. Pachter (2018) Many-on-One Pursuit October 13, / 64

61 Three Slow Pursuers Figure: Evader Surrounded by Pursuers whose Speed Ratio µ = 1 M. Pachter (2018) Many-on-One Pursuit October 13, / 64

62 References 1 Hugo Steinhaus: Definitions for a Theory of Games of Pursuit, Naval Research Logistics Quarterly, Vol. 7, No 2, pp , Rufus Isaacs: Differential Games: A Mathematical Theory with Applications to Warfare and Pursuit, Control and Optimization, Wiley 1965, pp E. Garcia, Z. Fuchs, D. Milutinovic, D. Casbeer, and M. Pachter: A Geometric Approach for the Cooperative Two-Pursuers One-Evader Differential Game, Proceedings of the 20th World Congress of IFAC, Toulouse, France, pp , July 9-14, M. Pachter: Isaacs Two-on-One Pursuit-Evasion Game, Submitted to the Annals of the ISDG. M. Pachter (2018) Many-on-One Pursuit October 13, / 64

63 References 1 E. Garcia, Z. Fuchs, D. Milutinovic, D. Casbeer, and M. Pachter: A Geometric Approach for the Cooperative Two-Pursuers One-Evader Differential Game, Proceedings of the 20th World Congress of IFAC, Toulouse, France, pp , July 9-14, M. Pachter, A. VonMoll, E. Garcia, D. Milutinovic, D.Casbeer: Two-on-One Pursuit, submitted to the AIAA J. of Guidance, Control and Dynamics. 3 H. Hagedorn and J. V. Breakwell: A Differential Game with Two Pursuers and One Evader, JOTA, Vol. 18, No. 1, January 1976, pp B. N. Pshenichnyi: Simple Pursuit By Several Objects, Cybernetics, May-June 1976, pp Gotfried T. Toussaint: Computing Largest Empty Circles with Location Constraints, Int. J. Computer and Information Sciences, Vol. 12, No. 5, 1983, pp M. Pachter (2018) Many-on-One Pursuit October 13, / 64

64 References 1 A. A. Chikrii and P. V. Prokopovich: Simple Pursuit of One Evader by a Group, Cybernetics and Systems Analysis, No. 3, May-June 1992, pp (in Russian). 2 S. Jin and Z. Qu: Pursuit-Evasion with Multi-Pursuer v.s. One Fast Evader, Proceedings of the 8th World Congress on Intelligent Control and Automation, July , Jinan, China, pp G. I. Ibragimov: A Game of Optimal Pursuit of One Object by Several, J. Appl. Maths. and Mechs., Vol. 62, No. 2, pp , S. Kopparty and C. V. Ravishankar: A Framework for Pursuit Evasion Games in R n, Information Processing Letters 96 (2005), pp Sergey S. Kumkov, Stephane Le Menec and Valerii S. Patsko: Zero-Sum Pursuit-Evasion Differential Games with Many Objects: Survey of Publications, Dynamic Games And Applications, 16 November 2016, DOI /s z. M. Pachter (2018) Many-on-One Pursuit October 13, / 64