Linear Differential Game With Two Pursuers and One Evader

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11000 10000 9000 8000 7000 Distance [m] 6000 5000 4000 3000 2000 1000 0 0 2 4 6 8 10 12 τ go [s] Figure 5 : NEZ extension in 2x1 game Figure 6 characterizes the saturations that apply in the evader optimal controls. Outside the 2x1 capture zone (always still considering E between the two pursuers) three different end games can happen: If z 1 f = 0 and z 2 f = 0 then the initial conditions belong to the 2x1 no-escape-zone. If z if ≠ 0 and z1 f = z2 f then the initial conditions belong to the 2x1 non-saturate zone (area in red in Figure 6). If z if ≠ 0 and z1 f ≠ z2 f then the initial conditions belong to the 2x1 saturate zone (case corresponding to the “trade off” evader controls overcoming the control bounds). 12000 10000 8000 ZEM [m] 6000 4000 2000 0 0 1 2 3 4 5 6 7 8 9 10 τ go [s] Figure 6 : Evader control saturation * This new limit, as well as the zone of v 2x1 saturation, have been investigated in linear and non-linear simulations in order to confirm the computation of τ 2x1. Up to now, the 2x1 game trajectories (Figure 3, Figure 5, Figure 6) have been plotted on a single ( ZEM , τ go ) representation. Figure 7 represents ZEM , τ go , ΔP1P2 . On the left side delimited by the the barrier of the 2x1 game in the state space ( ) red surface P2 intercepts E. On the right part (delimited by the blue surface) E is captured by P1. The capture zone extension due to the presence of two pursuers is depicted by green and purple lines. - 6 -
Moreover, in Figure 7 we draw an optimal trajectory (black line) starting (and remaining) on the new 2x1 boundary. 2x1 DGL/1 NEZ + optimal trajectory on the barrier 12000 10000 delta P1P2 [m] 8000 6000 4000 10 2000 0 -2000 0 2000 4000 6000 8000 10000 0 ZEM [m] 12000 5 Time to Go [sec] Figure 7 : 2x1 NEZ 3.4. Case of different time to go In this configuration, the two pursuers are launched at different times and so different time-to-go. The * previous expression of v 2x1 is no longer available and thus need to be generalized to take into account the difference in time-to-go. Note that in this new case, the evader is assumed to switch its command * to v 1x1 (and faces only one pursuer like in a 1x1 game) when it goes beyond the first opponent. The * optimal evader control, always called v 2x1 , should still lead to the equality of the final distances (in absence of evader control saturations): Z τ = = −Z τ 0 12000 ( 1 0) 2( 2 ) 2 2 ( ) τ = −z ( ) τ 1 = z1 0 P1 2 0 P2 10000 -z 1max z 2max 8000 Δ z 1 v* 2x1 6000 4000 2000 Δτ Δ z 2 v* 1x1 = -1 0 0 1 2 3 4 5 6 7 8 9 10 Figure 8 : Calculation of v with different time-to-go ( 2 τ 1 * 2x1 - 7 - τ < )
Page 1 and 2: Linear Differential Game With Two P
Page 3 and 4: The 2x1 game only makes a change in
Page 5: * ( μ 1H 11 − ε 1H 21 ) + ε 1H
Page 9 and 10: 12000 10000 8000 ZEM [m] 6000 4000

Linear Differential Game With Two Pursuers and One Evader

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