Linear Differential Game With Two Pursuers and One Evader

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ZEM z ( θ ) = 2 τ P a E max The Zero Effort Miss distance is given below for DGL/1: 2 2 ⎛θ ⎞ ZEM ( t) = y + y& t − && y τ h( θ ) + & y go P P E τ E h⎜ ⎟ ⎝ ε ⎠ Where y is the relative perpendicular miss, y& the relative perpendicular velocity, & y& P the (instantaneous) perpendicular pursuer acceleration and & y& E the perpendicular evader acceleration. The non-dimensional cost function is the normalized terminal miss distance subject to minimization by the pursuer and maximization by the evader. = z = z θ = 0 J f ( t) ( ) The (ZEM, Tgo) frame is divided into two regions, the regular area and the singular one. For some appropriated differential game parameters (pursuer to evader maximum acceleration ratio μ and evader to pursuer time lag ratio ε), the singular area plays the role of capture zone so called also NEZ (leading to zero terminal miss), whilst the regular area corresponds to the non capture zone. The NEZ can be bounded (closed) or unbounded (open). The natural optimal strategies are bang-bang controls corresponding to the sign of ZEM (some refinements exist when defining optimal controls inside the NEZ). We start the 2x1 DGL/1 analysis with unbounded 1x1 NEZ as pictured in Figure 1 (NEZ delimited by the two plain red lines, non capture zone corresponding to the state space filled with optimal trajectories in dot blue lines). Moreover, we first assume same Tgo in each DGL/1 game (same initial range, same velocity for each pursuer). 300 DGL/1, μ = 3.6, ε = 0.8, με = 2.88 200 100 z * + (τ) normalized z 0 -100 z * - (τ) -200 -300 0 5 10 15 normalized tgo Figure 1 : Unbounded NEZ (ZEM, Tgo frame) 3. Two pursuers one evader DGL/1 game 3.1. Criterion The outcome we consider in the 2x1 game is the minimum of the two terminal miss distances. With J the terminal miss, i ( u u , v) min { J ( u , v) , J ( u v) } J x , 2 1 1, 2 = 1 2 u the control of Pi , = { 1, 2} ( u v) ( u v) ( u u v) i and v the evader control. J 1 , : P1 E DGL/1 terminal miss J 2 , : P2 E DGL/1 terminal miss J 2 x 1 1, 2, : 2x1 DGL/1 outcome - 2 -
The 2x1 game only makes a change in the evader optimal command. The presence of a second pursuer doesn’t change the pursuers behaviour. Indeed, the optimal controls for each pursuer in the 2x1 game are the same as the ones in their respective 1x1 game. When considering two pursuers and one evader (in the framework of DGL representations), some cases are easy to solve. If E is “above” (or symmetrically “below”) the two pursuers (in ZEM, Tgo frame), the optimal evasion respect to each pursuer (considered alone) and according to both pursuers together is to turn right (to turn left when below) with maximum acceleration. In these cases (E “above” or “below”), there are no changes in evasion trajectory (optimal strategies) by adding a pursuer. ZEM P1 E P1E Reference P2 P2E Reference Figure 2 : 2x1 game with Evader between the two Pursuers (ZEM, Tgo frame) For the other initial conditions (see Figure 2) we have to refine the optimal evasion behaviour. If E is between P1 and P2 (as described in Figure 2) then the optimal 2x1 evasion is a trade off between the incompatible P1 E and P2 E optimal escapes. Notice that in Figure 2 the initial LOS (in each DGL/1 game) have been chosen parallel. 3.2. Trade off evader control * * * We compute the 2x1 DGL/1 game solution ( ( u u v ) J 2 x1 1 , 2 , 2x1 considering that the optimal evasion control is a constant value during the entire game and considering that the terminal miss distances respect to P1 and P2 have to be the same (for initial conditions outside the NEZ and when no control * * * saturation occurs). In the case of Figure 2, v = 1 (left turn), v 1 (right turn) and −1 ≤ v 1. J J J 2x1 * * ( u1 , v2 ) * * ( u2 , v1 ) * * ( u , u , v) 1 2 ≤ J 2x1 1 − 2 = * * * * * * * ( u , u , v ) = J ( u , v ) = J ( u , v ) 1 2 2x1 1 2x1 2 2x1 J ≤ J * * ( u1 , v1 ) * * ( u , v ) 2 2 2x1 ≤ According to notations and normalizations detailed in the previous section, we write the final distances equality condition in the following manner (where Z are the ZEM in each PE game) : ( t ) = Z ( t ) Z1 f − 2 Note that the minus sign is due to the y-axes orientation like in Figure 2. In normalized variable, we get (with τ Pi the Pursuer time lag constants) : 2 2 z τ = −z τ , z = z τ 0 i f ( ( )) 1 f P1 2 f P2 if i = Looking at Figure 3, the normalized zero-effort-miss at final time can be written (where distance diminutions due to application of sub optimal evader controls): Δz are miss - 3 -
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Linear Differential Game With Two Pursuers and One Evader

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