Ivancevic_Applied-Diff-Geom

234 Applied Differential Geometry: A Modern Introductionvehicle dynamics:ġ = gX, (3.64)with⎡ ⎤g 1 0 0g = ⎣ 0 g 2 0 ⎦ , X =0 0 g 3⎡⎣⎤X 1 0 00 X 2 0 ⎦ ,0 0 X 3where g is an element in the configuration space M = SE(2) × SE(2) ×SE(2) and X ∈ se(2) × se(2) × se(2) is linearly parameterised by inputs(ω 1 , v 1 ), (ω 2 , v 2 ) and (ω 3 , v 3 ).Now, the target set T is defined aswhereT = {g ∈ M|l 1 (g) < 0 ∨ l 2 (g) < 0 ∨ l 3 (g) < 0},l 1 (g) = min{(x 2 − x 1 ) 2 + (y 2 − y 1 ) 2 − ρ 2 , (x 3 − x 1 ) 2 + (y 3 − y 1 ) 2 − ρ 2 },l 2 (g) = min{(x 3 − x 2 ) 2 + (y 3 − y 2 ) 2 − ρ 2 , (x 1 − x 2 ) 2 + (y 1 − y 2 ) 2 − ρ 2 },l 3 (g) = min{(x 2 − x 3 ) 2 + (y 2 − y 3 ) 2 − ρ 2 , (x 1 − x 3 ) 2 + (y 1 − y 3 ) 2 − ρ 2 }.The control inputs u = (u 1 , u 2 , u 3 ) are the actions of vehicle 1, 2 and 3:where U i are defined asU i = ([ω miniu i = (ω i , v i ) ∈ U i ,, ω maxi ], [vimin , vi max ]).Clearly, this can be generalized to N vehicles.The cost functions J i (g, {u i (·)}, t) are defined asN∏N∏J i (g, {u i (·)}, t) : SE i (2) × U i × R − → R,i=1such that J i (g, {u i (·)}, t) = l i (g(0)).The simplest non–cooperative solution strategy is a so–called non–coopera-tive Nash equilibrium (see e.g., [Basar and Olsder (1995)]). A setof controls u ∗ i , (i = 1, ..., N) is said to be a Nash strategy, if for eachplayer modification of that strategy under the assumption that the othersi=1