Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 229in which g is an element in the configuration manifold M = SE(2)×SE(2),while the vector–fields X i ∈ se(2) × se(2) are linearly parameterised byvelocity inputs (ω 1 , v 1 ) ∈ R 2 and (ω 2 , v 2 ) ∈ R 2 .The goal of each vehicle is to maintain safe operation, meaning that(i) the vehicles remain outside of a specified target set T with boundary∂T , defined byT = {g ∈ M|l(g) < 0},where l(g) is a differentiable circular function,l(g) = (x 2 − x 1 ) 2 + (y 2 − y 1 ) 2 − ρ 2(with ρ denoting the radius of a circular protected zone) defines the minimumallowable lateral separation between vehicles; and(ii)dl(g) ≠ 0 on ∂T = {g ∈ M|l(g) = 0},where d represents the exterior derivative (a unique generalization of thegradient, divergence and curl).Now, due to possible uncertainty in the actions of vehicle 2, the safestpossible strategy of vehicle 1 is to drive along a trajectory which guaranteesthat the minimum allowable separation with vehicle 2 is maintainedregardless of the actions of vehicle 2. We therefore formulate this problemas a zero–sum dynamical game with two players: control vs. disturbance.The control is the action of vehicle 1,u = (ω 1 , v 1 ) ∈ U,and the disturbance is the action of vehicle 2,d = (ω 2 , v 2 ) ∈ D.Here the control and disturbance sets, U and D, are defined asU = ([ω min1 , ω max1 ], [v1 min , v1 max ]),D = ([ω min2 , ω max2 ], [v2 min , v2 max ])and the corresponding control and disturbance functional spaces, U and Dare defined as:U = {u(·) ∈ P C 0 (R 2 )|u(t) ∈ U, t ∈ R},D = {d(·) ∈ P C 0 (R 2 )|d(t) ∈ U, t ∈ R},