Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 2273.8.5 Application: Dynamical Games on SE(n)−GroupsIn this section we propose a general approach to modelling conflict resolutionmanoeuvres for land, sea and airborne vehicles, using dynamicalgames on Lie groups. We use the generic name ‘vehicle’ to representall planar vehicles, namely land and sea vehicles, as well asfixed altitude motion of aircrafts (see, e.g., [Lygeros et. al. (1998);Tomlin et. al. (1998)]). First, we elaborate on the two–vehicle conflictresolution manoeuvres, and after that discuss the multi–vehicle manoeuvres.We explore special features of the dynamical games solution when theunderlying dynamics correspond to left–invariant control systems on Liegroups. We show that the 2D (i.e., planar) motion of a vehicle may bemodelled as a control system on the Lie group SE(2). The proposed algorithmsurrounds each vehicle with a circular protected zone, while thesimplification in the derivation of saddle and Nash strategies follows fromthe use of symplectic reduction techniques [Marsden and Ratiu (1999)]. Tomodel the two–vehicle conflict resolution, we construct the safe subset ofthe state–space for one of the vehicles using zero–sum non–cooperative dynamicgame theory [Basar and Olsder (1995)] which we specialize to theSE(2) group. If the underlying continuous dynamics are left–invariant controlsystems, reduction techniques can be used in the computation of safesets.3.8.5.1 Configuration Models for Planar VehiclesThe configuration of each individual vehicle is described by an element ofthe Lie group SE(2) of rigid–body motions in R 2 . Let g i ∈ SE(2) denotethe configurations of vehicles labelled i, with i = 1, 2. The motion of eachvehicle may be modelled as a left–invariant vector–field on SE(2):ġ i = g i X i , (3.59)where the vector–fields X i belong to the vector space se(2), the Lie algebraassociated with the group SE(2).Each g i ∈ SE(2) can be represented in standard local coordinates(x i , y i , θ i ) as⎡⎤cos θ i − sin θ i x ig i = ⎣ sin θ i cos θ i y i⎦ ,0 0 1