Ivancevic_Applied-Diff-Geom

156 Applied Differential Geometry: A Modern IntroductionRelation (3.15), representing two autonomous dynamical systems, givenby two sets of ordinary differential equations (ODEs), geometrically definestwo velocity vector–fields: (i) robot velocity vector–field, v r ≡ v r (x r , t) :=ẋ r (x r , t); and human velocity vector–field, u h ≡ u h (q h , t) := ˙q h (q h , t). Recallthat a vector–field defines a single vector at each point x r (in somedomain U) of a manifold in case. Its solution gives the flow, consisting ofintegral curves of the vector–field, such that all the vectors from the vector–field are tangent to integral curves at different points x i ∈ U. Geometrically,a velocity vector–field is defined as a cross–section of the tangent bundleof the manifold. In our case, the robot velocity vector–field v r = ẋ r (x r , t)represents a cross–section of the robot tangent bundle T M r g , while the humanvelocity vector–field u h = ˙q h (q h , t) represents a cross–section of thehuman tangent bundle T N h a . In this way, two local velocity vector–fields,v r and u h , give local representations for the following two global tangentmaps,T C : T N h a → T M r g , and T F : T M r g → T N h a .To be able to proceed along the geometrodynamical line, we neednext to formulate the two corresponding acceleration vector–fields, a r ≡a r (x r , ẋ r , t) and w h ≡ w h (q h , ˙q h , t), as time rates of change of the two velocityvector–fields v r and u h . Now, recall that the acceleration vector–fieldis defined as the absolute time derivative, ˙¯v r = D dt vr , of the velocity vector–field. In our case, we have the robotic acceleration vector–field a r := ˙¯v rdefined on M r g bya r := ˙¯v r = ˙v r + Γ r stv s v t = ẍ r + Γ r stẋ s ẋ t , (3.16)and the human acceleration vector–field w h := ˙ū h defined on N h abyw h := ˙ū h = ˙u h + Γ h jku j u k = ẍ r + Γ h jk ˙q j ˙q k , (3.17)Geometrically, an acceleration vector–field is defined as a cross–section ofthe second tangent bundle of the manifold. In our case, the robot accelerationvector–field a r = ˙¯v r (x r , ẋ r , t), given by the ODEs (3.16), represents across–section of the second robot tangent bundle T T M r g , while the humanacceleration vector–field w h = ˙ū h (q h , ˙q h , t), given by the ODEs (3.17), representsa cross–section of the second human tangent bundle T T N h a . In thisway, two local acceleration vector–fields, a r and w h , give local representationsfor the following two second tangent maps,T T C : T T N h a → T T M r g , and T T F : T T M r g → T T N h a .