Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 155continuous and smooth, time–dependent human configuration coordinatesq h = q h (t). In other words, all robot coordinates, x r = x r (t), constitutethe smooth Riemannian manifold M r g (such that r = 1, ..., dim(M r g )), withthe positive–definite metric formg ↦→ ds 2 = g rs (x)dx r dx s (3.11)similarly, all human coordinates q h = q h (t), constitute a smooth Riemannianmanifold N h a (such that h = 1, ..., dim(N h a )) , with the positive–definitemetric forma ↦→ dσ 2 = a hk (q)dq h dq k . (3.12)In this Riemannian geometry settings, the feedforward command/controlaction of humans upon robots is defined by a smooth map,C : N h a → M r g ,which is in local coordinates given by a general (nonlinear) functional transformationx r = x r (q h ), (r = 1, ..., dim(M r g ); h = 1, ..., dim(N h a )), (3.13)while its inverse, the feedback map from robots to humans is defined by asmooth map,F = C −1 : M r g → N h a ,which is in local coordinates given by an inverse functional transformationq h = q h (x r ), (h = 1, ..., dim(N h a ); r = 1, ..., dim(M r g )). (3.14)Now, although the coordinate transformations (3.13) and (3.14) arecompletely general, nonlinear and even unknown at this stage, there issomething known and simple about them: the corresponding transformationsof differentials are linear and homogenous, namelydx r = ∂xr∂q h dqh , and dq h = ∂qh∂x r dxr ,which imply linear and homogenous transformations of robot and humanvelocities,ẋ r = ∂xr∂q h ˙qh , and ˙q h = ∂qh∂x r ẋr . (3.15)