Ivancevic_Applied-Diff-Geom

560 Applied Differential Geometry: A Modern IntroductionAlso R U M (q 0, ≤ T ) = ∪ 0≤t≤T R U M (q 0, t). Now, regarding the local controllability,we are only interested in points which can be reached withouttaking ‘large excursions’. Control problems which are local in this way havethe advantage that they can be characterized by Lie brackets. So, we wantto describe our reachable set R U M (q, ≤ T ) for the simple mechanical controlsystem (4.56). The system (4.56) is locally configuration accessible (LCA)at q if there exists T > 0 so that R U M (q, ≤ t) contains a non–empty opensubset of M for each neighborhood U of q and each t ∈]0, T ]. Also, (4.56)is locally configuration controllable (LCC) at q if there exists T > 0 so thatR U M (q, ≤ t) contains a neighborhood of q for each neighborhood U of qand each t ∈]0, T ]. Although sound very similar, the notions of local configurationaccessibility and local configuration controllability are genuinelydifferent (see Figure 4.7). Indeed, one need only look at the example ofthe robotic leg with the F 1 input. In this example one may show that thesystem is LCA, but is not LCC [Lewis (1998)].Fig. 4.7 Difference between the notions of local configuration accessibility (a), and localconfiguration controllability (b).Local Configuration AccessibilityThe accessibility problem is solved by looking at Lie brackets. For thiswe need to recall the definition of the vertical lift [Lewis (1998)]:verlift(Y (v q )) = d dt∣ (v q + tY (q)),t=0in local coordinates, if Y = Y i ∂ q i, then verlift(Y ) = Y i ∂ v i. Now we canrewrite (4.56) in the first–order form:˙v = Z(v) + u a verlift(Y a (v)),