Ivancevic_Applied-Diff-Geom

562 Applied Differential Geometry: A Modern Introductionwith initial velocity tangent to N remains on N. This can be weakened todistributions: a distribution D on M is geodesically invariant if for everygeodesic γ : [0, T ] → M, ˙γ(0) ∈ D γ(0) implies ˙γ(t) ∈ D γ(t) for t ∈]0, T ].D is geodesically invariant i it is closed under symmetric product [Lewis(1998)]. This Theorem says that the symmetric product plays for geodesicallyinvariant distributions the same role the Lie bracket plays for integrabledistributions. This result was key in providing the geometricaldescription of the reachable configurations.An integrable distribution is geodesically generated distribution if it isthe involutive closure of a geodesically invariant distribution. This basicallymeans that one may reach all points on a leaf with geodesics lying in somesubdistribution. The picture one should have in mind with the geometry ofthe reachable sets is a foliation of M by geodesically generated (immersed)submanifolds onto which the control system restricts if the initial velocity iszero. The idea is that when we start with zero velocity we remain on leavesof the foliation defined by C hor [Lewis and Murray (1997); Lewis (2000a)].Note that for cases when the affine connection possesses no geodesicallyinvariant distributions, the system (4.56) is automatically LCA. This istrue, for example, of S 2 with the affine connection associated with its roundmetric.Clearly C ver is the smallest geodesically invariant distribution containingspan{Y 1 , ..., Y m }. Also, C hor is geodesically generated by spanspan{Y 1 , ..., Y m }. Thus R U M is contained in, and contains a non–emptyopen subset of, the distribution geodesically generated by span{Y 1 , ..., Y m }.Note that the pretty decomposition we have for systems with no potentialenergy does not exist at this point for systems with potential energy.Local Configuration ControllabilityThe problem of configuration controllability is harder than the one ofconfiguration accessibility. Following [Lewis and Murray (1999); Lewis(2000a)], we will call a symmetric product in {Y 1 , ..., Y m } bad if it containsan even number of each of the input vector–fields. Otherwise we willcall it good. The degree is the total number of vector–fields. For example,〈〈Y a : Y b 〉 : 〈Y a : Y b 〉〉 is bad and of degree 4, and 〈Y a : 〈Y b : Y b 〉〉 is good andof degree 3. If each bad symmetric product at q is a linear combination ofgood symmetric products of lower degree, then (4.56) is LCC at q.Now, the single–input case can be solved completely: The system (4.56)with m = 1 is LCC iff dim(M) = 1 [Lewis and Murray (1999)].