Generic one-step bracket-generating distributions of rank four

On symmetries of a sub--Riemannian structure with growth vector $(4,7)$

“…By nilpotent control problems we mean invariant control problems on Carnot groups. There are several types of non-free (4, 7)-case, [5], and we focus on one special type.…”

Local Geometric Control of a Certain Mechanism with the Growth Vector (4,7)

“…Remark on dual curvature. Following [18,7], curvature of a distribution H on a manifold Q is a linear bundle map F :…”

“…To study extremal trajectories in the next section, we need the sub-Riemannian structure on the nilpotent approximation. We consider a control metric g in D = N 1 , N 2 , N 3 , N 4 such that the fields N i for i = 1, 2, 3, 4 are orthogonal and have the length one with respect to g. This clearly determines a left-invariant sub-Riemannian structure g of D (with respect to the action given by group structure (7) on N ).…”

“…We are interested in the modification corresponding to one or more prismatic joints such that the control distribution will be a that of the growth vector (4,7). Local controllability of such robot is given by the appropriate Pfaff system of ODEs.…”

Geometry of bracket-generating distributions of step 2 on graded manifolds

“…A smooth distribution D ⊂ T M is said to be bracket-generating if all iterated brackets among its sections generate the whole tangent space to the manifold M, [1,8]. D is a bracket-generating distribution of step 2 if D 2 = T M, where D 2 = D + [D, D].…”