Nonhomogeneous nilpotent approximations for systems with singularities

Marilena Vendittelli, Giuseppe Oriolo, Frédéric Jean and 
Jean-Paul Laumond
Publication type:
Paper in peer-reviewed journals
IEEE Transactions on Automatic Control, vol. 49(2), pp. 261-266
Nilpotent approximations are a useful tool for analyzing and controlling systems whose tangent linearization does not preserve controllability, such as nonholonomic mechanisms. However, conventional homogeneous approximations exhibit a drawback: in the neighborhood of singular points (where the system growth vector is not constant) the vector fields of the approximate dynamics do not vary continuously with the approximation point. The geometric counterpart of this situation is that the sub-Riemannian distance estimate provided by the classical Ball-Box Theorem is not uniform at singular points. With reference to a specific family of driftless systems, we show how to build a nonhomogeneous nilpotent approximation whose vector fields vary continuously around singular points. It is also proven that the privileged coordinates associated to such an approximation provide a uniform estimate of the distance.
    author={Marilena Vendittelli and Giuseppe Oriolo and Frédéric Jean 
           and Jean-Paul Laumond },
    title={Nonhomogeneous nilpotent approximations for systems with 
           singularities },
    doi={10.1109/TAC.2003.822872 },
    journal={IEEE Transactions on Automatic Control },
    year={2004 },
    volume={49(2) },