Local controllability of control-affine systems with quadractic drift and constant control-input vector fields

In this paper we study the small-time local controllability (STLC) property of polynomial control-affine systems whose drift vector field is a 2-homogeneous polynomial vector field and whose control-input vector fields are constant. Such systems arise in the study of controllability of mechanical control systems. Using control variations and rooted trees, we obtain a combinatorial expression for the Taylor series coefficients of a composition of flows of vector fields and use it to derive a high-order sufficient condition for STLC for these systems. The resulting condition is stated in terms of the image of the control-input subspace under the drift vector field and is therefore invariant under (linear) feedback transformations.