Is there a way to test for small time local controllability at non-equilibrium points?

Question

For a system of the form $x'(t) = f(x,u)$, define $R(x_0, T) = \{x(T):x'(t) = f(x,u)\text{ with } x(0) = x_0 \text{ and $u$ piecewise continuous on $[0,T]$}\}$. The control system is called small-time locally controllable (STLC) at $x_0$ if for $t> 0$ small enough, $x_0 \in \text{int}\ R(x_0, t)$. I have found several papers on sufficient conditions for STLC, (eg. this paper by Hermes, this one by Lewis, and another one by Krastanov). These papers assume $f$ to be of the form $f(x,u) = f_0(x) + \sum u_i f_i(x)$, which is not really a problem for me, but their assumption of $f_0(x_0) = 0$ is. Are there any results concerning STLC at a non-equilibrium point (i.e. $f(x_0, u) \neq 0$)? In particular, I am interested in systems that are state-affine $f(x,u) = F(u) x$ where $F: \Bbb R \to \Bbb R^{n\times n}$ is non-singular for $u\neq 0$. I would like to note that the $x_0$ I am interested in is not a null-eigenvector of $F(0)$. Alternatively, I would also be interested in results concerning piecewise constant controls $u$ s.t. $x(T) = x_0$ for some $T\in (0, \infty)$.