For polynomial (and immersible analytic) control systems, the set of accessibility singular points is algebraic, and the paper provides algorithms to compute it and the exact accessibility index.
Kawski, On the problem whether controllability is finitely determined, in: Proceedings of MTNS, volume 6, 2006
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Finite determination of accessibility and geometric structure of singular points for nonlinear systems
For polynomial (and immersible analytic) control systems, the set of accessibility singular points is algebraic, and the paper provides algorithms to compute it and the exact accessibility index.