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.
Risler, A bound for the degree of nonholonomy in the plane, Theoretical Computer Science 157 (1996) 129–136
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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.