Planning paths of minimal curvature

We consider the problem of planning curvature constrained paths amidst polygonal obstacles, connecting the given start and target configurations. Let the critical curvature R_c^-1 be the minimal curvature for which a constrained path exists. We describe an algorithm which approximates the critical curvature and finds a corresponding path. Further, we give an efficient decision procedure to determine if there exists a path satisfying the given curvature constraint R^-1, with running time polynomial in |R-R_c |/R