A path re-planning method is proposed based on a discretization of the state space, aiming at finding a collision-free path for the vehicle which is capable of forward and backward motion when changes occur in the environment. A Control Set with Turning Radius Constraint (CSTRC) is formulated and the feasibility of paths in CSTRC is also proved out. The A* search is applied to produce a feasible path considering the distance and angle between the vehicle and the target pose. Path re-planning can be carried out efficiently when the environment changes. Simulation results demonstrate that the method realizes path re-planning effectively and the path satisfy the turning radius constraint.