This paper proposes a local path planning method with a reachable set for Automated vertical Parking Systems (APS). First, given a parking lot layout with a goal position, we define an intermediate pose for the APS to accomplish reverse parking with a single maneuver, i.e., without changing the gear shift. Then, we introduce a reachable set which is a set of points consisting of the grid points of all possible intermediate poses. Once the APS approaches the goal position, it must select an intermediate pose in the reachable set. A minimization problem was formulated and solved to choose the intermediate pose. We performed various scenarios with different parking lot conditions. We used the Hybrid-A* algorithm for the global path planning to move the vehicle from the starting pose to the intermediate pose and utilized clothoid-based local path planning to move from the intermediate pose to the goal pose. Additionally, we designed a controller to follow the generated path and validated its tracking performance. It was confirmed that the tracking error in the mean root square for the lateral position was bounded within 0.06m and for orientation within 0.01rad.

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