## 1 Introduction  
It is an important working condition for driverless vehicles to complete safe parallel parking in narrow parking spaces, where it is more difficult to park than in ordinary parking conditions [1]. Planning a feasible and safe path with curvature continuity is crucial for autonomous parking yet remains to be fully solved [2, 3]. At present, the main methods of path planning in parallel parking are based on geometry, sampling, and numerical optimization. The path planning method based on sampling or numerical optimization can plan a feasible parking path in a complex environment and small parking space but with the disadvantage of a large amount of calculation. The method of path planning based on geometry has high computational efficiency, good real-time performance, and is convenient for engineering implementation [4]. Common parking path curves designed include fifth-order polynomial [5], arc [6], Bezier curve, etc. For example, the path in parallel parking is divided into the entry section to guide the car to approach the parking space and the adjustment section to guide the car to adjust the target pose. The constrained optimization problem of arc-line-cyclotron curve is solved to establish the entry section path, the adjustment section path is planned through the combination of arc-line, guiding the car to complete parking task safely and comfortably [7]. The path planning method of a non-parallel initial state based on global planning to establish a three-arc parking path has stronger robustness than the traditional minimum radius method [8]. In the path planning method based on the improved quintic polynomial and designed penalty function, the genetic algorithm is used to calculate the optimum parking path and minimum parking space to complete fast and effective parking with less vehicle damage and low space requirements [9]. The path planning problem is often converted to an optimization problem [10; 11]. In a guaranteed collision-free trajectory planning method, a dynamic optimization problem is built, including vehicle kinematics, collision avoidance constraints, and physical restraints. To describe collision avoidance constraints, the concepts of the virtual protection frame and the magnification parameter are introduced. Tests in three common scenarios and a complex scenario illustrate the effectiveness [1]. A method based on the Frenet coordinate system is proposed under more complex driving scenarios such as curving roads. To balance the driving comfort ability and safety of the path candidate, a sequential quadratic programming algorithm is employed [12]. A scene-dependent B-spline parallel parking path planning algorithm meets vehicle kinematics model constraints well. Here, the vehicle actuator's dynamic performance and parking path smoothness are key attributes [13]. To solve the multi-vehicle cooperative autonomous parking trajectory planning problem uniformly and effectively with a faster convergence speed, a vehicle kinematics model with dynamic constraints, namely endpoint and collision avoidance constraints, is established. The problem is transformed into an optimal control problem with the shortest parking completion time as the optimal cost function. An initial guess generation strategy is proposed, and the iteration method is used to solve by adding collision avoidance constraints to improve the convergence and robustness [14]. However, the method of integrating a neural network to optimize the comprehensive performances of the initial path planned by the geometric method needs further study. The rest of the paper is organized as follows. In Section 2, the composition and working process of an autonomous parking system for driverless vehicle are analyzed. Experiments concerning parking space detection based on an Arduino intelligent minicar and ultrasonic sensor are carried out. Based on the idea of simulating a human driver and the geometric method, the parallel parking path of arc-line-arc is initially planned.