## 1 Introduction  
With rapid advancements in automotive intelligence, vehicles now provide functionalities beyond basic transportation, driving increased user expectations for safety, convenience, and comfort. As a crucial component of intelligent vehicles, automatic reverse parking (ARP) has garnered extensive attention from both academic and industrial sectors. The effectiveness of ARP systems primarily depends on the accuracy of parking reference trajectory planning and the system's ability to track and control this trajectory [1]. To meet users' parking requirements, it is imperative during the ARP process to ensure that the vehicle does not come into contact with obstacles, while striving for a parking trajectory that is as short and smooth as possible [2]. A successful ARP operation entails the completion of three critical steps: assessing the feasibility constraints of parking (including detection of available garages), optimizingthe acquisition of reference trajectory for parking (trajectory optimization), and executing vehicle tracking control to follow the reference parking trajectory into the designated region (trajectory tracking) [3]. In recent years, scholars in the relevant field have achieved a series of significant research outcomes concerning the issue of ARP. For the detection of available parking garages, a parking garage identifying method based on deep learning was introduced in [4]. This approach utilized the Faster R-CNN model, known for its faster training and testing speeds, thus satisfying real-time detection requirements. However, this method could not detect obstacles inside parking garages. A parking detection method based on vehicle occurrence frequency was introduced in [5]. For automatic parking technology, there is significant focus on the parking detection and tracking control technologies that influence the success rate of parking. However, with an increase in vehicle intelligence, people have higher requirements for the quality of automatic parking trajectory, including collision avoidance, smoothness, and sufficiently short path length. Hence, parking trajectory has garnered increasing attention and has become progressively more mature. In the field of parking trajectory optimization, various methodologies including geometric programming methods, sampling-based approaches, search algorithms, and numerical optimization techniques have been systematically implemented. A parallel parking path planning framework was developed through three-phase curve interpolation, where cyclotron curves, circular arcs, and quintic polynomial curves were integrated to satisfy narrow spatial constraints in [6]. The computational efficiency limitations, curvature discontinuity issues, and strict initiating position requirements in automatic parking systems were comprehensively addressed through an enhanced RRT* algorithm with non-linear optimization techniques in [7]. Furthermore, critical challenges in path planning security and search effectiveness for unmanned parking systems were resolved by a safety-enhanced and efficiency-enhanced hybrid A* algorithm design in [8]. An optimization scheme for the parallel parking trajectory in automatic parking systems, utilizing the Gaussian collocation pseudospectral method subject to phase constraints, was introduced in [9]. An optimized parking trajectory planning approach grounded in mirror parking objectives was proposed to resolve challenges including low parking success rates, time-consuming operations, and undefined operational design domains (ODDs) in automatic parking systems for confined garages [10].