## 1 Introduction
This method calculates an appropriate safety distance to ensure timely braking when the actual distance falls below this threshold, effectively preventing collisions. Reference [21] described a system that uses lasers to measure the distance to the car ahead and automatically brakes to avoid rear-end collisions. Reference [22] proposed an algorithm for a rear-end collision warning and avoidance system that integrates tire-road friction estimation. Reference [23] proposed a novel model that considers the realistic deceleration behavior of vehicles during emergency braking maneuvers. Reference [24] designed a hierarchical AEB control system based on the safety distance, with a high-level rule-based supervisor, an intermediate-level switching algorithm, and a low-level control module. However, both the TTC model and safety distance model can compromise comfort. To improve this, integrating the two models using a time-safety distance fusion algorithm facilitates effective coordination in both the temporal and spatial domains, enhancing the braking comfort [25]. To increase the acceptance of AEB systems, adaptive TTC models (such as those that adjust the TTC threshold based on the road conditions [26] or modify the braking intensity according to the TTC [27]), along with adaptive safety distance models [28], are utilized to simultaneously address the needs for vehicle comfort and safety. Reference [26] compared how drivers felt about a regular automatic braking system and one that adjusts for slippery roads. Reference [27] created a system that automatically brakes to avoid crashes, but it also ensures that the ride is comfortable. Reference [28] proposed an adaptive safety distance that was adjusted along the uncertain braking duration of the object vehicle, resulting in an improvement in driving comfort. In addition to the aforementioned methods, employing emergency braking control strategies from professional drivers for automated vehicles can significantly enhance comfort during emergency braking, thereby increasing the acceptance of AEB systems [29; 30; 31; 32]. Reference [29] introduced a collision avoidance deceleration assistance system, designed based on expert driver braking behavior models, to enable smooth and safe braking. Our research team has identified a key challenge in AEB control: determining when and how to initiate braking. We propose an AEB algorithm that integrates risk perception and professional driver braking behavior to enhance rear-end collision avoidance [30; 31; 32]. Furthermore, vehicle-to-vehicle communication has also been utilized to improve the emergency braking performance [33; 34; 35]. To tackle the challenges posed by unreliable vehicle-to-everything (V2X) communication and its impact on traffic safety, Reference [33] devised a method to evaluate the reaction time required for a vehicle to avoid a collision caused by another vehicle. Reference [34] analyzed the safety performance of emergency braking using emerging vehicle-to-vehicle (V2V) communication compared to radar-based measurements in existing automatic emergency braking systems. The findings indicated that car-to-car communication enhances the safety more effectively than relying solely on radar sensors. Reference [35] analyzed the safety in emergency braking scenarios with multiple vehicles, showing how decentralized notification messages can reduce the safe following distances. This study's contribution is the development of an innovative automatic braking control method that redefines collision avoidance as a trajectory planning problem. Similar to trajectory planning for lane change avoidance [36; 37; 38; 39], this study proposes a seventh-degree polynomial-based automatic braking control method that considers various constraints at the start and end of braking. Compared to traditional TTC models and safety distance models, the proposed control method significantly enhances the braking comfort while ensuring effective collision avoidance. The structure of this study is organized as follows.