## 1 Introduction  
According to traffic reports in recent years, most traffic accidents are caused by unreasonable lane-changing behavior of vehicles, which are human-made [1]. Reasonable lane-changing behavior is one of the effective ways to ensure traffic safety and solve traffic congestion [2]. With the emergence of smart cars and the maturity of autonomous driving technology, traffic accidents caused by human factors have been greatly reduced. The lane-changing system of intelligent vehicles mainly includes the identification of the surrounding traffic environment, the corresponding decision-making, trajectory planning, and control [3]. For a lane change problem, the current research usually uses the rule method, utility theory method, Game theory method, and artificial intelligence method to make decisions [4]. The rule method was first proposed by [PERSON] [5], who made lane-changing decisions based on the minimum safe distance and comprehensive lane-changing intention. [PERSON] [6] considered the relationship between lane-changing interactive vehicles to design safe distances and improve the [PERSON] model. [PERSON] et al. [7] designed a Decision model for lane change with minimum safety distance considering the impact of self-owned vehicle safety and traffic flow. The utility theory method is widely studied in lane-changing problems. [PERSON] [8] incorporated the driver's personality into the utility theory and established a Decision model for lane changing. [PERSON] et al. [9] classified lane-changing issues into mandatory lane changing and autonomous lane changing and analyzed them for different scenarios. The Game theory method is often used in current research. [PERSON] [10]first used non-zero and non-cooperative games to study vehicle confluence. [PERSON] et al. [11] designed a more accurate and effective Decision model for lane changing based on utility theory and Game theory. Artificial intelligence methods have emerged with current developments, utilizing learning from data to obtain the best decisions in the current state. [PERSON] et al. [12] established a support vector machine Decision model for lane changing and conducted a comparative study with the rule method. [PERSON] et al. [13] utilized the proximal strategy optimization algorithm to simulate and learn the optimal strategy from the traffic environment. Lane-changing trajectory planning, as an indispensable condition for lane-changing, has always been one of the research focus and hotspots of scholars in related fields [14]. For lane-changing trajectory planning, the current mainstream research methods mainly include specific function-based trajectory planning methods, search-based trajectory planning methods, and some intelligent algorithms. Algorithms based on function curves include lane-changing trajectory of sine and cosine algorithms, arc lane-changing trajectory, positive and negative trapezoidal acceleration lane-changing trajectory, and polynomial-based lane-changing trajectory. Among the commonly used geometric curves, the polynomial curve is the most widely used in trajectory planning, which can plan a smooth curve with low computational cost and optimize the trajectory by adjusting the order of the polynomial. [PERSON] et al. [15] proposed the polynomial method for the first time; they used polynomial curves instead of arc curves to plan continuous curvature trajectories, which ensured the traceability of the trajectory. [PERSON] et al. [16] proposed a trajectory planning model based on cubic polynomial curves; they focused on the problem of trajectory re-planning in dynamic driving environments. [PERSON] et al. [17] increased the polynomial function to the quintic and solved the optimal trajectory by considering the travel time and travel distance as free variables.