# Trajectory Optimization with Dynamic Drivable Corridor-Based Collision Avoidance
## 1 Introduction
### Related Works  
Collision avoidance remains a challenge in trajectory planning, and its implementation depends on the chosen planning strategy. Sampling- and search-based methods [8; 9], such as State Lattice Planning [10], Rapidly exploring Random Trees and its variants [11; 12], Hybrid A*[13; 14], and the Dynamic Window Approach [15], typically achieve collision avoidance by discretizing the search space and conducting collision checks during node selection. These methods are advantageous due to their simplicity and computational efficiency in complex environments. However, they suffer from several limitations, including insufficient path smoothness arising from discretization, difficulties in ensuring global optimality, and sharply declining search efficiency in high-dimensional state spaces, especially when vehicle dynamic constraints are considered. Numerical optimization-based methods, in contrast, explicitly formulate collision avoidance as constrained optimization problems, typically through inequalities representing safety distances [16; 17; 18; 19]. Such methods produce smooth, precise trajectories and naturally incorporate multiple constraints for coordinated optimization. Nonetheless, their computational complexity can be substantially higher, particularly when addressing complex, non-convex collision avoidance constraints [20]. To balance computational efficiency and trajectory quality, researchers have proposed two main improvement strategies. The first involves hierarchical planning frameworks, which employ sampling- or search-based methods to quickly generate initial paths. These preliminary paths serve as initial guesses or reference trajectories for numerical optimization, significantly reducing the computational burden [21; 22]. The second strategy simplifies the formulation of collision constraints to a lower computational complexity. Aligned with the first strategy, this work adopts a two-stage trajectory planning approach: An initial coarse trajectory is rapidly generated using the Hybrid A* algorithm, providing a feasible starting solution. Subsequently, a numerical optimization method refines this trajectory to meet precision and constraint requirements. Many recent studies [23; 24; 25] have also pursued the second strategy--simplifying collision constraint formulations, as summarized in Table 1. A widely employed method is the Artificial Potential Field (APF), which implicitly achieves collision avoidance by combining attractive potentials (guiding towards goals) and repulsive potentials (repelling obstacles). Despite its simplicity, APF methods are susceptible to local minima, particularly when obstacles are near goal positions, limiting global optimality. [PERSON] et al. [26] introduced an alternative method approximating collision avoidance constraints as linear constraints. However, this approach faces challenges in precisely quantifying approximation errors, potentially compromising reliability. [PERSON] et al. [27; 28; 29] proposed the DC method to simplify complex obstacle avoidance constraints. Each drivable corridor (DC) defines a safe, constrained region within which the vehicle is expected to remain at a specific time instance. Throughout the planning horizon, \begin{table}
\begin{tabular}{c c c} \hline \hline
**Method** & **Advantages** & **Disadvantages** \\ \hline APF & Simple implementation and high computational efficiency & Susceptibility to local minima \\ \hline Linear Constraint Approximation & Linearized constraint formulation and reduced computational complexity & Approximation errors and reduced solution reliability \\ \hline DC Method & Reduction of nonlinearity in optimization and decoupling from environment complexity & High computational cost of corridor construction \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of collision constraint simplification methods. a sequence of such time-indexed rectangular regions collectively forms a spatiotemporal safety envelope, ensuring that the vehicle's trajectory stays within allowable boundaries at all times.