# Trajectory Optimization with Dynamic Drivable Corridor-Based Collision Avoidance
## 4 Simulation Results and Discussion
This occurs because lower resolutions correspond to larger grid sizes, enabling each grid to cover a broader area and thereby reducing the number of grids needed to represent environmental obstacles. This reduction directly decreases the computational load of collision detection during DC construction, significantly improving the efficiency of collision avoidance constraint modeling. Figure 10: Planned velocity, acceleration, steering angle, and angular velocity profiles over time in two typical benchmark cases: (**a**) first typical case and (**b**) second typical case. Dashed lines in each subplot represent upper and lower bounds that the vehicle must not exceed. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Method** & **Average CPU Time (s)** & **Max CPU Time (s)** \\ \hline Planner 2 & 1.9285 & 3.2293 \\ \hline This work (0.1 m resolution) & 0.0893 & 0.1688 \\ \hline This work (0.2 m resolution) & 0.0732 & 0.1454 \\ \hline This work (0.3 m resolution) & 0.0707 & 0.1476 \\ \hline This work (0.4 m resolution) & 0.0683 & 0.1269 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison of DC construction efficiency between the proposed method and Planner 2. To comprehensively validate the feasibility and effectiveness of our proposed method, it is necessary to evaluate it within the complete numerical optimization process. Statistical results in Table 4 demonstrate that, compared to Planner 1 and Planner 2, our planner exhibits superior computational efficiency, with substantially reduced total trajectory optimization time. Specifically, under a grid resolution of 0.1 m, our scheme achieves a 61.3% reduction in average solution time compared to Planner 2, while maintaining a 100% planning success rate. This validates the significant advantages of the proposed method in computational efficiency and robustness. However, as the resolution decreases, the planning success rate of the planner shows a declining trend. As can be seen from Table 4, when the resolution is reduced to 1 m, the success rate drops from 100% at resolutions of 0.05 to 0.2 m to 88%, indicating a significant decrease in planning success rate. Therefore, in the selection of grid resolution, it should neither be too large nor too small, and a balance between efficiency and success rate needs to be struck: Lower grid resolutions (larger grid sizes) reduce the number of occupied grids representing obstacles, thereby decreasing collision detection computational load and improving efficiency. However, a lower resolution may lead to insufficient precision in obstacle representation, resulting in a loss of passable space (theoretically, the maximum loss can be up to twice the grid resolution), especially in already narrow passages, which may further reduce the drivable area and increase the risk of planning failure. Conversely, a higher grid resolution can depict obstacles more finely, enhancing the accuracy of path planning, but the number of grids used to represent obstacles in the environment increases, leading to higher computational complexity and time consumption for collision detection. Thus, practical applications require an appropriate balance in grid resolution to ensure complementary improvements in planning accuracy and computational efficiency. As shown in Figure 11, the passable space between two obstacles is relatively narrow. When the grid size exceeds a certain threshold, this region fails to construct a valid DC. Since the initial solution passes through this region and the optimized trajectory maintains a homotopy [33] relationship with the initial solution, the optimized trajectory must also traverse this area. However, due to the low grid resolution, the obstacle representation lacks precision, resulting in the loss of passable space. Consequently, a valid DC cannot be constructed in this region, ultimately leading to solution failure.