# Trajectory Optimization with Dynamic Drivable Corridor-Based Collision Avoidance
## 4 Simulation Results and Discussion
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Description** & **Setting** \\ \hline \(\mathrm{L_{F}}\) & Front hang length of vehicle. & 0.96 m \\ \hline \(\mathrm{L_{W}}\) & Wheelbase of vehicle. & 2.80 m \\ \hline \(\mathrm{L_{R}}\) & Rear hang length of vehicle. & 0.929 m \\ \hline \(\mathrm{L_{B}}\) & Width of vehicle. & 1.942 m \\ \hline \(\mathrm{a_{max}}\) & Upper bound of \(\mid\!a(t)\!\mid\). & 4.0 m/s\({}^{2}\) \\ \hline \(\mathrm{v_{max}}\) & Upper bound of \(\mid\!\upsilon(t)\!\mid\). & 4.0 m/s \\ \hline \(\mathrm{\Phi_{max}}\) & Upper bound of \(\mid\!\phi(t)\!\mid\). & 0.85 rad \\ \hline \(\mathrm{\Omega_{max}}\) & Upper bound of \(\mid\!\omega(t)\!\mid\). & 1.0 rad/s \\ \hline \(\mathrm{N_{R}}\)+1 & Number of DCs for \(\mathrm{P_{r}}\) or \(\mathrm{P_{f}}\). & 100 \\ \hline \(\mathrm{\Delta s}\) & Unit step length in DC specification. & 0.1 m \\ \hline \(\mathrm{L_{limit}}\) & Maximum step length in DC specification. & 5.0 m \\ \hline \(\mathrm{N_{fe}}\) & Number of finite elements. & 100 \\ \hline resolution & Spatial resolution of the grid. & 0.1 m \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parametric settings regarding model and approach. Figure 10 illustrates the variation trends of the vehicle's velocity, acceleration, front-wheel steering angle, and angular velocity in the first two typical scenarios. The results indicate that all evaluated parameters (including acceleration, angular velocity, and steering angle) remain within the permissible limits, satisfying the constraints defined by (2). To further evaluate the performance of our approach, we introduce two comparative planners for experimental validation. The first planner (denoted as Planner 1) replaces within-DC constraints with full-size collision avoidance constraints [32], while the second (denoted as Planner 2) employs within-DC constraints but adopts a different DC construction methodology [27] from the proposed approach. All three planners share identical configurations except for their collision avoidance strategies. We test their performance across 100 scenarios, with experimental results statistically summarized in Tables 3 and 4 (DC expansion step size: \(\Delta\)s = 0.1 m). In Table 4, there are some points to note: Average CPU time represents the average duration per case across all successful scenarios; 99 th percentile CPU time indicates that in all successful scenarios, and 99% of the scenarios have an execution time less than or equal to this value; a success occurs when an NLP solution process ends with a converged optimum. Conversely, a failure refers to convergence to infeasibility or no convergence within default NLP iteration/CPU time. The advantages of employing DC-based strategies for collision avoidance over non-DC approaches, as well as the sensitivity of planning performance to the \(\Delta\)s parameter--the unit step length in DC specification--have been comprehensively  
Figure 8: Schematic illustration of paved DCs (blue boxes indicate front DCs, while green boxes represent rear DCs). Figure 9: Planned trajectories and corresponding vehicle footprints for representative benchmark cases: (**a**) Case 1; (**b**) Case 2; (**c**) Case 3; and (**d**) Case 4.  
investigated in previous work [27]. Therefore, neither a detailed comparison with Planner 1 nor a further sensitivity analysis on \(\Delta\)s is presented in this paper. Rather, we focus on evaluating the proposed planner against Planner 2, as both incorporate DC, to validate the effectiveness of our method. As shown in Table 3, our method significantly outperforms the approach used in Planner 2 for constructing the DC. Specifically, at a resolution of 0.1 m, the average time required for DC construction decreases from 1.9285 s in Planner 2 to 0.0893 s in our method, a reduction of 95.4%. As the resolution further decreases, the time required for DC construction gradually diminishes. When the resolution is reduced to 0.5 m, the reduction further improves to 96.5%.