# Efficient Optimization-Based Trajectory Planning for Truck-Trailer Systems
## 3 Results and Discussion
### Summary of Features of the Proposed Solution  
Several relevant indicators were selected for comparison of the proposed and reference solutions, as summarized in Table 2, which provides a numerical comparison between the proposed solution and reference solution across several performance metrics, including computational time, cost function value, total driving time, trajectory point count, distance traveled, and cumulative sum angle differences of the front wheels. The table highlights differences in efficiency and effectiveness, with a note indicating that the reference solution's computational time includes the use of an initial estimate. Finding the complete solution requires approximately 364 s, which represents a marked improvement over the reference solution. Although the authors did not specify exact times, they provided the Matlab code for their solution. Experimental testing of this code shows that it also reaches a solution in minutes (409 s), but only when the supplied initial guess is used. When this is omitted, the reference solution proved unsuccessful, as the provided code ran for over 12 h without convergence. Therefore, the authors' initial guess was adopted for comparison, though including it as part of the solution with the global SFS solver as the authors did may not be fully accurate. \begin{table}
\begin{tabular}{l c c} \hline \hline
**Comparison Indicator** & **Proposed Solution** & **Reference Solution** \\ \hline Computational time for solution [s] & 363.8 & 409.9 a  \\ The value of the cost function & 42.31 & 44.26 \\ Total driving time [s] & 42.35 & 31.58 \\ Number of points in the trajectory & 42 & 33 \\ Total distance traveled [m] & 44.42 & 44.26 \\ Cumulative sum angle diff of front wheels & 2.89 & 20.09 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Numerical comparison of the proposed solution and reference solution. Figure 7: Comparison of control inputs: proposed solution (**left**) and reference solution (**right**). Proposing multiple optimization phases proved beneficial for the problem addressed in this paper. To support this, we also tested the solution without Phase 4 in order to examine its importance. Although constraints related to the truck and its possible collision with obstacles as a part of Phase 4 are also included in Phase 5, incorporating Phase 4 makes the computation much faster. Omitting this led to an almost tenfold increase in computational time, as documented in Table 3. The times are shown in two columns, with and without Phase 4, indicated by a dash symbol. All computations and timing measurements were performed using the same hardware and software setup for consistency: Matlab R2021b on a laptop with an Intel Core i7-11390H processor and 16 GB of RAM. To ensure repeatability and validate results, additional tests were also conducted with different hardware and software configurations. The goal of testing on different hardware and software configurations was to analyze the following issues:  
* Repeatability and Validation: Here, the objective is to ensure reliability by making sure that the proposed solution consistently produces similar results across different configurations. * Hardware Limitations: The effect of RAM size on computation speed was tested, with no observed improvement for this specific task. However, a limitation of the proposed solution lies in its high RAM requirements as the number of nodes in [PERSON]'s algorithm increases. In the presented case study, a single node per meter (\(k_{g}=1\)) was used. If this number was increased to \(k_{g}=5\), then 64 GB of RAM would be required, with RAM demand growing exponentially as more nodes are added. This issue can be effectively addressed by implementing an adaptive grid, which reduces complexity by using a finer grid near obstacles and a coarser grid elsewhere, significantly lowering the computational load. This mainly relates to Phase 1. The computational load for the rest of phases significantly depends on CPU performance when considering our PC, Windows OS, and Matlab environment.