## 2 Materials and Methods
### Path Smoothing and Set Trajectory Calculation
To ensure that the paths remained both collision-free and dynamically feasible after smoothing, the following features were incorporated into the planning methods:  
* For A* and RRT*, morphological preprocessing (dilation and closing) is applied to artificially expand the obstacles before planning;
* In the APF method, repulsive forces activate within a 10 m radius around obstacles;
* The GA and PSO reject intermediate waypoints that are less than 10 m from any obstacle. ```
1:Given: grid map \(M\), start \((x_{s},y_{s})\), goal \((x_{g},y_{g})\), step size \(\delta\), neighbor radius \(r_{\text{nbr}}\), goal threshold \(d_{\text{goal}}\), max iterations \(I_{\text{max}}\)
2: Initialize tree \(\mathcal{T}\leftarrow\{(x_{s},y_{s},0,0)\}\)
3:for\(i=1\) to \(I_{\text{max}}\)do
4: With a prob. of \(0.1\), set \((x_{\text{rand}},y_{\text{rand}})\leftarrow(x_{g},y_{g})\); otherwise, sample uniformly in \(M\)
5: Find \((x_{\text{near}},y_{\text{near}})\gets argmin_{(x,y)\in\mathcal{T}}\|(x, y)-(x_{\text{rand}},y_{\text{rand}})\|\)
6: Compute the direction \(\theta\leftarrow\arctan 2(y_{\text{rand}}-y_{\text{near}},x_{\text{rand}}-x_{ \text{near}})\)
7: \((x_{\text{new}},y_{\text{new}})\leftarrow(x_{\text{near}}+\delta\cos\theta,y_ {\text{near}}+\delta\sin\theta)\)
8:if CollisionFree\((x_{\text{near}},y_{\text{near}},x_{\text{new}},y_{\text{new}},M)\)then
9:\(\mathcal{T}\leftarrow\mathcal{T}\cup(x_{\text{new}},y_{\text{new}},\text{ parent,cost})\)
10: Find \(\mathcal{N}_{\text{nbr}}\leftarrow\{(x,y)\in\mathcal{T}\mid\|(x,y)-(x_{\text{ new}},y_{\text{new}})\|<r_{\text{nbr}}\}\)
11: Set the parent \(\leftarrow\) index of \((x_{\text{near}},y_{\text{near}})\) with the minimal path cost
12: Update cost \(\leftarrow\) cost\((x_{\text{near}})+\|(x_{\text{near}},y_{\text{near}})-(x_{\text{new}},y_{ \text{new}})\|\)
13:for each \((x_{\text{nbr}},y_{\text{nbr}})\in\mathcal{N}_{\text{nbr}}\)do
14:if\(\text{cost}(x_{\text{new}})+\|(x_{\text{new}},y_{\text{new}})-(x_{\text{ nbr}},y_{\text{nbr}})\|<\text{cost}(x_{\text{nbr}})\)then
15:if CollisionFree\((x_{\text{new}},y_{\text{new}},x_{\text{nbr}},y_{\text{nbr}},M)\)then
16: Update the parent of \((x_{\text{nbr}},y_{\text{nbr}})\) to \((x_{\text{new}},y_{\text{new}})\)
17: Recompute the cost\((x_{\text{nbr}})\) and propagate to the descendants
18:endif
19:endif
20:endfor
21:endif
22:if\(\|(x_{\text{new}},y_{\text{new}})-(x_{g},y_{g})\|<d_{\text{goal}}\)then
23:break
24:endif
25:endfor
26:Find \((x_{\text{last}},y_{\text{last}})\leftarrow argmin_{(x,y)\in\mathcal{T}}\|(x, y)-(x_{g},y_{g})\|\)
27: Reconstruct the path by backtracking the parents from \((x_{\text{last}},y_{\text{last}})\) to \((x_{s},y_{s})\)
28: Append \((x_{g},y_{g})\) to the path if within the threshold
29:Output: Final route path
```  
**Algorithm 8** Rapidly Exploring Random Tree-Star (RRT*)  
The smoothed path was described by \(\big{(}x(t),\,y(t)\big{)}\) with the first and second derivatives denoted as \((x^{\prime}(t),\,y^{\prime}(t))\) and \((x^{\prime\prime}(t),\,y^{\prime\prime}(t))\), respectively. The curvature \(\kappa(t)\) was defined as  
\[\kappa(t)=\frac{\big{|}\,x^{\prime}(t)\,y^{\prime\prime}(t)-y^{\prime}(t)\,x^{ \prime\prime}(t)\big{|}}{\Big{(}x^{\prime}(t)^{2}+y^{\prime}(t)^{2}\Big{)}^{ \frac{3}{2}}} \tag{25}\]  
The maximum lateral acceleration \(a_{\text{lat}}\) imposes a curvature-based velocity limit:  
\[v_{\text{curv}}(t)=\sqrt{\frac{a_{\text{lat}}}{\kappa(t)}}\quad\text{(if }\kappa(t)>0) \tag{26}\]  
This condition dynamically constrains the reference velocity of the vehicle based on the instantaneous path curvature, effectively linking the allowable speed to the lateral acceleration induced by the maneuver. As a result, the vehicle slows down in segmentswith a high curvature, which prevents excessive deviation from the planned trajectory during turning. Under these conditions, inertial effects such as Coriolis and centripetal forces are inherently limited and have only a minor impact on the vehicle's motion.