*3.3. Analytical Expansion*

The discretized control actions achieve the expansion of the node during the forward node search. While using the optimal steering action, the moving object may not be able to precisely reach the predefined position and orientation of the goal node *T*. As a result, analytic expansions based on the RS curve would be used to guide the node search near the goal. The RS curve would also be checked for collision with the obstacles along the way. When the analytical expansion is employed, the algorithms perform an analysis that looks for an RS curve to the goal node *T* before conducting the node search from the node *t*. The

node search would be terminated if the algorithm could find a collision-free path to the destination based on the RS curve. The RS curve would be augmented to the searched path, and the overall path planning from *t* to *T* is accomplished.
