## 3 Related Work  
Planning paths of limited curvature has been addressed with various methodologies. The most relevant properties of selected known approaches have been gathered in Table 1. A fundamental result from Reference [5] characterizes the shortest paths of bounded curvature as particular combinations of line segments and circle arcs, i.e., Reeds-Shepp (RS) paths. Note that this result does not consider obstacles, which have been partially taken into account for Dubins paths in Reference [6]. The drive toward paths with continuous curvature and a continuous-curvature arc-length derivative has led to the development of the CC-steer method [7] and its \(\mathbb{G}^{3}\)-continuous variant introduced in Reference [8]. Our work can also be viewed as a \(\mathbb{G}^{3}\)-continuous extension of the CC-steer method. However, contrary to the approach from Reference [8], our transition segments are not represented by curvature profiles. We propose transition segments with explicit representation of \(x\) as a function of \(y\), which can easily be used with known path-following controllers (e.g., Reference [9]). We also devised a closed-form expression utilized to check for collisions between our \(\mathbb{G}^{3}\)-continuous path primitive and an obstacle, which leads to efficient and exact collision checking. There has also been some work on improving the computational cost of finding RS-like paths in Reference [10]; however, path-continuity and collision-checking issues for paths of high continuity were not explicitly addressed, to the best of our knowledge.  
One can also find various methods of generating paths with a different structure to RS paths. For example, polynomial spline-based path primitives such as \(\eta^{3}\)-splines [11] and \(\eta^{4}\)-splines [12] have been developed. Their advantages are generality, conceptual simplicity, and high continuity. However, collision and curvature-constraint checking must usually be done numerically in the case of such primitives, which leads to significant computational cost. Furthermore, the parameters of such primitives can be hard to tune, even though this was partially addressed in Reference [13]. Another group of path primitives and extend procedures relies on B-splines [14; 15; 16] providing a limited curvature, curvature continuity (but not \(\mathbb{G}^{3}\)-continuity), and parameters that are easy to tune. Collision checking is done numerically in this case as well.  
There are also various application-specific methods using the path primitives mentioned above, such as planning algorithms for environments with a roadlike structure similar to References [17; 18], a method used to design curvature profiles of paths passing through waypoints [19], a spline-based approach exploiting sum-of-squares optimization to handle exact collision checking, or an elastic-band-like algorithm [20]. Those methods share the various benefits and drawbacks of different path primitives. However, thanks to the proposed path primitive, our approach combines \(\mathbb{G}^{3}\)-path continuity with analytically guaranteed curvature limits, as well as fast and exact analytic collision checking. To the best of our knowledge, such a combination of features is not exhibited by most known path primitives, as shown in Table 1.  
Figure 1: (**a**) A carlike robot with a rectangular footprint. (**b**) A level-curve representation of a transition path segment \(T\), with path shown in red.