## 4 The \(\mathbb{G}^{3}\)-Continuous Extend Procedure  
### Main Concept  
We introduce the so called \(\mathbb{G}^{3}\)-continuous path primitive, which consists of a transition segment (further explained in Section 4.2), a circle arc, a reversed transition segment, and a line segment. Similarly to the form of path encoding taken from Reference [5], we introduce a language for encoding paths with three words corresponding to path segments:  
* \(T(\mathbf{w}_{1},\mathbf{w}_{2},\mu)\) denotes a transition segment connecting \(\mathbf{w}_{1}\) with \(\mathbf{w}_{2}\) (defined in Section 4.2),
* \(C(\mathbf{w}_{1},\mathbf{w}_{2})\) is a circular arc of radius \(1/\kappa_{c}\) connecting \(\mathbf{w}_{1}\) with \(\mathbf{w}_{2}\),
* \(S(\mathbf{w}_{1},\mathbf{w}_{2})\) corresponds to a straight line connecting \(\mathbf{w}_{1}\) with \(\mathbf{w}_{2}\),  
where \(\mathbf{w}_{k}\triangleq[w_{k\theta}\ w_{kx}\ w_{ky}]^{\top}=[w_{k\theta}\ \mathbf{w}_{k}^{\top}]^{\top}\) for \(k=1,2,3,4,5\) correspond to the reference vehicle-body configurations at endpoints of path segments. Using this encoding, one can describe the proposed \(\mathbb{G}^{3}\)-continuous path primitive connecting \(\mathbf{w}_{1}\) with \(\mathbf{w}_{5}\) as  
\[T(\mathbf{w}_{1},\mathbf{w}_{2},\mu_{1})C(\mathbf{w}_{2},\mathbf{w}_{3})T(\mathbf{w}_{4},\mathbf{w}_{ 3},\mu_{2})S(\mathbf{w}_{4},\mathbf{w}_{5}).\]  
The geometric interpretation of our \(\mathbb{G}^{3}\)-continuous path primitive is shown in Figure 2 (please, note the intentionally reversed order of the arguments in \(T(\mathbf{w}_{4},\mathbf{w}_{3},\mu_{2})\) in the above formula; it simply corresponds to a reversal of the segment's endpoints). Note that, by taking transition segments of zero length, one can obtain an RS path. Depending on a choice of parameters \(\mu_{1},\mu_{2}\), one can compromise between path length and smoothness resulting from longer transition segments. Since properties of RS paths are well known and problems such as collision checking or curvature limit checking are trivial in their case, we focused on transition segments in the sequel, and show how solutions to Problems 1-3 can be obtained with their help.  
The extend procedure utilizing the \(\mathbb{G}^{3}\)-continuous path primitive consists of the following main stages:  
1. Choose parameters \(\mu_{1}\), \(\mu_{2}\), and curvature \(\kappa_{c}\
eq 0\).
2. Find a sequence of \(\mathbb{G}^{3}\)-continuous path primitives connecting two prescribed vehicle-body configurations.
3. Check for collisions.
4. Return computed path or a collision signal.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Path Primitive** & **Continuity** & **Bounded \(\kappa\)** & **Collision Checking** & **Length Computation** \\ \hline Reeds–Shepp [5] & \(\mathbb{G}^{1}\) & yes & analytic & analytic \\ \(\eta^{3}\)-splines [11] & \(\mathbb{G}^{3}\) & no & numerical & numerical \\ \(\eta^{4}\)-splines [12] & \(\mathbb{G}^{4}\) & no & numerical & numerical \\ Clothoid-based & \(\mathbb{G}^{2}\) & yes & numerical & analytic \\ Fermat’s spiral [21] & \(\mathbb{G}^{2}\) & yes & numerical & numerical \\ Low-order B-Splines [14; 15] & \(\mathbb{G}^{2}\) & yes & numerical & numerical \\ Cubic curvature splines [8] & \(\mathbb{G}^{3}\) & yes & numerical & numerical \\ High-order B-splines & \(\mathbb{G}^{3+}\) & no & approx. analytic & numerical \\ HC-Steer [10] & \(\mathbb{G}^{1}\) & yes & analytic & numerical \\ \(\mathbb{G}^{3}\)**-continuous path primitive** & \(\mathbb{G}^{3}\) & **yes** & **analytic** & **numerical** \\ \hline \hline \end{tabular}
\end{table}
Table 1: Comparison of the proposed \(\mathbb{G}^{3}\)-continuous path primitive with known path primitives.