## 4 The \(\mathbb{G}^{3}\)-Continuous Extend Procedure
### Path Continuity and Curvature Limit Satisfaction Analysis  
Let us begin by showing that Curvature Limit (5) is satisfied. Since \(\kappa_{c}\in[-\kappa_{B},\kappa_{B}]\setminus\{0\}\), circle arcs and line segments satisfy the path curvature limit by construction. The curvature limit is also satisfied by both transition segments, since their curvature is limited by \(\kappa_{c}\), as proven in Property 2 in Reference [22].  
We now turn to path continuity. To ensure \(\mathbb{G}^{3}\)-continuity of the proposed primitive, one must guarantee that, for every transition segment \(T\), the following relations hold:  
\[\lim_{s\to s_{1}}\kappa_{d}(s)=0\quad\lim_{s\to s_{1}}\frac{dx_{d}}{ds}(s)=0, \quad\kappa_{d}(s_{2})=\kappa_{c},\quad\frac{dx_{d}}{ds}(s_{2})=0,\]  
where \(s_{1}\) and \(s_{2}\) are the values of \(s\), such that \(\bar{\mathbf{q}}_{d}(s_{1})=\mathbf{w}_{1}\) and \(\bar{\mathbf{q}}_{d}(s_{2})=\mathbf{w}_{2}\). Note that a limit is sufficient to ensure \(\mathbb{G}^{3}\)-continuity in the case of point \(s_{1}\), since at this point only a connection with line segments or boundary conditions of the path can occur, which guarantees continuity on the other side of the transition segment connection.  
Condition \(\kappa_{d}(s_{2})=\kappa_{c}\) is immediately satisfied since point \(w_{2y}^{1}\) is defined in such a way, that it corresponds to the point of maximal curvature for the transition segment (assuming \(\mu\in(0.5,1)\), which is satisfied in our case) as shown in the proof of Property 2 in Reference [22]. Since a curvature maximum occurs at \(s_{2}\), it also implies that \(\frac{dx_{d}}{ds}(s_{2})=0\) holds, because \(s_{2}\) is a stationary point of \(\kappa_{d}(s)\). In the proof of Property 2 from Reference [22], we have also shown that \(\lim_{y^{1}\to 0}\kappa_{f}(y^{1})=0\) for \(\mu\in(0.5,1)\), where \(\kappa_{f}\) denotes curvature of the transition segment as a function of \(y^{1}\). We conclude that this implies \(\lim_{s\to s_{1}}\kappa(s)=0\).  
Therefore, it remains to show that \(\lim_{s\to s_{1}}\frac{dx_{d}}{ds}(s)=0\). Let us recall that, by the definition of curve's curvature, a transition segment curve \(T\) has a curvature that can be represented as  
\[\kappa_{f}(y^{1})=\frac{\frac{d^{2}f^{1}}{d(y^{1})^{2}}}{\left(1+\left(\frac{ df}{dy^{1}}\right)^{2}\right)^{3/2}}. \tag{9}\]  
One can also compute the arc-length derivative of curvature \(\frac{dx_{f}}{ds}\) as follows:  
\[\frac{dx_{f}}{ds}=\frac{dx_{f}(y^{1})}{dy^{1}}\frac{dy^{1}}{ds}=\frac{dx_{f}(y ^{1})}{dy^{1}}sin\theta_{d}(s)=\frac{dx_{f}(y^{1})}{dy^{1}}\frac{m\operatorname {sgn}\left(\mu\right)y^{1}}{\sqrt{f(y^{1})^{2}+(y^{1})^{2}}}, \tag{10}\]  
where \(m=\operatorname{const}>0\), whereas the second equality results from the consideration of Equation (1) expressed in terms of parameter \(s\) (i.e., \(t=s\)), and the last equality results from the definition of \(\theta_{d}(s)\) tangent to the transition segment \(T\), which can be found from the definition of the VFO convergence vector field as shown in, e.g., Reference [22]. Then, differentiation of Equation (9) with respect to \(s\) and substitution of the result into the final form of Equation (10) results in a relation, which, after some tedious algebra, can be written as follows  
\[\frac{d\kappa_{f}}{ds} =\frac{8\,\mu\,\text{sgn}\left(w_{2x}^{1}\right)(1-5\mu+(y^{1})^{2 \mu}\,h_{1}+(y^{1})^{4\mu}\,h2+(y^{1})^{6\mu}\,h_{3})}{\left((y^{1})^{1-2\mu}((y ^{1})^{2\mu}+1)^{7/2}(1-\mu^{2})^{5}\right)},\] \[h_{1} =-6\mu^{4}+6\mu^{3}+3\mu^{2}-5\mu+3,\] \[h_{2} =-6\mu^{4}-5\mu^{3}+3\mu^{2}+5\mu+3,\] \[h_{3} =2\mu^{4}+7\mu^{3}+9\mu^{2}+5\mu+1.\]  
This clearly implies that, for \(\mu>0.5\), limit \(\lim_{y^{1}\to 0}\frac{ds}{ds}(s)=0\). As a consequence, \(\lim_{s\to s_{1}}\frac{d\kappa}{ds}(s)=0\), which concludes our analysis and proves that the proposed primitive is \(\mathbb{G}^{3}\)-continuous.