## 4 The \(\mathbb{G}^{3}\)-Continuous Extend Procedure
### Transition Segments and \(\mathbb{G}^{3}\)-Continuous Path Primitives  
Since computing the endpoints of a line segment and a circle arc given its radius and length is straightforward, we explicitly only define relations concerning transition segments \(T(\mathbf{w}_{1},\mathbf{w}_{2},\mu)\). Let us denote variables expressed in a local coordinate frame fixed at point \(\mathbf{w}_{1}\) by \((\cdot)^{1}\), that is, \(\mathbf{w}_{2}^{1}\) corresponds to \(\mathbf{w}_{2}\) expressed in coordinates of \(\mathbf{w}_{1}\). A transition segment connecting vehicle-body configuration \(\mathbf{w}_{1}\) with \(\mathbf{w}_{2}\) is defined by the following curve, expressed in the coordinates of endpoint \(\mathbf{w}_{1}\):  
\[x^{1} =f(y^{1})=\left\{ \begin{aligned} \frac{-\text{sgn}\left(w_{2x}^ {1}\right)\left|y^{1}\right|}{2}\left[\left(\frac{y^{1}}{p}\right)^{\mu}-\left( \frac{y^{1}}{p}\right)^{-\mu}\right]&\text{for}\quad y^{1}\
eq 0,\\ 0&\text{for}\quad y^{1}=0,\end{aligned} \right. \tag{7}\] \[p \triangleq w_{2y}^{1}\exp\left(\frac{\left|\text{arsinh}\left(w_{2x}^ {1}/w_{2y}^{1}\right)\right|}{\mu}\right),\quad w_{2y}^{1}=\frac{K}{\kappa_{c }}y^{*},\quad w_{2x}^{1}=\frac{K}{\kappa_{c}}x^{*}, \tag{8}\]  
with  
\[K =\frac{y^{*}\left(-\mu+\mu^{2}\frac{x^{*}}{r}\right)}{-\left(r \right)^{2}\left(1+\mu^{2}-2\mu\frac{x^{*}}{r}\right)^{3/2}},\quad r\triangleq \sqrt{\left(x^{*}\right)^{2}+\left(y^{*}\right)^{2}},\quad x^{*}=f(y^{*}),\] \[y^{*} =\left(\left(\frac{4}{3}p_{5}-p_{3}\right)^{1/3}-\frac{-6\mu^{3} +\mu^{2}+2\mu+3}{\left(6\mu+3\right)\left(\mu+1\right)^{2}}+p_{2}\right)^{1/2 \mu},\]  
Figure 2: (**a**) Structure and parameters of the proposed \(\mathbb{G}^{3}\)-continuous path primitive. (**b**) Collision checking between transition segment and a line segment connecting \(\mathbf{p}_{i}\) and \(\mathbf{r}_{i}\). Only points \(\mathbf{p}_{i}\) and \(\mathbf{n}_{i}\) have to be checked for collisions because point \(\mathbf{r}_{i}\) lies outside the domain of interest. whereas  
\[p_{1} =-6\mu^{4}-5\mu^{3}+3\mu^{2}+5\mu+3,\] \[p_{2} =\frac{4\mu^{2}\left(18\mu^{4}+3\mu^{3}-17\mu^{2}-11\mu+7\right)}{9 p_{1}\left(\mu+1\right)^{4}\left(2\mu+1\right)^{2}},\] \[p_{3} =\frac{(p_{1})^{3}}{27(2\mu+1)^{3}(\mu+1)^{9}}+\frac{(2\mu-1)(\mu- 1)^{3}}{(4\mu+2)(\mu+1)^{3}}-p_{4},\] \[p_{4} =\frac{p_{1}(-6\mu^{4}+5\mu^{3}+3\mu^{2}-5\mu+3)}{6(2\mu+1)^{2}( \mu+1)^{6}},\] \[p_{5} =\sqrt{\frac{\mu^{6}(\mu-1)^{3}(-36\mu^{4}+33\mu^{2}-29)}{(2\mu+1 )^{4}(\mu+1)^{9}}},\]  
where \(y^{*}\) is a rational function of \(\mu\in(0.5,1)\), \(\kappa_{c}\in[-\kappa_{B},\kappa_{B}]\setminus\{0\}\) denotes the curvature of adjacent circle arc \(C\), while \(\mu\in(0.5,1)\) is a design parameter influencing the supremum value of curvature arc-length derivative \(|dx_{d}(s)/ds|\) during the transition segment and its length, as shown in Figures 3 and 4. Figure 4: (**a**) Maximal arc-length derivative of transition segment curvature \(\kappa_{d}(s)\) as a function of parameter \(\mu\). (**b**) Evolution of transition segment curvature arc-length derivative for selected values of \(\mu\). \[K =\frac{y^{*}\left(-\mu+\mu^{2}\frac{x^{*}}{r}\right)}{-\left(r \right)^{2}\left(1+\mu^{2}-2\mu\frac{x^{*}}{r}\right)^{3/2}},\quad r\triangleq \sqrt{\left(x^{*}\right)^{2}+\left(y^{*}\right)^{2}},\quad x^{*}=f(y^{*}),\]0 (**a**) \(\mathbb{G}^{3}\)-continuous path primitives for selected values of parameter \(\mu\). (**b**) Curvature profiles of selected \(\mathbb{G}^{3}\)-continuous path primitives. Given a particular value of \(\mu\) and transition segment endpoint \(\mathbf{w}_{1}\), one can compute coordinates of the other endpoint \(\mathbf{w}_{2}\) from Equation (8). The curve representing a transition segment is given by Curve (7) with particular value of parameter \(p\) resulting from Equation (8). Curve (7) was derived and analyzed in Reference [22]. It corresponds to an integral curve of the convergence vector field from the Vector Field Orientation (VFO) control law for the waypoint-following task.