## 4 Methodology
### Homotopic Path Planning Approach
\right] \tag{29}\]  
where \(\mu_{\text{trans}0}\) is the curvature bound of the arcs in the curve; \(\mu_{a}\) and \(\mu_{b}\) are the curvatures of arc segments along the Dubins path; \(l\) is the length of the line segment along the Dubins path; \(l_{a}\) and \(l_{b}\) are the lengths of arc segments along the Dubins path. On this basis, \(H(u)\) indicates a path with a curvature vector \(\mathbf{c}=\left[ \begin{array}{cccc}\mu_{a}&\mu_{\text{trans}}&-\mu_{\text{trans} }&\mu_{\text{trans}}&\mu_{b}\end{array} \right]\) for the arbitrary \(u\in(0,1)\). To describe this path, a linear function between \(u\) and \(\mathbf{c}\) can be defined as  
\[\mu_{\text{trans}}=(1-u)\mu_{\text{trans}0}\,,\ \ \ u\in(0,1). \tag{30}\]  
Based on the definition of path homotopy 1 and the geometrical relationship, there exists \(\mu_{\text{trans}0}=4/l\). Since \(\mathbf{a}\), \(\mathbf{b}\) and the curvature vector \(\mathbf{c}\) are known variables, the length vector \(\mathbf{l}\) can be obtained by iteration. Moreover, path homotopy 1 in the lower half plane can also be constructed, and the paths within this path homotopy are described in a similar way. To cover all of the lengths of the curvature-bounded path, another path homotopy is proposed. As shown in Figure 7, path homotopy 2 consists of circles. Based on the geometrical relationship, the boundary conditions of path homotopy 2 in the upper half plane can be written as  
\[H(0)=\mathcal{T}(\mathbf{a},\mathbf{c}_{0},\mathbf{l}_{0}), \tag{31}\]  
Figure 6: Path homotopy 1. \[H(1)=\mathcal{T}(\mathbf{a},\mathbf{c}_{1},\mathbf{l}_{1}), \tag{32}\] \[\mathbf{c}_{0}=\Big{[} \begin{array}{cccc}\mu_{a}&0&\mu_{\text{trans}0} /2&0&\mu_{b}\end{array} \Big{]},\] (33) \[\mathbf{c}_{1}=\Big{[} \begin{array}{cccc}\mu_{a}&0&\mu_{\text{trans}0} &0&\mu_{b}\end{array} \Big{]},\] (34) \[\mathbf{l}_{0}=\Big{[} \begin{array}{cccc}l_{a}&l_{1}&4k\pi/\mu_{ \text{trans}}&l_{2}&l_{b}\end{array} \Big{]},k\in\mathbb{N}^{+},\] (35) \[\mathbf{l}_{1}=\Big{[} \begin{array}{cccc}l_{a}&l_{1}&2k\pi/\mu_{ \text{trans}}&l_{2}&l_{b}\end{array} \Big{]},k\in\mathbb{N}^{+},\] (36) \[\qquad\qquad\qquad l_{1}+l_{2}=l \tag{37}\]  
where \(l_{1}\) and \(l_{2}\) are the lengths of two line segments, and \(k\) represents the lap number of the circles. Similarly, \(H(u)\) indicates a path with a curvature vector \(\mathbf{c}=\Big{[} \begin{array}{cccc}\mu_{a}&0&\mu_{\text{trans}}&0&\mu_{b}\end{array}  \Big{]}\) for the arbitrary \(u\in(0,1)\). To describe this path, a linear function between \(u\) and \(\mathbf{c}\) can be defined as  
\[\mu_{\text{trans}}=\Big{(}\frac{u}{2}+\frac{1}{2}\Big{)}\mu_{\text{trans}0}\,\quad u\in(0,1). \tag{38}\]  
Since the UAVs are modeled as curvature-bounded vehicles, there exists \(\mu_{\text{trans}0}=\mu_{\text{max}}\). Then, the length vector \(l\) can be obtained by iteration as well. As before, path homotopy 2 in the lower half plane can be constructed and the paths within this path homotopy are described in a similar way. Based on the two path homotopy construction patterns, the curvature-bounded paths within the homotopy are described. In particular, the path varies monotonously based on \(u\), so that the path with the expected length can be planned by solving Equation (16) using the Newton-Raphson iterative method.