## 2 Problem Formulation
### Pythagorean Hodograph Path  
The Pythagorean hodograph (PH) curve was first introduced by [PERSON] and [PERSON] [26] in 1990. It is defined by a parameterized polynomial curve which has hodographs that satisfy a Pythagorean condition. The definition is given as:  
**Definition 1**. : (The PH curve [26]). _Suppose that \(\mathbf{r}(q)=\{x(q),y(q),z(q)\}\) is a polynomial curve parameterized by \(q\). \(\mathbf{r}(q)\) is a PH curve, if the first derivatives of its components satisfy the following Pythagorean condition:_  
\[xr^{2}(q)+yr^{2}(q)+zr^{2}(q)=\sigma^{2}(q), \tag{2}\]  
where \(x(q)\), \(y(q)\), \(z(q)\) and \(\sigma(q)\) are polynomials of \(q\). In view of numerical stability, PH curves can be written in the following Bernstein-Bezier form [16; 26]:  
\[\mathbf{r}(q)=\sum_{k=0}^{n}\mathbf{p}_{k}\left( \begin{array}{c}n\\ k\end{array} \right)(1-q)^{n-k}q^{k},\ \ \ \ \ q\in[0,1]\ \, \tag{3}\]  
where \(\mathbf{p}_{k}\) is known as the \"control point\" of the curve and \(n\) is the order of the curve. According to [27; 28], the lowest order of PH curve that has an inflection point is the fifth, called the quintic PH curve. The inflection point provides sufficient flexibility in path shape to be appropriate for path-planning. Hence, the quintic PH curve is applied for UAV path-planning in this paper. Generally, the poses at the initial and final locations, namely boundary conditions \(r(0)\),\(r(0)\),\(r(1)\) and \(r\prime(1)\), are known beforehand. Then, the first-order Hermite interpolation [29], combined with the complex vector method or with the quaternion method [29], can be used to solve the 2-D or 3-D quintic PH curve. Details about the solving process are introduced in [29; 30]. After getting the polynomial expression of the PH path \(\mathbf{r}(q)\), the path length of the PH curve can be obtained by integrating the polynomial \(\|\mathbf{r}(q)\|\), namely,  
\[L(q)=\int\|\mathbf{r}(q)\|\,\ \ \ q\in[0,1]\ . \tag{4}\]  
According to differential geometry theory [22], the curvature and torsion are fundamental properties of a curve, by which the smoothness of the curve is completely determined. In 2-D cases, the torsion is identically equal to zero. Curvature describes the bending extent of a curve, while torsion represents the twisting extent. For UAVs, these two properties also play an important role in the mechanics of a vehicle. The curvature is proved proportional to the lateral acceleration, while the torsion is proportional to the angular momentum. Thus, the UAV path should satisfy the maximum-curvature and maximum-torsion constraints, which are known as the kinematic constraints of a UAV. The curvature and torsion at any point on the path can be solved by following equations [22]:  
\[\kappa(q)=\frac{\left|\mathbf{r}(q)\times\mathbf{r}^{\prime\prime}(q)\right|} {\left|\mathbf{r}(q)\right|^{3}}, \tag{5}\]  
\[\tau(q)=\frac{\left(\mathbf{r}(q)\times\mathbf{r}^{\prime\prime}(q)\right) \cdot\mathbf{r}^{\prime\prime\prime}(q)}{\left|\mathbf{r}(q)\times\mathbf{r}^ {\prime\prime}(q)\right|^{2}}, \tag{6}\]  
where \(r\prime(q)\), \(\mathbf{r}^{\prime\prime}(q)\) and \(\mathbf{r}^{\prime\prime\prime}(q)\) are the first, second, and third derivatives of \(\mathbf{r}(q)\), respectively. Obviously, the curvature and torsion of a PH path are rational polynomials. To describe the smoothness of a PH path quantitatively, we introduce the definition of the elastic energy of a curve. **Definition 2**. : (The elastic energy [31]). _The elastic energy of a curve is defined as the integral of the sum of squares of curvature and torsion with respect to arc length, namely:_  
\[E=\int\left(\kappa^{2}+\tau^{2}\right)ds. \tag{7}\]  
In general, the lengths of direction vectors, \(m_{0}=\|\mathbf{r}(0)\|\) and \(m_{1}=\|\mathbf{r}(1)\|\), are considered to be the control variables. The lager the values of \(m_{0}\) and \(m_{1}\), the smoother the PH path, but the greater the path length.