## 2 Assumptions and Definitions  
In this paper, a simulated vehicle and simulation nodes are used to build a 3D path while considering a limited turn rate of the aerial vehicle. Considering the fixed-wing vehicle's limited turn rate, the simulated vehicle's maneuver is steered to reach the destination, computing a short path to the destination. Moreover, the size of the simulated vehicle is set considering the safety margin of the constructed path. This research applies two reference frames: a North-East-Depth (NED) reference frame \(\{NED\}\) and a body-fixed reference frame \(\{B\}\)[32]. The origin of \(\{NED\}\) is a 3D global location with three axes pointing north, east, and depth, respectively. In addition, \(\{B\}\) is fixed to the simulated vehicle's body such that the origin of \(\{B\}\) is at the vehicle's center. Let \(\phi\), \(\theta\), and \(\psi\) stand for _roll_, _pitch_ and _yaw_, respectively. Let \(c(\eta)\) stand for \(\cos(\eta)\). Let \(s(\eta)\) stand for \(\sin(\eta)\). Let \(arctan(y,x)\) stand for the angle of the complex number, for example, \(x+j*y\). Using [33], the matrix standing for the rotation of angle \(\psi\) with respect to the \(z\)-axis in \(\{B\}\) is  
\[\mathbf{R}_{z}(\psi)=\left( \begin{array}{ccc}c(\psi)&-s(\psi)&0\\ s(\psi)&c(\psi)&0\\ 0&0&1\end{array} \right). \tag{1}\]  
The matrix standing for the rotation of angle \(\theta\) with respect to the \(y\)-axis in \(\{B\}\) is  
\[\mathbf{R}_{y}(\theta)=\left( \begin{array}{ccc}c(\theta)&0&s(\theta)\\ 0&1&0\\ -s(\theta)&0&c(\theta)\end{array} \right). \tag{2}\]  
The matrix for the CC rotation of an angle \(\phi\) with respect to the \(x\)-axis in \(\{B\}\) is  
\[\mathbf{R}_{x}(\phi)=\left( \begin{array}{ccc}1&0&0\\ 0&c(\phi)&-s(\phi)\\ 0&s(\phi)&c(\phi)\end{array} \right). \tag{3}\]  
Using [33], the combined rotation matrix is constructed by multiplying (1), (2), and (3) as follows:  
\[\mathbf{R}(\psi,\theta,\phi)=\mathbf{R}_{z}(\psi)\mathbf{R}_{y}(\theta) \mathbf{R}_{x}(\phi). \tag{4}\]  
Let _destination_ stand for a 3D point in an open space. Let \(\mathbf{D}\in\mathcal{R}^{3}\) stand for the 3D global coordinates of the destination. It is assumed that \(\mathbf{D}\) is known in advance. Our goal is as follows: _form a safe 3D path to the destination, so that the fixed-wing aerial vehicle with a maximum turn rate constraint can safely reach the destination in a time-efficient manner._  
Consider the case where one generates a path for the aerial vehicle, whose safety margin is \(r\). This research utilizes one _simulated vehicle_\(R\) to plan the path of the real aerial vehicle. Considering the safety margin \(r\), let \(R\) stand for a sphere having radius \(r\) centered at \(\mathbf{R}\in\mathcal{R}^{3}\). Here, \(\mathbf{R}\in\mathcal{R}^{3}\) is the 3D global coordinates of \(R\). The simulated vehicle \(R\) is utilized to find a real aerial vehicle's safe path under computer simulations. One plans the path of \(R\) with radius \(r\), so that \(R\) avoids crossing an obstacle boundary while tracking along the path. We acknowledge that we can regard \(R\) as a point-mass, and then add safety margin \(r\) to all obstacles. However, adding \(r\) to all obstacles can be computationally expensive, considering the case where there are many arbitrary-shaped obstacles [34]. Thus, we consider \(R\) with radius \(r\), and collision can be checked by computing the distance between an obstacle boundary and the center of \(R\). The safety margin \(r\) can be set, considering the vehicle's size or the uncertainty in obstacle environments [34]. We consider a fixed-wing aerial vehicle with a constant speed, say \(v\). We use the motion model of the aerial vehicle in [35]. Let \(R_{m}\) denote the minimum turning radius of the vehicle in its horizontal plane. Using [35], \(R_{m}\) is  
\[R_{m}=\frac{v^{2}}{g\times\tan(\rho)} \tag{5}\]  
where \(v\) is the given speed of \(R\), and \(\rho\) denotes the bank angle. Also, \(g=9.8\) m/s\({}^{2}\) presents the gravity acceleration constant.