## 4 Trajectory Optimization Problem
Constraint C1\({}_{1}\) models the linearized kinematic motion model (where \(\mathcal{D}\mathcal{K}\mathcal{M}1\) and \(\mathcal{D}\mathcal{K}\mathcal{M}2\) can be used) to be plugged into OF1\({}_{2}\) and for calculating motion variables used in other constraints; C1\({}_{2}\) satisfies obstacle-free trajectory planning for crash avoidance navigation; C1\({}_{3,4,5}\) impose altitude, speed, and acceleration constraints (note that C1\({}_{5}\) is valid if \(\mathcal{D}\mathcal{K}\mathcal{M}2\) is used to calculate accelerations); and C1\({}_{6,7}\) linearize nonholonomic (rate of heading change) constraints as per the convex relaxation technique proposed in Lemma 1. Lemma 1: _The rate of heading change of the UAV in the 2D horizontal space can be bounded by applying the following constraints, \(\text{CL1}_{1,2}\), that limit the lower bounds and upper bounds of the \(v_{x}\) and \(v_{y}\) terms of the horizontal linear speed._  
Figure 2: The UAV (as RISeUAV or UAV-BS) navigates to provide aerial wireless communication support for IoVs in future 6G networks. \[CL1_{1}:= \begin{cases}v_{x}[k]-v_{x}[k-1]<\frac{\gamma v_{x}[k-1]}{V_{max}}\\ v_{y}[k]-v_{y}[k-1]<\frac{\gamma v_{x}[k-1]}{V_{max}}\end{cases}  \tag{17}\] \[CL1_{2}:= \begin{cases}v_{x}[k]-v_{x}[k-1]>-\frac{\gamma v_{x}[k-1]}{ V_{max}}\\ v_{y}[k]-v_{y}[k-1]>-\frac{\gamma v_{y}[k-1]}{V_{max}}\end{cases} \]  
Proof of Lemma 1.: In the horizontal 2D space, the velocity and heading are obtained as \(v_{xy}=\left\|v_{x},v_{y}\right\|_{2}\) and \(\psi=\tan^{-1}(v_{y}/v_{x})\). The rate of heading change is obtained by taking the first derivative of the heading, as follows:  
\[d\psi=\frac{v_{x}}{{{v_{x}}^{2}}+{v_{y}}^{2}}dv_{y}-\frac{v_{y}}{{{v_{x}}^{2}} +{{v_{y}}^{2}}}dv_{x}; \tag{18}\]  
which indicates that the heading rate is proportional to changes in velocity terms in the \(x\) and \(y\) directions. By limiting the lower/upper bounds of \(v_{x,y}\) in discrete time, as \(\Delta v_{x,y}=v_{x,y}[k]-v_{x,y}[k-1]\), \(\Delta\psi\) is linearly bounded. For this purpose, the upper bounds of \(v_{x,y}[k]\) can be constrained by limiting the difference between consecutive slots by \(CL1_{1}\), assuming that the largest possible value for linear horizontal speed is \(V_{max}\), which is modeled by \(\text{C}1_{6}\). Similarly, \(CL1_{2}\) limits lower bound of \(v_{x,y}\), which is modeled by \(\text{C}1_{7}\). Therefore, \(\text{C}1_{6,7}\) limit rapid heading change. \(\square\)  
The optimization problem \(\mathcal{P}1\) is an energy-efficient trajectory optimization in which the UAV kinematic model with motion (speed and nonholonomic) constraints is considered. Objectives and constraints associated with channel performance are not considered here. However, channel modeling adds more computational costs to trajectory optimization, such as power transmission, RIS phase shift control, and the maximization of channel performance, which are listed as future works in the Conclusion Section below. Nevertheless, there are some points related to solving \(\mathcal{P}1\) that are highlighted in the following Remarks. Remark 3.: \(\mathcal{P}1\) _is convex except in \(\text{C}1_{2}\), which is nonconvex in 3D space with obstacles (e.g., buildings and skyscrapers in dense urban environments), which makes \(\mathcal{P}1\) an NP-hard optimization problem. However, it can be relaxed through an iterative method, as given in Algorithm 1._  
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1. Choose kinodynamic model (i.e., \(\mathcal{D}\mathcal{K}\mathcal{M}1\) or \(\mathcal{D}\mathcal{K}\mathcal{M}2\)) based on the context. 2. Take input data (e.g., \(Z_{min}\), \(Z_{max}\), \(V_{max}\),\(A_{max}\), \(\mathbb{P}_{\text{Obstacle}}\), \(\tau_{1,2}\), \(\omega_{1_{2}}\))
3. Relax C1\({}_{2}\) (then \(\mathcal{P}1\) is a quadratically-constrained convex quadratic optimization problem)
4. Solve\(\mathcal{P}1\) using standard convex programming technique (e.g., interior point method)
5. Take the generated path from step 4. 6.