## 4 UAV Navigation and Control
### Path Planning and Trajectory Design
#### 4.2.4 Three-Dimensional Trajectory  
In the third group of works, which includes only a few studies, the UAV kinematic equation of motion has been used for trajectory design [135]. Also, obstacle-free navigation and motion constraints are modeled to make the trajectory feasible. Further, the motion primitives of input jerks generated by the trajectory optimization problem can be used for UAV control without an extra controller required for path tracking. In this section, we consider factors affecting the UAV trajectory design such as obstacle-free path planning for crash avoidance as well as motion constraints to make the generated trajectory feasible in practice. To the best of our knowledge, [88] is a rare work in wireless communication that has considered dynamic-aware 3D trajectory optimization. In this work, the conversion efficiency of electric motors, associated with quadrotors, has been considered in the trajectory optimization problem. For this reason, the energy consumption of the quadrotor for hovering and flight is obtained based on the power consumption of the rotors that is developed by using electromechanical formulations of rotors. Although accurate, the developed trajectory optimization method in [88] imposes prohibitive computational complexity for solving the trajectory optimization problem in real time, which is required for the autonomous navigation of the UAV. Additionally, the obstacle-free trajectory has not been considered, which makes the proposed technique ineffective for dense urban areas. We have considered this requirement in our previous works and studied the kinematic-aware crash avoidance RISeUAV trajectory design in dense urban environments for intelligent vehicles [41] and IoVs [33; 136]. However, a generic approach is required with a linear kinodynamic model and constraints for computationally efficient and real-time trajectory optimization for autonomous navigation applications. To bridge the gap, we have developed a kinodynamic-modeling-based trajectory optimization for UAV navigation in dense environments [117]. The basic obstacle-free trajectory optimization problem for a UAV-BS while satisfying kinodynamic motion constraints is modeled as \(\mathcal{P}2\) for each sampling time \(k\):  
In the case of the RISeUAV, where OF2\({}_{3}\) and OF2\({}_{4}\) may be used, the objective function is nonconvex, which is discussed in the solution in Section 6. 7. In formulating the \(min\) optimization problem \(\mathcal{P}2\), we have used the following technique. Instead of using a \(max\) problem for achievable rate maximization in (9) and (10) with double CLoS path model in the denominator, we used the \(min\) optimization problem by using OF2\({}_{2}\), OF2\({}_{3}\), and OF2\({}_{4}\) with the double CLoS path model in the nominator. These two \(min\) and \(max\) problems are equivalent, see the solutions in Section 6. 8. The optimization variables in \(\mathcal{P}2\) are input jerks (motion primitives) to the UAV motion system. Therefore, they can be directly used for UAV navigation without the requirement of an extra path-tasking or path-following controller. 9. Constraints related to channel performance and QoS (e.g., the minimum achievable rate for each UE or IoV) are not considered in \(\mathcal{P}2\). However, it is typical in alternating optimization methods (such as SCP and SCA) for problem objectives to be solved iteratively by relaxing other objectives or constraints. **Lemma 1**. : _UAV trajectory optimization with valid LoS paths between UAV and target nodes, which are randomly obstructed (e.g., by building), is an NP-hard problem for convex programming._  
Proof of Lemma 1.: When UAV waypoints experience LoS link obstructions due to obstacles (e.g., buildings), these obstructions can be modeled as fictitious obstacles that must be avoided. However, validating LoS links in such scenarios is computationally expensive.