## 1 Introduction
Their investigations ranged from utilizing optimized fuzzy deep learning for data classification to exploring cost-efficient strategies in software-defined networks [35; 36]. Furthermore, they delved into the integration of advanced technologies like blockchain for drone security in IoT and federated learning for drone authentication [37; 38]. These studies not only highlight the multifaceted approaches to system optimization but also underscore the importance of security and efficiency in autonomous systems, which are integral to our study's objectives. To summarize, as the number of power transmission lines and inspection missions increases, the energy requirements of inspection robots also increase significantly. The energy problem significantly limits the inspection efficiency and endurance of inspection robots. The FPTLIR provides a solution that enhances the autonomy and efficiency of its operation. Compared to travelling inspection missions, the FPTLIR consumes more energy during flight missions. The flight on-line and off-line missions are important preparatory steps for safely and efficiently deploying the FPTLIR to the ground line before starting the inspection process and landing it back on the ground at the end of the inspection. Reducing energy consumption and execution time during flight missions ensures more energy during inspection missions and longer inspection time, which is important for improving the overall efficiency of the inspection. Therefore, conducting a comprehensive analysis and optimization of the FPTLIR's energy requirements and mission execution time during the flight mission is crucial for improving inspection efficiency and achieving energy savings. This paper aims to address the above issues by proposing a multiobjective optimization approach that considers multiple constraints and integrates an energy consumption model, using the improved NSGA-II algorithm. The objective is to minimize the energy consumption of the system while ensuring the shortest mission execution time at the given energy consumption. The main contributions of this paper are as follows:  
1. A mathematical model is constructed to establish a relationship between the mission path and the energy consumption of the FPTLIR. This model takes into account a continuous derivation of the FPTLIR kinematics to the motor dynamics of the end effector and incorporates theories on blade units and aerodynamics. By considering these factors simultaneously, the accuracy and suitability of the model is improved, making it suitable for building a multiobjective optimization model;
2. An optimization model is developed with energy consumption and time as objectives, based on the energy consumption model. Complex constraint indices such as stability and safety during the FPTLIR missions are quantified and considered in the model. The influence of the uniform variable velocity flight state phase on energy consumption is also considered. This improves the adaptability of the model to actual mission requirements;
3. An optimization approach is proposed for energy consumption and execution time during the FPTLIR's both on-line and off-line mission paths. The approach uses an improved NSGA-II optimization algorithm that incorporates concepts from the Cauchy variation operator and the simulated annealing algorithm. This improvement improves the robustness and local search capability of the algorithm under complex constraints. In addition, the proposed approach effectively reduces energy consumption and ensures the shortest mission execution time under the corresponding energy consumption. The rest of this paper is organized as follows. Section 2 gives an overview of the FPTLIR system. Section 3 presents the inspection requirements and the working principle of the FPTLIR. Section 4 constructs an energy consumption model for the flight system, which establishes a relationship between the mission paths and the energy consumption of the FPTLIR.