## 5 Multiobjective Optimization Approach  
### Multiobjective Optimization Modeling  
#### 5.1.1 Mission Path Model  
The optimization of the FPTLIR flight energy consumption model for multiobjective purposes in both on-line and off-line mission paths consists of two main components: velocity curve optimization and time optimization. The main aim of the velocity curve optimization is to reduce the total energy consumption during mission intervals by adjusting the velocity commands of the FPTLIR in each path interval. The time optimization aims to reduce the total flight duration by adjusting the flight time of the FPTLIR in each mission path interval. This paper mainly focuses on reducing the energy consumption of the system while ensuring the shortest mission execution time under the corresponding energy consumption. A discrete time dimension is introduced to analyze the integrated optimization problem involving the FPTLIR's flight velocity curves and mission interval time. This extension transforms the physical network into a high-dimensional physical-spatiotemporal network. Figure 6 shows the spatiotemporal variation of the FPTLIR during on-line and off-line missions and depicts two mission intervals (#1,2,3) and (#4,5,6) comprising six zero-velocity Points (#1)-(#6), along with their corresponding discrete spatiotemporal networks. By discretizing the time \(T\) axis, Equation (21) represents the duration of these two mission intervals. \[T=\{t_{x}+n\tau|x=0,1;n=0,1,\ldots,M\} \tag{21}\]  
Figure 5: Integrated system energy consumption model block diagram. wherein \(t_{x}\), when the subscript \(x=0,1\) denotes the initial moments of the flight on-line and flight off-line missions, respectively, \(n\) is an integer, and \(\tau\) is a tiny time increment. The FPTLIR can arrive at the zero-velocity point or leave the zero-velocity point at any discrete moment on the \(T\)-axis. The spatiotemporal path \((i,j,t,t^{\prime})\) represents the FPTLIR leaving from point \(i\) at moment \(t\) and arriving at point \(j\) at moment \(t^{\prime}\). It is used to represent the spatiotemporal change process of the FPTLIR during the two mission intervals, and the interval flight time required for the process is \(t^{\prime}-t\). The flight mission path can be established after identifying the power lines and towers that need to be inspected. Based on the spatiotemporal network flight path model, the flight velocity curves are incorporated to represent the relationship between flight time and velocity in the FPTLIR mission interval. As shown in Figure 7, each spatiotemporal path \((i,j,t,t^{\prime})\) from Figure 6 corresponds to different specific flight velocity curves \(\left\{v_{ijtt^{\prime}}\middle|t<x<t^{\prime}\right\}\), thus forming a three-dimensional coordinate system for the FPTLIR's flight velocity curve. It is essential to optimize the flight state of the FPTLIR within each path interval to achieve the objective of energy and time conservation. The four path intervals are divided into state intervals based on different flight states, and the precision of these intervals significantly affects both optimization speed and effectiveness. Increasing the number of state intervals may improve the optimization results, but may lead to frequent flight state transitions that do not meet the stability requirements of the FPTLIR. Therefore, each path interval is divided into three sequential state intervals: acceleration, uniform velocity, and deceleration. Energy-saving and time-saving velocity curves can be generated for mission intervals through multiobjective optimization by optimizing eight flight state commands including vertical acceleration magnitude, vertical acceleration time, vertical uniform velocity time, vertical deceleration magnitude, horizontal acceleration magnitude, horizontal acceleration time, horizontal uniform velocity time, and horizontal deceleration magnitude within each path interval. Combined with the previ  
Figure 6: Flight path diagram based on space–time network.