## 5 Multiobjective Optimization Approach
### Multiobjective Optimization Modeling
#### 5.1.2 Constraints and Objective Functions  
Based on the design specifications and performance data of the FPTLIR, we carried out a comprehensive evaluation of its working conditions. To meet stability and safety requirements, kinematic constraints such as acceleration and velocity limitations need to be implemented. The maximum achievable horizontal acceleration for the FPTLIR is \(10\text{ m}/\text{s}^{2}\), while the maximum vertical acceleration is \(6\text{ m}/\text{s}^{2}\). Equation (26) is the expression of these acceleration constraints. \[\left\{ \begin{array}{l}0<a_{\text{hn1}},a_{\text{hn2}},a_{\text{hf1}},a_{ \text{h2}}\leq 10\text{ m}/\text{s}^{2}\\ 0<a_{\text{wn1}},a_{\text{vn2}},a_{\text{vf1}},a_{\text{vf2}}\leq 6\text{ m}/\text{s}^{2}\end{array} \right. \tag{26}\]  
FPTLIR has different upper limits for flight speed in various modes, and Equation (27) represents these limits. \[\left\{ \begin{array}{l}0\leq v\leq v_{\text{lim}}\\ v_{\text{lim}}=\{v_{\text{ascent\_max}},v_{\text{descent\_max}},v_{\text{ horizontal\_max}}\}\end{array} \right. \tag{27}\]  
where \(v\) is the flight velocity of the FPTLIR, \(v_{\text{lim}}\) is the maximum velocity limit, the maximum ascent velocity \(v_{\text{ascent\_max}}=10\text{ m}/\text{s}\), the maximum descent velocity \(v_{\text{descent\_max}}=7\text{ m}/\text{s}\), and the maximum horizontal flight velocity \(v_{\text{horizontal\_max}}=16.8\text{ m}/\text{s}\). According to the preceding section, it is evident that the instantaneous velocity of the FPTLIR at both its arrival and departure from the point of zero velocity is \(0\), which must adhere to the constraints presented in Equation (28). \[v_{ijtt^{\prime}}(t)=v_{ijtt^{\prime}}(t^{\prime})=0 \tag{28}\]  
The position accuracy metrics should be considered to ensure the success of both on-line and off-line missions. Specifically, these metrics need to be evaluated when the FPTLIR flight reaches the zero-velocity point, and their constraints can be expressed as follows:  
\[\left\{ \begin{array}{l}\left|S-S_{\text{target}}\right|\leq K_{\text{x}}; (x=1,2,3)\\ K_{4}\leq S-S_{\text{target}4}\end{array} \right. \tag{29}\]  
where \(S\) is the total flight distance under the flight path interval, \(S_{\text{target}}\) is the ideal zero-velocity point location distance under different flight path intervals, which is determined by the parameters of the power line to be inspected, \(S_{\text{target1}}\) is the ideal vertical flight distance at Point (#2), \(S_{\text{target2}}\) is the ideal horizontal flight distance at Point (#3), \(S_{\text{target3}}\) is the ideal horizontal flight distance at Point (#5), and \(S_{\text{target4}}\) is the ideal vertical flight distance at Point (#6). Correspondingly, \(K_{x}\) is the maximum allowable error, as shown in Figure 3, the maximum allowable errors at Points (#2) and (#3) in the on-line mission interval are \(K_{1}=0.1\) m and \(K_{2}=0.04\) m, respectively, and the maximum allowable errors at Points (#5) and (#6) in the off-line mission interval are \(K_{3}=0.04\) m and \(K_{4}=-0.1\) m. The optimization objective of the multiobjective optimization model is to achieve energy and time savings. The optimization model uses the energy consumption index from Section 3 as its objective function, presented in Equation (30). \[\left\{ \begin{array}{l}f_{\text{on1}}(a,v,t)=W_{1}(t)\\ W_{1}(t)=\sum\limits_{i=1}^{3}\left(W_{\text{vni}}+W_{\text{hni}}\right)\\ f_{\text{off1}}(a,v,t)=W_{2}(t)\\ W_{2}(t)=\sum\limits_{i=1}^{3}\left(W_{\text{hfi}}+W_{\text{vfi}}\right)\end{array} \right.