## 5 Multiobjective Optimization Approach
### Multiobjective Optimization Modeling
#### 5.1.3 Optimization Model  
To construct a multiobjective optimization model based on the flight energy model, we need to make some assumptions. These are:  
1. The mission path parameters, FPTLIR performance, and other characteristics are known;
2. There are no obstacles or wind disturbances along the FPTLIR's mission path;
3. The power consumption of the onboard auxiliary equipment for the FPTLIR remains constant throughout the mission. This paper aims to address the issue of energy consumption and mission execution time in the FPTLIR during two different mission intervals of on-line and off-line missions. We establish separate multiobjective optimization models for on-line and off-line mission intervals, considering several indicators such as stability, safety, position accuracy,energy consumption, and time savings. The comprehensive multiobjective optimization model for the on-line mission interval is presented in Equations (32)-(36). \[\min f(a_{\text{vn1}},t_{\text{vn1}},t_{\text{vn2}},a_{\text{vn2}},a_{\text{hn1} },t_{\text{hn1}},t_{\text{hn2}},a_{\text{hn2}})=\min[f_{\text{on1}}(a,v,t),f_{ \text{on2}}(t)] \tag{32}\]  
The objective function is expressed in Equation (33). \[\left\{ \begin{array}{l}f_{\text{on1}}(a,v,t)=W_{1}(t)\\ W_{1}(t)=\sum\limits_{i=1}^{3}\left(W_{\text{vni}}+W_{\text{hni}}\right)\\ f_{\text{on2}}(t)=\sum\limits_{i=1}^{3}\left(t_{\text{vni}}+t_{\text{hni}} \right)\\ \sum\limits_{i=1}^{3}\left(t_{\text{vni}}+t_{\text{hni}}\right)\leq t_{\text{ on}}\end{array} \right. \tag{33}\]  
The acceleration constraint is expressed as follows:  
\[\left\{ \begin{array}{l}0<a_{\text{vn1}},a_{\text{vn2}}\leq 6\;\text{m}/\text{s} ^{2}\\ 0<a_{\text{hn1}},a_{\text{hn2}}\leq 10\;\text{m}/\text{s}^{2}\end{array} \right. \tag{34}\]  
The velocity constraint is formulated in Equation (35). \[\left\{\begin{array}{l}v_{\text{vn}}-a_{\text{vn2}}t_{\text{vn3}}=v_{\text{ hn}}-a_{\text{hn2}}t_{\text{hn3}}=0\\ \left\{ \begin{array}{l}v_{\text{vn}}=a_{\text{vn1}}t_{\text{vn1}}=a_{\text{ vn2}}t_{\text{vn3}}\\ 0\leq v_{\text{vn}}\leq v_{\text{ascent\_max}}\\ 0<a_{\text{vn1}}t_{\text{vn1}}=a_{\text{vn2}}t_{\text{vn3}}\leq v_{\text{ascent\_ max}}\\ v_{\text{hn}}=a_{\text{hn1}}t_{\text{hn1}}=a_{\text{hn2}}t_{\text{hn3}}\\ 0\leq v_{\text{hn}}\leq v_{\text{horizontal\_max}}\\ 0<a_{\text{hn1}}t_{\text{hn1}}=a_{\text{hn2}}t_{\text{hn3}}\leq v_{\text{ horizontal\_max}}\end{array} \right. \tag{35}\]  
The positional accuracy constraints are defined in Equation (36). \[\left\{ \begin{array}{l}K_{\text{stop}}=\left|S_{1}-S_{\text{target}}\right|< K_{1}\\ S_{1}=\frac{1}{2}a_{\text{vn1}}t_{\text{vn1}}^{2}+v_{\text{vn}}t_{\text{vn2}}+v_{ \text{vn}}t_{\text{vn3}}-\frac{1}{2}a_{\text{vn2}}t_{\text{vn3}}^{2}\\ K_{\text{stop}}=\left|S_{2}-S_{\text{target}}\right|<K_{2}\\ S_{2}=\frac{1}{2}a_{\text{hn1}}t_{\text{hn1}}^{2}+v_{\text{hn}}t_{\text{hn2}}+ v_{\text{hn}}t_{\text{hn3}}-\frac{1}{2}a_{\text{hn2}}t_{\text{hn3}}^{2}\end{array} \right. \tag{36}\]  
Similarly, the comprehensive multiobjective optimization model for the off-line mission interval is presented in Equations (37)-(41). \[\min f(a_{\text{hfl}},t_{\text{hfl}},t_{\text{hfl}2},a_{\text{hfl}2},a_{\text{ vfl}},t_{\text{vfl}},t_{\text{vfl}},a_{\text{vfl}2})=\min[f_{\text{off1}}(a,v,t),f_{ \text{off2}}(t)] \tag{37}\]  
The objective function is derived as follows:  
\[\left\{ \begin{array}{l}f_{\text{off1}}(a,v,t)=W_{2}(t)\\ W_{2}(t)=\sum\limits_{i=1}^{3}\left(P_{\text{hfl}}T_{\text{hfl}}+P_{\text{vfl}}T _{\text{vfl}}\right)\\ \left\{\begin{array}{l}f_{\text{off2}}(t)=\sum\limits_{i=1}^{3}\left(t_{\text{ vfi}}+t_{\text{hfl}}\right)\\ \sum\limits_{i=1}^{3}\left(t_{\text{vfi}}+t_{\text{hfl}}\right)\leq t_{\text{off}} \end{array}\right.\end{array} \right. \tag{38}\]  
The acceleration constraint is expressed in Equation (39). \[\left\{ \begin{array}{l}0<a_{\text{hfl}},a_{\text{hfl}2}\leq 10\;\text{m}/\text{s} ^{2}\\ 0<a_{\text{vfl}},a_{\text{vfl}2}\leq 6\;\text{m}/\text{s}^{2}\end{array} \right. \tag{39}\]The velocity constraint is defined in Equation (40). \[