# An Efficient Trajectory Planning Method for High-Speed Interception of Invasive Drones
## 5 Trajectory Feasibility Assessment
### Thrust  
The interval \(\mathcal{T}\) is feasible relative to the thrust limit (7) if and only if:  
\[\underset{t\in\mathcal{T}}{\text{max}}f(t)^{2}\leq f_{\text{max}}^{2}\text{ and} \tag{36}\]  
\[\underset{t\in T}{\text{min}}f(t)^{2}\geq f_{\text{min}}^{2}. \tag{37}\]  
Square (18) obtained  
\[f_{t}^{2}=\|\mathbf{a}_{t}-\mathbf{g}\|^{2}=\sum_{k=1}^{3}(a_{k}-g_{k})^{2} \tag{38}\]  
where \(a_{k}\) represents the acceleration component of \(\mathbf{a}_{t}\) on axis \(k\), and \(g_{k}\) is the gravity component on axis \(k\). Combining (36)-(38), the thrust constraint can be explained as a spherical constraint on acceleration. Order:  
\[f_{k}(t)=a_{k}(t)-g_{k} \tag{39}\]  
By taking the extreme values of each axis of (38), the following boundaries are obtained:  
\[\underset{t\in\mathcal{T}}{\text{max}}f_{k}^{2}(t)\text{ for }k\in\{1,2,3\} \tag{40}\]  
\[\underset{t\in\mathcal{T}}{\text{max}}f_{t}^{2}(t)\leq\sum_{k=1}^{3}\underset {t\in\mathcal{T}}{\text{max}}f_{k}^{2}(t) \tag{41}\]  
\[\underset{t\in\mathcal{T}}{\text{min}}f_{t}^{2}(t)\geq\sum_{k=1}^{3}\underset {t\in\mathcal{T}}{\text{min}}f_{k}^{2}. \tag{42}\]  
If the left-hand side of (40) is greater than \(f_{\text{max}}\) then the interval is absolutely infeasible, while if the right-hand side of (41) is less than \(f_{\text{max}}\) and the right-hand side of (42) is greater than \(f_{\text{min}}\), then the interval is absolutely feasible relative to the thrust constraint.  
By solving the root of the differential \(j_{k}\) of \(a_{k}\) and comparing it with the endpoint values at the left and right boundaries of \(\mathcal{T}\), the maximum and minimum values of the thrust \(f_{k}\) in formula (39) can be found in a closed form (represented as \(\overline{f}_{k}\) and \(\underline{f}_{k^{\prime}}\), respectively):  
\[j_{k}(t^{*})=-\big{(}k_{rp}-1\big{)}\left(\frac{\big{(}C_{3}-C_{2}\text{e}^{ \underline{i}_{k}r^{*}}\big{)}}{\bar{k}_{rb}^{2}}+\frac{(C_{3}t^{*}+C_{1})}{ \bar{k}_{rb}}\right)=0 \tag{43}\]  
\[\begin{array}{ll}t^{*}=\begin{cases}\frac{\Delta_{1}-\text{W}_{0}(\Delta_{2 })}{\bar{k}_{rb}}&\Delta_{2}\in\left[-\frac{1}{e},+\infty\right)\\ \frac{\Delta_{1}-\text{W}_{-1}(\Delta_{2})}{\bar{k}_{rb}}&\Delta_{2}\in\left(- \frac{1}{e},0\right)\end{cases}\\ \text{where}\\ \Delta_{1}=-\frac{\text{C}_{3}+\text{C}_{1}\bar{k}_{rb}}{\text{C}_{3}}\\ \Delta_{2}=-\frac{\text{C}_{2}\text{e}^{\underline{i}_{1}}}{\text{C}_{3}}\end{cases} \tag{44}\]  
where \(j_{k}=\dot{a}_{k}\), \(\text{W}_{0}\) and \(\text{W}_{-1}\) are two branches of the Lambert W function.  
The extreme values of \(f_{k}^{2}\) are as follows:  
\[\underset{t\in\mathcal{T}}{\max}f_{k}^{2}(t)=\max\Bigl{\{}\overline{f}_{k^{\prime }}^{2},\underline{f}_{k}^{2}\Bigr{\}} \tag{45}\]  
\[\underset{t\in\mathcal{T}}{\min}f_{k}^{2}(t)=\begin{cases}\min\Bigl{\{} \overline{f}_{k^{\prime}}^{2},\underline{f}_{k}^{2}\Bigr{\}},&\text{if }\overline{f}_{k}\cdot \underline{f}_{k}\geq 0\\ 0,&\text{otherwise}\end{cases} \tag{46}\]  
where \(\overline{f}_{k}\cdot\underline{f}_{k}<0\) means that the symbol of \(f_{k}(t)\) in \(\mathcal{T}\) changes and there is a zero crossing point.  
Therefore, the sufficient condition for the infeasibility of the input thrust in formula (40) is:  
\[\max\[PERSON]{\{}\overline{f}_{k^{\prime}}^{2},\underline{f}_{k}^{2}\Bigr{\}}>f_{ \max}. \tag{47}\]  
Similarly, according to (41) and (42), the sufficient conditions for input thrust to be feasible are:  
\[\sum_{k=1}^{3}\max\Bigl{\{}\overline{f}_{k^{\prime}}^{2},\underline{f}_{k}^{2 }\Bigr{\}}\leq f_{\max}\text{and} \tag{48}\]  
\[\sum_{k=1}^{3}\min\Bigl{\{}\overline{f}_{k^{\prime}}^{2},\underline{f}_{k}^{2 }\Bigr{\}}\geq f_{\min} \tag{49}\]  
If conditions (47), (48), and (49) are not applicable, the thrust feasibility of this trajectory segment is uncertain.