# An Efficient Trajectory Planning Method for High-Speed Interception of Invasive Drones
## 3 Air Resistance Approximation and Motion Primitive Generation
### Motion Primitive Generation
#### 3.2.3 Single Axis Trajectory Generation  
By decoupling dynamics into three orthogonal inertial axes and using jerk as a control input, the problem of nonlinear trajectory generation is simplified. Subsequently, using Equations (18) and (21), the actual control inputs \(f_{t}\) and \(\mathbf{\omega}\) were inverted from the jerk input. The final state is determined by the target final state component \(\hat{\sigma}\) corresponding to each axis, and the trajectory duration \(T\) is specified. Decoupling the cost number \(J_{\Sigma}\) of three-dimensional motion into the cost per axis \(J_{k}\) through the integrand function in expansion (22):  
\[J_{\Sigma}=\sum_{k=1}^{3}J_{k},\quad\quad\text{where }J_{k}=\frac{1}{T}{ \int_{0}^{T}j_{k}(t)^{2}dt} \tag{25}\]  
Due to the consistency of the three axes in form, for the convenience of writing, the following text will analyze the single axis and ignore the subscript \(k\). Introduce state \(s=(p,v,a_{t})\), consisting of scalar position, velocity, and acceleration thrust components. Take the jerk \(j\) as input:  
\[\dot{s}=f_{s}(s,j)=(v,a_{t}-k_{rb}v-k_{rp}a_{t},j) \tag{26}\]By introducing the covariate variable \(\lambda=(\lambda_{1},\lambda_{2},\lambda_{3})\) and defining the Hamiltonian function \(H(s,j,\lambda)\), the optimal state trajectory can be directly solved using the Pontryagin minimum principle [31]. \[ \begin{array}{rl}H(s,j,\lambda)&=\frac{1}{7}j^{2}+\lambda^{T}f_{s}(s,j)\\ &=\frac{1}{7}j^{2}+\lambda_{1}v+\lambda_{2}(a_{t}-k_{rb}v-k_{rp}a_{t})+\lambda _{3}j\\ \end{array}  \tag{27}\]  
\[\dot{\lambda}=-\
abla_{s}H(s^{*},j^{*},\lambda)=\big{(}0,\;k_{rb}\lambda_{2}( t)-\lambda_{1}(t),\;\lambda_{2}(t)\big{(}k_{rp}-1\big{)}\big{)} \tag{28}\]  
Among them, \(j^{*}\) and \(s^{*}\) represent the optimal input and optimal state trajectories for a single axis, respectively. Solve the general solution of the co-state differential Equation (28), where \(C\) represents the undetermined constant:  
\[\lambda(t)=\frac{1}{T} \begin{bmatrix}C_{3}\\ \frac{C_{3}}{k_{rb}}-\frac{C_{2}e^{k_{rb}t}}{k_{rb}}\\ \frac{\big{(}C_{3}+C_{3}k_{rb}t-C_{2}e^{k_{rb}t}\big{)}\big{(}k_{rp}-1\big{)}} {k_{rb}^{2}}+\frac{C_{1}\big{(}k_{rp}-1\big{)}}{k_{rb}}\end{bmatrix}  \tag{29}\]  
By applying the optimal conditions to the Hamiltonian function \(H(s,\,j,\,\lambda)\), we can obtain:  
\[ \begin{array}{rl}j^{*}(t)&=\operatorname*{argmin}_{j(t)}H(s^{*}(t),j(t), \lambda(t))\\ &=\big{(}1-k_{rp}\big{)}\left(\frac{\big{(}C_{3}-C_{2}e^{k_{rb}t}\big{)}}{k_{rb }^{2}}+\frac{\big{(}C_{3}t+C_{1}\big{)}}{k_{rb}}\right)\end{array}  \tag{30}\]  
The optimal state trajectory can be obtained from Equation (26) and the initial condition \(s(0)=(p_{0},v_{0},a_{0})\):  
\[ \begin{array}{rl}s^{*}(t)=\big{(}k_{rp}-1\big{)}\left[\begin{array}{c}- \frac{e^{-k_{rb}t}\sigma_{1}-\sigma_{2}}{k_{rb}}-\frac{\sigma_{3}}{k_{rb}}\\ \frac{e^{-k_{rb}t}\sigma_{1}-\sigma_{2}}{k_{rb}}\\ \frac{a_{0}}{\left(k_{rp}-1\right)}+\frac{C_{2}\big{(}e^{k_{rb}t}-1\big{)}}{k_ {rb}^{3}}-\frac{t\big{(}C_{3}t+2C_{1}\big{)}}{2k_{rb}}-\frac{C_{3}t}{k_{rb}^{2 }}\end{array}\right]\\ \end{array}  \tag{31}\]  
where  
\[ \begin{array}{rl}\sigma_{1}=\frac{2v_{0}k_{rb}{}^{4}+\big{(}2a_{0}k_{rb}{}^{ 3}\big{)}(k_{rp}-1)-\big{(}C_{2}-2C_{1}k_{rb}\big{)}\big{(}k_{rp}-1\big{)}^{2}} {2k_{rb}{}^{3}\big{(}k_{rp}-1\big{)}}+\frac{e^{k_{rb}t}\big{(}C_{2}e^{k_{rb}t}- 2C_{3}k_{rb}t-2C_{1}k_{rb}\big{)}\big{(}k_{rp}-1\big{)}}{2k_{rb}{}^{3}}\\ \sigma_{2}=\frac{a_{0}k_{rb}{}^{3}-C_{2}\big{(}k_{rp}-1\big{)}}{k_{rb}{}^{3}}- \frac{\big{(}-2C_{2}e^{k_{rb}t}+(C_{3}k_{rb}t)(2+k_{rb}t)+2C_{1}k_{rb}t\big{)} \big{(}k_{rp}-1\big{)}}{2k_{rb}{}^{3}}\end{array}  \tag{32}\]  
Solve the remaining unknown constant \(C\) as a function of the expected end shift variable component \(\hat{\sigma}\) as shown in (10). For the case of fully defining the end effector translation state, make the expected end effector state equal to the position, velocity, and acceleration \(s(T)=(p_{f},v_{f},a_{f})\) defined by the variable \(\hat{\sigma}\).