## 6 Conclusions
\begin{array}{l}\delta\ddot{D}+\bigg{(}\frac{3D_{0}}{\upsilon_{0}}-\frac{2 \dot{D}_{0}}{D_{0}}\bigg{)}\delta\dot{D}\\ \quad+\bigg{[}3\dot{D}_{0}\bigg{(}\frac{\dot{D}_{0}}{D_{0}^{2}}-\frac{1}{ \upsilon_{0}}\bigg{)}+\frac{4D_{0}^{2}}{\upsilon_{0}^{2}}-\frac{1}{h_{s}} \bigg{(}\frac{\upsilon_{0}^{2}}{\upsilon_{0}}-g\bigg{)}-\frac{2\dot{D}_{0}}{ D_{0}}-\frac{\dot{c}_{D_{0}}^{2}}{\dot{C}_{D_{0}}^{2}}+\frac{\dot{c}_{D_{0}}}{ \dot{C}_{D_{0}}}\bigg{]}\delta D\\ \quad+\bigg{(}\frac{2D_{0}\upsilon_{0}}{\upsilon_{0}}-\frac{3\dot{D}_{0}D_{0} }{\upsilon_{0}^{2}}-\frac{8D_{0}^{3}}{\upsilon_{0}^{2}}+\frac{\dot{C}_{D_{0}} D_{0}^{2}}{C_{D_{0}}\upsilon_{0}^{2}}\bigg{)}\delta\upsilon\\ \quad=\quad-\frac{D_{0}^{2}}{h_{s}}\delta\bigg{(}\frac{L\cos\upsilon}{D} \bigg{)}+\frac{D_{0}}{C_{0}}\delta\ddot{C}_{D}+\bigg{(}\frac{D_{0}^{2}}{C_{D_{ 0}}\upsilon_{0}}-\frac{2D_{0}\dot{C}_{D_{0}}}{\dot{C}_{D_{0}}^{2}}\bigg{)} \delta\dot{C}_{D}\\ \quad+\bigg{(}\frac{2\dot{c}_{D_{0}}^{2}D_{0}}{\dot{C}_{D_{0}}^{2}}-\frac{\dot {C}_{D_{0}}D_{0}^{2}}{\dot{C}_{D_{0}}^{2}\upsilon_{0}}-\frac{\dot{C}_{D_{0}} D_{0}}{\dot{C}_{D_{0}}^{2}}\bigg{)}\delta C_{D}\end{array}  \tag{102}\]  
Normally, the angle of attack and bank angle are regarded as the two key control variables in the tracking guidance for a vehicle with the D-V profile method. For the purpose of simplification, the angle of attack can be designed as a segmental linear function of velocity. Therefore, tracking and guiding the vehicle can be accomplished by simply adjusting the bank angle. For further study, if the equations are expanded by perturbation at the current velocity point, some varying components are obtained directly as follows:  
\[\left\{ \begin{array}{l}\delta\dot{V}=0,\\ \delta\dot{C}_{D}=0,\\ \delta\ddot{C}_{D}=0.\end{array} \right. \tag{103}\]  
Thus, Equation (102) can be simplified to:  
\[ \begin{array}{l}\delta\ddot{D}+\bigg{(}\frac{3D_{0}}{\upsilon_{0}}-\frac{2 \dot{D}_{0}}{D_{0}}\bigg{)}\delta\dot{D}\\ \quad+\bigg{[}3\dot{D}_{0}\bigg{(}\frac{\dot{D}_{0}}{D_{0}^{2}}-\frac{1}{ \upsilon_{0}}\bigg{)}+\frac{4D_{0}^{2}}{\upsilon_{0}^{2}}-\frac{1}{h_{s}} \bigg{(}\frac{\upsilon_{0}^{2}}{\upsilon_{0}}-g\bigg{)}-\frac{2\dot{D}_{0}}{ D_{0}}-\frac{\dot{c}_{D_{0}}^{2}}{\dot{C}_{D_{0}}^{2}}+\frac{\dot{c}_{D_{0}}}{ \dot{C}_{D_{0}}}\bigg{]}\delta D\\ \quad+\bigg{(}\frac{2D_{0}\upsilon_{0}}{h_{s}r_{0}}-\frac{3\dot{D}_{0}D_{0}}{ \upsilon_{0}^{2}}-\frac{8D_{0}^{3}}{\upsilon_{0}^{3}}+\frac{\dot{C}_{D_{0}}D_{0 }^{2}}{C_{D_{0}}\upsilon_{0}^{2}}\bigg{)}\delta\upsilon\\ \quad=\quad-\frac{D_{0}^{2}}{h_{s}}\delta\bigg{(}\frac{L\cos\upsilon}{D}\bigg{)} \end{array}  \tag{104}\]Substituting Equation (A15) into it and collecting with some algebraic skills gives:  
\[ \begin{array}{l}0=\delta\bar{D}+\left(\frac{D_{0}^{2}}{h_{s}}f_{2}^{\prime}+ \frac{3D_{0}}{v_{0}}-\frac{2\dot{D}_{0}}{D_{0}}\right)\delta\dot{D}\\ \\ \quad+\left[\frac{D_{0}^{2}}{h_{s}}f_{1}^{\prime}+3\dot{D}_{0}\bigg{(}\frac{ \dot{D}_{0}}{D_{0}^{2}}-\frac{1}{v_{0}}\bigg{)}+\frac{4D_{0}^{2}}{V_{0}^{2}}- \frac{1}{h_{s}}\bigg{(}\frac{v_{0}^{2}}{v_{0}}-g\bigg{)}-\frac{2\dot{D}_{0}}{D_ {0}}-\frac{\dot{c}_{D_{0}}^{2}}{\dot{c}_{D_{0}}^{2}}+\frac{\bar{c}_{D_{0}}}{C_ {D_{0}}}\right]\delta D\\ \\ \quad+\left(\frac{D_{0}^{2}}{h_{s}}f_{3}^{\prime}+\frac{2D_{0}v_{0}}{h_{s} \cdot r_{0}}-\frac{3\dot{D}_{0}D_{0}}{v_{0}^{2}}-\frac{8D_{0}^{3}}{v_{0}^{3}}+ \frac{\dot{c}_{D_{0}}D_{0}^{2}}{C_{D_{0}}v_{0}^{2}}\right)\delta v\end{array} \] (A20)  
It is obvious that Equation (A20) is a second-order ordinary differential equation with an oscillatory form of the solution when \(\delta v=0\). By writing in the standard form:  
\[\delta\ddot{D}+2\xi\omega\delta\dot{D}+\omega^{2}\delta D=0\] (A21)  
where \(\omega\) and \(\xi\) are two designed constants. Combining Equations (A20) and (A21) and collecting with some algebraic skills, there is:  
\[\left\{