## 3 Flight Controller Design
### Proposed Analytic Form of the Proposed Position Integral Backstepping Controller
To obtain \(\ddot{\varphi}_{1,0}\) and \(\ddot{F}_{0}\), taking the time derivative of \(\dot{u}_{x,0}\) in (105) and \(\dot{u}_{z,0}\) in (104) as follows:  
\[\ddot{u}_{z,0} =\dot{u}_{z,0,0}+\triangle u_{z,1} \tag{119}\] \[\ddot{u}_{x,0} =\dot{u}_{x,0,0}+\triangle u_{x,1}\]  
where \(\ddot{u}_{z,0,0}\) is shown in (102), \(\ddot{u}_{x,0,0}\) is given by (101), and  
\[\triangle u_{z,1} =-k_{pz}\frac{d_{z}}{m}-k_{dz}\frac{\triangle F_{0}\varsigma_{ \theta}}{m} \tag{120}\] \[\triangle u_{x,1} =-k_{px}\frac{d_{x}}{m}-k_{dx}\frac{\triangle F_{0}\varsigma_{ \theta}}{m},\]  
respectively. Taking the time derivative of \(\dot{F}_{0}\) in (103) yields  
\[\dot{F}_{0} =\ddot{F}_{0,0}+F_{M}M+\triangle F_{1} \tag{121}\]  
where \(M\) is the AIBC to be determined later, \(\ddot{F}_{0,0}\) is shown as (100), and  
\[F_{M} =\frac{m\varsigma_{\theta}u_{z}}{\int\varsigma_{\theta}{}^{2}} \tag{122}\] \[\triangle F_{1} =m\frac{\triangle u_{z,1}}{\varsigma_{\theta}}+m\frac{d_{\theta} s_{\theta}u_{z}}{J\varsigma_{\theta}{}^{2}}\]  
Substituting \(\ddot{F}_{0}\) in (121) into the time derivative of \(\dot{\varphi}_{1,0}\) in (99) gives  
\[\ddot{\varphi}_{1,0} =m\left(\frac{\dot{u}_{x,0,0}+\triangle u_{x,1}}{F}-\frac{\dot{u }_{x,0}\ddot{F}_{0}}{F^{2}}-\frac{\dot{u}_{x,0}\triangle F_{0}}{F^{2}}\right)\] \[\quad-m\left(\frac{(\dot{u}_{x,0}+\triangle u_{x,0})\dot{F}_{0}+u _{x}\big{(}\ddot{F}_{0,0}+F_{M}M+\triangle F_{1}\big{)}}{F^{2}}-\frac{2u_{x} \ddot{F}_{0}\big{(}\ddot{F}_{0}+\triangle F_{0}\big{)}}{F^{3}}\right)\] \[=\ddot{\varphi}_{1,0,0}+\triangle\varphi_{1,1}-m\frac{u_{x}F_{M}M }{F^{2}} \tag{123}\]  
where \(\ddot{\varphi}_{1,0,0}\) is shown as (98) and  
\[\ddot{u}_{x,0} =\dot{u}_{x,0,0}+\triangle u_{x,1}\]0  
Taking the time derivative of \(\varphi_{2}\) in (96) gives\[\ddot{u}_{x,0} =\dot{u}_{x,0,0}+\triangle u_{x,1}\]1 \[= \frac{1}{\mathrm{c}_{\theta}}\bigg{(}\ddot{\varphi}_{1,0,0}+\bigtriangleup \varphi_{1,1}-m\frac{u_{x}F_{M}M}{F^{2}}-2\mathbf{e}_{x12}^{T}\mathbf{P}_{3x} \begin{bmatrix}0\\ \frac{F_{0}+\bigtriangleup F_{0}}{m}\end{bmatrix} \] \[\qquad-2\mathbf{g}_{x1}^{T}\mathbf{P}_{3x}\bigg{[}\frac{e_{x2}}{m }+\frac{d_{x}}{m}-\ddot{x}_{d}\bigg{]}-k_{z1}\big{(}\mathrm{c}_{\theta}\dot{ \theta}-\dot{\varphi}_{1,0}-\bigtriangleup\varphi_{1,0}\big{)}\bigg{)}\] \[+\frac{\mathrm{s}_{\theta}\dot{\theta}}{\mathrm{c}_{\theta}^{2}} \Big{(}\dot{\varphi}_{1,0}-2\mathbf{e}_{x12}^{T}\mathbf{P}_{3x}\mathbf{g}_{x 1}-k_{z1}z_{1}\Big{)}\] \[= \underbrace{\frac{1}{\mathrm{c}_{\theta}}\bigg{(}\ddot{\varphi}_ {1,0,0}-2\mathbf{e}_{x12}^{T}\mathbf{P}_{3x}\bigg{[}\frac{0}{\frac{F_{0}}{m}} \bigg{]}-2\mathbf{g}_{x1}^{T}\mathbf{P}_{3x}\bigg{[}\frac{e_{x2}}{\frac{F_{0} }{m}}-\ddot{x}_{d}\bigg{]}-k_{z1}\big{(}\mathrm{c}_{\theta}\dot{\theta}-\dot{ \varphi}_{1,0}\big{)}\bigg{)}}_{\triangleq\dot{\varphi}_{2,0}}\] \[+\underbrace{\frac{\mathrm{s}_{\theta}\dot{\theta}}{\mathrm{c}_{ \theta}^{2}}\Big{(}\dot{\varphi}_{1,0}-2\mathbf{e}_{x12}^{T}\mathbf{P}_{3x} \mathbf{g}_{x1}-k_{z1}z_{1}\Big{)}}_{\triangleq\dot{\varphi}_{2,0}}-m\frac{u_ {x}F_{M}M}{\mathrm{c}_{\theta}F^{2}}\] \[+\underbrace{\frac{1}{\mathrm{c}_{\theta}}\bigg{(}\bigtriangleup \varphi_{1,1}-2\mathbf{e}_{x12}^{T}\mathbf{P}_{3x}\bigg{[}\frac{0}{\bigtriangleup F _{0}}\bigg{]}-2\mathbf{g}_{x1}^{T}\mathbf{P}_{3x}\bigg{[}\frac{0}{\bigtriangleup \varphi_{1,0}}\bigg{]}+k_{z1}\bigtriangleup\varphi_{1,0}\bigg{)}}_{ \triangleq\Delta\varphi_{2,0}}\] \[= \dot{\varphi}_{2,0}+\bigtriangleup\varphi_{2,0}-m\frac{u_{x}F_{M}M }{\mathrm{c}_{\theta}F^{2}}\]  
where \(\dot{\varphi}_{2,0}=\dot{\varphi}_{2,0u}+\dot{\varphi}_{2,0b}\) is described in (95) and  
\[\bigtriangleup\varphi_{2,0}=\frac{1}{\mathrm{c}_{\theta}}\bigg{(}\bigtriangleup \varphi_{1,1}-2\mathbf{e}_{x12}^{T}\mathbf{P}_{3x}\bigg{[}\frac{0}{\bigtriangleup F _{0}}\bigg{]}-2\mathbf{g}_{x1}^{T}\mathbf{P}_{3x}\bigg{[}\frac{0}{\bigtriangleup F _{0}}\bigg{]}+k_{z1}\bigtriangleup\varphi_{1,0}\bigg{)} \tag{126}\]  
respectively. Consider the composite Lyapunov function candidate in (88).