## 3 Flight Controller Design
### Proposed Analytic Form of the Proposed Position Integral Backstepping Controller  
The analytic form of \(\phi_{1}\) and \(\phi_{2}\) will be obtained by introducing the system dynamics (9)-(11). Then, split the terms \(\phi_{1}\) and \(\phi_{2}\) into a disturbance-independent compensable term and a disturbance-dependent term, respectively. By utilizing the compensable terms to construct \(\dot{\phi}_{1}\) and \(\dot{\phi}_{2}\), the control chattering due to the differentiation of the noise can be avoided significantly. For readability, the results of the proposed analytic integral backstepping controller (AIBC) are summarized in the following theorem; afterward, the comments and proofs are provided. **Theorem 3**. : _Consider the systems (47)-(49). The IBC (54) presented in Theorem 2 is modified by the following analytic form, namely AIBC:_  
\[M=\frac{\int\!\varsigma_{\theta}^{3}F^{2}}{\varsigma_{\theta}^{3}F^{2}+m^{2} \mathbf{s}_{\theta}u_{x}u_{z}}\big{(}\dot{\varphi}_{2,0}-\varsigma_{\theta}( \mathbf{s}_{\theta}-\varphi_{1})-k_{22}\big{(}\dot{\theta}-\varphi_{2}\big{)}- w\,\mathrm{sign}\big{(}\dot{\theta}-\varphi_{2}\big{)}\big{)} \tag{94}\]  
_where_  
\[\dot{\varphi}_{2,0} = \tag{95}\] \[\dot{\varphi}_{1} =\] (96) \[\ddot{\varphi}_{1,0,0} = m\frac{\ddot{u}_{x,0}}{F}+m\frac{2u_{x}\dot{F}_{0}^{2}}{F^{3}}-m \frac{u_{x}\ddot{\Phi}_{0}}{F^{2}}-m\frac{2\dot{u}_{x,0}\ddot{\Phi}_{0}}{F^{2}}\] (98) \[\dot{\varphi}_{1,0} = m\frac{\dot{F}\dot{u}_{x,0}-u_{x}\dot{F}_{0}}{F^{2}}\] (99) \[\ddot{\ell}_{0} = m\bigg{(}\frac{\ddot{u}_{z,0}}{\varsigma_{\theta}}+\frac{\dot{ \theta}^{2}u_{z}}{\varsigma_{\theta}}+\frac{2\dot{\theta}^{2}\mathbf{s}_{ \theta}^{2}u_{z}}{\varsigma_{\theta}}+\frac{2\dot{\theta}\mathbf{s}_{\theta} \dot{u}_{z,0}}{\varsigma_{\theta}}\bigg{)}\] (100) \[\ddot{u}_{x,0,0} = x_{d}^{(4)}-k_{px}\bigg{(}\frac{F\mathbf{s}_{\theta}}{m}-\ddot{x }_{d}\bigg{)}-k_{iz}\varepsilon_{x2}-k_{dx}\bigg{(}\frac{\ddot{F}_{0}\mathbf{ s}_{\theta}+F\varsigma_{\theta}\dot{\theta}}{m}-x_{d}^{(3)}\bigg{)}\] (101) \[\ddot{u}_{z,0,0} = z_{d}^{(4)}-k_{px}\bigg{(}\frac{F\mathbf{c}_{\theta}}{m}-g-\ddot{ z}_{d}\bigg{)}-k_{iz}\varepsilon_{x2}-k_{dz}\bigg{(}\frac{\ddot{F}_{0}\mathbf{ c}_{\theta}-F\mathbf{s}_{\theta}\dot{\theta}}{m}-z_{d}^{(3)}\bigg{)}\] (102) \[\dot{F}_{0} = m\frac{\varsigma_{\theta}\dot{u}_{z,0}}{\varsigma_{\theta}}+ \frac{\dot{\theta}^{2}\dot{u}_{z}}{\varsigma_{\theta}}+\frac{2\dot{\theta}^{2 }\mathbf{s}_{\theta}^{2}u_{z}}{\varsigma_{\theta}}+\frac{2\dot{\theta} \mathbf{s}_{\theta}\dot{u}_{z,0}}{\varsigma_{\theta}^{2}}\bigg{)}\] (103) \[\ddot{u}_{z,0} = z_{d}^{(3)}-k_{px}\varepsilon_{x2}-k_{iz}\varepsilon_{z1}-k_{dz }\bigg{(}\frac{F\varsigma_{\theta}}{m}-g-\ddot{z}_{d}\bigg{)}\] (104) \[\dot{\varphi}_{2,0} = \tag{95}\]0  
_The terms \(\mathbf{e}_{x12}\), \(\mathbf{g}_{x1}\), \(\mathbf{P}_{3x}\), \(u_{z}\), and \(u_{x}\) are the same as given in (38), (53), (57), (58), and (59), respectively. The corresponding control gains \((k_{px},k_{iz},k_{dz},k_{px},k_{ix},k_{dx})>0\), and_  
\[\dot{\varphi}_{2,0} = \tag{95}\]1  
_where \(M^{+}>0\) is the upper bound of the absolute value of those disturbance-dependent non-compensable terms. Notice that \(\varphi_{2}\) in (96) is different from (55). The control law (94) guarantees that the extended error vector \(\mathbf{e}=[\mathbf{e}_{x}^{\mathrm{T}},z_{1}]^{\mathrm{T}}\) will approach a region \(\varepsilon_{xz0}\), i.e.,_  
\[\left\| \begin{bmatrix}\mathbf{e}_{x}\\ z_{1}\end{bmatrix}