## 3 Nonlinear Robust Control Laws
### Nonlinear Attitude Controller Based on Disturbance Observer
Equation (47) can be rewritten as  
\[\varsigma\dddot{X}=\varsigma\mathcal{A}\widetilde{X}+\mathcal{B}\dddot{\Xi} \Big{(}\Delta\Xi,J_{\Delta},L_{\Delta},\Delta_{2},o_{2},\ddot{\Theta},\dot{ \Theta},\xi_{1},e_{\xi},X,t\Big{)} \tag{50}\]  
Then, we redefine the error term as follows:  
\[\dddot{\Xi}_{\Xi}\Big{(}\Delta\Xi,J_{\Delta},L_{\Delta},\Delta_{2},o_{2}, \ddot{\Theta},\dot{\Theta},\dot{\zeta}_{1},e_{\xi},X,t\Big{)} \tag{51}\]  
Equation (52) is given by  
\[\varsigma\dddot{X}=\varsigma\mathcal{A}\widetilde{X}+\mathcal{B}\dddot{\Xi} _{\Xi}\Big{(}\Delta\Xi,J_{\Delta},L_{\Delta},\Delta_{2},o_{2},\ddot{\Theta}, \ddot{\Theta},\ddot{\xi}_{1},e_{\xi},\xi\ddot{X},t\Big{)} \tag{52}\]  
Choose the Lyapunov function as  
\[V(\xi_{1},e_{\xi},\Delta\Xi,\widetilde{X})= \frac{1}{2}\xi_{1}^{T}\xi_{1}+\frac{1}{2}e_{\xi}^{T}e_{\xi}+ \frac{1}{2}tr(\Delta\Xi^{T}\Delta\Xi)+\varsigma\widetilde{X}^{T}P\widetilde{X} \tag{53}\]  
where \(P=P^{T}>0\), and the time derivative of the Lyapunov function \(V(\xi_{1},e_{\xi},\Delta\Xi,\widetilde{X})\) is  
\[\dot{V}= \xi_{1}^{T}\xi_{1}+\xi_{\xi}^{T}e_{\xi}+tr(\Delta\Xi^{T}\Delta \Xi)+\widetilde{X}^{T}P\dot{\widetilde{X}}+\dot{\widetilde{X}}^{T}P\widetilde{X}\] \[= \xi_{1}^{T}\big{(}e_{\xi}+\xi_{2}^{*}\big{)}+e_{\xi}^{T}e_{\xi}+ tr(\Delta\Xi^{T}\Delta\Xi)+\varsigma\widetilde{X}^{T}P\dot{\widetilde{X}}+\varsigma \dot{\widetilde{X}}^{T}P\widetilde{X} \tag{54}\]  
Substituting \(\xi_{2}^{*}\) into (54), we have  
\[\dot{V}=-a_{1}\xi_{1}^{T}\xi_{1}+\xi_{1}^{T}e_{\xi}+e_{\xi}^{T} \Big{(}-R_{\Theta}J^{-1}S(\dot{\Theta})J\dot{\Theta}+R_{\Theta}J^{-1}LF^{b}+ \Xi^{T}W(\xi)-\ddot{\Theta}_{d}+\Delta_{\xi}\Big{)}+tr(\Delta\Xi^{T}\Delta \dot{\Xi})\] \[+\varsigma\widetilde{X}^{T}P\dot{\widetilde{X}}+\varsigma\dot{ \widetilde{X}}^{T}P\widetilde{X} \tag{55}\]Using \(F^{b}\) in (55), then we obtain  
\[\dot{V}\leq -a_{1}\xi_{1}^{T}\xi_{1}-a_{2}\epsilon_{\xi}^{T}\epsilon_{\xi}+ \epsilon_{\xi}^{T}\left(\Xi^{T}W(\xi)-\widehat{\Xi}^{T}W(\xi)+\Delta_{\xi}- \widehat{\Delta}_{\Xi}-\frac{1}{2\epsilon_{3}^{2}}\epsilon_{\xi}\right)+tr( \Delta\Xi^{T}\Delta\widehat{\Xi})+\varsigma\widehat{X}^{T}P\hat{\bar{X}}+ \varsigma\hat{\bar{X}}^{T}P\widetilde{\bar{X}}\] \[\varsigma\dddot{X}=\varsigma\mathcal{A}\widetilde{X}+\mathcal{B}\dddot{\Xi} \Big{(}\Delta\Xi,J_{\Delta},L_{\Delta},\Delta_{2},o_{2},\ddot{\Theta},\dot{ \Theta},\xi_{1},e_{\xi},X,t\Big{)} \tag{50}\]0 \[\varsigma\dddot{X}=\varsigma\mathcal{A}\widetilde{X}+\mathcal{B}\dddot{\Xi} \Big{(}\Delta\Xi,J_{\Delta},L_{\Delta},\Delta_{2},o_{2},\ddot{\Theta},\dot{ \Theta},\xi_{1},e_{\xi},X,t\Big{)} \tag{50}\]1 \[\varsigma\dddot{X}=\varsigma\mathcal{A}\widetilde{X}+\mathcal{B}\dddot{\Xi} \Big{(}\Delta\Xi,J_{\Delta},L_{\Delta},\Delta_{2},o_{2},\ddot{\Theta},\dot{ \Theta},\xi_{1},e_{\xi},X,t\Big{)} \tag{50}\]2 \[\varsigma\dddot{X}=\varsigma\mathcal{A}\widetilde{X}+\mathcal{B}\dddot{\Xi} \Big{(}\Delta\Xi,J_{\Delta},L_{\Delta},\Delta_{2},o_{2},\ddot{\Theta},\dot{ \Theta},\xi_{1},e_{\xi},X,t\Big{)} \tag{50}\]3  
where \(e_{\Delta}=\Delta_{\xi}-\widehat{\Delta}_{\Xi}\) and \(\mathcal{A}^{T}P+P\mathcal{A}=-Q\), \(Q>0\). Substituting inequalities  
\[ \begin{cases}\epsilon_{\xi}^{T}\epsilon_{\Delta}\leq\frac{1}{2 \epsilon_{3}^{2}}\epsilon_{\xi}^{T}\epsilon_{\xi}+\frac{\epsilon_{3}^{2}}{2 \epsilon_{\Delta}^{T}}\epsilon_{\Delta}\\ \widetilde{X}^{T}P\mathcal{B}\widetilde{\Delta}_{\Xi}\leq\frac{1}{2}\| \widetilde{X}\|^{2}+\frac{1}{2}\|P\|^{2}\|\mathcal{B}\|^{2}\|\widetilde{\Delta }_{\Xi}\|^{2}\leq\frac{1}{2}\|\widetilde{X}\|^{2}+\frac{1}{2}\|P\|^{2}\| \widetilde{\Delta}_{\Xi}\|^{2}\end{cases}