## Appendix B Existence Proof of Sliding Mode  
In the sense of the sliding mode control, the error variable \(z_{2}\) defined in (84) can be interpreted as the sliding surface. The control laws (54) and (94) are the sliding mode controller because they enable forcing the system states to reach the sliding surface \(z_{2}\). Consider the sliding dynamics (87), and select a Lyapunov function candidate as  
\[V_{z_{2}}=\frac{1}{2}z_{2}^{2} \tag{114}\]  
Taking the time derivative of (114) along the trajectory (87), it follows that  
\[\dot{V}_{z_{2}} =z_{2}\dot{z}_{2}\] \[=z_{2}\bigg{(}\frac{M}{J}+\frac{d_{\theta}}{J}-\dot{\varphi}_{2} \bigg{)} \tag{115}\]  
In what follows, the IBC (54) and AIBC (94) will be respectively substituted into (115) to show the approaching condition. First, substituting (54) into (115) obtains  
\[\dot{V}_{z_{2}} =z_{2}\bigg{(}\dot{\varphi}_{2}-\mathrm{c}_{\theta}(\mathbf{s}_{ \theta}-\varphi_{1})-k_{22}(\dot{\theta}-\varphi_{2})-w\,\text{sign}(z_{2})+ \frac{d_{\theta}}{J}-\dot{\varphi}_{2}\bigg{)}\] \[=z_{2}\bigg{(}-\mathrm{c}_{\theta}z_{1}-k_{22}z_{2}-w\,\text{sign }(z_{2})+\frac{d_{\theta}}{J}\bigg{)}\] \[=-k_{22}z_{2}^{2}-w|z_{2}|+z_{2}\bigg{(}-\mathrm{c}_{\theta}z_{1} +\frac{d_{\theta}}{J}\bigg{)}\] \[\leq-k_{22}z_{2}^{2}-w|z_{2}|+|z_{2}|\bigg{(}|\mathrm{c}_{\theta }z_{1}|+\frac{d_{\theta}^{+}}{J}\bigg{)}\] \[\leq-k_{22}z_{2}^{2}-|z_{2}|\bigg{(}w-|\mathrm{c}_{\theta}z_{1}|- \frac{d_{\theta}^{+}}{J}\bigg{)} \tag{116}\]  
Equation (116) indicates that the approaching condition is fulfilled by choosing the switching gain shown in (60), which \(\eta\) should be  
\[\eta=|\mathrm{c}_{\theta}z_{1}|+\widetilde{\eta}. \tag{117}\]  
Then, (116) becomes  
\[\dot{V}_{z_{2}}\leq-k_{22}z_{2}^{2}-\widetilde{\eta}|z_{2}| \tag{118}\]where \(\widetilde{\eta}>0\) is a parameter to be designed. Therefore, the sliding surface \(z_{2}\) exponentially converges to zero within the finite time \(t_{f}\). With the aid of (A14) and (A18), the finite time can be solved by  
\[\dot{V}_{z_{2}} =z_{2}\dot{z}_{2}\]0  
For (A17), it guarantees that the approaching condition is satisfied for any initial condition; then, the sliding dynamics (87) converge to zero within the finite time described in (19). In fact, the control parameter \(\eta>0\) can be arbitrarily assigned since the stability of the overall systems (47)-(49) has been guaranteed by (88) and (91). In the presence of mismatched disturbances, the switching gain (60) for any \(\eta>0\) does not guarantee the approaching condition \(\hat{z}_{2}z_{2}<0\). However, when the disturbances vanish, the state \(\theta\) and \(z_{1}\) will converge until the condition \(\eta-|c_{\theta}z_{1}|>0\) such that the approaching condition holds. Eventually, the sliding motion occurs. Next, we are going to analyze the approaching condition for the proposed AIBC (94). According to (125), replacing \(\dot{\varphi}_{2}\) in (15) with \(\dot{\varphi}_{2,0}+\triangle\varphi_{2,0}-m\frac{u_{x}F_{M}M}{c_{\theta}F^{2 }}\), it follows that  
\[\dot{V}_{z_{2}} =z_{2}\dot{z}_{2}\]1 \[\dot{V}_{z_{2}} =z_{2}\dot{z}_{2}\]2 \[\dot{V}_{z_{2}} =z_{2}\dot{z}_{2}\]3 \[\dot{V}_{z_{2}} =z_{2}\dot{z}_{2}\]4  
Substituting (94) into(A20) yields  
\[\dot{V}_{z_{2}} =z_{2}\dot{z}_{2}\]5 \[\dot{V}_{z_{2}} =z_{2}\dot{z}_{2}\]6 \[\dot{V}_{z_{2}} =z_{2}\dot{z}_{2}\]7 \[\dot{V}_{z_{2}} =z_{2}\dot{z}_{2}\]8 \[\dot{V}_{z_{2}} =z_{2}\dot{z}_{2}\]9  
Therefore, by designing \(\eta\) in (106) as  
\[=z_{2}\bigg{(}\frac{M}{J}+\frac{d_{\theta}}{J}-\dot{\varphi}_{2} \bigg{)} \tag{115}\]0  
then (A21) becomes  
\[=z_{2}\bigg{(}\frac{M}{J}+\frac{d_{\theta}}{J}-\dot{\varphi}_{2} \bigg{)} \tag{115}\]1  
for \(\widetilde{\eta}^{\prime}>0\). As mentioned above, (22) guarantees that the approaching condition is fulfilled for any initial condition. However, the systems (47)-(49) are still stable for any \(\eta>0\). It has been proven in (88) and (132). In the absence of mismatched disturbances, the system states converge.