## 6 Conclusions  
This paper presents the dynamic modeling, path following, and control allocation of a TRQ. Two types of SMC are proposed to enhance the robustness and capability: one is the first-order sliding mode for attitude following and the other is the second-order sliding mode for position following. Considering the parameter changes and external perturbation, we show the stability analysis based on the Lyapunov theory that the proposed control scheme can ensure the error dynamics' asymptotic stability for the position and attitude following. In the numerical simulation of the fully actuated mode, we demonstrated that the proposed SMC could achieve path following and attitude regulation goals in the presence of variable changes and external perturbations. The tilt-rotor quadrotor has more control degrees of freedom than the under-actuated quadrotor and, therefore, can make full use of the control redundancy to complete the simultaneous trajectory tracking and attitude control that a traditional quadrotor cannot do, and thus has a certain degree of actuator fault tolerance. In the future, we will integrate the sliding mode path following and control allocation into a fault-tolerant flight control.  
Figure 14: The rotor velocity with variable deviations and perturbations.  
Figure 13: The tilt angle path with variable deviations and perturbations.  
**Author Contributions:** Conceptualization, C.-C.Y. and S.-J.W.; methodology, C.-C.Y. and S.-J.W.; software, S.-J.W.; validation, C.-C.Y. and S.-J.W.; formal analysis, C.-C.Y.; investigation, S.-J.W.; writing--original draft preparation, C.-C.Y. and S.-J.W.; writing--review and editing, C.-C.Y.; visualization, S.-J.W. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This research received no external funding.  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** No new data were created or analyzed in this study. Data sharing is not applicable to this article.  
**Conflicts of Interest:** The authors declare no conflict of interest.  
**Nomenclature**  
\begin{tabular}{l l} \(B\) & quadrotor body \\ \(P_{i}\) & propeller \(i\) \\ \(F_{w}\) & inerita world frame \\ \(F_{B}\) & body frame \\ \(F_{p_{i}}\) & \(i\)th propeller frame \\ \(R_{T}\) & transform matrix from body angular rates to Euler rates \\ \(p\) & position of \(B\) in \(F_{w}\) \\ \(q\) & Euler angle of \(B\) in \(F_{w}\) \\ \(W_{R_{B}}\) & rotation matrix from \(F_{B}\) to \(F_{w}\) \\ \({}^{B}R_{p_{i}}\) & rotation matrix from \(F_{p_{i}}\)to \(F_{B}\) \\ \(w_{i}\) & \(i\)th propeller spinning velocity about \(Z_{p_{i}}\) \\ \(wp_{p_{i}}\) & the angular velocity in the ith propeller frame \\ \(a_{i}\) & \(i\)th propeller tilting angle about X\({}_{p_{i}}\) \\ \(T_{p_{i}}\) & the force in the ith propeller frame \\ \(w_{B}\) & angular velocity of \(B\) in \(F_{B}\) \\ \(v_{B}\) & velocity of \(B\) in \(F_{B}\) \\ \(\tau_{B}\) & torque in \(F_{B}\) \\ \(\tau_{p_{i}}\) & torque in \(F_{p_{i}}\) \\ \(\tau_{di}\) & \(i\)th propeller air drag torque about \(Z_{p_{i}}\) \\ \(T_{i}\) & \(i\)th propeller thrust along \(Z_{p_{i}}\) \\ \(\tau_{u_{i}}\) & motor torque along \(Z_{p_{i}}\) \\ \(m\) & total quadrotor mass \\ \(I_{pi}\) & inertia of the ith propeller \(P_{i}\) \\ \(I_{B}\) & inertia of the quadcopter body \(B\) \\ \(k_{f}\) & propeller thrust coefficient \\ \(k_{m}\) & propeller drag coefficient \\ \(L\) & distance of \(F_{p_{i}}\) to \(F_{B}\) \\ \(g\) & gravity constant \\ \end{tabular}