# Model-Free-Based Single-Dimension Fuzzy SMC Design for Underactuated Quadrotor UAV
## 3 Mathematical Model  
Underactuated quadrotor is one of the nonlinear unstable systems and comprises four brushless DC (BLDC) motors whose thrust is treated as an input to entire system. There are several factors that one must consider while deriving the dynamic model of underactuated quadrotor. These factors are thrust force generated via the rotation of propellers, torque, gravity component and gyroscopic factors. To derive the dynamic model using Newton Euler dynamics, one must consider the body frame of quadrotor as a rigid and symmetrical structure as one of the assumptions to derive appropriate dynamics. Table 1 shows all the symbols along with their description. The change in the underactuated quadrotor's attitude is indicated by roll, pitch, and yaw angles, which are denoted as \(\phi,\theta\) and \(\psi\). These are obtained relatively to the \(x\)-, \(y\)- and \(z\)-axes. Moreover, one may see some of the nearer and appropriate dynamical models for quadrotor in [57; 58]. Research contributions such as in [59; 60] exist, where researchers derived the Euler-lagrange technique to acquire the dynamics for underactuated quadrotor craft. Moreover, in the Equations (1)-(6), \(u_{1}=\mathrm{T}_{2}-\mathrm{T}_{4}=k_{\mathrm{thr}}\cdot\left(\Omega_{2}^{2} -\Omega_{4}^{2}\right)\), \(u_{2}=\mathrm{T}_{1}-\mathrm{T}_{3}=k_{\mathrm{thr}}\cdot\left(\Omega_{1}^{2} -\Omega_{3}^{2}\right)\), \(u_{3}=b_{\mathrm{drags}}\cdot\left(\Omega_{1}^{2}-\Omega_{2}^{2}+\Omega_{3}^{2 }-\Omega_{4}^{2}\right)\) and \(u_{4}=\sum_{i=1}^{4}\mathrm{F}_{i}=\sum_{i=1}^{4}\mathrm{T}_{i}\). This \(\mathrm{T}_{i}=\mathrm{T}_{1}+\mathrm{T}_{2}+\mathrm{T}_{3}+\mathrm{T}_{4}\) whereas \(\Omega\) is given as the sum \(=-\Omega_{1}+\Omega_{2}-\Omega_{3}+\Omega_{4}\). \[\ddot{\phi} =\left(lu_{1}-\dot{l}_{xx}\dot{\phi}\right)/I_{xx} \tag{1}\] \[\ddot{\theta} =\left(lu_{2}-\dot{l}_{yy}\dot{\theta}\right)/I_{yy}\] (2) \[\ddot{\psi} =\left(lu_{3}-\dot{l}_{zz}\dot{\psi}\right)/l_{zz}\] (3) \[\ddot{x} =-u_{4}\sin\theta/m\] (4) \[\ddot{y} =u_{4}\sin\varphi\cos\theta/m\] (5) \[\ddot{z} =u_{4}\cos\varphi\cos\theta/(m-g) \tag{6}\]The equation set (1) to (6) is derived via [PERSON] lagrange method, and it does not consider quadrotor as nonlinear system with very strong coupled factors. Thus, the dynamic model of underactuated quadrotor is derived using [PERSON] dynamic approach in [61] as given below:  
\[\ddot{\phi} =\frac{I_{yy}-I_{zz}}{I_{xx}}\dot{\phi}\dot{\psi}+\frac{l}{I_{xx}}u_{1} \tag{7}\] \[\ddot{\theta} =\frac{I_{zz}-I_{xx}}{I_{yy}}\dot{\phi}\dot{\psi}+\frac{l}{I_{yy}}u _{2}\] (8) \[\ddot{\psi} =\frac{I_{xx}-I_{yy}}{I_{zz}}\dot{\phi}\dot{\theta}+\frac{l}{I_{zz }}u_{3}\] (9) \[\ddot{x} =(u_{4}u_{x})/m\] (10) \[\ddot{\theta} =\left(lu_{2}-\dot{l}_{yy}\dot{\theta}\right)/I_{yy}\]0 (11) \[\ddot{\theta} =\left(lu_{2}-\dot{l}_{yy}\dot{\theta}\right)/I_{yy}\]1  
Moreover, in the equation set from (7) to (12), \(u_{1}=\mathrm{T}_{2}-\mathrm{T}_{4}\), \(u_{2}=\mathrm{T}_{1}-\mathrm{T}_{3}\), \(u_{3}=\mathrm{T}_{1}+\mathrm{T}_{3}-\mathrm{T}_{2}-\mathrm{T}_{4}\) and \(u_{\mathrm{x}}=\sin\psi\sin\theta\cos\varphi\), \(u_{y}=\sin\psi\sin\theta\cos\varphi-\cos\psi\sin\varphi\) and \(u_{4}=\sum_{l=1}^{4}\mathrm{F}_{\mathrm{i}}=\sum_{l=1}^{4}\mathrm{T}_{\mathrm{ i}}\). This \(\mathrm{T}_{\mathrm{i}}=\mathrm{T}_{1}+\mathrm{T}_{2}+\mathrm{T}_{3}+\mathrm{T }_{4}\). The shortcoming of this model is the ignorance of the torque factor that is produced by the revolutions of rotors during flight. To consider this one of the major factors, here we add \(-J_{r}\dot{\theta}\Omega/I_{xx}\) and \(J_{r}\dot{\theta}\Omega/I_{yy}\)[62; 63; 64].