# Consensus Tracking Control of Multiple Unmanned Aerial Vehicles Subject to Distinct Unknown Delays
## 2 System Dynamic Model  
In this article, the MAS consists of a group of quadrotors under a leader--follower scheme. Each of the quadrotors in the MAS is a system with six degrees of freedom that consists of the translation coordinates \(r_{i}(t)=(x_{i}(t),y_{i}(t),z_{i}(t))\); and three rotations, one in each of the axes \(\eta_{i}(t)=(\phi_{i}(t),\theta_{i}(t),\psi_{i}(t))\) called roll, pitch, and yaw, respectively. The MAS can be described with respect to an inertial reference \(I\), where the UAVs are considered mass points. The translational and rotational dynamics of the vehicles are defined by the second law of Newton and the external torque equation as follows [38]:  
\[m\ddot{r_{i}}(t) =\left[ \begin{array}{c}0\\ 0\\ -mg\end{array} \right]+R(\eta_{i})\left[ \begin{array}{c}0\\ 0\\ F_{i}(t)\end{array} \right] \tag{1}\] \[I_{r}(\eta_{i})\ddot{\eta_{i}} =\tau_{i}-\dot{\eta_{i}}\times I_{r}(\eta_{i})\dot{\eta_{i}}= \left[ \begin{array}{c}\tau_{i2}(t)\\ \tau_{i3}(t)\\ \tau_{i4}(t)\end{array} \right]+\left[ \begin{array}{c}C_{i1}\\ C_{i2}\\ C_{i3}\end{array} \right] \tag{2}\]  
where \(g\), \(m\), and \(\dot{r_{i}}(t)\) are the gravitational acceleration, mass, and lineal acceleration; \(\dot{\eta_{i}}(t)\) and \(\ddot{\eta_{i}}(t)\) are the angular velocity and angular acceleration of the vehicle, respectively. In addition, \(F_{i}(t)=F_{i1}(t)+F_{i2}(t)+F_{i3}(t)+F_{i4}(t)\) is the total thrust of the vehicle and \(F_{ip}(t)=k_{F}\omega_{ip}^{2}(t),p=1,2,3,4\). The term \(k_{F}\) is a constant, and \(\omega_{ip}(t)\) is the angular velocity of each rotor. The terms \(R(\eta_{i})\) represent the orientation relative to \(I\), and \(I_{r}(\eta_{i})\) is the inertia tensor of the body:  
\[R(\eta_{i})=\left[ \begin{array}{cc}s_{\theta_{i}}c_{\psi_{i}}+s_{\theta_{i}} c_{\theta_{i}}s_{\psi_{i}}\\ s_{\theta_{i}}s_{\psi_{i}}-s_{\theta_{i}}c_{\theta_{i}}c_{\psi_{i}}\\ c_{\phi_{i}}c_{\theta_{i}}\end{array} \right]\qquad I_{r}(\eta_{i})=\left[ \begin{array}{ccc}I_{xx}c_{\theta_{i}}&0&-I_{xx}c_{\phi_{i}}s_{\theta_{i}}\\ 0&I_{yy}&I_{yy}s_{\phi_{i}}\\ I_{zz}s_{\theta_{i}}&0&I_{zz}c_{\phi_{i}}c_{\theta_{i}}\end{array} \right]\]  
where the shorthand notations \(s_{\alpha_{i}}\) and \(c_{\alpha_{i}}\) represent the sine and cosine of the angle \(\alpha_{i}(t)\), respectively; \(\tau_{i}\) are the torques defined as \(\tau_{i2}(t)=L(F_{2}(t)-F_{i4}(t))\), \(\tau_{i3}(t)=L(F_{i3}(t)-F_{i1}(t))\), \(\tau_{i4}(t)=M_{i1}(t)-M_{i2}(t)+M_{i3}(t)-M_{i4}(t)\); \(L\) is the distance between the rotors and the center of mass, \(M_{ip}(t)=k_{M}\omega_{ip}^{2}(t)\) is the reaction torque and \(k_{M}\) is a constant. In addition, \(C_{i1}\), \(C_{i2}\), and \(C_{i3}\) have the following definitions:  
\[C_{i1}= \left[I_{xx}+I_{yy}-I_{zz}\right]\phi_{i}(t)\dot{\theta}_{i}(t)s_{ \theta_{i}}+\left[-I_{xx}+I_{yy}-I_{zz}\right]\phi_{i}(t)\dot{\psi}_{i}(t)s_{ \phi_{i}}s_{\theta_{i}}\] \[+ \left[I_{xx}+I_{yy}-I_{zz}\right]\dot{\theta}_{i}(t)\dot{\psi}_{i }(t)c_{\phi_{i}}c_{\theta_{i}}+\left[I_{yy}-I_{zz}\right]\dot{\psi}_{i}^{2}(t)s _{\phi_{i}}c_{\phi_{i}}c_{\theta_{i}}\] \[C_{i2}= \left[-I_{yy}+[-I_{xx}+I_{zz}]c_{\phi}^{2}\right]\dot{\phi}_{i}(t) \dot{\psi}_{i}(t)c_{\theta_{i}}+\left[-I_{xx}+I_{zz}\right]\left[\dot{\phi}_{i }^{2}(t)-\dot{\psi}_{i}^{2}(t)c_{\theta_{i}}^{2}\right]s_{\theta_{i}}c_{ \theta_{i}}\] \[C_{i3}= \left[I_{xx}-I_{yy}-I_{zz}\right]\dot{\phi}_{i}(t)\dot{\theta}_{i }(t)c_{\theta_{i}}+\left[I_{xx}-I_{yy}+I_{zz}\right]\dot{\phi}_{i}(t)\dot{\psi }_{i}(t)s_{\phi_{i}}c_{\theta_{i}}\] \[+ \left[-I_{xx}+I_{yy}+I_{zz}\right]\dot{\theta}_{i}(t)\dot{\psi}_{i }(t)c_{\theta_{i}}s_{\theta_{i}}+\left[-I_{xx}+I_{yy}\right]\dot{\psi}_{i}^{2} (t)s_{\phi_{i}}c_{\phi_{i}}s_{\theta_{i}}\]  
The Equations (1) and (2) are the nonlinear dynamic model of the quadrotor multi-agent system. We use the Taylor series and obtain the following expressions:  
\[m\dot{r}_{i}(t)= \left[ \begin{array}{c}0\\ 0\\ -mg\end{array} \right]+\left[