$(2)$ Consider the two$-$dimensional drone depicted below. In what follows you will develop the equations of motion and then you will linearize them. We will see the full development of the three$-$dimensional equations of motion and when linearized the three$-$dimensional equations of motion become uncoupled in three planes $($unless aerodynamic effects preclude this$)$ and, as we'll see, the equations of motion developed for the plane in this exercize are identical to those obtained by linearization of the three$-$dimensional equations of motion.

$x$

The drone is modelled as a central package with mass and moment of inertia $M$ and $J,$ respectively. The rotors are symetrically placed a distance $L$ from the central package, and are modelled as point masses with mass $m.$ The system moves with $x$ and $z-$direction velocities $u$ and $w,$ respectlively, and the system rotates with pitch angle $.$ The rotors provide thrust forces $F_l$ and $F_2$ and the system experiences an aerodynamic drag force in the x$-$direction modeled as $C_{u}u$ where $C_u$ is a constant coefficient. The system also experiences an aerodynamic moment modelled as $D_{u}u$ where $D_u$ is a constant coefficient. The coefficients $C_u<mn>0</mn>$ and $D_u>0$ and are related to stability derivatives $x_u$ and $M_u($defined later$).$ There is also a drag force in the $z$ direction with coefficient $E_w.$

$($i$)$ Define the rotor force $F=F_1+F_2$ and rotor torque $T=(F_1-F_2)L$ then apply Newton's Laws of Motion to show that the equations of motion are:

$(2m+M)u^{}=-$Fsin$+C_{u}u$

$(2m+M)w^{}=$Fcos$+E_{w}w-(2m+M)g$

$(2m{L}^2+J){}^{}=T+D_{u}u$