A quadrotor is performing maneuvers with roll rate $p(t),$ pitch rate $q(t),$ and yaw rate $r(t).$

The aerodynamic moment on the quadrotor due to the drag is modeled by ${?}^a{m}_B=-$

$u{}_B,$

where

$u$ is a constant $($drag$)$ coefficient. To achieve a desirable trajectory, the rotors apply

time$-$dependent control moments $L_c(t)=-p(t),M_c=-q(t),$ and $N_c=-(\frac{}{2})r(t),$ where

$$ is a constant coefficient. The mass of the quadrotor is $5$ kg $,$ resulting in moment of inertias

$I_x=I_y=I_z=2kg{m}^2.$ If the initial angular velocity of the quadrotor is ${}_B(0)=(2,3,2{)}^T$

$ra\frac{d}{s},$ answer the following questions.

$($a$)$ Solve for the time dependence of each angular velocity component. In other words, find

$p(t),q(t),$ and $r(t)$ as functions of

$u,,$ and $t.$

$($b$)$ TRUE or FALSE. Without any external disturbances, the quadrotor will eventually achieve

a steady$-$state orientation, i$.$e$.,$ the orientation of the copter approaches a fixed value as

$t.$ Justify your answer.