Problem $2(40$ points$)$

A quadrotor has a maximum thrust of $12$ N and a mass of $0\mathrm{.}8$ kg $.$

$($i$)$ While keeping the roll, pitch and yaw all $0$ degree, if the drone starts from the altitude of $60$ m with a velocity of $\backslash (0\backslash$mathrm$\{$~m$\}/\backslash$mathrm$\{$s$\}\backslash )$ and stop at the altitude of $100$ m with a velocity of $\backslash (0\backslash$mathrm$\{$~m$\}/\backslash$mathrm$\{$s$\}\backslash ),$ if in an ideal case without any error, what is the shortest time?

$(10$ marks$)$

$($ii$)$ While keeping the roll, pitch and yaw all $0$ degree, if the drone starts from the altitude of $60$ m with a velocity of $\backslash (0\backslash$mathrm$\{$~m$\}/\backslash$mathrm$\{$s$\}\backslash )$ and stops at the altitude of $20$ m with a velocity of $\backslash (0\backslash$mathrm$\{$~m$\}/\backslash$mathrm$\{$s$\}\backslash ),$ if in an ideal case without any error, what is the shortest time?

$(10$ marks$)$

$($iii$)$ If ignoring the transient states, while keeping the altitude, what is the shortest time for a drone to stop from a horizonal velocity of $\backslash (8\backslash$mathrm$\{$~m$\}/\backslash$mathrm$\{$s$\}\backslash )?$

$(10$ marks$)$

$($iv$)$ At time $\backslash ($ t$=0\backslash ),$ a quadrotor hovering in the air without a yaw is shown below, facing north. Then the drone $(1)$ moves $20$ meters up$,(2)200$ meters along the backward left direction, $(3)$ yaw $270$ degrees to face east. Please provide the sequence of motions that the drone needs to perform on adjustment of rotors' speed.

$(10$ marks$)$