Question $1$ In order to simplify control design we sometimes model a multirotor unmanned aerial

vehicle $($UAV$)$ in a vertical plane as shown in Figure $1.$ In this case the UAV has two inputs: total

thrust TinR which is a force vector applied at its centre of mass $($CoM$)$ and torque $inR$ which is

applied about the CoM. Note that thrust $T$ is always directed perpendular to the vehicle as shown

in Figure $1.$ We assume TinR and so both positive and negative thrust is possible. The position

of the UAV is denoted $p=(x,z{)}^{T}in{R}^2$ and the pitch angle the vehicle makes with the $x-$axis is

denoted $.$ We are interested in how the input $u=(T,)in{R}^2$ affects $p$ and $$ and this question

evaluates the tfs between $u$ and $p,.$ We asssume the UAV has a mass $m$ and moment of inertia $J$

about its CoM. The UAV flies in a gravitational field and hence is subject to a gravitational force

$mg$ which points in the $z-$direction.

$($i$)$ Draw the free body diagram $($FBD$)$ for the UAV showing all forces and moments. Be sure to

include the positive direction and origin of any coordinates you need.

$($ii$)$ Using your FBD$,$ apply Newton's law to obtain three $2$nd order ODEs for dependent variables

$p$ and $.$ Are these ODEs nonlinear? Explain briefly.

$($iii$)$ Using Part $($ii$)$ find all inputs $u$ which keep the UAV in equilibrium. By "equilibrium" we mean

solutions for $p$ and $$ which are constant. Denote these solutions $p_e,{}_e$ and their corresponding

equilirbrium input $u_e=(T_e,{}_e).$ Hint: you should find two distinct physical solutions for $u_e.$

For the $u_e$ you find, give the corresponding $p_e,{}_e.$

$($iv$)$ Define deviations away from equilibrium: $p=p-p_e,=-{}_e,u=u-u_e.$ Take a linear

approximation of the ODEs obtained in Part $($ii$)$ about $u_e$ and $p_e,{}_e$ to obtain ODEs for $p$

and $.$

That is$,$ suppose your ODE for $x$ is given by $x^{}=f(T,),$ then you can Taylor expand $f$ about

$(T_e,{}_e)$ and ignore higher order than linear terms in $T$ and $$ :

$f(T,)=f(T_e,{}_e)+T\frac{\text{d}\text{e}\text{l}\text{f}}{\text{d}\text{e}\text{l}\text{T}}(T_e,{}_e)+\frac{\text{d}\text{e}\text{l}\text{f}}{\text{d}\text{e}\text{l}}(T_e,{}_e)+$HOT

where "HOT" denotes higher order terms.

$($v$)$ Using Part iv obtain the transfer function from $u$ to $p$ and $.$

$($vi$)$ Using your linearized model, can you suggest a way to control the position $p$ of the UAV.