$2\mathrm{.}1$ Autonomous Landing of Rocket Stage

The autonomous landing of reusable rocket stages became a trending topic around the world in

the last decade after the successful flight missions of corporations like SpaceX, Blue Origin etc.

and started the new era of low$-$cost and commercial space missions worldwide.

Figure $1$: Reusable rocket stage and landing site.

In this simplified model, the equations of motion related to stage approaching to a landing

site is given. The rocket stage is assumed to have negligible aerodynamic effects and controlled

by thrust vectoring. Then, the equations become

$u^{}=-qw+\frac{Tcos}{m}-$gsin$,$

$w^{}=qu-\frac{Tsin}{m}+$gcos$,$

${}^{}=q,$

$q^{}=-\frac{x_{cg}Tsin}{I_{yy}},$

$m^{}=-\frac{T}{g_0{I}_{sp}}\mathrm{.}2\mathrm{.}1$ Autonomous Landing of Rocket Stage

The autonomous landing of reusable rocket stages became a trending topic around the world in

the last decade after the successful flight missions of corporations like SpaceX, Blue Origin etc.

and started the new era of low$-$cost and commercial space missions worldwide.

In this simplified model, the equations of motion related to stage approaching to a landing

site is given. The rocket stage is assumed to have negligible aerodynamic effects and controlled

by thrust vectoring. Then, the equations become

u$=$qw $+($T cos $\backslash$sigma $/$m$)$ g sin $\backslash$theta $,$

w$=$ qu $($T sin $\backslash$sigma $/$m$)+$ g cos $\backslash$theta $,$

$\backslash$theta $=$ q$,$

q$=($Xcg $*$ T sin $\backslash$sigma $)/$ Iyy

m$=$T $/$ g$0$Isp

$.$

Here, $u,w$ are the velocity components of rocket stage in body$-$fixed frame, $$ is the angular

displacement, $q$ is the angular rate and $m$ is the mass of stage. The motion of rocket stage is

controlled via thrust magnitude $T$ and thrust vector angle $.$ Note that the position of rocket

stage can be expressed by the integration of equations

$x^{}=$ucos$+$wsin$,$

$z^{}=-$usin$+$wcos$.$

The constant parameters related to problem is given as follows.

Table $1$: Problem specific parameters.

In the end, you have $5$ states $x=(u,w,,q,m)$ and $2$ inputs $u=(T,),$ and assume that

you can measure all the states.

a$)(40$ pts$.)$ Linearize the system around the conditions $x_0=(0,0,0,0,m_0)$ and $u_0=(m_{0}g,0)$

with given values, and obtain the linear state$-$space representation, that is$,$

$x^{}=Ax+Bu,$

$y=Cx.$

b$)(40$ pts$.)$ Obtain transfer functions $G_1(s)=u\frac{s}{T}(s)$ and $G_2(s)=w\frac{s}{}(s)$ using

formula

$G_i(s)=C_i(sI-A{)}^{-1}B,$ for $i=1,2,$

where $A,B,C_1,C_2$ are the corresponding state$-$space matrices you found at part $($a$).$

c$)(20$ pts$.)$ Draw the root locus $($either manually or by using Matlab rlocus function$)$

clearly, and comment on what types of controllers $($PD$,$ PI$,$ lead$-$lag etc.$)$ can be designed

for arbitrarily given complex conjugate pole locations.

d$)($Bonus$,40$ pts$.)$ Simulate the system by modeling the equations $(1)$ in Simulink. Run the

Simulink model with fixed step$-$size $t=0\mathrm{.}01($select ODE solver as Heun from Simulink

settings$)$ and draw the $(x,z)-$trajectory of rocket by integrating the inertial velocities $x^{},z^{}$

in $(2).$ Use initial conditions $(u_0,w_0,{}_0,q_0,m_0)=(-20,5,\frac{}{18},0,55000)$ for states and

$(x_0,z_0)=(30,200)$ for rocket position.

e$)($Bonus$,20$ pts$.)$ Implement a stabilizing controller of type in part $($c$)$ with suitable gains

$($no need to be found analytically, just trial$-$and$-$error$)$ and see the results.