ROSE-HULMAN INSTITUTE OF TECHNOLOGY
Department of Mechanical Engineering
ME 123
Comp Apps I
Mini-project 3 – Rocket landing
On April 8, 2016 SpaceX made history by successfully landing their first stage rocket on an
ocean platform 300 km west of their Florida launch site, while the second stage rocket delivered
its resupply payload to the International Space Station. This was a significant step in their goal
to reduce the costs of spaceflight by developing reusable first stage rockets.
Here is a video of the successful landing:https://www.youtube.com/watch?v=RPGUQySBikQ
Previous attempts were less successful, undergoing RUD (Rapid Unscheduled Disassembly):
https://www.youtube.com/watch?v=JergEf21HF0 We will attempt a small, simplified version of this task. We will focus on the phase of the flight
when the first stage rocket has completed its reentry burn and the aerodynamic guidance phase,
and is ready for the vertical landing (see Figure 1 below).
Figure 1. Floating platform recovery of the first stage Falcon 9 rocket.
(https://en.wikipedia.org/wiki/Falcon_9_booster_controlled‐
descent_and_landing_tests#/media/File:Falcon_9_First_Stage_Reusability_Graphic.jpg)
ROSE-HULMAN INSTITUTE OF TECHNOLOGY
Department of Mechanical Engineering
ME 123
Comp Apps I
At this point in its trajectory the first stage is about 1 km above the ocean and falling at 250 m/s.
It will reignite its engines in an attempt to land in the correct location on the barge with a
velocity of less than 1 m/s.
In this project, you will simulate the trajectory of a simplified rocket using Euler’s method, then
adjust the rocket thrust profile to match a nominal (good, desired) trajectory that will bring the
rocket in for a safe landing. To keep things simple we will assume that the rocket is already
directly over the barge—we are just trying to get it to go downwards.
This document explains the details of the rocket flight model, how to determine the thrust
profile, and how your trajectory prediction will be tested and graded. The deliverables for the
project are also discussed, and suggested steps are provided to give you some guidance.
1 — Nominal trajectory
We are going to try to get our rocket to follow a nominal trajectory. If we can keep the rocket
following its nominal trajectory then it will land safely.
Here are the things the nominal trajectory is trying to accomplish:
1. At time
0, we have
and
. That is, we want to match our nominal
to the initial altitude and velocity for the rocket. ( is the initial altitude and is the
initial velocity for the rocket. is positive upwards, so the initial velocity will be a
negative number.)
, we have
0 and
0. That is, when we get to the time when
2. At time
we reach the barge surface, we want to be going very slow (zero velocity). ( is the time
when we reach the barge surface.)
One of the easiest ways to create this profile is simply to have the altitude be a cubic function of
time and figure out the four coefficients to meet these four requirements. If we do that, we find
3
2
(1)
(2)
ROSE-HULMAN INSTITUTE OF TECHNOLOGY
Department of Mechanical Engineering
ME 123
Comp Apps I
with
2
2
(3)
3
(4)
(5)
(6)
Note: The altitude and velocity after time are supposed to stay zero. You will want to
pick a time limit
and make the nominal altitude and velocity trajectories zero between
and
. We recommend that you use
20s.
2 — Simulated trajectory
Once we have our nominal trajectory, we would
like to simulate the flight of the rocket and see if we
can create the thrust required to stay on that
nominal path.
The simplified equations of motion for the rocket
are actually quite close to those you used in your
homework for the flight of projectiles, except that
this time we have a rocket thrust. Figure 2 shows
the free-body diagram for the rocket.
Figure 2. Free-body diagram for
simplified rocket.
ROSE-HULMAN INSTITUTE OF TECHNOLOGY
Department of Mechanical Engineering
ME 123
Comp Apps I
The equations of motion we need to simulate are then
dV
d
d
d
(7)
(8)
d
d
(9)
where
1
2
4
(10)
The definitions for the various symbols are as follows:
=
=
=
=
=
=
=
=
=
=
=
=
=
Mass of the empty rocket
Mass of fuel
Rocket velocity
Time
Acceleration due to gravity
Drag proportionality constant
Thrust fraction
Maximum rocket thrust we can produce
Rocket altitude
Fuel consumption rate required for maximum rocket thrust
Air density
Rocket diameter
Coefficient of drag for the rocket
We also have initial conditions for all of our differential equations:
0
(11)
0
(12)
0
(13)
ROSE-HULMAN INSTITUTE OF TECHNOLOGY
Department of Mechanical Engineering
ME 123
Comp Apps I
Tables 1 and 2 give numerical values for selected variables.
Table 1. Numerical values for constants. (These values will not change.)
Constant
Description
Value with units
Mass of the empty rocket
18,000 kg
9.8 m/s2
Acceleration due to gravity
Maximum rocket thrust we can produce
800,000 N
Fuel consumption rate required for maximum rocket thrust
130 kg/s
Air density
1.2 kg/m3
Rocket diameter
3.66 m
Coefficient of drag for the rocket
0.51
Table 2. Typical numerical values for initial conditions.
(These values may be different on test day.)
Initial condition
Description
Value with units
Initial rocket altitude
1000 m
Initial rocket velocity
-250 m/s
Initial fuel mass
2200 kg
3 — Controlling the thrust fraction
At this point, we have a nominal trajectory and we can simulate the trajectory of the rocket for
any given thrust fraction. The thrust fraction can even be different at different times, which is
probably wise if we want a soft landing. The question now is, how do we choose the thrust
fraction over time so that we follow the nominal trajectories? To do this we will implement a
very simple control algorithm. (More about this in your sophomore and junior classes!)
ROSE-HULMAN INSTITUTE OF TECHNOLOGY
Department of Mechanical Engineering
ME 123
Comp Apps I
At every time inside your Euler simulation loop, we will find the error between the nominal
trajectories and the actual trajectories. The bigger the errors are, the more thrust we will apply:
(14)
However, is limited to the range 0.4 to 1, and if you run out of fuel, then
0. (It is actually
true that the rocket can only run in the range from 40% to 100% thrust.)
and
are constants
that you will need to pick to obtain good performance. ( has units of m and
has units of
s/m.)
Notice that the sign on the errors does matter, and for this to work properly you need to get the
signs correct in Equation (14).
Your goal for this project is to determine a set of values for , ,
and
so that your landing velocity is between 0 and -1 m/s. There
are many sets of values that will accomplish this goal, so the
“optimal” solution is the one that also minimizes thrust variability.
There are several constraints on the landing time and the controller constants that will affect your
analysis:
1. The desired landing time
cannot exceed the time limit
: 0
.
2. The controller constant
is positive and has an upper limit: 0
20m .
3. The controller constant
is positive and has an upper limit: 0
10s/m.
ROSE-HULMAN INSTITUTE OF TECHNOLOGY
Department of Mechanical Engineering
ME 123
Comp Apps I
4 — Thrust variability
Even if we land safely, for some of the choices of the controller constants the thrust fraction
over time is smooth and for others it varies wildly, as shown in Figure 3 below.
Thrust Fraction Over Time
Thrust Fraction Over Time
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
0
2
4
6
8
10
Time (s)
(a)
12
14
16
18
20
0
2
4
6
8
10
12
14
16
18
20
Time (s)
(b)
Figure 3. Thrust fraction over time for two successful landings.
The thrust profile in Figure 3b is going to be very tough on the rocket engine, while the thrust
profile in Figure 3a is going to be much better (but still may not be the best you can do). We
would like to quantify this intuition with a measure we will call thrust variability.
The thrust variability will be the rms (root mean square) value of the rate of change of the thrust
over time. If you store the thrust fraction ( ) in a vector f that has a time step size dt, then we
can write the thrust variability in MATLAB in just a single line:
thrust_variability = rms(diff(f)/dt);
The code diff(f)/dt generates a vector which is a very simple numerical derivative of f
with respect to time, and the rms function finds the root mean square value of that vector. You
can find out more about these functions from the MATLAB documentation (e.g., doc diff and
doc rms).
Notice that thrust_variability is a scalar. The thrust variability for the profile in Figure
3a is 1.82, while the thrust variability for the profile in Figure 3b is 4.48.
5 — The test exercise
ROSE-HULMAN INSTITUTE OF TECHNOLOGY
Department of Mechanical Engineering
ME 123
Comp Apps I
During class on Thursday of Week 10 (May 19, 2016), we will hold an exercise to test the
results of your program. The test exercise procedure is as follows:
1. A prediction card will be passed out to each team.
2. Your instructor will provide you with the rocket’s test-day , , and
. Teams will
have 20 minutes to complete Step 3. After that, no more submissions will be accepted.
3. Teams will use their MATLAB code to determine the values of , , and
that yield
the best trajectory, as well as the corresponding thrust variability. Teams will record
these results on their prediction card. Program results must be left on the teams’
computer screens.
4. Your instructor will verify that the reported values are the ones appearing on the teams’
computer screens.
5. The , , and
values predicted by each team will be tested by your instructor in a
“virtual experiment” (i.e., a computer simulation).
6 — Submitting your MATLAB code
By 5 pm on Friday of Week 10 (May 20, 2016), upload your team’s well-commented MATLAB
code (main routine and functions) as m-files (.m) to Moodle. Upload only ONE submission per
team. If your code failed to work at the test exercise, fix the code before you upload it to
Moodle.
7 — Project grading
Your grade for this project will be computed with the following weights:
40 pts
Test exercise performance
60 pts
MATLAB code (well-documented and easy to follow)
100 pts
Your test exercise prediction will be scored based on the actual value of the landing velocity and
the thrust variability: the closer you are to our (the ME 123 instructors’) minimum value of thrust
ROSE-HULMAN INSTITUTE OF TECHNOLOGY
Department of Mechanical Engineering
ME 123
Comp Apps I
variability, the higher your score. Your test exercise prediction score will be determined as
follows:
No submission , , and land the rocket with
0
1 m/s on your computer, but is out of range when we run it. , , and land the rocket with
0
1 m/s on our simulation. Score depends on the thrust variability following the rubric shown below. 0 pts 12 pts 24‐40 pts 100%
percent more variable
–
24
for
50%
rockettestperformancescore
40
0.32
50%
for0%
Rocket Test Perfomance Score (pts)
40
38
36
34
32
30
28
26
24
0
10
20
30
40
50
60
70
Percent More Variable (p)
80
90
100
Figure 4. Illustration of the scoring function for your test exercise prediction.
8 — Suggested steps
ROSE-HULMAN INSTITUTE OF TECHNOLOGY
Department of Mechanical Engineering
ME 123
Comp Apps I
This is a big project, so we suggest the following steps for you to use in breaking the project
down into more manageable pieces. You are not required to follow these steps, but you may find
them helpful. If you do take our advice, you are free to work faster than what is suggested, but
to keep from falling too far behind, you should try to keep up with this schedule.
Day 1 (Friday):
1. Create a function to generate
and
as a function of time for a given set of , ,
and values. Plot those curves and verify that they seem to behave the way you think
they should. You may want to label them nicely because they may be useful to you later
in assessing whether you have matched the nominal trajectories.
2. Create another function to generate the simulated rocket trajectory for given initial
conditions using Euler’s method. This is very similar to what you have done on your
homework. We suggest that you test the function using some constant thrust fractions
between 0.4 and 1 and plot the simulated trajectories on the same axes as the nominal
trajectories.
Day 2 (Monday):
1. Add the controller to the simulated rocket trajectory function. Test the controller with a
and
values. (For the suggested conditions in Table
few different combinations of
2, you should get a decent landing with
12s,
1m , and
4.3s/m.)
2. Now add in the calculation of the thrust variability. Verify that your thrust profile for the
case given for a decent landing looks like Figure 3a, and that the thrust variability is 1.82.
Day 3 (Tuesday):
You may have noticed that, when the time step for Euler’s method is small for accuracy, it takes
a little while to compute even one trajectory. It could take a very long time to run through 100
different combinations of launch conditions, never mind 1000 or so. Actually, there are far more
than 1000 possible combinations for this project, so we will definitely want to speed things up!
We can do this by pre-allocating the solution vectors.
In your code, it would not be surprising if you had something like the following structure, which
is code for a ball dropping under the influence of gravity:
ROSE-HULMAN INSTITUTE OF TECHNOLOGY
Department of Mechanical Engineering
ME 123
Comp Apps I
clc
clear variables
g = 9.8;
dt = 0.0001;
maxsteps = 1e7;
v(1) = 0;
x(1) = 0;
for i = 1:maxsteps
v(i+1) = v(i) - g*dt;
x(i+1) = x(i) + v(i)*dt;
end
When we start the for loop, the vectors v and x are only one element long, and each time we go
through the loop the vectors get longer by one element. The process of making the array longer
is very time consuming in MATLAB, especially when the vectors are large. This little program
takes about 5 seconds to run on my machine (yours may be faster or slower). If we could tell
MATLAB how big the vectors were going to get, it could make them that big to begin with and
the for loop would run much more quickly.
Now consider the code shown below:
clc
clear variables
g = 9.8;
dt = 0.0001;
maxsteps = 1e7;
v = zeros(1,maxsteps+1);
%
%
Makes a vector v with (maxsteps + 1)
zero elements
x = zeros(1,maxsteps+1);
%
%
Makes a vector x with (maxsteps + 1)
zero elements
v(1) = 0;
x(1) = 0;
% Not actually necessary now because v(1) = 0 already
% Not actually necessary now because x(1) = 0 already
for i = 1:maxsteps
v(i+1) = v(i) - g*dt;
x(i+1) = x(i) + v(i)*dt;
end
The two new lines are the “pre-allocation” of the vectors v and x; we set them up and then
execute the for loop. Running this now takes about 0.2 seconds! We strongly encourage you
to consider pre-allocating vectors for your trajectory code.
ROSE-HULMAN INSTITUTE OF TECHNOLOGY
Department of Mechanical Engineering
ME 123
Comp Apps I
The only drawback to this method is that you must estimate the length of the vectors you will
need. If you guess too small, your program will slow down again; if you guess too large, you
will get an “Out of Memory” error – there is not enough memory for you to use! Recall that the
time it takes the rocket to land is at most
; use that knowledge and the time step you have
selected to estimate how long the vectors need to be.
Once you have implemented vector pre-allocation in your program, think of a way to determine
the optimal combination of , , and . Of course, you may use a guess-and-check approach,
but if you could automate some or all of this process, then your search will likely be much faster.
This could be useful on the day of the test exercise when you have only 20 minutes to determine
the optimal set of launch conditions. Remember, there are many possible solutions – let
MATLAB help you find the optimal combination!