AAE 450 Project Bellerophon
A.6.2.2.1 Balloon and Ground Launch
p.g.1
A.6.2.2.1 Balloon and Ground Launch
The steering law is one of the most crucial problems in our project. A slight change in the
steering law affects the ∆Vdrag, ∆Vgravity and the eccentricity of the orbit; therefore, the rocket will
not be able to be in an acceptable orbit without a good sub-optimal steering law even if its
propulsion provides with enough ∆V.
Although we eventually incorporate the spherical Earth model, the aerodynamic drag due to the
atmosphere, and the gravity field as a function of altitude, the starting point of the construction of
the steering law was to consider the flat Earth problem without drag. This simplified problem is a
well-defined two point boundary value problem, which is analytically solvable by forming
Hamiltonian and applying Euler-Lagrange equations, Transversality condition and Weierstrass
condition. The optimal solution obtained is the Linear Tangent Steering Law1:
tan  at  b
(A.6.2.2.1.1)
where  is the steering angle, t is time, and a and b are the coefficients.
The Linear Tangent Steering Law is the optimal steering law for the flat Earth when there is no
atmosphere, that is, what we call “the flat Moon”. As the steering laws that private companies
and the governmental space agencies actually use are not published in public, we decide to apply
the Linear Tangent Steering Law for our ground and balloon launches. We recognize that it is
not the optimal steering law any more when we apply it to the spherical Earth model with
atmosphere; however, we also assume that the difference is small enough to treat the Linear
Tangent Steering Law as a good sub-optimal steering law.
We implement the steering law in our ordinary-differential-equations-solvers and numerically
integrate our equations of motion for each stage. Also, our rocket flies vertically without any
steering for the first ten seconds of the first stage, so we do not need to implement the steering
law for the very first part of the flight.
Figure A.6.2.2.1.1 shows how we measure the steering angles and depict the final steering angles
of each stage, which numerically define our steering law. Figure A.6.2.2.1.2 shows the steering
law versus time when the final steering angles at first, second and third stages are 40°, -20° and
Author: Junichi Kanehara
AAE 450 Project Bellerophon
A.6.2.2.1 Balloon and Ground Launch
p.g.2
-50° respectively. We should note that the initial steering angle is 88° rather than 90° since the
tangent function is undefined at 90°.
Fig. A.6.2.2.1.1: Schematic of the steering angle at the end of each stage.
(Amanda Briden)
Fig. A.6.2.2.1.2: Sample of the plot of steering law versus time.
(Amanda Briden)
Author: Junichi Kanehara
AAE 450 Project Bellerophon
A.6.2.2.1 Balloon and Ground Launch
p.g.3
By changing the final steering angles at each stage degree by degree, we are able to find the suboptimal steering law that makes it possible to attain the theoretical orbit with the eccentricity of
as small as 0.0055. In the section A.6.2.3 Optimization, we will discuss how we actually deal
with the computationally expensive process, which requires running the entire trajectory code
once for each set of the final steering angles, by using a normal PC in the year 2008 rather than
an expensive super computer.
In the process of choosing the launch type, ground launch or balloon launch, we needed to
compare the corresponding ∆Vdrag and ∆Vtotal. We had not had the final structural configurations
yet when we did the analysis on the week 5 of the project, and we did not have a sub-optimized
trajectory for each case either. However, the following results are still valid since the trend never
changes for our launch vehicles regardless of the modifications since the week 5.
Table A.6.2.2.1.1 Delta V Comparison
Payload [kg]
0.2
1.0
5.0
0.2
1.0
5.0
Launch type
Balloon
Balloon
Balloon
Ground
Ground
Ground
∆Vdrag
21
21
20
2,904
2,899
2,875
∆Vtotal
10,027
10,011
9,932
14,033
13,978
13,711
Units
m/s
m/s
m/s
m/s
m/s
m/s
Table A.6.2.2.1.1 shows ∆Vdrag of the ground launch is bigger than that of the balloon launch by
the factor of 150, and ∆Vtotal of the ground launch is significantly bigger since ∆Vdrag is the major
source of ∆Vtotal. Although, the accuracy in the numerical values is not perfect, as we use the
preliminary analysis on the week 5, it is obvious that the balloon launch has a very big advantage
in reducing ∆Vdrag.
Our final configurations of small, medium and big launch vehicle, whose payloads are 0.2 [kg], 1
[kg] and 5 [kg], for the balloon launch need ∆Vdrag of 6 [m/s], 6 [m/s] and 4 [m/s] respectively,
and they need ∆Vtotal of 9,313 [m/s], 9,379 [m/s] and 9,354 [m/s]. This result validates that our
preliminary analysis on the week 5 are numerically close enough to our final results; therefore,
Author: Junichi Kanehara
AAE 450 Project Bellerophon
A.6.2.2.1 Balloon and Ground Launch
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we confidently conclude that the balloon launch is better than the ground launch in terms of
∆Vtotal, which exponentially affects the total cost of our launch vehicles.
References:
1
Longuski, J.M. “AAE 508 Optimization in Aerospace Engineering Lectures,” Purdue
University, West Lafayette, IN, January 2008.
Author: Junichi Kanehara