Politecnico di Milano
Prova finale: Introduzione all’analisi di missioni spaziali
AA 2022-2023
Docente:
Colagrossi
Elaborato n.
A7
Autori:
Cod. Persona
10787552
10742312
10725944
Cognome
Dowell
Cerletti
Campi
Nome
Matteo
Elena
Anna
Data di consegna: 15/12/2022
Dowell, Cerletti, Campi
Table of contents
1.
Introduction
2.
Initial orbit characterization
2.1 Determining Keplerian parameters from position and velocity vectors
2.2 Characteristics of initial orbit
2.3 Plot 1: initial orbit
3.
Final orbit characterization
3.1 Determining position and velocity vectors from orbital elements
3.2 Characteristics of the final orbit
3.3 Plot 2: final orbit
4.
Transfer trajectory definition and analysis
4.1 Standard Trajectory
4.1.1 Description of the strategy
4.1.2 Comparison with alternatives
4.1.3 Plot 3: standard strategy and Plot 4: inverse standard 1
4.2 Alternative 1
4.2.1 Description of the strategy
4.2.2 Reasoning for the proposed strategy
4.2.3 Plot 5: first alternative trajectory
4.3 Alternative 2
4.3.1 Description of the strategy
4.3.2 Reasoning for the proposed strategy
4.3.3 Plot 6: second alternative strategy
5.
Conclusions
6.
Appendix
2
Dowell, Cerletti, Campi
1.
Introduction
The purpose of the presented report is that of creating and comparing different ways which may be adopted in
order to go from one point of an orbit to another of a different orbit. The proposed strategies were made with
the aim of minimizing the time interval necessary to arrive to the given point and/or to reduce the amount of
propellant and, thus, the cost needed to arrive to the given destination.
We started the study by analyzing the “standard” trajectory which can be achieved with a series of simple
maneuvers: an initial change in the plane of the starting orbit followed by a variation in the argument of the
pericenter and a final change in the shape of the orbit created with the use a bitangent maneuver. We then
examined a couple of other trajectory strategies, all of them consisting in a change of plane, shape and
pericenter anomaly, but performed in different chronological order. Lastly, we produced some less
conventional alternatives which appeared to be more convenient (in terms of propellant use or time taken)
compared to the other initial proposals.
In the whole work, all the maneuvers are considered to be impulsive. This implies that their cost is directly
associated to the ∆v (delta-v), or difference in velocity, of the spacecraft. With the use of different charts,
graphs and MATLAB functions, the data associated with the various strategies were compared and contrasted
to determine the most convenient solution in terms of ∆v and in terms of time taken to undertake the transition.
It must be noted that the proposed strategies are based on the assumptions and derivations of the Two-Body
problem, thus they represent an approximation of the actual motion for two bodies subject to the same
gravitational attraction. Though this model does not comprehend all of the forces playing a role in our analysis
sub-system, it still provides a good estimation of the motion of the bodies as the other effects are some orders
of magnitude lower. In addition, we will consider Kepler’s Problem or the Restricted Two–Body Problem,
which claims that, in a system with two bodies, if one of the two has a mass which is much lower than that of
the other body, then it does not affect its motion. Consequently, the heavier body can be considered at rest.
It must be noted that the above-mentioned hypothesis and assumptions prevent us from analyzing orbits with
distances from the Earth greater than 80000 km.
3
Dowell, Cerletti, Campi
2.
Initial orbit characterisation
MATLAB ® script: dati.m
2.1
To characterize the initial point and its orbit, we were given its ECI coordinates. With the appropriate
computations and using the MATLAB function rv2parorb.m, we were able to obtain its Keplerian parameters.
rri = [-7755.5213, -1168.7252, 2513.6369] km
vvi = [0.4443, -7.0170, -1.1680] km/s
𝑎𝑖 = 8666,3209 km
𝑒𝑖 = 0.0585
𝑖𝑖 = 0.3587 rad
ωi = 1.4983 rad
Ωi = 1.1749 rad
θi = 0.5892 rad
2.2
Calculating its radius of apogee, radius of perigee and
semimajor axis, it is observed that the initial orbit can be
classified as a MEO (Medium Earth Orbit, 8,000-20,000
km). In addition, the eccentricity of the orbit is very low,
meaning that it has an almost-circular shape: this translates
to less propellant required for changes in the anomaly of
pericenter of the orbit.
2.3
pi = ai ∙ (1 − ei 2 ) = 8636.6546 km
pi
rp,i =
= 8159.2736 km
(1+ ei )
pi
ra,i =
= 9173.3681 km
(1− ei )
Plot 1: Initial orbit
3.
Final orbit characterisation
MATLAB ® script: dati.m
3.1
For the final orbit on which the given destination point is found, we were given the Keplerian parameters
necessary to describe it. Thanks to these and using the MATLAB ® function parorb2rv.m we were also able to
identify the ECI coordinates of the final point.
𝑎𝑓 = 13330 km
𝑒𝑓 = 0.3315
𝑖𝑓 = 0.8162 rad
ωf = 0.4432 rad
Ωf = 1.9340 rad
θf = 1.2450 rad
rrf = [-6374.8989, -3767.0622, 7761.3311] km
vvf = [1.6610, -6.4080, 0.7700] km/s
3.2
Due to its radius of apogee and perigee and due to its semimajor
axis, the orbit can be classified as a MEO (Medium Earth Orbit).
Compared to the initial orbit, the value of the semi-latus rectum
increases (pf ≅ 1.37 ∙ pi) as well as that of the eccentricity, which
translates to more cost convenient changes in orbital inclination
close to the apocenter of the orbit.
3.3
pf = af ∙ (1 − ef 2 ) = 11865.1363 km
p
rp,f = (1+fe ) = 8911.105 km
f
p
ra,f = (1−fe ) = 17748.8950 km
f
4
Plot 2: Final orbit
Dowell, Cerletti, Campi
4.1 Standard and inverse standard trajectories
MATLAB ® script: standard_aapp.m , inverse1_aapp.m
4.1.1
As stated previously, the Standard orbital strategy consists in an initial change in the orbital plane, followed
by a change in pericenter anomaly and, lastly, a bitangent change in shape.
To reduce the time taken to arrive at the final point, the most convenient of the available solutions is one of
the possible variations of the Standard strategy, constituted by:
1. An initial change in orbital plane performed in the true anomaly θB = 2.7445 rad
(or θA = 5.8861 rad)
2. A change in the pericenter anomaly in θ = 1.3718 rad to bring ourselves on an orbit with ω= ω f+π
3. A periapsis-periapsis bitangent maneuver to bring ourselves on the final orbit
On the other hand, in terms of cost convenience, the strategy that seems the most advantageous is a variation
of the Standard obtained by performing the three maneuvers in a different order:
1. A periapsis-apoapsis bitangent shape variation
2. A change in orbital plane performed in θ = 2.7445 rad
3. A change in the anomaly of the pericenter performed in either one of the available points
(θa = 2.9426 rad, θb = 6.0842 rad)
Throughout the whole report, the true anomalies relative to the changes in periapsis argument are always
calculated for the orbit before the maneuver.
4.1.2
While analyzing the Standard strategy we also came up with and examined two others similar to the initial one
but with the maneuvers performed in different orders: the first of these (“Inverse Standard 1”) is constituted
by an initial bitangent change in shape, a change in plane and, lastly, in the pericenter argument; the remaining
one (“Inverse Standard 2”), on the other hand, is obtained by a change in the plane of the initial orbit,
followed by a bitangent change in its shape and a variation in the pericenter argument.
Utilizing the standard strategy and the “inverted standards”, we came up with the data shown in the following
tables.
Of the three strategies and of all the combinations associated with these, the two presented in paragraph 4.1.1
appear particularly advantageous in terms of propellant used and of time taken to arrive at the final point.
4.1.3
Plot 4: Inverse standard 1, change in pericenter anomaly in θa
Plot 3: Standard strategy, change plan in θB
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Dowell, Cerletti, Campi
LEGEND:
θA, θB = possible anomalies for plane change [rad]
θa, θb = possible anomalies for change in pericenter anomaly
ap = bitangent apoapsis-periapsis maneuver
pa= bitangent periapsis-apoapsis maneuver
aa= bitangent apoapsis-apoapsis maneuver
pp= bitangent apoapsis-periapsis maneuver
MATLAB ® script: standard_appa.m , standard_aapp.m
Tables 1-2: costs of Standard strategy
θA = 5.8861
STANDARD
dv [km/s]
dt [s]
θa=2.9426
ap
5,498346
24.748,53
pa
5,473543
31.450,21
θB=2.7445
θb=6.0842
ap
pa
5,498346
5,473543
16.837,93 31.568,65
θA = 5.8861
STANDARD
dv [km/s]
dt [s]
θa=1.3718
aa
pp
6,202189
6,219241
31.656,44 16.136,00
dv [km/s]
dt [s]
INVERSE1
dv [km/s]
dt [s]
dv [km/s]
dt [s]
θa=1.3718
aa
pp
5,770900
5,787952
31.656,44 16.136,00
θb=4.5134
aa
pp
5,770900
5,787952
24.213,44 16.722,03
aa
pp
θA = 5.8861
θB=2.7445
θA = 5.8861
θB=2.7445
θa=2.9426 θb=6.0842 θa=2.9426 θb=6.0842 θa=2.9426 θb=6.0842 θa=2.9426 θb=6.0842
6,422579
6,422579
4,337730
4,33773
6,439631
6,439631
4,354781
4,354781
33916,96
19950,88
33916,96
35267,27
41741,94
27775,86 26.425,55
27775,86
MATLAB ® script: inverse2_appa.m ,inverse2_aapp.m
θB=2.7445
ap
pa
ap
pa
θa=2.9426 θb=6.0842 θa=2.9426 θb=6.0842 θa=2.9426 θb=6.0842 θa=2.9426 θb=6.0842
6,100893
6,100893
6,076089
6,076089
5,669604
5,669604
5,644800
5,644800
30794,85
32145,17
37496,54
23530,46
22765,82
24116,13
37496,54
23530,46
θA = 5.8861
INVERSE2
pa
5,042253
23.421,18
MATLAB ® script: inverse1_appa.m ,inverse1_aapp.m
θA = 5.8861
dv [km/s]
dt [s]
ap
5,067057
16.719,49
θb=6.0842
ap
pa
5,067057
5,042253
16.837,93 31.568,65
ap
pa
θA = 5.8861
θB=2.7445
θA = 5.8861
θB=2.7445
θa=2.9426 θb=6.0842 θa=2.9426 θb=6.0842 θa=2.9426 θb=6.0842 θa=2.9426 θb=6.0842
6,340837
6,340837
4,255987
4,255987
6,316033
6,316033
4,231183
4,231183
38082,21
24116,13
22765,82
24116,13
37496,54
2353,046 37.496,54
38846,85
Tables 5-6: costs of Inverse standard 2 strategy
INVERSE2
θa=2.9426
θB=2.7445
θb=4.5134
aa
pp
6,202189
6,219241
32.242,47 24.751,07
Tables 3-4: costs of Inverse standard 1 strategy
INVERSE1
[rad]
θB=2.7445
pp
aa
pp
θa=2.9426 θb=6.0842 θa=2.9426 θb=6.0842 θa=2.9426 θb=6.0842 θa=2.9426 θb=6.0842
6,182635
6,182635
6,199687
6,199687
5,751346
5,751346
5,768398
5,768398
41945,99
27979,92
26425,55
27775,58
33916,96
19950,88
26425,55
27775,58
aa
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Dowell, Cerletti, Campi
4.2 First alternative trajectory
MATLAB ® script: risparmiot.m
4.2.1
The first alternative strategy that we adopted was created with the main objective of reducing the time
interval (∆t) necessary to arrive at the final point. To achieve this goal, three main maneuvers are undertaken:
1. A change in orbital plane made in θ = 2.7445 rad (the less time-consuming to reach of the two
possible points) to achieve if , Ωf
2. A change in the shape of the orbit and in the argument of the pericenter in θ = 3.2705 rad which
brings on a transfer orbit with its apogee in the coordinates of the final point and perigee on the initial
orbit
3. A second change in the shape and in the perigee argument of the transfer orbit in θ = 1.1934 rad to
bring ourselves on the final orbit
4.2.2
The expounded strategy was formulated after analyzing the standard strategy and its two inverse variations.
Upon realizing that the most convenient trajectories in terms of time taken were combinations of the standard
procedure, and not of its inverse versions, we attempted to create a similar alternative which would allow the
∆t to be further reduced.
Firstly, we performed a change in orbital plane in the closest point to the initial true anomaly of the starting
orbit.
Next, we attempted to find a more direct way compared to the standard strategy in order to arrive at the final
point, ultimately resorting for a transfer orbit with:
• pericenter on orbit 2 (initial orbit after change in orbital plane)
• apocenter in final point (final true anomaly of final orbit)
A series of geometric observations shown in Figure1 had to be undertaken in order to determine the transfer
orbit and its points of intersection with the final one, especially the point which did not coincide with the final
one.
The time required to complete the trajectory is 8555.98 s, with a ∆v of 6.1909 km/s.
4.2.3
ωt=ω2+θmm,i(1)
θm,f=θf+π+θm,t
θmm,i(1)= θpt2= θf+∆ω+π
θmm,i(2)= θpt2+ θmm,t(2)
Figure 1: Geometric observations
Plot 5: First alternative trajectory
7
Dowell, Cerletti, Campi
4.3 Second alternative trajectory
MATLAB ® script: risparmiov.m
4.3.1
The second alternative trajectory we created was made to minimize the amount of propellant used and, thus,
the ∆v necessary to reach the final point. Four main maneuvers were undertaken in the following order:
1. A change in the anomaly of the pericenter in θ = 2.9426 rad of the initial orbit
2. An initial tangent change in shape to create a transfer orbit with an apogee radius equal to the one of
the final orbit and a perigee radius equal to the one of the initial orbit
3. A change in orbital plane in θ = 3.1425 rad of the transfer orbit
4. A second tangent change in shape at the apogee of the orbit to arrive on the final orbit
4.3.2
As for the previous trajectory that we studied, we came up with this strategy by observing the data of the
standard and its two inverse strategies. Upon realizing that the most convenient costs were obtained with
certain combinations of the Inverse Standard 1, inspired by these we attempted to create a new procedure to
further minimize the quantity of propellant used.
Firstly, we realized that it would have been better to perform the change in the argument of pericenter
before any changes in the size of the orbit so as to not increase costs too much: this is due to the fact that the
initial orbit has a relatively low eccentricity factor (e < 0.1), meaning that the ∆v required for the maneuver
would be relatively low compared to if it were made after a change in shape and eccentricity of the orbit.
However, we also had to keep in mind that after the change in pericenter anomaly, a further variation in the
perigee argument would have occurred due to the change in orbital plane. For this reason, computations were
made in order to be sure that after the change in plane, the obtained argument of perigee would be equal to
the argument of pericenter calculated for the final orbit. It must be noted that the change in orbital plane
was made in the most distant to the Earth of the two available points to minimize the amount of propellant
used for the maneuver. The two changes in shape proved to be quite expensive but, as the ∆v necessary for the
change in orbital combined with the cost for the change in pericenter anomaly was a lot lower compared to the
Inverse Standard 1, the alternative trajectory appears to be more convenient in terms of propellent use.
For the trajectory to be completed, 3.4738 km/s of ∆v are required. The final point is reached in 38094.26s.
4.3.3
Plot 6: Second alternative trajectory
8
Dowell, Cerletti, Campi
5.
Conclusions
As part of the assignment, we proposed four main trajectories, each of them usable and useful depending on
the requirements and resources available for a determined mission.
Even if the Standard Strategy is less propellant-consuming compared to the Alternative Strategy 1, the
Alternative Trajectory requires a shorter interval of time to complete as it arrives at the final point in a
more direct manner.
For the Alternative Strategy 1, a 6.96% increase in ∆v can be seen compared to the standard and a 46,98%
decrease in time taken is noticed. While in the Standard Trajectory the amount of time necessary to arrive at
the destination is increased by the relatively long intervals of time for which the spacecraft has to stay on the
created transfer orbits (~11721.4s on the transfer orbit), in the First Alternative the less time is taken on the
transfer orbits, thus the trajectory can be considered more direct.
The Inverse Strategy 1, on the other hand, may seem quite useful in terms of required ∆v but, ultimately, is
outdone in cost convenience by the second Alternative Trajectory. Compared to the Standard Strategy, the
Inverse one requires a lower cost (-26.9%) but a longer interval of time to complete (+132,38%).
The Alternative Trajectory 2 seems the most convenient in terms of ∆v, with a 39.98% decrease in costs from
the Standard strategy, but a 136.08% increase in time taken. The Second Alternative is about 17.9% more
cost convenient than the presented Inverse Strategy: this is primarily due to the fact that, as stated earlier,
the Alternative Strategy performs part of its change in pericenter anomaly directly on the initial orbit, which
has an almost-circular shape and, thus, requires a lower ∆v to complete the maneuver.
In conclusion, the most useful of the proposed trajectories in terms of pure cost convenience is the Alternative
Trajectory 2, while the best in terms of time taken is the Alternative Trajectory 1.
Depending on the available resources and aims, any one of the presented strategies may be suited for a
particular mission depending on its main objectives. Between the two Alternative Strategies, however, the first
one may be considered more advantageous in general terms, as the time taken to arrive at the final point is
almost halved compared to the Standard Trajectory, while the required ∆v is increased by about 7%. The
second Alternative, on the other hand, would reduce propellant use by 39.98% but increase the time taken to
arrive at the destination of the spacecraft by more than 136%.
Figure 2: percentage change from Standard
9
Dowell, Cerletti, Campi
6.
Appendix
Transfer 1 (standard strategy)
t (s)
a (km)
e (-)
0
8666.3
0.0585
8666.3
0.0585
2774.0
8666.3
0.0585
8666.3
0.0585
8963.9
8666.3
0.0585
8666.3
0.0585
10571.65
8535.2
0.0440
8535.2
0.0440
14495.40
13330
0.3315
16136
13330
0.3315
i (rad)
0.3587
0.3587
0.8162
0.8162
0.8162
0.8162
0.8162
0.8162
0.8162
0.8162
Ω (rad)
1.1749
1.1749
1.9340
1.9340
1.9340
1.9340
1.9340
1.9340
1.9340
1.9340
ω (rad)
1.4983
1.4983
0.8412
0.8412
3.5848
3.5848
3.5848
3.5848
0.4432
0.4432
θ (rad)
0.5892
2.7445
2.7445
1.3718
4.9114
0
0
π
0
1.2450
Δv (km/s)
-
Inverse Standard 1
t (s)
a (km)
0
8666.3
8666.3
7356.22
12954.1
12954.1
14692.76
13330
13330
28235.57
13330
13330
29103.45
13330
37496.55 13330
e (-)
0.0585
0.0585
0.3701
0.3701
0.3315
0.3315
0.3315
0.3315
0.3315
0.3315
i (rad)
0.3587
0.3587
0.3587
0.3587
0.3587
0.3587
0.8162
0.8162
0.8162
0.8162
Ω (rad)
1.1749
1.1749
1.1749
1.1749
1.1749
1.1749
1.9340
1.9340
1.9340
1.9340
ω (rad)
1.4983
1.4983
1.4983
1.4983
1.4983
1.4983
0.8412
0.8412
0.4432
0.4432
θ (rad)
0.5892
0
0
π
π
2.7445
2.7445
2.9426
3.3406
1.2450
Δv (km/s)
-
First alternative strategy
t (s)
a (km)
0
8666.3
8666.3
2774.0
8666.3
8666.3
3526.5
9855.9
9855.9
6069.07
13330
8555.98
13330
e (-)
0.0585
0.0585
0.0585
0.0585
0.0884
0.0884
0.3315
0.3315
i (rad)
0.3587
0.3587
0.8162
0.8162
0.8162
0.8162
0.8162
0.8162
Ω (rad)
1.1749
1.1749
1.9340
1.9340
1.9340
1.9340
1.9340
1.9340
ω (rad)
1.4983
1.4983
0.8412
0.8412
4.8298
4.8298
0.4432
0.4432
θ (rad)
0.5892
2.7445
2.7445
3.2705
5.5651
1.1934
5.580
1.2450
Δv (km/s)
-
Second alternative strategy
t (s)
a (km)
e (-)
0
8666.3
0.0585
8666.3
0.0585
3056.5
8666.3
0.0585
8666.3
0.0585
6785.8
12954.1
0.3701
12954.1
0.3701
14126.7
12954.1
0.3701
12954.1
0.3701
28795.5
13330
0.3315
38094.3
13330
0.3315
i (rad)
0.3587
0.3587
0.3587
0.3587
0.3587
0.3587
0.8162
0.8162
0.8162
0.8162
Ω (rad)
1.1749
1.1749
1.1749
1.1749
1.1749
1.1749
1.9340
1.9340
1.9340
1.9340
ω (rad)
1.4983
1.4983
1.1003
1.1003
1.1003
1.1003
0.4432
0.4432
0.4432
0.4432
θ (rad)
0.5892
2.9426
3.3406
0
0
3.1425
3.1425
π
π
1.2450
Δv (km/s)
-
10
3.7811
0.7793
0.0493
1.1783
-
0.9903
0.1136
2.3675
0.7597
-
3.7811
0.5202
1.8896
-
0.1572
0.9903
2.2127
0.1136
-