Shing Chi Chan
December 2009 – February 2010
Laboratory Work
Hypersonic Re-entry Vehicle Team
In this project, our main objective is to contribute to the guidance of future hypersonic reentry vehicle. Our approach is first to understand the mechanics and control of the space vehicle
re-entry via modeling and simulation in Matlab. Our team is currently using the NASA
developed Orion Crew Exploration Vehicle or Orion CEV as a studying model. While the Orion
CEW employs a lot similarity of the Apollo vehicle, it also complies with the four basic forces as
any airplane.
Figure 1: Four basic forces being used in airplane: lift,
drag, weight, and thrust.
Figure 2: The Orion Crew Exploration Vehicle developed
by NASA
Equations for Lift and Drag:
L=
1 2
ρv SCL
2
D=
1 2
ρv SCD
2
Where L is lift force, D is drag force, ρ is the air density, v is the velocity, S is the reference area,
CL is the coefficient of lift, and CD is the coefficient of drag
While studying the mechanics of the space vehicle, we have the following equations of
motions to help us in understanding the dynamics of the re-entry vehicle.
1 cos  sin 1
r cos  D
1
1
 '   cos  sin
r
D
1
r '   sin 
D
1
1
1
L
 1
 ' 2
 sin    cos  cos tan   2 (tan  sin cos   sin  )
V cos   D
D
VD
 r
'
 '
1
V2
L

V 2  cos 
cos


g




r  D

D

1
  2 cos cos 
VD

Where θ is the longitude, ϕ is the latitude, r is the radius from the vehicle to the center of
the Earth, ψ is the heading angle, γ is the flight path angle, ω is the rotational speed of the Earth,
D is the drag force, and V is velocity of the vehicle, and α is the bank angle.
In order to be able to return from the moon at any time during the day, the vehicle needs
to have a large down range capability. One quick way to gain control of such ability is by
changing the bank angle. With bank angle alternation, the range of trajectory and performance of
the vehicle is significantly improved. In one of our recent simulation, it shows that the bank
angle α has an impact of the vehicle in terms of the skipping behavior. In fact, a skip is required
to achieve greater distances re-entry trajectory.
uL =
Lv Lcosα yields
CD
=
→
α = acos (uL )
D
D
CL
In our simulation conditions, we calculated the initial and final energy which correspond
to the altitude of 120km to 20km by the Energy Law,
1
μ
E = V2 −
2
r
120
110
100
Altitude, km
90
80
70
60
50
40
30
20
0
500
1000
1500
Downrange, km
2000
2500
3000
Figure 3: A zero degree bank angle induces a “skip”
behavior
120
110
100
Altitude, km
90
80
70
60
50
40
30
20
0
100
200
300
400
500
600
Downrange, km
700
800
900
Figure 4: A bank angle of 60 degree is enforced and
reduces the “skip” behavior
1000
In our mean of control study, the drag is chosen to be the trajectory control variable due
to its robustness.
ṡ = Vcos(γ)
ṙ = Vsin(γ)
Where ṡ is the horizontal velocity, 𝑉 is the velocity of the vehicle, γ is the flight path angle, and
ṙ is the vertical velocity.
s = ∫ Vcos(γ)dt
1
s = ∫ − dE
D
Where 𝑠 = horizontal distance, and 𝐷 is the drag force, and 𝐸 is the total energy.
Recently, a new step is added to the current study of the re-entry trajectory study. A
planning function is added to our current study as a reference trajectory which covers the desired
range without violating the path constraints.
Figure 5: The path constraints in terms of drag
The planning function also has the ability to recalculate the range and corresponding drag
profile in flight based on changes in atmospheric conditions. As a result of the planning function,
we have created an asymptotically stable tracking control.
250
Actual Drag
Reference Drag
Drag m/s 2
200
150
100
50
0
0
0.1
0.2
0.3
0.4
0.5
0.6
Normalized Energy
0.7
0.8
0.9
1
Figure 6: A comparison of the actual drag profile versus the reference drag profile