Katie Demko
September-December 2009
Laboratory Work
Hypersonic Reentry Vehicle Dynamics and Control
The ultimate goal of this project is to contribute to the guidance of future hypersonic reentry
vehicles by developing candidate guidance algorithms that enhance performance relative to the
state of the art. The objective for the students for the past several months has been to gain an
understanding of the dynamics of reentry vehicles and to learn methods for controlling the
trajectory of such vehicles.
Background
The forces acting on the capsule-shaped reentry vehicle during reentry include lift, drag, weight
and thrust from the reaction control system (RCS) thrusters. The vehicle is steered to the landing
site by changing the direction of the lift vector. This operation is carried out by the RCS
thrusters which rotate the lift vector by changing the bank angle. As seen in Figure 0.1, the
center of mass is offset from the vehicle axis of symmetry so that the vehicle generates lift which
can be rotated about the velocity vector.
Figure 0.1 - Vehicle Orientation with Axis of Symmetry (Bairstow, 2006)
Figure 0.2 - Rotation of Lift Vector (Bairstow, 2006)
The lift vector is an important means of control because it affects the range the vehicle flies. If
the lift vector is pointed toward the Earth, the vehicle has a rapid rate of descent. If the lift
vector is pointed upwards toward space the vehicle has a slower descent rate. Also by directing
some of the lift laterally, crossrange can be controlled.
Simulation
A Matlab simulation was created to gain a better understanding of the dynamics of hypersonic
reentry. The simulation is based on the equations of motion (Saraf, 2004), which are shown
below. There are five nonlinear differential equations, one for longitude (θ), one for latitude (Φ),
one for radius (r), one for the heading angle (ψ) and finally, one for flight path angle (γ). The
independent variable in these equations is energy because it is more important than time for
control purposes; ( )’ denotes differentiation with respect to energy. The other variables that
appear in these equations include the drag (D), the velocity (V) and the bank angle (σ).
(0.1)
(0.2)
(0.3)
(0.4)
(0.5)
The method of obtaining the initial conditions is shown below. To determine the vehicle’s initial
energy a radius of 647800 km (120km above the surface of the Earth), a velocity of 10.2 km/s
and the Earth’s gravitational constant µ were entered into the energy equation as follows
. To find the final energy at an altitude of 20 km, the corresponding velocity of
340m/s and the Earth’s gravitational constant were entered into the energy equation.
The vehicle specifications were modeled after Orion (Bairstow & Barton, 2007). The vehicle
cross sectional area is 23.8 m2. The vehicle mass is 9500 kg. The coefficient of lift is 0.44 and
the coefficient of drag is 1.25. The initial conditions for the equations of motion are latitude
θ(Ei)=0, longitude Φ(Ei)=0, radius r(Ei)=6498 km, heading angle Ψ(Ei)=0 degrees and flight path
angle γ(Ei)=-9 degrees. The initial latitude and longitude were taken to be zero to make it easier
to compute the downrange and crossrange distance along the equatorial plane. The initial
heading angle was also taken to be zero so that the vehicle would only travel along the equatorial
plane. The radius is measured from the center of the earth and therefore a radius of 6498km
corresponds to an altitude of 120km above the surface of the earth. This altitude was chosen
because it is the height at which the atmosphere begins. Finally, the flight path angle was chosen
to be -9 degrees based on previous similar simulations.
For the first simulation, shown below in Figure 0.1 and Figure 0.2 are plots of the altitude as a
function of downrange distance and the flight path angle as a function of time. Based on the first
plot we can see that with a bank angle of 0 degrees and without a controller, the vehicle has a
downrange distance of 2500km. We also note that due to the very low atmospheric density at an
altitude of 120km, the vehicle has a steep descent until the atmosphere becomes denser and lift
slows the descent of the vehicle.
120
110
100
Altitude, km
90
80
70
60
50
40
30
20
0
500
1000
1500
Downrange, km
2000
2500
3000
Figure 0.1 - Lifting entry with zero bank angle
10
5
Flight Path Angle, deg
0
-5
-10
-15
-20
-25
-30
-35
-40
0
100
200
300
Time, s
400
500
600
Figure 0.2 - Lifting entry with zero bank angle
The second plot from the simulation with zero bank angle shows that the flight path angle is
proportional to the change in altitude. This can be shown from the equation below.
r  V sin( )
(0.6)
The regions of positive slope in the plot of flight path angle as a function of time indicate a
slower descent rate in altitude and the regions of negative slope indicate a steeper descent rate.
In the second simulation with a bank angle of 60 degrees, the plots in Figure 0.3 and Figure 0.4
of altitude as a function of downrange distance shows that the downrange distance covered is
approximately 960km whereas without a bank angle the distance covered is 2500km. This
demonstrates that the greater the bank angle, the steeper the descent profile and the smaller the
downrange distance covered.
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
1000
Figure 0.3 - Lifting entry with σ=60 deg
5
0
Flight Path Angle, deg
-5
-10
-15
-20
-25
-30
-35
-40
-45
0
50
100
150
Time, s
200
250
Figure 0.4 - Lifting entry with σ=60 deg
300
As seen in the previous simulation, the plot of the flight path angle as a function of the time
shows the proportional relationship between the flight path angle and the descent rate.
To demonstrate the effect of having a constant non-zero bank angle in the simulation two
additional plots are shown below. Figure 0.5 illustrates that by adding a constant bank angle of
60 degrees the heading direction changes from the initial condition of zero. The bank angle is
defined such that a positive bank angle corresponds to a right turn. We can also see this from the
plot of the latitude and longitude that the vehicle veers to the South of the equatorial plane.
0
-0.2
Heading, rad
-0.4
-0.6
-0.8
-1
-1.2
-1.4
0
50
100
150
Time, s
200
250
Figure 0.5 - Lifting entry with σ=60 deg
300
0
-0.2
Latitude, deg
-0.4
-0.6
-0.8
-1
-1.2
-1.4
-1.6
0
1
2
3
4
5
Longitude, deg
6
7
8
9
Figure 0.6 - Lifting entry with σ=60 deg
One of the future efforts will be to add bank reversal maneuvers into the simulation to control
crossrange.
Finally, the reason these simulations were performed was to visualize the effect that different
bank angles have on the trajectory of a hypersonic reentry vehicle. We can confirm that the
larger the bank angle the smaller the downrange distance covered as seen in the Figure 0.7
below.
120
sigma=0
sigma=45deg
sigma=60deg
100
Altitude, km
80
60
40
20
0
-20
0
500
1000
1500
Downrange, km
2000
2500
3000
Figure 0.7 - Downrange distance for several bank angles
For this reason, our focus in the future will be to find ways to maximize the possible range the
vehicle can obtain by adding control logic for skip trajectories. This would be beneficial for the
control of Orion because it would be able to achieve a downrange distance potentially equal to
the circumference of the Earth.
Control Logic
The next objective is to add a guidance algorithm to the hypersonic reentry simulation. There
are two main steps involved to meet this objective. The first step of the process is to create a
planning function based on drag parameterization. Drag parameterization is used because the
reentry path length is related to the vehicle drag acceleration.
(0.1)
(0.2)
(0.3)
(0.4)
This is the same concept used in the U.S. Space Shuttle Orbiter entry guidance which determines
a reference drag profile based the range to the landing site as well as on the path constraints on
vehicle heating, acceleration and dynamic pressure (Leavitt, 2007). The equations for the path
constraints are shown below.
(0.5)
(0.6)
(0.7)
These three inequality equations can then be converted into equations for drag and graphed with
respect to normalized energy. Once the initial and final drag and energy are computed the drag
profile can be determined based on the range remaining to the landing site.
The bank angle is calculated using a separate algorithm. The algorithm that will be implemented
to find the bank angle is derived from the energy equation and the drag equation.
(0.8)
(0.9)
Initially, we estimate the value of the radius and solve for the velocity from the energy equation
since the energy is known from the drag profile. The calculated velocity is then inserted into the
drag equation and a second value for the radius is calculated. The radius is calculated from the
drag equation through the equation for the density shown below.
(0.10)
This process is repeated until the estimated radius from the energy equation is within a certain
tolerance of the radius calculated from the drag equation. Using this process, every drag and
energy pair can be matched to a bank angle and velocity pair to create the drag profile in terms of
bank angle prior to reentry.
The second step in the process is to create a tracking and correcting function. At the point when
the vehicle begins the reentry phase of flight a feedback linearization controller compares the
drag from the reference drag profile to the actual drag and implements bank angle corrections
using the same bank angle algorithm as above. Below are the equations for the feedback
linearization where D is the actual drag acceleration, Dref is the reference drag acceleration and σ
is the bank angle.
(0.11)
(0.12)
(0.13)
(0.14)
The process of calculating the new reference drag profile based on the range remaining to the
target, then calculating the corresponding bank angle and finally comparing the actual drag to the
reference drag and correcting the bank angle continues until the vehicle reaches the landing site.
References
A.1
Background
Orion, the crew exploration vehicle of the Constellation program, will require increased
performance during the reentry trajectory. Since Orion is required to be reusable, it will have to
execute accurate land landings instead of water landings such as Apollo. Land landings
generally require more precision than water landings because there are more constraints.
Another goal of the program is for Orion to be able to return from the moon at any time of day.
The combination of these two requirements means that the reentry trajectory for Orion could
involve covering greater distances than any previous mission. To achieve greater distances the
refinement of a control algorithm is necessary.
The forces acting on the capsule-shaped Orion during reentry include lift, drag, weight and thrust
from the reaction control system (RCS) thrusters. The vehicle is steered to the landing site by
changing the direction of the lift vector. This operation is carried out by the RCS thrusters which
rotate the lift vector by changing the bank angle. As seen in Figure 0.1, the center of mass is
offset from the vehicle axis of symmetry so that the vehicle generates lift which can be rotated
about the velocity vector.
Figure A.1.1 - Vehicle Orientation with Axis of Symmetry (Bairstow, 2006)
Figure A.1.2 - Rotation of Lift Vector (Bairstow, 2006)
The lift vector is an important means of control because it affects the range the vehicle flies. If
the lift vector is pointed toward the Earth, the vehicle has a rapid rate of descent. If the lift
vector is pointed upwards toward space the vehicle has a slower descent rate. Also by directing
some of the lift laterally, crossrange can be controlled.
A.2
Simulation
A Matlab simulation was created to gain a better understanding of the dynamics of hypersonic
reentry. The simulation is based on the equations of motion (Saraf, 2004), which are shown
below. There are five nonlinear differential equations, one for longitude (θ), one for latitude (Φ),
one for radius (r), one for the heading angle (ψ) and finally, one for flight path angle (γ). The
independent variable in these equations is energy because it is more important than time for
control purposes; ( )’ denotes differentiation with respect to energy. The other variables that
appear in these equations include the drag (D), the velocity (V) and the bank angle (σ).
(A.2.1)
(A.2.2)
(A.2.3)
(A.2.4)
(A.2.5)
The method of obtaining the initial conditions is shown below. To determine the vehicle’s initial
energy a radius of 647800 km (120km above the surface of the Earth), a velocity of 10.2 km/s
and the Earth’s gravitational constant µ were entered into the energy equation as follows
. To find the final energy at an altitude of 20 km, the corresponding velocity of
340m/s and the Earth’s gravitational constant were entered into the energy equation.
The vehicle specifications were modeled after Orion (Bairstow & Barton, 2007). The vehicle
cross sectional area is 23.8 m2. The vehicle mass is 9500 kg. The coefficient of lift is 0.44 and
the coefficient of drag is 1.25. The initial conditions for the equations of motion are latitude
θ(Ei)=0, longitude Φ(Ei)=0, radius r(Ei)=6498 km, heading angle Ψ(Ei)=0 degrees and flight path
angle γ(Ei)=-9 degrees. The initial latitude and longitude were taken to be zero to make it easier
to compute the downrange and crossrange distance along the equatorial plane. The initial
heading angle was also taken to be zero so that the vehicle would only travel along the equatorial
plane. The radius is measured from the center of the earth and therefore a radius of 6498km
corresponds to an altitude of 120km above the surface of the earth. This altitude was chosen
because it is the height at which the atmosphere begins. Finally, the flight path angle was chosen
to be -9 degrees based on previous similar simulations.
For the first simulation, shown below in Figure 0.1 and Figure 0.2 are plots of the altitude as a
function of downrange distance and the flight path angle as a function of time. Based on the first
plot we can see that with a bank angle of 0 degrees and without a controller, the vehicle has a
downrange distance of 2500km. We also note that due to the very low atmospheric density at an
altitude of 120km, the vehicle has a steep descent until the atmosphere becomes denser and lift
slows the descent of the vehicle.
120
110
100
Altitude, km
90
80
70
60
50
40
30
20
0
500
1000
1500
Downrange, km
2000
2500
3000
Figure A.2.1 - Lifting entry with zero bank angle
10
5
Flight Path Angle, deg
0
-5
-10
-15
-20
-25
-30
-35
-40
0
100
200
300
Time, s
400
500
600
Figure A.2.2 - Lifting entry with zero bank angle
The second plot from the simulation with zero bank angle shows that the flight path angle is
proportional to the change in altitude. This can be shown from the equation below.
r  V sin( )
(A.2.6)
The regions of positive slope in the plot of flight path angle as a function of time indicate a
slower descent rate in altitude and the regions of negative slope indicate a steeper descent rate.
In the second simulation with a bank angle of 60 degrees, the plots in Figure 0.3 and Figure 0.4
of altitude as a function of downrange distance shows that the downrange distance covered is
approximately 960km whereas without a bank angle the distance covered is 2500km. This
demonstrates that the greater the bank angle, the steeper the descent profile and the smaller the
downrange distance covered.
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
1000
Figure A.2.3 - Lifting entry with σ=60 deg
5
0
Flight Path Angle, deg
-5
-10
-15
-20
-25
-30
-35
-40
-45
0
50
100
150
Time, s
200
250
Figure A.2.4 - Lifting entry with σ=60 deg
300
As seen in the previous simulation, the plot of the flight path angle as a function of the time
shows the proportional relationship between the flight path angle and the descent rate.
To demonstrate the effect of having a constant non-zero bank angle in the simulation two
additional plots are shown below. Figure 0.5 illustrates that by adding a constant bank angle of
60 degrees the heading direction changes from the initial condition of zero. The bank angle is
defined such that a positive bank angle corresponds to a right turn. We can also see this from the
plot of the latitude and longitude that the vehicle veers to the South of the equatorial plane.
0
-0.2
Heading, rad
-0.4
-0.6
-0.8
-1
-1.2
-1.4
0
50
100
150
Time, s
200
250
Figure A.2.5 - Lifting entry with σ=60 deg
300
0
-0.2
Latitude, deg
-0.4
-0.6
-0.8
-1
-1.2
-1.4
-1.6
0
1
2
3
4
5
Longitude, deg
6
7
8
9
Figure A.2.6 - Lifting entry with σ=60 deg
One of the future efforts will be to add bank reversal maneuvers into the simulation to control
crossrange.
Finally, the reason these simulations were performed was to visualize the effect that different
bank angles have on the trajectory of a hypersonic reentry vehicle. We can confirm that the
larger the bank angle the smaller the downrange distance covered as seen in the Figure 0.7
below.
120
sigma=0
sigma=45deg
sigma=60deg
100
Altitude, km
80
60
40
20
0
-20
0
500
1000
1500
Downrange, km
2000
2500
3000
Figure A.2.7 - Downrange distance for several bank angles
For this reason, our focus in the future will be to find ways to maximize the possible range the
vehicle can obtain by adding control logic for skip trajectories. This would be beneficial for the
control of Orion because it would be able to achieve a downrange distance potentially equal to
the circumference of the Earth.
Control Logic
The next objective is to add a guidance algorithm to the hypersonic reentry simulation. There
are two main steps involved to meet this objective. The first step of the process is to create a
planning function based on drag parameterization. Drag parameterization is used because the
reentry path length is related to the vehicle drag acceleration.
(0.1)
(0.2)
(0.3)
(0.4)
This is the same concept used in the U.S. Space Shuttle Orbiter entry guidance which determines
a reference drag profile based the range to the landing site as well as on the path constraints on
vehicle heating, acceleration and dynamic pressure (Leavitt, 2007). The equations for the path
constraints are shown below.
(0.5)
(0.6)
(0.7)
These three inequality equations can then be converted into equations for drag and graphed with
respect to normalized energy. Once the initial and final drag and energy are computed the drag
profile can be determined based on the range remaining to the landing site.
The bank angle is calculated using a separate algorithm. The algorithm that will be implemented
to find the bank angle is derived from the energy equation and the drag equation.
(0.8)
(0.9)
Initially, we estimate the value of the radius and solve for the velocity from the energy equation
since the energy is known from the drag profile. The calculated velocity is then inserted into the
drag equation and a second value for the radius is calculated. The radius is calculated from the
drag equation through the equation for the density shown below.
(0.10)
This process is repeated until the estimated radius from the energy equation is within a certain
tolerance of the radius calculated from the drag equation. Using this process, every drag and
energy pair can be matched to a bank angle and velocity pair to create the drag profile in terms of
bank angle prior to reentry.
The second step in the process is to create a tracking and correcting function. At the point when
the vehicle begins the reentry phase of flight a feedback linearization controller compares the
drag from the reference drag profile to the actual drag and implements bank angle corrections
using the same bank angle algorithm as above. Below are the equations for the feedback
linearization where D is the actual drag acceleration, Dref is the reference drag acceleration and σ
is the bank angle.
(0.11)
(0.12)
(0.13)
(0.14)
The process of calculating the new reference drag profile based on the range remaining to the
target, then calculating the corresponding bank angle and finally comparing the actual drag to the
reference drag and correcting the bank angle continues until the vehicle reaches the landing site.
1) Bairstow, Sarah H., and Gregg H. Barton. "Reentry Guidance with Extended Range
Capability for Low L/D Spacecraft." AIAA Guidance, Navigation and Control Conference and
Exhibit Hilton Head, SC, August 2007.
2) Benito, Joel, and Kenneth D. Mease. "Nonlinear Predictive Controller for Drag tracking in
Entry Guidance." American Institute of Aeronautics and Astronautics :14.
3) Mease ,K.D., and Kremer, JP., “Shuttle Entry Guidance Revisited Using Nonlinear
Geometric Methods.” Journal of Guidance and Control, Vol. 17, No. 6, 1994 pp.1350-1356.
4) Harpold, J.C., and Graves, C. A., Jr., “Shuttle Entry Guidance,” Journal of the Astronautical
Sciences, Vol. 27, No. 3, 1979, pp.239-268.