A.3.2.6: Simulator Design
A.3.2.6: Simulator Design
We simulate the attitude and translational dynamics and control of the launch vehicle by
implementing a complex MATLAB SIMULINK model. This model inputs the nominal
trajectory and steering law, the initial conditions, and the physical parameters of the
launch vehicle. The model maintains the ability to output numerous parameters inside
many stages of calculation through SIMULINK scopes. However, we only need certain
outputs to understand the motion of the system. For the nominal cases, enough data was
recorded to graph the position and steering angles of the launch vehicle. For the Monte
Carlo simulation, only periapsis altitude, orbit eccentricity, specific impulses, and mass
data were recorded for each run. We consider the simulator to be the final proof-ofconcept for project Bellerophon, since all of the other groups supplied inputs for the
simulator. The success or failure of this project largely depends on the simulator’s results.
Ve(m/s)
Position Desired
Sim Position
Position
Xe(m)
Sim Position
Desired Position
Velocity
Nominal Trajectory
Sim Velocity
Euler Angles
v des
Vb
Sim Angles
Desired Angles
v sim
Angular Velocity
Sim Angles
Thruster Angles
Control Angles
Velocity In Body coords .
Angular Velocity
Angular Acceleration
Desired Angles
Angular Acceleration
Body Acceleration
Desired Angles
Desired_w
mu l h
Desired Velocity
Body Accelleration
Sim_w
Inertia Tensor
Desired Angular Velocity
Inertia Tensor
q
Controller
Attitude Requirements
Dynamics /EOMs
mu l h
q
Scopes and Output
Figure A.3.2.6.1: Main Dynamics and Control Simulator.
(Mike Walker, Alfred Lynam, and Adam Waite)
We partition the simulator into the following five interdependent subsystems: The
“Nominal Trajectory” subsystem, the “Attitude Requirements” subsystem, the
“Controller” subsystem, the “Dynamics/EOMs”, and the “Scopes and Output” subsystem.
Figure A.3.2.6.1 illustrates the general program flow of the simulator.
Author: Alfred Lynam
A.3.2.6: Simulator Design
The “Nominal Trajectory” and “Attitude Requirements” subsystems extract the steering
law and ephemeris data from MATLAB, and supply the data to the “Controller”
subsystem. The “Nominal Trajectory” subsystem interpolates the position and velocity
data from an ephemeris file for each of the integrator’s time steps. The interpolated data
is fed into “Controller” subsystem and the “Scopes and Outputs” subsystem.
x
1
Position
y
z
Clock
vx
2
Velocity
vy
vz
Figure A.3.2.6.2: Nominal Trajectory subsystem.
(Mike Walker and Adam Waite)
The “Attitude Requirements” subsystem uses a lookup table to interpolate the steering
law for any time step. The steering law determines the desired steering angles for the
“Controller” subsystem. We multiplex the desired steering law with two other Euler
angles that represent spin and out of orbit plane rotation. These desired Euler angles are
set to zero because there is no spin control and an out of plane component of thrust would
only waste propellant. A derivative block was used on the multiplexed desired angle
vector to calculate the Euler angle change rates. We next converted the Euler angles and
the Euler angle change rates into desired angular velocity via kinematic equations. The
desired angles and the desired angular velocity are output to the “Controller” subsystem.
Author: Alfred Lynam
A.3.2.6: Simulator Design
Time _of _flight _init
Constant 3
Clock
z
1
Desired Angles
0
zero
Derivative
du /dt
angles
w
angle rate
2
Desired Angular Velocity
xform angle rates to w
Figure A.3.2.6.3: Attitude Requirements subsystem.
(Alfred Lynam and Mike Walker)
The “Controller” subsystem determines the proper thrust vector angles used to control the
launch vehicle into orbit. The “Controller” subsystem inputs the feedback and desired
position, velocity, Euler angles, and angular velocity variables, along with the inertia
tensor. We split the “Controller” subsystem into two major portions, the guidance and the
autopilot. The purpose of the guidance system is to ensure that the launch vehicle remains
on course. The guidance system relies on extra position correction thrusters that are
controlled independently of the autopilot. The guidance system theory is explained in the
A.3.2.2 section of the report and by Yeh, Cheng, and Fu2. Since Project Bellerophon does
not use added thrusters, we determined that a navigation system was not feasible.
Author: Alfred Lynam
A.3.2.6: Simulator Design
1
Position Desired
2
v des
Sim Position
3
Position Desired
Sim Position
v sim
at
v des
Angles1
a
4
v sim
convert a to angles
Sim Angles
Guidance
rd
err mag
rs
 
5
Sim Angles
WE NEVER USE THIS GUIDANCE SECTION
Fail Mode 2
q
error quaternion
Need another degree of controllability to use it
!!!!
(ie extra thrusters )
Euler Angles to Quaternions
q
Desired Angles

6
q
Euler Angles to Quaternions 1
Desired _w
9
Inertia Tensor
J
8
Sim _w
w
Mu
Moment
Angles
1
Thruster Angles
Mv
Convert from M to angles
wd
7
Switch
Autopilot
Figure A.3.2.6.4: Controller subsystem.
(Adam Waite and Mike Walker)
The autopilot completely controls the launch vehicle. The autopilot ensures that the thrust
vector angles stay as close as possible to the steering law. We convert the feedback and
desired Euler angles to quaternions and we use a non-linear quaternion error feedback
controller to determine the required thruster moments to follow the steering law. The
autopilot control theory is also described in the A.3.2.2 section of the report and by Yeh,
Cheng, and Fu2.
3
w
wd
4
wd
we
q<>
t1
q4
s surface
2
J
1
q
J
term 1
<qx>1
J
qbar
w
q
q4
M
t3
1
Moment
wx
develop q 's
term 3
J
t2
w
we
So
term 2
qe
Develop Sliding Surface
S surface
Ta
gen _Ta
wx
w
wx
Figure A.3.2.6.5: Autopilot subsystem.
(Adam Waite and Mike Walker)
Author: Alfred Lynam
A.3.2.6: Simulator Design
The autopilot system outputs the thruster moments to the “Convert from M to angles”
subsystem. This subsystem inputs the two moment terms from the “Autopilot”
subsystem, the constant center of thrust position, and the constant nominal thrust. We
decompose the moments into their requisite transverse forces via the body center of thrust
position. We find the “precession” angle “κ” by taking the inverse tangent of the quotient
of the two transverse thrust components. The “nutation” angle “δ” is found, next. We
determine the axial component of the thrust from the total thrust magnitude and the two
transverse components via vector normalization theory. We determine the normalized
transverse component by taking the square root of the sum of the squares of the
transverse components. The “nutation” angle “δ” is found by taking the inverse tangent
of the quotient of the transverse component and the axial component. We output the
“precession” angle “κ” and the “nutation” angle “δ” from the “Controller” section to the
“Dynamics/EOMs” section.
1
Mu
Saturation 2
-1
Constant
square 4
Product
Constant 1
|u|
Saturation 1
1
2
2
|u|
square
Fu 2
|u|
|u|
square 2
Thrust
|u|
2
delta
Saturation
sqrt
sqrt
Thrust
Figure A.3.2.6.6: Convert from M to angles subsystem.
(Adam Waite and Mike Walker)
Author: Alfred Lynam
Fz
sqrt
2
Constant 2
atan
Add
sqrt1
square 3
Product 1
1
Angles
square 1MATLAB Fcn
Fv
Ct _Body (3)
2
Mv
MATLAB
Function
To Thrust Vector
Product 5
A.3.2.6: Simulator Design
We designed the “Dynamics/EOMs” section of the simulator around a built-in simulink
block named “Custom Variable Mass 6DoF ECEF (Quaternion)2,” as illustrated in Fig.
A.3.2.6.1. This block integrates the full six degree of freedom non-linear equations of
motion for a rigid body system. This block requires the following continuous time inputs:
The body-fixed forces in Newtons, the body-fixed moments in Newton-meters, the mass
flow rate in kilograms per second, the mass in kilograms, the change in moment of inertia
over time in kilogram-meters squared per second, and the moment of inertia in kilometermeters squared. The block also requires the initial geodetic latitude, longitude, and
altitude in degrees, degrees, and meters, respectively. In addition, the initial velocity in
body axes with units of meters per second is required. The initial Euler angle orientation
using roll, pitch, and yaw in radians is also required. Finally, the body angular velocity in
radians per second with respect to the North-East-Down(NED) frame is required.
V
F
xyz
(N)
ecef
(m/s)
Xecef (m)
 lh
M xyz (N-m)
ECEF
Quaternion
   (rad )
DCM bi
dm /dt (kg/s)
DCM
be
DCM
m (kg)
2
Custom Variable
Mass
dI /dt (kg-m /s)
ef
Vb (m/s)

rel
(rad /s)
 (rad /s)
d  /dt
I (kg-m 2)
2
A (m/s )
b
Custom Variable Mass 6DoF
ECEF (Quaternion )
Figure A.3.2.6.7: Custom Variable Mass 6DoF ECEF (Quaternion).
(Aerospace Toolbox2.)
Author: Alfred Lynam
A.3.2.6: Simulator Design
The block outputs the following data continuously: velocity in the Earth-Centered, EarthFixed(ECEF) frame in meters per second; position in the ECEF frame in meters; geodetic
latitude, longitude, and altitude in degrees, degrees, and meters, respectively; the Euler
angle orientation using roll, pitch, and yaw in radians; the direction cosine matrix of the
coordinate transformation from the Earth-Centered Inertial(ECI) frame to the body-fixed
axes; the direction cosine matrix of the coordinate transformation from the geodetic
frame to the body-fixed axes; the direction cosine matrix of the coordinate transformation
from the ECEF frame to the geodetic frame; velocity in body axes with units of meters
per second; the body angular velocity in radians per second with respect to the NorthEast-Down(NED) frame; the body angular velocity in radians per second with respect to
the ECI frame; the body angular acceleration in radians per second squared with respect
to the ECI frame; and the rectilinear acceleration in body axes with units of meters per
second squared. We sent the time histories of most of these variables to scopes in the
“Scopes and Outputs” section of the simulator. We fed some of these variables back to
other parts of the “Dynamics/EOMs” section or the “Controller” section so that the
required continuous time inputs to the solver could be calculated.
The first continuous time input is the resultant body-fixed force. We calculate the total
body-fixed force by taking the vector sum of the gravity force and the thrust force. The
gravity force vector was determined in the “Gravity Force Vector” subsystem by
applying the inverse square law.
F  m / r 2
(A.3.2.6.1)
where the force “F” is in Newtons. The radius “r” in meters is found by feeding back the
altitude and adding it to the radius of the Earth (assumed to be constant at 6356750
meters). The Earth’s gravitational parameter “  ” is also assumed to be constant at 3.986
x 1014 meters squared per second.
Author: Alfred Lynam
A.3.2.6: Simulator Design
The instantaneous mass in kilograms is fed from the “Mass Calculator” subsystem
described later. We multiply or divide these inputs together to calculate the magnitude of
the gravity force. We determine the direction of the gravity force, next. We first
determine the direction cosine matrix that transforms the gravity matrix from the NED
frame to the body frame. The direction cosine matrix is found using the instantaneous
Euler angles. We multiply the direction cosine matrix by a column vector with two zeros
as the first two terms and the magnitude of the gravity force as the third term. The result
of that multiplication is the final gravity force vector.
Gravity Force Vector calculator
TX 1
TX
2
mass
0
Constant 1
Matrix
Multiply
Product
1
1
Position
G
mu l h
Product 1
Constant
Subsystem
3
Euler Angles
   DCM be
Euler Angles to
Direction Cosine Matrix
1
Fxyz
grav _flag
Product 2
Constant 2
Figure A.3.2.6.8: Gravity Force Vector Calculator Subsystem.
(Mike Walker.)
We calculate the center of mass position for each time step using the simulink block from
the Aerospace toolbox called “Estimate Center of Gravity.” 2 This block inputs the initial
and final values of the center of mass, the instantaneous mass, and the instantaneous mass
flow rate. The block outputs the instantaneous center of gravity and the instantaneous
change rate of the center of gravity.
Author: Alfred Lynam
A.3.2.6: Simulator Design
Drift
off
3
h
Thrust
h
Product
1
Thruster Body Forces
Thrust Magnitude Calculator
Thrust
Thrust_Vector
F
Angles
1
Control Angles Spherical to Rectangular
2
Cg
CG
M
CP
2
Thruster Body Moments
Ct_Body
CtBody in Vector
Moments about CG
due to Thrust
Figure A.3.2.6.9: Thruster Subsystem.
(Mike Walker and Alfred Lynam)
The “thruster” subsystem requires the input of the thrust vector angles from the
controller, the instantaneous altitude, and the instantaneous center of mass position. The
magnitude of the vacuum thrust at any given time is determined by multiplying the
average specific impulse (Isp) of the engine by the effective mass flow rate of the engine.
We determine the actual thrust by subtracting the ambient pressure at that particular
altitude multiplied by the exit area from the vacuum thrust. The ambient pressure was
determined using the altitude by the simulink block in the Aerospace toolbox named
“COESA Atmosphere Model.” 2
Author: Alfred Lynam
A.3.2.6: Simulator Design
Thrust (N)
ISP
Isp
AV ISP
1
Thrust
eff _md
mdot
Product
Terminator
T (K )
a (m/s)
1
h
h (m)
Saturation
COESA
P (Pa )
Thrust (N)
3
 (kg/m )
COESA Atmosphere Model
Switch
0
zero
Ae
Product 1
Exit Area
Figure A.3.2.6.10: Thrust Magnitude Calculator Subsystem.
(Mike Walker.)
We determine the thrust vector by implementing the thrust vector angles as spherical
coordinates. We note that the thrust vector angles that the controller gives are saturated
such that the maximum thrust vector angle is 10 degrees. The conversion equations
implemented into our simulator are as follows:
Tx  T sin  cos 
(A.3.2.6.2)
Ty  T sin  sin 
(A.3.2.6.3)
Tz  T cos 
(A.3.2.6.4)
where Tx, Ty, and Tz are the body thrust vector elements in Newtons, T is the force
magnitude, δ is the thrust vector “nutation” angle, and κ is the thrust vector “precession”
angle. The values of Tx, Ty, and Tz were multiplexed into the final thrust vector and
output. We added the thrust force and the gravity force together and inputted the result
into the “Custom Variable Mass 6DoF ECEF (Quaternion)” block.
Author: Alfred Lynam
A.3.2.6: Simulator Design
The second continuous time input is the body fixed moments. The gravity moment is
assumed to be negligible, since the center of mass and the center of gravity are assumed
to be approximately coincident. Thus, the thruster moments and the aerodynamic
moments are the only moments taken into account. We calculate the thrust moment using
another simulink block from the Aerospace toolbox called “Moments about CG due to
Forces.” 2 This block inputs the thrust vector, the center of mass, and the position of the
thruster (named center of thrust), and outputs the thruster body moment vector.
The aerodynamic moment is calculated using the aerodynamic force, the center of
pressure, and the center of mass using the same block. We calculated the aerodynamic
force using a constant coefficient of drag for the x direction, and curve fits based on
Mach number for the y and z directions. We multiplied each coefficient of drag by the
dynamic pressure determined from the velocity of the rocket and the density of the
atmosphere determined by the “COESA Atmosphere Model” to calculate the drag force.
The lift was assumed to be negligible. Unfortunately, due to an oversight, the
aerodynamic forces were not output to the main program for the simulations, so they only
affected the calculation of aerodynamic moments. We added together the aerodynamic
and thruster moments to form the body-fixed moments inputted into the “Custom
Variable Mass 6DoF ECEF (Quaternion)” block.
Mass Calculator
Step
1
Mass Flow Rate
-dmdt
Constant
Product
1
s
Integrator Saturation
Figure A.3.2.6.11: Mass Calculator Subsystem.
(Mike Walker)
Author: Alfred Lynam
2
Mass
A.3.2.6: Simulator Design
The third continuous time input is mass flow rate, but it is not input because it adds an
additional thrust force inside the “Custom Variable Mass 6DoF ECEF (Quaternion)”
block that is redundant with the thrust force. The fourth continuous time input is the
mass. The mass is determined by integrating the mass flow rate for each individual time
step from an empty tank to a full tank. This value is output to the gravity force calculator
and the center of mass calculator in addition to the “Custom Variable Mass 6DoF ECEF
(Quaternion).” The fifth and sixth continuous time inputs are the moment of inertia
change rate and the moment of inertia. We calculated both of these values using the
simulink block in the Aerospace toolbox named “Estimate Inertia Tensor.” This block
inputs the instantaneous mass flow rate, the instantaneous mass, and the initial and final
inertia tensors. All of the previously described subsystems are connected together to form
the “Dynamics/EOM” subsystem of the simulator.
h
Vwind
h
Euler Angles
dm /dt
Mxyz
Vwind
Cg
Calculate Wind
mass
Aero Forces
Vxyz
V Free
qz
Sum Moments
V free
CG
Aerodynamic forces /moments
F
xyz
M
2 Ve(m/s)
Xe (m)
 lh
8
3
mu l h
   (rad )
DCM Euler Angles
X ecef (m)
Control Angles
Thruster Body Forces
Cg
Thruster Body Moments
h
Mass
Thruster
Mass Calculator
mass
I
dm /dt
dI /dt
Fxyz
Euler Angles
Gravity Force Vector
(N-m)
Sum Forces
Mass Flow Rate
Position
xyz
Estimate
Inertia Tensor
9
Inertia Tensor
1
Vecef (m/s)
(N)
dCG /dt
Estimate
Center of Gravity
1
Control Angles
mass
10
q
dm /dt (kg/s)
ECEF
Quaternion
Custom Variable
Mass
bi
DCM be
C
DCM ef To Workspace
m (kg)
4
V b (m/s)
Vb
 rel (rad /s)
5
dI /dt (kg-m 2/s)
 (rad /s)Angular Velocity
d /dt
6
I (kg-m 2)
A (m/s2) Angular Acceleration
b
7
Custom Variable Mass 6DoF
Body Acceleration
ECEF (Quaternion )
Figure A.3.2.6.12: Dynamics/EOM subsystem (Most connectors disconnected).
(Mike Walker, Alfred Lynam, and Adam Waite)
The “Scopes and Outputs” subsystem merely outputs values of interest to the user or to
the workspace using scopes. The most important outputs from this section are the ideal
and actual steering law angle “Xi”, the altitude “h”, the final velocity, and the final
position. The steering law output allows the fidelity of the simulated trajectory to be
analyzed. The altitude output allows the user to qualitatively determine if the rocket
catastrophically crashes into the Earth or incinerates in the atmosphere. The final velocity
and position provide sufficient conditions for orbit determination.
Author: Alfred Lynam
A.3.2.6: Simulator Design
9
Desired Velocity
Vx
w1
7
Angular Velocity
w2
1
Sim Velocity
2
Sim Position
X_Pos
3
Desired Position
Y_Pos
Vy
w3
sqrt
Dot Product
6
Velocity In Body coords .
4
Sim Angles
angle
r angle
Math
Function
Vz1
Vz
8 Terminator 3
Angular Acceleration
Z_Pos
l
11
mu l h
Vbx
Vby
du /dt
Vbz
Xi
Derivative
du /dt
Resolve Angle 2
Xidot
Fail Mode 2
q
Psi
Fail Mode 1
10
du /dt
Derivative 2
eta
h
q
12
Derivative 1 psidot
5
angle r angle
Desired Angles
Resolve Angle 1
mu
h
Body Accelleration
Body Accelleration 1
Eta
Figure A.3.2.6.13: Scopes and Output subsystem.
(Mike Walker, Alfred Lynam, and Adam Waite)
References:
1
Fu-Kuang Yeh, Kai-Yuan Cheng, and Li-Chen Fu “Rocket Controller Design with TVC
and DCS” National Taiwan University, Taipei, Taiwan 2003
2
Aerospace Toolkit and Aerospace Blockset are part of MATLAB and SIMULINK. Both
registered trademarks: 1984-2007 by The MathWorks, Inc.
Author: Alfred Lynam