Adam Waite
3/27/08
Dynamics and Control
Control Theory and Steering Law Modifications
AAE 450 Spring 2008
Control Theory
Attempted to design our own control method – caused massive instabilities
Used control theory based on paper from the National Taiwan University
Used autopilot system to control launch vehicle’s attitude
Did not need to use Guidance (Position) Control for this project
Method:
- Control Theory outputs a moment needed to follow the steering law
- Solved for thrust vector angles from given moment
- Fed these angles into the thruster model
- Adjusted gains in the gain matrix for tighter control as needed
- Modified the steering law to avoid corners in nominal steering law
 Result:
- Working controller that successfully guides launch vehicle to orbit
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AAE 450 Spring 2008
Dynamics and Control
Steering Law Modification
• Used polynomials to
approximate steering law
• Allows for stable transition to
third stage
• This configuration of steering
law is fed into the controller
• Output is adjusted to reflect
angles used by the trajectory
group
350
300
Steering Angle (degrees)
• Linear steering law in upper
stages creates a more
manageable constant change
in pitch angle for launch
vehicle to follow
Corner
5 kg Example
250
200
Original Steering Law
Section 1 Modification
150
Section 2 Modification
Section 3 Modification
100
Poly. (Section 1 Modification)
Poly. (Section 2 Modification )
50
Linear (Section 3 Modification)
0
0
100
200
300
400
Time (s)
Figure by Adam Waite
AAE 450 Spring 2008
Dynamics and Control
500
600
References
1.
2.
3.
4.
Fu-Kuang Yeh, Kai-Yuan Cheng, and Li Chen Fu “Rocket
Controller Design With TVC and DCS” National Taiwan
University, Taipei, Taiwan 2003.
Stevens, B. L., and F. L. Lewis, Aircraft Control and Simulation,
Second Edition, John Wiley & Sons, New York, 2003.
McFarland, Richard E., A Standard Kinematic Model for Flight
simulation at NASA-Ames, NASA CR-2497.
Main D&C Simulator Code
AAE 450 Spring 2008
Dynamics and Control
Autopilot System
1
Sim Angles
 
q
Euler Angles to Quaternions
Mu
Angles
Mv
Desired Angles
2

1
Thruster Angles
Convert from M to angles
q
q
J
Euler Angles to Quaternions 1
Moment
Desired _w
w
3
wd
4
Sim _w
angles
Autopilot
5
Inertia Tensor
un productive moment
Figure by Mike Walker, Alfred Lynam, and Adam Waite
• Figure shows the autopilot outputting the moment which is then
converted to angles
• These angles are shown to output to the thruster
AAE 450 Spring 2008
Autopilot Sub-System Block
3
w
4
wd
we
wd
M1
q<>
we
we1
t1
q4
s surface
J
2
J
term 1
M2
M
<qx>1
1
q
qbar
q
1
Moment
q4
develop q 's
J
t2
w
M3
term 2
we
So
qe
Develop Slidign Surface
J
t3
w
wx
term 3
wx
w
wx
S surface
Ta
gen _Ta
Figure by Mike Walker
AAE 450 Spring 2008
M4
Gain Matrix
• Gain Matrix is optimized so that the launch vehicle closely follows the trajectory
X
P   0
 0
0
Y
0
0
0 
Z 
X controls the emphasis on the steering (pitch) angle
Y controls the emphasis on the yaw angle
Z controls the emphasis on the spin angle
• Our gain matrices for each case have very large values for the X variable
• This tells the thruster to put most of its control towards making sure the
steering (pitch) angle of the launch vehicle closely follows the given trajectory
AAE 450 Spring 2008
200g Case
1kg Case
5kg Case
1st Stage
100 0 0 
P   0 10 0 
 0
0 10
10000 0 0 
P   0
10 0 
 0
0 10
1000 0 0
P   0
1 0
 0
0 1
2nd Stage
100 0 0
P   0 0 0
 0 0 0
1000 0 0
P   0
0 0
 0
0 0
1000 0 0
P   0
0 0
 0
0 0
1 0 0
P  0 0 0
0 0 0
1 0 0
P  0 0 0
0 0 0
1 0 0
P  0 0 0
0 0 0
3rd Stage
AAE 450 Spring 2008
1kg Example of Pitch and Yaw Angles
-6
1
4.8
4.6
x 10
0
Actual Angle Using Controller
Desired Angle
4.4
-1
Yaw Angle (rad)
Pitch Angle (rad)
4.2
4
3.8
-2
-3
3.6
-4
Yaw with Controller
Desired Yaw
3.4
-5
3.2
3
0
50
100
150
200
250
-6
0
Time (sec)
50
100
150
200
250
Time (sec)
Figure by Adam Waite
Figure by Adam Waite
• This figure shows the effect of high
emphasis on controlling the pitch angle
• This figure shows that the yaw angle
only varies by a very small amount even
with low emphasis placed on it
AAE 450 Spring 2008
1 kg Example of Spin Angle
-4
3
x 10
• The gain matrices were tested
with different values many times
before the final configurations
were chosen
Spin Angle with Controller
Desired Spin Angle
2
Spin Angle (rad)
1
• All three cases exhibit the trend
of very small deviations from the
desired yaw and spin angles
0
-1
-2
-3
0
50
100
150
200
250
Time (sec)
Figure by Adam Waite
• This graph of the spin angle also shows a small
variance even with low emphasis placed on it
AAE 450 Spring 2008
• Adjusting the gain matrix and
modifying the nominal steering
law are the two methods that
have the biggest impact on the
final orbit and periapsis
Final Steering Angle for 200g Case
100
Controlled Angle of Launch Vehicle
Modified Steering Law
Nominal Steering Law
80
Steer Angle (deg)
60
40
20
0
-20
-40
0
50
100
150
200
250
300
time (s)
350
400
450
Figure by Mike Walker and Adam Waite
AAE 450 Spring 2008
500
Final Steering Angle for 1kg Case
100
Controlled Angle of Launch Vehicle
Modified Steering Law
Nominal Steering Law
80
Steer Angle (deg)
60
40
20
0
-20
0
50
100
150
200
250
300
time (s)
350
400
Figure by Mike Walker and Adam Waite
AAE 450 Spring 2008
450
500
Final Steering Angle for 5kg Case
100
Controlled Angle of Launch Vehicle
Modified Steering Law
Nominal Steering Law
80
Steer Angle (deg)
60
40
20
0
-20
-40
0
100
200
300
time (s)
400
Figure by Mike Walker and Adam Waite
AAE 450 Spring 2008
500