adam waite 3/27/08 dynamics and control control theory and steering law modifications
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Adam Waite 3/27/08 Dynamics and Control Control Theory and Steering Law Modifications. Control Theory. Attempted to design our own control method – caused massive instabilities Used control theory based on paper from the National Taiwan University - PowerPoint PPT PresentationTRANSCRIPT
AAE 450 Spring 2008
Adam Waite3/27/08
Dynamics and ControlControl 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
Dynamics and Control
AAE 450 Spring 2008
Steering Law Modification
Dynamics and Control
0 100 200 300 400 500 6000
50
100
150
200
250
300
350
Original Steering Law
Section 1 Modification
Polynomial (Section 1 Modifica-tion)
Section 2 Modification
Polynomial (Section 2 Modification )
Section 3 Modification
Linear (Section 3 Modification)
Time (s)
Stee
ring
Angl
e (d
egre
es)
Figure by Adam Waite
5 kg Example• Used polynomials to approximate steering law
• Linear steering law in upper stages creates a more manageable constant change in pitch angle for launch vehicle to follow
• 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
Corner
AAE 450 Spring 2008
References1. Fu-Kuang Yeh, Kai-Yuan Cheng, and Li Chen Fu “Rocket
Controller Design With TVC and DCS” National Taiwan University, Taipei, Taiwan 2003.
2. Stevens, B. L., and F. L. Lewis, Aircraft Control and Simulation, Second Edition, John Wiley & Sons, New York, 2003.
3. McFarland, Richard E., A Standard Kinematic Model for Flight simulation at NASA-Ames, NASA CR-2497.
4. Main D&C Simulator Code
Dynamics and Control
AAE 450 Spring 2008
Thruster Angles1
un productive moment
angles
Euler Angles to Quaternions 1
q
Euler Angles to Quaternions
q
Convert from M to angles
Mu
MvAngles
Autopilot
q
J
w
wd
Moment
Inertia Tensor5Sim _w
4
Desired _w
3
Desired Angles
2
Sim Angles1
Autopilot System
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
Moment1
wx
w wx
we1
we
term 3
J
w
wx
t3
term 2
J
wt2
term 1
wd
q<>
we
q4
s surface
J
t1
gen _Ta
S surface Ta
develop q 's
q
<qx>1
qbar
q4
M4
M3
M2
M1
M
Develop Slidign Surface
we
qeSo
wd4
w3
J2
q1
Autopilot Sub-System Block
Figure by Mike Walker
AAE 450 Spring 2008
Gain Matrix• Gain Matrix is optimized so that the launch vehicle closely follows the trajectory
ZY
XP
000000
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
1st Stage
2nd Stage
3rd Stage
200g Case 1kg Case 5kg Case
1000010000100
P
00000000100
P
000000001
P
100001000010000
P
000000001000
P
000000001
P
100010001000
P
000000001000
P
000000001
P
AAE 450 Spring 2008
1kg Example of Pitch and Yaw Angles
0 50 100 150 200 2503
3.2
3.4
3.6
3.8
4
4.2
4.4
4.6
4.8
Time (sec)
Pitc
h A
ngle
(rad
)
Actual Angle Using ControllerDesired Angle
Figure by Adam Waite
• This figure shows the effect of high emphasis on controlling the pitch angle
0 50 100 150 200 250-6
-5
-4
-3
-2
-1
0
1x 10
-6
Time (sec)Y
aw A
ngle
(rad
)
Yaw with ControllerDesired Yaw
Figure by Adam Waite
• 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
0 50 100 150 200 250-3
-2
-1
0
1
2
3x 10
-4
Time (sec)
Spi
n A
ngle
(rad
)
Spin Angle with ControllerDesired Spin Angle
Figure by Adam Waite
• This graph of the spin angle also shows a small variance even with low emphasis placed on it
• The gain matrices were tested with different values many times before the final configurations were chosen
• All three cases exhibit the trend of very small deviations from the desired yaw and spin angles
• 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
AAE 450 Spring 2008
Final Steering Angle for 200g Case
0 50 100 150 200 250 300 350 400 450 500-40
-20
0
20
40
60
80
100
time (s)
Ste
er A
ngle
(deg
)
Controlled Angle of Launch VehicleModified Steering LawNominal Steering Law
Figure by Mike Walker and Adam Waite
AAE 450 Spring 2008
Final Steering Angle for 1kg Case
0 50 100 150 200 250 300 350 400 450 500-20
0
20
40
60
80
100
time (s)
Ste
er A
ngle
(deg
)
Controlled Angle of Launch VehicleModified Steering LawNominal Steering Law
Figure by Mike Walker and Adam Waite
AAE 450 Spring 2008
Final Steering Angle for 5kg Case
0 100 200 300 400 500-40
-20
0
20
40
60
80
100
time (s)
Ste
er A
ngle
(deg
)
Controlled Angle of Launch VehicleModified Steering LawNominal Steering Law
Figure by Mike Walker and Adam Waite