Adam Waite 3/27/08 Dynamics and Control Control Theory and Steering Law Modifications

1. Adam Waite3/27/08Dynamics and ControlControl Theory and Steering Law Modifications AAE 450 Spring 2008

2. 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 AAE 450 Spring 2008 Dynamics and Control

3. Steering Law Modification Corner 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 Figure by Adam Waite AAE 450 Spring 2008 Dynamics and Control

4. References • 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

5. 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

6. Autopilot Sub-System Block Figure by Mike Walker AAE 450 Spring 2008

7. Gain Matrix • Gain Matrix is optimized so that the launch vehicle closely follows the trajectory 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

8. 200g Case 1kg Case 5kg Case 1st Stage 2nd Stage 3rd Stage AAE 450 Spring 2008

9. 1kg Example of Pitch and Yaw Angles 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

10. 1 kg Example of Spin Angle • 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 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

11. Final Steering Angle for 200g Case Figure by Mike Walker and Adam Waite AAE 450 Spring 2008

12. Final Steering Angle for 1kg Case Figure by Mike Walker and Adam Waite AAE 450 Spring 2008

13. Final Steering Angle for 5kg Case Figure by Mike Walker and Adam Waite AAE 450 Spring 2008