Adam Waite 03/06/2008 Dynamics and Control Splines and 1 kg Case

1. Adam Waite03/06/2008Dynamics and ControlSplines and 1 kg Case AAE 450 Spring 2008

2. Splines • Splines attempt to optimize trajectory’s steering law • Uses Excel equations to create new trajectory for all three stages • Attempts to smooth the transition between stages • For 1kg case, used linear steering law in upper stages to create a constant change in pitch rate • Performance is very sensitive to changes in spline and control gains Figure by Adam Waite Thanks to D&C group AAE 450 Spring 2008 Dynamics and Control

3. 1 kg Case (115%) Red – Trajectory Blue - Actual • Results using all average values: • Periapsis: 365 km • Apoapsis: 2,392 km • Eccentricity: 0.131 Future Work • Optimize spline and gains for 5kg case • Run Monte Carlo for all cases Figure by Adam Waite Thanks to D&C group AAE 450 Spring 2008 Dynamics and Control

4. References • Dynamics and Control group • Trajectory Group • Main D&C code • Everybody else that contributed to design and sizing of rocket AAE 450 Spring 2008 Dynamics and Control

5. Excel Sheet for 1kg Case Black Line – Actual Path Yellow Line – Spline Blue Line – Trajectory’s Steering Law Figure by Adam Waite AAE 450 Spring 2008

6. Logic Behind Splines in Excel • First, trajectory’s steering law was plotted using arctangent • Sharp changes in steering law were noted • Those points were eliminated from the plot and a polynomial curve fit was used in its place • These polynomial equations are then input into the main configuration file for the rocket to follow with the times for each spline also being input • The upper stages undergo a scalar adjustment in the positive or negative direction to either increase or decrease the radial velocity • Periapsis is determined and the spline is adjusted accordingly AAE 450 Spring 2008

7. Earlier Test Spline for 200g Case Black line – Actual Path Yellow line – Trajectory’s Steering Law Blue line- Spline Figure by Adam Waite AAE 450 Spring 2008

8. Earlier Test Spline for 5kg Case Black Line – Actual Path Colored Lines - Spline Figure by Adam Waite AAE 450 Spring 2008

9. Alternative Orbit Views Red Line – Trajectory’s Orbit Blue Line – Actual Orbit Figures by Adam Waite with thanks to D&C group AAE 450 Spring 2008