Optimal Trajectory Design

1. Optimal Trajectory Design Xiaoli Bai Postdoctoral Research Associate Email:xiaolibai@tamu.edu (office)979-8624992 752 Bright Building

2. OUTLINE • Fundamentals of optimal control • After you solve the optimal control problem for one case, how to do the mission design? • Mission design study in 2008 • Mission design study in 2009 • A recent design option: low-thrust • Look at a hybrid approach

3. Fundamentals of optimal control Cost function: Dynamical equations: Boundary conditions: Other constraints on the states and control

4. Classical Lambert’s Problem

5. through optimality condition: (Indirect Approach) Boundary conditions Unknown variables: Pontryagin’s principle Introduce costates and Lagragian multipliers

6. Indirect Approach: an Chebyshev pseudospectral method Transform to a nonlinear programming problem

7. Fundamentals you need to have to • Optimal control theory: AERO 623: SPACECRAFT MANEUVERS by Dr. Vadali is very worthwhile! • Some numerical tool -MATLAB: [x,fval,exitflag,output,lambda,grad,hessian]= fmincon(fun,x0,A,b,Aeq,beq,lb,ub,nonlcon,options) -SNOPT -A self-developed tool will benefit you a lot!

8. Design in 2008 Contour of C3: Km2/s2

9. Contour of :Km/s

10. Multiple Criterion Tradeoff

11. Favorite Launch Window ( <2km/s) • 2011-10-21 to 2011-12-3: maybe more pressure on the current development • 2012-04-13 to 2012-04-30:smallest total delta-v • 2012-12-20 to 2013-01-20: rendezvous delta-v 1.9km/s-1.7km/s; shortest flight time(103-126days) • 2013-04-04 to 2013-05-26: rendezvous delta-v 1.9km/s-0.64km/s; although may put more pressure on the mitigation. (193-337days)

12. Design in 2009 Trajectory Design • We looked at launch opportunities from 2020-2028; • Ideal launch opportunity occurs Feb-10-2021 to March-3-2021 • We also looked at alternate launch opportunities Sep-14-2021 Feb-19-2021

13. Launch opportunities from 2020 to 2028; max C3: 50km^2/s^2

14. Launch opportunities from 2020 to 2028 max Vf: 10km/s

15. Launch opportunity from 2020 to 2028

16. Nominal launch Feb-10-2021 to March-3-2021;C3<5km^2/s^2

17. Nominal launch: Feb-10-2021 to March-3-2021;Vf<3.09km/s

18. Cruise Phase Sep-14-2021 TOF:208days; C3:4.3km^2/s^2; Vf:3km/s Feb-19-2021

19. A new approach: Low thrust propulsion Deep Space 1 Dawn Organization: NASA Major contractors Orbital Sciences, JPL, UCLA Mission type Flyby/Orbiter Satellite of Mars (flyby) Vesta then Ceres Launch date: September 27, 2007, Launch vehicle: Delta 7925H Mission duration:8 years Mass:1,250 kg (2,756 lb) xenonion thrusters Isp=3100s, T=90mN Total delta-v: 10km/s Dawn is NASA’s first purely exploratory mission to use ion propulsion engines Organization: NASA Mission type Testing of technologies, Flyby Satellite: of Sun. flew by Braille and Comet Borrelly Launch date October 24, 1998 Launch vehicle Delta II rocket Mission duration: ended 13 December2001 NSTAR ion thruster Isp:1000-2000s, 92mN at max Power fired for 678 total days No interference was found with other spacecraft systems, such as radio communications or the science instruments

22. Bang-bang control

23. Thrust angle

24. Hybrid Impulsive and Low Thrust Optimal Control • Optimization Goal: minimize the total impulse energy by designing • thrust angles!

25. Initial Impulse Contour of V1(km/s) using hybrid Contour of V1(km/s) using Lambert • For a launch vehicle C3(V1^2) less than 18km/s^2, the launch window for an Apophis mission spans a full year.

26. Rendezvous Impulse Contour of V2(km/s) using hybrid Contour of V2(km/s) using Lambert • The pork chop plot patterns remain similar, whereas the hybrid approach requires much less impulse.

27. Total Delta-v

28. I hope you have fun with this class, at least to the same level as I have had!-