1. Robert G. Melton Department of Aerospace Engineering Penn State University 6 th International Workshop and Advanced School, “Spaceflight Dynamics and Control” Covilhã, Portugal March 28-30, 2011 Numerical analysis of constrained time-optimal satellite reorientation

2. Gamma-Ray Bursts/ Swift First detected by Vela satellites in 1960’s Source: formation of black holes or neutron star collsions Intense gamma-ray burst, with rapidly fadiing afterglow (discovered by Beppo-SAX satellite) Swift detects burst with wide- FOV detector, then slews to align narrow-FOV telescopes (X-ray, UV/optical) – Sensor axis must avoid Sun, Earth, Moon (“keep-out” zones – constraint cones) 2

3. Unconstrained Time-Optimal Reorientation Bilimoria and Wie (1993) unconstrained solution NOT eigenaxis rotation spherically symmetric mass distribution independently and equally limited control torques bang-bang solution, switching is function of reorientation angle Others examined different mass symmetries, control architectures Bai and Junkins (2009) discovered different switching structure, local optima for magnitude-limited torque vector, solution IS eigenaxis rotation 3

4. Constrained Problem (multiple cones): No Boundary Arcs or Points Observed Example: 0.1 deg. gap between Sun and Moon cones 4

5. t f = 3.0659, 300 nodes, 8 switches Constrained Problem (multiple cones) 5

6. Keep-out Cone Constraint (cone axis for source A) (sensor axis) 6

7. Optimal Control Formulation  Resulting necessary conditions are analytically intractable 7

8. Numerical Studies 1.Sensor axis constrained to follow the cone boundary (forced boundary arc) 2.Sensor axis constrained not to enter the cone 3.Entire s/c executes -rotation about  A Legendre pseudospectral method used (DIDO software) Scaling: lie on constraint cone I 1 = I 2 = I 3 and M 1,max = M 2,max = M 3,max  lies along principal body axis b 1 final orientation of b 2, b 3 generally unconstrained 8

9. Case BA-1 (forced boundary arc)  A = 45 deg. (approx. the Sun cone for Swift) Sensor axis always lies on boundary Transverse body axes are free  = 90 deg.  9

10. Case BA-1 (forced boundary arc) t f = 1.9480, 151 nodes 10

11. Case BA-1 (forced boundary arc) 11

12. Case BA-2 (forced boundary arc)  A = 23 deg. (approx. the Moon cone for Swift) Sensor axis always lies on boundary Transverse body axes are free  = 70 deg. t f = 1.3020, 100 nodes 12

13. Case BA-2 (forced boundary arc) 13

14. Case BP-1 same geometry as BA-1 (  A = 45 deg.,  = 90 deg.) forced boundary points at initial and final times sensor axis departs from constraint cone t f = 1.9258 (1% faster than BA-1) 250 nodes Angle between sensor axis and constraint cone 14

15. Case BP-1 15

16. Case BP-1 16

17. same geometry as BA-2 (  A = 23 deg.,  = 70 deg.) forced boundary points at initial and final times sensor axis departs from constraint cone Case BP-2 t f = 1.2967 (0.4% faster than BA-2) 100 nodes Angle between sensor axis and constraint cone 17

18. Case BP-2 18

19. Case BP-2 19

20. Sensor axis path along the constraint boundary Constrained Rotation Axis Entire s/c executes -rotation sensor axis on cone boundary rotation axis along cone axis 20

21. Problem now becomes one-dimensional, with bang-bang solution Applying to geometry of: BA-1  t f = 2.1078 (8% longer than BA-1) BA-2  t f = 2.0966 (37% longer than BA-2) Constrained Rotation Axis 21

22. Practical Consideration Pseudospectral code requires 20 minutes <  t < 12 hours (if no initial guess provided) Present research involves use of two-stage solution: 1. approx soln S (via particle swarm optimizer) 2. S = initial guess for pseudospectral code (states, controls, node times at CGL points) Successfully applied to 1-D slew maneuver 22

23. 23 Dido No guess cpu time = 148 sec. With PSO guess cpu time = 76 sec,

24. Conclusions and Recommendations For independently limited control torques, and initial and final sensor directions on the boundary: trajectory immediately departs the boundary no interior BP’s or BA’s observed forced boundary arc yields suboptimal time Need to conduct more accurate numerical studies Bellman PS method Interior boundary points? (indirect method) Study magnitude-limited control torque case Implementation expand PSO+Dido to 3-D case 24

25. fin 25