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Numerical analysis of constrained time-optimal satellite reorientation

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### Numerical analysis of constrained time-optimal satellite reorientation

Robert G. Melton

Department of Aerospace Engineering

Penn State University

6th International Workshop and Advanced School,

“Spaceflight Dynamics and Control”

Covilhã, Portugal March 28-30, 2011

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

Constrained Problem (multiple cones):

No Boundary Arcs or Points Observed

Example:

0.1 deg. gap between Sun and Moon cones

Constrained Problem (multiple cones)

tf= 3.0659, 300 nodes, 8 switches

- Resulting necessary conditions are analytically intractable

Sensor axis constrained to follow the cone boundary (forced boundary arc)

Sensor axis constrained not to enter the cone

Entire s/c executes -rotation about A

Legendre pseudospectral method used

(DIDO software)

Scaling:

lie on constraint cone

- I1 = I2 = I3and M1,max = M2,max = M3,max
- lies along principal body axis b1
- final orientation of b2, b3generally unconstrained

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.

Case BA-1 (forced boundary arc)

tf = 1.9480, 151 nodes

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.

tf = 1.3020, 100 nodes

- same geometry as BA-1 (A = 45 deg., = 90 deg.)
- forced boundary points at initial and final times
- sensor axis departs from constraint cone

Angle between sensor axis and constraint cone

tf= 1.9258 (1% faster than BA-1)

250 nodes

- same geometry as BA-2 (A = 23 deg., = 70 deg.)
- forced boundary points at initial and final times
- sensor axis departs from constraint cone

Angle between sensor axis and constraint cone

tf= 1.2967 (0.4% faster than BA-2)

100 nodes

constraint boundary

Constrained Rotation Axis

- Entire s/c executes -rotation
- sensor axis on cone boundary
- rotation axis along cone axis

Problem now becomes one-dimensional,

with bang-bang solution

Applying to geometry of:

BA-1 tf = 2.1078 (8% longer than BA-1)

BA-2 tf = 2.0966 (37% longer than BA-2)

Practical Consideration

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

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

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