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. 6 th International Workshop and Advanced School, “Spaceflight Dynamics and Control” Covilh ã , Portugal March 28-30, 2011.

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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

Keep-out Cone Constraint

(cone axis for source A)

(sensor axis)

Optimal Control Formulation

• Resulting necessary conditions are analytically intractable

Numerical Studies

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

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

Angle between sensor axis and constraint cone

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

250 nodes

Case BP-2

• 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

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

Constrained Rotation 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

Dido

No guess

cpu time = 148 sec.

With PSO guess

cpu time = 76 sec,

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