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

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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


Gamma-Ray Bursts/ Swift


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-1 (forced boundary arc)


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 BA-2 (forced boundary arc)


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-1


Case BP-1


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


Case BP-2


Case BP-2


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


fin


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