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

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

Gamma-Ray Bursts/ Swift


Numerical analysis of constrained time optimal satellite reorientation

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


Numerical analysis of constrained time optimal satellite reorientation

Constrained Problem (multiple cones):

No Boundary Arcs or Points Observed

Example:

0.1 deg. gap between Sun and Moon cones


Numerical analysis of constrained time optimal satellite reorientation

Constrained Problem (multiple cones)

tf= 3.0659, 300 nodes, 8 switches


Numerical analysis of constrained time optimal satellite reorientation

Keep-out Cone Constraint

(cone axis for source A)

(sensor axis)


Numerical analysis of constrained time optimal satellite reorientation

Optimal Control Formulation

  • Resulting necessary conditions are analytically intractable


Numerical analysis of constrained time optimal satellite reorientation

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


Numerical analysis of constrained time optimal satellite reorientation

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.


Numerical analysis of constrained time optimal satellite reorientation

Case BA-1 (forced boundary arc)

tf = 1.9480, 151 nodes


Numerical analysis of constrained time optimal satellite reorientation

Case BA-1 (forced boundary arc)


Numerical analysis of constrained time optimal satellite reorientation

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


Numerical analysis of constrained time optimal satellite reorientation

Case BA-2 (forced boundary arc)


Numerical analysis of constrained time optimal satellite reorientation

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


Numerical analysis of constrained time optimal satellite reorientation

Case BP-1


Numerical analysis of constrained time optimal satellite reorientation

Case BP-1


Numerical analysis of constrained time optimal satellite reorientation

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


Numerical analysis of constrained time optimal satellite reorientation

Case BP-2


Numerical analysis of constrained time optimal satellite reorientation

Case BP-2


Numerical analysis of constrained time optimal satellite reorientation

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


Numerical analysis of constrained time optimal satellite reorientation

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

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


Numerical analysis of constrained time optimal satellite reorientation

Dido

No guess

cpu time = 148 sec.

With PSO guess

cpu time = 76 sec,


Numerical analysis of constrained time optimal satellite reorientation

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


Numerical analysis of constrained time optimal satellite reorientation

fin


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