Robert G. MeltonDepartment of Aerospace EngineeringPenn State University

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

Numerical analysis of constrained time-optimal satellite reorientation

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

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

Constrained Problem (multiple cones): No Boundary Arcs or Points Observed

Example:0.1 deg. gap between Sun and Moon cones

4

0 0.5 1 1.5 2 2.5 3 3.5-1

0

1

2Angular Velocity

time

i

1

2

3

0 0.5 1 1.5 2 2.5 3 3.5-1

0

1

2Euler Parameters

time

i

1

2

3

4

0 0.5 1 1.5 2 2.5 3 3.5-1

0

1Control Torques

time

Mi

M1

M2

M3

tf = 3.0659, 300 nodes, 8 switches

Constrained Problem (multiple cones)

5

AA ˆˆcos 1

A

A

Keep-out Cone Constraint

(cone axis for source A)

(sensor axis)

6

4321

321

4321

321

HftJ min

0ˆˆcos 1 AAC

CHH ~

0000

CC

if if

0C :conditiontangency plus

Optimal Control Formulation

Resulting necessary conditions are analytically intractable

7

maxmaxmax

max

MMM

III

MI i

ii

iii

Numerical Studies

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

2. Sensor axis constrained not to enter the cone3. Entire s/c executes -rotation about A

Legendre pseudospectral method used(DIDO software)

Scaling:

fi ˆ,ˆ lie on constraint cone

• I1 = I2 = I3 and M1,max = M2,max = M3,max • lies along principal body axis b1

• final orientation of b2, b3 generally unconstrained8

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.

A

A

i

f

9

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-1

0

1

time

i

1

2

3

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-1

0

1

2

time

i

1

2

3

4

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-1

0

1

time

Mi

M1M2M3

Case BA-1 (forced boundary arc)

tf = 1.9480, 151 nodes 10

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-1.04

-1.02

-1

-0.98

-0.96

time

Ham

ilton

ian

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-1500

-1000

-500

0

500

time

cost

ates

1

2

3

1

2

3

4

Case BA-1 (forced boundary arc)

11

0 0.2 0.4 0.6 0.8 1 1.2 1.4-1

-0.5

0

0.5

time

i

1

2

3

0 0.2 0.4 0.6 0.8 1 1.2 1.4-1

0

1

2

time

i

1

2

3

4

0 0.2 0.4 0.6 0.8 1 1.2 1.4-1

0

1

time

Mi

M1M2M3

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 12

0 0.2 0.4 0.6 0.8 1 1.2 1.4-1.04

-1.02

-1

-0.98

-0.96

time

Ham

ilton

ian

0 0.2 0.4 0.6 0.8 1 1.2 1.4-300

-200

-100

0

100

time

cost

ates

1

2

3

1

2

3

4

Case BA-2 (forced boundary arc)

13

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

dist

ance

from

"kee

p-ou

t" co

ne (r

ad)

time

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

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

250 nodes

Angle between sensor axis and constraint cone

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0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-1

0

1

time

i

1

2

3

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-1

0

1

2

time

i

1

2

3

4

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-1

0

1

time

Mi

M1M2M3

Case BP-1

15

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-1.08

-1.06

-1.04

-1.02

-1

-0.98

time

Ham

ilton

ian

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-3000

-2000

-1000

0

1000

2000

time

cost

ates

1

2

3

1

2

3

4

Case BP-1

16

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.002

0.004

0.006

0.008

0.01

0.012

dist

ance

from

"kee

p-ou

t" co

ne (r

ad)

time

• 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

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

100 nodes

Angle between sensor axis and constraint cone

17

0 0.2 0.4 0.6 0.8 1 1.2 1.4-1

-0.5

0

0.5

time

i

1

2

3

0 0.2 0.4 0.6 0.8 1 1.2 1.4-1

0

1

2

time

i

1

2

3

4

0 0.2 0.4 0.6 0.8 1 1.2 1.4-1

0

1

time

Mi

M1M2M3

Case BP-2

18

0 0.2 0.4 0.6 0.8 1 1.2 1.4-1.1

-1.05

-1

-0.95

-0.9

time

Ham

ilton

ian

0 0.2 0.4 0.6 0.8 1 1.2 1.4-15000

-10000

-5000

0

5000

time

cost

ates

1

2

3

1

2

3

4

Case BP-2

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Sensor axis path along the constraint boundary

i

A ˆˆ

f

Constrained Rotation Axis

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

20

maxmax /ˆ MM

),,max( 321max

MIt f 2,

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)

Constrained Rotation Axis

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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

0 0.5 1 1.5 2 2.5-2

-1.5

-1

-0.5

0

0.5

1

1.51-D Slew Maneuver

time

cont

rol t

orqu

e

PSO valuesChebyshev approx

0 0.5 1 1.5 2 2.5 3

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Dido Result -- 200 nodes, accurate mode

time

cont

rol t

orqu

e

Dido

No guesscpu time = 148 sec.

With PSO guesscpu 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 case24

fin

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