Tracking Control of Nanosatellites with Uncertain Time Varying Parameters

Problem StatementControl Formulation

Conclusions

Tracking Control of Nanosatelliteswith Uncertain Time Varying Parameters

Divya Thakur1 and Belinda G. Marchand2

Department of Aerospace Engineering and Engineering MechanicsThe University of Texas at Austin

AAS/AIAA Astrodynamics Specialist ConferenceJuly 31 - August 4, 2011 Girdwood, Alaska

1Graduate Student, Department of Aerospace Engineering2Assistant Professor, Department of Aerospace Engineering, AIAA Associate Fellow

Thakur, Marchand Tracking Control for Uncertain Time-Varying Matrix 1/ 18


Conclusions

MotivationError DynamicsModeling a Time-Varying Inertia Matrix

Motivation

◮ Spacecraft tracking problem is widely studied.

◮ Many adaptive control solutions for systems with constant uncertain inertiaparameters.

◮ Limited research in adaptive control of time-varying inertia matrix.

◮ Focus of study: Adaptation mechanism that maintains consistent trackingperformance in the face of uncertain time-varying inertia matrix.



Conclusions


Attitude-Tracking Error Dynamics

◮ Attitude-error dynamics:

qe0 = −12

qTevωe

qev=

12

(qe0I +

[qev

×])

ωe

Angular-velocity tracking error dynamics:

ωe = J−1(

−Jω − [ω×]Jω + u)

+ [ωe×]BCR(qe)ωr −BCR(qe)ωr

◮ Control objective: Find u(t) s.t. limt→∞

[qe, ωe

]= 0 for any

[qr(t),ωr(t)] for all [q(0),ω(0)], assuming full feedback[q(t),ω(t)] anduncertainty inJ(t).



Conclusions




qe0 = −12

qTevωe

qev=

12

(qe0I +

[qev

×])

ωe


ωe = J−1(

−Jω − [ω×]Jω + u)



[qe, ωe

]= 0 for any




Conclusions




qe0 = −12

qTevωe

qev=

12

(qe0I +

[qev

×])

ωe


ωe = J−1(

−Jω − [ω×]Jω + u)



[qe, ωe

]= 0 for any




Conclusions




qe0 = −12

qTevωe

qev=

12

(qe0I +

[qev

×])

ωe


ωe = J−1(

−Jω − [ω×]Jω + u)



[qe, ωe

]= 0 for any




Conclusions


Type of Inertia Matrix Considered

◮ Time-varying inertia matrix of the form

J(t) = JoΨ(t)

◮ Jo:(

Jo > 0, JTo = Jo

)

constant, unknown or uncertain

◮ Ψ(t):(

Ψ > 0, ΨT = Ψ)

time-varying, known

◮ Uncertainty itself is constant, multiplicative

◮ May be used to model spacecraft undergoing1. Thermal Variations2. Fuel slosh3. Appendage deployment (sensor booms, solar sails, antennas, etc.)



Conclusions


Example of a Time-Varying Inertia Matrix (1/2)◮ Consider a spacecraft undergoing boom deployment (e.g., GOES-R spacecraft):

◮ Boom extension rate controlled by miniature DC-torque motors◮ Initial mass of prism:m0◮ Mass of fully extended boom:αm0, 0 < α < 1

DEPLOYED BOOM

12l

SENSOR

SATELLITE

MAIN BODY

1l

13l

1l

STOWED

COLLABPSIBLE

BOOM

SATELLITE

MAIN BODYSTOWED

BOOM



Conclusions


Example of a Time-Varying Inertia Matrix (2/2)◮ Rod length, rod mass, and prism mass are (respectively)

r(t) =2l1τ

, mp(t) =αm0

τt, mc(t) = m0 − 2mp(t)

◮ Inertia matrix given by

Jo =

56m0l21 0

0 56m0l21 0

0 0 16m0l21

For 0≤ t ≤ τ ,

Ψ(t) =

1− 2α

τt 0 0

0 1− 75α

τt + 12

5α

τ2 t2 + 16

5α

τ3 t3 0

0 0 1+ α

τt + 12α

τ2 t2 + 16α

τ3 t3

,

for t > τ ,

Ψ(t) =

1− 2α 0 00 1− 7

5α+ 125 α+ 16

5 α 00 0 1+ α+ 12α+ 16α

.



Conclusions

Adaptive ControlNumerical Simulations

Control Formulation

◮ Control method based on the non-certainty equivalence (non-CE) adaptivecontrol results of Seo and Akella (2008)3.

◮ Provides superior performance over traditional CE based methods whenreference trajectory does not satisfy certain persistence of excitation (PE)conditions.

◮ Original result treats constant inertia matrix.

◮ Present investigation modifies original result to handle time-varying inertiamatrix of the specific form

J(t) = JoΨ(t).

3Seo, D. and Akella, M. R., High-Performance Spacecraft Adaptive Attitude-Tracking Control ThroughAttracting-Manifold Design,Journal of Guidance, Control, and Dynamics, Vol. 31, No. 4, 2008, pp. 884–891



Conclusions


Non-CE Adaptive Controller◮ For problem described by tracking-error equations and inertia matrix

J = JoΨ(t), control input is

u = Ψ

(

−W(

θ + δ)

+ WfΓWTf

(kp(qev

− ωef ) + ωe

))

˙θ = ΓWT

f

[(β + kv)ωef + kpqev

]− ΓWT

ωef

δ = ΓWTf ωef ,

◮ Regressor matrix

Wθ∗ = −Ψ

−1JoΨω −Ψ−1[ω×]JoΨω + Jo

(

[ω×]BCR(qe)ωr −BCR(qe)ωr

)

+ Jo

(kpβqev

+ kpqev+ kvωe

),

◮ Parameters:θ∗ = [Jo11, Jo12, Jo13, Jo22, Jo23, Jo33]T.

◮ Filter variables

ωef = −βωef + ωe

Wf = −βWf + W,



Conclusions




u = Ψ

(

−W(

θ + δ)

+ WfΓWTf

(kp(qev

− ωef ) + ωe

))

˙θ = ΓWT

f


]− ΓWT

ωef

δ = ΓWTf ωef ,


Wθ∗ = −Ψ


(


)

+ Jo

(kpβqev

+ kpqev+ kvωe

),




Wf = −βWf + W,



Conclusions




u = Ψ

(

−W(

θ + δ)

+ WfΓWTf

(kp(qev

− ωef ) + ωe

))

˙θ = ΓWT

f


]− ΓWT

ωef

δ = ΓWTf ωef ,


Wθ∗ = −Ψ


(


)

+ Jo

(kpβqev

+ kpqev+ kvωe

),




Wf = −βWf + W,



Conclusions




u = Ψ

(

−W(

θ + δ)

+ WfΓWTf

(kp(qev

− ωef ) + ωe

))

˙θ = ΓWT

f


]− ΓWT

ωef

δ = ΓWTf ωef ,


Wθ∗ = −Ψ


(


)

+ Jo

(kpβqev

+ kpqev+ kvωe

),




Wf = −βWf + W,



Conclusions




u = Ψ

(

−W(

θ + δ)

+ WfΓWTf

(kp(qev

− ωef ) + ωe

))

˙θ = ΓWT

f


]− ΓWT

ωef

δ = ΓWTf ωef ,


Wθ∗ = −Ψ


(


)

+ Jo

(kpβqev

+ kpqev+ kvωe

),




Wf = −βWf + W,



Conclusions




u = Ψ

(

−W(

θ + δ)

+ WfΓWTf

(kp(qev

− ωef ) + ωe

))

˙θ = ΓWT

f


]− ΓWT

ωef

δ = ΓWTf ωef ,


Wθ∗ = −Ψ


(


)

+ Jo

(kpβqev

+ kpqev+ kvωe

),




Wf = −βWf + W,



Conclusions


Numerical Simulations

◮ Two sets of simulations:1. Non-PE reference trajectory2. PE reference trajectory

◮ Simulation features

◮ Quantities used to calculateJo andΨ

m0 = 30 kg, l = 0.2 m, α = 0.1, τ = 200 s

◮ Uncertain parameter

Jo =

0.2 00 0.2 00 0 1.0

−→ θ∗ = [0.2, 0, 0, 0.2, 0, 1.0]T

◮ Initial parameter estimate:θ(0) + δ(0) = 1.3 θ∗

◮ Simulation period isτ = 200 seconds.



Conclusions


Non-PE Reference Trajectory (1/3)◮ ωr =

(

0.1 cos(t)(1− e0.01t2) + (0.08π + 0.006 sin(t))te−0.01t2)

· [1, 1, 1]T

◮ Gain valueskp = 0.08,kv = 0.07,Γ = diag{100, 0.01, 0.01, 200, 0.01, 100}.

0 50 100 150 2001e−007

1e−006

1e−005

0.0001

0.001

0.01

0.1

time (s)

Ang

ular

Vel

. Err

or V

ecto

r N

orm

Norm of angular velocity error vector‖ωe‖

0 50 100 150 2001e−007

1e−006

1e−005

0.0001

0.001

0.01

0.1

1

time (s)

Qua

tern

ion

Err

or V

ecto

r N

orm

Norm of quaternion error vector‖qev‖



Conclusions


Non-PE Reference Trajectory (2/3)

10−2

10−1

100

101

102

103

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

time(s)

Con

trol

Tor

que

Nor

m (

N−

m)

Norm of control vector‖u‖



Conclusions


Non-PE Reference Trajectory (3/3)

10−1

100

101

102

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

time(s)

J 0 Par

amet

ers

Jo(3,3)

Jo(1,1) = J

o(2,2)

Estimated

True

Parameter estimates converge to true values due to additionalpersistence of excitation introduced byΨ(t)



Conclusions


PE Reference Trajectory (1/3)◮ ωr =

[cos(t) + 2 5 cos(t) sin(t) + 2

]T

◮ Gain values:kp = 0.8, kv = 0.8,Γ = diag{1, 0.001, 0.001, 1, 0.001, 1}.

0 50 100 150 2000.0001

0.001

0.01

0.1

1

10

time (s)

Ang

ular

Vel

. Err

or V

ecto

r N

orm

Norm of angular velocity error vector‖ωe‖

0 50 100 150 2000.0001

0.001

0.01

0.1

1

time (s)

Qua

tern

ion

Err

or V

ecto

r N

orm

Norm of quaternion error vector‖qev‖



Conclusions


PE Reference Trajectory (2/3)

10−2

10−1

100

101

102

103

0

2

4

6

8

10

12

time(s)

Con

trol

Tor

que

Nor

m (

N−

m)

Norm of control vector‖u‖



Conclusions


PE Reference Trajectory (3/3)

10−1

100

101

102

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

time(s)

J 0 Par

amet

ers

Jo(3,3)

Jo(1,1) = J

o(2,2)

Estimated

True

Parameter estimates converge to true values



Conclusions

Conclusions

◮ A non-CE adaptive control law employed for spacecraft attitude tracking in thepresence of uncertain time-varying inertia matrix.

◮ Uncertainty has special multiplicative structure.

◮ Numerical simulations performed for PE and non-PE reference signals.

◮ Attitude and angular-velocity tracking errors converge to zero.

◮ Parameter estimates converge to true values even when reference signal isnon-PE.



Conclusions

Extra 1: Some Necessary Manipulations

◮ The following algebraic manipulations are necessary to enable the adaptivecontrol derivation

ωe = −kpβqev− kpqev

− kvωe︸︷︷︸

subtracted term

+ J−1o

Ψ

−1(

u − JoΨω − [ω×]JoΨω)

− Joφ+ Jo

(kpβqev

+ kpqev+ kvωe

)

︸︷︷︸

added term

,

whereφ =([ωe×]BCR(qe)ωr −

BCR(qe)ωr

)

◮ kp, kv > 0 andβ = kp + kv

◮ Note: Dynamics are unchanged



Conclusions

Extra 2: Initial Conditions for Simulations

◮ Initial conditions

q(0) =[

0.9487, 0.1826, 0.1826, 0.18268]T

ω(0) =[

0, 0, 0]T

rad/s

qr(0) = [1, 0, 0, 0]T

Wf (0) = 0 , ωf (0) =ωe(0) + kpqve

(0)

kp

◮ Initial filter-states1 areWf (0) = 0 andωf (0) =ωe(0)+kpqve (0)

kp.


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Tracking Control of Nanosatellites with Uncertain Time Varying Parameters

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