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Tracking Control of Nanosatellites with Uncertain Time Varying Parameters
1. Problem Statement
Control Formulation
Conclusions
Tracking Control of Nanosatellites
with Uncertain Time Varying Parameters
Divya Thakur 1
and Belinda G. Marchand 2
Department of Aerospace Engineering and Engineering Mechanics
The University of Texas at Austin
AAS/AIAA Astrodynamics Specialist Conference
July 31 - August 4, 2011 Girdwood, Alaska
1
Graduate Student, Department of Aerospace Engineering
2
Assistant Professor, Department of Aerospace Engineering, AIAA Associate Fellow
Thakur, Marchand Tracking Control for Uncertain Time-Varying Matrix 1/ 18
2. Problem Statement
Control Formulation
Conclusions
Motivation
Error Dynamics
Modeling a Time-Varying Inertia Matrix
Motivation
◮ Spacecraft tracking problem is widely studied.
◮ Many adaptive control solutions for systems with constant uncertain inertia
parameters.
◮ Limited research in adaptive control of time-varying inertia matrix.
◮ Focus of study: Adaptation mechanism that maintains consistent tracking
performance in the face of uncertain time-varying inertia matrix.
Thakur, Marchand Tracking Control for Uncertain Time-Varying Matrix 2/ 18
3. Problem Statement
Control Formulation
Conclusions
Motivation
Error Dynamics
Modeling a Time-Varying Inertia Matrix
Attitude-Tracking Error Dynamics
◮ Attitude-error dynamics:
˙qe0 = −
1
2
qT
ev
ωe
˙qev
=
1
2
qe0 I + qev
× ωe
Angular-velocity tracking error dynamics:
˙ωe = J−1
−˙Jω − [ω×]Jω + u + [ωe×]B
CR
(qe)ωr − B
CR
(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)] and
uncertainty in J(t).
Thakur, Marchand Tracking Control for Uncertain Time-Varying Matrix 3/ 18
4. Problem Statement
Control Formulation
Conclusions
Motivation
Error Dynamics
Modeling a Time-Varying Inertia Matrix
Attitude-Tracking Error Dynamics
◮ Attitude-error dynamics:
˙qe0 = −
1
2
qT
ev
ωe
˙qev
=
1
2
qe0 I + qev
× ωe
Angular-velocity tracking error dynamics:
˙ωe = J−1
−˙Jω − [ω×]Jω + u + [ωe×]B
CR
(qe)ωr − B
CR
(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)] and
uncertainty in J(t).
Thakur, Marchand Tracking Control for Uncertain Time-Varying Matrix 3/ 18
5. Problem Statement
Control Formulation
Conclusions
Motivation
Error Dynamics
Modeling a Time-Varying Inertia Matrix
Attitude-Tracking Error Dynamics
◮ Attitude-error dynamics:
˙qe0 = −
1
2
qT
ev
ωe
˙qev
=
1
2
qe0 I + qev
× ωe
Angular-velocity tracking error dynamics:
˙ωe = J−1
−˙Jω − [ω×]Jω + u + [ωe×]B
CR
(qe)ωr − B
CR
(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)] and
uncertainty in J(t).
Thakur, Marchand Tracking Control for Uncertain Time-Varying Matrix 3/ 18
6. Problem Statement
Control Formulation
Conclusions
Motivation
Error Dynamics
Modeling a Time-Varying Inertia Matrix
Attitude-Tracking Error Dynamics
◮ Attitude-error dynamics:
˙qe0 = −
1
2
qT
ev
ωe
˙qev
=
1
2
qe0 I + qev
× ωe
Angular-velocity tracking error dynamics:
˙ωe = J−1
−˙Jω − [ω×]Jω + u + [ωe×]B
CR
(qe)ωr − B
CR
(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)] and
uncertainty in J(t).
Thakur, Marchand Tracking Control for Uncertain Time-Varying Matrix 3/ 18
7. Problem Statement
Control Formulation
Conclusions
Motivation
Error Dynamics
Modeling a Time-Varying Inertia Matrix
Type of Inertia Matrix Considered
◮ Time-varying inertia matrix of the form
J(t) = JoΨ(t)
◮ Jo: Jo > 0, JT
o = 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 undergoing
1. Thermal Variations
2. Fuel slosh
3. Appendage deployment (sensor booms, solar sails, antennas, etc.)
Thakur, Marchand Tracking Control for Uncertain Time-Varying Matrix 4/ 18
8. Problem Statement
Control Formulation
Conclusions
Motivation
Error Dynamics
Modeling a Time-Varying Inertia Matrix
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 BODY
STOWED
BOOM
Thakur, Marchand Tracking Control for Uncertain Time-Varying Matrix 5/ 18
9. Problem Statement
Control Formulation
Conclusions
Motivation
Error Dynamics
Modeling a Time-Varying Inertia Matrix
Example of a Time-Varying Inertia Matrix (2/2)
◮ Rod length, rod mass, and prism mass are (respectively)
r(t) =
2l1
τ
, mp(t) =
αm0
τ
t, mc(t) = m0 − 2mp(t)
◮ Inertia matrix given by
Jo =
5
6
m0l2
1 0
0 5
6
m0l2
1 0
0 0 1
6
m0l2
1
For 0 ≤ t ≤ τ,
Ψ(t) =
1 − 2α
τ
t 0 0
0 1 − 7
5
α
τ
t + 12
5
α
τ2 t2
+ 16
5
α
τ3 t3
0
0 0 1 + α
τ
t + 12 α
τ2 t2
+ 16 α
τ3 t3
,
for t > τ,
Ψ(t) =
1 − 2α 0 0
0 1 − 7
5
α + 12
5
α + 16
5
α 0
0 0 1 + α + 12α + 16α
.
Thakur, Marchand Tracking Control for Uncertain Time-Varying Matrix 6/ 18
10. Problem Statement
Control Formulation
Conclusions
Adaptive Control
Numerical Simulations
Control Formulation
◮ Control method based on the non-certainty equivalence (non-CE) adaptive
control results of Seo and Akella (2008)3
.
◮ Provides superior performance over traditional CE based methods when
reference 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 inertia
matrix of the specific form
J(t) = JoΨ(t).
3
Seo, D. and Akella, M. R., High-Performance Spacecraft Adaptive Attitude-Tracking Control Through
Attracting-Manifold Design, Journal of Guidance, Control, and Dynamics, Vol. 31, No. 4, 2008, pp. 884–891
Thakur, Marchand Tracking Control for Uncertain Time-Varying Matrix 7/ 18
11. Problem Statement
Control Formulation
Conclusions
Adaptive Control
Numerical Simulations
Non-CE Adaptive Controller
◮ For problem described by tracking-error equations and inertia matrix
J = JoΨ(t), control input is
u = Ψ −W ˆθ + δ + Wf ΓWT
f kp(qev
− ωef ) + ωe
˙ˆθ = ΓWT
f (β + kv)ωef + kpqev
− ΓWT
ωef
δ = ΓWT
f ωef ,
◮ Regressor matrix
Wθ∗
= −Ψ−1
Jo
˙Ψω − Ψ−1
[ω×]JoΨω + Jo [ω×]B
CR
(qe)ωr − B
CR
(qe) ˙ωr
+ Jo kpβqev
+ kp ˙qev
+ kvωe ,
◮ Parameters: θ∗
= [Jo11 , Jo12 , Jo13 , Jo22 , Jo23 , Jo33 ]T
.
◮ Filter variables
˙ωef = −βωef + ωe
˙Wf = −βWf + W,
Thakur, Marchand Tracking Control for Uncertain Time-Varying Matrix 8/ 18
12. Problem Statement
Control Formulation
Conclusions
Adaptive Control
Numerical Simulations
Non-CE Adaptive Controller
◮ For problem described by tracking-error equations and inertia matrix
J = JoΨ(t), control input is
u = Ψ −W ˆθ + δ + Wf ΓWT
f kp(qev
− ωef ) + ωe
˙ˆθ = ΓWT
f (β + kv)ωef + kpqev
− ΓWT
ωef
δ = ΓWT
f ωef ,
◮ Regressor matrix
Wθ∗
= −Ψ−1
Jo
˙Ψω − Ψ−1
[ω×]JoΨω + Jo [ω×]B
CR
(qe)ωr − B
CR
(qe) ˙ωr
+ Jo kpβqev
+ kp ˙qev
+ kvωe ,
◮ Parameters: θ∗
= [Jo11 , Jo12 , Jo13 , Jo22 , Jo23 , Jo33 ]T
.
◮ Filter variables
˙ωef = −βωef + ωe
˙Wf = −βWf + W,
Thakur, Marchand Tracking Control for Uncertain Time-Varying Matrix 8/ 18
13. Problem Statement
Control Formulation
Conclusions
Adaptive Control
Numerical Simulations
Non-CE Adaptive Controller
◮ For problem described by tracking-error equations and inertia matrix
J = JoΨ(t), control input is
u = Ψ −W ˆθ + δ + Wf ΓWT
f kp(qev
− ωef ) + ωe
˙ˆθ = ΓWT
f (β + kv)ωef + kpqev
− ΓWT
ωef
δ = ΓWT
f ωef ,
◮ Regressor matrix
Wθ∗
= −Ψ−1
Jo
˙Ψω − Ψ−1
[ω×]JoΨω + Jo [ω×]B
CR
(qe)ωr − B
CR
(qe) ˙ωr
+ Jo kpβqev
+ kp ˙qev
+ kvωe ,
◮ Parameters: θ∗
= [Jo11 , Jo12 , Jo13 , Jo22 , Jo23 , Jo33 ]T
.
◮ Filter variables
˙ωef = −βωef + ωe
˙Wf = −βWf + W,
Thakur, Marchand Tracking Control for Uncertain Time-Varying Matrix 8/ 18
14. Problem Statement
Control Formulation
Conclusions
Adaptive Control
Numerical Simulations
Non-CE Adaptive Controller
◮ For problem described by tracking-error equations and inertia matrix
J = JoΨ(t), control input is
u = Ψ −W ˆθ + δ + Wf ΓWT
f kp(qev
− ωef ) + ωe
˙ˆθ = ΓWT
f (β + kv)ωef + kpqev
− ΓWT
ωef
δ = ΓWT
f ωef ,
◮ Regressor matrix
Wθ∗
= −Ψ−1
Jo
˙Ψω − Ψ−1
[ω×]JoΨω + Jo [ω×]B
CR
(qe)ωr − B
CR
(qe) ˙ωr
+ Jo kpβqev
+ kp ˙qev
+ kvωe ,
◮ Parameters: θ∗
= [Jo11 , Jo12 , Jo13 , Jo22 , Jo23 , Jo33 ]T
.
◮ Filter variables
˙ωef = −βωef + ωe
˙Wf = −βWf + W,
Thakur, Marchand Tracking Control for Uncertain Time-Varying Matrix 8/ 18
15. Problem Statement
Control Formulation
Conclusions
Adaptive Control
Numerical Simulations
Non-CE Adaptive Controller
◮ For problem described by tracking-error equations and inertia matrix
J = JoΨ(t), control input is
u = Ψ −W ˆθ + δ + Wf ΓWT
f kp(qev
− ωef ) + ωe
˙ˆθ = ΓWT
f (β + kv)ωef + kpqev
− ΓWT
ωef
δ = ΓWT
f ωef ,
◮ Regressor matrix
Wθ∗
= −Ψ−1
Jo
˙Ψω − Ψ−1
[ω×]JoΨω + Jo [ω×]B
CR
(qe)ωr − B
CR
(qe) ˙ωr
+ Jo kpβqev
+ kp ˙qev
+ kvωe ,
◮ Parameters: θ∗
= [Jo11 , Jo12 , Jo13 , Jo22 , Jo23 , Jo33 ]T
.
◮ Filter variables
˙ωef = −βωef + ωe
˙Wf = −βWf + W,
Thakur, Marchand Tracking Control for Uncertain Time-Varying Matrix 8/ 18
16. Problem Statement
Control Formulation
Conclusions
Adaptive Control
Numerical Simulations
Non-CE Adaptive Controller
◮ For problem described by tracking-error equations and inertia matrix
J = JoΨ(t), control input is
u = Ψ −W ˆθ + δ + Wf ΓWT
f kp(qev
− ωef ) + ωe
˙ˆθ = ΓWT
f (β + kv)ωef + kpqev
− ΓWT
ωef
δ = ΓWT
f ωef ,
◮ Regressor matrix
Wθ∗
= −Ψ−1
Jo
˙Ψω − Ψ−1
[ω×]JoΨω + Jo [ω×]B
CR
(qe)ωr − B
CR
(qe) ˙ωr
+ Jo kpβqev
+ kp ˙qev
+ kvωe ,
◮ Parameters: θ∗
= [Jo11 , Jo12 , Jo13 , Jo22 , Jo23 , Jo33 ]T
.
◮ Filter variables
˙ωef = −βωef + ωe
˙Wf = −βWf + W,
Thakur, Marchand Tracking Control for Uncertain Time-Varying Matrix 8/ 18
17. Problem Statement
Control Formulation
Conclusions
Adaptive Control
Numerical Simulations
Numerical Simulations
◮ Two sets of simulations:
1. Non-PE reference trajectory
2. PE reference trajectory
◮ Simulation features
◮ Quantities used to calculate Jo and Ψ
m0 = 30 kg, l = 0.2 m, α = 0.1, τ = 200 s
◮ Uncertain parameter
Jo =
0.2 0
0 0.2 0
0 0 1.0
−→ θ∗
= [0.2, 0, 0, 0.2, 0, 1.0]T
◮ Initial parameter estimate: ˆθ(0) + δ(0) = 1.3 θ∗
◮ Simulation period is τ = 200 seconds.
Thakur, Marchand Tracking Control for Uncertain Time-Varying Matrix 9/ 18
18. Problem Statement
Control Formulation
Conclusions
Adaptive Control
Numerical Simulations
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 values kp = 0.08, kv = 0.07, Γ = diag {100, 0.01, 0.01, 200, 0.01, 100}.
0 50 100 150 200
1e−007
1e−006
1e−005
0.0001
0.001
0.01
0.1
time (s)
AngularVel.ErrorVectorNorm
Norm of angular velocity error vector ωe
0 50 100 150 200
1e−007
1e−006
1e−005
0.0001
0.001
0.01
0.1
1
time (s)
QuaternionErrorVectorNorm
Norm of quaternion error vector qev
Thakur, Marchand Tracking Control for Uncertain Time-Varying Matrix 10/ 18
19. Problem Statement
Control Formulation
Conclusions
Adaptive Control
Numerical Simulations
Non-PE Reference Trajectory (2/3)
10
−2
10
−1
10
0
10
1
10
2
10
3
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
time(s)
ControlTorqueNorm(N−m)
Norm of control vector u
Thakur, Marchand Tracking Control for Uncertain Time-Varying Matrix 11/ 18
20. Problem Statement
Control Formulation
Conclusions
Adaptive Control
Numerical Simulations
Non-PE Reference Trajectory (3/3)
10
−1
10
0
10
1
10
2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
time(s)
J
0
Parameters
J
o
(3,3)
J
o
(1,1) = J
o
(2,2)
Estimated
True
Parameter estimates converge to true values due to additional
persistence of excitation introduced by Ψ(t)
Thakur, Marchand Tracking Control for Uncertain Time-Varying Matrix 12/ 18
21. Problem Statement
Control Formulation
Conclusions
Adaptive Control
Numerical Simulations
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 50 100 150 200
0.0001
0.001
0.01
0.1
1
10
time (s)
AngularVel.ErrorVectorNorm
Norm of angular velocity error vector ωe
0 50 100 150 200
0.0001
0.001
0.01
0.1
1
time (s)
QuaternionErrorVectorNorm
Norm of quaternion error vector qev
Thakur, Marchand Tracking Control for Uncertain Time-Varying Matrix 13/ 18
22. Problem Statement
Control Formulation
Conclusions
Adaptive Control
Numerical Simulations
PE Reference Trajectory (2/3)
10
−2
10
−1
10
0
10
1
10
2
10
3
0
2
4
6
8
10
12
time(s)
ControlTorqueNorm(N−m)
Norm of control vector u
Thakur, Marchand Tracking Control for Uncertain Time-Varying Matrix 14/ 18
23. Problem Statement
Control Formulation
Conclusions
Adaptive Control
Numerical Simulations
PE Reference Trajectory (3/3)
10
−1
10
0
10
1
10
2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
time(s)
J
0
Parameters
J
o
(3,3)
J
o
(1,1) = J
o
(2,2)
Estimated
True
Parameter estimates converge to true values
Thakur, Marchand Tracking Control for Uncertain Time-Varying Matrix 15/ 18
24. Problem Statement
Control Formulation
Conclusions
Conclusions
◮ A non-CE adaptive control law employed for spacecraft attitude tracking in the
presence 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 is
non-PE.
Thakur, Marchand Tracking Control for Uncertain Time-Varying Matrix 16/ 18
25. Problem Statement
Control Formulation
Conclusions
Extra 1: Some Necessary Manipulations
◮ The following algebraic manipulations are necessary to enable the adaptive
control derivation
˙ωe = −kpβqev
− kp ˙qev
− kvωe
subtracted term
+ J−1
o
Ψ−1
u − Jo
˙Ψω − [ω×]JoΨω − Joφ + Jo kpβqev
+ kp ˙qev
+ kvωe
added term
,
where φ = [ωe×]B
CR
(qe)ωr − B
CR
(qe) ˙ωr
◮ kp, kv > 0 and β = kp + kv
◮ Note: Dynamics are unchanged
Thakur, Marchand Tracking Control for Uncertain Time-Varying Matrix 17/ 18
26. Problem Statement
Control Formulation
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
are Wf (0) = 0 and ωf (0) =
ωe(0)+kpqve
(0)
kp
.
Thakur, Marchand Tracking Control for Uncertain Time-Varying Matrix 18/ 18