STEERING LAWS FOR CONTROL MOMENT GYROSCOPE SYSTEMS USED IN SPACECRAFTS ATTITUDE CONTROL
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Transcript of STEERING LAWS FOR CONTROL MOMENT GYROSCOPE SYSTEMS USED IN SPACECRAFTS ATTITUDE CONTROL
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Middle East Technical UniversityMiddle East Technical UniversityAeroAerospacespace Engineering Department Engineering Department
STEERING LAWS FOR CONTROL MOMENT GYROSCOPE SYSTEMS
USED IN SPACECRAFTS ATTITUDE CONTROL
by by Emre YAVUZOĞLUEmre YAVUZOĞLU
Supervisor: Supervisor: Assoc. Prof. Dr. Assoc. Prof. Dr. Ozan TEKİNALPOzan TEKİNALP
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Outline• Objectives
• Properties of SGCMGs
• Overview of Steering Laws
• Simulation Work I
• CMG based ACS Model
• Simulation Work II
• Conclusion
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• Investigation of the kinematic properties of SGCMGs ( + singularity problem)
• Steering laws:– Existing steering laws
– Development of new steering laws
– Comparison of steering laws through simulations
Objectives
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• Momentum exchange device
• A SGCMG consists of– Flywheel
(spinning at a constant rate)– Gimbal motor
(to change the direction of h)
SGCMGs
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The output torque is:
τ h δ h
τ h δ h
(Torque amplification)
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4-SGCMG Cluster in a Typical Pyramid Mounting Arrangement
• 3 CMGs to provide full 3-axis attitude control
• 1 CMG to provide extra degree of control
(min. redundancy for singularity)
β=54.73º to the horizontal
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• Total angular momentum for pyramid configuration:
4
i ii=1
h= h ( )
2 431
1 2 3 4
1 2 43
-cos coscos sincos sin
cos -cos sin -cos -cos sin
sin sin sin sin sin sinsin sin
-
h = + + +
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•Total output torque (time rate of change of total h)
1 2 3 4
1 2 3 4
1 2 3 4
cos cos sin cos cos sin
sin cos cos sin cos cos
sin cos sin cos sin cos sin cos
hJ(δ)
δ
h=J(δ)δ
However, in ACS part we need to determine gimbal rate, that provides the required torque. Thus, we need an inversion of torque equation:
where instantaneous system Jacobian matrix:
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• Minimum two-norm solution of this problem gives Moore Penrose pseudo inverse:
1
T TMPδ = J(δ) J(δ)J(δ) τ
• Most of the steering laws is pseudo inverse based. However, the main problem of these methods are SINGULARITY (Although CMG cluster is redundant).
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What is Singularity?
• Mathematically; When J loses rank (rank:2), (JJT)-
1 undefined.• Physically; all output torque vectors remain on the
same plane (rank:2). No output torque can be produced along direction, s, normal to this plane. (s: singularity direction). Three axis controllability is lost.
TJ s 0
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Singularity Measure
det( )m TJ(δ)J(δ)
(System is how much close to the singularity)
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• Singular states & directions produces singular surfaces in momentum envelope created by mapping of gimbal angle set to angular momentum space of the cluster.
• Singularity types seen in momentum envelope are summarized in a detailed fashion according to number of criteria in Chp 3 and Appendix A-3. The most dangerous ones are internal elliptic singularities.
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Overview of Steering Laws
2. SR Inverse
3. GSR Inverse
4. IG Method
1
T TMPδ = J(δ) J(δ)J(δ) τ
1. MP INVERSE (high possibility of encountering singular states)
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2. Singular Robust Inverse• Transition method adapted from robotic manipulators
(As singularity is approached small torque errors are permitted to transit through it.)
SR SRδ = J τ1T T a
SR 3J J JJ I
• α, the singularity avoidance parameter to be properly selected. It can be shown that the matrix within brackets is never singular.
• DIS: Although singularity measure never becomes zero, internal elliptic singularities still can not be passed with SR!
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3. Generalized Singular Robust Inverse• Modified version of SR inverse
As singularity is approached, deliberate deterministic dither signals of increasing amplitude are used to get out of singularity quickly:
T T -1G-SR [ ] δ J JJ E τ
3 2
3 1
2 1
1
= 1 0
1
E 0.01sin(0.5 )i it
0.01exp( 10 )m where
• DIS: Not suitable for precision tracking missions
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4. Inverse Gain • Previous particular solutions can be combined with
homogenous solution of torque equation to avoid singularities (=null motion):
particular homogenousδ = δ + δ
31 2
1 2 3 4
hh hn C ,C ,C ,C
δ δ δ
homogenous c.δ n
6
6
, 1
, 1
m mc
m m
• DIS: Null rates may become extremely high, even though system is away from singularity.
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New Steering Logic:Unified Steering Law
• Derived solving the following minimization problem:
err err err err
1
2T T
δmin δ Qδ τ Rτ
• Starting aim in the development was to find gimbal rates both satisfying torque commanded and, driving the gimbals to desired nonsingular configurations, spontaneously.
err desiredδ δ - δerrτ = Jδ - τ
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Resulted gimbal rate equation:
T Tq q -1
USL 4 desiredδ = I + J J δ + J τ
• Through simulations we have observed that selection of desired gimbal rate, and blending coefficient, q, are the key points in the utilization of the method. According to this selection, 2 approaches are proposed:
1. Preplanned Steering2. Online Steering
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1. Preplanned Steering
• h trajectory to be followed is known beforehand
• Gimbal angle solutions with higher m satisfying the h at discrete time points (=nodes) are computed using SA.
• Then, system is driven to desired gimbal solutions at these nodes by adjusting the gimbal rates as:
• DIS: Only requirement to steer desired gimbal set is that required h trajectory should be known priori.
( 1)
1,...,k
k t t k t
k pk t t
δ δ
δ
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2. Online Steering Approaches
For selection of desired gimbal rate in USL Eqn.:
a. Homogenous gimbal rates found by IG
b. Arbitrary constant vector
c. Intelligently selected constant vector
d. Dynamic vector with randomly changing elements
(White Noise)
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Simulations I• Constant torque study
0 0.5 1 1.5 2 2.5-0.2
0
0.2
0.4
0.6
0.8
1
Torque Realized
t (s)
T (
Nm
)
TxTyTz
0 ,0 ,0 ,0T oδ [1.155,0,0]T
des τ
0 0.5 1 1.5 2 2.5-0.5
0
0.5
1
1.5
2
2.5
3
Angular Momentum Trajectory
t (s)
h (
Nm
.s)
hxhyhz
Ideal Profiles
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MP & SR Fails at internal elliptic singularity!
0 0.5 1 1.5 2 2.5-0.5
0
0.5
1
1.5Angular Momentum Trajectory
t (s)
h (
Nm
.s)
hxhyhz
0 0.5 1 1.5 2 2.50
0.2
0.4
0.6
0.8
1
1.2
Singularity Measure
t (s)
m
0 0.5 1 1.5 2 2.5-0.2
0
0.2
0.4
0.6
0.8
1
Torque Realized
t (s)
T (
Nm
)
TxTyTz
0 0.5 1 1.5 2 2.5-100
-80
-60
-40
-20
0
20
40
60
80
100Gimbal Angles
t (s)
G (
deg
)
g1g2g3g4
90 ,0 ,90 ,0T
S δ 1.155,0,0T
S h
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GSR works as transition method(1.5 sec delay, high gimbal rates)
0 1 2 3 4-0.2
0
0.2
0.4
0.6
0.8
1
Torque Realized
t (s)
T (
Nm
)
TxTyTz
0 0.5 1 1.5 2 2.5 3 3.5 4 4.50
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2Singularity Measure
t (s)
m
0 1 2 3 4-0.5
0
0.5
1
1.5
2
2.5
3
Angular Momentum Trajectory
t (s)
h (
Nm
.s)
hxhyhz
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5-5
-2.5
0
2.5
5Gimbal Rates
t (s)
Gd
ot
(rad
/s)
g1dotg2dotg3dotg4dot
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USL Preplanned
0 0.5 1 1.5 2 2.5-0.2
0
0.2
0.4
0.6
0.8
1
Torque Realized
t (s)
T (
Nm
)
TxTyTz
•8 nodes are used. Successfully, desired torque is realized while accurately achieving desired gimbal set at nodes.
0 0.5 1 1.5 2 2.5
-100
-50
0
50
100
150
200Gimbal Angles
t (s)
G (
deg
) g1g2g3g4
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USL Online using Null vector
0 0.5 1 1.5 2 2.5-0.2
0
0.2
0.4
0.6
0.8
1
Torque Realized
t (s)
T (
Nm
)
TxTyTz
0 0.5 1 1.5 2 2.5-0.2
0
0.2
0.4
0.6
0.8
1
Torque Realized
t (s)
T (
Nm
)
TxTyTz
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USL Online Using Constant Vector (with dynamically adjusted blending coefficient q=0.5exp(-10m))
0 0.5 1 1.5 2 2.5 3 3.5-0.2
0
0.2
0.4
0.6
0.8
1
Torque Realized
t (s)
T (
Nm
)
TxTyTz
0 0.5 1 1.5 2 2.5-0.2
0
0.2
0.4
0.6
0.8
1
Torque Realized
t (s)
T (
Nm
)
TxTyTz
•Steering w. arbitrary vector [0,1,0,0]•Steering w. intelligently selected vector•Steering w. white noise
0 1 2 3 4-0.2
0
0.2
0.4
0.6
0.8
1
Torque Realized
t (s)
T (
Nm
)
TxTyTz
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USL Preplanned Corner maneuver
0 0.5 1 1.5-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1Torque Realized
t (s)
T (
Nm
)
TxTyTz
Repeatability maneuver
Cyclic Disturbance
0 0.5 1 1.5 2 2.5-0.2
0
0.2
0.4
0.6
0.8
1
1.2
Torque Realized
t (s)
T (
Nm
)
TxTyTz
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
-100
-50
0
50
100
150
200Gimbal Angles
t (s)
G (
deg
)
g1g2g3g4gdes1gdes2gdes3gdes4
T T1.15 0 0 0.115 [1 1 1] sin(2 )tdesiredτ =desired
1[1,1,0] , t<0.8344
21
[1, 1,0] , t>0.8344 2
T
T
τ
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CMG based ACS ModelThree main parts to be considered:
1. Spacecraft Dynamics
2. Quaternion Based Feedback Controller
3. CMG Steering Law
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Spacecraft Dynamics
Total angular momentum equation;
Corresponding rotational EoM of a rigid S/C equipped with momentum exchange actuators such as CMGs, in general given by;
Text: the external torque vector including the gravity gradient, solar pressure, and aerodynamic torques all expressed in the same S/C body axes.
S/C S/C extH ω H T
S/CH Jω h
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Combining these 2 equations, we simply obtain;
u: Internal control torque input generated by CMG and transferred to S/C
Rewriting equation in two parts;
By using last two equations, and combining them with S/C kinematics equations (such as quaternions), an ACS can be designed. Assuming S/C control torque input is known, the desired CMG momentum rate is selected as:
+ u- u extJω h ω Jω h T
extJω ω Jω T u
h ω h u
h ω h u
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Simulations II
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USL Preplanned Results
0 50 100 150 200 250 300-0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Angular Momentum Trajectory
t (s)
h (
Nm
.s)
hxhyhz
0 50 100 150 200 250 300-150
-100
-50
0
50
100Gimbal Angles
t (s)
G (
deg
)
g1g2g3g4gdes1gdes2gdes3gdes4
Desired h profile from ideal systemGimbal profiles obtained with USL
0 50 100 150 200 250 300-70
-60
-50
-40
-30
-20
-10
0
10Spacecraft Attitude Profile
t (s)
Ro
ll-P
itch
-Yaw
(
deg
.)
RollPitchYaw
Attitude profile obtained with USL
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USL Online Results
0 50 100 150 200 250-70
-60
-50
-40
-30
-20
-10
0
10Spacecraft Attitude Profile
t (s)
Ro
ll-P
itch
-Yaw
(
deg
.)
RollPitchYaw
0 50 100 150 200 2500
0.2
0.4
0.6
0.8
1
1.2
Singularity Measure
t (s)
m
0 50 100 150 200 250-250
-200
-150
-100
-50
0
50
100
150
200
250Gimbal Angles
t (s)
G (
deg
) g1g2g3g4
• Although simulation is started this time at internal elliptic singularity (i.e. [90, 0, -90, 0]deg), USL online method effectively takes the system away from singularity rapidly, and maneuver is completed on time!
• Arbitrarily selected vector [0,1,0,0] is used as desired gimbal rate with dynamically adjusted blending coefficient.
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Attitude Hold Maneuver
0.0025 sin(2 ) 0 0T
ntextT
A hypothetical cyclic disturbance torque, Text is given to the system:
Despite of the disturbance acting about one orbital period (~5400 s), the spacecraft is commanded to maintain its initial attitude of RPYinitial = [0˚,0˚,0˚] all the time.
•USL is used preplanned and online fashion for Attitude Hold maneuver. Both are successful and repeatable gimbal histories are observed.
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USL Attitude Hold Results
0 1000 2000 3000 4000 5000 6000-0.06
-0.04
-0.02
0
0.02
0.04
0.06Spacecraft Attitude Profile
t (s)
Ro
ll-P
itch
-Yaw
(
deg
.)RollPitchYaw
0 1000 2000 3000 4000 5000 6000-0.5
0
0.5
1
1.5
2
2.5Angular Momentum Trajectory
t (s)
h (
Nm
.s)
hxhyhz
Attitude profile obtained with USL online
0 1000 2000 3000 4000 5000 6000-100
-50
0
50
100
150Gimbal Angles
t (s)
G (
deg
)
g1g2g3g4
Gimbal History (Repeating pattern)
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CONCLUSION
• A new original robust steering law is presented. The steering law combines desired gimbal rates with torque requirements in a weighted fashion.
• The law can be employed in a both preplanned and spontaneous fashion. The repeatability of the approach is demonstrated.
• Singularity is not a problem with this method. Through simulations, it is demonstrated that it can replace all existing steering laws.
• This method may also be adapted to robotic manipulators as a future work.
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