Transcript STEERING LAWS FOR CONTROL MOMENT GYROSCOPE SYSTEMS USED IN SPACECRAFTS ATTITUDE CONTROL

Middle East Technical University
Aerospace Engineering Department
STEERING LAWS FOR
CONTROL MOMENT GYROSCOPE SYSTEMS
USED IN
SPACECRAFTS ATTITUDE CONTROL
by Emre YAVUZOĞLU
Supervisor: Assoc. Prof. Dr. Ozan TEKİNALP
Outline
•
•
•
•
•
•
•
Objectives
Properties of SGCMGs
Overview of Steering Laws
Simulation Work I
CMG based ACS Model
Simulation Work II
Conclusion
Objectives
• 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
SGCMGs
• Momentum exchange device
• A SGCMG consists of
– Flywheel
(spinning at a constant rate)
– Gimbal motor
(to change the direction of h)
τ  h  δh
The output torque is:
τ hδh
(Torque amplification)
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
• Total angular momentum for pyramid
configuration:
4
h=  hi(i )
i=1
- cos  sin    -cos 
 cos  sin   

cos

2
4
3 
1 

 

h =  cos1  + -cos  sin  2  +  -cos 3  + -cos  sin  4 
 

 
 

sin

sin

sin

sin

sin

sin

sin

sin


1  
2  
4 
3  
•Total output torque (time rate of change of total h)
h = J(δ)δ
where instantaneous system Jacobian matrix:
sin  2
  cos  cos 1
h 
J(δ) 
   sin 1
 cos  cos  2
δ
 sin  cos 1
sin  cos  2
cos  cos  3
sin  3
sin  cos  3
 sin  4

cos  cos  4 
sin  cos  4 
However, in ACS part we need to determine gimbal rate,
that provides the required torque. Thus, we need an
inversion of torque equation:
• Minimum two-norm solution of this problem gives Moore
Penrose pseudo inverse:
1
δMP = J(δ)  J(δ)J(δ)  τ
T
T
• Most of the steering laws is pseudo inverse based.
However, the main problem of these methods are
SINGULARITY (Although CMG cluster is redundant).
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.
J s0
T
Singularity Measure
(System is how much close to the singularity)
m  det(J(δ)J(δ) )
T
• 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.
Overview of Steering Laws
1. MP INVERSE
(high possibility of encountering singular states)
1
δMP = J(δ)  J(δ)J(δ)  τ
T
2. SR Inverse
3. GSR Inverse
4. IG Method
T
2. Singular Robust Inverse
• Transition method adapted from robotic manipulators
(As singularity is approached small torque errors are permitted to
transit through it.)
δSR = JSR τ
J SR  J  JJ  aI 3 
T
T
1
• 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!
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:
δG-SR  J T [JJ T   E]-1 τ
where
  0.01exp(10m)
 1 3 2 


E=   3 1  1   0
 

 2 1 1
 i  0.01sin(0.5 t  i )
• DIS: Not suitable for precision tracking missions
4. Inverse Gain
• Previous particular solutions can be combined with
homogenous solution of torque equation to avoid
singularities (=null motion):
δ = δparticular + δhomogenous
.
δhomogenous  cn
n
h1 h 2 h 3


  C1 ,C2 ,C3 ,C4 
δ
δ
δ
 m 6 , m  1
c   6
 m , m  1
• DIS: Null rates may become extremely high, even
though system is away from singularity.
New Steering Logic:
Unified Steering Law
• Starting aim in the development was to find gimbal rates both
satisfying torque commanded and, driving the gimbals to
desired nonsingular configurations, spontaneously.
• Derived solving the following minimization problem:

1 T
min
δerr Qδ err  τ Terr Rτ err
δ
2
δerr  δ - δdesired

τerr = Jδ - τ
Resulted gimbal rate equation:
-1
δUSL =  qI 4 + J J   qδdesired + J T τ 
T
• 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.
2.
Preplanned Steering
Online Steering
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:
δk  δ
δ
k t  t
(k  1)t    t  k t  
k  1,..., p
• DIS: Only requirement to steer desired gimbal set is
that required h trajectory should be known priori.
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)
Simulations I
• Constant torque study
δo   0, 0, 0, 0
T
τ des  [1.155, 0, 0]T
Torque Realized
Angular Momentum Trajectory
3
1
2.5
2
h (Nm.s)
T (Nm)
0.8
0.6
0.4
1.5
1
0.2
0.5
0
0
-0.2
0
0.5
1
-0.5
1.5 0
t (s)
hx
hy
hz
Tx
Ty
Tz
2 0.5
Ideal Profiles
2.5 1
1.5
t (s)
2
2.5
MP & SR Fails at internal elliptic singularity!
h S  1.155, 0, 0
T
δ S   90, 0,90, 0
T
Torque Realized
AngularSingularity
MomentumMeasure
Trajectory
Gimbal Angles
1.5
100
1.2
1
80
60
1 0.8
1
0.5
0.6
0.8
0.4
0.6
G (deg)
Tm
(Nm)
h (Nm.s)
40
20
0
0.2
0.4
-40
0
-60
0
0.2
Tx
Ty
Tz
-20
hx
hy
hz
-80
-0.2
-0.5 00 0
0
0.50.5
0.5 -1000
g1
g2
g3
g4
1
1
1
1.51.5
0.5 t1.5
(s)
t (s) 1
t (s)
2
2
2
1.5 2.5
t (s)
2.52.5
2
2.5
GSR works as transition method
(1.5 sec delay, high gimbal rates)
Gimbal Rates
5
Torque Realized
Singularity
Measure
Angular Momentum Trajectory
2
1.8
3
1.6
2.5
2.5
1
2
1.2
0.6
01
0.4
0.8
h (Nm.s)
T (Nm)
Gdot
(rad/s)
m
0.8
1.4
1.5
0.6
0.2
-2.5
0.4
0
0.2
0
-0.2
-5 0
00
0.5
g2dot
g3dot
g4dot
0
0.5 -0.51
0.5
101
hx
hy
hz
Tx
Ty
g1dot
Tz
1
1.5
1.5
2.5
222
2.5
1t t(s)
(s)
t (s)
3.5
333
3.5
2
t (s)
44 4
3
4.5
4.5
4
USL Preplanned
•8 nodes are used. Successfully, desired torque is realized
while accurately achieving desired gimbal set at nodes.
Gimbal Angles
Torque Realized
200
150
1
100
T (Nm)
G (deg)
0.8
g1
g2
g3
g4
0.650
0.4 0
Tx
Ty
Tz
0.2
-50
0
-100
-0.2 0
0
0.5
0.5
1
1
1.5
t 1.5
(s)
t (s)
2
2
2.5
2.5
USL Online using Null vector
Torque Realized
Torque Realized
1
T (Nm)
T (Nm)
0.8 1
0.6
0.8
0.4
0.6
0.2
0.4
Tx
Ty
Tz
Tx
0
0.2
Ty
Tz
-0.2 0
0
-0.2
0
0.5
0.5
1
1.5
t (s)
1
1.5
t (s)
2
2.5
2
2.5
USL Online Using Constant Vector
(with dynamically adjusted blending coefficient q=0.5exp(-10m))
Realized
Torque
Torque
Realized
Torque
Realized
1 11
0.6 0.6
0.6
(Nm)
TT (Nm)
T (Nm)
0.8
0.8 0.8
0.4 0.4
0.4
TxTx
TyTy
TzTz
0.2 0.2
0.2
Tx
Ty
Tz
0 00
-0.2-0.20
-0.2
0
0
0.5
0.5
1 1
1
1.5
1.52
t (s) 2
t (s)
t (s)
2.52
3
3
2.53.5
•Steering
w. arbitrary
vector
[0,1,0,0]
•Steering
w. intelligently
selected vector
•Steering
w. white
noise
4
USL Preplanned
Corner maneuver
Cyclic Disturbance
Torque Realized
1
Torque Realized
Gimbal Angles
0.8
200
1.2
0.4
150
1
0.2
100
-0.2
0.8
T (Nm)
0
G (deg)
T (Nm)
0.6
50
0.6
0.4
Tx
Ty
Tz
0
-0.4
-0.6
-0.8
0
0.2
-50
-100
0
Tx
Ty
Tz
0
0.5
0.5
-0.2
0
1
t1.5
(s)
1 0.5
2
2.5
t (s)
τ desired
g1
g2
g3
g4
gdes1
gdes2
gdes3
gdes4
1
3
1.5 1.5
3.5 t (s)4
2
2.5
4.5
 1
T
[1,1,
0]
,
t<0.8344
T
 2
τ desired = 1.15 0 0  0.115 [1 1 1]T sin(2t )

Repeatability maneuver
 1 [1, 1, 0]T , t>0.8344
 2
CMG based ACS Model
Three main parts to be considered:
1. Spacecraft Dynamics
2. Quaternion Based Feedback Controller
3. CMG Steering Law
Spacecraft Dynamics
Total angular momentum equation;
HS/C  Jω  h
Corresponding rotational EoM of a rigid S/C equipped with
momentum exchange actuators such as CMGs, in general given by;
H S/C  ω  H S/C  Text
Text: the external torque vector including the gravity gradient, solar
pressure, and aerodynamic torques all expressed in the same S/C
body axes.
Combining these 2 equations, we simply obtain;
Jω  h  ω   Jω  h   Text + u- u
u: Internal control torque input generated by CMG and
transferred to S/C
Rewriting equation in two parts;
h  ω  h  u
Jω  ω  Jω  Text  u
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:
h  ω  h  u
Simulations II
USL Preplanned Results
100
Angular
Momentum
Gimbal
AnglesTrajectory
Spacecraft Attitude Profile
1.4 10
1.2
0
50
0
Roll-Pitch-Yaw (deg.)
G (deg)
h (Nm.s)
1 -10
0.8 -20
0.6 -30
-50 0.4
0.2
-100
0
-40
-50
g1
g2
g3
g4
gdes1
gdes2
gdes3
gdes4
hx
Roll
hy
Pitch
hz
Yaw
-60
-0.2
0-70
50
100
150
200
250
300
-150
0 50
50 100 100 150
t (s)150 200 200 250 250 300 300
0
t (s) t (s)
Desired h profile from ideal system
Gimbal profiles
Attitude
obtained
profile
with
obtained
USL with USL
USL Online Results
m
Roll-Pitch-Yaw (deg.)
G (deg)
Spacecraft
Attitude
Profile
• Although
simulation
is
started
this
time at internal elliptic
Angles
Gimbal
Singularity Measure
10
250
singularity (i.e. [90, 0, -90, 0]deg), USL online method
200 0
effectively
takes
the system away from singularity rapidly,
1.2
150 -10
and maneuver
is completed on time!
1
100
-20 selected vector [0,1,0,0] is used as desired
• Arbitrarily
50
0.8
g1
gimbal -30
rate with
dynamically
adjusted
blending
g2
Roll
g3
0
Pitch
coefficient.
0.6
g4
-50
-100 -50
-150
Yaw
-40
-60
-200
-70
0
-250
0
0.4
0.2
0
0
50
50
50
100
100
100
150
t (s)
t (s)150
t (s)
150
200
200
200
250 250
250
Attitude Hold Maneuver
A hypothetical cyclic disturbance torque, Text is given
to the system:
T
Text  0.0025 sin(2nt ) 0 0
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.
USL Attitude Hold Results
Spacecraft Attitude Profile
2.5
150
Roll
hxPitch
hyYaw
hz
0.04
G (deg)
Roll-Pitch-Yaw (deg.)
2
0.06
Angular Momentum Trajectory
Gimbal Angles
100
0.02
h (Nm.s)
1.5
50
0
1
g1
g2
g3
g4
-0.02
0
0.5
-0.04
0
-50
-0.06
0
-0.5
0
-100
0
1000
1000
2000
3000
t (s)
4000
10002000 20003000 3000 40004000
t (s) t (s)
5000
6000
50005000 60006000
Attitude profile obtained with USL online
Gimbal History (Repeating pattern)
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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