MODELLING A SATELLITE CONTROL SYSTEM SIMULATOR

National Institute for Space Research – INPE
Space Mechanics and Control Division – DMC
São José dos Campos, SP, Brasil
MODELLING A SATELLITE
CONTROL SYSTEM SIMULATOR
Luiz C Gadelha Souza
[email protected]
3rd International Workshop and Advanced School - Spaceflight Dynamics and Control
1
October 88-10, 2007 - University of Beira Interior - Covilhã, Portugal.
Introduction
Placing a satellite in orbit is a risky and expensive process.
Space Projects must guarantee that satellite and/or its
equipments work properly.
Attitude Control System (ACS) should use new control
techniques to improve reliability and performance.
Experimental validation of new equipment and/or control
techniques through simulators (prototypes) is one way to
increase confidence and performance of the system.
2
Types of Simulators
Basically, there are two types of simulators:
The Planar one, with translational motion in one or two
directions
The spherical one, with rotation around one, two or three
axes.
The simulators consist of a platform supported on a plane or
a spherical air bearing.
The platform can accommodate various satellites
components: like sensors, actuators, computers and its
respective interface and electronic.
3
Example of Simulators
Planar Simulador - Stanford University robotic arm
4
Example of Simulators
Spherical Simulator - Georgia Institute of Technology (GIT)
5
DMC Lab ativities
Brazilian Data Collection Satellite Prototype for experimental
verification of its various sub systems
6
DMC Lab ativities
7
DMC Lab ativities
Attitude Maneuvers Software for the China Brasil Earth Remote
Sensing Satellite CBERS
8
DMC - Simulators
DMC is responsible for constructing two simulators to test and
implementing satellite ACS. A 1D simulator with rotation
around the vertical axis with gyro as sensor and reaction wheel as
actuator.
9
DMC - Simulators
A 3D simulator with
rotation around
three axes, over
which is possible to
put satellite ACS
components like
sensors, actuators,
computers, batteries
and etc.
10
DMC – Simulators
11
Objetives
This talk presents the development of a 3D Satellite Attitude Control
System Simulator Software Model.
This simulator model allows to
investigate fundamental aspects
of the satellite dynamics and
attitude control system.
12
Objetives
From the Simulator Model
One designs the simulator ACS based on a PD controller with gain
obtained by the pole allocation method
After that, using recursive least
squares method the platform inertia
parameters are estimated,
considering data from the Simulator
model.
13
Objetives
Once the recursive least squares
method has been checked.
One uses it to estimate the 1D
simulator inertia moment having
experimental data from gyro and
reaction wheel.
14
3D Platform Equations of Motion
The platform angular velocity is given by
W = pi + qj + rk
w3
W
The total angular moment is the sum of the
base and reaction wheels angular moment
rcg
mg
H=
3
ρ
r
×
(
W
×
r
)
dm
+
R
×
(
w
∑ ∫ i i × ρ i )dm
∫
B+RW
i =1
w1
w2
RW
Deriving the previously expression the equation of
motion of the platform is given by
3  3 
dH rcg ×(mg) = = (h)r +W×h + ∑(hi )r +W ×∑hi 
dt
i =1
 i=1 
15
3D Platform Equations of Motion
The reaction wheels equations of motion are
T 1 = I 1 [w 1 + p ]
T 2 = I 2 [w 2 + q ]
w3
W
T 3 = I 3 [w 3 + r ]
rcg
The kinematic equations considering Euler
angles (φ, θ, ψ) in the sequence 3-2-1 are
mg
w1
w2
φ = p + tan(θ )[q sin(φ ) + r cos(φ ) ]
θ = q cos(φ ) − r sin(φ )
ψ = sec(θ )[q sin(φ ) + r cos(φ ) ]
16
3D Platform Equations of Motion
Putting together the previous equations of motion in matrix form yields
 I xx
I
 xy
 I xz

 0
 0

 0
 1

 0
 0

I xy
I yy
I xz
I yz
0
0
0
0
0
0
I1
0
0
I2
I yz
0
I zz
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
1
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
0
0
1
0
1
0
0
0
0
0
 ( I xx − I zz )( qr ) + I xy ( pr ) − I xz ( pq ) + I yz ( r 2 − q 2 ) + I 2 ( w 2 r ) + 


 − I 3 ( w 3 q ) + mgr y cos( φ ) cos( θ ) − mgr z sin( φ ) cos( θ )

 ( I − I )( pr ) + I ( pq ) − I ( qr ) + I ( p 2 − r 2 ) − I ( w r ) + 
1
1
xx
yz
xy
xz
 zz

0   p   + I ( w p ) − mgr cos( φ ) cos( θ ) − mgr sin( θ )

3
3
x
z



2
2

0  q 
( I − I yy )( pq ) + I xz ( qr ) − I yz ( pr ) + I xy ( q − p ) + I 1 ( w 1 q ) + 
 xx

I 3   r   − I 2 ( w 2 p ) + mgr x sin( φ ) cos( θ ) + mgr y sin( θ )

  

0  φ 
p + tan( θ ) [q sin( φ ) + r cos( φ ) ]


  
0   θ  = 
q cos( φ ) − r sin( φ )

  

1
0  ψ
[q sin( φ ) + r cos( φ ) ]

  
cos( θ )
0   w 1  

  
T1

0   w 2  

I1

1   w 3  
T2




I2


T3




I3
17
Control Law Design
To design the control law, one needs the linear system. therefore, assuming small angles the equations
of motion for designing purpose are
 p
 q

 r
 φ
 θ

ψ

0
0




0
 = 

1

0



 0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
1
0
0
0
0
0
0
0
0

I
0 p 
 1

0   q 

0   r 

  + 
0φ 




0
θ

 

0  ψ 



X = AX + Bu
Y = CX
u = −KX
1
− I
0
0
0
1
− I
0
xx
I
2
0
0
0
0
0
0
0
0
yy
I3
1
− I
0
0
0
zz




  T1

 T 2
  T 3










The control gains are obtained applying the pole
allocation method
18
Simulation Results
TABLE I – Typical Platform data used in the simulations
Platform
Platform
Reaction
wheel
External
torque
Ixx=1.2
Ixy = 0.02
I1 = 0.0015
Mgrx =0.012
Iyy= 1.2
Ixz = -0.02
I2 = 0.0015
Mgry =0.035
Izz= 2.0
Iyz = 0.02
I2 = 0.0015
Mgrz =0.755
19
Simulation Results
Using pole allocation method, one has defined three sets of poles
p1,2,3 , in order to analyze the dynamic behavior of the system.
p1 = {−0.5+ i −0.5−i −0.3+ i −0.3−i −0.2 + i −0.2 −i}
p2 = {− 2 + 0.3i − 2 −0.3i − 2.25+ 0.3i − 2.25−0.3i − 2.5+ 0.3i − 2.5−0.3i}
p3 = {− 4 − 4 − 4.25 − 4.25 − 4.5 − 4.5}
The first set of poles p1 is closer to the imaginary axis than the
second set p2 and the third set p3 has only real part
20
Simulation Results
S imulac ao nao linear p x
t
35
polos (1)
polos (2)
polos (3)
30
25
v
e
l
o
c
i
d
a
d
e
(
d
e
g
/
s
)
20
15
10
5
0
-5
-10
0
2
4
6
8
t
10
(s )
12
S imulac ao nao linear q x
14
16
18
20
t
15
polos (1)
polos (2)
polos (3)
10
v
e
l
o
c
i
d
a
d
e
(
d
e
g
/
s
)
5
0
-5
-10
-15
-20
0
2
4
6
8
t
10
(s )
12
14
16
18
20
Simulac ao nao linear r x t
80
polos (1)
polos (2)
polos (3)
70
60
50
v
e
lo
c
id
a
d
e
(
d
e
g
/s
)
Figures show the
angular velocity
p , q and r of the
platform for the
three set of poles
p1 , p2 and p3 .
40
30
20
10
0
-10
0
2
4
6
8
10
t (s )
12
14
16
18
20
21
Simulation Results
Simulacao nao linear phi x t
15
polos (1)
polos (2)
polos (3)
10
5
a
n
g
u
lo
(
d
e
g
)
0
-5
-10
-15
-20
-25
0
2
4
6
8
10
t (s)
12
14
16
18
20
Simulacao nao linear theta x t
10
polos (1)
polos (2)
polos (3)
a
n
g
u
lo
(
d
e
g
)
5
0
-5
-10
-15
0
2
4
6
8
10
t (s)
12
14
16
18
20
Simulacao nao linear ps i x t
10
polos (1)
polos (2)
polos (3)
0
-10
a
n
g
u
lo
(
d
e
g
)
Figures show the
angles (φ
φ, θ, ψ) of
the platform for
the three set of
poles p1 , p2 and
p3 .
-20
-30
-40
-50
-60
0
2
4
6
8
10
t (s)
12
14
16
18
20
22
Simulation Results
Simulac ao nao linear w1 x t
4000
polos (1)
polos (2)
polos (3)
3000
v
e
lo
c
id
a
d
e
(
r
p
m
)
2000
1000
0
-1000
Figures show the
reaction wheel
rotation (ω
ω1, ω2 , ω3)
for the three set of
poles p1 , p2 and p3 .
-2000
-3000
0
2
4
6
8
10
t (s)
12
Simulac ao nao linear w2 x
14
16
18
20
t
2000
polos (1)
polos (2)
polos (3)
1500
v
e
lo
c
id
a
d
e
(
r
p
m
)
1000
500
0
-500
-1000
-1500
-2000
0
2
4
6
8
10
t (s)
12
14
16
18
20
Simulac ao nao linear w3 x t
2000
polos (1)
polos (2)
polos (3)
0
-2000
v
e
lo
c
id
a
d
e
(
r
p
m
)
-4000
-6000
-8000
-10000
-12000
-14000
-16000
0
2
4
6
8
10
t (s)
12
14
16
18
20
23
Simulation Results
Simulac ao nao linear T1 x t
2
polos (1)
polos (2)
polos (3)
1
0
-1
to
r
q
u
e
(
N
.m
)
0.05
-2
0.04
0.03
-3
0.02
0.01
-4
0
-0.01
-5
-0.02
15.95
-6
16
16.05
-7
-8
0
2
4
6
8
10
t (s)
12
Simulac ao nao linear T2 x
14
16
18
20
t
3
polos (1)
polos (2)
polos (3)
2.5
2
0.02
0.01
to
r
q
u
e
(
N
.m
)
1.5
0
-0.01
1
-0.02
-0.03
0.5
-0.04
15.9
16
16.1
0
-0.5
-1
0
2
4
6
8
10
t (s)
12
Simulac ao nao linear T3 x
14
16
18
20
t
5
polos (1)
polos (2)
polos (3)
0
-5
0.06
-10
to
r
q
u
e
(
N
.m
)
Figures show the torques
T1,2,3 applied by reaction
wheel for the three set of
poles p1 , p2 and p3 .
0.04
0.02
0
-15
-0.02
-0.04
-20
-0.06
15.98
16
16.02
-25
-30
-35
0
2
4
6
8
10
t (s)
12
14
16
18
20
24
Comments : Dynamics and Control
The first set of poles (red line) have the undesirable low damping rate
associated with great oscillation.
The third set of poles (blue line) although it shows short time for
damping the overshoots reach great values.
The second set of poles (green line), reduce the angular velocities and
angles in short time, with small overshoot and the reaction wheels
rotation are in acceptable levels.
In the sequel the 3D platform model with the control law designed with
the second set of poles are used to generated data to estimate the
platform inertia parameters.
25
Parameters Estimation
In the estimation process the vector X has the inertia parameters and the
location of the platform gravity center. The matrix G and vector Y contain
angles, angular velocities, sensor measures and reaction wheels inertia which
are known.
[G ]{X } = {Y }
[G ] =
K
 G1
G
 2
 
G K






{Y } =
K
 Y1
Y
 2

  Y K






26
Parameters Estimation
The recursive form of the least square method needs to satisfy the
following equations :
[LK] =[PK−1][GK] ([I] +[GK][PK−1][GK] )
[PK] =([I] −[LK][GK])[PK−1]
{XK} ={XK−1} +[LK]({YK} −[GK]{XK−1})
T
T −1
[P0 ] = ([G0 ][G0 ] )
T
{X0} = [P0 ][G0 ] {Y0}
T −1
27
Parameters Estimation
The matrices G, Y and X are given by
T
pK
pKr
− pKqK 



r
q
q
pq
−




rq
r
− pr


2
2
q
−
rp
p
+
rq
p
−
q


2
2
r − pq
q + r p 
[GK ] =  q −r


2
2
r
p
q
r
p
p
rq
+
−
−



0
cos(φ ) cos(θ) − sin(φ )cos(θ )


− sin(θ ) 
0
−cos(φ)cos(θ )
 sin(φ )cos(θ)

sin
θ
(
)
0


−I1w
1 +I2w2r −I3w3q


{YK} =−I2w2 −I1w1r +I3w3 p
−I w

+
I
w
q
−
I
w
p
 33 11 2 2 
X =mgrx, mgry , mgrz , Ixx, Iyy, Izz , Ixy, Ixz, Iyz
28
Parameters Estimation
The parameters are estimated with measures that have
been done in time interval of 5s for simulation of 20s.
The results are shown in the next Figures
29
Parameters Estimation
8
5
Mimimos quadrados recursivo
x 10
Ixx
Iyy
Izz
4
2.15
3
2.14
2.13
2
inercia(kg.m2)
2.12
1
2.11
2
4
6
8
10
12
14
16
18
20
2
4
6
8
10
12
14
16
18
20
2
4
6
8
10
t (s)
12
14
16
18
20
0
-1
1.18
-2
1.17
-3
-4
-5
1.16
1.15
0
Platform principal inertia moments estimation
30
Parameters Estimation
8
5
Mimimos quadrados recursivo
x 10
Ixy
Iyz
Ixz
4
3
inercia(kg.m
2)
2
1
0
-1
0.02
0.01
-2
0
-3
-0.01
-4
-5
-0.02
0
2
4
6
8
10
12
14
16
18
2
4
6
8
10
t (s)
12
14
16
18
20
Platform cross inertia moments estimation
31
Parameters Estimation
8
5
Mimimos quadrados recursivo
x 10
mgRx
mgRy
mgRz
4
0.77
3
0.765
Forçaxbraço(N
.m
)
2
0.76
0.755
1
0.75
5
10
15
20
0
0.04
-1
0.03
-2
0.02
-3
0.01
-4
-5
0
2
4
6
8
10
12
14
16
18
4
6
8
10
t (s)
12
14
16
18
20
External torque estimation
32
Comments : Parameters Estimation
From the previous result, one observes that the recursive
least square method is reliable.
Therefore, it will be used to estimate the 1D simulator
inertia parameter from experimental data.
33
Inertia estimation - 1D Platform
The previous recursive procedure is applied considering the simplification of
the 3D equation of motion for rotation around the vertical axis which is
given by
rI zz + w J = 0
W
[r]{I zz } = {− Jw }
[G ]{X } = {Y }
w
y1
x
rcg
mg
z
X
x1
y
z1
Z
Y
Where the experimental data come from gyros and reaction wheel.
34
Inertia estimation - 1D Platform
The equipments used to perform the experiments are :
The air baring platform diameters : 650mm
Sunspace reaction wheel
Angular rotation : -/+ 4200 rpm
Maximum torque : 50mNm
Maximum angular moment : 0.65Nms
Inertia moment : 1.5E-3 Kgm.m
Voltage : 12 Vdc
Sunspace Fiber Optics Gyroscope
Field of measure : -/+ 80º/s
Freeware Radio-Modem ; 908 – 950 MHz
Rate : 110Kbps with RS-232 protocol
Battery : 12Vdc
National Instruments PC 1.26GHz
Interface : RS-232/RS-485
35
Experiment Procedure
One stars with both angular velocities of the platform and reaction
wheel equal to zero.
Then one sends a commander to the reaction wheel so that it
increases its angular velocity up to a certain value.
That action makes the platform to move with opposite angular
velocity.
After that, one sends a commander to decrease the reaction wheel
angular velocity up to zero.
Again the platform will react with angular motion in opposite
direction.
During that process the platform is monitored by the gyroscope and
the reaction wheel angular velocity is also measured.
It is important to say that the platform friction has been neglected.
36
Experiment Procedure
The reaction wheel angular velocity
37
Experiment Procedure
The platform angular velocity
38
Inertia moment estimation
Izz = 0.495 Kgm.m
39
Summary
This talk presents a mathematical model of a platform that simulates
a satellite ACS in 3-D with three reaction wheels as actuators and
three gyros as sensors.
A control law based on a PD controller using poles allocation method
is designed and its performance is evaluated .
That model is used to generated data to estimate the inertia
parameters of the platform, using the least square recursive
method.
The simulations has shown that the recursive method is reliable for
the simulator objectives.
The 1D platform inertia moment is estimated using real data using
the recursive method.
40
National Institute for Space Research – INPE
Space Mechanics and Control Division – DMC
São José dos Campos, SP, Brasil
MODELLING A SATELLITE CONTROL SYSTEM SIMULATOR
Luiz C Gadelha Souza
[email protected]
Thank you …… !
3rd International Workshop and Advanced School - Spaceflight Dynamics and Control
41
October 88-10, 2007 - University of Beira Interior - Covilhã, Portugal.