Comparative Study Between Known and Novel
Attitude Controllers for an Underactuated Satellite1
Nadjim Mehdi HORRI
Stephen HODGART
Surrey Space Centre
University of Surrey
Guildford, Surrey, GU2 7XH,
United Kingdom
Abstract:— Failure of mechanical controllers onboard a satellite is a well known phenomenon that has
already been disastrous for several space missions. In the case studied here, we take as an example the minisatellite UoSat-12 built by SSTL at University of Surrey. The Z-axis reaction wheel of this satellite has failed
and the 3-axis control performance using magnetorquing is being very limited. The use of thrusters would
involve undesired fuel consumption. As an alternative, we present here some latest theory, which shows how
full 3-axis control can still be achieved, using the two remaining reaction wheels from a standard orthogonal
3-wheel configuration. Using a novel nonlinear time invariant and discontinuous approach, we show that the
attitude is more precisely and rapidly altered to the required reference, than using the only 3-axis stabilizing
controller proposed before using two wheels, based on a time varying approach. One consequence of these
results is that a fully redundant 3-axis control can be practically envisaged using a 3-wheel configuration.
Key-words: - Underactuated, Time varying, Discontinuous control, 3-axis stabilization, Reaction wheel,
Satellite.
satellites (generally with thrusters). A few papers
addressed the 3-axis attitude control problem with only
1 Introduction
two orthogonal reaction wheels. The stabilization of a
The control of underactuated satellites has turned
zero total angular momentum satellite has been shown
to be a research challenge in the recent years. We
in ref [2] using a time varying control law, but the
take as an example the low cost 320 Kg minisystem goes through transient oscillations. The only
satellite UoSAT-12 which was built by SSTL
attempt to stabilize a bias momentum satellite was in
(Surrey Satellite Technology Ltd). This satellite,
[3] but only roll and pitch stabilization has been proven
designed for high performance 3-axis attitude
in this case. In other references such as [4], [5], spin
determination and control, uses a 3-axis reaction
axis stabilization and yaw maneuvers have been
wheel configuration. One of the three onboard
obtained using smooth controllers, but the 3-axis
reaction wheels (located on the Z axis) is now in a
stabilization is no longer achieved.
failure mode. Therefore, full 3-axis attitude
In our work, a time-invariant discontinuous
control, applying conventional theory, using the
control
law for the 3-axis attitude stabilization of a zero
remaining reaction wheels is no longer possible.
total angular momentum satellite is presented, based on
The interesting question is how the X and Y
the Rodriguez parameterization of the attitude. Using
wheels can be used in this case to have 3-axis
this approach, we are able to bring the satellite from an
control.
‘upside down’ configuration to a standard 3 axis
We show here the application of new theory
stabilised Earth pointing mode without any transient
to an underactuated satellite in a so-called
oscillations. The initial angular momentum of the
nonholonomic system. Conventional smooth
satellite needs to be small (exactly how small
control laws cannot stabilize this system because it
demonstrated). Therefore there has to be some
fails the so-called Brockett’s necessary condition.
independent ‘de-tumbling’ manoeuvre using possible
Only non-smooth (discontinuous or time varying)
magnetorquers, if the initial momentum is too high.
control laws can be utilized in this case.
Practical issues such as the effect of an external
Recently, several papers have addressed the
disturbance on the Z-axis, and the influence of the
problem of the attitude stabilization using two
small non-zero satellite momentum will be discussed.
control torques for the case of underactuated
For the small non-zero initial total momentum case, it
UoSat : University of Surrey Satellite.
1
is also shown that we can bring the attitude to the
desired reference with residual constant amplitude
oscillations.
In this paper , the objective is to present a
novel two wheels 3-axis controller that improves
the results of the only 3-axis stabilizing controller
known to the author using two wheels (the time
varying controller presented in ref [3]).
where we define: Ni  h i , i=1,3.
Without any loss of generality, we assume one reaction
wheel failure on the z-axis (which is precisely the case
of UoSat-12). In this case, we have to set
N 3  0, h3  0 , and the corresponding dynamic
equation for the underactuated satellite is:
I11  I 2  I 3  2 3  N1   3 h2
I 2 2  I 3  I1  31  N 2   3 h1
2 Problem Formulation
(4)
I 3 3  I1  I 2 1 2  1h2   2 h1
We have first to derive the satellite dynamic and
kinematic models of the underactuated satellite.
2.3 Kinematic Model
2.1 Notations:
A IB : Direction cosine matrix from inertial
reference to body frame.
I = I1 , I 2 , I 3 T : Inertia tensor of the body of the
satellite about its centre of mass.
P =  p1, p2 , p3 T : Attitude of the satellite using
Rodriguez parameters.
ω = 1, 2 ,3 T : Vector of the angular velocity in
body fixed reference frame.
Iwi: Momentum of inertia of the ith wheel.
i , zi : Rotation angle, and unit vector of the ith
wheel rotational axis, respectively.
h  h1 , h2 , h3  : Angular momentum generated
by the wheels in the body frame.
H, L: Total angular momentum in the inertial and
body frames respectively.
Rodriguez parameters are used here for the
representation of the attitude kinematics. These
parameters are derived from the standard Euler
axis/angle representation
P  [ p1, p2 , p3 ]T  tan( / 2).e
where e is the Euler axis and  is the rotation about it.
Alternatively each pi= qi/q4 where the q’s are the
standard quaternions.
The kinematic equation of a rigid body (satellite) using
Rodriguez parameters is:
p 
T
2.2 Dynamic Model
If no external disturbance torque is assumed, we
have:
  ωL  0
L
(1)
And the equation of the total angular momentum
is:
L  Iω  h
2
L  A IB H  Iω   I wi ω   i z i 

(6)






1

2
 p 1  2 1  ( p 3  p1 p 2 ) 2  ( p 2  p1 p 3 ) 3  p1 1

1

2
 p 2   2  ( p 3  p1 p 2 )1  ( p1  p 2 p 3 ) 3  p 2  2
2

1

2
 p 3  2  3  ( p 2  p1 p 3 )1  ( p1  p 2 p 3 ) 2  p 3  3

(7)
We first assume a zero total angular momentum
satellite. We will show that in this case the kinematic
model using Rodriguez parameters will be considerably
simplified.
The zero total angular momentum H = 0 condition can
be written as:
(2)
Iω 
2
 I wi ω  α i z i   0
(8)
i1
By substituting L from equation (2), into the
equation (1), we obtain the general Euler’s
rotational equation using three reaction wheels:
I 3 3  I1  I 2 1 2  N 3  1h2   2 h1

1
~  pp T ω
Ip
2
which can also be written for each component as :
i 1
I11  I 2  I 3  2 3  N1   2 h3   3h2
I 2 2  I 3  I1  31  N 2   3h1  1h3
(5)
which is equivalent to:
ω  1x1   2 x 2
  1z1   2 z 2
(9)
(3)
I wi z i  (I  I w1  I w 2 )x i
(10)
In these equations 1  ̂1 and  2  ̂ 2 where the
2
change of symbol implies the demanded or target
spin rates. Then in an actual control system the
angular acceleration of the wheels is given by:

αi  k w α̂
i α
i

(11)
where α i are the actual spin rates and kw is a
constant of the control system.
The corresponding control torque is:
N i   I wii
(12)
By substituting equation (9) into equation (6) as it
has been shown in ref [3], we have:
p1  


1
1
1  p12  1   p3  p1 p2  2
2
2
(13)
 u1
p 2  


1
 p3  p1 p2 1  1 1  p22  2
2
2
(14)
 u2
1
1
(15)
p 3   p2  p1 p3 1   p1  p2 p3  2
2
2
And when the control law ( u1 ,u 2 ) is determined,
1, 2 are given by:
3 Problem Solution
We now need to control the underactuated satellite
having derived the appropriate kinematics. We show
here a non-linear discontinuous controller, based on
advanced nonlinear control theory that is designed to
achieve the 3-axis stabilization of the satellite.
3.1 Discontinuous controller
Discontinuous control has already been proposed in
order to stabilize nonlinear and nonholonomic systems
such as underactuated satellites using pairs of thrusters
(ref [1]). Our parameterization of the problem here in
the two reaction wheels case is however different.
For our two reactions wheels control system, we want
to design a controller that first stabilizes the unactuated
Z-axis.
p 3   p2u1  p1u 2


  1  p 22
 1 
2
  
2
2
2 
 2  1  p1  p 2  p3  p3  p1 p 2
  p3  p1 p 2   u1 
   (16)
 1  p12  u 2 


As a consequence of that, the variables u1,u2 can be
regarded as redefined control inputs.
In this case, it can be found by replacing the result
of equation (16) into equation (15), that the
kinematic model using Rodriguez parameter simply
reduces to the well-known Brockett integrator:
p1  u1
p 2  u2
  g p3
(19)
where g is a positive constant of the controller
The relation (19) is ensured using the control law:
u1  kp1  g
u 2  kp2  g
p2
p12
p3
 p22
p1
(20)
p3
p12  p22
where k is another positive constant of the controller
(17)
p 3   p2u1  p1u2
 i of the
Then the commanded rotation speeds ̂
wheels are deduced from (9) and to repeat are
regarded as target values.
The simple Laplace transform of (11) goes to:
i ( s ) 
A time varying controller has been designed for such a
system in reference [3]. However, the attitude of the
system in this case goes through important transient
oscillations before we can properly achieve the
stabilization.
kw ˆ
i ( s )
s  kw
Using physical intuition, we can roughly justify this
form of the control law by the superposition of a
continuous term in the feedbacks ( kp1,kp2 ) used to
stabilize the attitude on the actuated axes (X and Y),
and interconnection discontinuous terms used to
complete the stabilization of the unactuated axis (Zaxis).
This control law is discontinuous when p1  p2  0 .
(18)
from which the angular accelerations of the wheels
(equation (11)) can be obtained:
Finally the corresponding control torque is
computed from equation (12).
We have now to show that the above control law, not
only stabilizes the unactuated Z-axis, but also brings
the attitude along the X and Y axes to zero.
By substituting the control law from equation (20) into
equation (17), the resulting closed loop control system
is given by:
3
p1  kp1  g
p 2  kp2  g
p3
 p22
p1  h'1  u1
p 2  h'2  u2
p1
p 3  h'3   p2u1  p1u 2
p2
p12
p3
p12  p22
(21)
Where it can be shown that:
p 3   gp3
h  h1 , h2 , h3 T
The asymptotic stability of the variable p3 is
trivial. The convergence of p3 to zero is also trivial.
Before we can check the stability for the two
remaining variables, we can notice that any
trajectory
starting
in
3
2
2
  ( p1, p2 , p3 )  IR / p1  p2  0
will remain
defined in  .


Indeed if we consider a variable: V  p12  p22 ,
simple calculations yield:
V  kV
Therefore,
manifold
3
p12
p22

We can even bring the system to any other
equilibrium point just by replacing the attitude by
the attitude error in the expression of the
controller.
However, singularities may happen when p1, p2 are
simultaneously too small, we then have to consider
a saturated version of the controller as follows:
 p

u1  kp1  g sat 2 2 2 p3 , a 
p p

2
 1

 p

u 2  kp2  g sat 2 1 2 p3 , a 
p p

2
 1

(23)
(26)
where the total angular momentum H is a (in the free
disturbance case).
And we can show via simulations that the controller
designed for a zero total angular momentum satellite
still brings the system to a neighborhood of the
reference to follow, provided that the momentum of the
satellite can be made small enough initially.
a xa
xa
x  a
3.2 Time varying controller
We want to compare the results of the discontinuous
controller with those of the only 3-axis stabilizing time
varying controller presented in the recent literature
(reference [3]) , for the parameters of UoSat-12. The
proposed control law in [3]was:
u1  kp1  f1 ( p3 ) cos t
(27)
u2  kp2  f 2 ( p3 ) sin t
where:
f1   sign( p3 ) p3
(28)
f 2    p3
Stability proofs using the control law (27) have been
given in ref [3].
3.3 Numerical Simulations
The parameters of the mini-satellite UoSAT-12 are
used for the simulation are:
I1  40 .45 kg.m 2 , I 2  42 .09 kg.m 2 , I 3  40 .36 kg.m 2
Where the saturation function is given by:
x

Sat( x)   a
 a

1
 (I 33  p~  pp T )( I  I w1  I w 2 ) 1 A IB .H
2
(22)
defined
by:
D  ( p1 , p2 , p3 )  IR /

0 ,
is
also
exponentially
attractive
and
stability
is
consequently ensured for the complete attitude to
the zero equilibrium point.

the
(25)
I w1  8.10  3 kg.m 2 , I w2  7.7.10  3 kg.m 2
(24)
We use the saturations levels: -1, +1. (a=1)
The previous theory applies exactly to the case of
non-zero momentum. The effect of a small nonzero total angular momentum needs to be studied
by simulations. The required modifications to the
equations from (16) are:
The discontinuous controller parameters were
determined empirically to be best at: g = 0.1, k = 0.25
and kw = 1.
The time varying controller parameters have been
determined empirically to be best at: k= 0.1,  = 0.1,
and =1.
Initial attitude condition: p1(0) = 1, p2(0) = 1 and
p3(0) = 1, which represents the state of an ‘upside
down’ UoSat-12.
3.3.1 Zero total momentum case study
Simulations of the system (17), with the control law
4
(20) are presented in Fig (1). The corresponding
control torque being simulated is achievable by the
practical on=board reaction wheels (for
UoSAT12). We see that the attitude is precisely
and rapidly altered to the required attitude of p1(0)
= , p2(0) =  and p3(0) =  or ‘zero roll, pitch
and yaw’ which is the desired attitude of the
satellite.
3.3.2 Effect of the small momentum of the
satellite
Similar simulations are carried out for system (25),
which now allows for non-zero initial total
momentum, with
H =0.01.I 33 as in (Fig(2)),
and a much smaller value as in H=0.001.I 33
(Fig(3)). We observe in both cases that there is a
good beginning where the attitude goes quickly to
zero, as require but there is a left a persistent
oscillation in roll and pitch (amplitude = 4 in case
of fig (2) and 12 in the case of fig (3)). In both
cases there is a negligible amount of residual
oscillations in yaw.
3.3.3 Time varying controller
In Fig(4), and after 200 seconds we notice that the
stabilization in this case is still achieved but there are
undesired remaining, and slowly decreasing transient
oscillations for both the control torque and the
satellite’s attitude.
- The numerical simulations show that using the
proposed nonlinear discontinuous approach, we can
rapidly achieve the 3-axis stabilization of the satellite
after large angle maneuvers for a zero total momentum
satellite or simply by assuming that the momentum of
the satellite can be made small initially via detumbling.
When we compare the results in fig (4) and
fig(1), we can clearly conclude that the discontinuous
controller presented here highly improves the results of
the time varying approach using two wheels, which
was the only 3-axis stabilizing controller using two
wheels found in the recent literature.
Fig. 1 - Euler angles and control torque of UoSAT12 using the discontinuous controller.
(Case of a zero momentum mode)
Fig.2- Euler angles and control torque of UoSAT12 using the discontinuous controller.
(with H=H0=0.01.I
5
33 )
Fig. 3 - Euler angles and control torque of UoSAT12 using the discontinuous controller.
(with H=H0=0.001.I 33 )
Fig. 4 - Euler angles and control torque of UoSAT12 using the time varying controller.
(case of a zero momentum mode)
4 Conclusion
F.Terui, S.Motohashi, T.Fujiwara, N.Sako,
A.Nakasuka“ Attitude maneuver of a bias momentum
micro satellite using two wheels ”, 22nd International
Symposium on Space Technology and Science,
Morioka, Japan, 2000.
[3] K.Yamada, S.Yoshikawa, I.Yamagushi, “
Feedback attitude control of a spacecraft by two
reaction wheels ”, 21st International Symposium on
space Technology and Science, Omiya, Japan, 1998.
[4] C.Tillier, B.Agrawal, “Yaw steering for a LEO
Satellite using any two of three reaction wheels”, 51st
International Astronautical Congress, 2-6 Oct
2000/Rio de Janero Brazil.
[5] Sungpil Kim, Youdan Kim “ Spin-axis
stabilization of a rigid spacecraft using two reaction
wheels, AIAA Journal of Guidance, Control, and
Dynamics, Vol.24, No.5, pp.1046-1049, 2001.
[2]
Our work here shows that the application of special
nonlinear discontinuous control theory can be highly
effective in achieving attitude control of an underactuated satellite. With only two active wheels full
fast and decisive slewing on all 3 axes is achievable;
with the only proviso that on the unactuated axis (here
the Z- axis) there must be a further monitoring and
control of both initial angular momentum and external
bias
torque
(suggested
solution
low-cost
magnetorquing).
References:
[1] P.Tsiotras, V.Doutchmenko, “ Control of
spacecraft subject to actuator failures: State of the art
and open problems ”, R. H.Battin Astrodynamics
Conference, American Astronautical society, AAS 00264-2000.
6