World Journal of Mechanics, 2013, 3, 1-5
http://dx.doi.org/10.4236/wjm.2013.35A001 Published Online August 2013 (http://www.scirp.org/journal/wjm)
Attitude Control of an Axi-Symmetric Rigid Body Using
Two Controls without Angular Velocity Measurements
Paper
Tawfik El-Sayed Tawfik
Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, Egypt
Received March 23, 2013; revised April 23, 2013; accepted May 1, 2013
Copyright © 2013 Tawfik El-Sayed Tawfik. This is an open access article distributed under the Creative Commons Attribution Li-
cense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
This paper considers the problem of controlling the rotational motion of an axi-symmetric rigid body using two inde-
pendent control torques without angular velocity measurements. The control law which stabilizes asymptotically this
motion is obtained only in terms of the orientation parameters. Global asymptotic stability is shown by applying LaSalle
invariance principal. Numerical simulation is introduced.
Keywords: Attitude Control; Rigid Body; Stabilization; Two Controls
1. Introduction
A rigid body in general (non-symmetric) is controlled
with three independent controls without angular velocity
measurements [1-3]. If one of the controls is failure, the
rigid body is not controllable. Thus the attitude control of
a rigid body motion using two controls is an important
control problem.
The angular velocity along the symmetric axis of the
rigid body is fixed to its initial value. In this case, two
control torques are used to stabilize asymptotically the
rotational motion about the symmetric axis. Moreover,
the orientation of the symmetric axis is described using
stereographic coordinates form direction cosines [4].
Many authors have discussed the attitude control of a
rigid body motion using two controls that depend in
terms of the angular velocities of the rigid body and the
orientation parameters. The stabilization of a zero total
angular momentum satellite using two reaction wheels
has been shown in [5,6]. Two controls which stabilize
asymptotically a rigid body motion using matching con-
dition are obtained in terms of the angular velocities of
the rigid body [7]. Two controls which stabilize asymp-
totically an axi-symmetric rigid spacecraft are obtained
in terms of the angular velocities of the rigid body and
the orientation parameters [8-11]. The angular velocity
measurement is noisy. It contains high frequency and
random fluctuations. In this paper, two control torques
which stabilize asymptotically the rotational motion of an
axi-symmetric rigid body are obtained only in terms of
the orientation parameters.
The present paper is organized as follows: Section 2
presents dynamic and kinematic equations of an axi-
symmetric rigid body with two control torques. Section 3
is devoted to obtain the two control torques which stabi-
lize asymptotically the rotational motion of an axi-sym-
metric rigid body in terms of the orientation parameters.
The asymptotic stability of this motion is proved by ap-
plying LaSalle invariance principal. Section 4 contains
numerical simulation to illustrate the theoretical results
of the paper.
2. Dynamics and Kinematics
Consider the rotational dynamics of an axi-symmetric
rigid body controlled by two independent control torques.
Two reference frames are introduced. The first


123
ˆˆˆˆ ,,nnnn is an inertial reference frame, and the
second


123
ˆˆ
,,bb
1
ˆ
b
ˆˆ
bb is a body-fixed reference frame
and coincident with the principal axes of inertia of the
body. The unit vector 3 lies along the axis of symme-
try. Two control torques 1 and 2 u are applied along
the unit vectors and 2, respectively (Figure 1). Let
i
ˆ
b
ˆ
bu
A
and


1, 2,3
ii
  be the principal moments of iner-
tia of the rigid body and the components of the angular
velocity of the body referred to the frame, respec-
tively. The dynamic equations take the form:
ˆ
b
C
opyright © 2013 SciRes. WJM

T. EL-SAYED TAWFIK 2
Figure 1. Axi-symmetric rigid body with two controls.



11232 31
2231 312
3312 12
,
,
.
A
AA u
A
AA u
AAA



 
 




(2.1)
Since 12
A
AA
,
, if we let the initial condition
30

3
0

  3

will remain constant throughout the
maneuver. Equations (2.1) can rewrite as:


13302
23 301
,
.
1
2
A
AA u
A
AA u


 
 

 (2.2)
The orientation of an axi-symmetric rigid body is de-
scribed by using stereographic coordinates form direction
cosines [4]. Two orientation parameters and 2
can be used to describe the position of the 3 inertial
axis in the body fixed frame. These parameters sat-
isfy the differential equation:
1
W
ˆ nW
ˆ
b


22 1
1322121 2
22 2
23111221
1,
2
1.
2
WWWW WW
WWWW WW




 
 


(2.3)
equations (2.2) and (2.3) can be written in a vector form
as:



330 ,
A
AAS u   

 (2.4)
 
30
WS WFW

 
 (2.5)
where
 
TT
121 212
,,WWWuuu  
T
,


F
W is the 2 symmetric matrix
2

 
T
112
2
FWWW IWWT
(2.6)
and


30
S

is the skew-symmetric matrix
22

30
30
30
0.
0
S









 (2.7)
Equations (2.4) and (2.5) can be used to solve the prob-
lem of controlling the rotational motion of an axi-
symmetric rigid body, using Liapunov function tech-
nique.
The main objective is to determine the control law
in terms of the orientation parameters that will derive
u

and to zero. To derive this control law, we introduce
the new parameters
W
T
12
ˆˆˆ
WWW





which estimate the orientation parameters

T
12
,WWW
respectively. Also we suppose that the orientation pa-
rameters and their estimates satisfy the following auxil-
iary system of differential Equation:

30
ˆˆ.WWWS W
 
 (2.8)
Using the kinematic Equation (2.5) the auxiliary sys-
tem (2.8) can be written in the form:


FW

  
 (2.9)
where
ˆ.WW


 (2.10)
3. Stabilization Problem
The main object of this section is to determine the control
law which stabilizes asymptotically the system (2.4),
(2.5), (2.9). This control law depends upon the orientation
parameters only.
u
Theorem. The control law




TT
ukW FW    (3.1)
where stabilizes asymptotically the system (2.4),
(2.5), (2.9).
0k
Proof. Assume that, the Liapunov function in the form

TTT
2Φ.AkWW


   
(3.2)
This function is a positive definite with respect to stabi-
lize variables

, , and
W

. The time derivative of the
Liapunov function (3.2) using (2.4), (2.5), (2.9) and the
control law (3.1) takes the form




TTT
TTT
T
d
d
0.
AkWW
t
ukWFW k
k
 
T



 
 
 



(3.3)
The time derivative of the Liapunov function is a nega-

T. EL-SAYED TAWFIK 3
tive semi-definite function (constant sign function). Thus,
under the control law (3.1), the system is stable.
Now, we will prove the asymptotic stability of this sys-
tem using LaSalle Invariance Principle [12]. Define Ω
as the largest invariant set in








,, :0,, :0.WW    

On Ω we have that


0FW 


 from (2.9).
This implies that 0

 on Ω. Since Ω is invariant,
0


 in turn implies

T0kWF W
(from (2.4) and (3.1)). This implies that on 0WΩ.
Therefore
 
,, :0.WW   
4. Numerical Simulation
This section shows the effect of the value of the control
constant in control purposes. The Program used in
this numerical approach is MAPLE. We choose the iner-
tial moments of an axi-symmetric rigid body, the initial
angular velocities of the rigid body, the initial orientation
parameters and the initial error attitude parameters as
follows:
k
 
 
 
2
12 3
123
12
12
15,20 kgm,
00.2, 00.2, 00.1rads
00, 00,
00, 00.
AA A
WW






,
(4.1)
Figures 2(a)-(d) show the time response of the body
angular velocities, the orientation parameters, the error of
the orientation parameters and the control torques, re-
spectively for the control constant . 2 k
Figures 3(a)-(d) show the time response of the body
angular velocities, the orientation parameters, the error of
the orientation parameters and the control torques, re-
spectively for the control constant . 20 k
Figures 4(a)-(d) show the time response of the body
angular velocities, the orientation parameters, the error of
the orientation parameters and the control torques, re-
spectively for the control constant . 40 k
Based on the above numerical simulation study we
conclude that, increasing the value of the control constant
has the effect of decreasing the convergence behavior
of the system. This result is different from the result
when the control torques are obtained in terms of the
angular velocities of the rigid body and the orientation
parameters [11].
k
5. Conclusion
The angular velocity measurement contains high fre-
(a)
(b)
(c)
(d)
Figure 2. (a) Body angular velocities; (b) Orientation pa-
rameters; (c) Error orientation parameters; (d) Control
torques.

T. EL-SAYED TAWFIK 4
(a)
(b)
(c)
(d)
Figure 3. (a) Body angular velocities; (b) Orientation pa-
rameters; (c) Error orientation parameters; (d) Control
torques.
(a)
(b)
(c)
(d)
Figure 4. (a) Body angular velocities; (b) Orientation pa-
rameters; (c) Error orientation parameters; (d) Control
torques.

T. EL-SAYED TAWFIK 5
nd noises or random fluctuations. Two control
[1] A. AbdessamGlobal Traj
quency a
torques (3.1) which stabilize asymptotically the rotational
motion of an axi-symmetric rigid body are obtained in
terms of the orientation parameters without angular ve-
locity measurements. Global asymptotic stability is shown
by applying LaSalle invariance principal.
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