Attitude Control of an AxiSymmetric Rigid Body Using Two Controls without Angular Velocity Measurements Paper

This paper considers the problem of controlling the rotational motion of an axi-symmetric rigid body using two independent 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.


Introduction
A rigid body in general (non-symmetric) is controlled with three independent controls without angular velocity measurements [1][2][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 condition are obtained in terms of the angular velocities of the rigid body [7].Two controls which stabilize asymptotically an axi-symmetric rigid spacecraft are obtained in terms of the angular velocities of the rigid body and the orientation parameters [8][9][10][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 axisymmetric rigid body with two control torques.Section 3 is devoted to obtain the two control torques which stabilize asymptotically the rotational motion of an axi-symmetric rigid body in terms of the orientation parameters.The asymptotic stability of this motion is proved by applying LaSalle invariance principal.Section 4 contains numerical simulation to illustrate the theoretical results of the paper.

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    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 symmetry.Two control torques 1 and 2 u are applied along the unit vectors and 2 , respectively (Figure 1).
be the principal moments of inertia of the rigid body and the components of the angular velocity of the body referred to the frame, respectively.The dynamic equations take the form:  will remain constant throughout the maneuver.Equations (2.1) can rewrite as: The orientation of an axi-symmetric rigid body is described 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 satisfy the differential equation: 2) and (2.3) can be written in a vector form as: where Equations (2.4) and (2.5) can be used to solve the problem of controlling the rotational motion of an axisymmetric rigid body, using Liapunov function technique.
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 which estimate the orientation parameters respectively.Also we suppose that the orientation parameters and their estimates satisfy the following auxiliary system of differential Equation: Using the kinematic Equation (2.5) the auxiliary system (2.8) can be written in the form: (2.10)

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.
where stabilizes asymptotically the system (2.4), (2.5), (2.9).0 k  Proof.Assume that, the Liapunov function in the form This function is a positive definite with respect to stabilize 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 The time derivative of the Liapunov function is a nega-Copyright © 2013 SciRes.WJM 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 system using LaSalle Invariance Principle [12].Define Ω as the largest invariant set in On Ω we have that from (2.9).This implies that 0 (from (2.4) and (3.1)).This implies that on 0

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 inertial 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:  (a)-(d) show the time response of the body angular velocities, the orientation parameters, the error of the orientation parameters and the control torques, respectively 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].

Conclusion
The angular velocity measurement contains high fre-

Figures 2 (
Figures2(a)-(d)show the time response of the body angular velocities, the orientation parameters, the error of the orientation parameters and the control torques, respectively for the control constant . 2 k  Figures3(a)-(d)show the time response of the body angular velocities, the orientation parameters, the error of the orientation parameters and the control torques, respectively for the control constant .20 k  Figures4(a)-(d)show the time response of the body angular velocities, the orientation parameters, the error of the orientation parameters and the control torques, respectively 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