Asymptotic Stability of Quaternion-based Attitude Control System with Saturation Function

Received Dec 9, 2016 Revised May 28, 2017 Accepted Jun 12, 2017 In the design of attitude control, rotational motion of the spacecraft is usually considered as a rotation of rigid body. Rotation matrix parameterization using quaternion can represent globally attitude of a rigid body rotational motions. However, the representation is not unique hence implies difficulties on the stability guarantee. This paper presents asymptotically stable analysis of a continuous scheme of quaternion-based control system that has saturation function. Simulations run show that the designed system applicable for a zero initial angular velocity case and a non-zero initial angular velocity case due to utilization of deadzone function as an element of the defined constraint in the stability analysis. Keyword:


INTRODUCTION
Design of a spacecraft or satellite attitude control by analyzing rotational motion of a rigid body remains become a challenging research field until recent years. Ashok et al proposed control moment gyros based attitude control system to achieve time-optimal maneuver for agile (rigid) satellite [1]. Chabot and Schaub presented a spherical actuator for satellite attitude control that covers modeling, simulation of attitude control of a rigid body system motion as well as comparison with a configuration of three reaction wheels [2]. in [3], Stevenson and Schaub used rigid body approach in the attitude control development prior to do testbed experiment of remote electrostatic charge control. Rezanezhad in [4] presented Takagi-Sugeno fuzzybased attitude controller in order to reduce thruster fuel consumption and increase longevity of satellite. Particle swarm optimization algorithm is used to reduce limit cycle on the fuzzy system. Pirouzmand in [5] proposed a model reference adaptive system-based robust model predictive controller for three degree of freedom satellite attitude control system. Whilst the controller gain is obtained through solving a convex optimization problem using linear matrix inequality approach.
The kinematics of a rigid body rotational motion is represented by rotation matrix that is member of the special orthogonal three group, . Among parameterizations of the rotation matrix, a parameterization using quaternion is the only parameterization with four parameters. Hence, it can represent global attitude of a rigid body rotational motion. However, a physical attitude, which is represented in a unique rotation matrix value, is represented by two values in quaternion, i.e. a pair antipodal value. This fact implies difficulties on the stability guarantee of the quaternion based attitude control system [6].  [7] developed an adaptive neural network based attitude controller for satellite that is actuated by control moment gyros, where external disturbance as well as disturbance of the control moment gyros' tachometer are considered; Calvo et al. proposed an adaptive fuzzy controller for momentum wheel actuated satellite and its performance is compared with a custom PD controller [8]; in [9], Septanto et al proposed a continuous scheme of quaternion based controller that employs augmented dynamic; to name a few. In [10], a continuous scheme of quaternion-based control system that has saturation function is also proposed with boundedness of solution guarantee. In this paper, the system has been further analyzed to have stronger guarantee, i.e. asymptotically stable.
The organization of the paper is the following: mathematical preliminaries including modeling of the satellite dynamics and kinematic are presented in the next section. Section 3 presents the problem formulation and methodology. The main contribution of this paper will be presented in Section 4. Discussion and numerical simulations are also presented in Section 5. Section 6 provides the concluding remarks.

BACKGROUND AND PRELIMINARIES 2.1. Mathematical Preliminaries
Some mathematical notations will be used in the rest of paper. R is the set of all real numbers. Suppose there is a column matrix n C  R , hence C denotes 2-norm of C . Matrix 0 nm  denotes nm  matrix that all of its entry are zero.
In this paper, a vector is defined as in (1).
where: r denotes the vector of r , l F is a columns matrix consists of three unit vector 1 l , 2 l and 3 l that are associated with the inertial reference frame, and 3 l r  R is a column matrix whose three components of r that are expressed (decomposed) into the inertial reference frame, l F . Besides superscript l (subscript l ) that are associated with the inertial reference frame, this paper uses superscript b (subscript b ) and superscript d (subscript d ) associated with the satellite's fixed body frame and satellite's desired frame, respectively. For brevity, the satellite's fixed body frame, the inertial reference frame and the satellite's desired frame may be written as body frame, inertial frame and desired frame, respectively.

Dynamic and Kinematics of Spacecraft
In this paper, motion of a satellite is regarded as a rigid body motion. Dynamic of a rigid satellite is given by Euler Equation, [11]. The dynamic of a satellite that is expressed where the symmetric positive definite matrix 33 J   R is the satellite inertia moment about its center of mass that is located in the origin of b F , (kg.m 2 ); : Now, consider a kinematics Equation represented in quaternion given by (3)- (5).
q is also regarded as the attitude error between the satellite body frame b F and the satellite desired or target frame d F . Since the information from an attitude sensor is with respect to the inertial frame l F , i.e. bl q and the target attitude is also with respect to l F , i.e. dl q , hence the attitude error bd q is obtained form quaternion multiplication between bl q and dl q as given in (6)

PROBLEM DEFINITION AND METHODOLOGY
Consider the spacecraft (2) and the controller  (7) proposed in [11], where   The designed attitude control system is for the case of constant attitude tracking, i.e. . In [11], there is relation between rotation matrix bd R and quaternion bd q as given by Equation (9) then 0 E is the locally asymptotically stable set.

Remarks 1.
The stability notion in Definition 1 is not a standard stability in the sense of Lyapunov which is used to guarantee stability of a point. Instead, LaSalle's invariance principle theorem in conjunction with Lyapunov function properties is used to guarantee stability of the set. One may said that LaSalle's theorem extends Lyapunov's theorem since, naturally, it can be used for the system that has an equilibrium set [12].
The main objective of this paper is to find necessary conditions of the system that consists of the rigid spacecraft (2) and the controller (9) such that asymptotic stability guarantee as in Definition 1 is achieved. Through Lyapunov stability method, conditions of the attitude control system parameters will be resulted. In addition, some numerical simulations are done to illustrate its performance.

MAIN RESULT Theorem 1
Consider the quaternion-based spacecraft attitude control system consists of &2), (7) and Consider the Lyapunov function candidate given in Equation (11) to show set stability of 0 E .
where 0 V  , for all 0 k  . Note that, instead of bd q , only bd  is appeared in (11) since bd  is inherently in bd  . Furthermore, the time derivative of V is given by (12).
From the facts that If L is a symmetric definite positive matrix, then (14) is fulfilled.
Note that the constraint (10) implies Substituting these values to the system (2) and (7)

 
R . Hence, a prerequisite condition for Lyapunov theorem utilization is fulfilled.

Remarks 3
The stability analysis presented in this paper is no claim of global stability. In addition, global stability claim in [13] is incorrect since there are equilibrium points other than

DISCUSSION AND NUMERICAL EXAMPLES
The result presented in the previous section provides advancement into two directions. First, its asymptotic stability guarantee presents a stronger stability guarantee than the stability guarantee presented in [10], i.e. boundedness of solution guarantee. Second, this result corrects the global stability claim in [13], as stated in Remarks 3.
In addition to the theoretical result, the attitude control system's performance will be illustrated via simulations that are run through three scenarios presented in Table 1 and the rest arbitrary-chosen parameters are given in Table 2. Simulation results are represented in Figure 1and  interesting and shows that utilization of the deadzone function X  in the defined constraint (10) provides its advantage to allow the attitude control system to start in a non-zero angular velocity. A graphic in relating to the angular velocity b bl  trend is shown in Figure 2(a). In addition, it is also interesting to observe the unwinding phenomenon existence. Hence, euler angle trends from the Scenario 1 is compared to the one form Scenario 3. Note that all parameters of Scenario 1 and Scenario 3 are same, except the desired attitude dl q value where both actually represent a same physical condition. The Euler angle trends outlined in Figure 2(c) show that the designed system is regulated without demonstrating the unwinding phenomenon.

CONCLUDING REMARKS
Proposed stability analysis for a continuous scheme of quaternion-based control system that employs saturation function has been presented. This analysis results that the designed system has asymptotically stability guarantee. To verify and observe a designed attitude control system, three scenarios of simulation are run. Simulations run show that the designed system applicable for a zero initial angular velocity case as well as a non-zero initial angular velocity case. The deadzone function X  utilization in the defined constraint shows its benefit such that the designed system is also allowed for a condition that has non-zero initial angular velocity.