Abstract
Attitude stability analysis and robust control algorithms for spacecrafts orbiting irregular asteroids are investigated in the presence of model uncertainties and external disturbances. Rigid spacecraft nonlinear attitude models are considered and detailed attitude stability analysis of spacecraft subjected to the gravity gradient torque in an irregular central gravity field is included in retrograde orbits and direct orbits using linearized system model. The robust adaptive backstepping sliding mode control laws are designed to make the attitude of the spacecrafts stabilized and responded accurately to the expectation in the presence of disturbances and parametric uncertainties. Numerical simulations are included to illustrate the spacecraft performance obtained using the proposed control laws.
1. Introduction
SMALL bodies including mainly asteroids and comets are studied by scientists because of the insight they can give into the history of the solar system. NASA missions are as follows: Galileo to Jupiter via asteroids Gaspra and Ida in 1989, Near Earth Asteroid Rendezvous (NEAR) Shoemaker to asteroid 433 Eros in 1996 [1, 2], and NASA Flyby Mission Deep Space 1 to asteroid Braille in 1998, Genesis—NASA Discovery Solar Wind Sample Return Mission in 2001. Hayabusa (Muses-C) is the Japan Aerospace Exploration Agency Sample Return Mission to Asteroid 25143 Itokawa [3, 4], and Rosetta is the ESA Comet Mission, flew by asteroids Steins and Lutetia [5].
While there is an increasing interest in such missions, the necessity and importance of orbital and attitude dynamics analyses of the small solar system bodies as the critical success factors of those missions are rising as well. The oblateness torque effects can be ignored for studying the attitude motion of spacecrafts around planetary bodies, while asteroids and comets usually have irregular shapes which lead to the complicated orbital and attitude dynamics in comparison with approximately spherical bodies such as the Earth. An asteroid’s irregular shape, mass distribution, and the state of its rotation (rapid or slow) have significant effects on the evolution of spacecraft orbit and attitude motion. Scheeres and his coworkers have made a large number of contributions to the study of orbital motion about asteroids [6–9]. These effects especially may deteriorate the attitude performance significantly, which lead to unstable attitude motion and thereby failure of the space mission. Wang and Xu find that the attitude stability domain is modified significantly due to the significantly nonspherical shape and rapid rotation of the asteroid, and attitude stability subjected to the disturbance of the gravity gradient torque is generalized to a rigid spacecraft on a stationary orbit around an asteroid [10, 11]. In order to solve this problem, it is important to understand the attitude motion of spacecrafts orbiting asteroids by deriving the stability conditions and thereupon develop effective control laws to neutralize the effects of asteroid shape and mass distributions. Riverin and Misra have proposed the attitude motion of the spacecraft depending heavily on the shape of the asteroid and the rotational state [12]. Then Misra and Panchenko have found the radius for which resonant pitch oscillations, considering the general three-dimensional attitude motion in 2006 [13]. Riverin and Misra have examined the spacecraft pitch motion assuming the spacecraft is in an equatorial orbit but spacecraft attitude control algorithms have not been perfectly investigated. Kumar and Shah have set up the general formulation of the spacecraft equations of motion in an equatorial eccentric orbit using Lagrangian method and made some analysis about the stability. Then the control laws for three-axis attitude control of spacecrafts have been developed and a closed-form solution of the system has been derived in [14]. Mahmut et al. have designed Lyapunov-based nonlinear feedback laws to control the rotational and translational motion of the spacecraft for an asteroid orbiting spacecraft in [15]. However, in the above articles about orbiting attitude motion, external perturbations acting on the spacecraft are not taken into account and the control laws are not robust.
Backstepping is a systematic and recursive design methodology for nonlinear feedback control. The idea is to select recursively some appropriate functions of state variables as pseudocontrol inputs for lower dimension subsystems of the overall system. Each backstepping stage results in a new pseudocontrol design, expressed in terms of the pseudocontrol designs from preceding design stages. When the procedure terminates, a feedback design for the true control input is the result which achieves the original design objective by virtue of a final Lyapunov function, which is formed by summing up the Lyapunov functions associated with each individual design stage. Sliding mode control is a nonlinear robust control and applicable to solve the tracking of nonlinear system [16, 17]. The adaptive algorithm is adopted to estimate the external disturbances and uncertain parameters due to the highly complex environment in real time [18]. Moreover, owing to the robust control performance of adaptive backstepping control and sliding-mode control, many combined adaptive backstepping and sliding mode control schemes have appeared. Although good robust control strategies for uncertain nonlinear system and tracking problems have been proposed in [19, 20], adaptive backstepping sliding mode control is also effective and easier for implementation in real time. In this paper, the performance of spacecrafts orbiting irregular asteroids with perturbations is overall analysed, and attitude motion is influenced seriously. Moreover, the robust adaptive backstepping sliding mode control laws are proposed to compensate the uncertainties and perturbations and make the attitude angles decay and reach the null state, which ensure orbiting motion and space mission.
This paper is organized as follows. In Section 2, gravity gradient torque of spacecraft orbiting irregular asteroids is derived and three-dimensional attitude motion equations of the rigid spacecraft are first examined considering the perturbations, which is followed by deriving the linearized system model. In Section 3, the stability analysis about the spacecraft is presented in retrograde orbits and direct orbits with the orbital radius. Then in Section 4, effective adaptive backstepping sliding mode control schemes for a spacecraft orbiting the asteroid Eros 433 are developed to stabilize the system. Computer simulations are carried out to illustrate the effectiveness of the control laws. Conclusions are presented in Section 5.
2. System Equations of Motion
2.1. Coordinate Frames
At first, the following Coordinate Frames are set up to make the problem clear, which are shown in Figure 1.(1)Asteroid centered inertial frame : the origin of this frame is at the center of mass of the asteroid.(2)Asteroid-fixed frame : the origin of this frame is at the center of mass of the asteroid, the vectors are aligned along the three centroidal principal axis of the smallest, the intermediate, and the largest moment of inertia, respectively. The asteroid rotational state relates the two frames, the unit vector points in the same direction as . Asteroid-fixed frame is assumed to rotate with constant angular velocity .(3)The spacecraft orbital frame : the origin of this frame is at the center of mass of the spacecraft, points towards the center of mass of the asteroid, points towards the transverse direction in the orbital plane, and satisfied the right hand rule. For equatorial orbits, the orbital frame is obtained from the inertial frame by a single rotation through an angle equal to the true anomaly .(4)The spacecraft-fixed frame : aligned with the principal axes of the spacecraft, their orientation with respect to the orbital frame can be defined in terms of the attitude angles (roll, pitch, and yaw).
The sequence of rotations used here is yaw () around axis, followed by pitch () around axis, and then followed by roll () around axis.
2.2. Attitude Kinematics Model
The following assumptions are made in deriving the equations of motion firstly.(1)The spacecraft is rigid.(2)The gravitational attraction of the asteroid is the main disturbance force acting on the spacecraft, and the solar radiation and solar gravitation are considered perturbation force.(3)The rotation rate of the asteroid is constant.(4)The orbital motion of the spacecraft is not affected by attitude dynamics.(5)Moment of inertias is affected by the irregular gravitational force of small bodies.(6)Orbital motion of the spacecraft is fully described as a closed, planar, and periodic orbit.(7)The asteroid is assumed to be a rotating triaxial ellipsoid.
In view of the first assumption, the attitude motion can be described by Euler’s equations of motion for a rigid body: where are the principal moments of inertia of the spacecraft, are the components of the angular velocity along the principal axes in the spacecraft-fixed frame, are the relative angular velocity of the spacecraft with respect to the orbital frame expressed in the spacecraft-fixed frame, and can be calculated by the coordinate transformation matrix from the orbital frame to the spacecraft-fixed frame:
And , , and are the components of the external control moment, are the components of the gravitational field of the asteroid, are the components of the perturbation force, and is the instantaneous orbital rate. Therefore, the full nonlinear equations of the attitude motion have been obtained by (1a), (1b), (2a), and (2b).
In view of the fifth assumption, the gravitational field of the asteroid is the primary and complex effect term which needs to be discussed in detail.
2.3. Gravity Gradient Torque
The gravitational potential of any arbitrary primary can be written in spherical harmonic series [21, 22]: where is the distance of an orbiting particle from the center of mass of the primary and is the characteristic length of the primary, while and are, respectively, the latitude and longitude of the orbiting particle measured in an asteroid-fixed frame. The terms are Legendre polynomials of degree and order 0, and terms are associated Legendre polynomials of degree and order . The two kinds of terms are given as
The corresponding and are known as harmonic coefficients. When , they are called sectorial harmonic coefficients, and ; the corresponding are known as zonal harmonic coefficients of order 0. The coefficients specify the oblateness of the asteroid while characterize the ellipticity of the asteroid’s equator. For the Earth, is and the other coefficients are . However, for some familiar asteroids these coefficients can be as high as . Thus, the irregular shape of an asteroid can have a much stronger effect on attitude dynamics. We approximate the small body is a homogeneous triaxial ellipsoid with axes , , and in order to simplify the problem. We can calculate the coefficients as follows.
for all or , for or are odd and while other conditions is Kronecker symbol, and the value is For our purposes we have stopped the expansion of (3) to the second order, so we get the following coefficient:
The gravitational force acting on a particle of mass at a distance from the asteroid center of mass, having latitude and longitude , is given by where is given in (3), while are unit vectors associated with the spherical coordinate system , , as shown in Figure 1. The position vector of the element can be expressed as where is the position vector of the center of mass of the spacecraft relative to the asteroid center of mass, while is the position vector of the element in the spacecraft frame. We assume that and are much greater than . Clearly
In conclusion, the gravity gradient torque on the spacecraft can then be determined from
Evaluation of this torque involves expansion of the various powers of using the binomial theorem and neglecting terms involving third and higher powers of .
Let denote the gravity gradient torque in the spacecraft-fixed frame and let denote the inertia matrix for the spacecraft, which is given as
The unit vectors appearing in (8), (10), and (11) can now be expressed in terms of the yaw, pitch, and roll. The gravity-gradient torque components () in the spacecraft-fixed frame can be written as follows after some algebra: where are defined as follows, respectively:
2.4. Three-Dimensional Motion for Equatorial Orbits
Three-dimensional motion of a spacecraft in an equatorial orbit is considered, and the attitude motion is small. It is assumed that the asteroid is rotating with a constant angular velocity . Assuming the rotating orbit of the spacecraft is circular orbits, , where is constant and stands for the orbital angular velocity of the spacecraft. Therefore, the longitude of the center of mass of the spacecraft is then given by where the plus and minus signs apply for retrograde and direct orbits, respectively.
Furthermore, for small motion, the angular velocity components given in (1a), (1b), (2a), and (2b) become
Therefore, a set of linearized equations for small motion of spacecraft are obtained in (21)–(23) by introducing (14)–(19) into (1a) and (1b): where , , , , , are control accelerations in three directions, and , , are perturbation force accelerations consisting of gravitation higher order terms and solar radiation pressure, and so forth. Note that the pitch motion is decoupled from the roll and yaw motions, and this fact is similar to the case of a spacecraft orbiting symmetrically mass distributed planetary bodies.
3. Analysis of Motion for Orbiting Circular Orbits
3.1. Regular Resonance Analysis and Numerical Results of Pitch Motion
For circular orbits, , , . For the understanding of the pitch behavior, let us consider small motion. Equation (22) then reduces to
One can cast (24) which represents a harmonically excited system with periodic stiffness. If is negative, the pitch motion is normally unstable; hence, the case of positive is considered in this paper.
Since , , choosing the minus sign in (19), parametric resonance occurs when the spacecraft is in a retrograde orbit when the asteroid and orbital angular velocities are related [14] approximately by
Regular resonance takes place when ; that is,
3.2. Results for the Three-Dimensional Case
Equations (21)–(23) are quite complex and must be solved numerically with the initial conditions of roll, pitch, and yaw rad, ; rad, ; rad, .
Table 1 gives simulation parameters for three-dimensional motion about Eros 433. Figures 2, 3, 4, 5, 6, 7, 8, 9, and 10 give the three-dimensional motions of a spacecraft orbiting Eros in equatorial circular retrograde orbits at km, , , respectively, without taking into account perturbation torque.
The pitch motion is quite regular with amplitude of 0.1 rad at with the above initial conditions, and amplitudes of the roll and yaw motions are all steady. When orbital radius is decrease the three-dimensional motions especially pitch are irregular, and the irregularity is becoming apparent when the spacecraft is nearer to the asteroid. Similarly Figures 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, and 22 give the three-dimensional motions of a spacecraft orbiting Eros in equatorial circular direct orbits at km, km, km, km, respectively, without taking into account perturbation torques. The three-dimensional motion has the same trend with retrograde orbits when the orbital radius are decrease, but the roll and raw motions become instable when .
It is observed that irregularities of attitude angles are more obvious when the spacecraft is nearer to the small body, which make the vibration amplitude and frequency of the spacecrafts more strong. We can draw the conclusion from the simulation results that the irregular gravity-gradient torque of the asteroid has the primary and complex effect on the spacecraft orbiting motion. The spacecraft may get out of the orbit if external perturbations such as solar radiation pressure are taken into account. So it is essential to design the robust control algorithms to compensate the uncertainties and perturbations and stabilize the attitude angles.
4. Controller Design
In this section, we present adaptive sliding mode control laws based on backstepping which achieves three-axes stabilized nadir-pointing attitude. In other words, the control objective is to align the spacecraft-fixed axes with the orbital reference axes. The desired attitude angles yaw (), pitch (), and roll () are zero.
4.1. Backstepping Control
The basic idea of backstepping method is decomposition of a complicated nonlinear system, then designing Lyapunov function and suppositional control for the decomposed system. The final control laws are designed after backing to the overall system.
Regardless of perturbation, suppose (21), (22), and (23) are Define position error ; is the expected trajectory .
Assume the virtual control in Define and Lyapunov function , so (29) is obtained:
Introducing (28) into (29), is obtained.
If , then .
Define Lyapunov function ; then
The following control laws are obtained in
Thus, .
Spacecraft attitude angles pitch, roll, and yaw angles can reach regular resonance based on control law (31) with the initial conditions of roll, pitch, and yaw rad, rad; rad, rad; rad, rad. In the case of the simulation parameters described as Table 1, the desired attitude motion of the spacecraft can be specified by periodic functions with the desired amplitudes. The corresponding attitude motions are determined numerically in Figures 23, 24, and 25. Figures 26, 27, and 28 give the control accelerations of three-dimensional motions. Backstepping control laws can make roll, pitch, and yaw track the desired periodic trajectory without external disturbances when the spacecraft in circular retrograde or direct orbits. The desired attitude angles of spacecraft are adopted periodic function as in the experiments.
From the experimental results in Figures 23, 24, and 25, backstepping control law (31) can guarantee the output signals stable, tracking the desired attitude of spacecraft globally and asymptotically without external perturbances.
4.2. Adaptive Backstepping Sliding Mode Control
With external disturbances and uncertain parameters, adaptive backstepping sliding mode control schemes are developed, which have been applied in uncertain systems [23]. It introduces the sliding mode control in backstepping design to modify the last step of backstepping algorithm and simplify the design of controller.
Without loss of generality, suppose (21), (22), and (23) are is the whole external disturbances and uncertain parameters, and we suppose it changes slowly; that is .
To begin with, define the position error ; is the expected position: The stability term is , and is positive constant.
Define Lyapunov function to be and Then, .
Define Lyapunov function ; is the switching function.
Taking the derivative of (35) is obtained: The control laws are deduced as follows supposing is known: Here and are all positive constants.
It is not easy to obtain the boundary of external disturbances and uncertain parameters due to the highly complex space environment. The adaptive algorithm is adopted to estimate the external disturbances and uncertainties in order to retain from the boundary.
Define Lyapunov function as ; the error is , and is the estimated value of ; is a positive constant. Substituting into Equation as The adaptive controller laws are obtained as
The stability of the controller is proved as follows.
Take Rewriting (39) to is ensured to be a positive definite matrix while are appropriate values.
The desired attitude of spacecraft is adopted exponential function as
The higher order terms of gravitational potential are regarded as uncertainties and solar radiation pressures are regarded as disturbances which are assumed as the following equation:
To verify and visualize the efficacy of the developed control scheme, numerical simulations under external disturbances and uncertainties are conducted using (21)-(23) and control law (38). Parameters related to operating conditions are also given about Eros 433 in Table 1.
Some experimental results are provided to demonstrate the effectiveness of the proposed adaptive backstepping sliding mode control laws. Figures 29, 30, and 31 give the spacecraft pitch, roll, and yaw attitude angles response motion around Eros 433. We also can obtain the pitch controller response as Figure 32. Compared with the attitude stability analysis in references [10–13], closed-loop controllers are proposed to make the spacecraft attitude angles tracking the desired attitude as (42) and reach the null state as time increases. Moreover, from simulation results one can obtain the control law neutralizing the effects of asteroid shape and mass distributions and orbital eccentricity as well as external disturbances and uncertainties described as (43). The robust control performance of the proposed adaptive backstepping sliding-mode control system is obvious than references [14, 15], which ensure stable orbiting motion and space mission.
(a) Control of pitch
(b) Partial enlargement of control
(c) Partial enlargement of control
5. Conclusions
This paper has focused on the attitude dynamics and effect control algorithms for spacecraft orbiting rotating asteroids. Firstly, three-dimensional attitude motion of the spacecraft is examined considering the perturbation force. Then stability analysis is presented in retrograde orbits and direct orbits using linearized system model. It appears that the nonspherical shape and the rotational state of asteroids can have important effects on the attitude motion. The adaptive backstepping sliding mode control laws are designed to make the attitude angles decay and reach the null state. Computer simulations are carried out for the asteroid Eros 433 to illustrate the effectiveness of the control laws.
Nomenclature
| : | Major semiaxes of the asteroid considered as triaxial ellipsoid |
| : | Asteroid centered inertial frame |
| : | Asteroid-fixed frame |
| : | Spacecraft orbital frame |
| : | Spacecraft-fixed frame |
| : | Principal moments of inertia of the spacecraft |
| : | Components of the angular velocity along the principal axes in the spacecraft-fixed frame |
| : | Relative angular velocity of the spacecraft with respect to the orbital frame |
| : | Coordinate transformation matrix from the orbital frame to the spacecraft-fixed frame |
| , : | Components of the external control moment |
| : | Components of the gravitational field of the asteroid |
| Components of the perturbation force | |
| : | Instantaneous orbital rate |
| : | Characteristic length of the primary |
| , : | Latitude and longitude of the orbiting particle measured in an asteroid-fixed frame |
| : | Legendre polynomials of degree and order |
| , : | Harmonic coefficients |
| : | Spacecraft mass |
| : | Asteroid mass |
| : | Orbital radius |
| : | The distance of an orbiting particle from the center of mass of the primary |
| : | Spacecraft yaw, pitch, and roll angles |
| : | Asteroid’s spin rate |
| : | Density of the asteroid |
| : | Desired attitude angle of spacecraft |
| : | Asteroid’s gravitational constant parameter. |
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work is supported by National Basic Research Program of China (973 Program) under Grant 2012CB720000. The authors also gratefully acknowledge the helpful comments and suggestions of the reviewers, which have improved the presentation.