Quaternion-Based Attitude Control System Design of Single and Cooperative Spacecrafts: Boundedness of Solution Approach

Septanto, Harry; Bambang, Riyanto T.; Syaichu-Rohman, Arief; Eko Poetro, Ridanto; Ravi Ibrahim, Adrianto

doi:https://doi.org/10.1155/2014/652571

Modelling and Simulation in Engineering

On this page

Abstract Introduction Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2014 | Article ID 652571 | https://doi.org/10.1155/2014/652571

Quaternion-Based Attitude Control System Design of Single and Cooperative Spacecrafts: Boundedness of Solution Approach

Harry Septanto,^1,2Riyanto T. Bambang,¹Arief Syaichu-Rohman,¹Ridanto Eko Poetro,³and Adrianto Ravi Ibrahim¹

Academic Editor: Franco Ramírez

Received25 Dec 2013

Accepted21 Apr 2014

Published19 May 2014

Abstract

It is well known that single equilibrium orientation point in matrix rotation is represented by two equilibrium points in quaternion. This fact would imply nonefficient control effort as well as problem in guaranteeing stability of the two equilibrium points in quaternion. This paper presents a solution to design quaternion-based spacecraft attitude control system whose saturation element is in its control law such that those problems are overcome. The proposed feature of methodology is the consideration on boundedness of solution in the control system design even in the presence of unknown external disturbance. The same methodology is also used to design cooperative spacecrafts attitude control system. Through the proposed method, the most relaxed information-state topology requirement is obtained, that is, the directed graph that contains a directed spanning tree. Some numerical simulations demonstrate effectiveness of the proposed feature of methodology.

1. Introduction

Over the past decades, spacecraft developments have been done to bring wide-range missions, including earth observation and communication. Attitude control system plays an important role in these missions. Many research efforts on spacecraft attitude control design have been reported, that is, linear matrix inequality- (LMI-) based robust mixed / attitude control system design with linearization of dynamics and kinematics approach [1]; LMI-based nonlinear continuous attitude control design, [2]; proportional-derivative+ (PD+) type output feedback attitude control system with uniformly practically asymptotically stability guarantee for its equilibrium points [3]; attitude control system with discontinuous control law applying inverse cotangent function [4]; and hybrid attitude control system with property of robustness to measurement noise [5], to name a few.

From an attitude determination calculation, orientation of spacecraft is obtained as a rotation matrix that belongs to special orthogonal order-3 space, . However, some parameterizations are usually employed in a design of attitude control that is also employed in [1–5]—all of them employ quaternion parameterization, except [1], which uses Euler angle parameterization.

Because quaternion can represent spacecraft attitude globally, it is useful for spacecraft whose missions require doing reorientation maneuver over a large rotation angle. In contrast, Euler angle parameterization cannot represent spacecraft attitude globally due to its singularity property. Note that quaternion—also called Euler parameters—is the only parameterization consisting of four parameters and can represent attitude or orientation globally [6]. Nevertheless, its representation is not unique. There are two-antipodal values that correspond to a single physical orientation or a single orientation in matrix rotation representation. Therefore, single equilibrium orientation point in matrix rotation representation is represented by two equilibrium points in quaternion. This fact would imply nonefficient control effort called unwinding phenomenon when only one equilibrium point is regarded in quaternion-based attitude control design [7]. Besides, difficulty in guaranteeing stability appeared when the two equilibrium points in quaternion are regarded. There are many research results giving solution to this problem. They include discontinuous control approach [4], optimal control and finite time stability approach [8], backstepping-based attitude control design [9], hybrid mechanism attitude control [5, 10], and continuous nonlinear attitude control employing augmented dynamic [11], to name a few.

In addition to above problem, numerous spacecraft formation flying missions have been conceived and some of them have been launched, for example, interferometric synthetic aperture radar spacecrafts [12] and rendezvous technology experiment [13]. Cooperative control is a main issue for spacecraft formation control. In [14], Ren proposed proportional-derivative (PD) type attitude controller with several consensus algorithms for attitude formation. Hybrid attitude controller with connected and acyclic exchange of information-state topology for attitude formation has been reported in [15]. In [16], PD type controller with connected (undirected) exchange of information-state topology for attitude formation has been designed through Input to state stability approach where cooperative controller term is regarded as external input of each spacecraft.

In this paper, attitude controller appearing in [17] is utilized. Main contributions of this paper are attitude control system designed through boundedness of solution approach, that is, ultimately bounded solution and input to state stability for single attitude control case and boundedness of solution for cooperative attitude control case. Unlike the attitude control system employing PD controller that considers only one equilibrium point in quaternion representation, the attitude controller used in this paper considers two equilibrium points in quaternion representation. Less strict information-state topology requirement for attitude formation is also obtained, that is, topology that consists of a directed spanning tree. Preliminary result of this research has been presented in [18].

Overall, this paper is organized as follows. Section 2 discusses the system model. This section consists of notation, kinematics, and dynamics model of spacecraft and information-state exchange framework of the cooperative spacecrafts used in the rest of the paper. Section 3 describes controllers design and simulations. Finally, some concluding remarks are stated in Section 4.

2. Modeling

In this paper, a spacecraft is considered a rigid body. In the following descriptions, subscripts or/and superscripts , , and denote spacecraft’s fixed body frame, inertial reference frame, and spacecraft’s desired frame, respectively. For brevity, the spacecraft’s fixed body frame, the inertial reference frame, and the spacecraft’s desired frame may be written as body frame, inertial frame, and desired frame, respectively. Here, set of -dimension real column matrices, set of real matrices, and set of positive integers are denoted by , , and , respectively, where . Given and , is 2-norm of column matrix ; and are minimum and maximum Eigen value of , respectively; and are transpose matrix of and , respectively; means that is a positive definite matrix.

Following [19], vectors are defined as follows: where is a vectrix associated to the body frame, that is, a column matrix with three unit vectors , , and ; is a column matrix whose three components of are expressed or decomposed into the body frame; is skew-symmetric matrix of .

2.1. Spacecraft Kinematics and Dynamics Model

Consider the rotation matrix to transform a vector expressed in to be expressed in that satisfies (2): Taking the time derivative of (2) w.r.t. (with respect to) , is satisfied. Moreover, it implies (3), that is, the rotation matrix based kinematics equation of body frame w.r.t. desired frame: The corresponding quaternion-based kinematics equation of body frame w.r.t. desired frame is given by where Since quaternion is member of unit sphere order-3, , satisfies , where and ; is the corresponding Euler axis and is the corresponding Euler angle. Note that, here, this kinematics also represents attitude error.

Dynamic of the rigid body rotation system decomposed in is given by the Euler equation [19] (6): where positive definite matrix is a spacecraft moment of inertia about its center of mass located in the origin of (kg m²); is angular velocity of w.r.t. decomposed in , (rad/s); is the total external control torque about its center of mass located in the origin of (Nm); and is the external disturbance torque or total environmental effects, for example, gravity-gradient effects, solar radiation, magnetic field torques, and air-drag.

2.2. Cooperative Attitude Framework

Index of a spacecraft in a group composed by -spacecrafts is denoted by subscript (and sometimes both subscript and superscript) () and its neighbor is denoted by subscript (and sometimes both subscript and superscript) (), where , . For example, denotes body frame of spacecraft .

The dynamic of spacecraft in a group is represented by Quaternion-based kinematics equation that represents orientation of body frame of spacecraft , , w.r.t. body frame of spacecraft , , is where Information-state exchange between spacecraft in a group composed by -spacecraft is modeled by a directed graph topology where is the node set; is the edges set; and is the adjacency matrix of the graph , where , .

The entry of adjacency matrix if spacecraft receives information-state from spacecraft , where is the parent node and is child node and . If there is no information-state exchange from spacecraft to spacecraft , then . Self-edge is not allowed, that is, . In a directed graph, . If the graph has at least one node with a directed path to all other nodes then is called to have a directed spanning tree.

3. Control Systems Design and Simulations

Equations (12) and (13) [17] are the feedback term for (6) and (7), respectively, where and are tuning parameters. The function is a column matrix of saturation function that, defined element-wise, follows the scalar saturation function (14), with the saturation limit :

3.1. Problem Statements of the Control Systems Design

Next subsection would discuss the proposed attitude control design for single spacecraft case as well as cooperative spacecraft case. By utilizing Lyapunov stability theory, and have to be found such that the solution of single spacecraft attitude control system composed by (6) and (12) is ultimately bounded if and is input to state stable if . Similarly, a cooperative spacecraft attitude control has to be designed by determining and and information-state exchange topology such that the solution of each spacecraft in a group is ultimately bounded if and is input to state stable if . The authors suggest the readers should refer to [20] for definition of ultimately bounded solution and input to state stability.

3.2. Single Spacecraft Case

It is well known that Lyapunov stability theory requires the existence and uniqueness of a solution, for a given initial condition, for all future time. The following proposition states that the system composed by (6) and (12) satisfies existence and uniqueness solution requirement because the system is locally Lipschitz [19].

Proposition 1. Consider the system (6). Suppose . The spacecraft attitude control system composed by (4)–(6) and (12) is locally Lipschitz in .

Proof. For brevity, let , , , and , where . Then, consider the following: Since (5) are continuously differentiable, then the following inequalities can be obtained where .
Now consider as follows: Since and , where , are continuously differentiable functions, then the following inequalities can be obtained: where .
Now, recall (18) and (19), since the term with saturation function satisfies ≤ ≤ , where , then the following inequalities can be obtained: where symmetric matrix .
Recall (15), (16), and (20), then the following inequalities can be obtained: where and symmetric matrix .
The last inequality in (21) states that the system composed by (4)–(6) and (12) is locally Lipschitz in . It implies that the system has a unique solution for all future time.

Proposition 2. Consider the system (6). If is continuous and bounded for all , then the attitude control system composed by (4)–(6) and (12) satisfies local Lipschitz in .

Proof. Follow idea of the proof of Proposition 1.

Theorem 3. Consider the system (6) and suppose there is no external disturbance, that is, . If scalar is positive, matrix and moment of inertia are symmetric and positive definite, then solution of the quaternion-based attitude control system composed by (4)–(6) and (12) is ultimately bounded.

Proof. Consider , a set that consists of all , (22), and positive definite function (23): where To make sure that is a positive definite matrix, that is, , hence and symmetric matrix . If has three different eigenvalues, then will be bounded below by and bounded above by , where and are class functions given below: Now, take time derivative of (23). Since the desired angular velocity is zero, . In addition, note that (see Fact 3.10.1 in [21]). Following these, (26) is satisfied: is a nondecreasing function since is not always negative. Fortunately, the unit quaternion is bounded by the unit sphere order-3 property (27): Therefore, the solution is bounded above by as shown below: As a direct consequence of (28), is bounded above by , that is, In addition, for all positive scalar and positive definite symmetric matrix , if , then the solution is bounded, that is, Note that has to be a symmetric and positive definite matrix to make sure that all eigenvalue of are positive so that, along with , (30) is satisfied.
From (28)–(30), is bounded as follows: Consider (26); using property (29), the following inequalities are obtained: where and is given as follows: Therefore, inequality (34) is fulfilled for every satisfying (35): In according to Theorem 4.18 in [20], solution of the system is ultimately bounded with the ultimate bound as follows:

Remark 4. Note that in (35) clearly satisfies . This fact implies that in (35) is possible to be smaller when is getting smaller. Since is getting smaller, then the ultimate bound is smaller. Thus, if , then it will be equivalent to asymptotic stability of the set consisting of two equilibrium points .

The solid line in Figure 1 depicts the response of Euler angle attitude control system designed based on Theorem 3 using parameters in Table 1. The response is also compared by Euler Angle response of the attitude control system (4)–(6) using a well-known PD-like control law, —depicted by dash-dot line.

As seen from Figure 1, the attitude control system employing PD-like control law exhibits unwinding phenomenon because it is designed by considering only one equilibrium point in quaternion parameterization. Figure 2 shows more clearly that unlike the attitude control system employing PD-like control law, the designed control system converges to the closer equilibrium point, . Figure 3 shows the opposite direction of rotation between the two control systems. This situation implies the efficiency of energy consumption as shown in Figure 4. In addition, the responses depicted in Figures 2 and 3 show that the situation explained in Remark 4 is promising.

(a)

(b)

(c)

(d)

(a)

(b)

(c)

Now, suppose that the external disturbance of system (6) is continuous and bounded, that is,

Theorem 5. If scalar is positive and matrix is symmetric and positive definite, quaternion-based attitude control system composed by (4)–(6) and (12) is input to state stable.

Proof. Using the same energy-like function (23), its time derivative is given as follows: Note that, according to (37), the disturbance is bounded by . Following the way of proof of Theorem 3, for , (40) is satisfied: where satisfies , is the same as (29) and every satisfies (40): According to Theorem 4.19 in [20], the system is input to state stable with ultimate bound of the system’s solution being as follows:

To verify Theorem 5, a simulation is run using the same parameters in Table 1 and a disturbance function, as shown in Figure 5. The boundedness of solution properties of the designed attitude control system in the presence of disturbance is confirmed by Figures 6 and 7. Figure 9 shows the control action of the designed control system in order to have robustness property in the presence of disturbance as shown in inset of Figure 6. In addition, Figure 8 is presented as comparison control action without the presence of disturbance.

(a)

(b)

(c)

(a)

(b)

(c)

(d)

(a)

(b)

(c)

(d)

(e)

(f)

(a)

(b)

(c)

(d)

(e)

(f)

3.3. Cooperative Spacecraft Case

Bounded solution of cooperative spacecraft attitude control system involves attitude errors between spacecrafts, that is, . In this subsection it will be shown that through the boundedness of solution approach, cooperative attitude system could be designed.

Theorem 6. Consider the cooperative spacecraft attitude control system composed by (7)–(10) and (13). Suppose scalar is positive and matrix and moment of inertia are symmetric and positive definite, for . Suppose external disturbance is continuous and bounded function, , , for . If at least there is a directed spanning tree in the topology representing information-state exchange between spacecraft and spacecraft in a group composed by -spacecrafts, then solution of the cooperative system is bounded.

Proof. Consider the energy-like function (43): Then, its time derivatives can be written as follows: Since , then (45) is satisfied. Regarding property of spacecraft orientation and angular velocity, there is inequality as follows: Noting that , and the bound of then the following inequality can be obtained: where . And Therefore, there are and symmetric matrix such that implies that is bounded for , and .
Now, suppose is not bounded. Without loss of generality, taking , the following inequality is obtained: Then, there is a sufficiently large , where, inherently, also sufficiently large and , such that , and the increasing are resulting .
Let the information-state exchange topology consist of one directed spanning tree and ; therefore, the following inequality is obtained: This contradicts that, for , . Note that it implies that . Therefore, by contradiction, it is shown that is bounded.

Remark 7. Through the proof by contradiction described above, solution of the cooperative system is bounded for all and symmetric matrix where that only requires a directed spanning tree in information-state exchange topology.

Remark 8. Note that here the condition of information-state exchange topology is less-conservative condition, for example, directed spanning tree. In contrast, for example, [15] requires acyclic topology and [16] requires connected (undirected) topology.

Simulation of cooperative attitude control case is run using , , and as shown in Table 1 and the same disturbance function as used in the previous subsection. The simulation used three spacecrafts with different moment of inertia and three different initial attitudes as shown in Table 2. Meanwhile, initial angular velocities of all spacecrafts are zero. A cyclic directed graph representing information-state exchange is applied in the simulation (Figure 10). Figure 11 shows that each spacecraft converges to absolute attitude about 95°. It implies, as shown in Figure 12, that the relative attitudes between spacecrafts are regulated.

(a)

(b)

(c)

(d)

(e)

(f)

(a)

(b)

(c)

4. Concluding Remarks

Dynamics of spacecraft rotational motion based on Euler equation and quaternion-based kinematics describing the spacecraft attitudes have been presented. For cooperative spacecrafts case, information-state exchange is modeled by directed graph. Using these models, design of attitude control through boundedness of solution approach for single and cooperative spacecrafts have been proposed. The control designs are done by regarding two equilibrium points.

In single spacecraft attitude control system case, the attitude control system has been designed, for zero disturbances, via ultimately bounded solution and for nonzero disturbances, via input to state stability approach. For zero disturbances, if the ultimate bound is zero at , then it is equivalent to asymptotic stability. In cooperative spacecraft attitude control system case, the cooperative system has been designed via the boundedness of solution approach. If the information-state exchange topology has directed spanning tree, then the cooperative spacecraft attitude system is bounded. In addition, all theorems stated in this paper have been verified in simulations. The transient responses and steady state responses demonstrate effectiveness of the proposed methodology.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research is partly supported by Beasiswa Unggulan, BPKLN, Kemendikbud, Indonesia. The authors also gratefully acknowledge the referees for carefully reading the paper and many helpful comments.

References

E. R. Pinheiro and L. C. G. de Souza, “Design of the microsatellite attitude control system using the mixed $H_{2} / H_{\infty}$ method via LMI optimization,” Mathematical Problems in Engineering, vol. 2013, Article ID 257193, 8 pages, 2013.
View at: Publisher Site | Google Scholar
L.-L. Show, J.-C. Juang, and Y.-W. Jan, “An LMI-based nonlinear attitude control approach,” IEEE Transactions on Control Systems Technology, vol. 11, no. 1, pp. 73–83, 2003.
View at: Publisher Site | Google Scholar
R. Schlanbusch, A. Loria, R. Kristiansen, and P. J. Nicklasson, “PD+ based output feedback attitude control of rigid bodies,” IEEE Transactions on Automatic Control, vol. 57, no. 8, pp. 2146–2152, 2012.
View at: Publisher Site | Google Scholar
H. Septanto, B. Riyanto-Trilaksono, A. Syaichu-Rohman, and R. Eko Poetro, “On the stability analysis of a discontinuous quaternion attitude control system,” in Proceedings of the International Conference on System Engineering and Technology (ICSET '12), pp. 1–4, Bandung, Indonesia, September 2012.
View at: Publisher Site | Google Scholar
C. G. Mayhew, R. G. Sanfelice, and A. R. Teel, “Quaternion-based hybrid control for robust global attitude tracking,” IEEE Transactions on Automatic Control, vol. 56, no. 11, pp. 2555–2566, 2011.
View at: Publisher Site | Google Scholar
J. Stuelpnagel, “On the parameterization of the three-dimensional rotation group,” SIAM Review, vol. 6, no. 4, pp. 422–430, 1964.
View at: Publisher Site | Google Scholar
S. P. Bhat and D. S. Bernstein, “A topological obstruction to continuous global stabilization of rotational motion and the unwinding phenomenon,” Systems and Control Letters, vol. 39, no. 1, pp. 63–70, 2000.
View at: Google Scholar
S. Li, S. Ding, and Q. Li, “Global set stabilization of the spacecraft attitude control problem based on quaternion,” International Journal of Robust and Nonlinear Control, vol. 20, no. 1, pp. 84–105, 2010.
View at: Publisher Site | Google Scholar
R. Kristiansen, P. J. Nicklasson, and J. T. Gravdahl, “Satellite attitude control by quaternion-based backstepping,” IEEE Transactions on Control Systems Technology, vol. 17, no. 1, pp. 227–232, 2009.
View at: Publisher Site | Google Scholar
C. G. Mayhew, R. G. Sanfelice, and A. R. Teel, “On path-lifting mechanisms and unwinding in quaternion-based attitude control,” IEEE Transactions on Automatic Control, vol. 58, no. 5, pp. 1179–1191, 2013.
View at: Publisher Site | Google Scholar
H. Septanto, B. Riyanto-Trilaksono, A. Syaichu-Rohman, R. Eko Poetro, and A. R. Ibrahim, “The quaternion-based attitude control system with an augmented dynamic,” in Proceedings of the 3rd International Conference on Instrumentation, Communications, Information Technology, and Biomedical Engineering (ICICI-BME '13), pp. 190–193, Bandung, Indonesia, November 2013.
View at: Publisher Site | Google Scholar
G. Krieger, A. Moreira, H. Fiedler et al., “TanDEM-X: a satellite formation for high-resolution SAR interferometry,” IEEE Transactions on Geoscience and Remote Sensing, vol. 45, no. 11, pp. 3317–3341, 2007.
View at: Publisher Site | Google Scholar
S. Persson, S. Veldman, and P. Bodin, “PRISMA—a formation flying project in implementation phase,” Acta Astronautica, vol. 65, no. 9-10, pp. 1360–1374, 2009.
View at: Publisher Site | Google Scholar
W. Ren, “Distributed attitude alignment in spacecraft formation flying,” International Journal of Adaptive Control and Signal Processing, vol. 21, no. 2-3, pp. 95–113, 2007.
View at: Publisher Site | Google Scholar
C. G. Mayhew, R. G. Sanfelice, J. Sheng, M. Arcak, and A. R. Teel, “Quaternion-based hybrid feedback for robust global attitude synchronization,” IEEE Transactions on Automatic Control, vol. 57, no. 8, pp. 2122–2127, 2012.
View at: Publisher Site | Google Scholar
Q. Hu, Y. Lv, J. Zhang, and G. Ma, “Input-to-state stable cooperative attitude regulation of spacecraft formation flying,” Journal of the Franklin Institute, vol. 350, no. 1, pp. 50–71, 2012.
View at: Google Scholar
H. Septanto, B. Riyanto-Trilaksono, A. Syaichu-Rohman, and R. Eko Poetro, “The saturated quaternion control law with application for spacecraft formation flying,” in Proceedings of the IEEE Conference on Control, Systems & Industrial Informatics (ICCSII '12), pp. 59–63, Bandung, Indonesia, September 2012.
View at: Publisher Site | Google Scholar
H. Septanto, B. Riyanto-Trilaksono, A. Syaichu-Rohman, and R. Eko Poetro, “Analisis Solusi Sistem Kendali Sikap Satelit Berelemen Saturasi,” in Seminar Nasional Teknik Elektro Unjani, Indonesian National Conference, 2013.
View at: Google Scholar
P. C. Hughes, Spacecraft Attitude Dynamics, Dover, New York, NY, USA, 2004.
H. K. Khalil, Nonlinear Systems, Prentice Hall, Upper Saddle River, NJ, USA, 3rd edition, 2002.
D. S. Bernstein, Matrix Mathematics: Theory, Facts and Formulas, Princeton University Press, Princeton, NJ, USA, 2nd edition, 2009.
D. S. Roldugin and P. Testani, “Spin-stabilized satellite magnetic attitude control scheme without initial detumbling,” Acta Astronautica, vol. 94, no. 1, pp. 446–454, 2014.
View at: Publisher Site | Google Scholar

Copyright

Copyright © 2014 Harry Septanto et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation

Download other formats

Order printed copies

Views

2800

Downloads

1088

Citations