Adaptive Fuzzy Sliding-Mode Attitude Controller Design for Spacecrafts with Thrusters

Space exploration has often been used as a proxy competition for geopolitical rivalries. The early era of space exploration was driven by a "space race" among might countries. Nowadays, many advanced nations have entered the space arena and competed with one another for the space technology development as they recognize the importance of space technologies for national strength. In general, physical exploration of space is conducted both by human spaceflights and by robotic spacecrafts. To promote the scientific standard of space vehicle performance, stability, and control, it requires analysis of the six degrees of freedom of the vehicle's flight. One is the translational motion in three dimensional axes; the other is the orientation about the vehicle's center of mass in these axes, known as pitch, roll and yaw. In this paper, the main goal is to propose an adaptive fuzzy sliding-mode controller for spacecrafts with thrusters to follow the predetermined trajectory in outer space by use of employing the spacecraft attitude control. By using fuzzy inference mechanism, the upper bounds of the lumped uncertainty can be estimated, and the adaptive theory with center adaption of membership functions is designed to estimate optimal upper bounds of the lumped uncertainty, respectively. For the above reasons, we know that the attitude control of spacecrafts for the space exploration is essential to successfully develop the space activities [1].


Introduction
Space exploration has often been used as a proxy competition for geopolitical rivalries. The early era of space exploration was driven by a "space race" among might countries. Nowadays, many advanced nations have entered the space arena and competed with one another for the space technology development as they recognize the importance of space technologies for national strength. In general, physical exploration of space is conducted both by human spaceflights and by robotic spacecrafts. To promote the scientific standard of space vehicle performance, stability, and control, it requires analysis of the six degrees of freedom of the vehicle's flight. One is the translational motion in three dimensional axes; the other is the orientation about the vehicle's center of mass in these axes, known as pitch, roll and yaw. In this paper, the main goal is to propose an adaptive fuzzy sliding-mode controller for spacecrafts with thrusters to follow the predetermined trajectory in outer space by use of employing the spacecraft attitude control. By using fuzzy inference mechanism, the upper bounds of the lumped uncertainty can be estimated, and the adaptive theory with center adaption of membership functions is designed to estimate optimal upper bounds of the lumped uncertainty, respectively. For the above reasons, we know that the attitude control of spacecrafts for the space exploration is essential to successfully develop the space activities [1].
In order to deal with the nonlinear spacecraft attitude dynamics, we employ the quaternion representation [2][3][4] to model the equation of rotational motion and the time derivative of quaternion, so that the nonlinear attitude control is applicable. Using the quaternion representation, the global control effect can be fulfilled and the singularity problem, which will be faced with the discontinuity by the three-dimensional Euler's representation [5], can be avoided.
To cope with non-ideal factors surrounding the spacecraft under attitude control and to enhance the robustness property of the attitude tracking system, the sliding mode control has been employed by Hu et al. [6], which integrate both the command input shaping and sliding mode output feedback control techniques to investigate the vibration problem for a flexible spacecraft during attitude maneuvering, and by Yeh [7], which proposes two nonlinear attitude controllers for spacecrafts with thrusters to follow the predetermined trajectory to estimate parameters and eliminate disturbances. Servidia and Pena [8] present the attitude stabilization of a spacecraft using thrusters, considering from a practical point of view. Hu [9] proposes a dual-stage control system design scheme for rotational maneuvers and vibration stabilization of a flexible spacecraft in the presence of the parameter uncertainty and external disturbances as well as the control input saturation to actively suppress certain flexible modes. Xia et al. [10] present the adaptive law and the extended state observer for a spacecraft model that is nonlinear in dynamics with the inertia uncertainty and external disturbances to converge to the reference attitude states. Despite the popularity of such control technique, it is however well known that the chattering problem is worthy of more attention for the sake of practical deployment. Taking into consideration of the aforementioned reason, a guide to sliding mode control for practical implementation has been proposed by Young et al. [11]. Eker [12] proposes a second-order sliding mode control for uncertain plants using the equivalent control approach to improve the performance of control systems, in which second-order plant parameters are experimentally determined using input-output measured data.
A spacecraft equipped with thrusters can effectively control its acceleration direction [8,13], which in turn implies that the maneuverability/controllability of the spacecraft can be greatly enhanced during the stage when the spacecraft is flying in the outer space; whereas Janhunen et al. [14] propose a space propulsion concept of electric solar wind sail using the natural solar wind dynamic pressure for producing spacecraft thrust.
Estimation theory is used to deal with estimating values of parameters based on measured/empirical data that have a disturbance component. A parameter estimation approach called adaptive control has been developed by Slotine [15,16] to achieve accurate attitude tracking of a rigid spacecraft with large loads of unknown mass. Zou et al. [17] investigate the robust adaptive output feedback controller based on Chebyshev neural networks (CNN) for an uncertain spacecraft to counteract CNN approximation errors and external disturbances. Huang et al. [18] propose a robust adaptive PID-type controller incorporating a fuzzy logic system and a sliding-mode control action for compensating parameter uncertainties and the robust tracking performance. An adaptive fuzzy theory is generally employed to approximate unstructured uncertainties and dynamic disturbances, such as Tong et al. [19] discuss an adaptive fuzzy output feedback control approach for nonlinear systems to estimate unmeasured states; Islam and Liu [20] propose a robust adaptive fuzzy control system for the trajectory tracking control problem of robotic systems to approximate the certainty equivalent-based optimal controller and to cope with uncertainties.
In this paper, we investigate the adaptive fuzzy sliding-mode control for spacecrafts with thrusters, employing the fuzzy sliding-mode controller to estimate upper bounds of the lumped uncertainty, and the adaptive fuzzy sliding-mode controller with center adaption of membership functions to estimate optimal bounds of the lumped uncertainty, respectively. This paper is organized as follows. In Section 2, preliminaries for deriving three-degree-offreedom attitude models of a spacecraft equipped with thrusters. In Section 3, we respectively propose the sliding-mode, the fuzzy sliding-mode and the adaptive fuzzy sliding-mode attitude controllers aiming for tracking the predetermined trajectory in outer space. For tracking realization, three simulation results incorporating the so-called quaternion-based attitude control are developed in Section 4. To demonstrate the superior property of the proposed attitude controllers, three numerical simulations are provided in that Section. Finally, conclusions are drawn in Section 5.

Equations of rotational motion for spacecrafts
Assume the spacecraft is a rigid body; therefore, the Euler's equation of rotational motion is adopted with the following general form as where all the variables are defined in and please referring to the Nomenclature.
Assume that the movable nozzle is located at the center of the spacecraft tail, and the distance between the movable nozzle center and the spacecraft's center of gravity is  . Furthermore, we also assume that the spacecraft is equipped with a number of thrusters on the surface near the center of gravity that will produce a pure rolling moment whose direction is aligned with the vehicle axis, b X , referring to Fig. 1. Thus, the vector b L , defined as the relative displacement from the spacecraft's center of gravity to the center of the movable nozzle, satisfies b L   . Note that J is the moment of inertia matrix for a spacecraft with respect to the body coordinate frame, and hence is a 3 3  symmetric matrix.
After referring to Fig. 1 and Fig. 2, the torque exerted on the spacecraft can be expressed in the body coordinate frame as where n is the magnitude of the movable nozzle thrust, p d and y d are the pitch angle and yaw angle, respectively, of the movable nozzle, is the aforementioned variable moment in the axial direction of the spacecraft, and Tb F is the force produced by the movable nozzle in the body coordinate frame. Let the rotation matrix b B denote the transformation from the body coordinate frame to the inertial coordinate frame. Thus, the force exerted on the spacecraft observed in the inertial coordinate system is as follows: where   123 T Dddd  is a disturbance vector in the body coordinate frame.
Generally speaking, the attitude of a rigid body may be described in various ways, and "quaternion" is one of the means. According to Euler's rotation theory [21], there exist a unit vector U and an angle  such that U is perpendicular to the rotation plane with respect to a rotation angle  . Thus, for any quaternion, it can be defined as four parameters

Sliding-mode attitude controller design
Considering the presence of model uncertainties, parameter variations, and disturbances, we recognized that the sliding mode control is an effectively robust controller for various applications; in this paper, we consider a sliding-mode control to eliminate all variation influences of a spacecraft during the whole flying course for the practical controller design. We first design a sliding-mode attitude controller, which can compensate for the adverse effect owing to spacecraft variations.
The principal procedure to verify the stability and robustness of the sliding-mode attitude tracking problem consists of the sliding and reaching conditions, and that will be given in detail as follows.
Step 1: Choose the sliding surface such that the sliding condition will be satisfied and hence the origin of the error dynamic is exponentially stable.

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From the sliding mode theory, once the reaching condition is satisfied, the system is eventually forced to stay on the sliding surface, i.e., where P is a positive definite diagonal matrix. The system dynamics are then constrained by the following differential equations, referring to Eq. (7) By Lyapunov stability theory, it can be proved that e Q will be driven to zero when the system is constrained on the sliding mode dynamics and so will the error angular velocity e  . For the above reason, the system origin     31 31 ,0 , 0 ee Q   of the ideal system can be verified to be exponentially stable.
Step 2: Design the controller such that the reaching condition is satisfied.
Let us define the sliding surface as above equation (9) shown as Here, we make an assumption that J is a symmetric and positive definite matrix, and let the Lyapunov function candidate be set as www.intechopen.com where 0 s V  only when 0 S  . Then, the time derivative of s V can be derived as Substituting Eq. (7) and the time derivative of Eq. (9) into the above equation (14) is the minimum eigenvalue of s K , where s K is a positive definite diagonal matrix as above mentioned. Therefore, the reaching and sliding conditions of the sliding mode 0 S  are guaranteed. As a result, the exponential stability and robustness of the sliding mode attitude controller can be achieved.  (16) always suffers from the chattering problem. To alleviate such undesirable phenomenon, the sign function can be simply replaced by the saturation function. The system is now no longer forced to stay on the sliding surface but is constrained within the boundary layer i s   , where  is a small positive value. The cost of such substitution is a reduction in the accuracy of the desired performance.
To alleviate the chattering phenomenon, the saturation function may be employed to the control input of the sliding mode attitude control system.

Fuzzy sliding-mode attitude controller design
Upper bounds of the lumped uncertainty, which includes the external disturbances and internal perturbations, for the sliding-mode control need to be decided before the controller is using. In general, upper bounds of the lumped uncertainty are difficult to be obtained in advanced by computing, but always by the method of try and error. In this section, a fuzzy sliding-mode controller is proposed, in which a fuzzy inference mechanism is used to estimate upper bounds of the lumped uncertainty. We recognized that the prior expert knowledge of the fuzzy inference mechanism, which can be used to estimate upper bounds of the lumped uncertainty, is available effectively.
The following Lemma is introduced to enable the fuzzy sliding-mode controller such that the reaching condition of the switching surface is satisfied. By Lyapunov stability theory, the fuzzy sliding-mode attitude controller of the spacecraft can be proved as an exponentially stable system. The sufficient conditions for successful stability effect are stated in the following lemma.  Eq. (21). Then the exponential stability and robustness of the fuzzy sliding-mode attitude control system can be achieved.
Proof: See the Appendix 1. From the proof in the Appendix 1, the exponential stability and robustness of the fuzzy sliding-mode attitude controller can be guaranteed, so that the spacecraft attitude tracking system can be achieved completely. www.intechopen.com

Adaptive fuzzy sliding-mode attitude controller design
Since the sliding-mode attitude controller for a spacecraft requires estimating upper bounds (constant values) of the lumped uncertainty, so that the lumped uncertainty during the entire flying course can be always eliminated. If upper bounds are not chosen appropriately, the high-gain problem will be suffered. That is, the control input torque is over-large definitely and the implementation cost is increased as well. Optimal upper bounds si c , 1,2,3 i  , cannot be obtained exactly by pure sliding mode control owing to the unknown of uncertainties. Therefore, the adaptive fuzzy control algorithm based on the principle of sliding mode control is developed to estimate optimal upper bounds of the lumped uncertainty and to achieve the minimum control torque.
Assume there exists estimated upper bounds ˆs i c , 1,2,3 i  which can achieve minimum control torque and satisfy the sliding-mode condition of the switching surface, errors between estimated and optimal upper bounds are shown as , which can achieve minimum control torque and satisfy the sliding-mode condition of the switching surface, the error vector between the estimated and optimal upper-bound vectors are shown as The following theorem is introduced to enable the adaptive fuzzy sliding-mode controller such that optimal upper bounds can be obtained. By Lyapunov stability theory, the adaptive fuzzy sliding-mode attitude controller of a spacecraft can be proved as an exponentially stable system. The sufficient conditions for the successful stability effect are stated in the following theorem.     is a positive definite diagonal matrix. Then the exponential stability and robustness of the adaptively fuzzy sliding-mode attitude control system can be achieved.
Proof: To achieve the exponential stability and convergence of the spacecraft attitude tracking system, let us define the sliding surface as above equation (9) is the minimum eigenvalue of s K , where s K is a positive definite diagonal matrix as above mentioned. Therefore, the reaching and sliding conditions of the sliding mode 0 S  are guaranteed. As a result, the exponential stability and robustness of the adaptive fuzzy sliding mode attitude controller can be achieved. Therefore, the global stability of the attitude tracking system is guaranteed by the proposed adaptive fuzzy sliding-mode controller. In the following, let us in detail verify the convergence of the states e Q and e  and of the parameters s C  . . And because that a quaternion always satisfies the socalled unit-norm property, we have e Q , e  are bounded. Let us now detail the proof itself.
Since V  is negative or zero and V is lower bounded, and because S , e Q , e  are bounded. Now, given that S converges to zero only exponentially, the actual differential equation that governs e Q by some exponentially decaying term can be given as From Eqs. (7) is an exponentially stable system, implying that the stability and convergence of Eq. (32) are governed by 2 () e f Q . If 2 () 0 e fQ  exponentially as 0 t  , then the system is reduced to system Eq. (7), which is an exponentially stable system. From Eq.
(31),  is an exponentially decaying function, and because  

Simulations
To validate the proposed attitude tracking problem, we present three nonlinear controllers respectively consisting of the sliding-mode attitude controller design in Section 3.1, the fuzzy sliding-mode attitude controller design in Section 3.2 and the adaptive fuzzy sliding-mode attitude controller design in Section 3.3 for a rigid spacecraft, so as to demonstrate the performance and effectiveness of the respective controller designs.
For simulation the initial conditions of quaternion and angular velocity are set as     Fig. 3(e). In our simulation, some positive definite diagonal matrices of P (see Eq. (9)) for the sliding surface and s K (see Eq. (16)) for the control torque input will also be shown as follows.  (17)) to satisfy the better stability requirements of the overall system, such that external disturbances can be eliminated completely. www.intechopen.com

Simulation results of the sliding-mode attitude controller
The appealing effect of the sliding-mode attitude control presented in quaternion form is given in Fig. 5(a), which shows the present attitude and the desired one simultaneously. The solid line in each sub-figure denotes the current quaternion of the spacecraft, where the dashed line denotes the desired attitude in Fig. 5(a), we can see that the current and desired attitudes are coincident with each other, that is, the attitude tracking effect is fulfilled after about 3 seconds. This is to show feasible of the conclusion from Eq. (19) and to show well results of attitude tracking of the spacecraft. Here, the robustness and effectiveness can simultaneously be demonstrated by results shown in Fig. 3 for the attitude control system. The sliding surface and the torque input of the sliding-mode controller with varied desired attitude are shown in Figs. 5(b) and 5(c), respectively to demonstrate the practical effect. Variations of the moment of inertia matrix J and disturbances D , which are used in the proposed three nonlinear controllers, are shown in Figs. 3(d) and 3(e), respectively.

Simulation results of the fuzzy sliding-mode attitude controller
In this section, the revealing response of the fuzzy sliding-mode attitude control is given in Fig. 6(a), which also shows the present attitude and the desired one simultaneously. The solid line in each sub-figure denotes the current quaternion of the spacecraft, where the dashed line denotes the desired attitude in Fig. 6(a), we can see that the current and desired attitudes are coincident with each other. From Fig. 6(a), the attitude tracking effect is fulfilled almost totally after 3 seconds. This is to show feasible of the conclusion from Eq.

Simulation results of adaptive fuzzy sliding-mode attitude controller
The revealing response of the adaptive fuzzy sliding-mode attitude control also presented in quaternion form is given in Fig. 7(a), which also shows the present attitude and the desired one simultaneously. The solid line in each sub-figure denotes the current quaternion of the spacecraft, where the dashed line denotes the desired attitude in Fig. 7(a), we can see the current and desired attitudes are coincident with each other. From Fig. 7(a), the attitude tracking effect can be seen to be fulfilled after about 3 seconds. This is to show feasible of the conclusion from Eq. (30) and to show well results of spacecraft attitude tracking. Here, the robustness and effectiveness can simultaneously be demonstrated by the results shown in  estimation can be fulfilled finally. Therefore, for upper-bound estimation the persistent exciting of a spacecraft is rich enough for the proposed controller.

Conclusions
Since the attitude control and system stability are the key for space technology, we address the nonlinear attitude controller designs of a spacecraft, respectively consisting of the sliding-mode, the fuzzy sliding-mode and the adaptive fuzzy sliding-mode attitude controllers. The fuzzy sliding-mode controller is designed to estimate upper bounds of the lumped uncertainty, and the adaptive fuzzy sliding-mode controller with center adaption of membership functions is designed to estimate optimal upper bounds of the lumped uncertainty, respectively. We prove the exponential stability for the proposed attitude controllers for external disturbances and inertia uncertainties through the aids of the Lyapunov stability analysis and the Barbalat's lemma.
Extensive simulations have been adopted to verify the feasibility for three attitude tracking controllers corresponding to the lumped uncertainty. The system performance and its stability can also be demonstrated by use of the aforementioned theoretical derivations and the realistic simulations.  The book also provides a detailed overview on Spacecraft Systems including reliability of low-cost AOCS, sliding mode controlling and a new view on attitude controller design based on sliding mode, with thrusters. It also provides a technological roadmap for HVAC optimization. The book also gives an excellent overview of resolving the difficulties for interplanetary missions with the comparison of present technologies and new advancements. Overall, this will be very much interesting book to explore the roadmap of technological growth in spacecraft systems.