Nonlinear Back Stepping Adaptive Controller Design for Spacecraft Attitude Maneuver

Abstract:

The nonlinear backstepping adaptive controller is designed for the large angle fast attitude maneuver of spacecraft with uncertainties in the moment of inertia. Deviation between the actual position and command position is depicted by error quaternion. The nonlinear backstepping tracking function is employed to design the adaptive controller both of command torque and estimation of unknown moment inertia. And the stability is analyzed based on Lyapunovo function. By Matlab/Simulink programming, the simulation of spacecraft attitude manoeuver control is discussed, and its results demonstrate the effectiveness and feasibility of the proposed controller.

• Periodical: Applied Mechanics and Materials (Volume 574)
• Pages: 539-545
• DOI: https://doi.org/10.4028/www.scientific.net/AMM.574.539
• Online since: July 2014
• Authors: Wei Jie Wang, Yuan Ren, Yi Yong Li
• Keywords: Adaptive Control, Attitude Maneuver, Backstepping, Error Quaternion, Spacecraft

References

[1] where is unknown positive definite symmetrical inertia matrix of spacecraft.

[2] is spacecraft angular velocity vector in body coordinate relative to inertia coordinate , is spacecraft control torque vector. is skew symmetric matrix.

[3] In order to avoid the singular question of large angle maneuver described with Eulerian angle, the quaternion is used to represent spacecraft attitude. By rigid-body kinematics, reference frame only needs to rotate one time from initial position to target frame. A quaternion consisted of vector and scalar are employed to represent the rotation from one coordinate to other through some axis or vector.

[4] where is identity matrix. ( is dimensional Euclidean space) is actual quaternion which body frame is relative to inertia frame, satisfies normal constraint equation , where . is defined to be skew symmetric matrix.

[5] The Eulerian angles that spacecraft rotates principal axis of inertia are recorded as roll angle , pitch angle , yaw angle . As the time of agile spacecraft maneuver is quite short, the orbit angular velocity can be neglected, then the switching relationship between quaternion and Eulerian angles can be written as following.

[6] In the mission of spacecraft attitude maneuver, there is a fixed angle between the expected frame and inertia frame. The angle can be indicated by command quaternion , the command quaternion also satisfies the constraint . The error quaternion of attitude maneuver indicates the difference between actual quaternion and command quaternion . The relationship among them is.

DOI: 10.1201/9780429446580-12

[7] where represents quaternion multiplication. Additionally, error quaternion can be expressed as.

[8] The derivation of Eq. 8 is.

[9] The Design of Controller According to the kinematics described by Eq. (4), the following Lyapunov function is selected to stabilize the subsystem.

[10] On the basis of the constraint condition.

[11] To differentiate Eq. (11), there is.

[12] For the sake of Lyapunov stability, namely keeping , the following design steps are taken. With regard to the kinematics Eq. (9), the is taken as virtual control input, the stabilization function is designed to stabilize the kinematics system of Eq. (9). New variable is defined as following.

[13] .

[14] The nonlinear tracking function is proposed as[] K. S. Kim, and Y. Kim. Robust Backstepping Control for Slew Maneuver Using Nonlinear Tracking Function. IEEE Transactions on Control Systems Technology, Vol. 11, no. 6, p.822–829, 2003. ].

DOI: 10.1109/tcst.2003.815608

[15] where , are positive constant. Taking the Eq. (13) Eq. (14) into the Eq. (12), there is.

[16] Taking Eq. (15)into Eq. (16).

[17] When , , combining with character of the arctan function there is.

[18] Consequently is asymptotic stability. To differentiate Eq. (14).

[19] where can be derived from the derivation of equation Eq. (15).

[20] The new variable is defined as.

[21] Both side of Eq. (19) are left multiplied by , then bringing Eq. (1) Eq. (21)in, there is.

[22] Making.

[23] So Eq. (22)can be written as.

[24] where is unknown constant, representing the spacecraft actual moment of inertia.

[25] The expression of is as following.

[26] The estimation error of spacecraft moment of inertia is defined as.

[27] where is estimation value of moment of inertia, because is unknown constant, the derivation of Eq. (27)is.

[28] The system Lyapunov function can be derived by augmenting Eq. (10).

[29] where is estimation matrix of the moment of inertia, to differentiate Eq. (29).

[30] There is by Eq. (27), bringing , Eq. (24), Eq. (28)into Eq. (30)yields.

[31] Obviously.

[32] Bringing Eq. (17) Eq. (32)into Eq. (31), there is.

[33] In order to ensure , the adaptive controller is designed as following.

[34] .

[35] where is positive constant. Bring Eq. (34), Eq. (35)into Eq. (33)yields.

[36] Obviously, the system is global asymptotic stability. Simulation Analysis With Matlab/Simulink, simulation is carried out according to the following parameters [] N. Yan. Research on the Simulation and Experiment of Attitude Control Method for Agile Small Satellite. Beijing, Beihang University, 2011. ] in order to validate the effectiveness and superiority of the designed nonlinear backstepping adaptive controller. Spacecraft initial attitude angle . Spacecraft initial angle velocity . Spacecraft moment of inertia The estimation matrix of spacecraft moment of inertia Command quaternion . Control constants , ,. The nonlinear backstepping adaptive controller represented by Eq. (34) Eq. (35)is used to simulate, the simulation results are shown in Fig. 1-5. Fig. 1 gives the command torque time history. Fig. 2 shows the attitude angle time response. Fig. 3 is the attitude angle velocity time response. Fig. 4 depicts the quaternion time response. Fig. 5 describes the tracking error time response, and Fig. 6 draws the time response of spacecraft estimation moment of inertia. Fig. 1. The time history of command torque. Fig. 2. Time response of attitude angle. Fig. 3. The angle velocity time response. Fig. 4. Quaternion time response. Fig. 5. Tracking error time response. Fig. 6. The time response of estiation of inertia. From the aforesaid simulation results, the designed nonlinear backstepping adaptive controller fast reduced the peak of control torque, achieved the estimation of unknown moment of inertia, and ensured the closed loop stability of system. Conclusion The spacecraft large attitude maneuver based on nonlinear backstepping adaptive is studied in this paper. The non-singular quaternion is used to represent the mathematical model of spacecraft attitude maneuver. Aimed at the uncertainty of spacecraft moment of inertia, the nonlinear backstepping tracking function is adopted to design the adaptive controller both of control torque and the estimation of unknown moment of inertia, the system stability is analyzed by Lyapunovo function. The simulation study is carried out through Matlab/Simulink. The simulation results show that the designed nonlinear backstepping adaptive controller can fast reduce the peak of control torque, achieve the estimation of unknown moment of inertia, and ensure the closed loop stability of system. The controller is validated to be effective and feasible. Acknowledgement This work partially supported by National Natural Science Foundation and Youth Science and Technology Foundation. Reference.