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Quasi-Continuous Higher-Order Sliding Mode Controller Designs for Spacecraft Attitude Tracking Manoeuvres
C. Pukdeboon Department of Applied Mathematics, University of Sheffield, Sheffield, UK A. S. I. Zinober Department of Applied Mathematics, University of Sheffield, Sheffield, UK M.-W. L. Thein Department of Mechanical Engineering, University of New Hampshire, Durham, NH 03824, USA

Abstract—This paper studies high-order sliding mode control laws to deal with some spacecraft attitude tracking problems. Second and third order quasi-continuous sliding control are applied to quaternion-based spacecraft attitude tracking manoeuvres. A class of linear sliding manifolds is selected as a function of angular velocities and quaternion errors. The second method of Lyapunov theory is used to showthat tracking is achieved globally. An example of multiaxial attitude tracking manoeuvres is presented and simulation results are included to verify and compare the usefulness of the various controllers.

I. I NTRODUCTION In general spacecraft motion is governed by the so-called kinematics equations and dynamics equations [1]. These mathematical descriptions are highly nonlinear and thus linearfeedback control techniques are not suitable for the global controller design. First-order sliding mode control has been considered as a useful scheme for spacecraft attitude control. Vadeli [2] designed a variable structure attitude control law based on quaternion kinematics. A similar approach was later proposed in [3] where sliding mode controller was designed for spacecraft tracking problems.This was illustrated by an example of multiaxis attitude tracking manoeuvres. An adaptation of the sliding mode control technique was derived and applied to a quaternion-based spacecraft attitude tracking manoeuvres. This modified version presented in [4] is the smoothing model-reference sliding mode control (SMRSMC). This technique improves the transient response and reduces the chatterphenomenon. In [5] the (additive) quaternion-based tracking of spacecraft manoeuvres used sliding mode control in the sense of optimal control. McDuffie and Shtessel [6] designed a de-coupled sliding mode controller and observer for spacecraft attitude control. From the previous literature we conclude that sliding mode control can be used for quaternion-based spacecraft attitude tracking manoeuvres. Floquet[7] presented the stabilization of the angular velocity of rigid body via first-order and second-order sliding mode controllers but it has not been applied to spacecraft tracking problems. Higher-order sliding mode control has desired properties, such as robustness, similar to sliding mode control. It also may reduce chattering and provides better accuracy than first order sliding. Hence we willstudy spacecraft attitude tracking manoeuvres using higher-order sliding mode control.

This paper is organized as follows. Section II presents the kinematics and dynamic equations of a rigid spacecraft. In Section III the sliding manifold and first-order sliding mode control are presented for attitude tracking manoeuvres. In Section IV the sliding manifold and the second-order quasi-continuouscontroller [8] are presented. A first-order differentiator [9] is applied to estimate the time derivative of the sliding vector. Section V presents the design of thirdorder quasi-continuous controller. We add a precompensator (first-order lag) to the spacecraft model description to smooth the control signal, and use a second-order differentiator [7] to estimate the first and second time derivatives ofthe sliding vector. A numerical example of the multiaxial attitude tracking problem [4] is illustrated in Section VI to verify the usefulness the third-order quasi continuous controller. Section VII is the conclusion. II. S PACECRAFT M ODEL D ESCRIPTION We consider the general case of a rigid spacecraft rotating under the influence of body-fixed torquing devices. According to [10], the kinematics...
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