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 . 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  designed a variable structure attitude control law based on quaternion kinematics. A similar approach was later proposed in  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  is the smoothing model-reference sliding mode control (SMRSMC). This technique improves the transient response and reduces the chatterphenomenon. In  the (additive) quaternion-based tracking of spacecraft manoeuvres used sliding mode control in the sense of optimal control. McDufﬁe and Shtessel  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 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  are presented. A ﬁrst-order differentiator  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  to estimate the ﬁrst and second time derivatives ofthe sliding vector. A numerical example of the multiaxial attitude tracking problem  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 , the kinematics...