The stable orbit of a small satellite flying around the space station and the orbit maintenance are studied based on the Hill equation and a sliding-mode control (SMC) approach. First, the problem of the stable orbit of the small satellite flying around the space station is transformed into the sliding-mode control problem. Then, sliding-mode controllers are proposed to guarantee the existence of sliding-mode motion, the controller are relay-type controllers. Meanwhile, the motion of the small satellite will enter and maintain the sliding-mode in finite time, and the stable orbit of the small satellite flying around the space station is obtained. In order to economize the energy and restrain the chattering of control signals, the sliding-mode with dead region is proposed, and the corresponding controllers are derived to make the motion of the closed-loop system enter a defined region. A mathematical derivation and simulation study are carried out to prove the following three main points: (1) the condition for the stable orbit of the small satellite flying around the space station to be guaranteed by some sliding-mode motion; (2) the constant term in the chosen sliding-mode can be used to adjust the position of the center of the stable orbit; (3) sliding-mode with dead-region can also obtain the stable flying orbit. The total influence of the parameters in our controllers on the working performance of the closed-loop system is discussed by simulation.
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