This paper presents a generalization of conventional sliding mode control designs for systems in Euclidean spaces to fully actuated simple mechanical systems whose configuration space is a Lie group for the trajectory-tracking problem. A generic kinematic control is first devised in the underlying Lie algebra, which enables the construction of a Lie group on the tangent bundle where the system state evolves. A sliding subgroup is then proposed on the tangent bundle with the desired sliding properties, and a control law is designed for the error dynamics trajectories to reach the sliding subgroup globally exponentially. Tracking control is then composed of the reaching law and sliding mode, and is applied for attitude tracking on the special orthogonal group SO(3) and the unit sphere S3. Numerical simulations show the performance of the proposed geometric sliding-mode controller (GSMC) in contrast with two control schemes of the literature.

翻译：本文提出将传统欧氏空间滑模控制设计推广至完全驱动简单机械系统的轨迹跟踪问题，其配置空间为李群。首先在李代数中设计通用运动学控制，使得系统状态演化的切丛上可构造李群结构。随后在切丛上构建具有理想滑模特性的滑动子群，并设计控制律使误差动力学轨迹全局指数收敛至滑动子群。跟踪控制由趋近律与滑模组成，并应用于特殊正交群SO(3)与单位球面S3的姿态跟踪。数值仿真结果表明，所提几何滑模控制器（GSMC）相较于文献中两种控制方案具有更优性能。