中文摘要：

研究了三连杆平面欠驱动机械臂的轨迹跟踪问题．机械臂的第3个关节为被动关节，施加在自由运动连杆上的动力学约束是二阶非完整的．通过全局输入和坐标变换，系统的动力学方程被变换成高阶链式形式．基于后推法（backstepping）的思想推导出保证系统全局渐近收敛于参考轨迹的时变反馈控制规律．后推法将系统分解为低阶子系统来处理，利用中间虚拟控制变量和部分Lyapunov函数简化了控制器的设计．数值仿真结果显示系统能稳定地跟踪参考轨迹，也证明了控制器设计是有效的．

英文摘要：

The trajectory tracking control of a three-link planar under-actuated manipulator was presented. The third joint of the manipulator was passive. The dynamic constraint on the free link was shown to be second-order nonholonomic. Motion equations of the system were transformed into high order chained form using some global coordinate and input changes. A time-varying feedback controller was proposed to force the under-actuated manipulator to globally asymptotically track a reference trajectory. The control development was based on the current popular integrator backstepping technique. The method breaks down nonlinear system into low dimensional systems and simplifies the controller design using virtual control inputs and partial Lyapunov functions. Simulation results show the system can stably converge to a desired trajectory and demonstrate the effectiveness of the proposed control law.