Abstract：
In this paper, we address the swing-down control of the Acrobot, a two-link planar robot operating in a vertical plane with only the second joint being actuated. The control objective is to rapidly stabilize the Acrobot around the downward equilibrium point, with both links in the downward position, from almost all initial states. Under the conditions of no friction and measurability of only the angle and angular velocity of the actuated joint, we present a sinusoidal-derivative (SD) controller. This controller consists of a linear feedback of the sinusoidal function of the angle of the actuated joint and a linear feedback of its angular velocity. We prove that the control objective is achieved if the sinusoidal gain is greater than a negative constant and the derivative gain is positive. We establish crucial relationships between the relative stability of the Acrobot under the SD controller and its physical parameters, presenting analytically all optimal control gains. These gains minimize the real parts of the dominant poles of the linearized model of the resulting closed-loop system around the downward equilibrium point. We demonstrate that the resulting dominant closed-loop poles can be double complex conjugate poles, or a quadruple real pole, or a triple real pole, depending on the Acrobot\'s physical parameters. Simulation studies indicate that the proposed SD controller outperforms the derivative (D) controller in rapidly stabilizing the Acrobot at the downward equilibrium point.
摘要：
在本文中,我们解决了Acrobot的向下摆动控制，两连杆平面机器人在垂直平面中操作，仅第二关节被致动。控制目标是在向下平衡点附近快速稳定Acrobot，两个链接都处于向下位置，几乎所有的初始状态。在无摩擦且仅可测量致动关节的角度和角速度的条件下，我们提出了一个正弦微分（SD）控制器。该控制器由致动关节角度的正弦函数的线性反馈和其角速度的线性反馈组成。我们证明，如果正弦增益大于负常数并且导数增益为正，则可以实现控制目标。我们建立了SD控制器下Acrobot的相对稳定性与其物理参数之间的关键关系，分析提供所有最优控制增益。这些增益使所得到的闭环系统围绕向下平衡点的线性化模型的主导极点的实部最小化。我们证明了所得的主要闭环极点可以是双复共轭极点，或者四根真极，或者三重真极，取决于Acrobot的物理参数。仿真研究表明，所提出的SD控制器在快速稳定Acrobot在向下平衡点上的性能优于导数（D）控制器。
