The problem of orbital stabilization of underactuated mechanical systems with one passive degree-of-freedom (DOF) is revisited. Virtual holonomic constraints are enforced using a continuous controller; this results in a dense set of closed orbits on a constraint manifold. A desired orbit is selected on the manifold and a Poincare section is constructed at a fixed point on the orbit. The corresponding Poincare map is linearized about the fixed point; this results in a discrete linear time-invariant system. To stabilize the desired orbit, impulsive inputs are applied when the system trajectory crosses the Poincare section; these inputs can be designed using standard techniques such as LQR. The Impulse Controlled Poincare Map (ICPM) based control design has lower complexity and computational cost than control designs proposed earlier. The generality of the ICPM approach is demonstrated using the 2-DOF cart-pendulum and the 3-DOF tiptoebot.

中文翻译：

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使用虚拟完整约束和脉冲控制庞加莱映射对欠驱动系统进行轨道稳定

重新讨论了具有一个被动自由度 (DOF) 的欠驱动机械系统的轨道稳定问题。使用连续控制器强制执行虚拟完整约束；这会在约束流形上产生一组密集的闭合轨道。在流形上选择所需的轨道，并在轨道上的固定点构建庞加莱部分。对应的 Poincare 映射是关于不动点线性化的；这导致离散线性时不变系统。为了稳定所需的轨道，当系统轨迹穿过庞加莱部分时应用脉冲输入；这些输入可以使用标准技术（例如 LQR）进行设计。基于脉冲控制庞加莱映射 (ICPM) 的控制设计比之前提出的控制设计具有更低的复杂性和计算成本。