具有多个未驱动循环变量的机械系统中的阻尼诱导自恢复现象
On the Damping-Induced Self-Recovery Phenomenon in Mechanical Systems with Several Unactuated Cyclic Variables
摘要：
阻尼诱导的自恢复现象是指欠驱动机械系统的基本性质：如果未驱动的循环变量在粘性阻尼力的作用下，系统从静止开始，那么当驱动变量停止时，循环变量总是会回到其初始状态。规则的动量守恒现象可以被看作是阻尼诱导的自恢复现象的极限，在这个意义上，自恢复现象消失的阻尼趋于零。本文将以往关于阻尼诱导自恢复的结果推广到多个非驱动循环变量的情况。我们描述了一类外力，诱导新的守恒量，我们称之为阻尼诱导动量。在一定条件下，阻尼诱导动量使非驱动循环变量产生一阶渐近稳定动力学，从而诱导自恢复现象。它还表明，粘性阻尼力施加的范围内的未驱动的循环变量的轨迹的界限。两个例子来证明分析发现：平面摆与万向节驱动器和三连杆平面机械手在一个水平面上。
英文摘要：
The damping-induced self-recovery phenomenon refers to the fundamental property of underactuated mechanical systems: if an unactuated cyclic variable is under a viscous damping-like force and the system starts from rest, then the cyclic variable will always move back to its initial condition as the actuated variables come to stop. The regular momentum conservation phenomenon can be viewed as the limit of the damping-induced self-recovery phenomenon in the sense that the self-recovery phenomenon disappears as the damping goes to zero. This paper generalizes the past result on damping-induced self-recovery for the case of a single unactuated cyclic variable to the case of multiple unactuated cyclic variables. We characterize a class of external forces that induce new conserved quantities, which we call the damping-induced momenta. The damping-induced momenta yield first-order asymptotically stable dynamics for the unactuated cyclic variables under some conditions, thereby inducing the self-recovery phenomenon. It is also shown that the viscous damping-like forces impose bounds on the range of trajectories of the unactuated cyclic variables. Two examples are presented to demonstrate the analytical discoveries: the planar pendulum with gimbal actuators and the three-link planar manipulator on a horizontal plane.
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