This paper presents a general methodology for the analysis and synthesis of a positive semi-definite system described by mass, damping and stiffness matrices that is often encountered in impedance control in robotics research. This general methodology utilizes the fundamental kinematic concept of rigid-body and non-rigid-body motions of which all motions consist. The rigid-body mode results in no net change in the potential energy from the stiffness matrix of the multiple degree-of-freedom (DoF) discrete mechanical system. Example of an unconstrained discrete mechanical system is presented to illustrate the theoretical principle as applied in obtaining the free and forced vibration responses, as well as the dynamic characteristics of the system in natural frequency, $$\omega_n$$ and damping ratio, $$\zeta$$ . In addition, the methodology is applied to the impedance control of redundant robots. The rigid-body mode is equivalent to the motions of a redundant robot which result in no net change in potential energy, also called the zero-potential or ZP mode, of impedance control. Example of a redundant robot is used to demonstrate the application of the methodology in robotics. The dynamic characteristics of $$\omega _n$$ and $$\zeta$$ in the modal space are analyzed, which can be synthesized to modulate the damping of the system analytically.

中文翻译：

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无约束离散系统振动分析的分析方法及其在冗余机器人阻抗控制中的应用

本文提出了一种一般的方法，用于分析和综合由质量，阻尼和刚度矩阵描述的正半定系统，这在机器人技术的阻抗控制中经常遇到。这种通用方法利用了刚体运动和非刚体运动所组成的基本运动学概念。刚体模式不会导致多自由度（DoF）离散机械系统的刚度矩阵中的势能发生净变化。给出了无约束离散机械系统的示例，以说明用于获得自由振动和强迫振动响应的理论原理，以及系统在固有频率$$ \ omega_n $$和阻尼比$$中的动态特性。 \ zeta $$。此外，该方法应用于冗余机器人的阻抗控制。刚体模式等效于冗余机器人的运动，该运动不会导致阻抗控制的势能（也称为零势或ZP模式）发生净变化。以冗余机器人为例来说明该方法在机器人技术中的应用。分析了模态空间中的$$ \ omega _n $$和$$ \ zeta $$的动态特性，可以合成这些特性以分析地调节系统的阻尼。