This article presents an analytical method to modulate the dynamic response of a robotic manipulator interacting with its environment when performing impedance-related robotic tasks, through the choice of stiffness and damping parameters. By joint space analysis of vibration and experiments, we prove that in order to preserve a desired dynamic behavior of the robot in the Cartesian space, neither a stiffness nor damping matrix can be arbitrarily chosen; this choice has to meet the desired dynamic criteria for any given configuration. After mapping the parameters (matrices) into the joint space, we analyze the vibratory dynamics of the robot and identify the proper way of suppressing of the vibration modes by specific elements in the Cartesian damping matrix without need of trial and error. Our method is especially useful for redundant robots. We show and compare experimental results from two different 7 degrees of freedom robotic manipulators.

Implementing impedance control in Cartesian task space or directly at the joint level is a popular option for achieving desired compliance behavior for robotic manipulators performing tasks. The damping ratio is an important control criterion for modulating the dynamic response; however, tuning or selecting this parameter is not easy, and can be even more complicated in cases where the system cannot be directly solved at the joint space level. Our study proposes a novel methodology for calculating the local optimal damping ratio value and supports it with results obtained from five different scenarios. We carried out 162 different experiments and obtained the values of the inertia, stiffness, and damping matrices for each experiment. Then, data preprocessing was carried out to select the most significant variables using different criteria, reducing the seventeen initial variables to only three. Finally, the damping ratio values were calculated (predicted) using automatic regression tools. In particular, five-fold cross-validation was used to obtain a more generalized model and to assess the forecasting performance. The results show a promising methodology capable of calculating and predicting control parameters for robotic manipulation tasks.

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, $$\omega_n$$ω n and damping ratio, $$\zeta$$ζ . 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 $$\omega _n$$ω n and $$\zeta$$ζ in the modal space are analyzed, which can be synthesized to modulate the damping of the system analytically.

This article presents an experimental study with theory to identify quantitatively the zero-potential-energy (ZP) motion in Cartesian impedance control of redundant manipulators, based on a new analytical methodology. This ZP mode of motion, analogous to the rigid-body mode in classic mechanical systems, is a result of the redundancy of the robot. When subject to an external perturbation under impedance control, a redundant robot will assume a new equilibrium configuration determined by the ZP motion, governed by the least-energy principle. Consequently, this creates a steady-state deviation from its initial configuration after a perturbation and reaches a new equilibrium. We determine such ZP motion(s) by utilizing a closed-form solution based on vibration theory. Experiments were conducted on a 7-degrees-of-freedom (DoF) redundant Panda robot to determine the new equilibrium after a perturbation. The experimental results are compared with the theoretical prediction of the ZP motions to validate the theoretical results of the zero-potential-energy motions due to stiffness in impedance control. Furthermore, we demonstrated that the ZP motion due to redundancy can be eliminated by removing the redundancy through experimental validation by employing the null-space control, as expected.