Citation: Patiño, J.; Encalada-Dávila,
Á.; Sampietro, J.; Tutivén, C.;
Saldarriaga, C.; Kao, I. Damping
Ratio Prediction for Redundant
Cartesian Impedance-Controlled
Robots Using Machine Learning
Techniques. Mathematics 2023, 11,
1021. https://doi.org/10.3390/
math11041021
Academic Editors: Shuai Li, Dechao
Chen, Vasilios N. Katsikis, Predrag S.
Stanimirovic, Dunhui Xiao and
Mohammed Aquil Mirza
Received: 27 December 2022
Revised: 28 January 2023
Accepted: 10 February 2023
Published: 17 February 2023
Copyright: © 2023 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
Article
Damping Ratio Prediction for Redundant Cartesian
Impedance-Controlled Robots Using Machine
Learning Techniques
José Patiño
1
, Ángel Encalada-Dávila
1
, José Sampietro
2
, Christian Tutivén
1
, Carlos Saldarriaga
1,
*
and Imin Kao
3
1
Faculty of Mechanical Engineering and Production Sciences (FIMCP), ESPOL Polytechnic University, Escuela
Superior Politécnica del Litoral, Campus Gustavo Galindo, Km. 30.5 Vía Perimetral,
Guayaquil EC09015863, Ecuador
2
Facultad de Ingenierías, Universidad Ecotec, Km. 13.5 Samborondón, Samborondón EC092302 , Ecuador
3
Department of Mechanical Engineering, Stony Brook University, Stony Brook, NY 11794, USA
* Correspondence: cxsaldar@espol.edu.ec
Abstract:
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.
Keywords:
robotic manipulator; Cartesian impedance; MCK system; machine learning; XGBoost;
random forest; support vector regressor; LightGBM; CatBoost
MSC: 68T40; 68T07
1. Introduction
In order to achieve the desired compliant behavior for a robotic manipulator perform-
ing a task, a popular choice is implementing impedance control at either the task (Cartesian)
space or directly at the joint level. Among these common tasks are wiping, deburring,
polishing, and several others for human–robot interaction, in which safety becomes an
important feature that must be handled in a careful manner [1].
Usually, in the one-DoF (Degree of Freedom) mass-damper-spring case or in discrete
systems that can be decoupled, a set of control criteria such as damping ratios
ζ
or natu-
ral frequencies
ω
n
can be imposed according to particular (scalar) parameters. After the
system is no longer decoupled, the tuning or selection of the parameters is not straightfor-
ward, and it becomes much more complicated in cases of unconstrained discrete systems
or redundant serial kinematic chains, which are not directly solvable at the joint space level.
In this work, a new methodology is applied to perform a joint space analysis based on
the theory of mechanical vibrations of discrete unconstrained systems. As per the expansion
theorem, the solution of unconstrained systems such as the one shown in Figure 1 can be
Mathematics 2023, 11, 1021. https://doi.org/10.3390/math11041021 https://www.mdpi.com/journal/mathematics