Article
A Novel, Oriented to Graphs Model of Robot Arm Dynamics
George Boiadjiev *, Evgeniy Krastev , Ivan Chavdarov and Lyubomira Miteva
Citation: Boiadjiev, G.; Krastev, E.;
Chavdarov, I.; Miteva, L. A Novel,
Oriented to Graphs Model of Robot
Arm Dynamics. Robotics 2021, 10, 128.
https://doi.org/10.3390/
robotics10040128
Academic Editor: Dario Richiedei
Received: 10 October 2021
Accepted: 25 November 2021
Published: 28 November 2021
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Faculty of Mathematics and Informatics, Sofia University St. Kliment Ohridski, 1164 Sofia, Bulgaria;
eck@fmi.uni-sofia.bg (E.K.); ivannc@fmi.uni-sofia.bg (I.C.); llmiteva@fmi.uni-sofia.bg (L.M.)
* Correspondence: george@fmi.uni-sofia.bg; Tel.: +359-896-683-218
Abstract:
Robotics is an interdisciplinary field and there exist several well-known approaches to
represent the dynamics model of a robot arm. The robot arm is an open kinematic chain of links
connected through rotational and translational joints. In the general case, it is very difficult to obtain
explicit expressions for the forces and the torques in the equations where the driving torques of the
actuators produce desired motion of the gripper. The robot arm control depends significantly on
the accuracy of the dynamic model. In the existing literature, the complexity of the dynamic model
is reduced by linearization techniques or techniques like machine learning for the identification of
unmodelled dynamics. This paper proposes a novel approach for deriving the equations of motion
and the actuator torques of a robot arm with an arbitrary number of joints. The proposed approach for
obtaining the dynamic model in closed form employs graph theory and the orthogonality principle, a
powerful concept that serves as a generalization for the law of conservation of energy. The application
of this approach is demonstrated using a 3D-printed planar robot arm with three degrees of freedom.
Computer experiments for this robot are executed to validate the dynamic characteristics of the
mathematical model of motion obtained by the application of the proposed approach. The results
from the experiments are visualized and discussed in detail.
Keywords:
robot arm; open kinematic chain; equations of motion; graph theory; orthogonality
principle; law of conservation of energy
1. Introduction
There exist several different approaches for modeling the motion of robot arms in
the existing literature. Most frequently, the kinematic aspects of motion are expressed by
Denavit–Hartenberg parameters, Euler angles, or normalized quaternions [
1
]. It is usually
difficult to integrate these models with Newton-Euler, Lagrange and Hamilton methods
employed to represent the dynamic model [
2
,
3
]. This is especially true for robot arms
with redundant degrees of freedom, where the redundancy allows improving the quality
of motion by introducing optimization criteria. In the existing literature, these criteria
are defined on the level of kinematics or dynamics. Time optimization, manipulability,
or obstacle avoidance are some of the typical criteria for optimal path planning at the
kinematical level [
4
–
6
]. Singularity avoidance is another popular way to exploit redundant
degrees of freedom in executing a given task [
7
–
9
]. Criteria related to energy saving
represent a special interest if the level of dynamics is concerned [10,11].
The accuracy of motion control of a physical robot arm strongly depends on the
completeness of the model of dynamics. The dynamic system model describes the rela-
tionship between the forces and torques applied to the robot arm on the one hand and
the resulting robot arm motion in joint space or workspace coordinates on the other hand.
The equations of motion serve to solve the forward and the inverse dynamic problems,
respectively [
12
,
13
]. Given the motion in the joint space or workspace, these equations
determine the driving torques of the actuators in the joints or the forces of the gripper. In
the inverse problem, the motion in the joint space or workspace is computed knowing
Robotics 2021, 10, 128. https://doi.org/10.3390/robotics10040128 https://www.mdpi.com/journal/robotics
