10th International Congress of the Serbian Society of Mechanics (18-20. 06. 2025, Niš) [pp. 251-260]
AUTHOR(S) / AUTOR(I): Vuk Todorović, Nikola Nešić 
and Mihailo Lazarević
Download Full Pdf 
DOI: 10.46793/ICSSM25.251T
ABSTRACT / SAŽETAK:
In this paper, we used the Paden-Kahan subproblems and their extensions in the context of screw theory to solve the inverse kinematics of common robot manipulators: RR, RRR, RPR planar, and RPP spatial robot models. The inverse kinematics problem of a robot is fundamental to determining a robot’s mechanics, and among the different ways to solving it, we chose the Paden- Kahan subproblems approach for its effectiveness and simplicity. The canonical subproblems have been significantly extended and of particular interest to us are the inclusion of solutions to paral- lel screw axes for the second subproblem and prismatic joints instead of only revolute joints, also known as Pardos-Gotor subproblems, After solving the inverse kinematics of each robot model and representing the solutions via graphs, we then proceeded to demonstrate the correctness of our solutions against the desired configuration and found that the orientational and positional errors between the desired configuration and our solution were in the range of floating-point arithmetic errors. Taking into account the simplicity during solving the inverse kinematics and the number of operations performed, we concluded that, at least where applicable, the Paden-Kahan subproblems and their extensions provide significant advantages to other methods.
KEYWORDS / KLJUČNE REČI:
Inverse kinematics, Paden-Kahan subproblems, Pardos-Gotor subproblems, RR Robot, RRR Robot, RPR Robot, RPP Robot
ACKNOWLEDGEMENT / PROJEKAT:
This research has been supported by the research grant of the Serbian Ministry of Science, Technological Development and Innovations, grant No. 451-03-137/2025-03/ 200105 from 04.02.2025.
REFERENCES / LITERATURA:
- Lynch K., & Park F., Modern Robotics: Mechanics, Planning, and Control, Cambridge Uni- versity Press, UK, 2017.
- Murray R., Li Z., & Sastry S., A Mathematical Introduction to Robotic Manipulation, CRC Press, USA, 1994.
- Ceccarelli M., Screw Axis Defined by Giulio Mozzi in 1763 and Early Studies on Helicoidal Motion, Mechanism and Machine Theory 35(6), 761-770,
- Manocha D., & Canny J. F., Real Time Inverse Kinematics for General 6R Manipulators, IEEE International Conference on Robotics and Automation, 383-389, 1992.
- Paden B., KinematicsandControlofRobotManipulators, PhD thesis, Department of Electri- cal Engineering and Computer Sciences, University of California, USA, 1986.
- Kahan W., LecturesonComputationalAspectsofGeometry, Unpublished, 1983.
- Dimovski I., Trompeska M., Samak S., Dukovski V., & Cvetkoska D., Algorithmic Approach to Geometric Solution of Generalized Paden-Kahan Subproblem and its Extension, Interna- tional Journal of Advanced Robotic Systems 15(1), 2018.
- Pardos-Gotor J., Screw Theory in Robotics: an Illustrated and Practicable Introduction to Modern Mechanics, CRC Press, USA, 2022.
- Oliphant T., url: https: //numpy. org , 2005.
- Todorovic V., url: https: //github. com/ VuckoT/ mehanika_ robota , 2024.