Acta Mechanica Slovaca 2026, 30(1):34-41 | DOI: 10.21496/ams.2026.010

Numerical Kinematic and Dynamic Analysis of a 6-DOF Industrial Manipulator Using a Lagrangian Formulation

Martin Uzik1, *, Anton Hovana2
1 Vorstand des Berlin Institute of Finance, Innovation and Digitalization, Hochschule für Wirtschaft und Recht Berlin, Badensche Straße 52, 10825 Berlin
2 Department of Applied Mathematics and Informatics, Faculty of Mechanical Engineering, Technical University Ko¹ice, Letná 1/9, 042 00 Ko¹ice-Sever

This paper presents a numerical kinematic and dynamic analysis of a six-degree-of-freedom (6-DOF) industrial manipulator based on the Denavit-Hartenberg convention and the Euler-Lagrange formulation. The objective of the study was to combine analytical modeling with numerical implementation in order to evaluate joint torque requirements relevant for actuator dimensioning and control system design. The forward kinematic model was derived using homogeneous transformation matrices and validated through MATLAB-based simulation. A predefined circular spatial trajectory was applied to determine joint positions, velocities, and accelerations. Based on the Lagrangian dynamic formulation, inverse dynamic analysis was performed to compute the required actuator torques for the given motion. Simulation results revealed dominant torque loading in proximal joints, particularly Joint 2, due to cumulative gravitational and inertial effects. Peak torque values exceeded 40 Nm, while RMS torque values provided additional insight into continuous actuator loading conditions. Increased trajectory acceleration resulted in a significant rise in peak torque demand, confirming the nonlinear nature of dynamic loading. The results demonstrate that dynamic modeling is essential not only for motion control and mechanical design, but also for identifying productivity-related sensitivities and lifecycle-oriented operating regimes in industrial robotic systems.

Keywords: Industrial robot; kinematics; dynamics; Euler-Lagrange; UR5; MATLAB; inverse dynamics

Received: February 11, 2026; Revised: March 5, 2026; Accepted: March 11, 2026; Published: April 1, 2026  Show citation

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Uzik, M., & Hovana, A. (2026). Numerical Kinematic and Dynamic Analysis of a 6-DOF Industrial Manipulator Using a Lagrangian Formulation. Acta Mechanica Slovaca, 30(1), 34-41. doi: 10.21496/ams.2026.010
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    References

    1. Spong, M. W., Hutchinson, S., Vidyasagar, M. (2020). Robot Modeling and Control. John Wiley & Sons, Hoboken.
    2. Craig, J. J. (2005). Introduction to Robotics: Mechanics and Control. Pearson Prentice Hall, Upper Saddle River.
    3. Siciliano, B., Sciavicco, L., Villani, L., Oriolo, G. (2010). Robotics: Modelling, Planning and Control. Springer, London. Go to original source...
    4. Khalil, W., Dombre, E. (2002). Modeling, Identification and Control of Robots. Butterworth-Heinemann, Oxford. Go to original source...
    5. Murray, R. M., Li, Z., Sastry, S. S. (1994). A Mathematical Introduction to Robotic Manipulation. CRC Press, Boca Raton.
    6. Featherstone, R. (2008). Rigid Body Dynamics Algorithms. Springer, New York. Go to original source...
    7. Angeles, J. (2014). Fundamentals of Robotic Mechanical Systems: Theory, Methods, and Algorithms. Springer, New York. Go to original source...
    8. Siciliano, B., Khatib, O. (2016). Springer Handbook of Robotics. Springer, Cham. Go to original source...
    9. Khalil, W., Kleinfinger, J. F. (1986). A new geometric notation for open and closed-loop robots. International Journal of Robotics Research, 5(3), 52-63. Go to original source...
    10. Walker, M. W., Orin, D. E. (1982). Efficient dynamic computer simulation of robotic mechanisms. Journal of Dynamic Systems, Measurement, and Control, 104(3), 205-211. Go to original source...
    11. Hollerbach, J. M. (1980). A recursive Lagrangian formulation of manipulator dynamics. IEEE Transactions on Systems, Man, and Cybernetics, 10(11), 730-736. Go to original source...
    12. Khalil, W., Gautier, M. (1991). Identification of robot dynamics. International Journal of Robotics Research, 10(4), 368-381.
    13. MathWorks. Robotics System Toolbox User's Guide, from https://www.mathworks.com/help/robotics/ (accessed January 2026).
    14. Universal Robots. UR5e Technical Specifications, from https://www.universal-robots.com/ (accessed January 2026).
    15. Denavit, J., Hartenberg, R. S. (1955). A kinematic notation for lower-pair mechanisms based on matrices. Journal of Applied Mechanics, 22(2), 215-221. Go to original source...

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