Kinematic Analysis of the Influence of Hexapod Reconfiguration on Its Workspace
DOI:
https://doi.org/10.32515/2414-3820.2026.56.256-270Keywords:
hexapod, parallel kinematics, workspace, singularity, Jacobian matrix, integral workspace, condition numberAbstract
Parallel kinematics machine tools, particularly hexapods based on the Gough-Stewart platform, offer high structural stiffness and positioning accuracy but suffer from a limited workspace and a high susceptibility to singularities, especially under five-axis machining conditions with significant spindle tilt angles.
This paper introduces a comprehensive mathematical framework and a numerical simulation method for evaluating the "integral workspace" of a reconfigurable hexapod featuring controllable sliding base joints. To ensure kinematic stability and eliminate dimensionality mismatch, the row elements of the kinematic Jacobian matrix are normalized by a characteristic length and mapped directly to the tool tip rather than the platform's geometric center. A dedicated computational script written in Python, utilizing the NumPy and Matplotlib libraries, was developed to implement a polar coordinate space-scanning algorithm combined with Singular Value Decomposition (SVD). The algorithm filters out unstable and singular zones using a strict threshold for the matrix condition number (). Results. Quantitative analysis of the cross-sectional areas at heights ranging from 200 to 600 mm demonstrates the exceptional efficiency of structural reconfiguration. At a vertical spindle orientation (), the total effective cross-sectional area increases by 12.7%, while all potential singularity zones are completely excluded from the functional workspace. Under five-axis simulation scenarios, the integration of 64 discrete binary boundary configurations yields a 46.3% area increase at a tilt, and a 76.6% increase at a tilt.
The proposed reconfiguration strategy successfully mitigates the severe workspace reductions typical of traditional parallel robots. Poses that are completely unusable in classical hexapods are effectively restored for stable mechanical processing, laying a solid theoretical foundation for adaptive structural control in advanced machine tool designs.
References
1. Valiavskyi, I. A., Lysenko, O. V., & Lysenko, I. A. (2023). Technological equipment with parallel kinematics: A textbook. CNTU [in Ukrainan].
2. Stewart, D. (1965). A platform with six degrees of freedom. Proceedings of the Institution of Mechanical Engineers, 180(1), 371–386. https://doi.org/10.1243/PIME_PROC_1965_180_029_02 DOI: https://doi.org/10.1243/PIME_PROC_1965_180_029_02
3. Merlet, J.-P. (2006). Parallel robots (2nd ed.). Springer. INRIA, Sophia-Antipolis. https://doi.org/10.1007/1-4020-4133-0 DOI: https://doi.org/10.1007/1-4020-4133-0
4. Gosselin, C., & Angeles, J. (1990). Singularity analysis of closed-loop kinematic chains. IEEE Transactions on Robotics and Automation, 6(3), 281–290. https://doi.org/10.1109/70.56660 DOI: https://doi.org/10.1109/70.56660
5. Zlatanov, D., Bonev, I. A., & Gosselin, C. M. (2002). Constraint singularities of parallel mechanisms. Proceedings of the 2002 IEEE International Conference on Robotics and Automation (ICRA 2002), 496–502. https://doi.org/10.1109/ROBOT.2002.1013408 DOI: https://doi.org/10.1109/ROBOT.2002.1013408
6. Merlet, J.-P. (1994). Designing a parallel manipulator for a specific workspace. The International Journal of Robotics Research, 16(4), 545–556. https://doi.org/10.1177/027836499701600407 DOI: https://doi.org/10.1177/027836499701600407
7. Huang, Z., Li, Q., & Ding, H. (2013). Theory of parallel mechanisms. Springer. https://doi.org/10.1007/978-94-007-4201-7 DOI: https://doi.org/10.1007/978-94-007-4201-7
8. Pavlenko, I. I., Valiavskyi, I. A., & Hnatiuk, A. O. (2010). Integral workspace of hexapod machine tools. Machinery in agricultural production, industrial engineering, automation: coll. of science avenue of Kirovohrad National Technical University, 23, 100–107 [in Ukrainian].
9. Dasgupta, B., & Mruthyunjaya, T. S. (2000). The Stewart platform manipulator: A review. Mechanism and Machine Theory, 35(1), 15–40. https://doi.org/10.1016/S0094-114X(99)00006-3 DOI: https://doi.org/10.1016/S0094-114X(99)00006-3
10. Dasgupta, B., & Mruthyunjaya, T. S. (1998). Singularity-free path planning for the Stewart platform manipulator. Mechanism and Machine Theory, 33(6), 711–725. https://doi.org/10.1016/S0094-114X(97)00095-5 DOI: https://doi.org/10.1016/S0094-114X(97)00095-5
11. Lin, C.-J., & Chen, C.-T. (2016). Reconfiguration for the maximum dynamic wrench capability of a parallel robot. Applied Sciences, 6(3), Article 80. https://doi.org/10.3390/app6030080 DOI: https://doi.org/10.3390/app6030080
12. Dash, A. K., Chen, I.-M., Yeo, S. H., & Yang, G. (2003). Singularity-free path planning of parallel manipulators using clustering algorithm and line geometry. Proceedings of the 2003 IEEE International Conference on Robotics and Automation, 1, 761–766. https://doi.org/10.1109/ROBOT.2003.1241685 DOI: https://doi.org/10.1109/ROBOT.2003.1241685
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Copyright (c) 2026 Ivan Valiavskyi, Oleksandr Lysenko, Olexandr Skibinskyi, Anton Aparakin
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