Geodesic path planning characteristics of the reconfigurable 1-S robot workspaces for hyperbolic, elliptical, and Euclidean geometries

• Autorzy: Sahin Haydar

Identyfikatory

DOI

10.24425/acs.2024.153102

• Języki publikacji: EN

Abstrakty

EN

The path-planning strategies are implemented by establishing the Riemann curvature tensor and geodesic equations of the 1-S robot workspace. This paper’s originality lies in formulation of the parametric 1-S robot workspace for path planning, which is based on the differential geometry of the geodesic and Riemann curvature equations. The novel results in defining the path plan with diffeomorphic and expandable trajectories with zero and negative sectional curvatures are encouraging, as shown in the research article’s result sections. The constant negative, constant positive, and zero sectional curvatures produce hyperbolic, elliptical, and Euclidean geometries. The workspace equation, derived using Lie algebra, defines the parameters of 𝑢₁, ₂, 𝑢₃, and 𝑢₄ to obtain the shortest distances in path planning. The geodesic equations determine the shortest distances in the context of Riemann curvature tensor equations. These parameters from the workspace equation (𝛼₁, 𝛼₂, 𝜃₁, 𝑟₁) are used in the geodesic and Riemann curvature tensor equations. The results show that one needs to choose the most convenient parameters of the mechanism for path-planning capabilities. Both the topology of the mechanism, which is 1-S herein and the parameters of the workspaces should be selected for the pre-defined trajectories of the path planning, as shown in the results. The reconfigurable robots have many mechanism topologies to transform.

Słowa kluczowe

EN

Riemann curvature tensor; geodesic equations; robot workspace; path planning; Dirac vector; Clifford algebra

• Wydawca: Polish Academy of Sciences, Committee of Automatic Control and Robotics
• Czasopismo: Archives of Control Sciences
• Rocznik: 2024
• Tom: Vol. 34, No. 4
• Strony: 777--803
• Opis fizyczny: Bibliogr. 15 poz., rys., tab., wzory

Twórcy

autor

Sahin Haydar

• İstanbul Gedik University, Mechatronics Engineering, 34953 Kartal Türkiye

• https://orcid.org/0000-0001-8435-9448

Bibliografia

• [1] J.M. Lee: Riemannian Manifolds, An Introduction to Curvature. Springer, 1997.
• [2] M.M. Postnikov: Encyclopedia of Mathematical Sciences. 91 Geometry VI, Riemannian Geometry. Springer, 2001.
• [3] J. Gallier and J. Quaintance: Differential Geometry and Lie Groups, A Computational Perspective. Springer, 2020.
• [4] S. Hassani: Mathematical Physics. Springer, 2013.
• [5] H. Sahin: The modular non-overlapping grasp workspaces and dynamics for the grippers using the micro and macro C-manifold design. Journal of Scientific and Industrial Research, 80(9), (2021), 766-776. DOI: 10.56042/jsir.v80i09.47040
• [6] L. Godinho and J. Natário: An Introduction to Riemannian Geometry with Applications to Mechanics and Relativity. Springer. 2014.
• [7] N. Islam: Tensors and their Applications. New Age International (P) Ltd., 2006.
• [8] Y. Gu: Space-Time Geometry and Some Applications of Clifford Algebra in Physics. Advances in Applied Clifford Algebras, 28 (2018), 1-19. DOI: 10.1007/s00006-018-0896-1
• [9] E. Marsch and Y. Narita: Connecting in the Dirac Equation the Clifford Algebra of Lorentz Invariance with the Lie Algebra of SU(N) Gauge Symmetry, Symmetry, 13(3), (2021), 475, 1-9. DOI: 10.3390/sym13030475
• [10] J. Snygg: A New Approach to Differential Geometry using Clifford’s Geometric Algebra. Springer, 2012.
• [11] M. Gromov and H.B. Lawson: Positive scalar curvature and the Dirac operator on complete Riemannian manifolds. Publications Mathematiques del’IHES, 28(58), (1983), 83-196. DOI: 10.1007/BF02953774
• [12] S. Sommer, T. Fletcher, and X. Pennec: Introduction to differential and Riemannian geometry. Riemannian Geometric Statistics in Medical Image Analysis, 28 (2020), 3-37, DOI: 10.1016/b978-0-12-814725-2.00008-x
• [13] V. Berestovskii and Y. Nikonorov: Riemannian Manifolds and Homogeneous Geodesics. Springer, 2020.
• [14] H. Sahin: Algorithmic workspace programming of the collaborative multi-robots. Osmaniye Korkut Ata Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 5(1), (2022), 325-341, DOI: 10.47495/okufbed.1030575
• [15] H. Sahin: Robot grasping and regrasping kinematics using Lie algebra, the geodesic, and Riemann curvature tensor. Archives of Control Sciences, 33(1), (2023), 5-23, DOI: 10.24425/acs.2023.145111

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa nr POPUL/SP/0154/2024/02 w ramach programu "Społeczna odpowiedzialność nauki II" - moduł: Popularyzacja nauki (2025)