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

Sahin Haydar

Archives of Control Sciences

2024

Vol. 34, No. 4

777--803

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.

Robot grasping and regrasping kinematics using Lie algebra, the geodesic, and Riemann curvature tensor

Sahin Haydar

Archives of Control Sciences

2023

Vol. 33, No. 1

5--23

Differential geometry is a strong and highly effective mathematical subject for robot gripper design when grasping within the predetermined trajectories of path planning. This study in grasping focuses on differential geometry analysis utilizing the Lie algebra, geodesic, and Riemann Curvature Tensors (RCT). The novelty of this article for 2RR robot mechanisms lies in the approach of the body coordinate with the geodesic and RCT. The importance of this research is significant especially in grasping and regrasping objects with varied shapes. In this article, the types of workspaces are clarified and classified for grasping and regrasping kinematics. The regrasp has not been sufficiently investigated of body coordinate systems in Lie algebra. The reason for this is the difficulty in understanding relative coordinates in Lie algebra via the body coordinate system. The complexity of the equations has not allowed many researchers to overcome this challenge. The symbolic mathematics toolbox in the Maxima, on the other hand, aided in the systematic formulation of the workspaces in Lie algebra with geodesic and RCT. The Lie algebra se(3) equations presented here have already been developed for robot kinematics from many references. These equations will be used to derive the following workspace types for grasping and regrasping. Body coordinate workspace, spatial coordinate workspace with constraints, body coordinate workspace with constraints, spatial coordinate workspace with constraints are the workspace types. The RCT and geodesic solutions exploit these four fundamental workspace equations derived using Lie algebra.