The Modular Nonoverlapping Grasp Workspaces and Dynamics for the Grippers using the Micro and Macro C-Manifold Design

#### Abstract

The toolbox for the gripper workspace analyses using Lie algebra is developed for shape variables (α_{1-4 }− θ_{1,2}) of the skew revolute joints. The unique methodology for grippers comprises to enable the variety of manifold analyses for kinematics and dynamics using symbolic mathematics. The Controllable Instantaneous Screw Axes (C-ISA) are defined through the shape variables considering the twists of the skew revolute joints **se(3)**. The derivation and analyses of the kinematics and dynamics equations are made possible using the developed methodology with the defined constraints for gripper mechanisms. The Modular Gripper with Lie Algebra Toolbox (M-GLAT) is developed for the defined constraints of the angle between C-ISA 1 and C-ISA 2. The novelty subject of this article is the development of the M-GLAT method for derivation of the constraint based workspaces with the shape variables (α_{1-4 }− θ_{1,2}) in the field of the spatial 2-RR gripper mechanisms. The gripper dynamics with constraint based workspaces of the skew revolute joints are developed for varied configurations of α_{1-4 }with ICs of θ_{1,2}. The modular rule-based workspaces are analyzed for the shape variables of the (α_{1‑4 }− θ_{1,2}) with the task spaces. This design produces dexterity with the modular grasp workspaces for the gripper fingers with skew revolute joints. One can select a combination of C-manifolds of (π/20, π/40, π/80) for the requirement of the nonoverlapping workspaces of the gripper finger designs as the grasp surfaces to control. The modular nonoverlapping workspace design with dynamics herein is based on the shape variables (α_{1-4 }− θ_{1,2}) using skew revolute joint which produce the high dexterity for the grasping capability of the grippers. The modular micro and macro C-manifold designs obtained the constraint based workspace algorithms of the 2-RR gripper which is expandable into the higher modular revolute joints of the n-R for the grippers. The n-R modular expandable grippers are increasing the precision and power grasping capability.

#### Keyword(s)

Lie algebra, Lie group theory, Shape variables of skew revolute, Spatial robot kinematics and dynamics, Task space

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