# Intrinsic Geometry in Screw Algebra and Derivative Jacobian and Their Uses in the Metamorphic Hand

Student thesis: Doctoral ThesisDoctor of Philosophy

### Abstract

Line geometry is a foundation of screw algebra in line coordinates that were created by Plücker as ray coordinates taking a line as a ray between two points and axis coordinates taking a line as the intersection of two planes. This Thesis reveals the geometrical meaning and intrinsic relationship between these ray coordinates and axis coordinates, leading to an in-depth understanding of conformability and duality of these two sets of screw coordinates, and their related vector space and dual vector space.

Based on the study of screw algebra, the resultant twist of a serial manipulator is presented geometrically by an assembly of unit joint screws with the corresponding velocity amplitudes. This leads to the geometrical interpretation for the resultant twist with its instantaneous screw axis (ISA) that is formulated by a combination of weighted position vectors of joint screws. The screw-based Jacobian is then derived after recognizing the resultant twist of a serial manipulator. The case leads to a revelation for the first time the relationship of a Jacobian matrix acquired by using screw algebra and a derivative Jacobian matrix using differential functions, and to an in-depth investigation of transformation between these two Jacobians.

To extend the application of screw algebra and this derivative Jacobian, kinematics analysis of a novel reconfigurable base-integrated parallel mechanism is proposed and its screw-based Jacobian is derived, leading to its equivalent model, the Metamorphic hand with a reconfigurable palm. The method is then applied to the investigation of the Metamorphic Hand, while manipulating an object, based on the product sub-manifolds and the exponential method. Evaluation of the functionality of the Metamorphic hand is further analysed, with the Anthropomorphism Index (AI) and palmar shape modulation as the criteria, evaluating performance enhancement of the Metamorphic hand in comparison to other robotic hands with a fixed palm.

The Thesis presents novel discoveries in the intrinsic geometry of screw coordinates and the coherent connection between Jacobian formed by screw algebra and the Jacobian using the derivative method. This intrinsic geometry insight is then used to investigate for the first time the parallel mechanism with a reconfigurable base, paving a way for an in-depth investigation of the Metamorphic hand on its reconfigurability and grasp affordability and for the first time using Anthropomorphic Index to evaluate the Metamorphic hand.
Date of Award 2017 English King's College London Jian Dai (Supervisor) & Hak-Keung Lam (Supervisor)

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