Screw-Based and Differential-Based Jacobians and Their Uses in the Bio-Like Mechanisms

Student thesis: Doctoral ThesisDoctor of Philosophy


In the vast realm of robotics, kinematics plays a pivotal role, underpinning the understanding and analysis of mechanism motion. The dissertation embarks on a comprehensive exploration into the differential-based and screw-based Jacobians, their interrelationships, and their applications to the kinematic analysis of bio-like origami mechanisms. The applications encompass forward kinematics, inverse kinematics, singularity analysis, and bifurcation analysis.
The dissertation commences with a deep dive into the historical evolution of screw theory and the Denavit-Hartenberg (D-H) method. These are the mathematical foundations of the screw-based Jacobian and the differential-based Jacobian, respectively. Their geometric and physical implications are elucidated. A notable gap in the geometric interpretation of these fundamental concepts, despite their widespread application in robotic modelling, is identified. This research bridges this gap, offering profound insights into the geometric meanings of both D-H parameters and screws.

The Jacobian matrix, emblematic of first-order kinematics, acts as a bridge, translating joint rates into corresponding angular and linear velocities of the end effector. Two primary approaches exist for the formulation of the Jacobian matrix, each grounded in distinct mathematical concepts. The first method relies on the differentiation of D-H homogeneous transformation, resulting in the differential-based Jacobian, while the second is founded on screw theory, leading to the screw-based Jacobian. A novel differential method for Jacobian derivation is presented, leveraging the D-H transformation matrix. This method presents the correlation between the homogenous transformation matrix and the Jacobian matrix. Further, by investigating the geometrical implications of the D-H homogeneous transformation matrix and screws, the relationship and transformation between the differential-based Jacobian and the screw-based Jacobian are established for the first time. This investigation also reveals their physical connections and distinctions.

The focus then shifts to bio-like origami mechanisms, which offer versatile and innovative solutions to various engineering challenges, drawing inspiration from nature’s origami-like folding principles. However, the inherent complexity in their design is acknowledged, underscoring the challenges in comprehending their motion behaviours. The study probes into the kinematic behaviour of these mechanisms, particularly focusing on the applications of the differential-based and screw-based Jacobians, encompassing Jacobian analysis, singularity analysis, and bifurcation analysis.

In the exploration of the bio-like origami mechanism, the research introduces a mathematical model of the ladybird wing, by using the principle of equivalent mechanisms. This model leads to the mathematical description of its folding movement and velocity. A novel algebraic approach, rooted in Sylvester’s dialytic elimination method, is proposed for solving the relations between different joint angles in the spherical multi-bar linkage. Incorporating with the results from the algebraic approach, the product of exponentials and the screw-based Jacobian determine the location and instantaneous velocity of any desired point in this ladybird wing origami mechanism, respectively. The proposed algebraic approach has broad applicability in the kinematic analysis of bio-like origami mechanisms.

Additionally, this dissertation derives a novel double spherical 7R mechanism with dual loops, inspired by origami art mimicking bat wing-flapping motions. Both forward and inverse kinematic analyses of this mechanism are conducted, utilising screw algebra and the concept of equivalent mechanisms. Singular configurations of the mechanism are identified using two modified Singular Value Decomposition (SVD) numerical methods, based on the differential-based Jacobian and the screw-based Jacobian, respectively. The research culminates in a bifurcation analysis of the bat-origami-like mechanism, using a modified systematic approach that integrates high-order kinematic analysis with the SVD numerical method. This approach, based exclusively on the screw-based Jacobian, discerns four motion branches at the fully extended configuration and delineates the entire motion paths and singular value curves for each branch.

In conclusion, this comprehensive exploration into Jacobian matrices, their geometric implications, and their application in bio-like origami mechanisms offers a novel perspective on robotic kinematics. The findings and methodologies presented hold significant potential for advancing research in this domain, providing a robust foundation for future endeavours.
Date of Award1 Jan 2024
Original languageEnglish
Awarding Institution
  • King's College London
SupervisorJian Dai (Supervisor) & Hak-Keung Lam (Supervisor)

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