Repelling-Screw-Based Mechanism Analysis and the Relevance of Finite Screw Representations in Accordance with Dual Euler-Rodrigues Formula

Student thesis: Doctoral ThesisDoctor of Philosophy


Screw theory provides an integrated geometric description of both rotational and translational quantities that arise in the robotic mechanics, multibody dynamics, and computational geometry, which significantly simplifies the design and analysis of mechanisms and robots. This thesis presents a comprehensive study on screw theory, especially focuses on repelling screw system and the intrinsic relevance of finite screw representations, as well as their uses in mechanism theory. The theoretical background and the fundamentals of both instantaneous screw theory and finite screw theory are first introduced, based on which new theoretical findings as well as novel mechanical designs are proposed, leading to the development and extension of screw algebra.

Using linear algebra, an approach for the construction of repelling screw system is proposed, upon which the thesis provides a geometrical insight into the dualities of compliant mechanisms. By means of screw system algebra, both orthogonal and dual properties of twist and wrench spaces are studied and extended, upon which the kinematics, statics, and stiffness/compliance of both full- and limited-mobility compliant mechanisms are analyzed. The internal correlations between the repelling screw system and the dualities of mechanisms are investigated, upon which a novel approach for the configuration transformation between compliant parallel and serial mechanisms is developed. Moreover, with the proposed approach, one can judiciously select a better mechanism geometry to achieve a given stiffness/compliance. The thesis extends the concept of dual elastic mechanisms to more general cases, including rank deficient compliances and constraints. The spatial elastic behaviors having constraints and/or rank deficiencies can be realized by either a compliant parallel or serial mechanism, without the decomposition of the stiffness/compliance matrix.

Subsequently, the thesis presents a unified and generalized approach for the construction of Jacobian matrices for general, nonredundant parallel manipulators with serial and/or mixed-topology limbs, based on the concept of repelling screw system. The effect of the linear combinations of the screws in a screw system on its corresponding repelling screw system is revealed, and further used to determine the constraint wrenches/twists of a parallel manipulator and its limbs, without using the displacement manifold method, solving linear algebraic equations, or calculating the reciprocal screw system twice for each actuator/constraint - this stands in contrast to previous works. By bridging the kinematics and statics of mechanisms with repelling screws, the underlying correlations amongst the screws (twists and wrenches) that correspond to both the per-mitted and constrained motions of general mechanisms are identified and analyzed. The derived multilevel hierarchical Jacobian contains complete kinematics, statics, and constraint information at both limb and platform levels, thus distinguishing itself from other works on this topic.

To demonstrate the application of the generalized Jacobian analysis, an analytical approach is presented to evaluate the transmission performance of planar 3-RRR manipulators, upon which an optimum design considering the shape parameters of platforms, along with a sensitivity analysis of the influences of the design parameters on the overall performance of the manipulators, are conducted.

With regard to the representations of finite screw motions, the thesis examines the variations and derivations of the dual Euler-Rodrigues formula from various mathematical forms, including the matrix in 6×6, the dual matrix, Lie group 𝑆𝐸(3) of the exponential map of Lie algebra 𝑠𝑒(3), and the dual quaternion conjugation, and investigates their intrinsic connections. Based on the dual Euler-Rodrigues formula, the axis, the dual rotation angle, and the new traces are obtained by using the properties of the skew-symmetric matrices. The thesis relates the finite displacement screw matrix, the exponential map, and the dual quaternion conjugation to the dual Euler-Rodrigues formula and reveals their connection with the Mozzi-Chasles axis screw. Further, using the Mozzi-Chasles axis screw, the thesis presents a complete geometrical interpretation, including both the translation and rotation, and associates it with the algebraic presentation. The intrinsic connection between various presentations of rigid body transfor-mations is revealed by formulating them into the dual Euler-Rodrigues formula, and the relation between the exponential map of the Mozzi-Chasles axis screw to the finite displacement screw matrix and the dual Euler-Rodrigues formula is presented.

Finally, to demonstrate the design and application of metamorphic mechanisms, the thesis presents the Origaker, a novel multi-mimicry quadruped robot. Based on a single-loop spatial metamorphic mechanism, the Origaker is able to transform between different working modes, as the reptile-, arthropod-, and mammal-like modes, without disassembly and reassembly. The combination of the metamorphic mechanism and the quadruped robot enables the Origaker to pitch vertically, twist horizontally, and change the positional correlation between the trunk and legs. In consideration of its reconfigurability and structure adaptability, gaits and movement strategies, namely, fast spinning gait, stair climbing gait, self-recovery, packaging, and crawling through narrow spaces and right-angled bends, are proposed and analyzed, demonstrating that the metamorphic mechanism provides the robot with enhanced maneuverability. Unlike other quadruped robots, the metamorphic robot integrates multiple working modes into a single mechanical structure, and these modes can be rearranged in a controllable way during its body reconfiguration so as to maintain better adaptability and have more movement options on different terrains.
Date of Award1 Jul 2023
Original languageEnglish
Awarding Institution
  • King's College London
SupervisorJian Dai (Supervisor) & Emmanouil Spyrakos Papastavridis (Supervisor)

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