A Closed Form of Higher-Order Cayley Transforms and Generalized Rodrigues Vectors Parameterization of Rigid Motion

Daniel Condurache (), Mihail Cojocari and Ioan-Adrian Ciureanu ()
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Daniel Condurache: Department of Theoretical Mechanics, “Gheorghe Asachi” Technical University of Iasi, D. Mangeron 59, 700050 Iasi, Romania
Mihail Cojocari: Department of Theoretical Mechanics, “Gheorghe Asachi” Technical University of Iasi, D. Mangeron 59, 700050 Iasi, Romania
Ioan-Adrian Ciureanu: Department of Preventive Medicine and Interdisciplinarity, “Grigore T. Popa” University of Medicine and Pharmacy Iasi, 700116 Iasi, Romania

Mathematics, 2024, vol. 13, issue 1, 1-19

Abstract: This paper introduces a novel closed-form coordinate-free expression for the higher-order Cayley transform, a concept that has not been explored in depth before. The transform is defined by the Lie algebra of three-dimensional vectors into the Lie group of proper orthogonal Euclidean tensors. The approach uses only elementary algebraic calculations with Euclidean vectors and tensors. The analytical expressions are given by rational functions by the Euclidean norm of vector parameterization. The inverse of the higher-order Cayley map is a multi-valued function that recovers the higher-order Rodrigues vectors (the principal parameterization and their shadows). Using vector parameterizations of the Euler and higher-order Rodrigues vectors, we determine the instantaneous angular velocity (in space and body frame), kinematics equations, and tangent operator. The analytical expressions of the parameterized quantities are identical for both the principal vector and shadows parameterization, showcasing the novelty and potential of our research.

Keywords: Cayley map; higher-order Rodrigues vector; kinematic equation; tangent operator (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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