A Closed Form of Higher-Order Cayley Transforms and Generalized Rodrigues Vectors Parameterization of Rigid Motion
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 ap...
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2024-12-01
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| author | Daniel Condurache Mihail Cojocari Ioan-Adrian Ciureanu |
| author_facet | Daniel Condurache Mihail Cojocari Ioan-Adrian Ciureanu |
| author_sort | Daniel Condurache |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-644cfb3108514173b905bb6d8616c757 |
| institution | Kabale University |
| issn | 2227-7390 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj-art-644cfb3108514173b905bb6d8616c7572025-01-10T13:18:17ZengMDPI AGMathematics2227-73902024-12-0113111410.3390/math13010114A Closed Form of Higher-Order Cayley Transforms and Generalized Rodrigues Vectors Parameterization of Rigid MotionDaniel Condurache0Mihail Cojocari1Ioan-Adrian Ciureanu2Department of Theoretical Mechanics, “Gheorghe Asachi” Technical University of Iasi, D. Mangeron 59, 700050 Iasi, RomaniaDepartment of Theoretical Mechanics, “Gheorghe Asachi” Technical University of Iasi, D. Mangeron 59, 700050 Iasi, RomaniaDepartment of Preventive Medicine and Interdisciplinarity, “Grigore T. Popa” University of Medicine and Pharmacy Iasi, 700116 Iasi, RomaniaThis 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.https://www.mdpi.com/2227-7390/13/1/114Cayley maphigher-order Rodrigues vectorkinematic equationtangent operator |
| spellingShingle | Daniel Condurache Mihail Cojocari Ioan-Adrian Ciureanu A Closed Form of Higher-Order Cayley Transforms and Generalized Rodrigues Vectors Parameterization of Rigid Motion Mathematics Cayley map higher-order Rodrigues vector kinematic equation tangent operator |
| title | A Closed Form of Higher-Order Cayley Transforms and Generalized Rodrigues Vectors Parameterization of Rigid Motion |
| title_full | A Closed Form of Higher-Order Cayley Transforms and Generalized Rodrigues Vectors Parameterization of Rigid Motion |
| title_fullStr | A Closed Form of Higher-Order Cayley Transforms and Generalized Rodrigues Vectors Parameterization of Rigid Motion |
| title_full_unstemmed | A Closed Form of Higher-Order Cayley Transforms and Generalized Rodrigues Vectors Parameterization of Rigid Motion |
| title_short | A Closed Form of Higher-Order Cayley Transforms and Generalized Rodrigues Vectors Parameterization of Rigid Motion |
| title_sort | closed form of higher order cayley transforms and generalized rodrigues vectors parameterization of rigid motion |
| topic | Cayley map higher-order Rodrigues vector kinematic equation tangent operator |
| url | https://www.mdpi.com/2227-7390/13/1/114 |
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