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Dual-Manipulator Testing Technique
Published in Chunguang Xu, Robotic Nondestructive Testing Technology, 2022
An important concept associated with Lie group is Lie algebra. The tangential space of the identity element in a Lie group is called the Lie algebra of this Lie group, represented by “g”. The Lie algebra, together with a bidirectional mapping, is called Lie parenthesis, which constitutes a vector space. The bilinear mapping [x, y] satisfies: Antisymmetry: [x,y]=−[y,x];Jacobi identity: [x,[y,z]]+[z,[x,y]]+[y,[z,x]]=0.
Introduction
Published in Sabiha Wadoo, Pushkin Kachroo, Autonomous Underwater Vehicles, 2017
Let G be a Lie group with identity I, and let XI be a tangent vector to G at I. For any g ∈ G, define the left translation by g to be a map Lg: G → G such that Lg(x) = gx, where x ∈ G. Since G is a Lie group, Lg is a diffeomorphism of G for each g. Diffeomorphism is a smooth inveritable function that maps one differential manifold to another. Taking the differential of Lg at e results in a map from the tangent space of G at e to the tangent space of G at g: dLg:TgG→TgG such that Xg=dLg(Xg)
Harmonic Analysis on Groups
Published in Gregory S. Chirikjian, Alexander B. Kyatkin, Engineering Applications of Noncommutative Harmonic Analysis, 2021
Gregory S. Chirikjian, Alexander B. Kyatkin
The representation theory of Lie groups has many physical applications. This theory evolved with the development of quantum mechanics, and today serves as a natural language in which to articulate the quantum theory of angular momentum. There is also a very close relationship between Lie group representations and the theory of special functions [17, 57, 81, 83]. It is this connection that allows for our concrete treatment of harmonic analysis on the rotation and motion groups in Chapters 9 and 10.
Minimal control effort and time Lie-group synchronisation design based on proportional-derivative control
Published in International Journal of Control, 2022
A smooth manifold that is also an algebraic group is termed a Lie group. An algebraic group is a structure () where m : is an associative binary operation (such as a multiplication), is an identity element with respect to m and i : is the function which represents the inversion with respect to the operation m, so that for each . A left translation on a Lie group is denoted by and is defined as .
A review on some classes of algebraic systems
Published in International Journal of Control, 2020
Víctor Ayala, Heriberto Román-Flores
A homomorphism between two Lie groups G and H is called a Lie group homomorphism. A bijective Lie group homomorphism of G with itself is called a Lie group automorphism. If G is connected, the set of G-automorphisms is a Lie group with Lie algebra (Warner, 1971).
Modelling and control of a spherical pendulum via a non–minimal state representation
Published in Mathematical and Computer Modelling of Dynamical Systems, 2021
Ricardo Campa, Israel Soto, Omar Martínez
A group which is also a differential manifold is called a Lie group. So unit quaternions form a 3–dimensional Lie group, which is a double cover of the orientation manifold. An implication of this fact is that quaternion multiplication is closely related to the composition of rotations in 3D space.