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Harmonic Analysis on the Euclidean Motion Groups
Published in Gregory S. Chirikjian, Alexander B. Kyatkin, Engineering Applications of Noncommutative Harmonic Analysis, 2021
Gregory S. Chirikjian, Alexander B. Kyatkin
The Euclidean motion group, SE(N),1 is the semidirect product of ℝN with the special orthogonal group, SO(N), That is, SE(3)= xi ⋊φSO(3). We denote elements of SE(N) as g = (a. A) ∈SE(N) where A∈SO(N) and a ∈ℝN. For any g = (a. A) and h = (r, R) ∈ SE(N), the group law is written as g ○h = (a + Ar, AR), and g−1= (−ATa, AT). Alternately, one may represent any element of SE(N) as an (N + 1) ×(N +1) homogeneous transformation matrix of the form H(g)=(Aa0T1).
Derivative-based Optimization: Lie Algebra Method
Published in Kenichi Kanatani, 3D Rotations, 2020
We can think of this situation as follows. The set of all rotations can be thought of as a “surface” defined by the “nonlinear” constraints R⊤R = I and |R| = 1 in the 9D space of the nine elements of 3 × 3 matrices. This is called the special orthogonal group4 of dimension 3, or the group of rotations for short, and denoted by SO(3). It has the algebraic property that products and inverses of its elements are defined; such a set is called a group. It also has the geometric (or topological) property that the elements constitute a curved surface, or a manifold to be precise, in a high dimensional space. A set that has these two propperties is called a Lie group; SO(3) is a typical Lie group. See Appendix C for groups, manifolds, and Lie groups.
Nonlinear Control Design
Published in Jitendra R. Raol, Ramakalyan Ayyagari, Control Systems, 2020
Jitendra R. Raol, Ramakalyan Ayyagari
Most mechanical systems, robotic examples in this chapter and aerospace examples in Chapter 14, are governed by second order differential equations and evolve on nonlinear manifolds. At a superficial level, with more linear theory behind, typical control techniques for these systems rely on the introduction of local coordinates so as to set up a smooth one-to-one correspondence between the actual state space and Cartesian space within an admissible range. Although this converts control on a manifold to control in Cartesian space, the engineer would have enjoyed the intuition. The issue of singularity exists for any choice of local coordinates on certain nonlinear manifolds, such as the special orthogonal group SO(3). We have purposefully avoided the kinematics, inverse kinematics, and the possibility of singularities in this text, while placing emphasis on touching upon the controller desgin. Interested readers may look into books [17–24] on dynamics and control.
A Robust Adaptive Nonlinear Control Design via Geometric Approach for a Quadrotor
Published in IETE Journal of Research, 2022
Awatif Guendouzi, Mustapha Hamerlain, Nadia Saadia
We consider a quadrotor UAV, as depicted in Figure 1. Let is the position vector of the center of mass of the aircraft relative to the inertial frame . The rotation matrix representing the orientation of the body frame with respect to the inertial frame , where is an orthogonal matrix defined by . The configuration space of the quadrotor UAV with respect to the inertial frame can be observed as an element of the Special Euclidean Group SE(3), i.e.. Then, the equations of motion are given as follows [19]: where is the total mass of the aircraft, are the angular velocity vector and the linear velocity vector, respectively, the inertia matrix, is the gravity acceleration and is the standard basis of . is a mapping from to so (3) (the Lie algebra of SO(3)), i.e. is the skew-symmetric matrix of :