China 
Africa 
Explore chapters and articles related to this topic
Group Theory
Published in Gregory S. Chirikjian, Alexander B. Kyatkin, Engineering Applications of Noncommutative Harmonic Analysis, 2021
Gregory S. Chirikjian, Alexander B. Kyatkin
A basis for the Lie algebra of the scale-Euclidean group SIM (2) is X1=(001000000);X2=(000001000);X3=(0−10100000);andX4=(100010000).
Study of Three Different Philosophies to Automatic Target Recognition
Published in Jitendra R. Raol, Ajith K. Gopal, Mobile Intelligent Autonomous Systems, 2016
Vishal C. Ravindra, Venkatesh K. Madyastha, Girija Gopalratnam
where det(⋅) denotes the determinant operator† and H denotes Hermitian or complex conjugate transpose. The 2nd equation in 30.2 can be easily seen by taking the determinant on both sides of the 1st equation of 30.2, that is, det(OOH) = det(O) det(OH) = (det(O))2 = det(I) = 1. The type of rotation matrices, that form a subgroup called special orthogonal group, for which det(O) = +1 are also called as proper rotation matrices. Another interesting property of groups in the scene building context is that one transformation s1 applied before another transformation s2 has the combined effect of a third transformation s3 = s1⋅ s2 applied alone. Translation and rotation can be combined to form a single transformation that is a member of the special Euclidean group SE(n). Consider a point x ∈ ℜ2 on a template in an image. If a translation operation p ∈ ℜ2 is applied, it results in new coordinates x + p. In addition, if a rotation O ∈SO(2) is applied, it results in the point x being rotated to result in new coordinates Ox. Together, the operation could be combined and represented by Ox + p. Generally, the joint translation–rotation can be represented in matrix form as follows: for any n × n matrix U ∈ ℜn×n such that () U=[Op01]
Algebraic Structures and Applications
Published in Richard L. Shell, Ernest L. Hall, Handbook of Industrial Automation, 2000
We briefly describe the group of all rigid motions (the Euclidean group). This is the group of all distance-preserving functions from ℝn→ℝn. When n = 2or 3, these groups are of great practical importance and contain translations, rotations, and reflections. Here ℝn = {(x1, x2, …, xn) | xi ∈ℝ, 1 ≤ i ≤ n} consists of all ordered n-tuples of real numbers, and the distance between two points x = (x1, x2, …, xn) and y = (y1, y2, …,yn) is defined by d(x,y) where d(x,y)=‖x−y‖=∑i=1n(xi−yi)2. A function T : ℝn → ℝn is distance preserving if d(Tx, Ty) = ||Tx −Ty|| = ||x−y||=d(x,y) for all x,y ∈ ℝn. All distance-preserving functions from ℝn to ℝn form a group under composition of functions. Such functions are called rigid motions of ℝn. When n = 2, ℝ2 defines a plane and when n = 3, ℝ3 defines the usual three-dimensional space.
Vision-based trajectory tracking control of quadrotors using super twisting sliding mode control
Published in Cyber-Physical Systems, 2020
The coordinate of frame is expressed as . For arbitrary two frames, such as frame and frame , the transformation matrix from frame to frame is , where refers to the special Euclidean group. The transformation can be divided into two parts, that is the translation between two frames and the rotation between two frames , where represents the special orthogonal group. The relationship can be described as
Consensus-based formation control for nonholonomic vehicles with parallel desired formations
Published in International Journal of Control, 2021
Special Euclidean group with two dimensions, i.e. SE(2), is a kind of Lie group, which is generally employed to describe the motion of planar rigid bodies. The group element , representing the configuration, is denoted by where (Special Orthogonal group with two dimensions) is the attitude matrix, and is the position vector.