China
Africa
Explore chapters and articles related to this topic
The Magnetic Polarizability Dyadic and Point Symmetry
Published in Carl E. Baum, Detection and Identification of Visually Obscured Targets, 2019
The full orthogonal group in three dimensions O3 allows all rotations and reflections which keep r invariant, where we have the usual spherical coordinates (r, θ, ϕ) with () ψ=rsin(θ),z=rcos(θ)
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]
Anisotropic and conewise elasticity
Published in Benjamin Loret, Fernando M. F. Simões, Biomechanical Aspects of Soft Tissues, 2017
Benjamin Loret, Fernando M. F. Simões
The proper orthogonal group 𝒮𝒪(3) involves only rotations as defined in Section 2.5.5. The larger the symmetry group, the more the material displays symmetries, and the smaller is the number of independent coefficients required to define the elastic stiffness. If 𝒢 is the complete orthogonal group 𝒪(3), then the material response is said to be isotropic. If 𝒢 is the proper orthogonal group 𝒮𝒪(3), then the material response is said to be hemitropic.
Observer-based hybrid control for global attitude tracking on SO(3) with input quantisation
Published in International Journal of Control, 2023
Seyed Hamed Hashemi, Naser Pariz, Seyed Kamal Hosseini Sani
The problem of global attitude stabilisation of a rigid body has received increasing attention over the last decade. This is due to various applications ranging from spacecraft, robotics to autonomous underwater vehicles (AUVs) (Nian et al., 2020). The state-space of attitude kinematics: the special orthogonal group of order three is a non-Euclidean space. This configuration space is a compact manifold without boundary. Therefore, has the topological structure of non-contractibility (Hashemi et al., 2021b). Consequently, a given reference set cannot be globally stabilised by any continuous feedback. Accordingly, the best that one can attain with continuous feedback law is almost global stabilisation, where the basin of attraction of a given set in is excluded from a set of Lebesgue measure zero (Zhou et al., 2017).
Minimal control effort and time Lie-group synchronisation design based on proportional-derivative control
Published in International Journal of Control, 2022
The Lie group considered for the quadcopter mathematical model is the special orthogonal group whose Lie algebra reads The symbol represents a identity matrix. The rotational motion of the master drone is governed by the equations with denoting the master's state-pair, while the rotational motion of the controlled slave drone is governed by the equations where denotes the slave's state-pair and is a control field to be designed to make the two twin systems sync in time. The system (4) will be further analysed in Subsection 4.3 to verify the physical realisability of the designed controller.
Enclosing a moving target with an optimally rotated and scaled multiagent pattern
Published in International Journal of Control, 2021
Miguel Aranda, Youcef Mezouar, Gonzalo López-Nicolás, Carlos Sagüés
A way to address the design of a controller to achieve the objective expressed in (3) is by defining the following general cost function: with being a positive scalar, an angle, and , a rotation matrix in the Special Orthogonal group of dimension two . We note that the norm considered in this paper is the Euclidean one. This cost function is a sum of squared distances that expresses how separated the agents are from a configuration that represents a rotated and scaled version of the prescribed pattern, with the target at its centroid. We can now precisely specify our control goal, as follows.