Related papers: The Geometry and Kinematics of the Matrix Lie Grou…
The irreducible representations of the cubic space group are described and used to determine the mapping of continuum states to lattice states with non-zero linear momentum. The Clebsch-Gordan decomposition is calculated from the character…
Three-dimensional almost contact B-metric manifolds are constructed by a three-parametric family of Lie groups. It is established the class of the investigated manifolds which has an important geometrical interpretation. It is determined…
In many real-world applications of regression, conditional probability estimation, and uncertainty quantification, exploiting symmetries rooted in physics or geometry can dramatically improve generalization and sample efficiency. While…
We give very flexible, concrete constructions of discrete and faithful epresentations of right-angled Artin groups into higher-rank Lie groups. Using the geometry of the associated symmetric spaces and the combinatorics of the groups, we…
In the subdivision approach to robot path planning, we need to subdivide the configuration space of a robot into nice cells to perform various computations. For a rigid spatial robot, this configuration space is $SE(3)=\mathbb{R}^3\times…
This thesis is devoted to algorithmic aspects of the implementation of Cartan's moving frame method to the problem of the equivalence of submanifolds under a Lie group action. We adopt a general definition of a moving frame as an…
Robotic assembly of complex, non-convex geometries with tight clearances remains a challenging problem, demanding precise state estimation for successful insertion. In this work, we propose a novel framework that relies solely on contact…
We study the question of whether the topological quotient of a real linear representation of a simple three-dimensional compact Lie group is a manifold. We obtain an upper bound for the dimension of a representation whose quotient is a…
Lie-symmetry methods are used to determine the symmetry group of reduced magnetohydrodynamics. This group allows for arbitrary, continuous transformations of the fields themselves, along with space-time transformations. The derivation…
To achieve robust and accurate state estimation for robot navigation, we propose a novel Visual Inertial Odometry(VIO) algorithm with line features upon the theory of invariant Kalman filtering and Cubature Kalman Filter (CKF). In contrast…
We present a concise pedagogic introduction to group representation theory motivated by the historical developments surrounding the advent of the Eightfold Way. Abstract definitions of groups and representations are avoided in favour of the…
Extended Kalman filter (EKF) does not guarantee consistent mean and covariance under linearization, even though it is the main framework for robotic localization. While Lie group improves the modeling of the state space in localization, the…
Vision based localization is a popular approach to carry out manoeuvres particularly in GPS-restricted indoor environments, because vision can complement other activities performed by the robot. The objective is to estimate the current…
There is a decomposition of a Lie algebra for open matrix chains akin to the triangular decomposition. We use this decomposition to construct unitary irreducible representations. All multiple meson states can be retrieved this way.…
Proper group actions are ubiquitous in mathematics and have many of the attractive features of actions of compact groups. In this survey, we discuss proper actions of Lie groups on smooth manifolds. If the group dimension is sufficiently…
In this paper we give detailed construction of $G$-equivariant Kuranishi chart of moduli spaces of pseudo-holomorphic curves to a symplectic manifold with $G$-action, for an arbitrary compact Lie group $G$. The proof is based on the…
We demonstrate model-based, visual robot manipulation of linear deformable objects. Our approach is based on a state-space representation of the physical system that the robot aims to control. This choice has multiple advantages, including…
Successful control of a rigid-body rotating in three dimensional space requires accurate estimation of its attitude. The attitude dynamics are highly nonlinear and are posed on the Special Orthogonal Group $SO(3)$. In addition, measurements…
Continuum robots, known for their high flexibility and adaptability, offer immense potential for applications such as medical surgery, confined-space inspections, and wearable devices. However, their non-linear elastic nature and complex…
This paper addresses the problem of Simultaneous Localization and Mapping (SLAM) for rigid body systems in three-dimensional space. We introduce a new matrix Lie group SE_{3+n}(3), whose elements are composed of the pose, gravity, linear…