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Human body representation in terms of SE(3)/SE(2)-groups of rigid-body motion, with the vertebral column represented as a chain of 26 flexibly-coupled SE(3)-groups.
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These lecture notes in Lie Groups are designed for a 1--semester third year
or graduate course in mathematics, physics, engineering, chemistry or biology.
This landmark theory of the 20th Century mathematics and physics gives a
rigorous foundation to modern dynamics, as well as field and gauge theories in
physics, engineering and biomechanics. We g...
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... within the car, and (ii) their “soft-hard characteristics”, similar to the helmet characteristics described above. B. In case of spinal injury (see Figure 6), the contribution of our universal Jolt theory is the following: 1. The spinal injury is always localized at the certain vertebral or inter-vertebral point; 2. In case of severe translational injuries (vertebral fractures or discus herniae) they can be identified using X-ray or other medical imaging scans; in case of microscopic rotational injuries (causing the back-pain syndrome) they cannot be identified using current medical imaging scans; 3. There is no spinal injury without one of the following two causes: a. Impulsive rotational + translational loading caused by either fast human movements or various crashes/impacts; and/or b. Static eccentricity from the normal physiological spinal form, caused by external loading; c. Any spinal injury is caused by a combination of the two points above: impulsive rotational + translational loading and static eccentricity. This is a straightforward consequence of our universal Jolt theory. We cannot separately analyze translational and rotational spinal injuries. Also, there are no “static injuries” without eccentricity. Indian women have for centuries carried bulky loads on their heads without any spinal injuries; they just prevented any load eccentricities and any jerks in their motion. The currently used “Principal loading hypothesis” that describes spinal injuries in terms of spinal tension, compression, bending, and shear, covers only a small subset of all spinal injuries covered by our universal Jolt theory. To prevent spinal injuries we need to develop spinal jolt awareness: ability to control all possible impulsive spinal loadings as well as static eccentricities. C. In case of general musculo-skeletal injury (see Figure 7 for the particular case of knee injury), the contribution of our universal Jolt theory is the following: 1. The injury is always localized at the certain joint or bone and caused by an impulsive loading, which hits this particular joint/bone in several coupled degrees- of-freedom simultaneously; 2. Injury happens when most of the body mass is hanging on that joint; for example, in case of a knee injury, when most of the body mass is on one leg ...
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The phylogenetic Mean Pairwise Distance (MPD) is one of the most popular measures for computing the phylogenetic distance between a given group of species. More specifically, for a phylogenetic tree and for a set of species R represented by a subset of the leaf nodes of , the MPD of R is equal to the average cost of all possible simple p...
Citations
... There exists an exponential map from each element in the Lie algebra to an element in its corresponding Lie group [21,22]; that is, every rotation and pose matrix representation can be written as the exponential of some matrix. We can better appreciate this mapping by considering the kinematics of rotations and poses. ...
Locating vehicles, targets and objects in three-dimensional space is key to many fields of science and engineering such as robotics, aerospace, computer vision and graphics. Rotations and poses (position plus orientation) of bodies can be expressed in a variety of ways. Rotation matrices constitute one of the classic matrix Lie groups, the special orthogonal group— SO ( 3 ) . Poses can likewise be represented by matrices. One such representation is embodied in a 4 × 4 matrix establishing another famous matrix Lie group, the special Euclidean group— SE ( 3 ) . An alternative representation of pose uses 6 × 6 matrices and is referred to as the group of pose adjoints— Ad ( SE ( 3 ) ) . The eigenstructures of these representations reveal much about them from Euler’s theorem for rotations to the Mozzi–Chasles theorem for the general displacement of a rigid body. While the eigenstructure of SO ( 3 ) has been extensively studied, those of SE ( 3 ) and Ad ( SE ( 3 ) ) have hardly received the same scrutiny yet their structure is much richer. Motivated by their importance in kinematics and dynamics, we provide here a complete characterization of rotations and poses in terms of the eigenstructure of their matrix Lie group representations. An eigendecomposition of pose matrices reveals that they can be cast into a form similar to that of rotations although the structure of the former can vary depending on the nature of the pose involved. In particular, the pose matrices of SE ( 3 ) and Ad ( SE ( 3 ) ) cannot generally be diagonalized as can rotation matrices but they of course do yield to a Jordan normal form, from which we can identify a principal-axis pose in much the same manner that we can a principal-axis rotation. We also address the minimal polynomials for poses and derive a novel expression for the Jacobian in Ad ( SE ( 3 ) ) . Finally, we argue that the true counterpart to SO ( 3 ) for poses is not SE ( 3 ) but Ad ( SE ( 3 ) ) .
... D.1 PROOF OF EQUIVARIANCE PROPERTY (7) FOR LAYER DEFINITION (8) As each roto-translation can be built from a rotation followed by a translation (Ivancevic & Ivancevic, 2011) it is enough to prove equivariance under rotations and translations (of positions) independently. ...
Convolutional networks are successful, but they have recently been outperformed by new neural networks that are equivariant under rotations and translations. These new networks work better because they do not struggle with learning each possible orientation of each image feature separately. So far, they have been proposed for 2D and 3D data. Here we generalize them to 6D diffusion MRI data, ensuring joint equivariance under 3D roto-translations in image space and the matching 3D rotations in $q$-space, as dictated by the image formation. Such equivariant deep learning is appropriate for diffusion MRI, because microstructural and macrostructural features such as neural fibers can appear at many different orientations, and because even non-rotation-equivariant deep learning has so far been the best method for many diffusion MRI tasks. We validate our equivariant method on multiple-sclerosis lesion segmentation. Our proposed neural networks yield better results and require fewer scans for training compared to non-rotation-equivariant deep learning. They also inherit all the advantages of deep learning over classical diffusion MRI methods. Our implementation is available at https://github.com/philip-mueller/equivariant-deep-dmri and can be used off the shelf without understanding the mathematical background.
... As discussed in Section III-A, the rotation between two reference frames is represented as an element of SO(3). The special orthogonal group SO(3) can be described in terms of quadratic constraints [28]: ...
... Similar to rotations, the rigid transformation between two rotated and translated reference frames F − → b and F − → a is an element of the special Euclidean group SE(3) and is also defined by quadratic constraints [28]: ...
... As discussed in Section III-A, the rotation between two reference frames is represented as an element of SO(3). The special orthogonal group SO(3) can be described in terms of quadratic constraints [29]: ...
... Similar to rotations, the rigid transformation between two rotated and translated reference frames F − → b and F − → a is an element of the special Euclidean group SE(3) and is also defined by quadratic constraints [29]: ...
Correct fusion of data from two sensors is not possible without an accurate estimate of their relative pose, which can be determined through the process of extrinsic calibration. When two or more sensors are capable of producing their own egomotion estimates (i.e., measurements of their trajectories through an environment), the 'hand-eye' formulation of extrinsic calibration can be employed. In this paper, we extend our recent work on a convex optimization approach for hand-eye calibration to the case where one of the sensors cannot observe the scale of its translational motion (e.g., a monocular camera observing an unmapped environment). We prove that our technique is able to provide a certifiably globally optimal solution to both the known- and unknown-scale variants of hand-eye calibration, provided that the measurement noise is bounded. Herein, we focus on the theoretical aspects of the problem, show the tightness and stability of our solution, and demonstrate the optimality and speed of our algorithm through experiments with synthetic data.
... Additionally, an explicit formula for the exponential map exp : se(3) → SE(3) exists (see [35] and [36] for example). This gives an exact way to solve (67). ...
This paper develops a new mathematical framework that enables nonparametric joint semantic/appearance and geometric representation of continuous functions using data. The joint semantic and geometric embedding is modeled by representing the processes in a reproducing kernel Hilbert space. The framework allows the functions to be defined on arbitrary smooth manifolds where the action of a Lie group is used to align them. The continuous functions allow the registration to be independent of a specific signal resolution and the framework is fully analytical with a closed-form derivation of the Riemannian gradient and Hessian. We study a more specialized but widely used case where the Lie group acts on functions isometrically. We solve the problem by maximizing the inner product between two functions defined over data, while the continuous action of the rigid body motion Lie group is captured through the integration of the flow in the corresponding Lie algebra. Low-dimensional cases are derived with numerical examples to show the generality of the proposed framework. The high-dimensional derivation for the special Euclidean group acting on the Euclidean space showcases the point cloud registration and bird's-eye view map registration abilities. A specific derivation and implementation of this framework for RGB-D cameras outperform the state-of-the-art robust visual odometry and performs well in texture and structure-scares environments.
... Additionally, an explicit formula for the exponential map exp : se(3) → SE(3) exists (see [20] and [27] for example). This gives an exact way to solve (31). ...
... Additionally, an explicit formula for the exponential map exp : se(3) → SE(3) exists (see [17] and [23] for example). This gives an exact way to solve (31). ...
This paper reports on a novel formulation and evaluation of visual odometry from RGB-D images. Assuming a static scene, the developed theoretical framework generalizes the widely used direct energy formulation (photometric error minimization) technique for obtaining a rigid body transformation that aligns two overlapping RGB-D images to a continuous formulation. The continuity is achieved through functional treatment of the problem and representing the process models over RGB-D images in a reproducing kernel Hilbert space; consequently, the registration is not limited to the specific image resolution and the framework is fully analytical with a closed-form derivation of the gradient. We solve the problem by maximizing the inner product between two functions defined over RGB-D images, while the continuous action of the rigid body motion Lie group is captured through the integration of the flow in the corresponding Lie algebra. Energy-based approaches have been extremely successful and the developed framework in this paper shares many of their desired properties such as the parallel structure on both CPUs and GPUs, sparsity, semi-dense tracking, avoiding explicit data association which is computationally expensive, and possible extensions to the simultaneous localization and mapping frameworks. The evaluations on experimental data and comparison with the energy-based formulation of the problem confirm the effectiveness of the proposed technique, especially, when the lack of structure and texture in the environment is evident.
... elements of the special Euclidean group SE(3) (see ref. [43]), i.e., rotations and translations. Although one might favor Poincaré transformations as they include Lorentz boosts, which are certainly required for interfacing external interaction models, this would require to add a time-like coordinate to all geometric objects. ...
A large scientific community depends on the precise modeling of complex processes in particle cascades in various types of matter. These models are used most prevalently in cosmic ray physics, astrophysical-neutrino physics, and gamma ray astronomy. In this white paper, we summarize the necessary steps to ensure the evolution and future availability of optimal simulation tools. The purpose of this document is not to act as a strict blueprint for next-generation software, but to provide guidance for the vital aspects of its design. The topics considered here are driven by physics and scientific applications. Furthermore, the main consequences of implementation decisions on performance are outlined. We highlight the computational performance as an important aspect guiding the design, since future scientific applications will heavily depend on an efficient use of computational resources.
MyJay is an open-source robot designed to facilitate play between children with and without physical disabilities. The robot acts as a proxy for children with upper limb challenges, allowing them to participate in physical games with their peers. Our design was inspired by the FIRST Robotics Competition, which involves teleoperating robots to manipulate objects. Taking a user-centred perspective, we consulted therapists and conducted remote interviews with children with disabilities and their guardians at various stages of the design process. We then conducted an in-person feasibility study with 18 typically developing children in a school setting. The study involved children teleoperating the robot to pick up and throw balls into a designated goal, and the interaction was evaluated using the user experience questionnaire and the Robotic Social Attributes Scale. The results of the study show great potential for MyJay to act as a play mediator in various scenarios, and the response from the children was positive. The ultimate aim of our research agenda is to pave the way towards creating more inclusive play environments through robot-mediated interactions, breaking barriers posed by physical limitations.
Robotised Non Destructive Testing (NDT) presents multifaceted advantages, saving time and reducing repetitive manual workloads for highly skilled Ultrasonic Testing (UT) operators. Due to the requisite accuracy and reliability of the field, robotic NDT has traditionally relied on digital twins for complex path planning procedures enabling precise deployment of NDT equipment. This paper presents a multi-scale and collision-free path planning and implementation methodology enabling rapid deployment of robotised NDT with commercially available sensors. Novel algorithms are developed to plan paths over noisy and incomplete point clouds from low-cost sensors without the need for surface primitives. Further novelty is introduced in online path corrections utilising laser and force feedback while applying a Conformable-Wedge probe UT sensor. Finally, a novel source of data beneficial to automated NDT is introduced by collecting frictional forces of the surface informing the operator of the surface preparation quality. The culmination of this work is a new path-planning free, single-shot automated process removing the need for complex operator-driven procedures with a known surface, visualising collected data for the operator as a three-dimensional C-scan model. The dynamic robotic control enables a move to the industry 4.0 model of adaptive online path planning. Experimental results indicate the flexible and streamlined pipeline for robotic deployment, and demonstrate intuitive data visualisation to aid highly skilled operators in a wide field of industries.
