Homogeneous Rigid Body Mechanics with Elastic Coupling

Preface Contributors Part I. Algebra and Geometry Point Groups and Space Groups in Geometric Algebra (D. Hestenes) The Inner Products of Geometric Algebra (L. Dorst) Unification of Grassmann's Progressive and Regressive Products using the Principle of Duality (S. Blake) From Unoriented Subspaces to Blade Operators (T.A. Bouma) Automated Theorem Proving in the Homogeneous Model with Clifford Bracket Algebra (H. Li) Rotations in n Dimensions as Spherical Vectors (W.E. Baylis/S. Hadi) Geometric and Algebraic Canonical Forms (N. Gordon) Functions of Clifford Numbers or Square Matrices (J. Snygg) Compound Matrices and Pfaffians: A Representation of Geometric Algebra (U. Prells/M.I. Friswell/S.D. Garvey) Analysis Using Abstract Vector Variables (F. Sommen) A Multivector Data Structure for Differential Forms and Equations (J.A. Chard/V. Shapiro) Jet Bundles and the Formal Theory of Partial Differential Equations (R. Baker/C. Doran) Imaginary Eigenvalues and Complex Eigenvectors Explained by Real Geometry (E.M.S. Hitzer) Symbolic Processing of Clifford Numbers in C++ (J.P. Fletcher) Clifford Numbers and their Inverses Calculated using the Matrix Representation (J.P. Fletcher) A Toy Vector Field Based on Geometric Algebra (A. Rockwood/S. Binderwala) Quadratic Transformations in the Projective Plane (G. Georgiev) Annihilators of Principal Ideals in the Grassmann Algebra (C. Koc/S. Esin) Part II. Applications to Physics Homogeneous Rigid Body Mechanics with Elastic Coupling (D. Hestenes/E.D. Fasse) Analysis of One and Two Particle Quantum Systems using Geometric Algebra (R. Parker/C. Doran) Interaction and Entanglement in the Multiparticle Spacetime Algebra (T.F. Havel/C.J.L. Doran) Laws of Reflection from Two or More Plane Mirrors in Succession (M. Derome) Exact Kinetic Energy Operators for Polyatomic Molecules (J. Pesonen) Geometry of Quantum Computing by Hamiltonian Dynamics of Spin Ensembles (T. Schulte-Herbruggen/K. Huper/U. Helmke/S.J. Glaser) Is the Brain a 'Clifford Algebra Quantum Computer'? (V. Labunets/E. Rundblad/J. Astola) A Hestenes Spacetime Algebra Approach to Light Polarization (Q.M. Sugon/D. McNamara) Quaternions, Clifford Algebra and Symmetry Groups (P.R. Girard) Part III. Computer Vision and Robotics A Generic Framework for Image Geometry (J.J. Koenderink) Color Edge Detection Using Rotors (E. Bayro-Corrochano/S. Flores) Numerical Evaluation of Versors with Clifford Algebra (C.B.U. Perwass/G. Sommer) The Role of Clifford Algebra in Structure-Preserving Transformations for Second-Order Systems (S.D. Garvey/M.I. Friswell/U. Prells) Applications of Algebra of Incidence in Visually Guided Robotics (E. Bayro-Corrochano/P. Lounesto/L.R. Lozano) Monocular Pose Estimation of Kinematic Chains (B. Rosenhahn/O. Granert/G. Sommer) Stabilization of 3D Pose Estimation (W. Neddermeyer/M. Schnell/W. Winkler/A. Lilienthal) Inferring Dynamical Information from 3D Position Data using Geometric Algebra (H. Udugama/G.S. Sajeewa/J. Lasenby) Clifford Algebra Space Singularities of Inline Planar Platforms (M.A. Baswell/R. Ablamowicz/J.N. Anderson) Part IV. Signal Processing and Other Applications Fast Quantum Fourier--Heisenberg--Weyl Transforms (V. Labunets/E. Rundblad/J. Astola) The Structure Multivector (M. Felsberg/G. Sommer) The Application of Clifford Algebra to Calculations of Multicomponent Chemical Composition (J.P. Fletcher) An Algorithm to Solve the Inverse IFS-Problem (E. Hocevar) Fast Quantum n-D Fourier and Radon Transforms (V. Labunets/E. Rundblad/J. Astola)

The dynamics of rotations and translations in molecular crystals is derived from a microscopic Hamiltonian. The dynamical variables are the lattice displacements and their conjugate momenta for the translations and, in the case of orientationally disordered phases, the appropriate symmetry-adapted functions for the rotations. Coupled dynamical equations for the oscillatory lattice displacements and the orientational molecular relaxation are obtained. The coupling between these equations is expressed in terms of static susceptibilities which are calculated from the Hamiltonian. The coherent inelastic neutron scattering law, including interference between translations and rotations, is derived. Our results are relevant for inelastic neutron, Brillouin, and Raman scattering experiments. As a concrete example we refer to KCN.

A position and attitude control law is developed using geometric algebra (GA). GA is a powerful representational and computational system for geometry. The rigid body motion can be represented by the versor product in GA framework. Using the kinematics of the motor (the versor which represents the rigid body motion in GA), the control law of the rigid body motion can be developed. This paper provides a GA-based position and attitude control law by using the negative feedback of the motor. The stability of the control law is validated by the Lyapunov theorem and the numerical simulation.

In Sect. 9.1, following [1], we study the problems of motion of a free rigid body carrying a movable mass connected with the body by an elastic coupling in the presence of viscous friction. The cases of complete dynamic symmetry of the body and the case of the axially dynamically symmetric body are considered.

Using a reverse engineering capability to quantify the factors that control the rigid body mechanics of the wrist, a mathematical forward animation capability and model of wrist motion that allows the carpus to move under its own rules is being developed. This characterises the isometric connections, from which was developed the Stable Central Column Theory of Carpal Mechanics - which incorporates the Law of Rules Based Motion.This work has now advanced to the ability to reapply the extracted rules to allow rules-based rigid body reanimation of an individual wrist. As each wrist is unique, there is a given reality that each reanimation must be based on an individual wrist's unique rules, and the aspiration to create a standard or normal wrist is unrealistic.Using True Life Anatomy (Adelaide, Aust) analysis software, the specific rules (morphology / connectivity / interaction / loading) of individual wrists have been characterised, and then reapplied in a rigid body reanimation environment using Adams (MSC Software, U.S.) software.In the preliminary application of this biomechanics environment, by using the reverse engineering / forward reanimation process, wrist motion can be recreated - based purely on the unique rules, extracted from individual wrists. Instability of the proximal scaphoid was evident in several of the animations, and there was confirmation that the spatial attachment points of the isometric constraints are very exacting. The actual attachment and specific morphology of the carpal bones varied between individual wrists.Using a reverse engineering and then forward reanimation process, we have been able to recreate wrist motion using the rigid body mechanics based on the Law of Rules Based Motion. Further work is required, but the potential to apply “what if” virtual surgery options to an individual injured wrist and more precisely characterise and test solutions to wrist dysfunction are becoming realised.

This study proposes a finite-time tracking control system for an unmanned aerial vehicle (UAV) in the presence of dynamic payloads and external disturbances. Changes in payload result in time-varying parametric uncertainties, making traditional control strategies ineffective. A sliding mode control (SMC) scheme is designed to handle uncertainties in both translational and rotational dynamics. A finite-time SMC (FTSMC) approach is developed to handle unknown time-varying mass and external disturbances in the translational dynamics. Furthermore, differences in the payload influence the inertia characteristics of the UAV, introducing further uncertainties in the feedback control design. A finite-time sliding mode controller is also implemented for rotational dynamics to guarantee the attitude stabilization under unknown inertia parameter variations. The proposed method has the objective to attenuate the chattering effect associated with traditional SMC within a finite time frame. Theoretical stability analysis with rigorous mathematical proofs is presented to confirm the effectiveness of the control scheme. Numerical simulations are provided to validate its performance under dynamic operating conditions.

This article presents a unified synchronization framework with application to precision formation flying spacecraft. Central to the proposed innovation, in applying synchronization to both translational and rotational dynamics in the Lagrangian form, is the use of the distributed stability and performance analysis tool, called contraction analysis that yields exact nonlinear stability proofs. The proposed decentralized tracking control law synchronizes the attitude of an arbitrary number of spacecraft into a common time-varying trajectory with global exponential convergence. Moreover, a decentralized translational tracking control law based on phase synchronization is presented, thus enabling coupled translational and rotational maneuvers. While the translational dynamics can be adequately controlled by linear control laws, the proposed method permits highly nonlinear systems with nonlinearly coupled inertia matrices such as the attitude dynamics of spacecraft whose large and rapid slew maneuvers justify the nonlinear control approach. The proposed method integrates both the trajectory tracking and synchronization problems in a single control framework.

Dynamics of freely falling perforated disks

Mechanical structural calculus can be a complex task, considering quality and quantity of treated aspects. Mechanics of rigid body and structural analysis are two mechanical engineering well defined fields. Amazing, and in many cases favorable, calculus aspects of rigid body mechanics and strength of materials are interconnecting one each other, and so they validate the problem formulation and the solution. A complex application, floating crane, illustrates the above mentioned aspects.

While conventional Single Particle Tracking (SPT) techniques acquire 2D or 3D trajectories of particle probes, we have developed Single Particle Orientation and Rotational Tracking (SPORT) techniques to extract orientation and rotational information. Combined with DIC microscopy, the SPORT technique has been applied in biophysical studies, including membrane diffusion and intracellular transport. The rotational dynamics of nanoparticle vectors on live cell membranes was recorded and its influence on the fate of these nanoparticle vectors was elucidated. The rotational motions of gold nanorods with various surface modifiers were tracked continuously at a temporal resolution of 5 ms under a DIC microscope. We found that the rotational behaviors of gold nanorod vectors are strongly related to their surface charge, specific surface functional groups, and the availability of receptors on cell membranes. The study of rotational Brownian motion of nanoparticles on cell membranes will lead to a better understanding of the mechanisms of drug delivery and provide guidance in designing surface modification strategies for drug delivery vectors under various circumstances. To characterize the rotation mode of surface functionalized gold nanorods on cell membranes, the SPORT technique is combined with the correlation analysis of the bright and dark DIC intensities. The unique capabilities of visualizing and understanding rotational motions of functionalized nanoparticles on live cell membranes allow us to correlate rotational and translational dynamics in unprecedented detail and provide new insights for complex membrane processes, including electrostatic interactions, ligand-receptor binding, and lateral (confined and hopping) diffusion of membrane receptors. Surface-functionalized nanoparticles interact with the membrane in fundamentally different ways and exhibit distinct rotational modes. The early events of particle-membrane approach and attachment are directly visualized for the first time. The rotational dynamics of cargos in both active directional transport and pausing stages of axonal transport was also visualized using high-speed SPORT with a temporal resolution of 2 ms. Both long and short pauses are imaged, and the correlations between the pause duration, the rotational behaviour of the cargo at the pause, and the moving direction after the pause are established. Furthermore, the rotational dynamics leading to switching tracks are visualized in detail. These first-time observations of cargo's rotational dynamics provide new insights on how kinesin and dynein motors take the cargo through the alternating stages of active directional transport and pause. To improve the localization precision of the SPT technique with DIC microscopy, a precise three-dimensional (3D) localization method of spherical gold nanoparticle probes using model-based correlation coefficient mapping was introduced. To accomplish this, a stack of sample images at different z-positions are acquired, and a 3D intensity profile of the probe serving as the model is used to map out the positions of nanoparticles in the sample. By using this model-based correlation imaging method, precise localization can be achieved in imaging techniques with complicated point spread functions (PSF) such as differential interference contrast (DIC) microscopy. The 3D superlocalization method was applied to tracking gold nanospheres during live endocytosis events. Finally, a novel dual-modality imaging technique has been developed to super-localize a single gold nanorod while providing its orientation and rotational information. The super-localization of the gold nanorod can be accomplished by curve fitting the modified bright-field images generated by one of the two beams laterally shifted by the first Nomarski prism in a DIC microscope. The orientation and rotational information is derived from the DIC images of gold nanorods. The new imaging setup has been applied to study the steric hindrance induced by relatively large cargos in the microtubule gliding assay and to track nanocargos in the crowded cellular environment.

In this paper, we discuss a generalization of Vieta theorem (Vieta’s formulas) to the case of Clifford geometric algebras. We compare the generalized Vieta’s formulas with the ordinary Vieta’s formulas for characteristic polynomial containing eigenvalues. We discuss Gelfand – Retakh noncommutative Vieta theorem and use it for the case of geometric algebras of small dimensions. The results can be used in symbolic computation and various applications of geometric algebras in computer science, computer graphics, computer vision, physics, and engineering.

In this paper, we discuss characteristic polynomials in (Clifford) geometric algebras $$\mathcal {G}_{p,q}$$ of vector space of dimension $$n=p+q$$ . We present basis-free formulas for all characteristic polynomial coefficients in the cases $$n\le 6$$ , alongside with a method to obtain general form of these formulas. The formulas involve only the operations of geometric product, summation, and operations of conjugation. All the formulas are verified using computer calculations. We present an analytical proof of all formulas in the case $$n=4$$ , and one of the formulas in the case $$n=5$$ . We present some new properties of the operations of conjugation and grade projection and use them to obtain the results of this paper. We also present formulas for characteristic polynomial coefficients in some special cases. In particular, the formulas for vectors (elements of grade 1) and basis elements are presented in the case of arbitrary n, the formulas for rotors (elements of spin groups) are presented in the cases $$n\le 5$$ . The results of this paper can be used in different applications of geometric algebras in computer graphics, computer vision, engineering, and physics. The presented basis-free formulas for characteristic polynomial coefficients can also be used in symbolic computation.

This paper analyzes in detail the dynamics in a neighborhood of a Génot--Brogliato point, colloquially termed the G-spot, which physically represents so-called dynamic jam in rigid body mechanics with unilateral contact and Coulomb friction. Such singular points arise in planar rigid body problems with slipping point contacts at the intersection between the conditions for onset of lift-off and for the Painlevé paradox. The G-spot can be approached in finite time by an open set of initial conditions in a general class of problems. The key question addressed is what happens next. In principle, trajectories could, at least instantaneously, lift off, continue in slip, or undergo a so-called impact without collision. Such impacts are nonlocal in momentum space and depend on properties evaluated away from the G-spot. The answer is obtained via an analysis that involves a consistent contact regularization with a stiffness proportional to $1/\varepsilon^2$ for some $\varepsilon$. Taking a singular limit as $\varepsilon \to 0$, one finds an inner and an outer asymptotic zone in the neighborhood of the G-spot. Matched asymptotic analysis then enables continuation from the G-spot in the limit $\varepsilon \to 0$ and also reveals the sensitivity of trajectories to $\varepsilon$. The solution involves large-time asymptotics of certain generalized hypergeometric functions, which leads to conditions for the existence of a distinguished smoothest trajectory that remains uniformly bounded in $t$ and $\varepsilon$. Such a solution corresponds to a canard that connects stable slipping motion to unstable slipping motion through the G-spot. Perturbations to the distinguished trajectory are then studied asymptotically. Two distinct cases are distinguished according to whether the contact force becomes infinite or remains finite as the G-spot is approached. In the former case it is argued that there can be no such canards and so an impact without collision must occur. In the latter case, the canard trajectory acts as a dividing surface between trajectories that momentarily lift off and those that do not before taking the impact. The orientation of the initial condition set leading to each eventuality is shown to change each time a certain positive parameter $\beta$ passes through an integer. Finally, the results are illustrated in a particular physical example, namely the frictional impact oscillator first studied by Leine, Brogliato, and Nijmeijer.

In rigid body mechanics, a body's configuration consisting of orientation and position have to be described, as well as angular velocity and velocity. There are several different ways to do this, leading to different configuration spaces. Popular choices include SO(3) × ℝ3, SE(3) or unit dual quaternions. All these configuration spaces possess a Lie group structure. We present the novel approach 𝕊 ⋉ ℝ3, which is a configuration space similar to SE(3), but describes the orientation in terms of unit quaternions instead of rotation matrices. Furthermore we show its relation to and advantages over other configuration spaces. (© 2017 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Stabilization of the non-trivial relative equilibria of a gyrostat with an elastic element in a circular orbit