Dual-complex quaternion representation of gravitoelectromagnetism

In this paper, we propose the generalized description of electromagnetism and linear gravity based on the combined dual numbers and complex quaternion algebra. In this approach, the electromagnetic and gravitational fields can be considered as the components of one combined dual-complex quaternionic field. It is shown that all relations between potentials, field strengths and sources can be formulated in the form of compact quaternionic differential equations. The alternative reformulation of equations of gravitoelectromagnetism based on formalism of [Formula: see text] matrices is also discussed. The results reveal the similarity and isomorphism of distinctive algebraic structures.

Geometric control of quadrotor UAVs using integral backstepping

The traditional quadcopter control systems should deal with two common problems. Namely, the singularities related to the inverse kinematics and the ambiguity linked to the quaternion representation of the dynamic model. Moreover, the stability problem due to the system nonlinearity and high degree of coupling. This paper provides a solution to the two issues by employing a geometrical integral-backstepping control system. The integral terms were added to improve system ability to track desired trajectories. The high-level control laws are considered as a virtual control and transmitted to the low-level to track the high-level commands. The proposed control system along with the quadcopter dynamic model were expressed in the special Euclidean group SE(3). Finally, the control system robustness against mismatching parameters was studied while tracking various paths.

Color image encryption scheme based on quaternion discrete multi-fractional random transform and compressive sensing

A color image compression-encryption algorithm by combining quaternion discrete multi-fractional random transform with compressive sensing is investigated, in which the chaos-based fractional orders greatly improve key sensitivity. The original color image is compressed and encrypted with the assistance of compressive sensing, in which the partial Hadamard matrix adopted as a measurement matrix is constructed by iterating Chebyshev map instead of utilizing the entire Guassian matrix as a key. The sparse images are divided into 12 sub-images and then represented as three quaternion signals, which are modulated by the quaternion discrete multi-fractional random transform. The image blocking and the quaternion representation make the proposed cryptosystem avoid additional data extension existing in many transform-based methods. To further improve the level of security, the plaintext-related key streams generated by the 2D logistic-sine-coupling map are adopted to diffuse and confuse the intermediate results simultaneously. Consequently, the final ciphertext image is attained. Simulation results reveal that the proposed cryptosystem is feasible with high security and has strong robustness against various attacks.

Foundations of the Quaternion Quantum Mechanics

We show that quaternion quantum mechanics has well-founded mathematical roots and can be derived from the model of the elastic continuum by French mathematician Augustin Cauchy, i.e., it can be regarded as representing the physical reality of elastic continuum. Starting from the Cauchy theory (classical balance equations for isotropic Cauchy-elastic material) and using the Hamilton quaternion algebra, we present a rigorous derivation of the quaternion form of the non- and relativistic wave equations. The family of the wave equations and the Poisson equation are a straightforward consequence of the quaternion representation of the Cauchy model of the elastic continuum. This is the most general kind of quantum mechanics possessing the same kind of calculus of assertions as conventional quantum mechanics. The problem of the Schrödinger equation, where imaginary ‘i’ should emerge, is solved. This interpretation is a serious attempt to describe the ontology of quantum mechanics, and demonstrates that, besides Bohmian mechanics, the complete ontological interpretations of quantum theory exists. The model can be generalized and falsified. To ensure this theory to be true, we specified problems, allowing exposing its falsity.

Foundations of the Quaternion Quantum Mechanics

We show that the quaternion quantum mechanics has well-founded mathematical roots and can be derived from the model of elastic continuum by French mathematician Augustin Cauchy, i.e., it can be regarded as representing physical reality of elastic continuum. Starting from the Cauchy theory (classical balance equations for isotropic Cauchy-elastic material) and using the Hamilton quaternion algebra we present a rigorous derivation of the quaternion form of the non- and relativistic wave equations. The family of the wave equations and the Poisson equation are a straightforward consequence of the quaternion representation of the Cauchy model of the elastic continuum. This is the most general kind of quantum mechanics possessing the same kind of calculus of assertions as conventional quantum mechanics. The problem of the Schrödinger equation, where imaginary ‘i’ should emerge, is solved. This interpretation is a serious attempt to describe the ontology of quantum mechanics, and demonstrates that, besides Bohmian mechanics, the complete ontological interpretations of quantum theory exists. The model can be generalized and falsified. To ensure this theory to be true, we specified problems allowing exposing its falsity.

Joint-Space Characterization of a Medical Parallel Robot Based on a Dual Quaternion Representation of SE(3)

The paper proposes a mathematical method for redefining motion parameterizations based on the joint-space representation of parallel robots. The study parameters of SE(3) are used to describe the robot kinematic chains, but, rather than directly analyzing the mobile platform motion, the joint-space of the mechanism is studied by eliminating the Study parameters. From the loop equations of the joint-space characterization, new parameterizations are defined, which enable the placement of a mobile frame on any mechanical element within the parallel robot. A case study is presented for a medical parallel robotic system in which the joint-space characterization is achieved and based on a new defined parameterization, the kinematics for displacement, velocities, and accelerations are studied. A numerical simulation is presented for the derived kinematic models, showing how the medical robot guides the medical tool (ultrasound probe) on an imposed trajectory.

Quaternions and Cauchy Classical Theory of Elasticity

Developed by French mathematician Augustin-Louis Cauchy, the classical theory of elasticity is the starting point to show the value and the physical reality of quaternions. The classical balance equations for the isotropic, elastic crystal, demonstrate the usefulness of quaternions. The family of wave equations and the diffusion equation are a straightforward consequence of the quaternion representation of the Cauchy model of the elastic solid. Using the quaternion algebra, we present the derivation of the quaternion form of the multiple wave equations. The fundamental consequences of all derived equations and relations for physics, chemistry, and future prospects are presented.