We introduce a total order and an absolute value function for dual numbers. The of numbers takes number values, has properties similar to those the real define magnitude quaternion, as number. Based upon these, we extend 1-norm, $$\infty$$ -norm, 2-norm quaternion vectors.

The problem of integrating rotations and angular velocity from a known timeand space-dependent curvature is met in the dynamic analysis of non-linear spatial beams, if the numerical formulation is based on curvature, as in [1]. To avoid singularities and make the formulation more convenient, rotational quaternions q̂ are used for the parametrization of rotations. The relation between the curvatu...

Expander graphs are graphs in which the neighbors of any given “not too large” set of vertices X form a large set relative to the size of X—rumors tend to spread very fast. Among those, the Ramanujan graphs are extremal in their expansion properties. To be precise, the eigenvalues of the adjacency matrix have an extremal property that guarantees good expansion properties. Expanders, and hence R...

The ideas of instantaneous amplitude and phase are well understood for signals with real-valued samples, based on the analytic signal which is a complex signal with one-sided Fourier transform. We extend these ideas to signals with complex-valued samples, using a quaternion-valued equivalent of the analytic signal obtained from a one-sided quaternion Fourier transform which we refer to as the h...

A new spacecraft attitude estimation approach based on the Unscented Filter is derived. For nonlinear systems the Unscented Filter uses a carefully selected set of sample points to more accurately map the probability distribution than the linearization of the standard Extended Kalman Filter, leading to faster convergence from inaccurate initial conditions in attitude estimation problems. The fi...

We propose the digital signature scheme on non-commutative quaternion ring over finite fields in this paper. We generate the multivariate function of high degree F(X) . We construct the digital signature scheme using F(X). Our system is immune from the Gröbner bases attacks because obtaining parameters of F(X) to be secret keys arrives at solving the multivariate algebraic equations that is one...

Position and attitude set-point regulation of autonomous underwater vehicles (AUVs) in 6 degrees of freedom (DOF) is presented. Euler parameters are used in the representation of global attitude. Three quaternion feedback controllers are derived based on vector quaternion, Euler rotation and Rodrigues parameter feedback, respectively. Global asymp-totic stability is proven in the rst and second...

This paper describes new issues of the Mathematica standard package Quaternions for implementing Hamilton’s Quaternion Algebra. This work attempts to endow the original package with the ability to perform operations on symbolic expressions involving quaternion-valued functions. A collection of new functions is introduced in order to provide basic mathematical tools necessary for dealing with re...

In this paper, we generalize the linear canonical transform (LCT) to quaternion-valued signals, known as the quaternionic linear canonical transform (QLCT). Using the properties of the LCT we establish an uncertainty principle for the QLCT. This uncertainty principle prescribes a lower bound on the product of the effective widths of quaternion-valued signals in the spatial and frequency domains...