The well known conjugate gradient algorithm (cg-algorithm), introduced by Hestenes & Stiefel, [1952] intended for real, symmetric, positive definite matrices works as well for complex matrices and has the same typical convergence behavior. It will also work, not generally, but in many cases for hermitean, but not necessarily positive definite matrices. We shall show, that the same behavior is s...

An extended Kalman filter is derived for estimating the relative position and attitude of a pair of uninhabited air vehicles, designated leader and follower. All leader states are assumed known, while the relative states are estimated using line-of-sight measurements between the vehicles along with angular rate and acceleration measurements of the follower. Noise is present on all measurements,...

The well known conjugate gradient algorithm (cg-algorithm), introduced by Hestenes & Stiefel, [1952] intended for real, symmetric, positive definite matrices works as well for complex matrices and has the same typical convergence behavior. It will also work, not generally, but in many cases for hermitean, but not necessarily positive definite matrices. We shall show, that the same behavior is s...

Quaternions can be used as an alternative to model the fundamental patterns of electroencephalographic (EEG) signals in the time domain. Thus, this article presents a new quaternion-based technique known as quaternion-based signal analysis (QSA) to represent EEG signals obtained using a brain-computer interface (BCI) device to detect and interpret cognitive activity. This quaternion-based signa...

In this paper, a novel blind color image watermarking technique using Quaternion and Joint QSVD Blocks is proposed to protect the intellectual property rights of color images. The proposed method tries to insert the watermark in the joint blocks .In this method, the color image is considered as an array of pure quaternion numbers. Then the array of pure quaternion is divided into non-overlappin...

In this paper, we present results on the implementation of a hierarchical quaternion based attitude and trajectory controller for manual and autonomous flights of quadrotors. Unlike previous papers on using quaternion representation, we use the nonlinear complementary filter that estimates the attitude in quaternions and as such does not involve Euler angles or rotation matrices. We show that f...

We introduce the notion of even Clifford structures on Riemannian manifolds, which for rank r = 2 and r = 3 reduce to almost Hermitian and quaternion-Hermitian structures respectively. We give the complete classification of manifolds carrying parallel rank r even Clifford structures: Kähler, quaternion-Kähler and Riemannian products of quaternion-Kähler manifolds for r = 2, 3 and 4 respectively...

The celebrated Four Squares Theorem of Lagrange states that every positive integer is the sum of four squares of integers. Interest in this Theorem has motivated a number of different demonstrations. While some of these demonstrations prove the existence of representations of an integer as a sum of four squares, others also produce the number of such representations. In one of these demonstrati...

This paper considers a robust strong tracking nonlinear filtering problem in the case there are model uncertainties including the model mismatch, unknown disturbance and status mutation in the spacecraft attitude estimation system with quaternion constraint. Two multiple fading factor matrices are employed to regulate the prediction error covariance matrix, which guarantees its symmetry. The sp...

It is well-known that every two dimensional rotation around the origin in the plane R can be represented by the multiplication of the complex number e = cos θ + i sin θ, 0 ≤ θ < 2π. Similarly, every three dimensional rotation in the space R can be represented by the multiplications of the quaternion q from the left-hand side and its conjugate q̄ from the right-hand side, where q = cos(θ/2) + α s...