Quaternion Rotation Research Articles

REQUEST - A recursive QUEST algorithm for sequential attitude determination

In order to find the attitude of a spacecraft with respect to a reference coordinate system, vector measurements are taken. The vectors are pairs of measurements of the same generalized vector, taken in the spacecraft body coordinates, as well as in the reference coordinate system. We are interested in finding the best estimate of the transformation between these coordinate system.s The algorithm called QUEST yields that estimate where attitude is expressed by a quarternion. Quest is an efficient algorithm which provides a least squares fit of the quaternion of rotation to the vector measurements. Quest however, is a single time point (single frame) batch algorithm, thus measurements that were taken at previous time points are discarded. The algorithm presented in this work provides a recursive routine which considers all past measurements. The algorithm is based on on the fact that the, so called, K matrix, one of whose eigenvectors is the sought quaternion, is linerly related to the measured pairs, and on the ability to propagate K. The extraction of the appropriate eigenvector is done according to the classical QUEST algorithm. This stage, however, can be eliminated, and the computation simplified, if a standard eigenvalue-eigenvector solver algorithm is used. The development of the recursive algorithm is presented and illustrated via a numerical example.

Recursive methods for the estimation of rotation quaternions

A general framework for analyzing kinematics of a rigid body is presented. We concentrate on different types of rotational motion of an object, namely the constant rotation, the constant angular acceleration about a fixed axis and the precessional motion. Quaternions are used to formulate the motion analysis problem, assuming the availability of a sequence of position vectors at a uniformly sampled time interval. Computationally efficient recursive algorithms have been developed to estimate the relevant parameters defining the evolution of the attitude of the object. Simulation results indicate good performance of these algorithms even in the presence of high sensor noise. Another advantage of the proposed methods is that they allow a varying number of feature points at different time instants, enabling the algorithms to handle long term occlusion of features in machine vision applications.

Response of a coupled two-spin system to on-resonance amplitude modulated RF pulses

A weakly scalar-coupled two-spin system subjected to two amplitude modulated RF pulses on exact resonance is treated by means of the rotation operator approach. The theory presented here enables coherence evolution to be evaluated by the routine procedure and to be expressed in analytical form. The evolution behaviour from the equilibrium state is discussed in some detail. It is shown that the application of rotation matrix and quaternion elements clarifies evolution expressions. The numerical calculation is performed by way of quaternions. Examples of BURP (band-selective, uniform response, purephase) and sinc-shaped RF pulses are given and the case of time-symmetrical RF pulses is analysed further.

Visualizing quaternion rotation

Quaternions play a vital role in the representation of rotations in computer graphics, primarily for animation and user interfaces. Unfortunately, quaternion rotation is often left as an advanced topic in computer graphics education due to difficulties in portraying the four-dimensional space of the quaternions. One tool for overcoming these obstacles is the quaternion demonstrator, a physical visual aid consisting primarily of a belt. Every quaternion used to specify a rotation can be represented by fixing one end of the belt and rotating the other. Multiplication of quaternions is demonstrated by the composition of rotations, and the resulting twists in the belt depict visually how quaternions interpolate rotation. This article introduces to computer graphics the exponential notation that mathematicians have used to represent unit quaternions. Exponential notation combines the angle and axis of the rotation into concise quaternion expression. This notation allows the article to present more clearly a mechanical quaternion demonstrator consisting of a ribbon and a tag, and develop a computer simulation suitable for interactive educational packages. Local deformations and the belt trick are used to minimize the ribbon's twisting and simulate a natural-appearing interactive quaternion demonstrator.

Three-dimensional analysis of strongly curved saccades elicited by double-step stimuli

Earlier recordings of eye position in three dimensions have revealed that Listing's law is obeyed in reasonable approximation, both statically and dynamically. This implies that all eye positions are confined to a plane when using a rotation vector or quaternion representation. The orientation of the angular velocity axis is crucial in order to preserve the law. For a single-axis rotation, the eye's angular velocity axis has to tilt out of Listing's plane, otherwise the law cannot be preserved in eccentric saccades. Experimental work has confirmed that normal, visually-guided saccades resemble single-axis rotations whose angular velocity axis tilts by the right amount. We investigated how well the saccadic system implements Listing's law when the trajectory of the eyes is more complicated, as in a non-single-axis rotation where the angular velocity vector depends on instantaneous eye position. Eye position was measured in three dimensions using the magnetic scleral search coil method for five subjects. Non-single-axis rotations of the eye were evoked with a double-step paradigm. We found that Listing's law is obeyed equally well during fixations, single-axis saccades and in non-single-axis saccades. Some deviations from the law were found in both curved and single-axis eye movements, but we demonstrated that the net torsional component of eye position of these saccades is negligible compared to that expected if the angular velocity axis did not tilt at all. In addition, analysis of the angular velocity signals in the curved movements showed strong similarity to the computed signal required for implementing Listing's law. Our results show that the observed deviations from Listing's law reflect only minor failures in the mechanism underlying its dynamic implementation. We conclude that single-axis rotations are not a necessary condition for the implementation of Listing's law in saccades. Our results are compatible with the notion that the implementation of Listing's law relies upon internal feedback. Various suggestions of how models can be reconciled with recent data on the three-dimensional control of saccades are discussed.

Attitude Determination from Vector Observations: Quaternion Estimation

Two recursive estimation algorithms, which use pairs of measured vectors to yield minimum variance estimates of the quaternion of rotation, are presented. The nonlinear relations between the direction cosine matrix and the quaternion are linearized, and a variant of the extended Kalman filter is used to estimate the difference between the quaternion and its estimate. With each measurement this estimate is updated and added to the whole quaternion estimate. This operation constitutes a full state reset in the estimation process. Filter tuning is needed to obtain a converging filter. The second algorithm presented uses the normality property of the quaternion of rotation to obtain, in a straightforward design, a filter which converges, with a smaller error, to a normal quaternion. This algorithm changes the state but not the covariance computation of the original algorithm and implies only a partial reset. Results of Monte-Carlo simulation runs are presented which demonstrate the superiority of the normalized quaternion.

A sequel to: Nonlinear finite element analysis of elastic systems under nonconservative loading—Natural formulation. Part I. Quasistatic problems

The present paper addresses a weak form quadrature element formulation for the geometrically exact thin shell model in which the Kirchhoff-Love hypothesis is adopted. The displacement derivative continuity conditions are enforced by the reconstruction of rotation variables at the edges of elements. By the utilization of rotation quaternions, a total Lagrange updating scheme is implemented for edge constraint director rotations. Several numerical examples are presented to illustrate the effectiveness of the proposed formulation and the significant reduction in the number of degrees of freedom in geometrically nonlinear thin shell analysis with large displacements and rotations.

Singularity-free extraction of a quaternion from a direction-cosine matrix

A computer algorithm for extracting a quaternion from a direction-cosine matrix (DCM) is described. The quaternion provides a four-parameter representation of rotation, as against the nine-parameter representation afforded by a DCM. Commanded attitude in space shuttle steering is conveniently computed by DCM, while actual attitude is computed most compactly as a quaternion, as is attitude error. The unit length of the rotation quaternion, and interchangeable of a quaternion and its negative, are used to advantage in the extraction algorithm. Protection of the algorithm against square root failure and division overflow are considered. Necessary and sufficient conditions for handling the rotation vector element of largest magnitude are discussed

Comments on "Optimum Normalization of a Computed Quaternion of Rotation"

Comments on "Optimum Normalization of a Computed Quaternion of Rotation"

Optimum Normalization of a Computed Quaternion of Rotation

The normalization technique usually employed in "analytic platforms" in which quaternions are used to update the orientation of the vehicle is the one which divides the components of the quaternion by its magnitude. In this correspondence, it is analytically shown that this is the "best" normalization technique. A geometrical interpretation of this result is also given.

Dispersion of Sequences for Generating a Robust Enciphering System

This paper introduces a new paradigm for dispersion of information into three-dimensional space to generate a strenuous crypto system. The dispersion of the information is carried out by quaternion rotation matrix which is called quadripartite public-key (QPK). We considered a private key as a unit quaternion which has an arbitrary quadruple of real numbers. The public key has three different components which are constructed using quaternion rotation matrix QRM to generate a secure key, which significantly eliminates the risk of eavesdropping. Furthermore, quaternion has capability to provide a meditative encryption system for wireless images or voice transmissions. A computer-based simulation conducted to scrutinize the capability of QPK carefully in insuring the highest level of security is reported.